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 BoS. 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!

U2 Tim CP U2 Jane CP 2,1 0,0 0,0 1,2 Tim U2 CP U2 Jane CP 2,0 0,2 0,1 1,0 Prob = 1/2 (Jane wants to Meet) 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

nodes satisfying: for a player is a collection of decision – 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 BoS. We could model the game as Tim facing two different players (ie, the two types of Jane) U2 Tim CP U2 Jane CP 2,1 0,0 0,0 1,2 Prob = 1/2 (Meet) Tim U2 CP U2 Jane CP 2,0 0,2 0,1 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 U2 and the type that would avoid him would choose CP, then by choosing U2 he obtains a payoff of 0.5*2 + 0.5*0 = 1.

U2 Tim CP (U2,U2) 2 0 (U2,CP) Jane 1 1/2 (CP,U2) 1 1/2 (CP,CP) 0 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 knows 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 (U2,(U2,CP)). If Jane plays (U2,CP), clearly a best response, given his beliefs, is for Tim to play U2. Suppose Tim plays U2, then if Jane wants to meet, she should play U2; and if she wants to avoid, she plays CP. So we have BNE.

• Instead of (U2,CP) being “actions” of Jane, you can also think of them as (correct) “beliefs” of Tim. Ie, if Tim plays U2, then Tim thinks that if Jane wants to meet, she will also play U2, and Tim thinks that if she wants to avoid him, she will play CP.

Complete versus Incomplete Information

For Tim (table below): Tim has a unique best response to each of Jane’s strategies. For Jane (original game): If Tim plays U2, then Jane should play (U2,CP). If Tim plays CP, then Jane should play (CP,U2). Tim U2 CP (U2,U2) 2 0 (U2,CP) Jane (CP,U2) 1 1 1/2 1/2 (CP,CP) 0 1 So there is a unique (Bayesian) Nash Equilibium at (U2, (U2,CP)).

Bayesian Games Definition: a

Bayesian

game consists of

– a set of players – a set of actions for each player

i

, A i – a set of types for each player

i

, T i – a belief for each player

i

, p i – a payoff function for each player

i

, u i ( a 1 , a 2 ,…, a n ;t i )

Bayesian Games

Timing: 1. Nature draws a type vector t = (t 1 ,t 2 ,...,t n ), where t i is drawn from the set of possible types T i 2. Nature reveals t i player to player

i

but not to any other 3. The players simultaneously choose actions, player

i

choosing a i 4. Payoffs u i ( a 1 , a 2 ,…, from the set A a n ;t i i ) 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)*P(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 ,...,t n ) according to the prior probability distribution

p

(t). We often assume types are INDEPENDENT! In general, when nature reveals t i to player

i

, he can compute the belief

p

i (t -i | t i ) using Bayes’ rule:

p i



t



i i

 

p t



i

,  

i t i

 

t

 

i



T p



i



t p i

  ,

t t



i i

, 

t



i

Bayesian Games

Definition: The (pure) strategy profile s*=(s 1 *,s 2 *,…,s n *) is a Bayesian Nash equilibrium of a Bayesian game if for each player

i

and for each of

i’

s types t i in T i , s i *(t i ) solves

Max

{

a i



A i t



i

 

T



u i i

(

s

1 * (

t

1 ), ...,

a i

, ...,

s n

* (

t n

) |

t i

) *

p i

(

t



i

|

t i

) } 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 (page 282.1 in Osborne).

• First Price Sealed Bid Auction (N=2).

• Cournot game (page 285 in Osborne).