Bayes correlated equilibrium

Let $I$ be a set of players, and $\backslash Theta$ a set of possible states of the world. A game is defined as a tuple $G\; =\; \backslash langle\; (A\_i,\; u\_i)\_\{i\; \backslash in\; I\},\; \backslash Theta,\; \backslash psi\; \backslash rangle$, where $A\_i$ is the set of possible actions (with $A\; =\; \backslash prod\_\{i\; \backslash in\; I\}\; A\_i$) and $u\_i\; :\; A\backslash times\; \backslash Theta\; \backslash rightarrow\; \backslash mathbb\{R\}$ is the utility function for each player, and $\backslash psi\; \backslash in\; \backslash Delta\_\{++\}\; (\backslash Theta)$ is a full support common prior over the states of the world.

An information structure is defined as a tuple $S\; =\; \backslash langle\; (T\_i)\_\{i\; \backslash in\; I\},\; \backslash pi\; \backslash rangle$, where $T\_i$ is a set of possible signals (or types) each player can receive (with $T\; =\; \backslash prod\_\{i\; \backslash in\; I\}\; T\_i$), and $\backslash pi\; :\; \backslash Theta\; \backslash rightarrow\; \backslash Delta\; (T)$ is a signal distribution function, informing the probability $\backslash pi\; (t\; |\; \backslash theta)$ of observing the joint signal $t\; \backslash in\; T$ when the state of the world is $\backslash theta\; \backslash in\; \backslash Theta$.

By joining those two definitions, one can define $\backslash Gamma\; =\; (G,\; S)$ as an incomplete information game.{{cite journal |last1=Gossner |first1=Olivier |title=Comparison of Information Structures |journal=Games and Economic Behavior |date=2000 |volume=30 |issue=1 |pages=44–63 |doi=10.1006/game.1998.0706 |url=https://www.sciencedirect.com/science/article/pii/S0899825698907060|hdl=10230/596 |hdl-access=free }} A decision rule for the incomplete information game $\backslash Gamma\; =\; (G,\; S)$ is a mapping $\backslash sigma:\; T\; \backslash times\; \backslash Theta\; \backslash rightarrow\; \backslash Delta\; (A)$. Intuitively, the value of decision rule $\backslash sigma\; (a\; |\; t,\; \backslash theta)$ can be thought of as a joint recommendation for players to play the joint mixed strategy $\backslash sigma\; (a)\; \backslash in\; \backslash Delta(A)$ when the joint signal received is $t\; \backslash in\; T$ and the state of the world is $\backslash theta\; \backslash in\; \backslash Theta$.

A Bayes correlated equilibrium (BCE) is defined to be a decision rule $\backslash sigma$ which is obedient: that is, one where no player has an incentive to unilaterally deviate from the recommended joint strategy, for any possible type they may be. Formally, decision rule $\backslash sigma$ is obedient (and a Bayes correlated equilibrium) for game $\backslash Gamma\; =\; (G,\; S)$ if, for every player $i\; \backslash in\; I$, every signal $t\_i\; \backslash in\; T\_i$ and every action $a\_i\; \backslash in\; A\_i$, we have

:$\backslash sum\_\{a\_\{-i\},\; t\_\{-i\},\; \backslash theta\}\; \backslash psi\; (\backslash theta)\; \backslash pi\; (t\_i,\; t\_\{-i\}\; |\; \backslash theta)\; \backslash sigma\; (a\_i,\; a\_\{-i\}\; |\; t\_i,\; t\_\{-i\}\; ,\backslash theta)\; u\_i(a\_i,\; a\_\{-i\},\; \backslash theta)$

:$\backslash geq\; \backslash sum\_\{a\_\{-i\},\; t\_\{-i\},\; \backslash theta\}\; \backslash psi\; (\backslash theta)\; \backslash pi\; (t\_i,\; t\_\{-i\}\; |\; \backslash theta)\; \backslash sigma\; (a\_i,\; a\_\{-i\}\; |\; t\_i,\; t\_\{-i\}\; ,\backslash theta)\; u\_i(a\text{'}\_i,\; a\_\{-i\},\; \backslash theta)$

for all $a\text{'}\_i\; \backslash in\; A\_i$.

That is, every player obtains a higher expected payoff by following the recommendation from the decision rule than by deviating to any other possible action.

Every Bayesian Nash equilibrium (BNE) of an incomplete information game can be thought of a as BCE, where the recommended joint strategy is simply the equilibrium joint strategy.

