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<title>Perfect Bayesian equilibrium</title>

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Perfect Bayesian equilibrium</span></span>

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</style><table class="infobox"><tbody><tr><th colspan="2" class="infobox-above" style="border-bottom: solid sienna 1px; font-size: 125%;">Perfect Bayesian Equilibrium</th></tr><tr><td colspan="2" class="infobox-subheader"><a href="Solution_concept" title="Solution concept">Solution concept</a> in <a href="Game_theory" title="Game theory">game theory</a></td></tr><tr><th colspan="2" class="infobox-header" style="background-color: #fffdec; border-top: solid silver 1px; border-bottom: solid silver 1px;">Relationship</th></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Subset of</th><td class="infobox-data"><a href="Bayesian_Nash_equilibrium" class="mw-redirect" title="Bayesian Nash equilibrium">Bayesian Nash equilibrium</a></td></tr><tr><th colspan="2" class="infobox-header" style="background-color: #fffdec; border-top: solid silver 1px; border-bottom: solid silver 1px;">Significance</th></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Proposed by</th><td class="infobox-data">Cho and Kreps</td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Used for</th><td class="infobox-data">Dynamic <a href="Bayesian_game" title="Bayesian game">Bayesian games</a></td></tr><tr><th scope="row" class="infobox-label" style="white-space: nowrap;">Example</th><td class="infobox-data"><a href="Signaling_game" title="Signaling game">signaling game</a></td></tr></tbody></table>

<p>In <a href="Game_theory" title="Game theory">game theory</a>, a <b>Perfect Bayesian Equilibrium</b> (PBE) is a solution with Bayesian probability to a turn-based game with incomplete information. More specifically, it is an <a href="Equilibrium_concept" class="mw-redirect" title="Equilibrium concept">equilibrium concept</a> that uses Bayesian updating to describe player behavior in <a href="Dynamic_game" class="mw-redirect" title="Dynamic game">dynamic games</a> with <a href="Incomplete_information" class="mw-redirect" title="Incomplete information">incomplete information</a>. Perfect Bayesian equilibria are used to solve the outcome of games where players take turns but are unsure of the "type" of their opponent, which occurs when players don't know their opponent's preference between individual moves. A classic example of a dynamic game with types is a war game where the player is unsure whether their opponent is a risk-taking "<a href="Hawk-Dove" class="mw-redirect" title="Hawk-Dove">hawk</a>" type or a pacifistic "<a href="Hawk-Dove" class="mw-redirect" title="Hawk-Dove">dove</a>" type. Perfect Bayesian Equilibria are a refinement of <a href="Bayesian_Nash_equilibrium" class="mw-redirect" title="Bayesian Nash equilibrium">Bayesian Nash equilibrium</a> (BNE), which is a solution concept with Bayesian probability for non-turn-based games.

</p><p>Any perfect Bayesian equilibrium has two components -- <i>strategies</i> and <i>beliefs</i>:

</p>

<ul><li>The <b>strategy</b> of a player in a given information set specifies his choice of action in that information set, which may depend on the history (on actions taken previously in the game). This is similar to a <a href="Sequential_game" title="Sequential game">sequential game</a>.</li>

<li>The <b>belief</b> of a player in a given information set determines what node in that information set he believes the game has reached. The belief may be a <a href="Probability_distribution" title="Probability distribution">probability distribution</a> over the nodes in the information set, and is typically a probability distribution over the possible <i>types</i> of the other players. Formally, a belief system is an assignment of probabilities to every node in the game such that the sum of probabilities in any information set is 1.</li></ul>

<p>The strategies and beliefs also must satisfy the following conditions:

</p>

<ul><li><b>Sequential rationality</b>: each strategy should be optimal in expectation, given the beliefs.</li>

<li><b>Consistency</b>: each belief should be updated according to the equilibrium strategies, the observed actions, and <a href="Bayes'_rule" class="mw-redirect" title="Bayes' rule">Bayes' rule</a> on every path reached in equilibrium with positive probability. On paths of zero probability, known as <i>off-equilibrium paths</i>, the beliefs must be specified but can be arbitrary.</li></ul>

<p>A perfect Bayesian equilibrium is always a Nash equilibrium.

</p>

<div class="mw-heading mw-heading2"><h2 id="Examples_of_perfect_Bayesian_equilibria">Examples of perfect Bayesian equilibria</h2></div>

<p>Consider the following game:

</p>

<ul><li>The sender has two possible types: either a "friend" (with probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span>) or an "enemy" (with probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">

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<annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation>

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<li>The receiver has only one type, and two strategies: either accept the gift, or reject it.</li>

<li>The sender's utility is 1 if his gift is accepted, -1 if his gift is rejected, and 0 if he does not give any gift.</li>

<li>The receiver's utility depends on who gives the gift:

<ul><li>If the sender is a friend, then the receiver's utility is 1 (if he accepts) or 0 (if he rejects).</li>

<li>If the sender is an enemy, then the receiver's utility is -1 (if he accepts) or 0 (if he rejects).</li></ul></li></ul>

<p>For any value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p,}">

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<annotation encoding="application/x-tex">{\displaystyle p,}</annotation>

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</math></span><img src="./393fcf18074cb42eafb26b76c515a1e93e17512c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.906ex; height:2.009ex;" alt="{\displaystyle p,}" loading="lazy"></span> Equilibrium 1 exists, a <a href="Pooling_equilibrium" title="Pooling equilibrium">pooling equilibrium</a> in which both types of sender choose the same action:

</p>

<dl><dd><i>Equilibrium 1.</i> Sender: <i>Not give</i>, whether they are the friend type or the enemy type. Receiver: <i>Do not accept</i>, with the beliefs that <i>Prob(Friend|Not Give) = p</i> and <i>Prob(Friend|Give) = x,</i> choosing a value <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\leq .5.}">

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<annotation encoding="application/x-tex">{\displaystyle x\leq .5.}</annotation>

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</math></span><img src="./17f1671fd79521d5224faefce9813b7a1d1a4794.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.884ex; height:2.343ex;" alt="{\displaystyle x\leq .5.}" loading="lazy"></span></dd></dl>

<p>The sender prefers the payoff of 0 from not giving to the payoff of -1 from sending and not being accepted. Thus, <i>Give</i> has zero probability in equilibrium and Bayes's Rule does not restrict the belief <i>Prob(Friend|Give)</i> at all. That belief must be pessimistic enough that the receiver prefers the payoff of 0 from rejecting a gift to the expected payoff of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x(1)+(1-x)(-1)=2x-1,}">

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<annotation encoding="application/x-tex">{\displaystyle x(1)+(1-x)(-1)=2x-1,}</annotation>

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</math></span><img src="./4e3a9ed6eb3ed2261515984e0bfd7879f5eee4f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.304ex; height:2.843ex;" alt="{\displaystyle x(1)+(1-x)(-1)=2x-1,}" loading="lazy"></span> from accepting, so the requirement that the receiver's strategy maximize his expected payoff given his beliefs necessitates that <i>Prob(Friend|Give)</i><span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \leq .5.}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle \leq .5.}</annotation>

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</math></span><img src="./8c86144bfdd4503a1b6c6220a12ad3d358477083.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:4.909ex; height:2.343ex;" alt="{\displaystyle \leq .5.}" loading="lazy"></span> On the other hand, <i>Prob(Friend|Not give) = p</i> is required by Bayes's Rule, since both types take that action and it is uninformative about the sender's type.

