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data-event-name="pinnable-header.vector-toc.pin">move to sidebar</button> <button class="vector-pinnable-header-toggle-button vector-pinnable-header-unpin-button" data-event-name="pinnable-header.vector-toc.unpin">hide</button> </div> <ul class="vector-toc-contents" id="mw-panel-toc-list"> <li id="toc-mw-content-text" class="vector-toc-list-item vector-toc-level-1"> <a href="#" class="vector-toc-link"> <div class="vector-toc-text">(Top)</div> </a> </li> <li id="toc-Things_that_can_be_divided" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Things_that_can_be_divided"> <div class="vector-toc-text"> <span class="vector-toc-numb">1</span> <span>Things that can be divided</span> </div> </a> <ul id="toc-Things_that_can_be_divided-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Definitions_of_fairness" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Definitions_of_fairness"> <div class="vector-toc-text"> <span class="vector-toc-numb">2</span> <span>Definitions of fairness</span> </div> </a> <ul id="toc-Definitions_of_fairness-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Additional_requirements" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Additional_requirements"> <div class="vector-toc-text"> <span class="vector-toc-numb">3</span> <span>Additional requirements</span> </div> </a> <ul id="toc-Additional_requirements-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Procedures" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Procedures"> <div class="vector-toc-text"> <span class="vector-toc-numb">4</span> <span>Procedures</span> </div> </a> <ul id="toc-Procedures-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Extensions" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Extensions"> <div class="vector-toc-text"> <span class="vector-toc-numb">5</span> <span>Extensions</span> </div> </a> <ul id="toc-Extensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-History" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#History"> <div class="vector-toc-text"> <span class="vector-toc-numb">6</span> <span>History</span> </div> </a> <ul id="toc-History-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-In_popular_culture" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#In_popular_culture"> <div class="vector-toc-text"> <span class="vector-toc-numb">7</span> <span>In popular culture</span> </div> </a> <ul id="toc-In_popular_culture-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-See_also" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#See_also"> <div class="vector-toc-text"> <span class="vector-toc-numb">8</span> <span>See also</span> </div> </a> <ul id="toc-See_also-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-References" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#References"> <div class="vector-toc-text"> <span class="vector-toc-numb">9</span> <span>References</span> </div> </a> <ul id="toc-References-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Text_books" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Text_books"> <div class="vector-toc-text"> <span class="vector-toc-numb">10</span> <span>Text books</span> </div> </a> <ul id="toc-Text_books-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Survey_articles" class="vector-toc-list-item vector-toc-level-1 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class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Problem of sharing resources</div> <p><b>Fair division</b> is the problem in <a href="/wiki/Game_theory" title="Game theory">game theory</a> of dividing a set of <a href="/wiki/Resources" class="mw-redirect" title="Resources">resources</a> among several people who have an <a href="/wiki/Entitlement_(fair_division)" title="Entitlement (fair division)">entitlement</a> to them so that each person receives their due share. That problem arises in various real-world settings such as division of <a href="/wiki/Inheritance" title="Inheritance">inheritance</a>, partnership <a href="/wiki/Dissolution_(law)" title="Dissolution (law)">dissolutions</a>, <a href="/wiki/Divorce_settlement" title="Divorce settlement">divorce settlements</a>, electronic <a href="/wiki/Frequency_allocation" title="Frequency allocation">frequency allocation</a>, <a href="/wiki/Air_traffic_management" title="Air traffic management">airport traffic management</a>, and exploitation of <a href="/wiki/Earth_Observation_Satellite" class="mw-redirect" title="Earth Observation Satellite">Earth observation satellites</a>. It is an active research area in <a href="/wiki/Mathematics" title="Mathematics">mathematics</a>, <a href="/wiki/Economics" title="Economics">economics</a> (especially <a href="/wiki/Social_choice_theory" title="Social choice theory">social choice theory</a>), and <a href="/wiki/Dispute_resolution" title="Dispute resolution">dispute resolution</a>. The central tenet of fair division is that such a division should be performed by the players themselves, without the need for external <a href="/wiki/Arbitration" title="Arbitration">arbitration</a>, as only the players themselves really know how they value the goods. </p><p>The archetypal fair division <a href="/wiki/Algorithm" title="Algorithm">algorithm</a> is <a href="/wiki/Divide_and_choose" title="Divide and choose">divide and choose</a>. It demonstrates that two agents with different tastes can divide a cake such that each of them believes that he got the best piece. The research in fair division can be seen as an extension of this procedure to various more complex settings. </p><p>There are many different kinds of fair division problems, depending on the nature of goods to divide, the criteria for fairness, the nature of the players and their preferences, and other criteria for evaluating the quality of the division. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Things_that_can_be_divided">Things that can be divided</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=1" title="Edit section: Things that can be divided"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Formally, a fair division problem is defined by a set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> (often called "the cake") and a group of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> players. A division is a partition of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> into <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> disjoint subsets: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=X_{1}\sqcup X_{2}\sqcup \cdots \sqcup X_{n}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⊔</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>⊔</mo> <mo>⋯</mo> <mo>⊔</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=X_{1}\sqcup X_{2}\sqcup \cdots \sqcup X_{n}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ba97ac7d43bcf1055fc588844ad619a0f54c155c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:24.436ex; height:2.509ex;" alt="{\displaystyle C=X_{1}\sqcup X_{2}\sqcup \cdots \sqcup X_{n}}" /></span>, one subset per player. </p><p>The set <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> can be of various types: </p> <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> may be a finite set of indivisible items, for example: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C=\{{\text{piano}},{\text{car}},{\text{apartment}}\}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> <mo>=</mo> <mo fence="false" stretchy="false">{</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>piano</mtext> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>car</mtext> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext>apartment</mtext> </mrow> <mo fence="false" stretchy="false">}</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C=\{{\text{piano}},{\text{car}},{\text{apartment}}\}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8b93c947db35b0face006878b2efca42bafaa302" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.52ex; height:2.843ex;" alt="{\displaystyle C=\{{\text{piano}},{\text{car}},{\text{apartment}}\}}" /></span>, such that each item should be given entirely to a single person.</li> <li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> may be an infinite set representing a divisible resource, for example: money, or a cake. Mathematically, a divisible resource is often modeled as a subset of a real space, for example, the section [0,1] may represent a long narrow cake, that has to be cut into parallel pieces. The <a href="/wiki/Unit_disk" title="Unit disk">unit disk</a> may represent an apple pie.