A Computational Analysis of the Effect of Hard Choices on the Individuation of Values

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} div.type-section h2 { font-size: 20px; line-height: 26px; font-weight: 300; } div.type-section h3 { margin-left: 15px; margin-bottom: 0px; font-weight: 300; } .journal-tabs .tab-title.active a { } </style> <link rel="stylesheet" href="https://pub.mdpi-res.com/assets/css/slick.css?f38b2db10e01b157?1738315387"> <meta name="title" content="A Computational Analysis of the Effect of Hard Choices on the Individuation of Values"> <meta name="description" content="Background/Objectives: Experimental studies show that when an individual makes choices, they affect future decisions. Future choices tend to be consistent with past ones. This tendency matters in the context of ambivalent situations because they may not lead to clear choices, often leading people to make “arbitrary” decisions. Thus, because of choice consistency with the past, people’s decision-making values diverge. Thus, hard choices may contribute to the individuation of values. Methods: Here, we develop a Bayesian framework for the effects of cognitive choice consistency on decision-making. This framework thus extends earlier cognitive-science Bayesian theories, which focus on other tasks, such as inference. The minimization of total surprisals considering the history of stimuli and chosen actions implements choice consistency in our framework. We then use a computational model based on this framework to study the effect of hard choices on decision-making values. Results: The results for action selection based on sensory stimuli show that hard choices can cause the spontaneous symmetry breaking of the decision-making space. This spontaneous symmetry breaking is different across individuals, leading to individuation. If in addition, rewards are given to certain choices, then the direction of the symmetry breaking can be guided by these incentives. Finally, we explore the effects of the parametric complexity of the model, the number of choices, and the length of choice memory. Conclusions: Considering the brain’s mechanism of choice consistency and the number of hard choices made in life, we hypothesize that they contribute to individuality. We assess this hypothesis by placing our study in the context of the cognition-of-individuality literature and proposing experimental tests of our computational results." > <link rel="image_src" href="https://pub.mdpi-res.com/img/journals/brainsci-logo.png?8600e93ff98dbf14" > <meta name="dc.title" content="A Computational Analysis of the Effect of Hard Choices on the Individuation of Values"> <meta name="dc.creator" content="Norberto M. Grzywacz"> <meta name="dc.type" content="Article"> <meta name="dc.source" content="Brain Sciences 2025, Vol. 15, Page 131"> <meta name="dc.date" content="2025-01-29"> <meta name ="dc.identifier" content="10.3390/brainsci15020131"> <meta name="dc.publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="dc.rights" content="http://creativecommons.org/licenses/by/3.0/"> <meta name="dc.format" content="application/pdf" > <meta name="dc.language" content="en" > <meta name="dc.description" content="Background/Objectives: Experimental studies show that when an individual makes choices, they affect future decisions. Future choices tend to be consistent with past ones. This tendency matters in the context of ambivalent situations because they may not lead to clear choices, often leading people to make “arbitrary” decisions. Thus, because of choice consistency with the past, people’s decision-making values diverge. Thus, hard choices may contribute to the individuation of values. Methods: Here, we develop a Bayesian framework for the effects of cognitive choice consistency on decision-making. This framework thus extends earlier cognitive-science Bayesian theories, which focus on other tasks, such as inference. The minimization of total surprisals considering the history of stimuli and chosen actions implements choice consistency in our framework. We then use a computational model based on this framework to study the effect of hard choices on decision-making values. Results: The results for action selection based on sensory stimuli show that hard choices can cause the spontaneous symmetry breaking of the decision-making space. This spontaneous symmetry breaking is different across individuals, leading to individuation. If in addition, rewards are given to certain choices, then the direction of the symmetry breaking can be guided by these incentives. Finally, we explore the effects of the parametric complexity of the model, the number of choices, and the length of choice memory. Conclusions: Considering the brain’s mechanism of choice consistency and the number of hard choices made in life, we hypothesize that they contribute to individuality. We assess this hypothesis by placing our study in the context of the cognition-of-individuality literature and proposing experimental tests of our computational results." > <meta name="dc.subject" content="cognition" > <meta name="dc.subject" content="choice consistency" > <meta name="dc.subject" content="decision-making" > <meta name="dc.subject" content="individuation of values" > <meta name="dc.subject" content="brain’s reward system" > <meta name="dc.subject" content="memory" > <meta name="dc.subject" content="Bayesian theory" > <meta name="dc.subject" content="surprisals" > <meta name="dc.subject" content="symmetry breaking" > <meta name ="prism.issn" content="2076-3425"> <meta name ="prism.publicationName" content="Brain Sciences"> <meta name ="prism.publicationDate" content="2025-01-29"> <meta name ="prism.volume" content="15"> <meta name ="prism.number" content="2"> <meta name ="prism.section" content="Article" > <meta name ="prism.startingPage" content="131" > <meta name="citation_issn" content="2076-3425"> <meta name="citation_journal_title" content="Brain Sciences"> <meta name="citation_publisher" content="Multidisciplinary Digital Publishing Institute"> <meta name="citation_title" content="A Computational Analysis of the Effect of Hard Choices on the Individuation of Values"> <meta name="citation_publication_date" content="2025/2"> <meta name="citation_online_date" content="2025/01/29"> <meta name="citation_volume" content="15"> <meta name="citation_issue" content="2"> <meta name="citation_firstpage" content="131"> <meta name="citation_author" content="Grzywacz, Norberto M."> <meta name="citation_doi" content="10.3390/brainsci15020131"> <meta name="citation_id" content="mdpi-brainsci15020131"> <meta name="citation_abstract_html_url" content="https://www.mdpi.com/2076-3425/15/2/131"> <meta name="citation_pdf_url" content="https://www.mdpi.com/2076-3425/15/2/131/pdf?version=1738140596"> <link rel="alternate" type="application/pdf" title="PDF Full-Text" href="https://www.mdpi.com/2076-3425/15/2/131/pdf?version=1738140596"> <meta name="fulltext_pdf" content="https://www.mdpi.com/2076-3425/15/2/131/pdf?version=1738140596"> <meta name="citation_fulltext_html_url" content="https://www.mdpi.com/2076-3425/15/2/131/htm"> <link rel="alternate" type="text/html" title="HTML Full-Text" href="https://www.mdpi.com/2076-3425/15/2/131/htm"> <meta name="fulltext_html" content="https://www.mdpi.com/2076-3425/15/2/131/htm"> <link rel="alternate" type="text/xml" title="XML Full-Text" href="https://www.mdpi.com/2076-3425/15/2/131/xml"> <meta name="fulltext_xml" content="https://www.mdpi.com/2076-3425/15/2/131/xml"> <meta name="citation_xml_url" content="https://www.mdpi.com/2076-3425/15/2/131/xml"> <meta name="twitter:card" content="summary" /> <meta name="twitter:site" content="@MDPIOpenAccess" /> <meta name="twitter:image" content="https://pub.mdpi-res.com/img/journals/brainsci-logo-social.png?8600e93ff98dbf14" /> <meta property="fb:app_id" content="131189377574"/> <meta property="og:site_name" content="MDPI"/> <meta property="og:type" content="article"/> <meta property="og:url" content="https://www.mdpi.com/2076-3425/15/2/131" /> <meta property="og:title" content="A Computational Analysis of the Effect of Hard Choices on the Individuation of Values" /> <meta property="og:description" content="Background/Objectives: Experimental studies show that when an individual makes choices, they affect future decisions. Future choices tend to be consistent with past ones. This tendency matters in the context of ambivalent situations because they may not lead to clear choices, often leading people to make “arbitrary” decisions. Thus, because of choice consistency with the past, people’s decision-making values diverge. Thus, hard choices may contribute to the individuation of values. Methods: Here, we develop a Bayesian framework for the effects of cognitive choice consistency on decision-making. This framework thus extends earlier cognitive-science Bayesian theories, which focus on other tasks, such as inference. The minimization of total surprisals considering the history of stimuli and chosen actions implements choice consistency in our framework. We then use a computational model based on this framework to study the effect of hard choices on decision-making values. Results: The results for action selection based on sensory stimuli show that hard choices can cause the spontaneous symmetry breaking of the decision-making space. This spontaneous symmetry breaking is different across individuals, leading to individuation. If in addition, rewards are given to certain choices, then the direction of the symmetry breaking can be guided by these incentives. Finally, we explore the effects of the parametric complexity of the model, the number of choices, and the length of choice memory. Conclusions: Considering the brain’s mechanism of choice consistency and the number of hard choices made in life, we hypothesize that they contribute to individuality. We assess this hypothesis by placing our study in the context of the cognition-of-individuality literature and proposing experimental tests of our computational results." /> <meta property="og:image" content="https://pub.mdpi-res.com/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g001-550.jpg?1738140684" /> <link rel="alternate" type="application/rss+xml" title="MDPI Publishing - Latest articles" href="https://www.mdpi.com/rss"> <meta name="google-site-verification" content="PxTlsg7z2S00aHroktQd57fxygEjMiNHydKn3txhvwY"> <meta name="facebook-domain-verification" content="mcoq8dtq6sb2hf7z29j8w515jjoof7" /> <script id="Cookiebot" data-cfasync="false" src="https://consent.cookiebot.com/uc.js" data-cbid="51491ddd-fe7a-4425-ab39-69c78c55829f" type="text/javascript" async></script>  <script type="text/plain" data-cookieconsent="statistics"> (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start': new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0], j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src= 'https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f); 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href="/2076-3425/15/2/131/pdf?version=1738140596" data-name="A Computational Analysis of the Effect of Hard Choices on the Individuation of Values" data-journal="brainsci"> <i class="material-icons custom-download"></i> Download PDF </a> <div class='nav-btn'> <i class='material-icons'>settings</i> </div> <a href="/2076-3425/15/2/131/reprints" id="js-button-reprints" class="button button--color-inversed"> Order Article Reprints </a> </div> <div class='html-article-menu'> <div class='html-first-step row'> <div class='html-font-family large-6 medium-6 small-12 columns'> <div class='row'> <div class='html-font-label large-4 medium-4 small-12 columns'> Font Type: </div> <div class='large-8 medium-8 small-12 columns'> <span class="html-article-menu-option"><i style='font-family:Arial, Arial, Helvetica, sans-serif;' data-fontfamily='Arial, Arial, Helvetica, sans-serif'>Arial</i></span> <span class="html-article-menu-option"><i style='font-family:Georgia1, Georgia, serif;' 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medium-8 small-12 columns'> <div class="html-article-menu-option html-nav-bg html-nav-bright" data-bg="bright"> <i class="fa fa-file-text"></i> </div> <div class="html-article-menu-option html-nav-bg html-nav-dark" data-bg="dark"> <i class="fa fa-file-text-o"></i> </div> <div class="html-article-menu-option html-nav-bg html-nav-creme" data-bg="creme"> <i class="fa fa-file-text"></i> </div> </div> </div> </div> </div> </div> </div> <article ><div class='html-article-content'> <span itemprop="publisher" content="Multidisciplinary Digital Publishing Institute"></span><span itemprop="url" content="https://www.mdpi.com/2076-3425/15/2/131"></span> <div class="article-icons"><span class="label openaccess" data-dropdown="drop-article-label-openaccess" aria-expanded="false">Open Access</span><span class="label articletype">Article</span></div> <h1 class="title hypothesis_container" itemprop="name"> A Computational Analysis of the Effect of Hard Choices on the Individuation of Values </h1> <div class="art-authors hypothesis_container"> by <span class="inlineblock "><div class='profile-card-drop' data-dropdown='profile-card-drop13735223' data-options='is_hover:true, hover_timeout:5000'> Norberto M. Grzywacz</div><div id="profile-card-drop13735223" data-dropdown-content class="f-dropdown content profile-card-content" aria-hidden="true" tabindex="-1"><div class="profile-card__title"><div class="sciprofiles-link" style="display: inline-block"><div class="sciprofiles-link__link"><img class="sciprofiles-link__image" src="/profiles/2966786/thumb/Norberto_M._Grzywacz.png" style="width: auto; height: 16px; border-radius: 50%;"><span class="sciprofiles-link__name">Norberto M. Grzywacz</span></div></div></div><div class="profile-card__buttons" style="margin-bottom: 10px;"><a href="https://sciprofiles.com/profile/2966786?utm_source=mdpi.com&utm_medium=website&utm_campaign=avatar_name" class="button button--color-inversed" target="_blank"> SciProfiles </a><a href="https://scilit.com/scholars?q=Norberto%20M.%20Grzywacz" class="button button--color-inversed" target="_blank"> Scilit </a><a href="https://www.preprints.org/search?search1=Norberto%20M.%20Grzywacz&field1=authors" class="button button--color-inversed" target="_blank"> Preprints.org </a><a href="https://scholar.google.com/scholar?q=Norberto%20M.%20Grzywacz" class="button button--color-inversed" target="_blank" rels="noopener noreferrer"> Google Scholar </a></div></div><sup> 1,2</sup><span style="display: inline; margin-left: 5px;"></span><a class="toEncode emailCaptcha visibility-hidden" data-author-id="13735223" href="/cdn-cgi/l/email-protection#4d622e2329602e2a2462216228202c2421603d3f2239282e392422236e7d7d7d7c7b287c2e7d2e7d2f7c2e7c2c7d7c7f287d7f7c2f7d29797d7d2f7d2c7c2f"><sup><i class="fa fa-envelope-o"></i></sup></a><a href="https://orcid.org/0000-0003-4615-1560" target="_blank" rel="noopener noreferrer"><img src="https://pub.mdpi-res.com/img/design/orcid.png?0465bc3812adeb52?1738315387" title="ORCID" style="position: relative; width: 13px; margin-left: 3px; max-width: 13px !important; height: auto; top: -5px;"></a></span> </div> <div class="nrm"></div> <span style="display:block; height:6px;"></span> <div></div> <div style="margin: 5px 0 15px 0;" class="hypothesis_container"> <div class="art-affiliations"> <div class="affiliation "> <div class="affiliation-item"><sup>1</sup></div> <div class="affiliation-name ">Department of Psychology, Loyola University Chicago, Chicago, IL 60660, USA</div> </div> <div class="affiliation "> <div class="affiliation-item"><sup>2</sup></div> <div class="affiliation-name ">Department of Cognitive Science, Johns Hopkins University, Baltimore, MD 21218, USA</div> </div> </div> </div> <div class="bib-identity" style="margin-bottom: 10px;"> <em>Brain Sci.