Kelly Criterion凯利准则 - Wizard of Odds概率巫师

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<div class="breadcrumbs-holder"> <ul itemscope="itemscope" itemtype="https://schema.org/BreadcrumbList" class="breadcrumbs-list"> <li itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem" class="breadcrumbs-item"> <a href="/" itemprop="item"> <span itemprop="name" > 首页</span> <meta itemprop="position" content="1"> </a> <span class="breadcrumbs-separator">›</span> </li> <li itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem" class="breadcrumbs-item"> <a href="/gambling/" itemprop="item"> <span itemprop="name" > 赌博</span> <meta itemprop="position" content="2"> </a> <span class="breadcrumbs-separator">›</span> </li> <li itemprop="itemListElement" itemscope="" itemtype="https://schema.org/ListItem" class="breadcrumbs-item"> <span itemprop="name" class="last" >Kelly Criterion凯利准则</span> <meta itemprop="position" content="3"> </li> </ul> </div> <time class="article-time">最近更新 Sep 29, 2024</time> </div> <div class="d-lm-none"> <div class="sideNav-block short"> <h3>在这一页<i class="fa-solid fa-caret-down"></i></h3> </div> </div> <h1>Kelly Criterion凯利准则</h1> <div class="d-none-below-lm"> <div class="sideNav-block short"> <h3>在这一页<i class="fa-solid fa-caret-down"></i></h3> </div> </div> <h2>简介</h2> <div class="article-image"> <a class="trans" href="/media/modules/article/images/55746/balancing_both_risk_and_reward.jpg"> <img class="lcp-mobile" src="/media/modules/article/images/55746/balancing_both_risk_and_reward.jpg" alt="Kelly Criterion凯利准则" width="300" height="228"> </a> </div> <body><p>Kelly Criterion凯利准则是⼀种押注⾦额的技巧、针对优势赌客对于risk⻛险与reward回报两者之间的平衡。相同的原理也可运⽤在任何能够获利的投资上⾯。针对赌客/投资者俱备平均的投注资本与固定的押注⾦额, 在⼀笔押注之后的预期资⾦成⻓为:</p> <p><img alt="kelly4" src="/wizfiles/img/986/kelly4.png"></p> <p>例如, 假设⼀家赌场推⾏Craps花旗骰的促销活动, 总点数为2点⽀付3⽐1 的赔率、12点⽀付4⽐1的赔率。 3点, 4点, 9点, 10点, 11点还是维持⽀付1 ⽐1的赔率, 其他的总点数则算输。 2点或12点的机率各为1/36, 任何even money等额注⾦赔率的赢注机率为14/36, 输注的机率为20/36. 假设玩家每⼀笔押注⾦额为其资⾦的1%. 则每⼀笔押注预期的资⾦增⻓为:</p> <p>(1 + (0.01*3))^(1/36) * (1 + (0.01*1)^(14/36) * (1 + (0.01*-1))^(20/36) * (1 + (0.01*4))^(1/36)) - 1 = 0.00019661.</p> <p>这项乘积是Kelly betting凯利押注的最⼤值。 Kelly betting凯利押注也最⼩化达到加倍资⾦所需要的押注次数, 押注⾦额总是以⺫前资⾦的⽐例来做出调整。</p> <p>Kelly押注的⾦额是针对赌客的平均赢注机率、得出能够最⼤化预期资⾦成⻓的最佳押注⾦额。当押注多过Kelly的押注⾦额时、相较于刚好押注 Kelly⾦额的状况, 过多押注所增加的不稳定性将会导致⻓期资⾦成⻓的下降。