</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq 1/2}">

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</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\geq 1/2}</annotation>

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</math></span><img src="./0bfcef07f9c94f4e428cc30b9ce8f1ae43469fed.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-left: -0.089ex; width:7.845ex; height:2.843ex;" alt="{\displaystyle p\geq 1/2}" loading="lazy"></span>, a second pooling equilibrium exists as well as Equilibrium 1, based on different beliefs:

</p>

<dl><dd><i>Equilibrium 2.</i> Sender: <i>Give</i>, whether they are the friend type or the enemy type. Receiver: <i>Accept,</i> with the beliefs that <i>Prob(Friend|Give) = p</i> and <i>Prob(Friend|Not give) = x</i>, choosing any value for <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x.}</annotation>

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</math></span><img src="./d07e9f568a88785ae48006ac3c4b951020f1699a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.977ex; height:1.676ex;" alt="{\displaystyle x.}" loading="lazy"></span></dd></dl>

<p>The sender prefers the payoff of 1 from giving to the payoff of 0 from not giving, expecting that his gift will be accepted. In equilibrium, Bayes's Rule requires the receiver to have the belief <i>Prob(Friend|Give) = p</i>, since both types take that action and it is uninformative about the sender's type in this equilibrium. The out-of-equilibrium belief does not matter, since the sender would not want to deviate to <i>Not give</i> no matter what response the receiver would have.

</p><p>Equilibrium 1 is perverse if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p\geq .5.}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p\geq .5.}</annotation>

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</math></span><img src="./26cd90304be800758e52c82e27bc62503d156356.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:6.813ex; height:2.509ex;" alt="{\displaystyle p\geq .5.}" loading="lazy"></span> The game could have <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p=.99,}">

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<annotation encoding="application/x-tex">{\displaystyle p=.99,}</annotation>

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</math></span><img src="./32832f8e611ae253cd82cbb4402e1f581e298b6e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:7.976ex; height:2.509ex;" alt="{\displaystyle p=.99,}" loading="lazy"></span> so the sender is very likely a friend, but the receiver still would refuse any gift because he thinks enemies are much more likely than friends to give gifts. This shows how pessimistic beliefs can result in an equilibrium bad for both players, one that is not <a href="Pareto_efficient" class="mw-redirect" title="Pareto efficient">Pareto efficient</a>. These beliefs seem unrealistic, though, and game theorists are often willing to reject some perfect Bayesian equilibria as implausible.

</p><p>Equilibria 1 and 2 are the only equilibria that might exist, but we can also check for the two potential <a href="Separating_equilibrium" title="Separating equilibrium">separating equilibria</a>, in which the two types of sender choose different actions, and see why they do not exist as perfect Bayesian equilibria:

</p>

<ol><li>Suppose the sender's strategy is: <i>Give</i> if a friend, <i>Do not give</i> if an enemy. The receiver's beliefs are updated accordingly: if he receives a gift, he believes the sender is a friend; otherwise, he believes the sender is an enemy. Thus, the receiver will respond with <i>Accept</i>. If the receiver chooses <i>Accept</i>, though, the enemy sender will deviate to  <i>Give</i>, to increase his payoff from 0 to 1, so this cannot be an equilibrium.</li>

<li>Suppose the sender's strategy is: <i>Do not give</i> if a friend, <i>Give</i> if an enemy. The receiver's beliefs are updated accordingly: if he receives a gift, he believes the sender is an enemy; otherwise, he believes the sender is a friend. The receiver's best-response strategy is <i>Reject.</i> If the receiver chooses <i>Reject</i>, though, the enemy sender will deviate to  <i>Do not give</i>, to increase his payoff from -1 to 0, so this cannot be an equilibrium.</li></ol>

<p>We conclude that in this game, there is <i>no</i> separating equilibrium.

</p>

<p>In the following example,<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> the set of PBEs is strictly smaller than the set of SPEs and BNEs. It is a variant of the above gift-game, with the following change to the receiver's utility:

</p>

<ul><li>If the sender is a friend, then the receiver's utility is 1 (if they accept) or 0 (if they reject).</li>

<li>If the sender is an enemy, then the receiver's utility is <b>0</b> (if they accept) or <b>-1</b> (if they reject).</li></ul>

<p>Note that in this variant, accepting is a weakly <a href="Dominant_strategy" class="mw-redirect" title="Dominant strategy">dominant strategy</a> for the receiver.

</p><p>Similarly to example 1, there is no separating equilibrium. Let's look at the following potential pooling equilibria:

</p>

<ol><li>The sender's strategy is: always give. The receiver's beliefs are not updated: they still believe in the a-priori probability, that the sender is a friend with probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> and an enemy with probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1-p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 1-p}</annotation>

</semantics>

</math></span><img src="./9633a8692121eedfa99cace406205e5d1511ef8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.172ex; height:2.509ex;" alt="{\displaystyle 1-p}" loading="lazy"></span>. Their payoff from accepting is always higher than from rejecting, so they accept (regardless of the value of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

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<li>The sender's strategy is: never give. Suppose the receiver's beliefs when receiving a gift is that the sender is a friend with probability <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span>, where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span> is any number in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,1]}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle [0,1]}</annotation>

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</math></span><img src="./738f7d23bb2d9642bab520020873cccbef49768d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,1]}" loading="lazy"></span>. Regardless of <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle q}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle q}</annotation>

</semantics>

</math></span><img src="./06809d64fa7c817ffc7e323f85997f783dbdf71d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.07ex; height:2.009ex;" alt="{\displaystyle q}" loading="lazy"></span>, the receiver's optimal strategy is: accept. This is NOT a PBE, since the sender can improve their payoff from 0 to 1 by giving a gift.</li>

<li>The sender's strategy is: never give, and the receiver's strategy is: reject. This is NOT a PBE, since for <i>any</i> belief of the receiver, rejecting is not a best-response.</li></ol>

<p>Note that option 3 is a Nash equilibrium. If we ignore beliefs, then rejecting can be considered a best-response for the receiver, since it does not affect their payoff (since there is no gift anyway). Moreover, option 3 is even a SPE, since the only subgame here is the entire game. Such implausible equilibria might arise also in games with complete information, but they may be eliminated by applying <a href="Subgame_perfect_Nash_equilibrium" class="mw-redirect" title="Subgame perfect Nash equilibrium">subgame perfect Nash equilibrium</a>. However, Bayesian games often contain non-singleton information sets and since <a href="Subgame" title="Subgame">subgames</a> must contain complete information sets, sometimes there is only one subgame—the entire game—and so every Nash equilibrium is trivially subgame perfect. Even if a game does have more than one subgame, the inability of subgame perfection to cut through information sets can result in implausible equilibria not being eliminated.