</li></ul> <p>Additionally, the set to be divided may be: </p> <ul><li>homogeneous – such as money, where only the amount matters, or</li> <li>heterogeneous – such as a cake that may have different ingredients, different icings, etc.</li></ul> <p>Finally, it is common to make some assumptions about whether the items to be divided are: </p> <ul><li>goods – such as a car or a cake, or</li> <li>bads – such as house chores.</li></ul> <p>Based on these distinctions, several general types of fair division problems have been studied: </p> <ul><li><a href="/wiki/Fair_item_assignment" class="mw-redirect" title="Fair item assignment">Fair item assignment</a> – dividing a set of <i>indivisible and heterogeneous</i> goods.</li> <li>Fair resource allocation – dividing a set of <i>divisible and homogeneous</i> goods. A special case is <a href="/wiki/Fair_division_of_a_single_homogeneous_resource" title="Fair division of a single homogeneous resource">fair division of a single homogeneous resource</a>.</li> <li><a href="/wiki/Fair_cake-cutting" title="Fair cake-cutting">Fair cake-cutting</a> – dividing a <i>divisible, heterogeneous good</i>. A special case is when the cake is a circle; then the problem is called <a href="/wiki/Fair_pie-cutting" title="Fair pie-cutting">fair pie-cutting</a>.</li> <li>Fair <a href="/wiki/Chore_division" title="Chore division">chore division</a> – dividing a <i>divisible, heterogeneous bad.</i></li></ul> <p>Combinations and special cases are also common: </p> <ul><li><a href="/wiki/Housemates_problem" class="mw-redirect" title="Housemates problem">Rental harmony</a> (aka the housemates problem) – dividing a set of <i>indivisible heterogeneous goods</i> (e.g., rooms in an apartment), and simultaneously a <i>homogeneous divisible bad</i> (the rent on the apartment).</li> <li><a href="/wiki/Fair_river_sharing" title="Fair river sharing">Fair river sharing</a> – dividing waters flowing in an international river among the countries along its stream.</li> <li><a href="/wiki/Fair_random_assignment" title="Fair random assignment">Fair random assignment</a> – dividing lotteries over divisions – is especially common when allocating indivisible goods.</li></ul> <div class="mw-heading mw-heading2"><h2 id="Definitions_of_fairness">Definitions of fairness</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=2" title="Edit section: Definitions of fairness"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Most of what is normally called a fair division is not considered so by the theory because of the use of <a href="/wiki/Arbitration" title="Arbitration">arbitration</a>. This kind of situation happens quite often with mathematical theories named after real life problems. The decisions in the <a href="/wiki/Talmud" title="Talmud">Talmud</a> on <a href="/wiki/Entitlement_(fair_division)" title="Entitlement (fair division)">entitlement</a> when an estate is <a href="/wiki/Bankrupt" class="mw-redirect" title="Bankrupt">bankrupt</a> reflect the development of complex ideas regarding fairness.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> However, they are the result of legal debates by rabbis rather than divisions according to the valuations of the claimants. </p><p>According to the <a href="/wiki/Subjective_theory_of_value" title="Subjective theory of value">subjective theory of value</a>, there cannot be an objective measure of the value of each item. Therefore, <i>objective fairness</i> is not possible, as different people may assign different values to each item. Empirical experiments on how people define the concept of fairness have given inconclusive results.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> </p><p>Therefore, most current research on fairness focuses on concepts of <i>subjective fairness</i>. Each of the <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" /></span> people is assumed to have a personal, subjective <i>utility function</i> or <i>value function</i>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f300b83673e961a9d48f3862216b167f94e5668c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.155ex; height:2.509ex;" alt="{\displaystyle V_{i}}" /></span>, which assigns a numerical value to each subset of <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span>. Often the functions are assumed to be normalized, so that every person values the empty set as 0 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}(\emptyset )=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi mathvariant="normal">∅</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}(\emptyset )=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/77fbac9bb5ab97fd24a15298966e5d18efae90a6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.388ex; height:2.843ex;" alt="{\displaystyle V_{i}(\emptyset )=0}" /></span> for all i), and the entire set of items as 1 (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}(C)=1}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mn>1</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}(C)=1}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f70779067fb898dfaa0b8ebf17c5512894097d75" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.991ex; height:2.843ex;" alt="{\displaystyle V_{i}(C)=1}" /></span> for all i) if the items are desirable, and -1 if the items are undesirable. Examples are: </p> <ul><li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> is the set of indivisible items {piano, car, apartment}, then <a href="/wiki/Alice_and_Bob" title="Alice and Bob">Alice</a> may assign a value of 1/3 to each item, which means that each item is important to her just the same as any other item. <a href="/wiki/Alice_and_Bob" title="Alice and Bob">Bob</a> may assign the value of 1 to the set {car, apartment}, and the value 0 to all other sets except X; this means that he wants to get only the car and the apartment together; the car alone or the apartment alone, or each of them together with the piano, is worthless to him.</li> <li>If <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span> is a long narrow cake (modeled as the interval [0,1]), then, Alice may assign each subset a value proportional to its length, which means that she wants as much cake as possible, regardless of the icings. Bob may assign value only to subsets of [0.4, 0.6], for example, because this part of the cake contains cherries and Bob only cares about cherries.</li></ul> <p>Based on these subjective value functions, there are a number of widely used criteria for a fair division. Some of these conflict with each other but often they can be combined. The criteria described here are only for when each player is entitled to the same amount: </p> <ul><li>A <a href="/wiki/Proportional_division" title="Proportional division">proportional division</a> means that every player gets at least their due share <i>according to their own value function</i>. For instance if three people divide up a cake, each gets at least a third by their own valuation, i.e. each of the <i>n</i> people gets a subset of <i><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>C</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle C}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fc55753007cd3c18576f7933f6f089196732029" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" /></span></i> which he values as at least 1/<i>n</i> of the total value: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}(X_{i})\geq V_{i}(C)/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}(X_{i})\geq V_{i}(C)/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/8632385834b824750530fe14e6b7e9d78f68ed48" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.074ex; height:2.843ex;" alt="{\displaystyle V_{i}(X_{i})\geq V_{i}(C)/n}" /></span> for all i.</li></ul></li> <li>A <a href="/wiki/Super-proportional_division" class="mw-redirect" title="Super-proportional division">super-proportional division</a> is one where each player receives strictly more than 1/<i>n.</i> (Such a division exists only if the players have different valuations.): <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}(X_{i})>V_{i}(C)/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>></mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <mi>C</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}(X_{i})>V_{i}(C)/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0db2751914612cb92fd9cadec8e9501eb8bfcbb6" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.074ex; height:2.843ex;" alt="{\displaystyle V_{i}(X_{i})>V_{i}(C)/n}" /></span> for all <i>i</i>.