押注双倍的Kelly⾦额会导致预期资⾦的零成⻓。任何⼤过双倍Kelly ⾦额的押注、将会导致预期资⾦的减少。最常⻅的是押注少于Kelly⾦额的状况, 如此会降低预期的成⻓, 也会降低资⾦的不稳定性。例如, 押注⼀半的Kelly⾦额, 会降低50%的资⾦不稳定性, 不过成⻓却只有25%.</p> <p>针对只有两种结果的简单押注, 最佳的Kelly bet凯利押注是优势除以「to one⽐⼀」赔率基准的押注⽀付。⾄于超过⼀种可能结果的押注, 最佳的 Kelly wager凯利押注是最⼤化押注之后的资⾦log对数值。然⽽, 针对超过⼀种可能结果的押注, 可能很难得出结果。⼤部分的赌客都是运⽤ advantage优势/variance变异数来当作估计值, 那是相当好的⼀种估值。例如, 如果⼀种押注有2%的优势, ⽽variance变异数有4种, 赌客运⽤「full Kelly完全凯利押注⾦额」将会押注 0.02/4 = 0.5% 的资⾦在那次赌局上。记住variance变异数为standard deviation标准均⽅差的平⽅, 这个数值在我的<a href="/gambling/house-edge/">Game Comparison Guide游戏⽐较指南</a>当中列出的许多游戏⾥都有。</p> <p>让我们查看以下三种范例:</p> <p>范例1: ⼀位算牌者在算牌时可以得知1%的优势。从我的Game Comparison Guide游戏⽐较指南, 我们知道Blackjack⼆⼗⼀点的standard deviation标准均⽅差为1.15（这个数值会随着规则与算牌⽽有变化）。如果standard deviation标准均⽅差为1.15, 则variance变异数为 1.152 = 1.3225. 押注相对于资⾦的占⽐为 0.01 / 1.3225 = 0.76%.</p> <p>范例2: 城⾥⼀家赌场提供video poker视频扑克5X点数的促销活动。通常在slot club⽼⻁机俱乐部free play免费试玩在2/9⽼⻁机当中⽀付1%. 所以在5X, ⽼⻁机俱乐部⽀付1.11%. 最佳的游戏是「9/6 Jacks or Better」视频扑克、其回报率为99.54%. 在⽼⻁机俱乐部的点数之后, 其回报率为 99.54% + 1.11% = 100.65%, 或者是0.65%的优势。在「Game Comparison Guide游戏⽐较指南」当中显⽰「9/6 Jacks or Better」的 standard deviation标准均⽅差为4.42, 所以variance变异数为19.5364. 押注相对于资⾦的占⽐为0.0065 / 19.5364 = 0.033% . 顺便⼀提的是, 当我写作本⽂时的2007年9⽉2⽇和3⽇, Wynn永利赌场刚好正在提供这种促销活动。</p> <p>范例3: ⼀种运动投注有20%的赢注机会, ⽀付赔率为9⽐2. 其优势为 0.2×4.5 + 0.8×-1 = 0.1. 最佳的Kelly wager凯利押注为 0.1/4.5 = 2.22%.</p> <p>以下为范例3的数学运算。假设 x 为最佳的Kelly bet凯利押注, 押注之前的资⾦为1. 预期的押注之后的资⾦log对数值为...</p> <p>f(x) = 0.2 × log(1+4.5x) + 0.8 × log(1-x)</p> <p>要最⼤化f(x), 将函数设为零来推算。</p> <p>f'(x) = 0.2 × 4.5 / (1+4.5x) - 0.8 / (1-x) = 0</p> <p>0.9 / (1+4.5x) = 0.8/(1-x)</p> <p>0.9 - 0.9x = 0.8 + 3.6 x</p> <p>4.5x = 0.1</p> <p>x = .1/4.5 = 1/45 = 2.22%</p> <p>当超过⼀种可能结果时、数学推算会更加繁杂, 例如在video poker视频扑克的情况。⽅法还是相同, 不过要得出 x 会更加困难。在这种情况要解出x 的最简单⽅法, 依我所⻅, 是利⽤不同的数值加以实验, 运⽤较⾼或较低的技巧（像是「Price is Right价钱是正确的」Clock Game时钟游戏） , 直到f'(x)趋近于零。</p> <p>我针对回报率超过100%的两种普遍video poker视频扑克游戏来做这个运算。针对「<a href="/games/video-poker/strategy/deuces-wild/full-pay/optimal/">Full Pay Deuces Wild满付2点⺩牌</a>」、回报率为100.76%, 最佳押注⾦额为资⾦的0.0345%. 针对「<a href="https://wizardofodds.com/games/video-poker/strategy/double-bonus/10-7/">10/7 Double Bonus加倍红利</a>」、回报率为100.17%, 最佳押注⾦额为资⾦的0.0062637%. 我听过⼀种实⽤⽅法是, 你在视频扑克要运⽤它、应该准备同花⼤顺奖⾦的3到5倍资⾦去玩。如果是玩「Full Pay Deuces Wild满付2点⺩牌」, 则资⾦是3.66倍的同花⼤顺奖⾦。针对「10/7 Double Bonus加倍红利」, 则是19.96倍的同花⼤顺奖⾦。