</p><p>To summarize: in this variant of the gift game, there are two SPEs: either the sender always gives and the receiver always accepts, or the sender always does not give and the receiver always rejects. From these, only the first one is a PBE; the other is not a PBE since it cannot be supported by any belief-system.

</p>

<div class="mw-heading mw-heading3"><h3 id="More_examples">More examples</h3></div>

<p>For further examples, see <a href="Signaling_game#Examples" title="Signaling game">signaling game#Examples</a>. See also <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> for more examples. There is a recent application of this concept in Poker, by Loriente and Diez (2023).<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>

</p>

<div class="mw-heading mw-heading2"><h2 id="PBE_in_multi-stage_games">PBE in multi-stage games</h2></div>

<p>A <a href="Multi-stage_game" title="Multi-stage game">multi-stage game</a> is a sequence of simultaneous games played one after the other. These games may be identical (as in <a href="Repeated_game" title="Repeated game">repeated games</a>) or different.

</p>

<div class="mw-heading mw-heading3"><h3 id="Repeated_public-good_game">Repeated public-good game</h3></div>

</td>

<td style="width:33%; border-bottom: solid black 1px;">Build

</td>

<td style="width:33%; border-bottom: solid black 1px;">Don't

</td></tr>

<tr>

<td style="border-right: solid black 1px; text-align: right;">Build

</td>

<td style="border-right: solid black 1px; border-bottom: solid black 1px; background:white; font-size:120%; white-space:nowrap;">1-C1, 1-C2

</td>

<td style="border-right: solid black 1px; border-bottom: solid black 1px; background:white; font-size:120%; white-space:nowrap;">1-C1, 1

</td></tr>

<tr>

<td style="border-right: solid black 1px; text-align: right;">Don't

</td>

<td style="border-right: solid black 1px; border-bottom: solid black 1px; background:white; font-size:120%; white-space:nowrap;">1, 1-C2

</td>

<td style="border-right: solid black 1px; border-bottom: solid black 1px; background:white; font-size:120%; white-space:nowrap;">0,0

</td></tr>

<tr>

<td style="font-size: 90%;" colspan="3"><i>Public good game</i>

</td></tr></tbody></table>

<p>The following game<sup id="cite_ref-ft91_4-0" class="reference"><a href="#cite_note-ft91-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: section 6.2">: section 6.2 </span></sup> is a simple representation of the <a href="Free-rider_problem" title="Free-rider problem">free-rider problem</a>. There are two players, each of whom can either build a <a href="Public_good_(economics)" class="mw-redirect" title="Public good (economics)">public good</a> or not build. Each player gains 1 if the public good is built and 0 if not; in addition, if player <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> builds the public good, they have to pay a cost of <span class="mwe-math-element mwe-math-element-inline"><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_{i}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}}</annotation>

</semantics>

</math></span><img src="./cc49dc02c0ec8c86b67e7d10518ac791eda0bf22.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.461ex; height:2.509ex;" alt="{\displaystyle C_{i}}" loading="lazy"></span>. The costs are <i>private information</i> - each player knows their own cost but not the other's cost. It is only known that each cost is drawn independently at random from some probability distribution. This makes this game a <a href="Bayesian_game" title="Bayesian game">Bayesian game</a>.

</p><p>In the one-stage game, each player builds if-and-only-if their cost is smaller than their expected gain from building. The expected gain from building is exactly 1 times the probability that the other player does NOT build. In equilibrium, for every player <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i}</annotation>

</semantics>

</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span>, there is a threshold cost <span class="mwe-math-element mwe-math-element-inline"><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_{i}^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}^{*}}</annotation>

</semantics>

</math></span><img src="./040caa905f445f493a96313b27a9a991d44717da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{i}^{*}}" loading="lazy"></span>, such that the player contributes if-and-only-if their cost is less than <span class="mwe-math-element mwe-math-element-inline"><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_{i}^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle C_{i}^{*}}</annotation>

</semantics>

</math></span><img src="./040caa905f445f493a96313b27a9a991d44717da.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.852ex; height:2.843ex;" alt="{\displaystyle C_{i}^{*}}" loading="lazy"></span>. This threshold cost can be calculated based on the probability distribution of the players' costs. For example, if the costs are distributed uniformly on <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [0,2]}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [0,2]}</annotation>

</semantics>

</math></span><img src="./120ef5837b0c64a40a2333f5aefd3c36fc458e91.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.653ex; height:2.843ex;" alt="{\displaystyle [0,2]}" loading="lazy"></span>, then there is a symmetric equilibrium in which the threshold cost of both players is 2/3. This means that a player whose cost is between 2/3 and 1 will not contribute, even though their cost is below the benefit, because of the possibility that the other player will contribute.

</p><p>Now, suppose that this game is repeated two times.<sup id="cite_ref-ft91_4-1" class="reference"><a href="#cite_note-ft91-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup class="reference nowrap"><span title="Page / location: section 8.2.3">: section 8.2.3 </span></sup> The two plays are independent, i.e., each day the players decide simultaneously whether to build a public good in that day, get a payoff of 1 if the good is built in that day, and pay their cost if they built in that day. The only connection between the games is that, by playing in the first day, the players may reveal some information about their costs, and this information might affect the play in the second day.

</p><p>We are looking for a symmetric PBE. Denote by <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

</math></span><img src="./8417e85ae7f4eaee7df31347ce488f85c8884b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {c}}}" loading="lazy"></span> the threshold cost of both players in day 1 (so in day 1, each player builds if-and-only-if their cost is at most <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

</math></span><img src="./8417e85ae7f4eaee7df31347ce488f85c8884b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {c}}}" loading="lazy"></span>). To calculate <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

</p>

<ol><li>In day 1, no player built. So now both players know that their opponent's cost is above <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

</math></span><img src="./8417e85ae7f4eaee7df31347ce488f85c8884b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {c}}}" loading="lazy"></span>. They update their belief accordingly, and conclude that there is a smaller chance that their opponent will build in day 2. Therefore, they increase their threshold cost, and the threshold cost in day 2 is <span class="mwe-math-element mwe-math-element-inline"><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^{00}>{\hat {c}}}">

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c^{00}>{\hat {c}}}</annotation>

</semantics>

</math></span><img src="./13faa56b0975ba58db9c1f8a6410867e0070b810.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.273ex; height:2.676ex;" alt="{\displaystyle c^{00}>{\hat {c}}}" loading="lazy"></span>.</li>

<li>In day 1, both players built. So now both players know that their opponent's cost is below <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

</math></span><img src="./8417e85ae7f4eaee7df31347ce488f85c8884b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {c}}}" loading="lazy"></span>. They update their belief accordingly, and conclude that there is a larger chance that their opponent will build in day 2. Therefore, they decrease their threshold cost, and the threshold cost in day 2 is <span class="mwe-math-element mwe-math-element-inline"><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^{11}<{\hat {c}}}">

<msup>

</mrow>

</msup>

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c^{11}<{\hat {c}}}</annotation>

