Bayesian-Nash Equilibrium

Transcription

1 Bayesian-Nash Equilibrium A Bayesian game models uncertainty over types of opponents a player faces The game was defined in terms of players, their types, their available actions A player s beliefs about others players types, and payoffs (defined over actions and types) Last week focused on simple examples with finitely many types for each player Today: model situations where it is more natural to have a continuum of types Recall a Bayesian strategy is a function mapping types to actions, hence Find the Bayesian-Nash equilibria (Nash equilibria in Bayesian strategies) In particular, focus on two kinds of equilibria: cut-off and linear equilibria EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 1

2 (Yet Another) Noisy Battle-of-the-Sexes Player 1 does not know what payoffs player receives from playing C In fact, there is some noise on player s payoff, δ Likewise player 1 s payoff to S is perturbed by some ε, from the perspective of player Suppose δ, ε U[0, a] S C S 3+δ 4 + ε 0+δ 0+ε C 1 + δ 1 + ε 4 + δ 3+ε A Bayesian strategy for player maps all their types (of δ) to an action (S or C) A Bayesian strategy for player 1 maps all their types (of ε) to an action (S or C) EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM

3 A Cut-Off Strategy A simple cut-off strategy specifies a single value of a player s type, below which one action is chosen, and above which another action is chosen eg Consider the following cut-off strategy for player characterized by some δ [0, a]: s (δ) = { S if δ < δ, and C if δ δ Notice this is a pure strategy It specifies a pure action for all δ It is possible to specify much more complicated (including mixed) strategies However, from player 1 s perspective, this strategy looks like mixing EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 3

4 Player s Strategy Player s strategy is to play S if δ < δ and C if δ is big enough seems sensible f(δ) = 1 a δ 0 a 1 a Player s type: δ Play S Play C Recall δ U[0, a]: density is f(δ) = 1 a For some value of δ, the probability that player plays S is Pr[δ < δ] ie the area below δ Equal to δ/a Thus, Pr[ plays C] = 1 δ/a EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 4

5 Player 1 s Best Response Player 1 needs to compare the payoffs from S and C Expected utility from S u 1 (S; s (δ)) = Pr[ plays S] (4 + ε) + Pr[ plays C] (1 + ε) = Pr[δ < δ] (4 + ε) + Pr[δ δ] (1 + ε) = (4 + ε) δ a + (1 + ε)( 1 δ ) a = 4 δ a + ( 1 a) δ + ε A similar calculation for the expected utility received from playing C u 1 (C; s (δ)) = Pr[ plays S] 0 + Pr[ plays C] 3 = Pr[δ δ] 3 = 3 ( 1 δ a) So u 1 (S; s (δ)) = 4 δ a + ( 1 δ ) ( a + ε 3 1 δ a) = u1 (C; s (δ)) EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 5

6 Is a Cut-Off Strategy So player 1 prefers to play S rather than play C if and only if the payoff is larger: 4 δ a + ( 1 δ ) ( ) a + ε 3 1 δ a ε 6 δ a ε This is itself a cut-off strategy for player 1, with cut-off ε = 6 δ/a So s 1 (ε) = { S if ε ε, and C if ε < ε Now, given that player 1 plays this strategy, what should player do? Player will play C if and only if u (C; s 1 (ε)) u (S; s 1 (ε)) EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 6

7 Player s Best Response Calculate the probabilities with which player 1 plays S and C from s perspective: f(ε) = 1 a ε 0 a 1 a Player 1 s type: ε Play C Play S Player 1 plays S if ε ε, and note that: Pr[ε ε] = 1 ε/a So s payoff from playing S is 3(1 ε/a) From C it s (1 + δ)(1 ε/a) + (4 + δ) ε/a, Play C if δ 6 ε a δ EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 7

8 Solving for the Equilibrium So, two equations in two unknowns ( δ and ε) Solve to find these values: 6 ε a = δ and 6 δ a = ε δ = ε = a 6 + a Player 1 plays S if ε is bigger than this Player plays C if δ is bigger than this This is a Bayesian-Nash equilibrium in pure cut-off strategies Note that, from an outside observer s perspective, it appears that the players are mixing: Pr[1 plays S] = Pr[ε ε] = 4 + a 6 + a and Pr[ plays S] = Pr[δ < δ] = 6 + a EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 8