</em> <b>2025</b>, <em>15</em>(2), 131; <a href="https://doi.org/10.3390/brainsci15020131">https://doi.org/10.3390/brainsci15020131</a> </div> <div class="pubhistory" style="font-weight: bold; padding-bottom: 10px;"> <span style="display: inline-block">Submission received: 19 December 2024</span> / <span style="display: inline-block">Revised: 21 January 2025</span> / <span style="display: inline-block">Accepted: 24 January 2025</span> / <span style="display: inline-block">Published: 29 January 2025</span> </div> <div class="belongsTo" style="margin-bottom: 10px;"> (This article belongs to the Section <a href="/journal/brainsci/sections/Social_Cognitive_Affective_Neuroscience">Cognitive, Social and Affective Neuroscience</a>)<br/> </div> <div class="highlight-box1"> <div class="download"> <a class="button button--color-inversed button--drop-down" data-dropdown="drop-download-1579556" aria-controls="drop-supplementary-1579556" aria-expanded="false"> Download <i class="material-icons">keyboard_arrow_down</i> </a> <div id="drop-download-1579556" class="f-dropdown label__btn__dropdown label__btn__dropdown--button" data-dropdown-content aria-hidden="true" tabindex="-1"> <a class="UD_ArticlePDF" href="/2076-3425/15/2/131/pdf?version=1738140596" data-name="A Computational Analysis of the Effect of Hard Choices on the Individuation of Values" data-journal="brainsci">Download PDF</a> <br/> <a id="js-pdf-with-cover-access-captcha" href="#" data-target="/2076-3425/15/2/131/pdf-with-cover" class="accessCaptcha">Download PDF with Cover</a> <br/> <a id="js-xml-access-captcha" href="#" data-target="/2076-3425/15/2/131/xml" class="accessCaptcha">Download XML</a> <br/> <a href="/2076-3425/15/2/131/epub" id="epub_link">Download Epub</a> <br/> </div> <div class="js-browse-figures" style="display: inline-block;"> <a href="#" class="button button--color-inversed margin-bottom-10 openpopupgallery UI_BrowseArticleFigures" data-target='article-popup' data-counterslink = "https://www.mdpi.com/2076-3425/15/2/131/browse" >Browse Figures</a> </div> <div id="article-popup" class="popupgallery" style="display: inline; line-height: 200%"> <a href="https://pub.mdpi-res.com/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g001.png?1738140683" title=" <strong>Figure 1</strong><br/> <p>Bayesian update of the parameters of choice consistency. The figure shows three moments of parametric updates indicated in red, blue, and green. The first moment at time k (red) begins with a stimulus (Stim k) drawn from the probability distribution of stimuli (P(S)). With this stimulus, the brain calculates an action (Act k) from the probability distribution of actions given stimuli (P(A|S)). This calculation uses the set of parameters (Par k − 1) calculated at time k − 1. A reward (Rew k) then arrives from the probability distribution of rewards given stimuli and actions (P(R|A,S)). These stimulus and action are added to the histories of these values (Stim Hist and Act Hist). Given these histories and the new reward Rew k, a new parameter set (Par k) is computed, maximizing the Bayesian expected reward and action consistency. With this new set, one can repeat the process again at time k + 1 (blue). This process leads to the computation of a new parameter set (Par k + 1) that triggers the process again (green) and so on.</p> "> </a> <a href="https://pub.mdpi-res.com/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g002.png?1738140684" title=" <strong>Figure 2</strong><br/> <p>Computer simulation of our Bayesian theory of choice consistency with the standard parameters (<a href="#brainsci-15-00131-t001" class="html-table">Table 1</a>). (<b>A</b>) Choices of two actions for different stimuli over time. An example of such an action is buying a shirt with this or that pattern. In this figure, every dot stands for a choice (color) for the given sampled stimulus at the given time. (<b>B</b>) Running average (5 points) of the choices in panel (<b>A</b>). (<b>C</b>) Choice-consistency loss as a function of time. (<b>D</b>) Temporal evolution of the two parameters of the model. These time courses reveal that the choices separate spontaneously, with an apparent phase transition in loss and parameters.</p> "> </a> <a href="https://pub.mdpi-res.com/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g003.png?1738140686" title=" <strong>Figure 3</strong><br/> <p>Elimination of choices. (<b>A</b>) Longer simulations with standard parameters (<a href="#brainsci-15-00131-t001" class="html-table">Table 1</a>) show that eventually, choice consistency may cause one of the choices to eliminate the others. The conventions in this figure are the same as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A. (<b>B</b>) Distribution of times of choice elimination for two values of memory length, namely, Δ.</p> "> </a> <a href="https://pub.mdpi-res.com/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g004.png?1738140686" title=" <strong>Figure 4</strong><br/> <p>Eight consecutive simulations of actions in response to sensory stimuli with the standard set of parameters, using the conventions of <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A. Each simulation yielded a unique pattern of behavior. After a first random phase, the most common behaviors were such that positive sensory stimuli tended to yield Action 1 (Panels <b>D</b>,<b>G</b>,<b>H</b>) or Action 2 (Panels <b>A</b>,<b>C</b>,<b>E</b>). In these behaviors, negative sensory stimuli tended to yield the opposite actions. Occasionally, we also saw a behavior that was more mixed (Panel <b>B</b>). More rarely, we saw a behavior in which an action happened for positive stimuli earlier and negative ones later (<b>F</b>).</p> "> </a> <a href="https://pub.mdpi-res.com/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g005.png?1738140688" title=" <strong>Figure 5</strong><br/> <p>Increasing the number of choice parameters boosts the individuality arising from the model. In these simulations, we substituted <math display="inline"><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mn>1.25,0</mn> <mo>,</mo> <mn>1.75,0</mn> </mrow> </mfenced> </mrow> </semantics></math> for the standard value in <a href="#brainsci-15-00131-t001" class="html-table">Table 1</a> and thus, we had four choice parameters instead of two. (<b>A</b>–<b>C</b>) Examples of choices (with conventions as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A). (<b>D</b>–<b>F</b>) Temporal evolution of parameters in the simulations of (<b>A</b>, <b>B</b>, and <b>C</b>) respectively. Most simulations with 4 choice parameters yielded behaviors like those in <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>. However, some simulations yielded different behaviors, as illustrated in this figure. (<b>A</b>,<b>D</b>) Examples of not discriminating actions by stimuli. (<b>B</b>,<b>E</b>) Examples of switching stimulus dependence of choices. (<b>C</b>,<b>F</b>) Examples of Action 1 sandwiched between two stimulus locations of Action 2.</p> "> </a> <a href="https://pub.mdpi-res.com/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g006.png?1738140689" title=" <strong>Figure 6</strong><br/> <p>Outcome of the simulations with three or four choices instead of two. (<b>A</b>) Choices of three actions for different stimuli over time (conventions as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A). (<b>B</b>) Running average (5 points) of the choices in Panel (<b>A</b>). These panels show that choice consistency organizes the three actions in the space of stimuli. However, eventually one action dominates (Action 3 in this example), with one of the other actions stopping first (Action 2 in this example) and then the other (Action 1). (<b>C</b>) Scatter plot of the stoppage times of the losing actions. They tend to stop almost at the same time. (<b>D</b>) Running average (5 points) of a simulation with four choices.</p> "> </a> <a href="https://pub.mdpi-res.com/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g007.png?1738140691" title=" <strong>Figure 7</strong><br/> <p>The interaction between rewards and choice consistency with standard parameters, except for variations of <math display="inline"><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. (<b>A</b>) Simulation with only rewards (<math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). As expected from the choice of the standard parameters, Action 2 is chosen for positive stimuli and vice versa for Action 1. (<b>B</b>) Example of simulation with <math display="inline"><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> in which positive stimuli elicit Action 1 despite the rewards favoring the opposite. (<b>C</b>) Percentage of simulations for which Action 2 stimuli converge to values larger than those for Action 1 as a function of <math display="inline"><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. (<b>D</b>) Mean stoppage time of the losing actions as a function of <math display="inline"><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. Error bars in (<b>C</b>,<b>D</b>) are standard errors. As <math display="inline"><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> increases, we obtain more Action 2 because of the rewards, and the stoppage time rises because the influence of choice consistency diminishes.</p> "> </a> </div> <a class="button button--color-inversed" href="/2076-3425/15/2/131/notes">Versions Notes</a> </div> </div> <div class="responsive-moving-container small hidden" data-id="article-counters" style="margin-top: 15px;"></div> <div class="html-dynamic"> <section> <div class="art-abstract art-abstract-new in-tab hypothesis_container"> <p> <div><section class="html-abstract" id="html-abstract"> <h2 id="html-abstract-title">Abstract</h2><b>:</b> <div class="html-p">Background/Objectives: Experimental studies show that when an individual makes choices, they affect future decisions. Future choices tend to be consistent with past ones. This tendency matters in the context of ambivalent situations because they may not lead to clear choices, often leading people to make “arbitrary” decisions. Thus, because of choice consistency with the past, people’s decision-making values diverge. Thus, hard choices may contribute to the individuation of values. Methods: Here, we develop a Bayesian framework for the effects of cognitive choice consistency on decision-making. This framework thus extends earlier cognitive-science Bayesian theories, which focus on other tasks, such as inference. The minimization of total surprisals considering the history of stimuli and chosen actions implements choice consistency in our framework. We then use a computational model based on this framework to study the effect of hard choices on decision-making values. Results: The results for action selection based on sensory stimuli show that hard choices can cause the spontaneous symmetry breaking of the decision-making space. This spontaneous symmetry breaking is different across individuals, leading to individuation. If in addition, rewards are given to certain choices, then the direction of the symmetry breaking can be guided by these incentives. Finally, we explore the effects of the parametric complexity of the model, the number of choices, and the length of choice memory. Conclusions: Considering the brain’s mechanism of choice consistency and the number of hard choices made in life, we hypothesize that they contribute to individuality. We assess this hypothesis by placing our study in the context of the cognition-of-individuality literature and proposing experimental tests of our computational results.</div> </section> <div id="html-keywords"> <div class="html-gwd-group"><div id="html-keywords-title">Keywords: </div><a href="/search?q=cognition">cognition</a>; <a href="/search?q=choice+consistency">choice consistency</a>; <a href="/search?q=decision-making">decision-making</a>; <a href="/search?q=individuation+of+values">individuation of values</a>; <a href="/search?q=brain%E2%80%99s+reward+system">brain’s reward system</a>; <a href="/search?q=memory">memory</a>; <a href="/search?q=Bayesian+theory">Bayesian theory</a>; <a href="/search?q=surprisals">surprisals</a>; <a href="/search?q=symmetry+breaking">symmetry breaking</a></div> <div> </div> </div> </div> </p> </div> </section> </div> <div class="hypothesis_container"> <ul class="menu html-nav" data-prev-node="#html-quick-links-title"> </ul> <div class="html-body"> <section id='sec1-brainsci-15-00131' type='intro'><h2 data-nested='1'> 1. Introduction</h2><div class='html-p'>When an individual makes choices, they affect future ones because future decisions tend to be consistent with those from the past [<a href="#B1-brainsci-15-00131" class="html-bibr">1</a>,<a href="#B2-brainsci-15-00131" class="html-bibr">2</a>]. This consistency between past and future choices leads to the refinement or development of new preferences, a phenomenon referred to as “preference learning” [<a href="#B3-brainsci-15-00131" class="html-bibr">3</a>]. In addition, this temporal consistency of choices has been shown to have important consequences for perception. For example, this consistency tends to lead to aesthetic stability [<a href="#B4-brainsci-15-00131" class="html-bibr">4</a>]. Merely making a choice leads to a “spreading of alternatives”, whereby the two options become further apart in the preference domain, leading to improved detection [<a href="#B4-brainsci-15-00131" class="html-bibr">4</a>]. Another consequence of choice consistency is increased confidence [<a href="#B5-brainsci-15-00131" class="html-bibr">5</a>,<a href="#B6-brainsci-15-00131" class="html-bibr">6</a>,<a href="#B7-brainsci-15-00131" class="html-bibr">7</a>]. Self-consistency refers to an agreement between the current perceptual choice and the most frequent ones made for a given sensory stimulus and decision-making situation [<a href="#B7-brainsci-15-00131" class="html-bibr">7</a>]. Perceptual confidence thus becomes an estimation of the probability that one would make the same decision given the same situation and physical stimulus. Consistently, emerging theories of decision-making and economics predict that as people gain experience with a decision-making task, their apparent internal decision noise decreases [<a href="#B8-brainsci-15-00131" class="html-bibr">8</a>,<a href="#B9-brainsci-15-00131" class="html-bibr">9</a>].</div><div class='html-p'>Surprisingly, however, although past choices tend to stabilize future ones, they may also contribute to the individuation of values. This tendency is especially important in the context of hard, ambivalent situations because they may not lead to clear choices, often leading people to make different, “arbitrary” choices. Thus, because of choice consistency with the past, people’s decision-making values may begin to diverge. The same does not happen in easy situations, that is, those for which choices are clear. In these situations, people have clear preferences on how to act, rarely making “arbitrary” decisions. Thus, preference learning may ensue only with hard choices. The role of preference learning for people’s individuality has been discussed elsewhere [<a href="#B10-brainsci-15-00131" class="html-bibr">10</a>,<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>,<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>]. The emphasis of these studies has been on the reinforcement learning of aesthetic values of individuals under different exteroceptive (sensory) and interoceptive (body) environments [<a href="#B2-brainsci-15-00131" class="html-bibr">2</a>,<a href="#B13-brainsci-15-00131" class="html-bibr">13</a>,<a href="#B14-brainsci-15-00131" class="html-bibr">14</a>]. Therefore, decision-making values would be learned to improve reward estimation, thus making it better. However, given the role of choices in preference learning, we must generalize these reward-based theories. The simplest way is to consider good past choices as rewards themselves. In this article, we propose a way to add past choices to the existing theoretical framework and analyze the consequences of the new emerging theory.</div><div class='html-p'>To add choice consistency to the current theoretical frameworks for preference learning, we must start with a deeper understanding of how they work. They have been proposed and analyzed in conditions of fixed [<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>,<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>] and varying [<a href="#B15-brainsci-15-00131" class="html-bibr">15</a>] sensory environments. The basic idea of these frameworks is that at every instant, an individual receives sensory and body signals, and must decide on what action to take. A good decision depends on the correct estimation of reward for the various possible actions. After taking an action and receiving the reward, if the estimation is poor, the brain learns new parameters for its internal model of recompense expectation. In the case of aesthetic values, they are taken to be the expectation of reward. One can model such a process for aesthetic values with different mathematical tools. In this article, we take the approach of Bayesian theories of reinforcement learning [<a href="#B16-brainsci-15-00131" class="html-bibr">16</a>,<a href="#B17-brainsci-15-00131" class="html-bibr">17</a>,<a href="#B18-brainsci-15-00131" class="html-bibr">18</a>]. These Bayesian theories have been especially useful in accounting for the process of inference [<a href="#B16-brainsci-15-00131" class="html-bibr">16</a>]. Here, we expand them by adding choice consistency to the Bayesian framework by incorporating the recent history of the stimuli and decisions into the information provided to the individual. We assume that decisions made with choice consistency are more rewarding or equivalently, less discomforting.</div><div class='html-p'>Analysis of such theoretical frameworks has already shown that even without choice consistency, they tend to produce high individuality. Part of it appears from the exposure of individuals to the different statistics of distinct cultures and environments [<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>]. However, even if the statistics are similar, the stochasticity of the incoming signals causes variability in the values that we learn. This variability is magnified by a redundant space of values, that is, different parameters of the internal model of reward expectation predict the same rewards [<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>]. Our hypothesis is that choice consistency will magnify these individuation tendencies even further because random, hard decisions will separate people more.