</li></ul></li> <li>An <a href="/wiki/Envy-free" class="mw-redirect" title="Envy-free">envy-free</a> division guarantees that no-one will want somebody else's share more than their own, i.e. every person values their own share at least as much as all other shares: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}(X_{i})\geq V_{i}(X_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>≥</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}(X_{i})\geq V_{i}(X_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61936fbad1d6771e2bed461b083894d9d16c6e81" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.585ex; height:3.009ex;" alt="{\displaystyle V_{i}(X_{i})\geq V_{i}(X_{j})}" /></span> for all i and j.</li></ul></li> <li>A <a href="/wiki/Group-envy-free" class="mw-redirect" title="Group-envy-free">group-envy-free</a> division guarantees that no subset of agents envies another subset of the same size; this is a stronger condition than envy-freeness.</li> <li>An <a href="/wiki/Equity_(economics)" title="Equity (economics)">equitable</a> division means every player’s valuation of their own slice is equal, i.e. each receives equal value, or “experiences equal happiness”. This is a difficult aim as players need not be truthful if asked their valuation. <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}(X_{i})=V_{j}(X_{j})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}(X_{i})=V_{j}(X_{j})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/da0cd913509f6b29a605364bc8f58018fa72a00c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.695ex; height:3.009ex;" alt="{\displaystyle V_{i}(X_{i})=V_{j}(X_{j})}" /></span> for all i and j.</li></ul></li> <li>An <a href="/wiki/Exact_division" class="mw-redirect" title="Exact division">exact division</a> (aka consensus division) is one where all players agree on the value of each share: <ul><li><span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V_{i}(X_{i})=V_{j}(X_{i})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>V</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo stretchy="false">(</mo> <msub> <mi>X</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle V_{i}(X_{i})=V_{j}(X_{i})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/837bc3e5dcc3ba4c02bbd833c979ef3d59b79fbc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.585ex; height:3.009ex;" alt="{\displaystyle V_{i}(X_{i})=V_{j}(X_{i})}" /></span> for all i and j.</li></ul></li></ul> <p>All the above criteria assume that the participants have equal <a href="/wiki/Entitlement_(fair_division)" title="Entitlement (fair division)">entitlements</a>. If different participants have different entitlements (e.g., in a partnership where each partner invested a different amount), then the fairness criteria should be adapted accordingly. See <a href="/wiki/Proportional_cake-cutting_with_different_entitlements" title="Proportional cake-cutting with different entitlements">proportional cake-cutting with different entitlements</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Additional_requirements">Additional requirements</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=3" title="Edit section: Additional requirements"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>In addition to fairness, it is sometimes desired that the division be <a href="/wiki/Pareto_optimal" class="mw-redirect" title="Pareto optimal">Pareto optimal</a>, i.e., no other allocation would make someone better off without making someone else worse off. The term efficiency comes from the <a href="/wiki/Economics" title="Economics">economics</a> idea of the <a href="/wiki/Efficient_market" class="mw-redirect" title="Efficient market">efficient market</a>. A division where one player gets everything is optimal by this definition so on its own this does not guarantee even a fair share. See also <a href="/wiki/Efficient_cake-cutting" title="Efficient cake-cutting">efficient cake-cutting</a> and the <a href="/wiki/Price_of_fairness" title="Price of fairness">price of fairness</a>. </p> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Berlin_Blockade-map.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Berlin_Blockade-map.svg/200px-Berlin_Blockade-map.svg.png" decoding="async" width="200" height="153" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/19/Berlin_Blockade-map.svg/300px-Berlin_Blockade-map.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/19/Berlin_Blockade-map.svg/400px-Berlin_Blockade-map.svg.png 2x" data-file-width="774" data-file-height="593" /></a><figcaption>Berlin divided by the <a href="/wiki/Potsdam_Conference" title="Potsdam Conference">Potsdam Conference</a> </figcaption></figure> <p>In the real world people sometimes have a very accurate idea of how the other players value the goods and they may care very much about it. The case where they have complete knowledge of each other's valuations can be modeled by <a href="/wiki/Game_theory" title="Game theory">game theory</a>. Partial knowledge is very hard to model. A major part of the practical side of fair division is the devising and study of procedures that work well despite such partial knowledge or small mistakes. </p><p>An additional requirement is that the fair division procedure be <a href="/wiki/Strategyproofness" title="Strategyproofness">strategyproof</a>, i.e. it should be a dominant strategy for the participants to report their true valuations. This requirement is usually very hard to satisfy, especially in combination with fairness and Pareto-efficiency. As a result, it is often weakened to <a href="/wiki/Incentive_compatibility" title="Incentive compatibility">incentive compatibility</a>, which only requires players to report their true valuations if they behave according to a specified <a href="/wiki/Solution_concept" title="Solution concept">solution concept</a>. </p> <div class="mw-heading mw-heading2"><h2 id="Procedures">Procedures</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=4" title="Edit section: Procedures"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>A fair division <a href="/wiki/Algorithm" title="Algorithm">procedure</a> lists actions to be performed by the players in terms of the visible data and their valuations. A valid procedure is one that guarantees a fair division for every player who acts rationally according to their valuation. Where an action depends on a player's valuation the procedure is describing the <a href="/wiki/Strategy" title="Strategy">strategy</a> a rational player will follow. A player may act as if a piece had a different value but must be consistent. For instance if a procedure says the first player cuts the cake in two equal parts then the second player chooses a piece, then the first player cannot claim that the second player got more. </p><p>What the players do is: </p> <ul><li>Agree on their criteria for a fair division</li> <li>Select a valid procedure and follow its rules</li></ul> <p>It is assumed the aim of each player is to maximize the minimum amount they might get, or in other words, to achieve the <a href="/wiki/Minimax" title="Minimax">maximin</a>. </p><p>Procedures can be divided into <i>discrete</i> vs. <i>continuous</i> procedures. A discrete procedure would for instance only involve one person at a time cutting or marking a cake. Continuous procedures involve things like one player <a href="/wiki/Moving-knife_procedure" title="Moving-knife procedure">moving a knife</a> and the other saying "stop". Another type of continuous procedure involves a person assigning a value to every part of the cake. </p><p>For a list of fair division procedures, see <a href="/wiki/Category:Fair_division_protocols" title="Category:Fair division protocols">Category:Fair division protocols</a>. </p><p>No finite protocol (even if unbounded) can guarantee an envy-free division of a cake among three or more players, if each player is to receive a single connected piece.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> However, this result applies only to the model presented in that work and not for cases where, for example, a mediator has full information of the players' valuation functions and proposes a division based on this information.