Game Theory Static Bayesian Games

1. Game Theory Static Bayesian Games Univ. Prof.dr. M.C.W. Janssen University of Vienna Winter semester 2011-12 Week 48, 49 (November 29, December 7,8)

2. In Many Situations Players do not have pay-off relevant information • Buyer and Seller negotiating about price • Willingness to pay, cost (willingness to sell) • Oligopoly competition • Cost of other firms in Bertrand, Cournot • Auctions • Value of other bidders • …. • Here, we consider static games • Later, sequential games and updating (actions may then reveal private information)

3. Framework • Players can be of different types: θiΘi • A firm can have different cost, a buyer may have different willingness to pay • Strategy player i; si: θiAi • can condition action on private information • Prior probability of types F(θ1,.., θn) • Updating if types are correlated: p(θ-i / θi ) • If types are uncorrelated, knowing your own type does not reveal information about types of other players: posterior prob. = prior prob.

4. Example: Auction • Players’ valuations are uniformly and independently distributed over interval [0,1]: this is the prior distribution, vi [0, 1] • Valuations are private information and the action each player chooses is her bid bi. • Strategy is function si: [0,1]bi ; bi(vi) • What is then an equilibrium?

5. Bayes-Nash Equilibrium definition • For def. 1 we define u’i (si (.),s-i (.))= Eθui (si (.),s-i (.); θi ) • Def 1. A strategy combination (s*i (.),s*-i (.)) is a Bayes-Nash equilibrium if u’i (s*i (.),s*-i (.)) u’i (si (.), s*-i (.))for all i and all si Si • Def 2: A strategy combination (s*i (.),s*-i (.)) is a Bayes-Nash equilibrium if given s*-i each type θi chooses an optimal action (action that maximizes expected profits taking p(θ-i / θi )) • Definitions are equivalent • Profits of types are independent of each other

6. Examples: Battle of the Sexes game I • Consider battle of the sexes game where there is some uncertainty about both players pay-off • t, t’  [0,x] • What is a strategy? Choose B or F depending on value of t, t’ • Proposal: player 1 chooses B iff t > t; player 2 chooses F iff t’ > t’ • Reasonable proposal?

7. Examples: Battles of the Sexes game II • For player 2 it looks as if player 1 chooses B with probability (x-t)/x • Equilibrium? Check for player 2 • Eπ(B) = (x-t)/x * 1 + t/x * 0 • Eπ(F) = (x-t)/x * 0 + t/x * (2+t’) • B optimal iff t’ < -3 + x/t • This has to be equal to t’ • Similarly, player 1 B optimal if t > -3 + x/t’ = t • t is solution to t2 + 3t – x = 0

8. Purification theorem for Mixed Strategy equilibrium – an illustration • If t=0, mixed strategy equilibrium where player 1 plays B with prob 2/3 • In Bayes-Nash eq. of private info game t, t’  [0,x], player 1 plays B is if t > t, with prob. • What happens when x becomes very small? Probability converges to 2/3

9. Purification theorem for Mixed Strategy equilibrium: set-up • Mixed strategy can be considered as the limit of a pure strategy Bayes-Nash equilibrium where uncertainty disappears. • Is this a general phenomenon? • Harsanyi (1973): yes • Harsanyi’s set-up: perturb pay-offs as follows: • θsi random variable with range [-1,1]; ε is positive number (going to 0). • Then u’i (s,θi ) = ui (s)+ εθsi; • Pi denotes the probability distribution for θi

10. Purification theorem for Mixed Strategy equilibrium: result • Best reply is essentially unique and in pure strategies • If θi is continuously distributed it is rare event that best replies coincide for interval of θi values • Any equilibrium for unperturbed pay-offs ui (s) is the limit as ε→0 of a sequence of pure strategy equilibria of the perturbed game u’i (s) • Holds for pure and mixed strategy equilibria, • But not for pure strategy equilibria in weakly dominated strategies

11. First-Price Sealed-Bid Auction • Highest bidder wins and pays his own bid. • Players’ valuations are uniformly and independently distributed over interval [0,1]. • What is equilibrium with n players? • Pay-off to player i is (vi-bi)Pr(bi > max bj) • Suppose strategies are linear, then Pr(bi > bj) = Pr(bi > α+βvj) = Pr(vj < (bi – α)/β) = (bi – α)/β and Pr(bi > max bj) = {(bi – α)/β}n-1 • Maximizing pay-off wrt bi yields - {(bi – α)/β}n-1 + (n-1)* {(bi – α)/β}n-2 (vi-bi)/β = 0 • Solving yields (bi – α) = (n-1)(vi-bi) or bi = ((n-1)vi+α))/n • Linearity requires α=0, β=(n-1)/n

12. Second-price sealed-bid auction • This we already discussed (week 2): • Strategies are also bids dependent on valuations and dominant strategy is to bid valuation: bi = vi • This is thus also a Bayes-Nash equilibrium

13. Double Auction: set-up • Seller’s (Player 1) cost is c; buyer’s valuation (Player 2) is v. Both are uniformly distributed over [0,1]. Both make a bid. If b1≤b2 then they trade at price p = (b1+b2)/2; otherwise no trade • Pay-offs • Seller : (b1+b2)/2 – c if b1≤b2 otherwise 0 • Buyer : v - (b1+b2)/2 if b1≤b2 otherwise 0 • Without asymmetric information, continuum of equilibria where both players bid t  [c,v]

14. Double Auction: asymmetric info I • Buyer chooses p2 to maximize • (v-[b2+E(b1|b1<b2)]/2) * Pr (b1 < b2) • Seller chooses p1 to maximize • ([b1+E(b2 |b2 >b1)]/2 - c) * Pr (b1 < b2) • First, consider again linear strategies b1 = α1 + ß1c so that E(b1|b1 < b2) = (α1+b2)/2 • Buyer’s problem reduces to max • (v-[b2 +(α1+b2)/2]/2) * (b2 - α1)/ß1 • Solution: b2 = (2v+ α1)/3 • Similarly for seller, take b2 = α2+ ß2v, and we get b1 = (2c+(α2+ ß2))/3

15. Double Auction: asymmetric info II • Solution for buyer: b2 = (2v+ α1)/3 implies that ß2 = 2/3 and α2 = α1/3 • Solution for seller: b1 = (2c+(α2+ ß2))/3 implies that ß1 = 2/3 and α1= (α2+2/3)/3 • Thus, α2=1/12 and α1=1/4 • In linear equilibrium trade occurs if v ≥ c +¼ • Inefficiency • Other Bayes-Nash equilibria exist, such as • Buyer: offer price p if v ≥ p; otherwise offer 0 • seller : offer price p if p ≥ c; otherwise offer 1 • All other equilibria are also inefficient