Formally, let $\backslash Gamma\; =\; (G,\; S)$ be an incomplete information game, and let $s\; :\; T\; \backslash rightarrow\; \backslash Delta(A)$ be an equilibrium joint strategy, with each player $i$ playing $s\_i\; (a\_i\; |\; t\_i)\; \backslash in\; \backslash Delta\; (A\_i)$. Therefore, the definition of BNE implies that, for every $i\; \backslash in\; I$, $t\_i\; \backslash in\; T\_i$ and $a\_i\; \backslash in\; A\_i$ such that $s\_i\; (a\_i\; |\; t\_i)\; >\; 0$, we have

:$\backslash sum\_\{a\_\{-i\},\; t\_\{-i\},\; \backslash theta\}\; \backslash psi\; (\backslash theta)\; \backslash pi\; (t\_i,\; t\_\{-i\}\; |\; \backslash theta)\; \backslash left(\backslash prod\_\{j\; \backslash neq\; i\}\; s\_j\; (a\_j\; |\; t\_j)\; \backslash right)\; u\_i(a\_i,\; a\_\{-i\},\; \backslash theta)$

:$\backslash geq\; \backslash sum\_\{a\_\{-i\},\; t\_\{-i\},\; \backslash theta\}\; \backslash psi\; (\backslash theta)\; \backslash pi\; (t\_i,\; t\_\{-i\}\; |\; \backslash theta)\; \backslash left(\backslash prod\_\{j\; \backslash neq\; i\}\; s\_j\; (a\_j\; |\; t\_j)\; \backslash right)\; u\_i(a\text{'}\_i,\; a\_\{-i\},\; \backslash theta)$

for every $a\text{'}\_i\; \backslash in\; A\_i$.

If we define the decision rule $\backslash sigma$ on $\backslash Gamma$ as $\backslash sigma\; (a\; |\; t,\; \backslash theta)\; =\; s(a\; |\; t)\; =\; \backslash prod\_\{i\}\; s\_i\; (a\_i\; |\; t\_i)$ for all $t\; \backslash in\; T$ and $\backslash theta\; \backslash in\; \backslash Theta$, we directly get a BCE.

If there is no uncertainty about the state of the world (e.g., if $\backslash Theta$ is a singleton), then the definition collapses to Aumann's correlated equilibrium solution.{{cite journal |last1=Aumann |first1=Robert J. |author1-link=Robert J. Aumann |title=Correlated Equilibrium as an Expression of Bayesian Rationality |journal=Econometrica |date=1987 |volume=55 |issue=1 |pages=1–18 |doi=10.2307/1911154 |jstor=1911154 |url=https://www.jstor.org/stable/1911154}} In this case, $\backslash sigma\; \backslash in\; \backslash Delta\; (A)$ is a BCE if, for every $i\; \backslash in\; I$, we have{{cite journal |last1=Bergemann |first1=Dirk |last2=Morris |first2=Stephen |author1-link=Dirk Bergemann |author2-link=Stephen Morris (economist) |title=Information Design: A Unified Perspective |journal=Journal of Economic Literature |date=2019 |volume=57 |issue=1 |pages=44–95 |doi=10.1257/jel.20181489 |jstor=26673203 |url=https://www.jstor.org/stable/26673203|url-access=subscription }}

:$\backslash sum\_\{a\_\{-i\}\; \backslash in\; A\{-i\}\}\; \backslash sigma\; (a\_i,\; a\_\{-i\})\; u\_i(a\_i,\; a\_\{-i\})\; \backslash geq\; \backslash sum\_\{a\_\{-i\}\; \backslash in\; A\{-i\}\}\; \backslash sigma\; (a\_i,\; a\_\{-i\})\; u\_i(a\text{'}\_i,\; a\_\{-i\})$

for every $a\text{'}\_i\; \backslash in\; A\_i$, which is equivalent to the definition of a correlated equilibrium for such a setting.

Additionally, the problem of designing a BCE can be thought of as a multi-player generalization of the Bayesian persuasion problem from Emir Kamenica and Matthew Gentzkow.{{Cite journal |last1=Kamenica |first1=Emir |last2=Gentzkow |first2=Matthew |date=2011-10-01 |title=Bayesian Persuasion |url=https://www.aeaweb.org/articles?id=10.1257/aer.101.6.2590 |journal=American Economic Review |language=en |volume=101 |issue=6 |pages=2590–2615 |doi=10.1257/aer.101.6.2590 |issn=0002-8282|url-access=subscription }} More specifically, let $v\; :\; A\; \backslash times\; \backslash Theta\; \backslash rightarrow\; \backslash mathbb\; R$ be the information designer's objective function. Then her ex-ante expected utility from a BCE decision rule $\backslash sigma$ is given by:

:$V(\backslash sigma)\; =\; \backslash sum\_\{a,\; t,\; \backslash theta\}\; \backslash psi\; (\backslash theta)\; \backslash pi(t\; |\; \backslash theta)\; \backslash sigma\; (a\; |\; t,\; \backslash theta)\; v(a,\; \backslash theta)$

If the set of players $I$ is a singleton, then choosing an information structure to maximize $V(\backslash sigma)$ is equivalent to a Bayesian persuasion problem, where the information designer is called a Sender and the player is called a Receiver.

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Category:Game theory equilibrium concepts