9 A Comparison with the Complete-Information Game Recall the complete-information battle-of-the-sexes game with a = 0 ε = δ = 0: S C S C Mixed Nash at 4q + (1 q) = 3(1 q) q = 1 3 ; and p + 4(1 p) = 3p p = 3 In the Bayesian-Nash equilibrium the probability S is played when a 0 is lim Pr[1 plays S] = lim a 0 a 0 ( ) 4 + a 6 + a ( ) = 3, lim Pr[ plays S] = lim a 0 a a = 1 3 EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 9

10 A Trading Game A buyer s value for a good is v b U[0, 1] A seller s value is v s U[0, 1] They each observe their own value, but not the other player s They each simultaneously announce a price (p b and p s ) If p b p s a sale takes place at a price half way between: (p b + p s )/ Otherwise, no sale Players The players are the buyer and seller N = {b, s} Strategies A function p i : [0, 1] R assigning a price p i (v i ) to each of i s types Payoffs Both the players receive 0 if no sale is made (p b < p s ), otherwise u b (p b ; p s ) = v b p b + p s and u s (p b ; p s ) = p b + p s v s EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 10

11 Linear Strategies Suppose seller plays a linear strategy, so p s (v s ) = α + βv s What will buyer do? U b (p b ; p s (v s )) = Pr[p b p s ] { v b p b + E [p s p b p s ] } α p s α + β p b 1/β (α + p b )/ Note that p s = α + βv s U[α, α + β] The probability that p b p s is shaded: Pr[p b p s ] = (p b α)/β Expectation of p s given p b p s shown: E [p s p b p s ] = (α + p b )/ EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 11

12 The Buyer s Optimization Problem Substituting for this probability and expectation in the expression for U b gives U b (p b ; p s (v s )) = p b α β [ v b 1 { p b + α + p }] b Differentiating this (quadratic) function with respect to p b, and setting to zero: 1 β [ v b 1 { p b + α + p }] b = 3 4 { } pb α β p b (v b ) = 1 3 α + 3 v b Note this optimal strategy for the buyer is linear If the seller uses a linear strategy, so should the buyer Now consider the seller s optimization problem EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 1

13 The Seller s Problem Suppose buyer plays a linear strategy, so p b (v b ) = γ + δv b What will seller do? U s (p s ; p b (v b )) = Pr[p s p b ] { ps + E [p b p s p b ] v s } γ γ + δ p s 1/δ (p s + γ + δ)/ p b Note that p b = γ + δv b U[γ, γ + δ] The probability that p s p b is shaded: Pr[p s p b ] = (γ + δ p s )/δ Expectation of p b given p s p b shown: E [p s p s p b ] = (p s + γ + δ)/ EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 13

14 The Seller s Optimal Strategy Substituting for this probability and expectation in the expression for U s gives U s (p s ; p b (v b )) = γ + δ p s δ [ 1 { p s + p } ] s + γ + δ v s Differentiating this (quadratic) function with respect to p s, and setting to zero: 1 δ [ 1 { p s + γ + δ + p } ] s v s = 3 4 { } γ + δ ps δ p s (v s ) = 1 3 (γ + δ) + 3 v s So, again, the seller s best response to a linear strategy is also a linear strategy p s (v s ) = α + βv s = 1 3 (γ + δ) + 3 v s and p b (v b ) = γ + δv b = 1 3 α + 3 v b EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 14

15 Solving for the Equilibrium For these two strategies to be a Bayesian-Nash equilibrium, require the following: α = 1 3 (γ + δ), and β = 3, and γ = 1 3 α, and δ = 3 These solve to give α = 1 4 and γ = 1 1 So the linear Bayesian-Nash equilibrium is p b (v b ) = v b and p s (v s ) = v s Trade occurs whenever p s p b or, from the equilibrium strategies, whenever v s v b However, there is mutually beneficial trade opportunity whenever v s v b Therefore the linear equilibrium is inefficient EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 15

16 Illustrating the Inefficiency Some mutually beneficial trades do not take place In particular, whenever v s v b < v s, there is an inefficient lack of trade Plot this problem Seller s value: v s Buyer s value: vb A B v s = v b v s = v b In area A there is efficient trade In area B there is efficient lack of trade (seller s value higher than buyer s) Shaded area, inefficient lack of trade There are many other equilibria But this is the most efficient! EC3043 LECTURE 13: BAYESIAN-NASH EQUILIBRIUM 16