</div><div class='html-p'>In this article, we develop our Bayesian theoretical framework by adding to it the choice-consistency part. Bayesian theory has been influential in cognitive science, covering areas such as inference, perception, decision-making, and reinforcement learning [<a href="#B16-brainsci-15-00131" class="html-bibr">16</a>,<a href="#B17-brainsci-15-00131" class="html-bibr">17</a>,<a href="#B18-brainsci-15-00131" class="html-bibr">18</a>,<a href="#B19-brainsci-15-00131" class="html-bibr">19</a>,<a href="#B20-brainsci-15-00131" class="html-bibr">20</a>,<a href="#B21-brainsci-15-00131" class="html-bibr">21</a>,<a href="#B22-brainsci-15-00131" class="html-bibr">22</a>,<a href="#B23-brainsci-15-00131" class="html-bibr">23</a>,<a href="#B24-brainsci-15-00131" class="html-bibr">24</a>]. The addition of choice consistency into the Bayesian framework allows for the possible emergence of cognitive individuation. We then use this expanded framework to develop a computational model to perform simulations. This model is rich enough to allow us to reach conclusions about how choice consistency may interact with reward mechanisms to contribute to individuality.</div></section><section id='sec2-brainsci-15-00131' type=''><h2 data-nested='1'> 2. Theoretical Framework</h2><div class='html-p'>We have split the description of the theory into two subsections, physical and mathematical. The “Physical Description” section (<a href="#sec2dot1-brainsci-15-00131" class="html-sec">Section 2.1</a>) has an account of the ideas without any equations. Our goal in that section is to help the reader understand the elements of the theory at an intuitive level. That section may allow readers to skip the equations (<a href="#sec2dot2-brainsci-15-00131" class="html-sec">Section 2.2</a> and <a href="#sec3-brainsci-15-00131" class="html-sec">Section 3</a>) at a first read and go directly to the “Results” (<a href="#sec4-brainsci-15-00131" class="html-sec">Section 4</a>). We also place the mathematical developments in the appendices, leaving only the main equations in the “Mathematical Description” section (<a href="#sec2dot2-brainsci-15-00131" class="html-sec">Section 2.2</a>) to simplify the explanation.</div><section id='sec2dot1-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 2.1. Physical Description</h4><div class='html-p'>The core of a model of choice is that given information, for example, a sensory stimulus, the person must decide what action to take. The person uses a parametric internal model for this decision, with the parameters based on experience involving evolution (genes), development, learning, and choice consistency. How do the parameters change in time? In an optimal situation, parameter evolution should use a Bayesian process, recognizing that stimuli, choices, and rewards tend to be probabilistic. <a href="#brainsci-15-00131-f001" class="html-fig">Figure 1</a> illustrates how the Bayesian update of parameters works.</div><div class='html-p'>Without loss of generality, we will mostly refer in this article to sensory stimuli because all experimental evidence so far has only addressed them. However, as explained in the next section, our theory is compatible with body signals as well. We will address their importance in the Discussion.</div></section><section id='sec2dot2-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 2.2. Mathematical Description</h4><div class='html-p'>At time <math display='inline'><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>, a person receives sensory and/or body signals and must decide what action <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> to take. The best action must balance the maximization of reward, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, and the minimization of the deviation from past actions given similar stimuli. The stimulus <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> is sampled from the probability distribution of inputs, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> </mrow> </semantics></math> (<a href="#brainsci-15-00131-f001" class="html-fig">Figure 1</a>). From that, we use an internal model, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>, to decide on the action to take, where <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> is the vector of parameters learned at time <math display='inline'><semantics> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>. From the action sampled from the model, we can sample the reward <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> from the probability distribution of rewards given actions and stimuli, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <msub> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>.</div><div class='html-p'>In the next step, we update the parameters of the model before the next sampling begins. As mentioned above, this update must simultaneously optimize expected reward and consistency with past actions. So, we must perform an optimization that balances these two constraints. We do this by first finding the Bayesian expected loss for each constraint and then treating the problem as a Compromise Decision Problem [<a href="#B25-brainsci-15-00131" class="html-bibr">25</a>,<a href="#B26-brainsci-15-00131" class="html-bibr">26</a>].</div><div class='html-p'>The key decision in this step is the choice of parameters. To calculate the Bayesian expected choice-consistency loss as a function of the candidate parameters, we need the histories of the sampled stimuli and actions (<a href="#brainsci-15-00131-f001" class="html-fig">Figure 1</a>). We limit these histories to a finite number of time bins, that is, to a finite memory length. For each value of parameters, we can calculate the loss due to errors of choice consistency with these histories. The best parameters maximize the probability that from each stimulus in the past, we obtain the action seen in history. If we take the minus logarithm of this maximal probability, the best parameters minimize the total amount of surprisals in the history of the stimuli and actions. Thus, we choose the parameters that emphasize the most common stimulus–action pairs, weeding out the surprises. This choice yields the Bayesian expected choice-consistency loss <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>. The value of the parameters also affects the Bayesian expected reward loss through the chosen action, and the probability distribution of rewards given actions and stimuli. We denote the reward loss as <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>.</div><div class='html-p'>Finally, we propose working with Kempthorne’s <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>-Bayes-based compromise problem [<a href="#B25-brainsci-15-00131" class="html-bibr">25</a>] to propose that the brain minimizes<div class='html-disp-formula-info' id='FD1-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>λ</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mi>λ</mi> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>+</mo> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>−</mo> <mi>λ</mi> </mrow> </mfenced> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(1)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>λ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>. The parameter <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> simply balances the competition between rewards and choice consistency in decision-making. Equation (1) and the parameter <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> are like what Bayesian cognitive models of cue integration have proposed [<a href="#B27-brainsci-15-00131" class="html-bibr">27</a>,<a href="#B28-brainsci-15-00131" class="html-bibr">28</a>,<a href="#B29-brainsci-15-00131" class="html-bibr">29</a>,<a href="#B30-brainsci-15-00131" class="html-bibr">30</a>].</div><div class='html-p'>Any model based on Equation (1) must specify three mathematical quantities: <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>a</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> </mrow> </semantics></math>. To specify the three probability functions, we must begin with the domains of the variable. For simplicity, the stimulus and reward variables will be one-dimensional with an infinite domain, and we consider a discrete and finite action space. Furthermore, for simplicity, we base the probability distributions on Normal distributions. To begin, we take the distribution of the stimuli to be Normal, with zero mean and a given standard deviation <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD2-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mfrac> <mrow> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msubsup> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </msup> </mrow> <mrow> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(2)</label> </div> </div> Next, we represent the probabilistic relationship between stimulus and action as a sum of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> </semantics></math> Normal distributions with a standard deviation <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </semantics></math>, where the vector of parameters is <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>0</mn> </mrow> </mfenced> </mrow> </msubsup> <mo>,</mo> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>0</mn> </mrow> </mfenced> </mrow> </msubsup> <mo>,</mo> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> </msubsup> </mrow> </mfenced> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD3-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mrow> <munderover> <mo stretchy="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> </munderover> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mfenced separators="|"> <mrow> <mi>a</mi> <mo>−</mo> <mfenced separators="|"> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>0</mn> </mrow> </mfenced> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> </msubsup> <mi>s</mi> </mrow> </mfenced> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msubsup> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mstyle> </mrow> </msup> </mrow> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(3)</label> </div> </div> Hence, the components of the vector of parameters map linearly to the means of the Normal distributions selecting the actions. Finally, we also make the reward function a Normal distribution, standing for the probability of reward of each decision:<div class='html-disp-formula-info' id='FD4-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>a</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mfenced separators="|"> <mrow> <mi>r</mi> <mo>−</mo> <mi>α</mi> <mfenced separators="|"> <mrow> <mi>a</mi> <mo>−</mo> <mi>β</mi> </mrow> </mfenced> <mo>∗</mo> <mi>s</mi> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msubsup> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mstyle> </mrow> </msup> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(4)</label> </div> </div> where the parameters <math display='inline'><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math> set the relationship between the stimulus and the action to the mean reward for that pair, and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> is the standard deviation.</div></section></section><section id='sec3-brainsci-15-00131' type='methods'><h2 data-nested='1'> 3. Methods</h2><section id='sec3dot1-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 3.1. Algorithm of the Computer Simulations</h4><div class='html-p'>The simulation uses discrete time steps, labeled <math display='inline'><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>. At time <math display='inline'><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>, the model begins with the last estimation of the choice parameters, namely, <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math>. From this time, the simulations continue as follows:</div><div class='html-p'><ul class='html-alpha-lower'><li><div class='html-p'>Sample <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> (Equation (2)).</div></li><li><div class='html-p'>Add <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> to the history of stimuli (<a href="#secAdot1-brainsci-15-00131" class="html-sec">Appendix A.1</a>).</div></li><li><div class='html-p'>Sample <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> (Equation (3)).</div></li><li><div class='html-p'>Add <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> to the history of rewards (<a href="#secAdot1-brainsci-15-00131" class="html-sec">Appendix A.1</a>).</div></li><li><div class='html-p'>Sample <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> (Equation (4)).</div></li><li><div class='html-p'>If not enough time has passed to accumulate enough history, go back to Step a.</div></li><li><div class='html-p'>Otherwise, calculate <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> by minimizing Equation (1).</div></li><li><div class='html-p'>Go back to Step a.</div></li></ul></div><div class='html-p'>The minimization in Step g uses the simplex search method of Lagarias et al. [<a href="#B31-brainsci-15-00131" class="html-bibr">31</a>].</div></section><section id='sec3dot2-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 3.2. Parameters of the Simulations</h4><div class='html-p'>In this article, we report on simulations with different parameter sets to explore the model (see the end of <a href="#secAdot2-brainsci-15-00131" class="html-sec">Appendix A.2</a> for a complete list). We have chosen one of these sets as our standard set. We also show simulations with other parameter sets to illustrate individual differences and analyze the various behaviors of the model. <a href="#brainsci-15-00131-t001" class="html-table">Table 1</a> below shows the parameters of the standard simulations; we discuss others when presenting the results.</div><div class='html-p'>We have set <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> in this standard set because the emphasis of this article is on choice consistency, not rewards. Consequently, most simulations here are without the interference of rewards on the effects of choice consistency.</div></section></section><section id='sec4-brainsci-15-00131' type='results'><h2 data-nested='1'> 4. Results</h2><section id='sec4dot1-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 4.1. Algorithm of the Computer Simulations</h4><div class='html-p'>The goal of this article is to estimate whether people’s past choices may affect current ones significantly. The experimental observation is that choices of the past and present tend to be consistent. We have proposed a Bayesian theory for this consistency phenomenon. In this theory, choice consistency appears to adhere statistically to the recent history of stimuli and actions (<a href="#brainsci-15-00131-f001" class="html-fig">Figure 1</a>). Mathematically, such statistical adherence arises through the minimization of the total amount of surprisals in this history by judicial choice of the model parameters (Equation (1)). A typical computer simulation of a simple model based on such a total-surprisal theory of consistency appears in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>.</div><div class='html-p'>The results of the computer simulations help us to understand how choice consistency develops over time. In the beginning, if choices are hard, they stay statistically split between the possible actions (<a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A). However, because the choices are initially random, a small statistical imbalance appears as a function of the stimuli. The proposed minimization of total surprisals then takes over, slowly expanding this imbalance and breaking the symmetry. In <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A,B, this expansion causes Action 1 to occur mostly for positive stimuli and Action 2 for negative ones. This separation between the choices is initially slow but then reveals the characteristics of phase transition. A way to understand this phase transition is by seeing the total consistency loss (<a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>C—Equation (3)). After a first climb resulting from the addition of stimuli and actions, the loss collapses, reaching a low plateau at around <math display='inline'><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>55</mn> </mrow> </semantics></math>. This collapse is accompanied by a sudden change in the second model parameter (<a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>D). Other simulations yield comparable results, although the time of the phase transition and what parameters change can vary.