</semantics>

</math></span><img src="./994ec72edb3e39867d1cb82f5a06497defdff228.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.273ex; height:2.676ex;" alt="{\displaystyle c^{11}<{\hat {c}}}" loading="lazy"></span>.</li>

<li>In day 1, exactly one player built; suppose it is player 1. So now, it is known that the cost of player 1 is below <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

</math></span><img src="./8417e85ae7f4eaee7df31347ce488f85c8884b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {c}}}" loading="lazy"></span> and the cost of player 2 is above <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

<p>It is possible to calculate the expected payoff of the "threshold player" (a player with cost exactly <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

</math></span><img src="./8417e85ae7f4eaee7df31347ce488f85c8884b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {c}}}" loading="lazy"></span>) in each of these situations. Since the threshold player should be indifferent between contributing and not contributing, it is possible to calculate the day-1 threshold cost <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\hat {c}}}">

<mover>

</mover>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\hat {c}}}</annotation>

</semantics>

</math></span><img src="./8417e85ae7f4eaee7df31347ce488f85c8884b93.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.292ex; height:2.176ex;" alt="{\displaystyle {\hat {c}}}" loading="lazy"></span>. It turns out that this threshold is <i>lower</i> than <span class="mwe-math-element mwe-math-element-inline"><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^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c^{*}}</annotation>

</semantics>

</math></span><img src="./e0e1c40aab3bd8af2d5fc091d658f8e0034a635f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.061ex; height:2.343ex;" alt="{\displaystyle c^{*}}" loading="lazy"></span> - the threshold in the one-stage game. This means that, in a two-stage game, the players are <i>less</i> willing to build than in the one-stage game. Intuitively, the reason is that, when a player does not contribute in the first day, they make the other player believe their cost is high, and this makes the other player more willing to contribute in the second day.

</p>

<div class="mw-heading mw-heading3"><h3 id="Jump-bidding">Jump-bidding</h3></div>

<p>In an open-outcry <a href="English_auction" title="English auction">English auction</a>, the bidders can raise the current price in small steps (e.g. in $1 each time). However, often there is <a href="Jump_bidding" title="Jump bidding">jump bidding</a> - some bidders raise the current price much more than the minimal increment. One explanation to this is that it serves as a signal to the other bidders. There is a PBE in which each bidder jumps if-and-only-if their value is above a certain threshold. See <a href="Jump_bidding#signaling" title="Jump bidding">Jump bidding#signaling</a>.

</p>

<ul><li><a href="Sequential_equilibrium" title="Sequential equilibrium">Sequential equilibrium</a> - a refinement of PBE, that restricts the beliefs that can be assigned to off-equilibrium information sets to "reasonable" ones.</li>

<li><a href="Intuitive_criterion" title="Intuitive criterion">Intuitive criterion</a> and <a href="Divine_equilibrium" title="Divine equilibrium">Divine equilibrium</a> - other refinements of PBE, specific to <a href="Signaling_game" title="Signaling game">signaling games</a>.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><div class="reflist">

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.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("./mw/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("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/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}}

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</style><cite id="CITEREFJames_Peck" class="citation web cs1">James Peck. <a rel="nofollow" class="external text" href="https://www.asc.ohio-state.edu/peck.33/Econ5001/econ601l15.pdf">"Perfect Bayesian Equilibrium"</a> <span class="cs1-format">(PDF)</span>. Ohio State University<span class="reference-accessdate">. Retrieved <span class="nowrap">6 December</span> 2021</span>.</cite></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"><cite id="CITEREFZack_Grossman" class="citation web cs1">Zack Grossman. <a rel="nofollow" class="external text" href="http://econ.ucsb.edu/~grossman/teaching/Econ171/Perfect_Bayesian_Equilibrium-ho.pdf">"Perfect Bayesian Equilibrium"</a> <span class="cs1-format">(PDF)</span>. University of California<span class="reference-accessdate">. Retrieved <span class="nowrap">2 September</span> 2016</span>.</cite></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">Loriente, Martín Iñaki & Diez, Juan Cruz (2023). <a rel="nofollow" class="external text" href="https://repositorio.udesa.edu.ar/jspui/bitstream/10908/23530/1/%5BP%5D%5BW%5D%20T.%20L.%20Eco.%20Diez%2C%20Juan%20Cruz%20y%20Loriente%2C%20Mart%C3%ADn%20I%C3%B1aki.pdf">"Perfect Bayesian Equilibrium in Kuhn Poker"</a>. Universidad de San Andres. </span>

</li>

<li id="cite_note-ft91-4"><span class="mw-cite-backlink">^ <a href="#cite_ref-ft91_4-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-ft91_4-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><cite id="CITEREFFudenbergTirole1991" class="citation book cs1"><a href="Drew_Fudenberg" title="Drew Fudenberg">Fudenberg, Drew</a>; <a href="Jean_Tirole" title="Jean Tirole">Tirole, Jean</a> (1991). <a rel="nofollow" class="external text" href="http://www-mitpress.mit.edu/book-home.tcl?isbn=0262061414"><i>Game Theory</i></a>. Cambridge, Massachusetts: <a href="MIT_Press" title="MIT Press">MIT Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780262061414</bdi>.</cite> <a rel="nofollow" class="external text" href="https://books.google.com/books?id=pFPHKwXro3QC&pg=PA18">Book preview.</a></span>

</li>

</ol></div></div>

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</style><div id="Game_theory698" style="font-size:114%;margin:0 4em"><a href="Game_theory" title="Game theory">Game theory</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><a href="Glossary_of_game_theory" title="Glossary of game theory">Glossary</a></li>

<li><a href="List_of_game_theorists" title="List of game theorists">Game theorists</a></li>

<li><a href="List_of_games_in_game_theory" title="List of games in game theory">Games</a></li></ul>

</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Traditional_game_theory698" style="font-size:114%;margin:0 4em">Traditional <a href="Game_theory" title="Game theory">game theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Game_theory#Basic_concepts" title="Game theory">Definitions</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Asynchrony_(game_theory)" title="Asynchrony (game theory)">Asynchrony</a></li>

<li><a href="Bayesian_regret" title="Bayesian regret">Bayesian regret</a></li>

<li><a href="Best_response" title="Best response">Best response</a></li>

<li><a href="Bounded_rationality" title="Bounded rationality">Bounded rationality</a></li>

<li><a href="Cheap_talk" title="Cheap talk">Cheap talk</a></li>

<li><a href="Coalition" title="Coalition">Coalition</a></li>

<li><a href="Complete_contract" title="Complete contract">Complete contract</a></li>

<li><a href="Complete_information" title="Complete information">Complete information</a></li>

<li><a href="Complete_mixing" title="Complete mixing">Complete mixing</a></li>

<li><a href="Confrontation_analysis" title="Confrontation analysis">Confrontation analysis</a></li>

<li><a href="Conjectural_variation" title="Conjectural variation">Conjectural variation</a></li>

<li><a href="Contingent_cooperator" title="Contingent cooperator">Contingent cooperator</a></li>

<li><a href="Coopetition" title="Coopetition">Coopetition</a></li>

<li><a href="Cooperative_game_theory" title="Cooperative game theory">Cooperative game theory</a></li>

