5.1b Belief Updating

Belief Updating

Belief updating in Bayesian games involves players revising their beliefs about other players’ types as the game progresses and new information becomes available. If a player $i\in N$ wants to know the probability of player $j\in N$ playing some type $t_j\in T_j$, then

• The prior distribution over the types of player $j$ from the perspective of $i$ is a probability distribution $P_j:t_j\to [0,1]$ that represents $i$’s initial beliefs about the types of $j$ before any actions are observed or any information is revealed.
• The posterior distribution over the types of player $j$ from the perspective of $i$ is a probability distribution $P_j(\cdot |t_i):t_j\to [0,1]$ that represents $i$’s updated beliefs about $j$‘s after observing actions or receiving information.

Properties

(Independence) If $P_j(t_j|t_i)=P_j(t_j)$, then $t_i$ gives no information about $t_j$.