Bayes correlated equilibrium | |
Supersetof: | Correlated equilibrium, Bayesian Nash equilibrium |
Discoverer: | Dirk Bergemann, Stephen Morris |
In game theory, a Bayes correlated equilibrium is a solution concept for static games of incomplete information. It is both a generalization of the correlated equilibrium perfect information solution concept to bayesian games, and also a broader solution concept than the usual Bayesian Nash equilibrium thereof. Additionally, it can be seen as a generalized multi-player solution of the Bayesian persuasion information design problem.
Intuitively, a Bayes correlated equilibrium allows for players to correlate their actions in a way such that no player has an incentive to deviate for every possible type they may have. It was first proposed by Dirk Bergemann and Stephen Morris.[1]
Let
I
\Theta
G=\langle(Ai,ui)i,\Theta,\psi\rangle
Ai
A=\prodiAi
ui:A x \Theta → R
\psi\in\Delta++(\Theta)
An information structure is defined as a tuple
S=\langle(Ti)i,\pi\rangle
Ti
T=\prodiTi
\pi:\Theta → \Delta(T)
\pi(t|\theta)
t\inT
\theta\in\Theta
By joining those two definitions, one can define
\Gamma=(G,S)
\Gamma=(G,S)
\sigma:T x \Theta → \Delta(A)
\sigma(a|t,\theta)
\sigma(a)\in\Delta(A)
t\inT
\theta\in\Theta
A Bayes correlated equilibrium (BCE) is defined to be a decision rule
\sigma
\sigma
\Gamma=(G,S)
i\inI
ti\inTi
ai\inAi
\sum | |
a-i,t-i,\theta |
\psi(\theta)\pi(ti,t-i|\theta)\sigma(ai,a-i|ti,t-i,\theta)ui(ai,a-i,\theta)
\geq
\sum | |
a-i,t-i,\theta |
\psi(\theta)\pi(ti,t-i|\theta)\sigma(ai,a-i|ti,t-i,\theta)ui(a'i,a-i,\theta)
for all
a'i\inAi
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
\Gamma=(G,S)
s:T → \Delta(A)
i
si(ai|ti)\in\Delta(Ai)
i\inI
ti\inTi
ai\inAi
si(ai|ti)>0
\sum | |
a-i,t-i,\theta |
\psi(\theta)\pi(ti,t-i|\theta)\left(\prodjsj(aj|tj)\right)ui(ai,a-i,\theta)
\geq
\sum | |
a-i,t-i,\theta |
\psi(\theta)\pi(ti,t-i|\theta)\left(\prodjsj(aj|tj)\right)ui(a'i,a-i,\theta)
for every
a'i\inAi
If we define the decision rule
\sigma
\Gamma
\sigma(a|t,\theta)=s(a|t)=\prodisi(ai|ti)
t\inT
\theta\in\Theta
If there is no uncertainty about the state of the world (e.g., if
\Theta
\sigma\in\Delta(A)
i\inI
\sum | |
a-i\inA{-i |
for every
a'i\inAi
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.[5] More specifically, let
v:A x \Theta → R
\sigma
V(\sigma)=\suma,\psi(\theta)\pi(t|\theta)\sigma(a|t,\theta)v(a,\theta)
If the set of players
I
V(\sigma)