<li><a href="Dynamic_inconsistency" title="Dynamic inconsistency">Dynamic inconsistency</a></li>

<li><a href="Escalation_of_commitment" title="Escalation of commitment">Escalation of commitment</a></li>

<li><a href="Farsightedness_(game_theory)" title="Farsightedness (game theory)">Farsightedness</a></li>

<li><a href="Game_semantics" title="Game semantics">Game semantics</a></li>

<li><a href="Hierarchy_of_beliefs" title="Hierarchy of beliefs">Hierarchy of beliefs</a></li>

<li><a href="Imperfect_information" class="mw-redirect" title="Imperfect information">Imperfect information</a></li>

<li><a href="Incomplete_information" class="mw-redirect" title="Incomplete information">Incomplete information</a></li>

<li><a href="Information_set_(game_theory)" title="Information set (game theory)">Information set</a></li>

<li><a href="Move_by_nature" title="Move by nature">Move by nature</a></li>

<li><a href="Mutual_knowledge" title="Mutual knowledge">Mutual knowledge</a></li>

<li><a href="Non-cooperative_game_theory" title="Non-cooperative game theory">Non-cooperative game theory</a></li>

<li><a href="Non-credible_threat" title="Non-credible threat">Non-credible threat</a></li>

<li><a href="Outcome_(game_theory)" title="Outcome (game theory)">Outcome</a></li>

<li><a href="Perfect_information" title="Perfect information">Perfect information</a></li>

<li><a href="Perfect_recall_(game_theory)" title="Perfect recall (game theory)">Perfect recall</a></li>

<li><a href="Preference_(economics)" title="Preference (economics)">Preference</a></li>

<li><a href="Rationality" title="Rationality">Rationality</a></li>

<li><a href="Sequential_game" title="Sequential game">Sequential game</a></li>

<li><a href="Simultaneous_action_selection" title="Simultaneous action selection">Simultaneous action selection</a></li>

<li><a href="Spite_(game_theory)" title="Spite (game theory)">Spite</a></li>

<li><a href="Strategic_complements" title="Strategic complements">Strategic complements</a></li>

<li><a href="Strategic_dominance" title="Strategic dominance">Strategic dominance</a></li>

<li><a href="Strategic_form" class="mw-redirect" title="Strategic form">Strategic form</a></li>

<li><a href="Strategic_interaction" class="mw-redirect" title="Strategic interaction">Strategic interaction</a></li>

<li><a href="Strategic_move" title="Strategic move">Strategic move</a></li>

<li><a href="Strategy_(game_theory)" title="Strategy (game theory)">Strategy</a></li>

<li><a href="Subgame" title="Subgame">Subgame</a></li>

<li><a href="Succinct_game" title="Succinct game">Succinct game</a></li>

<li><a href="Topological_game" title="Topological game">Topological game</a></li>

<li><a href="Tragedy_of_the_commons" title="Tragedy of the commons">Tragedy of the commons</a></li>

<li><a href="Uncorrelated_asymmetry" title="Uncorrelated asymmetry">Uncorrelated asymmetry</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Economic_equilibrium" title="Economic equilibrium">Equilibrium<br>concepts</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Backward_induction" title="Backward induction">Backward induction</a></li>

<li><a href="Bayes_correlated_equilibrium" title="Bayes correlated equilibrium">Bayes correlated equilibrium</a></li>

<li><a href="Bayesian_efficiency" title="Bayesian efficiency">Bayesian efficiency</a></li>

<li><a href="Bayesian_game" title="Bayesian game">Bayesian game</a></li>

<li><a href="Bayesian_Nash_equilibrium" class="mw-redirect" title="Bayesian Nash equilibrium">Bayesian Nash equilibrium</a></li>

<li><a href="Berge_equilibrium" title="Berge equilibrium">Berge equilibrium</a></li>

<li><a href="Bertrand%E2%80%93Edgeworth_model" title="Bertrand–Edgeworth model">Bertrand–Edgeworth model</a></li>

<li><a href="Coalition-proof_Nash_equilibrium" title="Coalition-proof Nash equilibrium">Coalition-proof Nash equilibrium</a></li>

<li><a href="Correlated_equilibrium" title="Correlated equilibrium">Correlated equilibrium</a></li>

<li><a href="Cursed_equilibrium" title="Cursed equilibrium">Cursed equilibrium</a></li>

<li><a href="Edgeworth_price_cycle" title="Edgeworth price cycle">Edgeworth price cycle</a></li>

<li><a href="Epsilon-equilibrium" title="Epsilon-equilibrium">Epsilon-equilibrium</a></li>

<li><a href="Gibbs_measure" title="Gibbs measure">Gibbs equilibrium</a></li>

<li><a href="Incomplete_contracts" title="Incomplete contracts">Incomplete contracts</a></li>

<li><a href="Inequity_aversion" title="Inequity aversion">Inequity aversion</a></li>

<li><a href="Individual_rationality" class="mw-redirect" title="Individual rationality">Individual rationality</a></li>

<li><a href="Iterated_elimination_of_dominated_strategies" class="mw-redirect" title="Iterated elimination of dominated strategies">Iterated elimination of dominated strategies</a></li>

<li><a href="Markov_perfect_equilibrium" title="Markov perfect equilibrium">Markov perfect equilibrium</a></li>

<li><a href="Mertens-stable_equilibrium" title="Mertens-stable equilibrium">Mertens-stable equilibrium</a></li>

<li><a href="Nash_equilibrium" title="Nash equilibrium">Nash equilibrium</a></li>

<li><a href="Open-loop_model" title="Open-loop model">Open-loop model</a></li>

<li><a href="Pareto_efficiency" title="Pareto efficiency">Pareto efficiency</a></li>

<li><a href="Payoff_dominance" class="mw-redirect" title="Payoff dominance">Payoff dominance</a></li>

<li><a href="Price_of_anarchy" title="Price of anarchy">Price of anarchy</a></li>

<li><a href="Program_equilibrium" title="Program equilibrium">Program equilibrium</a></li>

<li><a href="Proper_equilibrium" title="Proper equilibrium">Proper equilibrium</a></li>

<li><a href="Quantal_response_equilibrium" title="Quantal response equilibrium">Quantal response equilibrium</a></li>

<li><a href="Quasi-perfect_equilibrium" title="Quasi-perfect equilibrium">Quasi-perfect equilibrium</a></li>

<li><a href="Rational_agent" title="Rational agent">Rational agent</a></li>

<li><a href="Rationalizability" class="mw-redirect" title="Rationalizability">Rationalizability</a></li>

<li><a href="Rationalizable_strategy" title="Rationalizable strategy">Rationalizable strategy</a></li>

<li><a href="Satisfaction_equilibrium" title="Satisfaction equilibrium">Satisfaction equilibrium</a></li>

<li><a href="Self-confirming_equilibrium" title="Self-confirming equilibrium">Self-confirming equilibrium</a></li>

<li><a href="Sequential_equilibrium" title="Sequential equilibrium">Sequential equilibrium</a></li>