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Extensions">Extensions</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=5" title="Edit section: Extensions"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Recently, the model of fair division has been extended from individual agents to <i>families</i> (pre-determined groups) of agents. See <a href="/wiki/Fair_division_among_groups" title="Fair division among groups">fair division among groups</a>. </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=6" title="Edit section: History"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>According to <a href="/wiki/Sol_Garfunkel" title="Sol Garfunkel">Sol Garfunkel</a>, the cake-cutting problem had been one of the most important open problems in 20th century mathematics,<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> when the most important variant of the problem was finally solved with the <a href="/wiki/Brams-Taylor_procedure" class="mw-redirect" title="Brams-Taylor procedure">Brams-Taylor procedure</a> by <a href="/wiki/Steven_Brams" title="Steven Brams">Steven Brams</a> and <a href="/wiki/Alan_D._Taylor" title="Alan D. Taylor">Alan Taylor</a> in 1995. </p><p><a href="/wiki/Divide_and_choose" title="Divide and choose">Divide and choose</a>'s origins are undocumented. The related activities of <a href="/wiki/Bargaining" title="Bargaining">bargaining</a> and <a href="/wiki/Barter" title="Barter">barter</a> are also ancient. <a href="/wiki/Negotiation" title="Negotiation">Negotiations</a> involving more than two people are also quite common, the <a href="/wiki/Potsdam_Conference" title="Potsdam Conference">Potsdam Conference</a> is a notable recent example. </p><p>The theory of fair division dates back only to the end of the second world war. It was devised by a group of <a href="/wiki/Poland" title="Poland">Polish</a> mathematicians, <a href="/wiki/Hugo_Steinhaus" title="Hugo Steinhaus">Hugo Steinhaus</a>, <a href="/wiki/Bronis%C5%82aw_Knaster" title="Bronisław Knaster">Bronisław Knaster</a> and <a href="/wiki/Stefan_Banach" title="Stefan Banach">Stefan Banach</a>, who used to meet in the <a href="/wiki/Scottish_Caf%C3%A9" title="Scottish Café">Scottish Café</a> in Lvov (then in Poland). A <a href="/wiki/Proportional_(fair_division)" class="mw-redirect" title="Proportional (fair division)">proportional (fair division)</a> division for any number of players called 'last-diminisher' was devised in 1944. This was attributed to Banach and Knaster by Steinhaus when he made the problem public for the first time at a meeting of the <a href="/wiki/Econometric_Society" title="Econometric Society">Econometric Society</a> in Washington, D.C., on 17 September 1947. At that meeting he also proposed the problem of finding the smallest number of cuts necessary for such divisions. </p><p>For the history of envy-free cake-cutting, see <a href="/wiki/Envy-free_cake-cutting#Short_history" title="Envy-free cake-cutting">envy-free cake-cutting</a>. </p> <div class="mw-heading mw-heading2"><h2 id="In_popular_culture">In popular culture</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=7" title="Edit section: In popular culture"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>The <a href="/wiki/17-animal_inheritance_puzzle" title="17-animal inheritance puzzle">17-animal inheritance puzzle</a> involves the fair division of 17 camels (or elephants, or horses) into the proportions 1/2, 1/3, and 1/9. It is a popular <a href="/wiki/Mathematical_puzzle" title="Mathematical puzzle">mathematical puzzle</a>, often claimed to have an ancient origin, but its first documented publication was in 18th-century Iran.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup></li> <li>In <i><a href="/wiki/Numb3rs" class="mw-redirect" title="Numb3rs">Numb3rs</a></i> season 3 episode "One Hour", Charlie talks about the cake-cutting problem as applied to the amount of money a kidnapper was demanding.</li> <li><a href="/wiki/Hugo_Steinhaus" title="Hugo Steinhaus">Hugo Steinhaus</a> wrote about a number of variants of fair division in his book <i>Mathematical Snapshots</i>. In his book he says a special three-person version of fair division was devised by G. Krochmainy in Berdechów in 1944 and another by Mrs L Kott.<sup id="cite_ref-7" class="reference"><a href="#cite_note-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup></li> <li><a href="/wiki/Martin_Gardner" title="Martin Gardner">Martin Gardner</a> and <a href="/wiki/Ian_Stewart_(mathematician)" title="Ian Stewart (mathematician)">Ian Stewart</a> have both published books with sections about the problem.<sup id="cite_ref-8" class="reference"><a href="#cite_note-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> Martin Gardner introduced the chore division form of the problem. Ian Stewart has popularized the fair division problem with his articles in <i><a href="/wiki/Scientific_American" title="Scientific American">Scientific American</a></i> and <i><a href="/wiki/New_Scientist" title="New Scientist">New Scientist</a></i>.</li> <li>A <i><a href="/wiki/Dinosaur_Comics" title="Dinosaur Comics">Dinosaur Comics</a></i> strip is based on the cake-cutting problem.<sup id="cite_ref-10" class="reference"><a href="#cite_note-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup></li> <li>In the Israeli movie <a href="/wiki/Saint_Clara_(film)" title="Saint Clara (film)">Saint Clara</a>, a Russian immigrant asks an Israeli math teacher, how a circular cake can be divided fairly among 7 people? His answer is to make 3 straight cuts through its middle, making 8 equal pieces. Since there are only 7 people, one piece should be discarded, in the spirit of communism.</li></ul> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=8" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fair_division_experiments" title="Fair division experiments">Fair division experiments</a></li> <li><a href="/wiki/List_of_unsolved_problems_in_fair_division" title="List of unsolved problems in fair division">List of unsolved problems in fair division</a></li> <li><a href="/wiki/Online_fair_division" title="Online fair division">Online fair division</a></li> <li><a href="/wiki/Strategic_fair_division" title="Strategic fair division">Strategic fair division</a></li> <li><a href="/wiki/Apportionment_(politics)" title="Apportionment (politics)">Apportionment</a> and <a href="/wiki/Mathematics_of_apportionment" title="Mathematics of apportionment">mathematics of apportionment</a></li> <li><a href="/wiki/Equity_(economics)" title="Equity (economics)">Equity (economics)</a></li> <li><a href="/wiki/International_trade" title="International trade">International trade</a></li> <li><a href="/wiki/Justice_(economics)" class="mw-redirect" title="Justice (economics)">Justice (economics)</a></li> <li><a href="/wiki/Knapsack_problem" title="Knapsack problem">Knapsack problem</a></li> <li><a href="/wiki/Nash_bargaining_game" class="mw-redirect" title="Nash bargaining game">Nash bargaining game</a></li> <li><a href="/wiki/Pizza_theorem" title="Pizza theorem">Pizza theorem</a></li> <li><a href="/wiki/Price_of_fairness" title="Price of fairness">Price of fairness</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=9" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFAumannMaschler1985" class="citation journal cs1">Aumann, Robert J.; Maschler, Michael (1985). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20060220022042/http://www.elsevier.com/framework_aboutus/Nobel/Nobel2005/nobel2005pdfs/aum16.pdf">"Game Theoretic Analysis of a bankruptcy Problem from the Talmud"</a> <span class="cs1-format">(PDF)</span>. <i>Journal of Economic Theory</i>. <b>36</b> (2): <span class="nowrap">195–</span>213. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0022-0531%2885%2990102-4">10.1016/0022-0531(85)90102-4</a>. Archived from <a rel="nofollow" class="external text" href="http://www.elsevier.com/framework_aboutus/Nobel/Nobel2005/nobel2005pdfs/aum16.pdf">the original</a> <span class="cs1-format">(PDF)</span> on 2006-02-20.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Economic+Theory&rft.atitle=Game+Theoretic+Analysis+of+a+bankruptcy+Problem+from+the+Talmud&rft.volume=36&rft.issue=2&rft.pages=%3Cspan+class%3D%22nowrap%22%3E195-%3C%2Fspan%3E213&rft.date=1985&rft_id=info%3Adoi%2F10.1016%2F0022-0531%2885%2990102-4&rft.aulast=Aumann&rft.aufirst=Robert+J.&rft.au=Maschler%2C+Michael&rft_id=http%3A%2F%2Fwww.elsevier.com%2Fframework_aboutus%2FNobel%2FNobel2005%2Fnobel2005pdfs%2Faum16.