</div><div class='html-p'>How stable is the division of choices such as those in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>? On one hand, one may expect major stability because after a division occurs it could self-perpetuate through the minimization of surprisals (Equation (1)). On the other hand, random sampling of choice could continue to bias one choice over another. This bias would then become stronger by symmetry-breaking mechanisms. Thus, one should not be surprised if eventually one choice swamps the other, taking over in perpetuity. <a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a> assesses this idea by lengthening the simulation.</div><div class='html-p'><a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a>A shows that with enough time, one choice may swamp the others. In this simulation, the equilibrium between the choices of Actions 1 and 2 persists until about <math display='inline'><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>30</mn> </mrow> </semantics></math>. After that time, choices of Action 2 for positive stimuli start breaking the symmetry. By <math display='inline'><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>100</mn> </mrow> </semantics></math>, most choices now go to Action 2, even in situations of slightly negative stimuli. Finally, after <math display='inline'><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math>, the choice of Action 1 is eliminated, with the individual always picking Action 2. With forty simulations as in <a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a>, Action 1 or Action 2 always wins out, and the mean time at which one of the actions disappears as a choice is <math display='inline'><semantics> <mrow> <mn>160</mn> <mo>±</mo> <mn>80</mn> </mrow> </semantics></math> (standard deviation). Hence, the division of choices is not stable, and even with consistency, the brain may choose to drop some of them. The broad range of choice-disappearance times (<a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a>; blue bins) also implies individuality in the stability of decision-making based on the history of selections.</div><div class='html-p'>The length of memory (<math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math>) should influence stability and thus, the time when one choice swamps the other. One would expect that the longer this length is, the more stable the choice dependence on stimulus becomes. This stability should stem from the increased evidence for a particular discrimination of choices. One would expect that the stability would increase with the square root of <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math>. Such dependence should hold because as a rule of thumb, the variability of the signal decreases by roughly the square root of the number of points averaged [<a href="#B32-brainsci-15-00131" class="html-bibr">32</a>]. To test this prediction, we run the simulations with larger lengths of memory.</div><div class='html-p'>The results in <a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a>B show a significant choice-stabilization effect as the length of memory increases. The red bins (<math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>) extend to longer times than the blue ones (<math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>). The mean time when one choice takes over the other with <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math> is <math display='inline'><semantics> <mrow> <mn>450</mn> <mo>±</mo> <mn>240</mn> </mrow> </semantics></math>. This time is statistically significantly longer than that reported above for <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math> (one-sided <span class='html-italic'>t</span>-test, <math display='inline'><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>5.32</mn> </mrow> </semantics></math>, 78 degrees of freedom, <math display='inline'><semantics> <mrow> <mi>p</mi> <mo><</mo> <msup> <mrow> <mn>5</mn> <mo>×</mo> <mn>10</mn> </mrow> <mrow> <mo>−</mo> <mn>7</mn> </mrow> </msup> </mrow> </semantics></math>). The difference in stoppage times is <math display='inline'><semantics> <mrow> <mo>≈</mo> </mrow> </semantics></math>290 iterations despite <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math> increasing by only <math display='inline'><semantics> <mrow> <mn>20</mn> </mrow> </semantics></math>. If one were to use the square root law theorized above, the increase in stoppage time by raising <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math> by a factor of <math display='inline'><semantics> <mrow> <mn>2</mn> </mrow> </semantics></math> should have been only <math display='inline'><semantics> <mrow> <mn>160</mn> <mo>∗</mo> <msqrt> <mn>2</mn> </msqrt> <mo>−</mo> <mn>160</mn> <mo>≈</mo> <mn>70</mn> </mrow> </semantics></math>. Consequently, the effect of <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math> on stoppage time is much more than simply increasing sampling. In contrast, the percentage of time in which the full discrimination of actions by stimuli occurs is <math display='inline'><semantics> <mrow> <mn>73</mn> <mo>%</mo> <mo>±</mo> <mn>7</mn> <mo>%</mo> </mrow> </semantics></math> for <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo>=</mo> <mn>40</mn> </mrow> </semantics></math>. This percentage is not statistically significantly different from that seen for <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>.</div></section><section id='sec4dot2-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 4.2. Individuation of Values from Choice Consistency</h4><div class='html-p'>The spontaneous symmetry breaking with choice consistency revealed in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a> and <a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a> suggests the possibility of another surprising phenomenon. Given that spontaneous symmetry breaking starts with small statistical imbalances, if these were distinct for different individuals, they may display completely different patterns of choices over time. Thus, random first hard choices may lead to later individuation even for identical observers. To evaluate this individuation-by-choice idea, we repeated the simulations of <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a> and <a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a> multiple times. The results of these simulations and their implications for individuation appear in <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>.</div><div class='html-p'><a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a> shows that the symmetry breaking induced by choice consistency can lead to strong individuality. This individuality expresses itself in multiple forms. The most usual form (<math display='inline'><semantics> <mrow> <mn>80</mn> <mo>%</mo> <mo>±</mo> <mn>6</mn> <mo>%</mo> </mrow> </semantics></math> in forty simulations; standard error) is that positive stimuli eventually give rise to Action 1 (<a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>D,G,H) or Action 2 (<a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>A,C,E), and vice versa for negative stimuli. Another form is that the chosen actions stay intermingled until one of them takes over (<a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>B—<math display='inline'><semantics> <mrow> <mn>12</mn> <mo>%</mo> <mo>±</mo> <mn>5</mn> <mo>%</mo> </mrow> </semantics></math>). Finally, an action may occasionally correspond to positive stimuli early and negative ones later (<a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>F—<math display='inline'><semantics> <mrow> <mn>8</mn> <mo>%</mo> <mo>±</mo> <mn>4</mn> <mo>%</mo> </mrow> </semantics></math>). Another important observation is that the time of separation of actions by stimuli varies, occurring early (<a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>A) or late (<a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>D). Therefore, early choices can cause strong individuality that may remain for a long time.</div><div class='html-p'>Another key observation is that the stimuli for which the model makes a choice are not constant but drift continuously. As mentioned above, one can see this in <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>F. However, even when the order of the stimuli yielding an action does not change, one can see a drift in the space of choices. For example, in <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>A,C, the stimuli giving rise to Action 1 become increasingly negative. Consequently, the selection of actions under choice consistency shows temporal instability.</div><div class='html-p'>We have also reasoned that the individuation may become more pronounced if the number of choice parameters (<math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>—Equation (3)) increases. The logic is that with more parameters, the set of choices becomes larger. This logic follows from a recent study showing that parametric redundancy increases individuality when performing reinforcement learning of aesthetic values with stochastic inputs [<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>]. Alternatively, an increase in the number of parameters may instead lead to more chaos and thus, apparently less organized choices. <a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a> shows a test of these alternatives. This test uses standard parameters (<a href="#brainsci-15-00131-t001" class="html-table">Table 1</a>), except that we have four choice parameters (<math display='inline'><semantics> <mrow> <msub> <mrow> <mn>2</mn> <mo>×</mo> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> <mo>)</mo> </mrow> </semantics></math> instead of two (<math display='inline'><semantics> <mrow> <msub> <mrow> <mn>2</mn> <mo>×</mo> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math>.</div><div class='html-p'>The increase in the number of choice parameters from two to four boosts individuation by adding new choice behaviors. Most simulations (out of forty) still give behaviors as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>, <a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a> and <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a> (<math display='inline'><semantics> <mrow> <mn>70</mn> <mo>%</mo> <mo>±</mo> <mn>7</mn> <mo>%</mo> </mrow> </semantics></math>). However, three new behaviors appear with more parameters. The first, and the rarest, is the continued intermingling of Actions 1 and 2 regardless of the stimuli (<math display='inline'><semantics> <mrow> <mn>2.5</mn> <mo>%</mo> <mo>±</mo> <mn>2.5</mn> <mo>%</mo> </mrow> </semantics></math> of the simulations—<a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>A). The second is cases in which one choice shifts to different stimuli during the simulation (<math display='inline'><semantics> <mrow> <mn>7.5</mn> <mo>%</mo> <mo>±</mo> <mn>4</mn> <mo>%</mo> </mrow> </semantics></math> of the simulations). Thus, in <a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>B, Action 2 tends to occur with negative stimuli until about <math display='inline'><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>60</mn> </mrow> </semantics></math>, but then shifts to positive ones after that. Finally, the third is a behavior in which a choice at intermediate stimuli (<math display='inline'><semantics> <mrow> <mo>≈</mo> <mn>0</mn> </mrow> </semantics></math> in <a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>C) has the other choice for higher and lower stimuli (<math display='inline'><semantics> <mrow> <mo>≈</mo> </mrow> </semantics></math>−1 and <math display='inline'><semantics> <mrow> <mo>≈</mo> </mrow> </semantics></math>1 in <a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>C). We have even seen such a sandwich behavior with four bands, for example, Action 1, Action 2, Action 1, and Action 2, as the stimulus increases. This sandwich behavior happens in <math display='inline'><semantics> <mrow> <mn>20</mn> <mo>%</mo> <mo>±</mo> <mn>6</mn> <mo>%</mo> </mrow> </semantics></math> of the simulations.</div><div class='html-p'>This increase in individuation can be understood in terms of the larger variability of the choice parameters. The intermingling, non-discriminating behavior in <a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>A results from the dominance of one of the parameters over the others (<a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>D). Such dominance reduces the ability to make discriminating choices. In turn, the sudden shift in the choice of Action 2 in <a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>B results from a phase transition in three of the four parameters (<a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>E). Finally, the sandwich behavior (<a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>C) arises from a delicate balance between the four choice parameters (<a href="#brainsci-15-00131-f005" class="html-fig">Figure 5</a>F). Such richness of behaviors is not possible with just two choice parameters because they have a more limited repertoire (<a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>D).</div></section><section id='sec4dot3-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 4.3. The Effect of the Number of Choices</h4><div class='html-p'>So far, the simulations have focused on two choices (<math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math>) because that is the most common situation in laboratory settings [<a href="#B7-brainsci-15-00131" class="html-bibr">7</a>]. However, in the real world, the number of choices is often larger. We thus asked in what ways this number would affect the results. Our first prediction was that in a constrained space of stimuli, we could not achieve full discrimination of choices when the number of actions became high. <a href="#brainsci-15-00131-f006" class="html-fig">Figure 6</a> shows a test of this prediction by using standard parameters (<a href="#brainsci-15-00131-t001" class="html-table">Table 1</a>), except that we have more than two choices (<math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mo>></mo> <mn>2</mn> </mrow> </semantics></math>).</div><div class='html-p'>The results show that with the standard parameters, except for three choices (<math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math>), the model can segment the actions into different regions of the stimulus space (<a href="#brainsci-15-00131-f006" class="html-fig">Figure 6</a>A,B). In forty simulations, such three-way segmentation happens often (<math display='inline'><semantics> <mrow> <mn>68</mn> <mo>%</mo> <mo>±</mo> <mn>7</mn> <mo>%</mo> </mrow> </semantics></math>). Hence, the incidence of three-way segmentation is not statistically significantly different from that seen with two choices. When the segmentation does not happen, we see either the intermingling of choices (as in <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>B; <math display='inline'><semantics> <mrow> <mn>25</mn> <mo>%</mo> <mo>±</mo> <mn>7</mn> <mo>%</mo> </mrow> </semantics></math>) or a two-way segmentation with two choices combined (<math display='inline'><semantics> <mrow> <mn>7</mn> <mo>%</mo> <mo>±</mo> <mn>4</mn> <mo>%</mo> </mrow> </semantics></math>). When the three-way segmentation happens, one of the actions comes to dominate eventually (<a href="#brainsci-15-00131-f006" class="html-fig">Figure 6</a>A,B), as for two choices (<a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a>). The stoppage times of the two losing actions are almost identical (<a href="#brainsci-15-00131-f006" class="html-fig">Figure 6</a>C). Plotting the second stoppage time against the first reveals an almost perfect correlation (<math display='inline'><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>0.996</mn> <mo>±</mo> <mn>0.003</mn> </mrow> </semantics></math>), with linear regression giving an intercept of <math display='inline'><semantics> <mrow> <mn>21</mn> <mo>±</mo> <mn>4</mn> </mrow> </semantics></math> (standard error) and a slope of <math display='inline'><semantics> <mrow> <mn>1.00</mn> <mo>±</mo> <mn>0.01</mn> </mrow> </semantics></math>. Therefore, the two stoppage times are about a constant 20 iterations apart. Moreover, these two stoppage times tend to occur later than when the stimulations had two choices (<a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a>). The first stoppage time with three choices is at <math display='inline'><semantics> <mrow> <mn>220</mn> <mo>±</mo> <mn>160</mn> </mrow> </semantics></math>, while the second is at <math display='inline'><semantics> <mrow> <mn>240</mn> <mo>±</mo> <mn>150</mn> </mrow> </semantics></math>.