<li><a href="Shapley_value" title="Shapley value">Shapley value</a></li>

<li><a href="Strong_Nash_equilibrium" title="Strong Nash equilibrium">Strong Nash equilibrium</a></li>

<li><a href="Subgame_perfect_equilibrium" title="Subgame perfect equilibrium">Subgame perfect equilibrium</a></li>

<li><a href="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="Strategy_(game_theory)" title="Strategy (game theory)">Strategies</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Appeasement" title="Appeasement">Appeasement</a></li>

<li><a href="Bid_shading" title="Bid shading">Bid shading</a></li>

<li><a href="Cheap_talk" title="Cheap talk">Cheap talk</a></li>

<li><a href="Collusion" title="Collusion">Collusion</a></li>

<li><a href="Commitment_device" title="Commitment device">Commitment device</a></li>

<li><a href="De-escalation" title="De-escalation">De-escalation</a></li>

<li><a href="Deterrence_theory" title="Deterrence theory">Deterrence</a></li>

<li><a href="Conflict_escalation" title="Conflict escalation">Escalation</a></li>

<li><a href="Fictitious_play" title="Fictitious play">Fictitious play</a></li>

<li><a href="Focal_point_(game_theory)" title="Focal point (game theory)">Focal point</a></li>

<li><a href="Grim_trigger" title="Grim trigger">Grim trigger</a></li>

<li><a href="Hobbesian_trap" title="Hobbesian trap">Hobbesian trap</a></li>

<li><a href="Markov_strategy" title="Markov strategy">Markov strategy</a></li>

<li><a href="Max-dominated_strategy" title="Max-dominated strategy">Max-dominated strategy</a></li>

<li><a href="Strategy_(game_theory)#Mixed_strategy" title="Strategy (game theory)">Mixed strategy</a></li>

<li><a href="Strategy_(game_theory)" title="Strategy (game theory)">Pure strategy</a></li>

<li><a href="Win%E2%80%93stay%2C_lose%E2%80%93switch" title="Win–stay, lose–switch">Win–stay, lose–switch</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="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-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="All-pay_auction" title="All-pay auction">All-pay auction</a></li>

<li><a href="Battle_of_the_sexes_(game_theory)" title="Battle of the sexes (game theory)">Battle of the sexes</a></li>

<li><a href="Bargaining_problem" class="mw-redirect" title="Bargaining problem">Nash bargaining game</a></li>

<li><a href="Bertrand_competition" title="Bertrand competition">Bertrand competition</a></li>

<li><a href="Blotto_game" title="Blotto game">Blotto game</a></li>

<li><a href="Centipede_game" title="Centipede game">Centipede game</a></li>

<li><a href="Coordination_game" title="Coordination game">Coordination game</a></li>

<li><a href="Cournot_competition" title="Cournot competition">Cournot competition</a></li>

<li><a href="Deadlock_(game_theory)" title="Deadlock (game theory)">Deadlock</a></li>

<li><a href="Dictator_game" title="Dictator game">Dictator game</a></li>

<li><a href="Dictator_game#Trust_game" title="Dictator game">Trust game</a></li>

<li><a href="Unscrupulous_diner's_dilemma" title="Unscrupulous diner's dilemma">Diner's dilemma</a></li>

<li><a href="Dollar_auction" title="Dollar auction">Dollar auction</a></li>

<li><a href="El_Farol_Bar_problem" title="El Farol Bar problem">El Farol Bar problem</a></li>

<li><a href="Electronic_mail_game" title="Electronic mail game">Electronic mail game</a></li>

<li><a href="Gift-exchange_game" title="Gift-exchange game">Gift-exchange game</a></li>

<li><a href="Guess_2/3_of_the_average" title="Guess 2/3 of the average">Guess 2/3 of the average</a></li>

<li><a href="Keynesian_beauty_contest" title="Keynesian beauty contest">Keynesian beauty contest</a></li>

<li><a href="Kuhn_poker" title="Kuhn poker">Kuhn poker</a></li>

<li><a href="Lewis_signaling_game" title="Lewis signaling game">Lewis signaling game</a></li>

<li><a href="Matching_pennies" title="Matching pennies">Matching pennies</a></li>

<li><a href="Obligationes" title="Obligationes">Obligationes</a></li>

<li><a href="Optional_prisoner's_dilemma" title="Optional prisoner's dilemma">Optional prisoner's dilemma</a></li>

<li><a href="Pirate_game" title="Pirate game">Pirate game</a></li>

<li><a href="Prisoner's_dilemma" title="Prisoner's dilemma">Prisoner's dilemma</a></li>

<li><a href="Public_goods_game" title="Public goods game">Public goods game</a></li>

<li><a href="Rendezvous_problem" title="Rendezvous problem">Rendezvous problem</a></li>

<li><a href="Rock_paper_scissors" title="Rock paper scissors">Rock paper scissors</a></li>

<li><a href="Stackelberg_competition" title="Stackelberg competition">Stackelberg competition</a></li>

<li><a href="Traveler's_dilemma" title="Traveler's dilemma">Traveler's dilemma</a></li>

<li><a href="Ultimatum_game" title="Ultimatum game">Ultimatum game</a></li>

<li><a href="Volunteer's_dilemma" title="Volunteer's dilemma">Volunteer's dilemma</a></li>

<li><a href="War_of_attrition_(game)" title="War of attrition (game)">War of attrition</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Game_theory#Theorems" title="Game theory">Theorems</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Arrow's_impossibility_theorem" title="Arrow's impossibility theorem">Arrow's impossibility theorem</a></li>

<li><a href="Aumann's_agreement_theorem" title="Aumann's agreement theorem">Aumann's agreement theorem</a></li>

<li><a href="Brouwer_fixed-point_theorem" title="Brouwer fixed-point theorem">Brouwer fixed-point theorem</a></li>

<li><a href="Competitive_altruism" title="Competitive altruism">Competitive altruism</a></li>

<li><a href="Folk_theorem_(game_theory)" title="Folk theorem (game theory)">Folk theorem</a></li>

<li><a href="Gibbard%E2%80%93Satterthwaite_theorem" title="Gibbard–Satterthwaite theorem">Gibbard–Satterthwaite theorem</a></li>

<li><a href="Gibbs_lemma" title="Gibbs lemma">Gibbs lemma</a></li>

<li><a href="Glicksberg's_theorem" title="Glicksberg's theorem">Glicksberg's theorem</a></li>

<li><a href="Kakutani_fixed-point_theorem" title="Kakutani fixed-point theorem">Kakutani fixed-point theorem</a></li>

<li><a href="Kuhn's_theorem" title="Kuhn's theorem">Kuhn's theorem</a></li>

<li><a href="One-shot_deviation_principle" title="One-shot deviation principle">One-shot deviation principle</a></li>

<li><a href="Prim%E2%80%93Read_theory" title="Prim–Read theory">Prim–Read theory</a></li>

<li><a href="Rational_ignorance" title="Rational ignorance">Rational ignorance</a></li>

<li><a href="Rational_irrationality" title="Rational irrationality">Rational irrationality</a></li>