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></span> </li> <li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYaariBar-Hillel1984" class="citation journal cs1">Yaari, M. E.; Bar-Hillel, M. (1984). "On dividing justly". <i>Social Choice and Welfare</i>. <b>1</b>: 1. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00297056">10.1007/BF00297056</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:153443060">153443060</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Social+Choice+and+Welfare&rft.atitle=On+dividing+justly&rft.volume=1&rft.pages=1&rft.date=1984&rft_id=info%3Adoi%2F10.1007%2FBF00297056&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A153443060%23id-name%3DS2CID&rft.aulast=Yaari&rft.aufirst=M.+E.&rft.au=Bar-Hillel%2C+M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></span> </li> <li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFStromquist2008" class="citation journal cs1">Stromquist, Walter (2008). <a rel="nofollow" class="external text" href="https://eudml.org/doc/129749">"Envy-free cake divisions cannot be found by finite protocols"</a>. <i>The Electronic Journal of Combinatorics</i>. <b>15</b>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://doi.org/10.37236%2F735">10.37236/735</a></span><span class="reference-accessdate">. Retrieved <span class="nowrap">October 26,</span> 2022</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+Electronic+Journal+of+Combinatorics&rft.atitle=Envy-free+cake+divisions+cannot+be+found+by+finite+protocols&rft.volume=15&rft.date=2008&rft_id=info%3Adoi%2F10.37236%2F735&rft.aulast=Stromquist&rft.aufirst=Walter&rft_id=https%3A%2F%2Feudml.org%2Fdoc%2F129749&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAumannDombb2010" class="citation conference cs1">Aumann, Yonatan; Dombb, Yair (2010). <a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-642-17572-5_3">"The Efficiency of Fair Division with Connected Pieces"</a>. <i>Internet and Network Economics</i>. International Workshop on Internet and Network Economics. Springer. pp. <span class="nowrap">26–</span>37. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-3-642-17572-5_3">10.1007/978-3-642-17572-5_3</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=conference&rft.atitle=The+Efficiency+of+Fair+Division+with+Connected+Pieces&rft.btitle=Internet+and+Network+Economics&rft.pages=%3Cspan+class%3D%22nowrap%22%3E26-%3C%2Fspan%3E37&rft.pub=Springer&rft.date=2010&rft_id=info%3Adoi%2F10.1007%2F978-3-642-17572-5_3&rft.aulast=Aumann&rft.aufirst=Yonatan&rft.au=Dombb%2C+Yair&rft_id=https%3A%2F%2Flink.springer.com%2Fchapter%2F10.1007%2F978-3-642-17572-5_3&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text">Sol Garfunkel. More Equal than Others: Weighted Voting. For All Practical Purposes. COMAP. 1988</span> </li> <li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFAgeron2013" class="citation journal cs1 cs1-prop-foreign-lang-source">Ageron, Pierre (2013). <a rel="nofollow" class="external text" href="https://ageron.users.lmno.cnrs.fr/17chameaux.pdf">"Le partage des dix-sept chameaux et autres arithmétiques attributes à l'immam 'Alî: Mouvance et circulation de récits de la tradition musulmane chiite"</a> <span class="cs1-format">(PDF)</span>. <i>Revue d'histoire des mathématiques</i> (in French). <b>19</b> (1): <span class="nowrap">1–</span>41.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Revue+d%27histoire+des+math%C3%A9matiques&rft.atitle=Le+partage+des+dix-sept+chameaux+et+autres+arithm%C3%A9tiques+attributes+%C3%A0+l%27immam+%27Al%C3%AE%3A+Mouvance+et+circulation+de+r%C3%A9cits+de+la+tradition+musulmane+chiite&rft.volume=19&rft.issue=1&rft.pages=%3Cspan+class%3D%22nowrap%22%3E1-%3C%2Fspan%3E41&rft.date=2013&rft.aulast=Ageron&rft.aufirst=Pierre&rft_id=https%3A%2F%2Fageron.users.lmno.cnrs.fr%2F17chameaux.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span>; see in particular pp. 13–14.</span> </li> <li id="cite_note-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-7">^</a></b></span> <span class="reference-text">Mathematical Snapshots. H.Steinhaus. 1950, 1969 <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-19-503267-5" title="Special:BookSources/0-19-503267-5">0-19-503267-5</a></span> </li> <li id="cite_note-8"><span class="mw-cite-backlink"><b><a href="#cite_ref-8">^</a></b></span> <span class="reference-text">aha! Insight. Martin. Gardner, 1978. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-7167-1017-2" title="Special:BookSources/978-0-7167-1017-2">978-0-7167-1017-2</a></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text">How to cut a cake and other mathematical conundrums. Ian Stewart. 2006. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-19-920590-5" title="Special:BookSources/978-0-19-920590-5">978-0-19-920590-5</a></span> </li> <li id="cite_note-10"><span class="mw-cite-backlink"><b><a href="#cite_ref-10">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.qwantz.com/index.php?comic=1345">"Dinosaur Comics!"</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Dinosaur+Comics%21&rft_id=http%3A%2F%2Fwww.qwantz.com%2Findex.php%3Fcomic%3D1345&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="Text_books">Text books</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=10" title="Edit section: Text books"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFYoung1995" class="citation book cs1">Young, Peyton H. (1995). <i>Equity: in theory and practice</i>. Princeton University Press.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Equity%3A+in+theory+and+practice&rft.pub=Princeton+University+Press&rft.date=1995&rft.aulast=Young&rft.aufirst=Peyton+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBramsTaylor1996" class="citation book cs1">Brams, Steven J.; Taylor, Alan D. (1996). <i>Fair division: from cake-cutting to dispute resolution</i>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-55644-9" title="Special:BookSources/0-521-55644-9"><bdi>0-521-55644-9</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fair+division%3A+from+cake-cutting+to+dispute+resolution&rft.pub=Cambridge+University+Press&rft.date=1996&rft.isbn=0-521-55644-9&rft.aulast=Brams&rft.aufirst=Steven+J.&rft.au=Taylor%2C+Alan+D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFRobertsonWebb1998" class="citation book cs1">Robertson, Jack; Webb, William (1998). <i>Cake-Cutting Algorithms: Be Fair If You Can</i>. Natick, Massachusetts: A. K. Peters. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-1-56881-076-8" title="Special:BookSources/978-1-56881-076-8"><bdi>978-1-56881-076-8</bdi></a>. <a href="/wiki/LCCN_(identifier)" class="mw-redirect" title="LCCN (identifier)">LCCN</a> <a rel="nofollow" class="external text" href="https://lccn.loc.gov/97041258">97041258</a>. <a href="/wiki/OL_(identifier)" class="mw-redirect" title="OL (identifier)">OL</a> <a rel="nofollow" class="external text" href="https://openlibrary.org/works/OL2730675W">2730675W</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Cake-Cutting+Algorithms%3A+Be+Fair+If+You+Can&rft.place=Natick%2C+Massachusetts&rft.pub=A.+K.+Peters&rft.date=1998&rft_id=https%3A%2F%2Fopenlibrary.org%2Fworks%2FOL2730675W%23id-name%3DOL&rft_id=info%3Alccn%2F97041258&rft.isbn=978-1-56881-076-8&rft.aulast=Robertson&rft.aufirst=Jack&rft.au=Webb%2C+William&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHerve_Moulin2004" class="citation book cs1">Herve Moulin (2004). <i>Fair Division and Collective Welfare</i>. Cambridge, Massachusetts: MIT Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780262134231" title="Special:BookSources/9780262134231"><bdi>9780262134231</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Fair+Division+and+Collective+Welfare&rft.place=Cambridge%2C+Massachusetts&rft.pub=MIT+Press&rft.date=2004&rft.isbn=9780262134231&rft.au=Herve+Moulin&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarbanel2005" class="citation book cs1">Barbanel, Julius B. (2005). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=aBC5VkegvDkC&pg=PA2"><i>The geometry of efficient fair division</i></a>. Introduction by Alan D. Taylor. Cambridge: Cambridge University Press. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FCBO9780511546679">10.1017/CBO9780511546679</a>. