Chapter 9 Games with Imperfect Information Bayesian Games

Complete versus Incomplete Information • So far we have assumed that players hold the correct belief about the other players’ actions. • In many situations, this is not realistic. Players’ payoffs may NOT be common knowledge. You may be participating in an auction where you do not know the valuations of the other bidders. Firms may not know their competitors’ cost functions. • In Bayesian games, we analyze situations in which each player is imperfectly informed about an aspect of her environment that is relevant to her choice of action.

Complete versus Incomplete Information • Variant of Bo. S. Suppose player I (Tim) does not know the type of player he is facing. With equal probability, player 2 (Jane) may wish to meet with Tim or Jane may wish in fact to avoid Tim at all costs! Jane U 2 CP Tim U 2 2, 1 0, 0 CP 0, 0 1, 2 Prob = 1/2 (Jane wants to Meet) Jane U 2 2, 0 Tim CP 0, 1 CP 0, 2 1, 0 Prob = 1/2 (Jane wants to Avoid)

Complete versus Incomplete Information • Perfect Information: at each move in the game the player with the move knows the full history of the game thus far. • Imperfect information: at some move of the game the player with the move does Not know the history of the game.

Complete versus Incomplete Information Definition: an information set for a player is a collection of decision nodes satisfying: – The player has the move at every node in the information set, and – When the play of the game reaches a node in the information set, the player with the move does not know which node in the information set has (or has not) been reached

Equivalent Definitions • Perfect Information: every information set is a singleton. • Imperfect Information: there is at least one nonsingleton information set.

Incomplete Information Definition: a subgame in an extensive game (account for imperfect information) – Begins at a decision x node that is a singleton information set – Includes all the decision and terminal nodes following x in the game tree (but no nodes that do not follow x), and – Does not cut any information set.

Complete versus Incomplete Information • So back to our example of Bo. S. We could model the game as Tim facing two different players (ie, the two types of Jane) Jane U 2 CP Tim U 2 2, 1 0, 0 CP 0, 0 1, 2 Prob = 1/2 (Meet) Jane U 2 2, 0 Tim CP 0, 1 CP 0, 2 1, 0 Prob = 1/2 (Avoid)

Complete versus Incomplete Information • Tim thinks that Jane is one of two possible types. For example if he thinks the type that would like to meet him would choose U 2 and the type that would avoid him would choose CP, then by choosing U 2 he obtains a payoff of 0. 5*2 + 0. 5*0 = 1. (U 2, U 2) Tim (U 2, CP) Jane (CP, U 2) (CP, CP) U 2 2 1 1 0 CP 0 1/2 1 So here we have Jane’s strategies are an (X, Y) pair. X is what she does if she wants to meet Tim and Y is what she does if she wants to avoid Tim. The payoffs listed are Tim’s EXPECTED payoff. What seems wrong with this type of analysis?

Complete versus Incomplete Information • Jane know’s her type!! • But to analyze the game, we need to model Jane as one of two possible types because Tim is unsure of his opponent’s type. • So Tim needs to have beliefs about what action each type of Jane will take, and in equilibrium, we will impose the condition that these beliefs are CORRECT! • One Nash equilibrium is (U 2, CP)). If Jane plays (U 2, CP), clearly a best response, given his beliefs, is for Tim to play U 2. Suppose Tim plays U 2, then if Jane wants to meet, she should play U 2; and if she wants to avoid, she plays CP. So we have BNE. • Instead of (U 2, CP) being “actions” of Jane, you can also think of them as (correct) “beliefs” of Tim. Ie, Tim thinks that if he plays U 2; then if Jane wants to meet, she will also play U 2 and if she wants to avoid him, she will play CP.

Complete versus Incomplete Information So player Tim has a unique best response to each of Jane’s strategies. If Tim plays U 2, then Jane should play (U 2, CP). If Tim plays CP, then Jane should play (CP, U 2). II (U 2, U 2) I (U 2, CP) (CP, U 2) (CP, CP) U 2 2 1 1 0 CP 0 1/2 1 So there is a unique (Bayesian) Nash Equilibium at (U 2, CP)).

Bayesian Games Definition: a Bayesian game consists of – a set of players – a set of actions for each player i, Ai – a set of types for each player i, Ti – a belief for each player i, pi – a payoff function for each player i, ui(a 1, a 2, …, an; ti)

Bayesian Games Timing: • Nature draws a type vector t = (t 1, t 2, . . . , tn), where ti is drawn from the set of possible types Ti • Nature reveals ti to player i but not to any other player • The players simultaneously choose actions, player i choosing ai from the set Ai • Payoffs ui(a 1, a 2, …, an; ti) are received

Bayesian Games Recall Bayes Rule: Suppose we have three events, A, B, and C. Then P(A|B) = P(A, B) / P(B) = P(B|A)*Pr(A) / [P(B|A)*P(A) + P(B|C)*P(C)] We will assume it is common knowledge that in step 1 nature draws a type vector t = (t 1, t 2, . . . , tn) according to the prior probability distribution p(t). We assume types are INDEPENDENT! When nature then reveals ti to player i, he can compute the belief pi(t-i| ti) using Bayes’ rule:

Bayesian Games Definition: The (pure) strategy profile s*=(s 1*, s 2*, …, sn*) is a Bayesian Nash equilibrium of a Bayesian game if for each player i and for each of i’s types ti in Ti, si*(ti) solves That is, no player wants to change his strategy, even if the change involves one action by one type

Bayesian Games • So in a strategic game, (simultaneous with perfect information), each player chose an action. In a Bayesian game, each type of each player chooses an action. In a NE of a Bayesian game, the action chosen by each type of each player is optimal, given the actions chosen by every type of every other player. • Fighting an Opponent of Unknown Strength (282. 1) • First Price Sealed Bid Auction (N=2) • Cournot game pg 285.