</div><div class='html-p'>One of the most interesting results in <a href="#brainsci-15-00131-f006" class="html-fig">Figure 6</a>A is that the stimulus range for Action 2 is narrow. In all simulations, the actions sandwiched between the other two have a narrow range. This narrowness is not surprising because different from the outer actions, the inner one has little room to expand. The narrowness of the inner action allows us to inspect in more detail the instability of the choices. In <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>, the instability appears as choices corresponding to stimuli that are more negative over time. However, the narrowness of the range of stimuli of Action 2 in <a href="#brainsci-15-00131-f006" class="html-fig">Figure 6</a>A,B reveals that the instability can show richer behavior. In these figures, the stimuli yielding Action 2 rise slowly initially and then fall rapidly. In multiple simulations, we have seen different rise-and-fall behaviors for the inner action.</div><div class='html-p'>We can still achieve full discrimination of actions by stimuli when the simulations run with more than three choices (<math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mo>></mo> <mn>3</mn> </mrow> </semantics></math>—<a href="#brainsci-15-00131-f006" class="html-fig">Figure 6</a>D). However, full discrimination becomes rarer as the number of choices increases. With forty simulations, the percentage of time to achieve full discrimination falls to <math display='inline'><semantics> <mrow> <mn>53</mn> <mo>%</mo> <mo>±</mo> <mn>5</mn> <mo>%</mo> </mrow> </semantics></math> with four choices (<math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>). This percentage collapses with five choices or more, becoming <math display='inline'><semantics> <mrow> <mn>10</mn> <mo>%</mo> <mo>±</mo> <mn>5</mn> <mo>%</mo> </mrow> </semantics></math> with <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mn>0</mn> <mo>%</mo> </mrow> </semantics></math> with <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>. We do not achieve any full discrimination with six choices or more (<math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mo>></mo> <mn>6</mn> </mrow> </semantics></math>) with our standard parameters. This lack of discrimination is consistent with our first prediction that in a constrained space of stimuli, one cannot achieve full discernment of choices when the number of actions is high.</div></section><section id='sec4dot4-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 4.4. The Interaction Between Reward and Choice Consistency</h4><div class='html-p'>The simulations have focused so far on choice consistency. However, in the real world, choices can be affected by both their consistency and their elicited rewards [<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>,<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>,<a href="#B15-brainsci-15-00131" class="html-bibr">15</a>]. We expect that if a reward favors the relation relationship between certain stimuli and an action, then this incentive would tend to bias the final discrimination. Consequently, we expect a reduction in the contribution to individuation by choice consistency. This reduction would not mean less total individuality because rewards can cause individuation by themselves [<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>,<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>]. To test this predicted competition between rewards and choice consistency, we use the standard parameters, except for varying the value of <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>, the parameter weighing the impact of these two factors (<a href="#brainsci-15-00131-f007" class="html-fig">Figure 7</a>). The design of the standard parameters is such that Actions 1 and 2 will obtain rewards for negative and positive stimuli, respectively.</div><div class='html-p'><a href="#brainsci-15-00131-f007" class="html-fig">Figure 7</a>A shows that if only rewards affect the choices (<math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>), then they will be such as to maximize the rewards. For the standard parameters, this maximization means that negative and positive stimuli will yield Action 1 and Action 2, respectively. Furthermore, different from what happens with choice consistency alone, no action dominates the other even after a long time. However, as <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> falls, the percentage of simulations for which action-2 stimuli converge to values larger than those for Action 1 decreases (<a href="#brainsci-15-00131-f007" class="html-fig">Figure 7</a>B,C). This curve has a sigmoidal shape as a function of <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>, with the change in dominance from Action 1 to Action 2 occurring between <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>≈</mo> <mn>0.4</mn> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>≈</mo> <mn>0.9</mn> </mrow> </semantics></math> for the standard parameters. Another important result is that the stoppage time of the losing action increases with <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> (<a href="#brainsci-15-00131-f007" class="html-fig">Figure 7</a>B,D). For the standard parameters, the stoppage time curve rises rapidly after about <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math>. This rise tends towards infinity because no action ever stops for <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. Hence, in the competition between reward and choice consistency, the model picks the action as a hybrid between them.</div></section></section><section id='sec5-brainsci-15-00131' type='discussion'><h2 data-nested='1'> 5. Discussion</h2><section id='sec5dot1-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 5.1. Contribution to Individuation by Symmetry Breaking in the Minimization of Surprisals</h4><div class='html-p'>We have proposed that the brain implements choice consistency by a minimization of the sum of surprisals [<a href="#B33-brainsci-15-00131" class="html-bibr">33</a>]. Thus, we propose minimizing unexpected actions given both the current stimulus and the history of stimuli and actions. Such a minimization has been used in other cognitive-science contexts, such as in perception [<a href="#B34-brainsci-15-00131" class="html-bibr">34</a>,<a href="#B35-brainsci-15-00131" class="html-bibr">35</a>] and active inference [<a href="#B36-brainsci-15-00131" class="html-bibr">36</a>]. Other models are possible for choice consistency, of course, but minimization of surprisal appears naturally from the Bayesian framework. We will discuss these alternate models below.</div><div class='html-p'>Our results show that such a surprisal-minimization mechanism of choice consistency can contribute to the individuation of values. Individuality means variance within and across people [<a href="#B37-brainsci-15-00131" class="html-bibr">37</a>]. Consequently, because individuality means that different people are distinct and because the simulations all have the same initial conditions, the individual uniqueness implies that the system undergoes spontaneous symmetry breaking [<a href="#B38-brainsci-15-00131" class="html-bibr">38</a>,<a href="#B39-brainsci-15-00131" class="html-bibr">39</a>,<a href="#B40-brainsci-15-00131" class="html-bibr">40</a>]. In physical systems, spontaneous symmetry breaking implies equations of motion obeying symmetries, as well as a lowest-energy state without the same symmetries. Our equations of motion are symmetric in that stochastic sampling allows any action to connect with any stimulus. However, when a connection between certain neighbor stimuli and an action becomes statistically stronger than others, the minimization of surprisals makes future choices compatible with this link, reinforcing it. This is positive feedback because this reinforcement makes the connection itself stronger. Such positive feedback is not unique to surprisal-minimization theories. But whatever the theory used, choice consistency will reinforce the connection between certain past stimuli and an action. Such a spontaneous symmetry breaking of choices with positive feedback is bound to generate phase transitions [<a href="#B39-brainsci-15-00131" class="html-bibr">39</a>]. We see phase-transition behavior in how the choice parameters change in our simulations.</div><div class='html-p'>Another result is that increasing the number of choice parameters strengthens the individuation tendency. Parametric redundancy has been studied recently as a way to increase system reliability [<a href="#B41-brainsci-15-00131" class="html-bibr">41</a>]. Such redundancy allows the automatic re-assignment of tasks performed by a basic element to a backup one. From our perspective, redundancy allows for the generation of optimal parametric surfaces instead of points [<a href="#B42-brainsci-15-00131" class="html-bibr">42</a>]. Therefore, because solutions can exist at different points on the surfaces, we can achieve individuality. As such, the individuation of values achieved with our minimal-surprisal theory of choice consistency has a relation to that achieved with reinforcement learning of rewards, which also has parametric redundancy [<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>].</div></section><section id='sec5dot2-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 5.2. Why a Choice Always Eventually Dominates</h4><div class='html-p'>We have already seen that spontaneous symmetry breaking causes different chosen actions to occupy distinct portions of the stimulus space. However, spontaneous symmetry breaking has another effect, namely, in every simulation, a choice ends up dominating the others after enough time has elapsed. To understand this effect, let us begin with the simplest example, that of <a href="#brainsci-15-00131-f003" class="html-fig">Figure 3</a>A. In this example, the model initially samples the action probabilistically. With the initial condition of the choice parameters, the mean of the underlying probability distribution is exactly in the middle between the two possible actions. However, over time, the mean moves statistically in the direction of one of the actions. This movement causes the parameters to shift towards the action, producing positive feedback. It, in turn, causes a phase transition that ends up dropping the alternate action. Its removal is a form of spontaneous symmetry breaking because the equations of motion do not favor any action in the beginning.</div><div class='html-p'>This spontaneous symmetry-breaking process can also account for two features of the times at which the losing actions stop: First, the stoppage times have a broad range. Such a range occurs because the model is stochastic and must reach the phase transition point. The distribution of first-passage times through this point is broad when the stimuli do not have bounds, as in our case [<a href="#B43-brainsci-15-00131" class="html-bibr">43</a>,<a href="#B44-brainsci-15-00131" class="html-bibr">44</a>,<a href="#B45-brainsci-15-00131" class="html-bibr">45</a>]. Second, when more than two choices are available, the losing actions tend to die at similar times. The positive-feedback process for choice elimination explained in the last paragraph works here as well. As the parameters move towards the winning choice, the process reinforces the motion itself in a positive-feedback loop. Hence, the winning choice becomes stronger, ending the others quickly in a phase transition.</div></section><section id='sec5dot3-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 5.3. How to Stabilize the Effect of Choice Consistency</h4><div class='html-p'>Can one change the theory or model to stabilize choices such that actions do not stop after a while? From the simulations, the strongest factor stabilizing the actions is the length of memory. Consequently, a way to stabilize the choices is to make the length of memory infinitely long. Of course, the brain cannot remember all the pairs of choices and stimuli in the distant past. However, an alternative is to accumulate past choices by using them to adjust the parameters of a model, such as in reinforcement learning [<a href="#B46-brainsci-15-00131" class="html-bibr">46</a>,<a href="#B47-brainsci-15-00131" class="html-bibr">47</a>]. Such adjustment can consider as much of the past as one wishes.</div><div class='html-p'>We emphasize that the instability is in part due to the use of a continuous stimulus space. In the laboratory, the number of stimuli is typically finite [<a href="#B5-brainsci-15-00131" class="html-bibr">5</a>,<a href="#B6-brainsci-15-00131" class="html-bibr">6</a>,<a href="#B7-brainsci-15-00131" class="html-bibr">7</a>]. With a finite set of well-separated stimuli, an action cannot easily invade the space of another one. This separation then results in more stability that we see in our simulations. Future experiments with longer durations and/or with continuous stimulus variables can test the instability predictions of this article.</div></section><section id='sec5dot4-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 5.4. How to Get More Choices Represented</h4><div class='html-p'>With our model and the standard parameters, only up to five choices could fully be discriminated by the stimuli eliciting them. Can we change the theory or the model to allow for the discrimination of more choices? The simplest such change is the reduction in the standard deviation of the connection between stimuli and choices. Such a reduction would separate the actions more with respect to the stimuli. Another way to improve the discrimination of actions is to use a nonlinear model for the relationship between stimuli and the mean of the distribution. If well designed, such a nonlinearity could allow a better separation between an action and its neighbor. Finally, the model could use a non-Normal distribution for the stimuli to counter the tendency of Normal distributions to bunch up the outputs around the mean. Instead, leptokurtotic distributions tend to have fatter, longer tails, allowing for more discrimination of actions [<a href="#B48-brainsci-15-00131" class="html-bibr">48</a>]. Such distributions occur in natural [<a href="#B49-brainsci-15-00131" class="html-bibr">49</a>,<a href="#B50-brainsci-15-00131" class="html-bibr">50</a>,<a href="#B51-brainsci-15-00131" class="html-bibr">51</a>] and human-made environments [<a href="#B52-brainsci-15-00131" class="html-bibr">52</a>].</div></section><section id='sec5dot5-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 5.5. Reward Versus Choice Consistency</h4><div class='html-p'>The competition between reward and choice-consistency mechanisms in the context of individuation merits a discussion. Reward mechanisms have been shown to elicit individuation in at least three ways: First, reinforcement learning of prediction of reward leads individuals from diverse cultures to develop different values [<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>,<a href="#B53-brainsci-15-00131" class="html-bibr">53</a>,<a href="#B54-brainsci-15-00131" class="html-bibr">54</a>]. Second, interoceptive (body) inputs to the brain modulate the learning of aesthetic values in ways that are individual [<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>,<a href="#B13-brainsci-15-00131" class="html-bibr">13</a>]. Third, the parametric redundancy intrinsic to reinforcement learning of rewards leads to individuation (<a href="#sec5dot1-brainsci-15-00131" class="html-sec">Section 5.1</a>). In this article, we show that parametric redundancy also helps magnify the individuality generated by choice consistency. And although we have emphasized sensory stimuli, the theory for choice consistency is also compatible with interoceptive signals (<a href="#sec2-brainsci-15-00131" class="html-sec">Section 2</a>). This compatibility adds further individuation power to choice-consistency mechanisms.</div><div class='html-p'>The question for us is as follows: are reward and choice mechanisms for individuation independent, adding to each other linearly, or do they interact in a nonlinear way? Our results show that this interaction is nonlinear. The more the model considers rewards, the more biased the choices become, to the point of occasionally disregarding consistency with the past. Thus, if choice consistency is akin to avoiding cognitive dissonance, we may avoid it if the price is right. However, we can also see this competition from the opposite perspective to reach a well-known startling conclusion. People often have so much discomfort with cognitive dissonance that they may make choices that are irrational from a reward (utility) perspective [<a href="#B55-brainsci-15-00131" class="html-bibr">55</a>,<a href="#B56-brainsci-15-00131" class="html-bibr">56</a>,<a href="#B57-brainsci-15-00131" class="html-bibr">57</a>]. To bring back the discussion to individuality, different people have distinct degrees of aversion to cognitive dissonance [<a href="#B58-brainsci-15-00131" class="html-bibr">58</a>,<a href="#B59-brainsci-15-00131" class="html-bibr">59</a>,<a href="#B60-brainsci-15-00131" class="html-bibr">60</a>]. Such a difference in degrees of aversion implies an inter-individual variability in the weights that people use to balance reward versus choice consistency. Therefore, although rewards may reduce the individuation due to choice consistency, the competition between these factors may vary across individuals.