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-84248-4" title="Special:BookSources/0-521-84248-4"><bdi>0-521-84248-4</bdi></a>. <a href="/wiki/MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=2132232">2132232</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+geometry+of+efficient+fair+division&rft.place=Cambridge&rft.pub=Cambridge+University+Press&rft.date=2005&rft_id=https%3A%2F%2Fmathscinet.ams.org%2Fmathscinet-getitem%3Fmr%3D2132232%23id-name%3DMR&rft_id=info%3Adoi%2F10.1017%2FCBO9780511546679&rft.isbn=0-521-84248-4&rft.aulast=Barbanel&rft.aufirst=Julius+B.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DaBC5VkegvDkC%26pg%3DPA2&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span> <small>Short summary is available at: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBarbanel2010" class="citation journal cs1">Barbanel, J. (2010). "A Geometric Approach to Fair Division". <i>The College Mathematics Journal</i>. <b>41</b> (4): 268. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4169%2F074683410x510263">10.4169/074683410x510263</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=The+College+Mathematics+Journal&rft.atitle=A+Geometric+Approach+to+Fair+Division&rft.volume=41&rft.issue=4&rft.pages=268&rft.date=2010&rft_id=info%3Adoi%2F10.4169%2F074683410x510263&rft.aulast=Barbanel&rft.aufirst=J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></small></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFSteven_J._Brams2008" class="citation book cs1">Steven J. Brams (2008). <i>Mathematics and Democracy: Designing Better Voting and Fair-Division Procedures</i>. Princeton, NJ: Princeton University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780691133218" title="Special:BookSources/9780691133218"><bdi>9780691133218</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mathematics+and+Democracy%3A+Designing+Better+Voting+and+Fair-Division+Procedures&rft.place=Princeton%2C+NJ&rft.pub=Princeton+University+Press&rft.date=2008&rft.isbn=9780691133218&rft.au=Steven+J.+Brams&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></li></ul> <div class="mw-heading mw-heading2"><h2 id="Survey_articles">Survey articles</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=11" title="Edit section: Survey articles"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>Vincent P. Crawford (1987). "fair division," <i>The <a href="/wiki/New_Palgrave:_A_Dictionary_of_Economics" class="mw-redirect" title="New Palgrave: A Dictionary of Economics">New Palgrave: A Dictionary of Economics</a></i>, v. 2, pp. 274–75.</li> <li><a href="/wiki/Hal_R._Varian" class="mw-redirect" title="Hal R. Varian">Hal Varian</a> (1987). "fairness," <i>The New Palgrave: A Dictionary of Economics</i>, v. 2, pp. 275–76.</li> <li>Bryan Skyrms (1996). <i>The Evolution of the Social Contract</i> Cambridge University Press. <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-55583-8" title="Special:BookSources/978-0-521-55583-8">978-0-521-55583-8</a></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFHill2000" class="citation journal cs1">Hill, T.P. (2000). "Mathematical devices for getting a fair share". <i>American Scientist</i>. <b>88</b> (4): <span class="nowrap">325–</span>331. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2000AmSci..88..325H">2000AmSci..88..325H</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1511%2F2000.4.325">10.1511/2000.4.325</a>. <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:221539202">221539202</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=American+Scientist&rft.atitle=Mathematical+devices+for+getting+a+fair+share&rft.volume=88&rft.issue=4&rft.pages=%3Cspan+class%3D%22nowrap%22%3E325-%3C%2Fspan%3E331&rft.date=2000&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A221539202%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1511%2F2000.4.325&rft_id=info%3Abibcode%2F2000AmSci..88..325H&rft.aulast=Hill&rft.aufirst=T.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFBrandtConitzerEndrissLang2016" class="citation book cs1">Brandt, Felix; Conitzer, Vincent; Endriss, Ulle; Lang, Jérôme; Procaccia, Ariel D. (2016). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=nMHgCwAAQBAJ"><i>Handbook of Computational Social Choice</i></a>. Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781107060432" title="Special:BookSources/9781107060432"><bdi>9781107060432</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Computational+Social+Choice&rft.pub=Cambridge+University+Press&rft.date=2016&rft.isbn=9781107060432&rft.aulast=Brandt&rft.aufirst=Felix&rft.au=Conitzer%2C+Vincent&rft.au=Endriss%2C+Ulle&rft.au=Lang%2C+J%C3%A9r%C3%B4me&rft.au=Procaccia%2C+Ariel+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DnMHgCwAAQBAJ&rfr_id=info%3Asid%2Fen.wikipedia.org%3AFair+division" class="Z3988"></span>, chapters 11–13.</li> <li><a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-90-481-9097-3_12">Fair Division</a> by Christian Klamler – in Handbook of Group Decision and Negotiation pp 183–202.</li> <li><a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-662-47904-9_7">Cake-Cutting: Fair Division of Divisible Goods</a> by Claudia Lindner and Jörg Rothe – in Economics and Computation pp 395–491.</li> <li><a rel="nofollow" class="external text" href="https://link.springer.com/chapter/10.1007/978-3-662-47904-9_8">Fair division of indivisible goods</a> by Jérôme Lang and Jörg Rothe – in Economics and Computation pp 493–550.</li></ul> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=Fair_division&action=edit&section=12" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a rel="nofollow" class="external text" href="https://web.archive.org/web/20091022113055/http://www.colorado.edu/education/DMP/fair_division.html">Fair Division</a> from the Discrete Mathematics Project at the University of Colorado at Boulder.</li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/SocialScience/Markers.shtml">Fair Division: Method of Markers</a></li> <li><a rel="nofollow" class="external text" href="http://www.cut-the-knot.org/Curriculum/SocialScience/SealedBids.shtml">Fair Division: Method of Sealed Bids</a></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol 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href="/wiki/First-player_and_second-player_win" title="First-player and second-player win">First-player and second-player win</a></li> <li><a href="/wiki/Game_complexity" title="Game complexity">Game complexity</a></li> <li><a href="/wiki/Graphical_game_theory" title="Graphical game theory">Graphical game</a></li> <li><a href="/wiki/Hierarchy_of_beliefs" title="Hierarchy of beliefs">Hierarchy of beliefs</a></li> <li><a href="/wiki/Information_set_(game_theory)" title="Information set (game theory)">Information set</a></li> <li><a href="/wiki/Normal-form_game" title="Normal-form game">Normal-form game</a></li> <li><a href="/wiki/Perfect_recall_(game_theory)" title="Perfect recall (game theory)">Perfect recall</a></li> <li><a href="/wiki/Preference_(economics)" title="Preference (economics)">Preference</a></li> <li><a href="/wiki/Sequential_game" title="Sequential game">Sequential game</a></li> <li><a href="/wiki/Simultaneous_game" title="Simultaneous game">Simultaneous game</a></li> <li><a href="/wiki/Simultaneous_action_selection" title="Simultaneous action selection">Simultaneous action selection</a></li> <li><a href="/wiki/Solved_game" title="Solved game">Solved game</a></li> <li><a href="/wiki/Succinct_game" title="Succinct game">Succinct game</a></li> <li><a href="/wiki/Mechanism_design" title="Mechanism design">Mechanism design</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Economic_equilibrium" title="Economic equilibrium">Equilibrium</a><br /><a href="/wiki/Solution_concept" title="Solution concept">concepts</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bayes_correlated_equilibrium" title="Bayes correlated equilibrium">Bayes correlated equilibrium</a></li> <li><a href="/wiki/Bayesian_Nash_equilibrium" class="mw-redirect" title="Bayesian Nash equilibrium">Bayesian Nash equilibrium</a></li> <li><a href="/wiki/Berge_equilibrium" title="Berge equilibrium">Berge equilibrium</a></li> <li><a href="/wiki/Core_(game_theory)" title="Core (game theory)"> Core</a></li> <li><a href="/wiki/Correlated_equilibrium" title="Correlated equilibrium">Correlated equilibrium</a></li> <li><a href="/wiki/Coalition-proof_Nash_equilibrium" title="Coalition-proof Nash equilibrium">Coalition-proof Nash equilibrium</a></li> <li><a href="/wiki/Epsilon-equilibrium" title="Epsilon-equilibrium">Epsilon-equilibrium</a></li> <li><a href="/wiki/Evolutionarily_stable_strategy" title="Evolutionarily stable strategy">Evolutionarily stable strategy</a></li> <li><a href="/wiki/Gibbs_measure" title="Gibbs