<li><a href="Sperner's_lemma" title="Sperner's lemma">Sperner's lemma</a></li>

<li><a href="Zermelo's_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%">Subfields</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algorithmic_game_theory" title="Algorithmic game theory">Algorithmic game theory</a></li>

<li><a href="Behavioral_game_theory" title="Behavioral game theory">Behavioral game theory</a></li>

<li><a href="Behavioral_strategy" title="Behavioral strategy">Behavioral strategy</a></li>

<li><a href="Compositional_game_theory" title="Compositional game theory">Compositional game theory</a></li>

<li><a href="Contract_theory" title="Contract theory">Contract theory</a></li>

<li><a href="Drama_theory" title="Drama theory">Drama theory</a></li>

<li><a href="Graphical_game_theory" title="Graphical game theory">Graphical game theory</a></li>

<li><a href="Heresthetic" title="Heresthetic">Heresthetic</a></li>

<li><a href="Mean-field_game_theory" title="Mean-field game theory">Mean-field game theory</a></li>

<li><a href="Negotiation_theory" title="Negotiation theory">Negotiation theory</a></li>

<li><a href="Quantum_game_theory" title="Quantum game theory">Quantum game theory</a></li>

<li><a href="Social_software_(research_field)" title="Social software (research field)">Social software</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key people</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Albert_W._Tucker" title="Albert W. Tucker">Albert W. Tucker</a></li>

<li><a href="Alvin_E._Roth" title="Alvin E. Roth">Alvin E. Roth</a></li>

<li><a href="Amos_Tversky" title="Amos Tversky">Amos Tversky</a></li>

<li><a href="Antoine_Augustin_Cournot" title="Antoine Augustin Cournot">Antoine Augustin Cournot</a></li>

<li><a href="Ariel_Rubinstein" title="Ariel Rubinstein">Ariel Rubinstein</a></li>

<li><a href="David_Gale" title="David Gale">David Gale</a></li>

<li><a href="David_K._Levine" title="David K. Levine">David K. Levine</a></li>

<li><a href="David_M._Kreps" title="David M. Kreps">David M. Kreps</a></li>

<li><a href="Donald_B._Gillies" title="Donald B. Gillies">Donald B. Gillies</a></li>

<li><a href="Drew_Fudenberg" title="Drew Fudenberg">Drew Fudenberg</a></li>

<li><a href="Eric_Maskin" title="Eric Maskin">Eric Maskin</a></li>

<li><a href="Harold_W._Kuhn" title="Harold W. Kuhn">Harold W. Kuhn</a></li>

<li><a href="Herbert_A._Simon" title="Herbert A. Simon">Herbert Simon</a></li>

<li><a href="Herbert_Scarf" title="Herbert Scarf">Herbert Scarf</a></li>

<li><a href="Herv%C3%A9_Moulin" title="Hervé Moulin">Hervé Moulin</a></li>

<li><a href="Jean_Tirole" title="Jean Tirole">Jean Tirole</a></li>

<li><a href="Jean-Fran%C3%A7ois_Mertens" title="Jean-François Mertens">Jean-François Mertens</a></li>

<li><a href="Jennifer_Tour_Chayes" title="Jennifer Tour Chayes">Jennifer Tour Chayes</a></li>

<li><a href="Ken_Binmore" class="mw-redirect" title="Ken Binmore">Ken Binmore</a></li>

<li><a href="Kenneth_Arrow" title="Kenneth Arrow">Kenneth Arrow</a></li>

<li><a href="Leonid_Hurwicz" title="Leonid Hurwicz">Leonid Hurwicz</a></li>

<li><a href="Lloyd_Shapley" title="Lloyd Shapley">Lloyd Shapley</a></li>

<li><a href="Martin_Shubik" title="Martin Shubik">Martin Shubik</a></li>

<li><a href="Melvin_Dresher" title="Melvin Dresher">Melvin Dresher</a></li>

<li><a href="Merrill_M._Flood" title="Merrill M. Flood">Merrill M. Flood</a></li>

<li><a href="Olga_Bondareva" title="Olga Bondareva">Olga Bondareva</a></li>

<li><a href="Oskar_Morgenstern" title="Oskar Morgenstern">Oskar Morgenstern</a></li>

<li><a href="Paul_Milgrom" title="Paul Milgrom">Paul Milgrom</a></li>

<li><a href="Peyton_Young" title="Peyton Young">Peyton Young</a></li>

<li><a href="Reinhard_Selten" title="Reinhard Selten">Reinhard Selten</a></li>

<li><a href="Robert_Aumann" title="Robert Aumann">Robert Aumann</a></li>

<li><a href="Robert_Axelrod_(political_scientist)" title="Robert Axelrod (political scientist)">Robert Axelrod</a></li>

<li><a href="Robert_B._Wilson" title="Robert B. Wilson">Robert B. Wilson</a></li>

<li><a href="Roger_Myerson" title="Roger Myerson">Roger Myerson</a></li>

<li><a href="Samuel_Bowles_(economist)" title="Samuel Bowles (economist)">Samuel Bowles</a></li>

<li><a href="Suzanne_Scotchmer" title="Suzanne Scotchmer">Suzanne Scotchmer</a></li>

<li><a href="Thomas_Schelling" title="Thomas Schelling">Thomas Schelling</a></li>

<li><a href="William_Vickrey" title="William Vickrey">William Vickrey</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Combinatorial_game_theory698" style="font-size:114%;margin:0 4em"><a href="Combinatorial_game_theory" title="Combinatorial game theory">Combinatorial game theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Core<br>concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Combinatorial_explosion" title="Combinatorial explosion">Combinatorial explosion</a></li>

<li><a href="Determinacy" title="Determinacy">Determinacy</a></li>

<li><a href="Disjunctive_sum" title="Disjunctive sum">Disjunctive sum</a></li>

<li><a href="First-player_and_second-player_win" title="First-player and second-player win">First-player and second-player win</a></li>

<li><a href="Game_complexity" title="Game complexity">Game complexity</a></li>

<li><a href="Impartial_game" title="Impartial game">Impartial game</a></li>

<li><a href="Mis%C3%A8re" title="Misère">Misère</a></li>

<li><a href="Partisan_game" title="Partisan game">Partisan game</a></li>

<li><a href="Solved_game" title="Solved game">Solved game</a></li>

<li><a href="Sprague%E2%80%93Grundy_theorem" title="Sprague–Grundy theorem">Sprague–Grundy theorem</a></li>

<li><a href="Strategy-stealing_argument" title="Strategy-stealing argument">Strategy-stealing argument</a></li>

<li><a href="Zugzwang" title="Zugzwang">Zugzwang</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Games</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chess" title="Chess">Chess</a></li>

<li><a href="Chomp" title="Chomp">Chomp</a></li>

<li><a href="Clobber" title="Clobber">Clobber</a></li>

<li><a href="Domineering" title="Domineering">Domineering</a></li>

<li><a href="Hackenbush" title="Hackenbush">Hackenbush</a></li>

<li><a href="Notakto" title="Notakto">Notakto</a></li>

<li><a href="Subtract_a_square" title="Subtract a square">Subtract a square</a></li>