</div></section><section id='sec5dot6-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 5.6. Practical Applications</h4><div class='html-p'>The results described in this article on the effects of rewards and choice consistency may have practical applications for society. The next section will address implications for cognitive psychology, while here, we provide three examples of real-world applications: First, in the marketing and behavioral economics fronts, rewards would bias initial consumer choices that would then linger for a long time [<a href="#B3-brainsci-15-00131" class="html-bibr">3</a>,<a href="#B8-brainsci-15-00131" class="html-bibr">8</a>,<a href="#B9-brainsci-15-00131" class="html-bibr">9</a>,<a href="#B61-brainsci-15-00131" class="html-bibr">61</a>,<a href="#B62-brainsci-15-00131" class="html-bibr">62</a>,<a href="#B63-brainsci-15-00131" class="html-bibr">63</a>]. Second, choice consistency has been used in clinical decision-making [<a href="#B64-brainsci-15-00131" class="html-bibr">64</a>], especially for individuals with intellectual and developmental disabilities [<a href="#B64-brainsci-15-00131" class="html-bibr">64</a>]. Third, what will happen when we start producing artificial intelligence (AI) systems that learn autonomously? Currently, such systems depend on their programmers. However, imagine, for example, an AI system in a spaceship exploring outer space. Because in these situations, one does not know what one may find, perhaps these systems should make their own choices, learning from them. And as for humans on Earth, perhaps choice consistency and free reinforcement learning may make sense as a part of how these systems learn. A consequence of this learning will be the breaking of symmetry and individuation of these AI systems.</div></section><section id='sec5dot7-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 5.7. Relationship with Other Studies of Individuation and Experimental Predictions</h4><div class='html-p'>The brain has mechanisms to allow the individuation of cognitive values to continue throughout life [<a href="#B65-brainsci-15-00131" class="html-bibr">65</a>,<a href="#B66-brainsci-15-00131" class="html-bibr">66</a>,<a href="#B67-brainsci-15-00131" class="html-bibr">67</a>,<a href="#B68-brainsci-15-00131" class="html-bibr">68</a>]. They are complex and not compatible with trait models of individuality [<a href="#B69-brainsci-15-00131" class="html-bibr">69</a>]. Thus, these mechanisms clash with the idea that personality is made up of stable characteristics that influence how people think, feel, and behave in different situations [<a href="#B70-brainsci-15-00131" class="html-bibr">70</a>,<a href="#B71-brainsci-15-00131" class="html-bibr">71</a>,<a href="#B72-brainsci-15-00131" class="html-bibr">72</a>]. Instead, individuation is associated with value instability [<a href="#B3-brainsci-15-00131" class="html-bibr">3</a>,<a href="#B4-brainsci-15-00131" class="html-bibr">4</a>]. In this article and elsewhere, we propose that these differences are due in part to brain mechanisms of stochastic learning [<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>,<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>,<a href="#B73-brainsci-15-00131" class="html-bibr">73</a>]. The brain valuation system has been extensively studied [<a href="#B74-brainsci-15-00131" class="html-bibr">74</a>] and the involvement of reward-learning systems, such as the basal ganglia, has been established [<a href="#B13-brainsci-15-00131" class="html-bibr">13</a>]. Such reward-learning systems may also help in learning choice consistency. This is because choice inconsistency may be related to cognitive dissonance (<a href="#sec5dot5-brainsci-15-00131" class="html-sec">Section 5.5</a>), which can be viewed as a negative reward. Other areas involved in choice inconsistency include the ventromedial prefrontal cortex, anterior cingulate cortex, and posterior cingulate cortex [<a href="#B75-brainsci-15-00131" class="html-bibr">75</a>]. Another result of interest related to the cognition of choice consistency is that taxing cognitive capacities reduces choice consistency [<a href="#B76-brainsci-15-00131" class="html-bibr">76</a>]. An explanation for this reduction is that under cognitive taxation, people make impulsive, irrational choices [<a href="#B76-brainsci-15-00131" class="html-bibr">76</a>,<a href="#B77-brainsci-15-00131" class="html-bibr">77</a>,<a href="#B78-brainsci-15-00131" class="html-bibr">78</a>]. Alternatively, people change their decision strategies to simpler ones when the cognitive load is higher [<a href="#B76-brainsci-15-00131" class="html-bibr">76</a>,<a href="#B79-brainsci-15-00131" class="html-bibr">79</a>].</div><div class='html-p'>This discussion on the cognitive mechanisms of choice consistency suggests new experiments to assess the computational models of this article. In these experiments, we imagine subjects first performing a continuous Likert rating [<a href="#B80-brainsci-15-00131" class="html-bibr">80</a>] of multiple stimuli organized according to a variable (for example, degree of complexity). Afterwards, we perform two alternative forced choices of pairs of these stimuli, especially those similarly liked during the Likert rating. We then return later to the Likert rating to see if the forced choice breaks the symmetry. Moreover, we repeat the rating days or weeks later to evaluate if the individuality is “permanent”. This general rating technique allows us to answer other questions. For example, does taxing cognitive capacities affect the learning of the choices or just the later decisions? Similarly, does taxing cognitive capacities cause impulsive choices or a simplified decision strategy? To answer these questions, we can repeat the later measurements to study whether the choices are inconsistent with the past but internally consistent. Finally, we can use modified versions of these kinds of measurements to evaluate the interaction between reward and choice consistency.</div></section><section id='sec5dot8-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> 5.8. Do Hard Choices Have an Effect on the Individuation of Values?</h4><div class='html-p'>In this section, we go back to the issue raised by the title of the article, namely, “the Effect of Hard Choices on the Individuation of Values”. To start, let us consider what hard choices are. They emerge in four main situations [<a href="#B81-brainsci-15-00131" class="html-bibr">81</a>,<a href="#B82-brainsci-15-00131" class="html-bibr">82</a>]: First, one often makes choices with incomplete information. Second, hard choices often arise from complex challenges, that is, those for which calculating an ideal action is difficult. Third, hard choices also arise from incomplete preferences, often resulting from novel situations, for which people have not yet developed clear values. Research has shown that behavior that is indicative of incomplete preferences is empirically associated with deliberate randomization as we use in our simulations [<a href="#B83-brainsci-15-00131" class="html-bibr">83</a>]. Fourth, when making choices, people must regularly juggle competing values, and no decision satisfies them all. Sometimes choice conflicts arise between one’s important values, causing negative emotions [<a href="#B84-brainsci-15-00131" class="html-bibr">84</a>]. Such negative emotions can contribute to perceived decision difficulty [<a href="#B85-brainsci-15-00131" class="html-bibr">85</a>,<a href="#B86-brainsci-15-00131" class="html-bibr">86</a>,<a href="#B87-brainsci-15-00131" class="html-bibr">87</a>].</div><div class='html-p'>Do these four situations occur often enough in everyday life to make hard choices significant factors in the individuation of values? According to some estimates, a person makes thousands of choices a day [<a href="#B88-brainsci-15-00131" class="html-bibr">88</a>,<a href="#B89-brainsci-15-00131" class="html-bibr">89</a>]. These choices include, for example, what to eat [<a href="#B90-brainsci-15-00131" class="html-bibr">90</a>], what to wear [<a href="#B91-brainsci-15-00131" class="html-bibr">91</a>], and when to slow down when driving a car [<a href="#B92-brainsci-15-00131" class="html-bibr">92</a>]. Such choices are often hard because of one or more of the four situations described above. For example, making choices with incomplete information is common in tasks [<a href="#B62-brainsci-15-00131" class="html-bibr">62</a>,<a href="#B93-brainsci-15-00131" class="html-bibr">93</a>,<a href="#B94-brainsci-15-00131" class="html-bibr">94</a>].</div><div class='html-p'>Therefore, given that hard choices are common, we can now address the issue in the title of this article. We begin by admitting that one cannot fully clarify this issue without experimentation (but see <a href="#sec5dot7-brainsci-15-00131" class="html-sec">Section 5.7</a>). However, the theoretical work described here strongly suggests that the answer is yes. Our theoretically optimal implementation of choice consistency leads us to believe that making hard choices makes our values diverge from those of others. Hence, we argue that hard choices contribute to our individuality.</div></section></section> </div> <div class="html-back"> <section class='html-notes'><h2>Funding</h2><div class='html-p'>This research received no external funding.</div></section><section class='html-notes'><h2 >Institutional Review Board Statement</h2><div class='html-p'>Not applicable.</div></section><section class='html-notes'><h2 >Informed Consent Statement</h2><div class='html-p'>Not applicable.</div></section><section class='html-notes'><h2 >Data Availability Statement</h2><div class='html-p'>The original contributions presented in this study are included in the article. Further inquiries can be directed to the author.</div></section><section id='html-ack' class='html-ack'><h2 >Acknowledgments</h2><div class='html-p'>I thank Pascal Mamassian for discussing the ideas of this article with me during various phases of the work. These discussions took place during my sabbatical leave at the Laboratoire des Systèmes Perceptifs (LSP), which he directs. I also thank LSP members Balkis Cadi, Claudia Lunghi, Peter Neri, Alain de Cheveigné, Cristian Lorenzi, Richard McWalter, Sophie-Cohen Bodenes, Izel Sari, Pierre Lélievre, Guilhem Marion, and Antoine Prosper, among others. They made my experience at the LSP enriching. Thanks go also to the École Normale Supérieur (ENS), Paris, France, the home of the LSP. Other parts of the work were performed at Loyola University Chicago and Johns Hopkins University. In these institutions, I thank Adrienne Bohl, Michelle Bukowski, Missy Kirby, and Ndella Seck for administrative support. Lastly, I thank Raymond Dye Jr. for creating an excellent work environment at the Department of Psychology of Loyola University Chicago.</div></section><section class='html-notes'><h2 >Conflicts of Interest</h2><div class='html-p'>I declare no conflicts of interest.</div></section><section><section id='app1-brainsci-15-00131' type=''><h2 data-nested='1'> Appendix A</h2><section id='secAdot1-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> Appendix A.1. A Bayesian Description of the Framework</h4><div class='html-p'>At time <math display='inline'><semantics> <mrow> <mi>k</mi> </mrow> </semantics></math>, a person receives sensory and/or body signals <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, and must decide what action <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> to take. The best action must balance the maximization of reward, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, and the minimization of the deviation from past actions given similar stimuli. The stimulus <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> is sampled from the probability distribution of inputs, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> </mrow> </semantics></math> (<a href="#brainsci-15-00131-f001" class="html-fig">Figure 1</a>). From that, we use an internal model, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>, to decide on the action, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, to take. For this model, we use the vector of parameters <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </semantics></math> learned at time <math display='inline'><semantics> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </semantics></math>. From the <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> outputted by the model, we can sample the reward <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>r</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> from the probability distribution of rewards given actions and stimuli, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <msub> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>,</mo> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>. The presence of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> in this probability distribution follows the same rationale given elsewhere that the reward depends on both the stimulus and the action taken [<a href="#B11-brainsci-15-00131" class="html-bibr">11</a>,<a href="#B12-brainsci-15-00131" class="html-bibr">12</a>].</div><div class='html-p'>In the next step, we update the parameters of the model before the next sampling begins. As mentioned above, this update must simultaneously optimize expected reward and consistency with past actions. So, we must perform an optimization that balances these two constraints. We do this by first finding the expected loss for each constraint and then treating the problem as a Compromise Decision Problem [<a href="#B25-brainsci-15-00131" class="html-bibr">25</a>,<a href="#B26-brainsci-15-00131" class="html-bibr">26</a>].</div><div class='html-p'>The key decision in this step is the choice of <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>. To calculate the expected choice-consistency loss as a function of the candidate parameter <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, we need the histories of the sampled stimuli and actions (<a href="#brainsci-15-00131-f001" class="html-fig">Figure 1</a>). We denote the number of time bins used in the histories as <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math>. Therefore, <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math> stands for how long the memory of choices last. For each value of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, we can calculate the loss due to errors of choice consistency with these histories. The best <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> maximizes the probability that from each <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> we obtain the <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> observed in history, that is, maximizes <math display='inline'><semantics> <mrow> <mrow> <msubsup> <mo stretchy="false">∏</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="sans-serif">Δ</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msubsup> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </mfenced> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mrow> </mrow> </semantics></math>. If we take the minus logarithm of this quantity, the best <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> obeys<div class='html-disp-formula-info' id='FD5-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>s</mi> </mrow> <mo>ˇ</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>a</mi> </mrow> <mo>ˇ</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">argmin</mi> </mrow> <mrow> <msup> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>*</mi> </mrow> </msup> </mrow> </msub> </mrow> <mo>⁡</mo> <mrow> <mo>−</mo> <mrow> <munderover> <mo stretchy="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="sans-serif">Δ</mi> </mrow> <mrow> <mi>k</mi> </mrow> </munderover> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">log</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>⁡</mo> <mrow> <mfenced separators="|"> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </mfenced> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msup> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>*</mi> </mrow> </msup> </mrow> </mfenced> </mrow> </mfenced> </mrow> </mrow> </mrow> </mrow> </mrow> </mrow> <mo>,</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A1)</label> </div> </div> where, for simplicity, we denote the histories with the notation [<a href="#B95-brainsci-15-00131" class="html-bibr">95</a>,<a href="#B96-brainsci-15-00131" class="html-bibr">96</a>,<a href="#B97-brainsci-15-00131" class="html-bibr">97</a>,<a href="#B98-brainsci-15-00131" class="html-bibr">98</a>]<div class='html-disp-formula-info' id='FD6-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>x</mi> </mrow> <mo>ˇ</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="sans-serif">Δ</mi> </mrow> </msub> <mo>,</mo> <mo> </mo> <mo>⋯</mo> <mo>,</mo> <msub> <mrow> <mi>x</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math> </div> <div