measure">Gibbs equilibrium</a></li> <li><a href="/wiki/Mertens-stable_equilibrium" title="Mertens-stable equilibrium">Mertens-stable equilibrium</a></li> <li><a href="/wiki/Markov_perfect_equilibrium" title="Markov perfect equilibrium">Markov perfect equilibrium</a></li> <li><a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a></li> <li><a href="/wiki/Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li> <li><a href="/wiki/Perfect_Bayesian_equilibrium" title="Perfect Bayesian equilibrium">Perfect Bayesian equilibrium</a></li> <li><a href="/wiki/Proper_equilibrium" title="Proper equilibrium">Proper equilibrium</a></li> <li><a href="/wiki/Quantal_response_equilibrium" title="Quantal response equilibrium">Quantal response equilibrium</a></li> <li><a href="/wiki/Quasi-perfect_equilibrium" title="Quasi-perfect equilibrium">Quasi-perfect equilibrium</a></li> <li><a href="/wiki/Risk_dominance" title="Risk dominance">Risk dominance</a></li> <li><a href="/wiki/Satisfaction_equilibrium" title="Satisfaction equilibrium">Satisfaction equilibrium</a></li> <li><a href="/wiki/Self-confirming_equilibrium" title="Self-confirming equilibrium">Self-confirming equilibrium</a></li> <li><a href="/wiki/Sequential_equilibrium" title="Sequential equilibrium">Sequential equilibrium</a></li> <li><a href="/wiki/Shapley_value" title="Shapley value">Shapley value</a></li> <li><a href="/wiki/Strong_Nash_equilibrium" title="Strong Nash equilibrium">Strong Nash equilibrium</a></li> <li><a href="/wiki/Subgame_perfect_equilibrium" title="Subgame perfect equilibrium">Subgame perfection</a></li> <li><a href="/wiki/Trembling_hand_perfect_equilibrium" title="Trembling hand perfect equilibrium">Trembling hand equilibrium</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Strategy_(game_theory)" title="Strategy (game theory)">Strategies</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Appeasement" title="Appeasement">Appeasement</a></li> <li><a href="/wiki/Backward_induction" title="Backward induction">Backward induction</a></li> <li><a href="/wiki/Bid_shading" title="Bid shading">Bid shading</a></li> <li><a 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title="Strategy (game theory)">Mixed strategy</a></li> <li><a href="/wiki/Strategy-stealing_argument" title="Strategy-stealing argument">Strategy-stealing argument</a></li> <li><a href="/wiki/Tit_for_tat" title="Tit for tat">Tit for tat</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Category:Game_theory_game_classes" title="Category:Game theory game classes">Classes<br />of games</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Auction" title="Auction">Auction</a></li> <li><a href="/wiki/Bargaining_problem" class="mw-redirect" title="Bargaining problem">Bargaining problem</a></li> <li><a href="/wiki/Differential_game" title="Differential game">Differential game</a></li> <li><a href="/wiki/Global_game" title="Global game">Global game</a></li> <li><a href="/wiki/Intransitive_game" title="Intransitive game">Intransitive game</a></li> <li><a href="/wiki/Mean-field_game_theory" title="Mean-field game theory">Mean-field game</a></li> <li><a href="/wiki/N-player_game" title="N-player game"><i>n</i>-player game</a></li> <li><a href="/wiki/Perfect_information" title="Perfect information">Perfect information</a></li> <li><a href="/wiki/Poisson_games" class="mw-redirect" title="Poisson games">Large Poisson game</a></li> <li><a href="/wiki/Potential_game" title="Potential game">Potential game</a></li> <li><a href="/wiki/Repeated_game" title="Repeated game">Repeated game</a></li> <li><a href="/wiki/Screening_game" title="Screening game">Screening game</a></li> <li><a href="/wiki/Signaling_game" title="Signaling game">Signaling game</a></li> <li><a href="/wiki/Strictly_determined_game" title="Strictly determined game">Strictly determined game</a></li> <li><a href="/wiki/Stochastic_game" title="Stochastic game">Stochastic game</a></li> <li><a href="/wiki/Symmetric_game" title="Symmetric game">Symmetric game</a></li> <li><a href="/wiki/Zero-sum_game" title="Zero-sum game">Zero-sum game</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/List_of_games_in_game_theory" title="List of games in game theory">Games</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Go_(game)" title="Go (game)">Go</a></li> <li><a href="/wiki/Chess" title="Chess">Chess</a></li> <li><a href="/wiki/Infinite_chess" title="Infinite chess">Infinite chess</a></li> <li><a href="/wiki/Draughts" class="mw-redirect" title="Draughts">Checkers</a></li> <li><a href="/wiki/All-pay_auction" title="All-pay auction">All-pay auction</a></li> <li><a href="/wiki/Prisoner%27s_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li> <li><a href="/wiki/Gift-exchange_game" title="Gift-exchange game">Gift-exchange game</a></li> <li><a href="/wiki/Optional_prisoner%27s_dilemma" title="Optional prisoner's dilemma">Optional prisoner's dilemma</a></li> <li><a href="/wiki/Traveler%27s_dilemma" title="Traveler's dilemma">Traveler's dilemma</a></li> <li><a href="/wiki/Coordination_game" title="Coordination game">Coordination game</a></li> <li><a href="/wiki/Chicken_(game)" title="Chicken (game)">Chicken</a></li> <li><a href="/wiki/Centipede_game" title="Centipede game">Centipede game</a></li> <li><a href="/wiki/Lewis_signaling_game" title="Lewis signaling game">Lewis signaling game</a></li> <li><a href="/wiki/Volunteer%27s_dilemma" title="Volunteer's dilemma">Volunteer's dilemma</a></li> <li><a href="/wiki/Dollar_auction" title="Dollar auction">Dollar auction</a></li> <li><a href="/wiki/Battle_of_the_sexes_(game_theory)" title="Battle of the sexes (game theory)">Battle of the sexes</a></li> <li><a href="/wiki/Stag_hunt" title="Stag hunt">Stag hunt</a></li> <li><a href="/wiki/Matching_pennies" title="Matching pennies">Matching pennies</a></li> <li><a href="/wiki/Ultimatum_game" title="Ultimatum game">Ultimatum game</a></li> <li><a href="/wiki/Electronic_mail_game" title="Electronic mail game">Electronic mail game</a></li> <li><a href="/wiki/Rock_paper_scissors" title="Rock paper scissors">Rock paper scissors</a></li> <li><a href="/wiki/Pirate_game" title="Pirate game">Pirate game</a></li> <li><a href="/wiki/Dictator_game" title="Dictator game">Dictator game</a></li> <li><a href="/wiki/Public_goods_game" title="Public goods game">Public goods game</a></li> <li><a href="/wiki/Blotto_game" title="Blotto game">Blotto game</a></li> <li><a href="/wiki/War_of_attrition_(game)" title="War of attrition (game)">War of attrition</a></li> <li><a href="/wiki/El_Farol_Bar_problem" title="El Farol Bar problem">El Farol Bar problem</a></li> <li><a class="mw-selflink selflink">Fair division</a></li> <li><a href="/wiki/Fair_cake-cutting" title="Fair cake-cutting">Fair cake-cutting</a></li> <li><a href="/wiki/Bertrand_competition" title="Bertrand competition">Bertrand competition</a></li> <li><a href="/wiki/Cournot_competition" title="Cournot competition">Cournot competition</a></li> <li><a href="/wiki/Stackelberg_competition" title="Stackelberg competition">Stackelberg competition</a></li> <li><a href="/wiki/Deadlock_(game_theory)" title="Deadlock (game theory)">Deadlock</a></li> <li><a href="/wiki/Unscrupulous_diner%27s_dilemma" title="Unscrupulous diner's dilemma">Diner's dilemma</a></li> <li><a href="/wiki/Guess_2/3_of_the_average" title="Guess 2/3 of the average">Guess 2/3 of the average</a></li> <li><a href="/wiki/Kuhn_poker" title="Kuhn poker">Kuhn poker</a></li> <li><a href="/wiki/Bargaining_problem" class="mw-redirect" title="Bargaining problem">Nash bargaining game</a></li> <li><a href="/wiki/Induction_puzzles" title="Induction puzzles">Induction puzzles</a></li> <li><a href="/wiki/Dictator_game#Trust_game" title="Dictator game">Trust game</a></li> <li><a href="/wiki/Princess_and_monster_game" title="Princess and monster game">Princess and monster game</a></li> <li><a href="/wiki/Rendezvous_problem" title="Rendezvous problem">Rendezvous problem</a></li> <li><a href="/wiki/Pursuit%E2%80%93evasion" title="Pursuit–evasion">Pursuit game</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theorems</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Aumann%27s_agreement_theorem" title="Aumann's agreement theorem">Aumann's agreement theorem</a></li> <li><a href="/wiki/Folk_theorem_(game_theory)" title="Folk theorem (game theory)">Folk theorem</a></li> <li><a href="/wiki/Minimax" title="Minimax">Minimax theorem</a></li> <li><a href="/wiki/Nash_equilibrium" title="Nash equilibrium">Nash's