5.1a Bayes-Nash Equilibrium

Bayes-Nash Equilibrium

A Bayes-Nash Equilibrium of a static Bayesian game is a strategy profile in which each type $t_i\in T_i$ of every player $i\in N$ uses their best response to the strategies of all other players given $i$‘s beliefs $t_i$ about $t_{-i}$.

Intuition

(Harsanyi’s Theorem) If the Bayesian game is finite, then a BNE must exist.

Examples

Contributions Game

Consider our standard contributions game of two players who have two choices: contribute or not contribute. They also have two types: a high contributions cost type $c_H=\frac{3}{4}$, and a low contributions cost type $c_L=\frac{1}{4}$. Note that these types are independent of one another, and therefore there is no posterior belief updating.

To find the BNE, we must use our intuition. There is no definitive method like in other equilibria. Intuitively, suppose there is a BNE if $c_L$ types contribute, and $c_H$ types do not. Then, from the perspective of player $i$:

• If $P_i$ contributes, their utility is $U_i(\text{C})=1-t_i$.
• If $P_i$ does not contribute, then $U_i(\text{NC})=1\times P_j(\text{C})+0\times P_j(\text{NC})=P_j(\text{C}),$ where $P_j(\text{C})$ is the probability of player $j$ contributing (or of $t_j=c_L$, since players contribute only when their type is low cost).

From $P_i$‘s perspective, if:

• $P_i$‘s type is $t_i=c_H$, then $P_i$ will prefer to not contribute if $P_j(\text{C})\ge 1-\frac{3}{4}.$
  • This is because $P_i$ will choose $\text{NC}$ only if $U_i(\text{NC})\ge U_i(\text{C})$.
• $P_i$’s type is $t_i=c_L$, then $P_i$ will prefer to contribute if $1-\frac{1}{4}\ge P_j(\text{C}).$
  • This is because $P_i$ will choose $\text{C}$ only if $U_i(\text{C})\ge U_i(\text{NC})$.

Combining the two inequalities, we see that a BNE where $i$ contributes appears to happen only between $\frac{1}{4}\le P_j(\text{C})\le \frac{3}{4}$.

Since the BNE above does not cover all $p\in [0,1]$, there are more equilibria!

Can we identify more symmetric BNEs?

You may be tempted to try an approach where

Jury Voting Games