class='l'> <label >(A2)</label> </div> </div> Equation (1) has an important interpretation because the <math display='inline'><semantics> <mrow> <mo>−</mo> <mi mathvariant="normal">l</mi> <mi mathvariant="normal">o</mi> <mi mathvariant="normal">g</mi> </mrow> </semantics></math> terms are the surprisals of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> given <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> [<a href="#B33-brainsci-15-00131" class="html-bibr">33</a>]. Consequently, the optimal <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> minimizes the total amount of surprisals in the history of the stimuli and actions. Thus, we choose the <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> to emphasize the most common (<math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math>) pairs, weeding out the surprises. This choice yields the expected choice-consistency loss as<div class='html-disp-formula-info' id='FD7-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mo>−</mo> <mrow> <munderover> <mo stretchy="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mi>k</mi> <mo>+</mo> <mn>1</mn> <mo>−</mo> <mi mathvariant="sans-serif">Δ</mi> </mrow> <mrow> <mi>k</mi> </mrow> </munderover> <mrow> <mrow> <mrow> <msub> <mrow> <mi mathvariant="normal">log</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>⁡</mo> <mrow> <mfenced separators="|"> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </mfenced> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> </mrow> </mrow> </mrow> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A3)</label> </div> </div> Because we sample <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>a</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>s</mi> </mrow> <mrow> <mi>i</mi> </mrow> </msub> </mrow> </semantics></math> according to their own probability, we can write the corresponding loss function [<a href="#B99-brainsci-15-00131" class="html-bibr">99</a>,<a href="#B100-brainsci-15-00131" class="html-bibr">100</a>]<div class='html-disp-formula-info' id='FD8-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <mi>s</mi> <mo>,</mo> <mi>a</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mrow> <mrow> <msub> <mrow> <mo>−</mo> <mi mathvariant="normal">log</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msub> </mrow> <mo>⁡</mo> <mrow> <mfenced separators="|"> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </mfenced> <mo>,</mo> <mo> </mo> </mrow> </mrow> </mrow> </semantics></math> </div> <div class='l'> <label >(A4)</label> </div> </div> namely, the surprisal itself.</div><div class='html-p'>The value of <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> also affects the expected reward loss through the chosen action and the function <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>a</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> </mrow> </semantics></math>. The expected reward is <math display='inline'><semantics> <mrow> <mrow> <mo stretchy="false">∭</mo> <mrow> <mi>r</mi> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>a</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> <mi>d</mi> <mi>s</mi> <mo> </mo> <mi>d</mi> <mi>a</mi> <mo> </mo> <mi>d</mi> <mi>r</mi> </mrow> </mrow> </mrow> </semantics></math>. The reward loss is then<div class='html-disp-formula-info' id='FD9-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mo>−</mo> <mrow> <mo stretchy="false">∭</mo> <mrow> <mi>r</mi> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>a</mi> <mo>,</mo> <mi>s</mi> <mo> </mo> </mrow> </mfenced> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> <mi>d</mi> <mi>s</mi> <mo> </mo> <mi>d</mi> <mi>a</mi> <mo> </mo> <mi>d</mi> <mi>r</mi> </mrow> </mrow> <mo>.</mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A5)</label> </div> </div> Again, because the decision is on <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math>, the expectation is through <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> </mrow> </semantics></math>, making the loss function<div class='html-disp-formula-info' id='FD10-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>L</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mo>−</mo> <mrow> <mo stretchy="false">∫</mo> <mrow> <mi>r</mi> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mi>d</mi> <mi>r</mi> </mrow> </mrow> <mo> </mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A6)</label> </div> </div> after integrating by <math display='inline'><semantics> <mrow> <mi>a</mi> </mrow> </semantics></math>. The interpretation of this loss function is simple: find the parameter that increases the reward for the expected stimuli.</div><div class='html-p'>Finally, we propose working with Kempthorne’s <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>-Bayes-based compromise problem to propose that the brain minimizes<div class='html-disp-formula-info' id='FD11-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>λ</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mi>λ</mi> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo>+</mo> <mfenced separators="|"> <mrow> <mn>1</mn> <mo>−</mo> <mi>λ</mi> </mrow> </mfenced> <msub> <mrow> <mi>l</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> <mfenced separators="|"> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> <mo> </mo> <mo>,</mo> <mo> </mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A7)</label> </div> </div> where <math display='inline'><semantics> <mrow> <mn>0</mn> <mo>≤</mo> <mi>λ</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>. See Kempthorne’s theorems 3.1 and 3.2 describing the properties of the <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>-based compromise problem [<a href="#B25-brainsci-15-00131" class="html-bibr">25</a>].</div></section><section id='secAdot2-brainsci-15-00131' type=''><h4 class='html-italic' data-nested='2'> Appendix A.2. A Simple Model to Explore the Properties of the Theory</h4><div class='html-p'>Any model based on Equations (1)–(7) must specify six mathematical quantities: <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>a</mi> <mo>,</mo> <mi>s</mi> <mo> </mo> </mrow> </mfenced> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. To specify the three probability functions, we must begin with the domains of <math display='inline'><semantics> <mrow> <mi>s</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>a</mi> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <mi>r</mi> </mrow> </semantics></math>. For simplicity, the stimulus and reward variables will be one-dimensional, such that <math display='inline'><semantics> <mrow> <mi>s</mi> <mo>,</mo> <mi>r</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </semantics></math>, and we consider a discrete and finite action space (with <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </semantics></math> possible actions) represented as<div class='html-disp-formula-info' id='FD12-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <mi>a</mi> <mo>∈</mo> <mfenced separators="|"> <mrow> <mn>1,2</mn> <mo>,</mo> <mo> </mo> <mo>…</mo> <mo>,</mo> <mo> </mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </mfenced> <mo> </mo> <mo>.</mo> <mo> </mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A8)</label> </div> </div> Again, for simplicity, we base <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </mfenced> </mrow> </semantics></math>, and <math display='inline'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>a</mi> <mo>,</mo> <mi>s</mi> <mo> </mo> </mrow> </mfenced> </mrow> </semantics></math> on Normal distributions. To begin, we take the distribution of <math display='inline'><semantics> <mrow> <mi>s</mi> </mrow> </semantics></math> to be Normal, with zero mean and a given standard deviation <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD13-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mi>s</mi> </mrow> </mfenced> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mfrac> <mrow> <msup> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msubsup> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>s</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mrow> </msup> </mrow> <mrow> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mo> </mo> <mo>.</mo> <mo> </mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A9)</label> </div> </div> Next, we represent the probabilistic relationship between <math display='inline'><semantics> <mrow> <mi>s</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>a</mi> </mrow> </semantics></math> as a sum of <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> </semantics></math> Normal distributions, where <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>0</mn> </mrow> </mfenced> </mrow> </msubsup> <mo>,</mo> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <mn>1</mn> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> </msubsup> <mo>,</mo> <mo>…</mo> <mo>,</mo> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>0</mn> </mrow> </mfenced> </mrow> </msubsup> <mo>,</mo> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>,</mo> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> </msubsup> </mrow> </mfenced> </mrow> </semantics></math> with equal standard deviation <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </semantics></math>:<div class='html-disp-formula-info' id='FD14-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>a</mi> </mrow> </mfenced> <mi>s</mi> <mo>,</mo> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> </mrow> </msub> </mrow> </mfenced> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <mrow> <munderover> <mo stretchy="false">∑</mo> <mrow> <mi>i</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> </munderover> <mrow> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mfenced separators="|"> <mrow> <mi>a</mi> <mo>−</mo> <mfenced separators="|"> <mrow> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>0</mn> </mrow> </mfenced> </mrow> </msubsup> <mo>+</mo> <msubsup> <mrow> <mi>w</mi> </mrow> <mrow> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mi>i</mi> </mrow> <mrow> <mfenced separators="|"> <mrow> <mn>1</mn> </mrow> </mfenced> </mrow> </msubsup> <mi>s</mi> </mrow> </mfenced> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msubsup> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>a</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mstyle> </mrow> </msup> </mrow> </mrow> <mo> </mo> <mo>.</mo> <mo> </mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A10)</label> </div> </div> Hence, the components of <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mi>k</mi> </mrow> </msub> </mrow> </semantics></math> map linearly to the means of the Normal distributions, selecting the actions. Finally, we also make the reward function a Normal distribution, standing for the probability of reward of each possible decision:<div class='html-disp-formula-info' id='FD15-brainsci-15-00131'> <div class='f'> <math display='block'><semantics> <mrow> <mi>P</mi> <mfenced separators="|"> <mrow> <mfenced open="" close="⌋" separators="|"> <mrow> <mi>r</mi> </mrow> </mfenced> <mi>a</mi> <mo>,</mo> <mi>s</mi> </mrow> </mfenced> <mo>=</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <mn>1</mn> </mrow> <mrow> <msqrt> <mn>2</mn> <mi>π</mi> </msqrt> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </mfrac> </mstyle> <msup> <mrow> <mi>e</mi> </mrow> <mrow> <mo>−</mo> <mstyle scriptlevel="0" displaystyle="true"> <mfrac> <mrow> <msup> <mrow> <mfenced separators="|"> <mrow> <mi>r</mi> <mo>−</mo> <mi>α</mi> <mfenced separators="|"> <mrow> <mi>a</mi> <mo>−</mo> <mi>β</mi> </mrow> </mfenced> <mo>∗</mo> <mi>s</mi> </mrow> </mfenced> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>2</mn> <msubsup> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>r</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msubsup> </mrow> </mfrac> </mstyle> </mrow> </msup> <mo> </mo> <mo>,</mo> <mo> </mo> </mrow> </semantics></math> </div> <div class='l'> <label >(A11)</label> </div> </div> where the parameters <math display='inline'><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math> set the relationship between the stimulus <math display='inline'><semantics> <mrow> <mi>s</mi> </mrow> </semantics></math> and the action <math display='inline'><semantics> <mrow> <mi>a</mi> </mrow> </semantics></math> to the mean reward for that pair, and <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math> is the standard deviation.</div><div class='html-p'>To summarize, the parameters that control the outcome of the simulations of the model are <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </semantics></math>, <math display='inline'><semantics> <mrow> <mi>α</mi> </mrow> </semantics></math> and <math display='inline'><semantics> <mrow> <mi>β</mi> </mrow> </semantics></math>. Without loss of generality, we can set <math display='inline'><semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>s</mi> </mrow> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and control the simulations with the rest of the parameters. Thus, we have nine parameters with which to explore the predictions of the theory.</div></section></section></section><section id='html-references_list'><h2>References</h2><ol class='html-xxx'><li id='B1-brainsci-15-00131' class='html-x' data-content='1.'>Chen, M.K.; Risen, J.L. How choice affects and reflects preferences: Revisiting the free-choice paradigm. <span class='html-italic'>J. Personal. Soc. Psychol.</span> <b>2010</b>, <span class='html-italic'>99</span>, 573. 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The figure shows three moments of parametric updates indicated in red, blue, and green. The first moment at time k (red) begins with a stimulus (Stim k) drawn from the probability distribution of stimuli (P(S)). With this stimulus, the brain calculates an action (Act k) from the probability distribution of actions given stimuli (P(A|S)). This calculation uses the set of parameters (Par k − 1) calculated at time k − 1. A reward (Rew k) then arrives from the probability distribution of rewards given stimuli and actions (P(R|A,S)). These stimulus and action are added to the histories of these values (Stim Hist and Act Hist). Given these histories and the new reward Rew k, a new parameter set (Par k) is computed, maximizing the Bayesian expected reward and action consistency. With this new set, one can repeat the process again at time k + 1 (blue). This process leads to the computation of a new parameter set (Par k + 1) that triggers the process again (green) and so on.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_brainsci-15-00131-f001"> <div class="html-caption"> <b>Figure 1.</b> Bayesian update of the parameters of choice consistency. The figure shows three moments of parametric updates indicated in red, blue, and green. The first moment at time k (red) begins with a stimulus (Stim k) drawn from the probability distribution of stimuli (P(S)). With this stimulus, the brain calculates an action (Act k) from the probability distribution of actions given stimuli (P(A|S)). This calculation uses the set of parameters (Par k − 1) calculated at time k − 1. A reward (Rew k) then arrives from the probability distribution of rewards given stimuli and actions (P(R|A,S)). These stimulus and action are added to the histories of these values (Stim Hist and Act Hist). Given these histories and the new reward Rew k, a new parameter set (Par k) is computed, maximizing the Bayesian expected reward and action consistency. With this new set, one can repeat the process again at time k + 1 (blue). This process leads to the computation of a new parameter set (Par k + 1) that triggers the process again (green) and so on.</div> <div class="html-img"><img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g001.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g001.png" alt="Brainsci 15 00131 g001" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g001.png" /></div> </div> <div class="html-fig-wrap" id="brainsci-15-00131-f002"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f002"> <img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g002.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g002.png" alt="Brainsci 15 00131 g002" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g002-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f002"></a> </div> </div> <div class="html-fig_description"> <b>Figure 2.</b> Computer simulation of our Bayesian theory of choice consistency with the standard parameters (<a href="#brainsci-15-00131-t001" class="html-table">Table 1</a>). (<b>A</b>) Choices of two actions for different stimuli over time. An example of such an action is buying a shirt with this or that pattern. In this figure, every dot stands for a choice (color) for the given sampled stimulus at the given time. (<b>B</b>) Running average (5 points) of the choices in panel (<b>A</b>). (<b>C</b>) Choice-consistency loss as a function of time. (<b>D</b>) Temporal evolution of the two parameters of the model. These time courses reveal that the choices separate spontaneously, with an apparent phase transition in loss and parameters.