theorem</a></li> <li><a href="/wiki/Negamax" title="Negamax">Negamax theorem</a></li> <li><a href="/wiki/One-shot_deviation_principle" title="One-shot deviation principle">One-shot deviation principle</a></li> <li><a href="/wiki/Purification_theorem" title="Purification theorem">Purification theorem</a></li> <li><a href="/wiki/Revelation_principle" title="Revelation principle">Revelation principle</a></li> <li><a href="/wiki/Sprague%E2%80%93Grundy_theorem" title="Sprague–Grundy theorem">Sprague–Grundy theorem</a></li> <li><a href="/wiki/Zermelo%27s_theorem_(game_theory)" title="Zermelo's theorem (game theory)">Zermelo's theorem</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key<br />figures</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Albert_W._Tucker" title="Albert W. Tucker">Albert W. Tucker</a></li> <li><a href="/wiki/Amos_Tversky" title="Amos Tversky">Amos Tversky</a></li> <li><a href="/wiki/Antoine_Augustin_Cournot" title="Antoine Augustin Cournot">Antoine Augustin Cournot</a></li> <li><a href="/wiki/Ariel_Rubinstein" title="Ariel Rubinstein">Ariel Rubinstein</a></li> <li><a href="/wiki/Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li> <li><a href="/wiki/Daniel_Kahneman" title="Daniel Kahneman">Daniel Kahneman</a></li> <li><a href="/wiki/David_K._Levine" title="David K. Levine">David K. Levine</a></li> <li><a href="/wiki/David_M._Kreps" title="David M. Kreps">David M. Kreps</a></li> <li><a href="/wiki/Donald_B._Gillies" title="Donald B. Gillies">Donald B. Gillies</a></li> <li><a href="/wiki/Drew_Fudenberg" title="Drew Fudenberg">Drew Fudenberg</a></li> <li><a href="/wiki/Eric_Maskin" title="Eric Maskin">Eric Maskin</a></li> <li><a href="/wiki/Harold_W._Kuhn" title="Harold W. Kuhn">Harold W. Kuhn</a></li> <li><a href="/wiki/Herbert_A._Simon" title="Herbert A. Simon">Herbert Simon</a></li> <li><a href="/wiki/Herv%C3%A9_Moulin" title="Hervé Moulin">Hervé Moulin</a></li> <li><a href="/wiki/John_Conway" class="mw-redirect" title="John Conway">John Conway</a></li> <li><a href="/wiki/Jean_Tirole" title="Jean Tirole">Jean Tirole</a></li> <li><a href="/wiki/Jean-Fran%C3%A7ois_Mertens" title="Jean-François Mertens">Jean-François Mertens</a></li> <li><a href="/wiki/Jennifer_Tour_Chayes" title="Jennifer Tour Chayes">Jennifer Tour Chayes</a></li> <li><a href="/wiki/John_Harsanyi" title="John Harsanyi">John Harsanyi</a></li> <li><a href="/wiki/John_Maynard_Smith" title="John Maynard Smith">John Maynard Smith</a></li> <li><a href="/wiki/John_Forbes_Nash_Jr." title="John Forbes Nash Jr.">John Nash</a></li> <li><a href="/wiki/John_von_Neumann" title="John von Neumann">John von Neumann</a></li> <li><a href="/wiki/Kenneth_Arrow" title="Kenneth Arrow">Kenneth Arrow</a></li> <li><a href="/wiki/Kenneth_Binmore" title="Kenneth Binmore">Kenneth Binmore</a></li> <li><a href="/wiki/Leonid_Hurwicz" title="Leonid Hurwicz">Leonid Hurwicz</a></li> <li><a href="/wiki/Lloyd_Shapley" title="Lloyd Shapley">Lloyd Shapley</a></li> <li><a href="/wiki/Melvin_Dresher" title="Melvin Dresher">Melvin Dresher</a></li> <li><a href="/wiki/Merrill_M._Flood" title="Merrill M. Flood">Merrill M. Flood</a></li> <li><a href="/wiki/Olga_Bondareva" title="Olga Bondareva">Olga Bondareva</a></li> <li><a href="/wiki/Oskar_Morgenstern" title="Oskar Morgenstern">Oskar Morgenstern</a></li> <li><a href="/wiki/Paul_Milgrom" title="Paul Milgrom">Paul Milgrom</a></li> <li><a href="/wiki/Peyton_Young" title="Peyton Young">Peyton Young</a></li> <li><a href="/wiki/Reinhard_Selten" title="Reinhard Selten">Reinhard Selten</a></li> <li><a href="/wiki/Robert_Axelrod_(political_scientist)" title="Robert Axelrod (political scientist)">Robert Axelrod</a></li> <li><a href="/wiki/Robert_Aumann" title="Robert Aumann">Robert Aumann</a></li> <li><a href="/wiki/Robert_B._Wilson" title="Robert B. Wilson">Robert B. Wilson</a></li> <li><a href="/wiki/Roger_Myerson" title="Roger Myerson">Roger Myerson</a></li> <li><a href="/wiki/Samuel_Bowles_(economist)" title="Samuel Bowles (economist)"> Samuel Bowles</a></li> <li><a href="/wiki/Suzanne_Scotchmer" title="Suzanne Scotchmer">Suzanne Scotchmer</a></li> <li><a href="/wiki/Thomas_Schelling" title="Thomas Schelling">Thomas Schelling</a></li> <li><a href="/wiki/William_Vickrey" title="William Vickrey">William Vickrey</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Search optimizations</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">Alpha–beta pruning</a></li> <li><a href="/wiki/Aspiration_window" title="Aspiration window">Aspiration window</a></li> <li><a href="/wiki/Principal_variation_search" title="Principal variation search">Principal variation search</a></li> <li><a href="/wiki/Max%5En_algorithm" title="Max^n algorithm">max^n algorithm</a></li> <li><a href="/wiki/Paranoid_algorithm" title="Paranoid algorithm">Paranoid algorithm</a></li> <li><a href="/wiki/Lazy_SMP" title="Lazy SMP">Lazy SMP</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Miscellaneous</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bounded_rationality" title="Bounded rationality">Bounded rationality</a></li> <li><a href="/wiki/Combinatorial_game_theory" title="Combinatorial game theory">Combinatorial game theory</a></li> <li><a href="/wiki/Confrontation_analysis" title="Confrontation analysis">Confrontation analysis</a></li> <li><a href="/wiki/Coopetition" title="Coopetition">Coopetition</a></li> <li><a href="/wiki/Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</a></li> <li><a href="/wiki/Glossary_of_game_theory" title="Glossary of game theory">Glossary of game theory</a></li> <li><a href="/wiki/List_of_game_theorists" title="List of game theorists">List of game theorists</a></li> <li><a href="/wiki/List_of_games_in_game_theory" title="List of games in game theory">List of games in game theory</a></li> <li><a href="/wiki/No-win_situation" title="No-win situation">No-win situation</a></li> <li><a href="/wiki/Topological_game" title="Topological game">Topological game</a></li> <li><a href="/wiki/Tragedy_of_the_commons" title="Tragedy of the commons">Tragedy of the commons</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?useformat=desktop&type=1x1&usesul3=0" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Fair_division&oldid=1263734056">https://en.wikipedia.org/w/index.php?title=Fair_division&oldid=1263734056</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" title="Help:Category">Categories</a>: <ul><li><a href="/wiki/Category:Fair_division" title="Category:Fair division">Fair division</a></li><li><a href="/wiki/Category:Game_theory" title="Category:Game theory">Game theory</a></li><li><a href="/wiki/Category:Welfare_economics" title="Category:Welfare economics">Welfare economics</a></li></ul></div><div id="mw-hidden-catlinks" class="mw-hidden-catlinks mw-hidden-cats-hidden">Hidden categories: <ul><li><a href="/wiki/Category:CS1_French-language_sources_(fr)" title="Category:CS1 French-language sources (fr)">CS1 French-language sources (fr)</a></li><li><a href="/wiki/Category:Articles_with_short_description" title="Category:Articles with short description">Articles with short description</a></li><li><a href="/wiki/Category:Short_description_matches_Wikidata" title="Category:Short description matches Wikidata">Short description matches Wikidata</a></li></ul></div></div> </div> </main> </div> <div class="mw-footer-container"> <footer id="footer" class="mw-footer" > <ul id="footer-info"> <li id="footer-info-lastmod"> This page was last edited on 18 December 2024, at 08:58<span class="anonymous-show"> (UTC)</span>.</li> <li id="footer-info-copyright">Text is available under the <a href="/wiki/Wikipedia:Text_of_the_Creative_Commons_Attribution-ShareAlike_4.0_International_License" title="Wikipedia:Text of the Creative Commons Attribution-ShareAlike 4.0 International License">Creative Commons Attribution-ShareAlike 4.0 License</a>; additional terms may apply. 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cdx-button--action-progressive" id="ca-addsection-sticky-header" tabindex="-1" data-event-name="addsection-sticky-header"><span class="vector-icon mw-ui-icon-speechBubbleAdd-progressive mw-ui-icon-wikimedia-speechBubbleAdd-progressive"></span> <span>Add topic</span> </a> </div> <div class="vector-sticky-header-icon-end"> <div class="vector-user-links"> </div> </div> </div> </div> </div> <div class="mw-portlet mw-portlet-dock-bottom emptyPortlet" id="p-dock-bottom"> <ul> </ul> </div> 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