<li><a href="Sylver_coinage" title="Sylver coinage">Sylver coinage</a></li>

<li><a href="Toads_and_Frogs" title="Toads and Frogs">Toads and Frogs</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Mathematical<br>tools</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Nimber" title="Nimber">Nimber</a></li>

<li><a href="On_Numbers_and_Games" title="On Numbers and Games">On Numbers and Games</a></li>

<li><a href="Surreal_number" title="Surreal number">Surreal number</a></li>

<li><a href="Winning_Ways_for_Your_Mathematical_Plays" title="Winning Ways for Your Mathematical Plays">Winning Ways for Your Mathematical Plays</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Search<br>algorithms</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Alpha%E2%80%93beta_pruning" title="Alpha–beta pruning">Alpha–beta pruning</a></li>

<li><a href="Expectiminimax" title="Expectiminimax">Expectiminimax</a></li>

<li><a href="Minimax" title="Minimax">Minimax</a></li>

<li><a href="Monte_Carlo_tree_search" title="Monte Carlo tree search">Monte Carlo tree search</a></li>

<li><a href="Negamax" title="Negamax">Negamax</a></li>

<li><a href="Paranoid_algorithm" title="Paranoid algorithm">Paranoid algorithm</a></li>

<li><a href="Principal_variation_search" title="Principal variation search">Principal variation search</a></li></ul>

<ul><li><a href="Claude_Shannon" title="Claude Shannon">Claude Shannon</a></li>

<li><a href="John_Conway" class="mw-redirect" title="John Conway">John Conway</a></li>

<li><a href="John_von_Neumann" title="John von Neumann">John von Neumann</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Evolutionary_game_theory698" style="font-size:114%;margin:0 4em"><a href="Evolutionary_game_theory" title="Evolutionary game theory">Evolutionary game theory</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Core<br>concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Bishop%E2%80%93Cannings_theorem" title="Bishop–Cannings theorem">Bishop–Cannings theorem</a></li>

<li><a href="Evolution_and_the_Theory_of_Games" title="Evolution and the Theory of Games">Evolution and the Theory of Games</a></li>

<li><a href="Evolutionarily_stable_set" title="Evolutionarily stable set">Evolutionarily stable set</a></li>

<li><a href="Evolutionarily_stable_state" title="Evolutionarily stable state">Evolutionarily stable state</a></li>

<li><a href="Evolutionarily_stable_strategy" title="Evolutionarily stable strategy">Evolutionarily stable strategy</a></li>

<li><a href="Replicator_equation" title="Replicator equation">Replicator equation</a></li>

<li><a href="Risk_dominance" title="Risk dominance">Risk dominance</a></li>

<li><a href="Stochastically_stable_equilibrium" title="Stochastically stable equilibrium">Stochastically stable equilibrium</a></li>

<li><a href="Weak_evolutionarily_stable_strategy" title="Weak evolutionarily stable strategy">Weak evolutionarily stable strategy</a></li></ul>

<ul><li><a href="Chicken_(game)" title="Chicken (game)">Chicken</a></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Applications</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Cultural_group_selection" title="Cultural group selection">Cultural group selection</a></li>

<li><a href="Fisher's_principle" title="Fisher's principle">Fisher's principle</a></li>

<li><a href="Mobbing_(animal_behavior)" title="Mobbing (animal behavior)">Mobbing</a></li>

<li><a href="Terminal_investment_hypothesis" title="Terminal investment hypothesis">Terminal investment hypothesis</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Key people</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="John_Maynard_Smith" title="John Maynard Smith">John Maynard Smith</a></li>

<li><a href="Robert_Axelrod_(political_scientist)" title="Robert Axelrod (political scientist)">Robert Axelrod</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table><div></div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks mw-collapsible mw-collapsed navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Mechanism_design698" style="font-size:114%;margin:0 4em"><a href="Mechanism_design" title="Mechanism design">Mechanism design</a></div></th></tr><tr><td colspan="2" class="navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Core<br>concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Algorithmic_mechanism_design" title="Algorithmic mechanism design">Algorithmic mechanism design</a></li>

<li><a href="Bayesian-optimal_mechanism" title="Bayesian-optimal mechanism">Bayesian-optimal mechanism</a></li>

<li><a href="Incentive_compatibility" title="Incentive compatibility">Incentive compatibility</a></li>

<li><a href="Market_design" title="Market design">Market design</a></li>

<li><a href="Monotonicity_(mechanism_design)" title="Monotonicity (mechanism design)">Monotonicity</a></li>

<li><a href="Participation_constraint_(mechanism_design)" title="Participation constraint (mechanism design)">Participation constraint</a></li>

<li><a href="Revelation_principle" title="Revelation principle">Revelation principle</a></li>

<li><a href="Strategyproofness" title="Strategyproofness">Strategyproofness</a></li>

<li><a href="Vickrey%E2%80%93Clarke%E2%80%93Groves_mechanism" title="Vickrey–Clarke–Groves mechanism">Vickrey–Clarke–Groves mechanism</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" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Myerson%E2%80%93Satterthwaite_theorem" title="Myerson–Satterthwaite theorem">Myerson–Satterthwaite theorem</a></li>

<li><a href="Revenue_equivalence" title="Revenue equivalence">Revenue equivalence</a></li></ul>

<ul><li><a href="Digital_goods_auction" title="Digital goods auction">Digital goods auction</a></li>

<li><a href="Knapsack_auction" title="Knapsack auction">Knapsack auction</a></li>

<li><a href="Truthful_cake-cutting" title="Truthful cake-cutting">Truthful cake-cutting</a></li></ul>

<ul><li><a href="Bertrand_paradox_(economics)" title="Bertrand paradox (economics)">Bertrand paradox</a></li>

<li><a href="Chainstore_paradox" title="Chainstore paradox">Chainstore paradox</a></li>

<li><a href="Computational_complexity_of_games" class="mw-redirect" title="Computational complexity of games">Computational complexity of games</a></li>

<li><a href="Helly_metric" title="Helly metric">Helly metric</a></li>

<li><a href="Multi-agent_system" title="Multi-agent system">Multi-agent system</a></li>

<li><a href="PPAD_(complexity)" title="PPAD (complexity)">PPAD-complete</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><td class="navbox-abovebelow" colspan="2"><div>

<ul><li><b><span class="nowrap"><span class="skin-invert-image noviewer" typeof="mw:File"></span> </span><a href="Portal%3AMathematics" title="Portal:Mathematics">Mathematics portal</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Commons page"></span></span><b><a href="https://commons.wikimedia.org/wiki/Category:Game_theory" class="extiw external" title="commons:Category:Game theory">Commons</a></b></li>

<li><span class="noviewer" typeof="mw:File"><span title="WikiProject"></span></span><b>WikiProject</b></li>

<li><span class="noviewer" typeof="mw:File"><span title="Category"></span></span><b>Category</b></li></ul>

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This article is issued from <a class="external text" title="Last edited on 2024-09-18" href="https://en.wikipedia.org/wiki/?title=Perfect_Bayesian_equilibrium&oldid=1246432894">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

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