1 / 15

Game Theory Static Bayesian Games

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). In Many Situations Players do not have pay-off relevant information. Buyer and Seller negotiating about price

Download Presentation

Game Theory Static Bayesian Games

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related