Game theory solution
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.[1]
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.[2]
Formal definition
Preliminaries
Let
be a set of players, and
a set of possible states of the world. A game is defined as a tuple
, where
is the set of possible actions (with
) and
is the utility function for each player, and
is a full support common prior over the states of the world.
An information structure is defined as a tuple
, where
is a set of possible signals (or types) each player can receive (with
), and
is a signal distribution function, informing the probability
of observing the joint signal
when the state of the world is
.
By joining those two definitions, one can define
as an incomplete information game.[3] A decision rule for the incomplete information game
is a mapping
. Intuitively, the value of decision rule
can be thought of as a joint recommendation for players to play the joint mixed strategy
when the joint signal received is
and the state of the world is
.
Definition
A Bayes correlated equilibrium (BCE) is defined to be a decision rule
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
is obedient (and a Bayes correlated equilibrium) for game
if, for every player
, every signal
and every action
, we have
![{\displaystyle \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\sigma (a_{i},a_{-i}|t_{i},t_{-i},\theta )u_{i}(a_{i},a_{-i},\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6b53105bda9dcc6504e85462d039d98daf6a9b3b)
![{\displaystyle \geq \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\sigma (a_{i},a_{-i}|t_{i},t_{-i},\theta )u_{i}(a'_{i},a_{-i},\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a791d5ef0a708f7938da5b2af66fd1cf97cedac)
for all
.
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.
Relation to other concepts
Bayesian Nash equilibrium
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.[2]
Formally, let
be an incomplete information game, and let
be an equilibrium joint strategy, with each player
playing
. Therefore, the definition of BNE implies that, for every
,
and
such that
, we have
![{\displaystyle \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\left(\prod _{j\neq i}s_{j}(a_{j}|t_{j})\right)u_{i}(a_{i},a_{-i},\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e46edd5b2f054eabd48f5c460a52da168d0f8c8)
![{\displaystyle \geq \sum _{a_{-i},t_{-i},\theta }\psi (\theta )\pi (t_{i},t_{-i}|\theta )\left(\prod _{j\neq i}s_{j}(a_{j}|t_{j})\right)u_{i}(a'_{i},a_{-i},\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d823e9fa0f83b4807674a4b749304a6f81771cb4)
for every
.
If we define the decision rule
on
as
for all
and
, we directly get a BCE.
Correlated equilibrium
If there is no uncertainty about the state of the world (e.g., if
is a singleton), then the definition collapses to Aumann's correlated equilibrium solution.[4] In this case,
is a BCE if, for every
, we have[1]
![{\displaystyle \sum _{a_{-i}\in A{-i))\sigma (a_{i},a_{-i})u_{i}(a_{i},a_{-i})\geq \sum _{a_{-i}\in A{-i))\sigma (a_{i},a_{-i})u_{i}(a'_{i},a_{-i})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/939d6d8e50bf27cb7727d9a0229fe27cee6cbdd5)
for every
, which is equivalent to the definition of a correlated equilibrium for such a setting.
Bayesian persuasion
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
be the information designer's objective function. Then her ex-ante expected utility from a BCE decision rule
is given by:[1]
![{\displaystyle V(\sigma )=\sum _{a,t,\theta }\psi (\theta )\pi (t|\theta )\sigma (a|t,\theta )v(a,\theta )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b86d9f24ec5a068f1f302bbe593c4bb3f066c9dd)
If the set of players
is a singleton, then choosing an information structure to maximize
is equivalent to a Bayesian persuasion problem, where the information designer is called a Sender and the player is called a Receiver.