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_brainsci-15-00131-f002"> <div class="html-caption"> <b>Figure 2.</b> Computer simulation of our Bayesian theory of choice consistency with the standard parameters (<a href="#brainsci-15-00131-t001" class="html-table">Table 1</a>). (<b>A</b>) Choices of two actions for different stimuli over time. An example of such an action is buying a shirt with this or that pattern. In this figure, every dot stands for a choice (color) for the given sampled stimulus at the given time. (<b>B</b>) Running average (5 points) of the choices in panel (<b>A</b>). (<b>C</b>) Choice-consistency loss as a function of time. (<b>D</b>) Temporal evolution of the two parameters of the model. These time courses reveal that the choices separate spontaneously, with an apparent phase transition in loss and parameters.</div> <div class="html-img"><img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g002.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g002.png" alt="Brainsci 15 00131 g002" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g002.png" /></div> </div> <div class="html-fig-wrap" id="brainsci-15-00131-f003"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f003"> <img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g003.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g003.png" alt="Brainsci 15 00131 g003" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g003-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f003"></a> </div> </div> <div class="html-fig_description"> <b>Figure 3.</b> Elimination of choices. (<b>A</b>) Longer simulations with standard parameters (<a href="#brainsci-15-00131-t001" class="html-table">Table 1</a>) show that eventually, choice consistency may cause one of the choices to eliminate the others. The conventions in this figure are the same as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A. (<b>B</b>) Distribution of times of choice elimination for two values of memory length, namely, Δ.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_brainsci-15-00131-f003"> <div class="html-caption"> <b>Figure 3.</b> Elimination of choices. (<b>A</b>) Longer simulations with standard parameters (<a href="#brainsci-15-00131-t001" class="html-table">Table 1</a>) show that eventually, choice consistency may cause one of the choices to eliminate the others. The conventions in this figure are the same as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A. (<b>B</b>) Distribution of times of choice elimination for two values of memory length, namely, Δ.</div> <div class="html-img"><img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g003.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g003.png" alt="Brainsci 15 00131 g003" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g003.png" /></div> </div> <div class="html-fig-wrap" id="brainsci-15-00131-f004"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f004"> <img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g004.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g004.png" alt="Brainsci 15 00131 g004" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g004-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f004"></a> </div> </div> <div class="html-fig_description"> <b>Figure 4.</b> Eight consecutive simulations of actions in response to sensory stimuli with the standard set of parameters, using the conventions of <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A. Each simulation yielded a unique pattern of behavior. After a first random phase, the most common behaviors were such that positive sensory stimuli tended to yield Action 1 (Panels <b>D</b>,<b>G</b>,<b>H</b>) or Action 2 (Panels <b>A</b>,<b>C</b>,<b>E</b>). In these behaviors, negative sensory stimuli tended to yield the opposite actions. Occasionally, we also saw a behavior that was more mixed (Panel <b>B</b>). More rarely, we saw a behavior in which an action happened for positive stimuli earlier and negative ones later (<b>F</b>).  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_brainsci-15-00131-f004"> <div class="html-caption"> <b>Figure 4.</b> Eight consecutive simulations of actions in response to sensory stimuli with the standard set of parameters, using the conventions of <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A. Each simulation yielded a unique pattern of behavior. After a first random phase, the most common behaviors were such that positive sensory stimuli tended to yield Action 1 (Panels <b>D</b>,<b>G</b>,<b>H</b>) or Action 2 (Panels <b>A</b>,<b>C</b>,<b>E</b>). In these behaviors, negative sensory stimuli tended to yield the opposite actions. Occasionally, we also saw a behavior that was more mixed (Panel <b>B</b>). More rarely, we saw a behavior in which an action happened for positive stimuli earlier and negative ones later (<b>F</b>).</div> <div class="html-img"><img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g004.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g004.png" alt="Brainsci 15 00131 g004" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g004.png" /></div> </div> <div class="html-fig-wrap" id="brainsci-15-00131-f005"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f005"> <img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g005.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g005.png" alt="Brainsci 15 00131 g005" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g005-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f005"></a> </div> </div> <div class="html-fig_description"> <b>Figure 5.</b> Increasing the number of choice parameters boosts the individuality arising from the model. In these simulations, we substituted <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mn>1.25,0</mn> <mo>,</mo> <mn>1.75,0</mn> </mrow> </mfenced> </mrow> </semantics></math> for the standard value in <a href="#brainsci-15-00131-t001" class="html-table">Table 1</a> and thus, we had four choice parameters instead of two. (<b>A</b>–<b>C</b>) Examples of choices (with conventions as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A). (<b>D</b>–<b>F</b>) Temporal evolution of parameters in the simulations of (<b>A</b>, <b>B</b>, and <b>C</b>) respectively. Most simulations with 4 choice parameters yielded behaviors like those in <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>. However, some simulations yielded different behaviors, as illustrated in this figure. (<b>A</b>,<b>D</b>) Examples of not discriminating actions by stimuli. (<b>B</b>,<b>E</b>) Examples of switching stimulus dependence of choices. (<b>C</b>,<b>F</b>) Examples of Action 1 sandwiched between two stimulus locations of Action 2.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_brainsci-15-00131-f005"> <div class="html-caption"> <b>Figure 5.</b> Increasing the number of choice parameters boosts the individuality arising from the model. In these simulations, we substituted <math display='inline'><semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo>=</mo> <mfenced separators="|"> <mrow> <mn>1.25,0</mn> <mo>,</mo> <mn>1.75,0</mn> </mrow> </mfenced> </mrow> </semantics></math> for the standard value in <a href="#brainsci-15-00131-t001" class="html-table">Table 1</a> and thus, we had four choice parameters instead of two. (<b>A</b>–<b>C</b>) Examples of choices (with conventions as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A). (<b>D</b>–<b>F</b>) Temporal evolution of parameters in the simulations of (<b>A</b>, <b>B</b>, and <b>C</b>) respectively. Most simulations with 4 choice parameters yielded behaviors like those in <a href="#brainsci-15-00131-f004" class="html-fig">Figure 4</a>. However, some simulations yielded different behaviors, as illustrated in this figure. (<b>A</b>,<b>D</b>) Examples of not discriminating actions by stimuli. (<b>B</b>,<b>E</b>) Examples of switching stimulus dependence of choices. (<b>C</b>,<b>F</b>) Examples of Action 1 sandwiched between two stimulus locations of Action 2.</div> <div class="html-img"><img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g005.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g005.png" alt="Brainsci 15 00131 g005" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g005.png" /></div> </div> <div class="html-fig-wrap" id="brainsci-15-00131-f006"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f006"> <img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g006.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g006.png" alt="Brainsci 15 00131 g006" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g006-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f006"></a> </div> </div> <div class="html-fig_description"> <b>Figure 6.</b> Outcome of the simulations with three or four choices instead of two. (<b>A</b>) Choices of three actions for different stimuli over time (conventions as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A). (<b>B</b>) Running average (5 points) of the choices in Panel (<b>A</b>). These panels show that choice consistency organizes the three actions in the space of stimuli. However, eventually one action dominates (Action 3 in this example), with one of the other actions stopping first (Action 2 in this example) and then the other (Action 1). (<b>C</b>) Scatter plot of the stoppage times of the losing actions. They tend to stop almost at the same time. (<b>D</b>) Running average (5 points) of a simulation with four choices.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_brainsci-15-00131-f006"> <div class="html-caption"> <b>Figure 6.</b> Outcome of the simulations with three or four choices instead of two. (<b>A</b>) Choices of three actions for different stimuli over time (conventions as in <a href="#brainsci-15-00131-f002" class="html-fig">Figure 2</a>A). (<b>B</b>) Running average (5 points) of the choices in Panel (<b>A</b>). These panels show that choice consistency organizes the three actions in the space of stimuli. However, eventually one action dominates (Action 3 in this example), with one of the other actions stopping first (Action 2 in this example) and then the other (Action 1). (<b>C</b>) Scatter plot of the stoppage times of the losing actions. They tend to stop almost at the same time. (<b>D</b>) Running average (5 points) of a simulation with four choices.</div> <div class="html-img"><img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g006.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g006.png" alt="Brainsci 15 00131 g006" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g006.png" /></div> </div> <div class="html-fig-wrap" id="brainsci-15-00131-f007"> <div class='html-fig_img'> <div class="html-figpopup html-figpopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f007"> <img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g007.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g007.png" alt="Brainsci 15 00131 g007" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g007-550.jpg" /> <a class="html-expand html-figpopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#fig_body_display_brainsci-15-00131-f007"></a> </div> </div> <div class="html-fig_description"> <b>Figure 7.</b> The interaction between rewards and choice consistency with standard parameters, except for variations of <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. (<b>A</b>) Simulation with only rewards (<math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). As expected from the choice of the standard parameters, Action 2 is chosen for positive stimuli and vice versa for Action 1. (<b>B</b>) Example of simulation with <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> in which positive stimuli elicit Action 1 despite the rewards favoring the opposite. (<b>C</b>) Percentage of simulations for which Action 2 stimuli converge to values larger than those for Action 1 as a function of <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. (<b>D</b>) Mean stoppage time of the losing actions as a function of <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. Error bars in (<b>C</b>,<b>D</b>) are standard errors. As <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> increases, we obtain more Action 2 because of the rewards, and the stoppage time rises because the influence of choice consistency diminishes.  </div> </div> <div class="html-fig_show mfp-hide" id="fig_body_display_brainsci-15-00131-f007"> <div class="html-caption"> <b>Figure 7.</b> The interaction between rewards and choice consistency with standard parameters, except for variations of <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. (<b>A</b>) Simulation with only rewards (<math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>). As expected from the choice of the standard parameters, Action 2 is chosen for positive stimuli and vice versa for Action 1. (<b>B</b>) Example of simulation with <math display='inline'><semantics> <mrow> <mi>λ</mi> <mo>=</mo> <mn>0.5</mn> </mrow> </semantics></math> in which positive stimuli elicit Action 1 despite the rewards favoring the opposite. (<b>C</b>) Percentage of simulations for which Action 2 stimuli converge to values larger than those for Action 1 as a function of <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. (<b>D</b>) Mean stoppage time of the losing actions as a function of <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math>. Error bars in (<b>C</b>,<b>D</b>) are standard errors. As <math display='inline'><semantics> <mrow> <mi>λ</mi> </mrow> </semantics></math> increases, we obtain more Action 2 because of the rewards, and the stoppage time rises because the influence of choice consistency diminishes.</div> <div class="html-img"><img data-large="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g007.png" data-original="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g007.png" alt="Brainsci 15 00131 g007" data-lsrc="/brainsci/brainsci-15-00131/article_deploy/html/images/brainsci-15-00131-g007.png" /></div> </div> <div class="html-table-wrap" id="brainsci-15-00131-t001"> <div class="html-table_wrap_td"> <div class="html-tablepopup html-tablepopup-link" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href='#table_body_display_brainsci-15-00131-t001'> <img data-lsrc="https://pub.mdpi-res.com/img/table.png" /> <a class="html-expand html-tablepopup" data-counterslinkmanual = "https://www.mdpi.com/2076-3425/15/2/131/display" href="#table_body_display_brainsci-15-00131-t001"></a> </div> </div> <div class="html-table_wrap_discription"> <b>Table 1.</b> Standard set of parameters. </div> </div> <div class="html-table_show mfp-hide " id="table_body_display_brainsci-15-00131-t001"> <div class="html-caption"><b>Table 1.</b> Standard set of parameters.</div> <table > <thead ><tr ><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Parameter(s)</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Equation</th><th align='center' valign='middle' style='border-top:solid thin;border-bottom:solid thin' class='html-align-center' >Values</th></tr></thead><tbody ><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mrow> <mover accent="true"> <mrow> <mi>w</mi> </mrow> <mo>→</mo> </mover> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mfenced separators="|"> <mrow> <mn>1.5,0</mn> </mrow> </mfenced> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>w</mi> </mrow> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>1</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mrow> <mi>N</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >A8</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>2</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mi mathvariant="sans-serif">Δ</mi> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >A2</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>20</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mi>λ</mi> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >1</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>0</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>a</mi> </mrow> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >3</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>1</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <msub> <mrow> <mi>σ</mi> </mrow> <mrow> <mi>r</mi> </mrow> </msub> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mn>1</mn> </mrow> </semantics> </math></td></tr><tr ><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' ><math display='inline'> <semantics> <mrow> <mfenced open="[" close="]" separators="|"> <mrow> <mi>α</mi> <mo>,</mo> <mi>β</mi> </mrow> </mfenced> </mrow> </semantics> </math></td><td align='center' valign='middle' style='border-bottom:solid thin' class='html-align-center' >4</td><td align='center' valign='middle' 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A Computational Analysis of the Effect of Hard Choices on the Individuation of Values