</p> <h2>模拟</h2> <p>为了要证明我的论点、Kelly最⼩化押注的次数以达到加倍的资⾦, 我假设⼀种even money等额注⾦赔率的押注、俱有51%的赢注机会, 也就是2% 的优势, 和2%的Kelly押注⾦额。这⾥是在不同的押注⾦额、平均需要押注多少次才能让资⾦加倍。如果在⼀笔赢注的押注⾦额、押注者必须押注超过加倍资⾦所需要的⾦额时, 那么他只要押注刚好能够加倍资⾦的⾦额即可。</p> <div class="box desk-50percent"> <h3 class="box-title">加倍资⾦的平均押注次数</h3>  <div class="box-content"> <table border="0" cellpadding="0" cellspacing="0" class="data"> <tbody> <tr> <th class="data-heading">押注⾦额</th> <th class="data-heading">平均押注次数</th> </tr> <tr> <td>0.5%</td> <td>7,901</td> </tr> <tr> <td>1%</td> <td>4,617</td> </tr> <tr> <td>2%</td> <td>3,496</td> </tr> <tr> <td>3%</td> <td>4,477</td> </tr> </tbody> </table> </div> </div>  <h2>Kelly Vs. 最佳视频扑克策略</h2> <p>在我的 <a href="https://wizardofodds.com/ask-the-wizard/196/">2007年9⽉20⽇Ask the Wizard请问巫师专栏</a> 当中, 我建议Kelly押注者有时候不要按照最佳视频扑克策略去玩牌。我的理由在那⾥有解释过。</p> <h2>连结到Kelly</h2> <p><a href="http://richtigspielen.com/das_kelly_kriteriumd.htm" rel="nofollow">德⽂版翻译</a> 本篇专⻚的.</p> <p><a href="http://www.amazon.com/exec/obidos/ASIN/0809045990/theWizardofodds" target="_blank">Fortune’s Formula</a> by William Poundstone. <a href="/gambling/book-reviews/">阅读我的评论</a> 。 </p> <p><a href="http://www.amazon.com/exec/obidos/ASIN/0818402997/theWizardofodds" target="_blank">针对赢家的赌场博弈</a></p> <p>Kelly的⼀种不错的材料来源, 特别是关于Blackjack⼆⼗⼀点, 是 <a href="http://www.amazon.com/exec/obidos/ASIN/0910575045/theWizardofodds" target="_blank">Blackjack Attack⼆⼗⼀点攻击</a> by Don Schlesinger.</p> <p>SBRForum.com 针对Kelly有不错的材料, 包括⽂章《A Quantitative Introduction to the Kelly Criterion凯利准则的量化介绍》, <a href="http://forum.sbrforum.com/players-talk/29009-expected-value-vs-expected-growth-kelly-criterion-part-i.html" rel="nofollow" target="_blank">part I</a> 和 <a href="http://forum.sbrforum.com/players-talk/29841-maximizing-expected-growth-kelly-criterion-part-ii.html" rel="nofollow" target="_blank">part II</a>, 还有⼀种 <a href="http://www.sbrforum.com/Betting+Tools/Kelly+Calculator.aspx" rel="nofollow" target="_blank">Kelly calculator计算器</a> 。</p> <p>Kelly Criterion凯利准则的 <a href="http://en.wikipedia.org/wiki/Kelly_criterion" target="_blank">Wikipedia维基百科</a>.</p> </body> <p class="article-author">撰写者: <a href="https://cn.wizardofodds.com/site/about/">Michael Shackleford</a></p> </div> </div> <div class="col-lm-24"> <div class="banner-block 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