Outline for Dynamic Games of Complete Information

Transcription

1 Outline for Dynamic Games of Complete Information I. Examples of dynamic games of complete info: A. equential version of attle of the exes. equential version of Matching Pennies II. Definition of subgame-perfect equilibrium III. ackward induction in finitely-repeated games: A. The finitely-repeated Prisoner s Dilemma / finitely-repeated ertrand Oligopoly. The Chain-tore Paradox IV. tackelberg Model of Duopoly Copyright 2004 by Lawrence M. Ausubel Outline for Dynamic Games of Complete Information V. argaining: A. Alternating-offer bargaining (Rubinstein, 982). Digression on the Nash bargaining solution C. Digression on the Coase Theorem VI. Infinitely-repeated games and trigger strategies VII. The Folk Theorem VIII. Maximally-collusive equilibria

2 ubgame-perfect Equilibrium Definition: In an n-player dynamic game of complete information, an n-tuple of strategies is said to form a subgame-perfect equilibrium (PE) if the strategies constitute Nash equilibria in every subgame. ( Game : An inverted tree.) ( ubgame : What is left of the original game after some moves have been played.) Finitely Repeated Prisoners Dilemma Cooperate Defect Cooperate 3, 3 0,4 Defect 4, 0,

3 Finitely Repeated ertrand Price Competition Chain-tore Paradox (elten, 978) Entrant tay out Enter (a, 0) Incumbent Entrant Fight Incumbent Accommodate a, b, c > 0 (0, c) (a 2, b)

4 Chain-tore Paradox (elten, 978) Entrant tay out Enter (a, 0) Incumbent Entrant Fight Incumbent Accommodate a, b, c > 0 (0, c) (a 2, b) Chain-tore Paradox (elten, 978) Entrant tay out Enter (a, 0) Incumbent Entrant Fight Incumbent Accommodate a, b, c > 0 (0, c) (a 2, b)

5 tackelberg Model of Duopoly elects q Firm elects q 2 Firm 2 (q [P(q + q 2 ) c], q 2 [P(q + q 2 ) c])

6 olution of tackelberg Model of Duopoly First, observe that firm 2 solves: q 2 * = arg max{q 2 [P(q + q 2 ) c]}. Consequently, given q, can calculate q 2 by: a q c q *( q ) =. 2 2 Hence, firm solves: { q[ a q q2 q c] } max *( ) = q Giving a first-order condition of: a q c q + = 0 a q c { q a c q 2 } a q c { q } = max q = max. q Thus, we conclude: q* = a c ; q *( q *) = a c

7 argaining # of Troops # of Troops NATO Proposal Warsaw Pact Proposal 9,000 6,000 4,000 20,000 argaining # of Troops # of Troops NATO Proposal Warsaw Pact Proposal Final Agreement 9,000 6,000 3,000 4,000 20,000 7,000

8 Alternating Offer argaining (p, p ) (δp 2, δ[ p 2 ]) Offers p Offers p 2 Offers p 3 (δ 2 p 3, δ 2 [ p 3 ]) (0, 0) Alternating Offer argaining Offers p 3 (δ 2 p 3, δ 2 [ p 3 ]) (0, 0)

9 Alternating Offer argaining Offers p 3 (δ 2 p 3, δ 2 [ p 3 ]) (0, 0) Alternating Offer argaining (δp 2, δ[ p 2 ]) Offers p 2 Offers p 3 (δ 2 p 3, δ 2 [ p 3 ]) (0, 0)

10 Alternating Offer argaining (δp 2, δ[ p 2 ]) Offers p 2 Offers p 3 (δ 2 p 3, δ 2 [ p 3 ]) (0, 0) Alternating Offer argaining (p, p ) (δp 2, δ[ p 2 ]) Offers p Offers p 2 Offers p 3 (δ 2 p 3, δ 2 [ p 3 ]) (0, 0)

11 Alternating Offer argaining (p, p ) (δp 2, δ[ p 2 ]) Offers p Offers p 2 Offers p 3 (δ 2 p 3, δ 2 [ p 3 ]) (0, 0) The unique PE of the infinite-horizon, alternating-offer bargaining game In any period in which it is the seller s turn to make an offer: The seller offers a price of. + δ The buyer accepts any price p and + δ rejects any price p >. + δ In any period in which it is the buyer s turn to make an offer: δ The buyer offers a price of. + δ δ The seller accepts any price p and + δ δ rejects any price p <. + δ

12 haked-utton (84) Proof of Uniqueness of ubgame-perfect Equilibrium in Alternating-Offer argaining uppose there exists at least one PE in: (a) the alternating-offer game where the seller moves first; (b) the alternating-offer game where the buyer moves first Let M be the supremum of seller payoffs over all PEs in the game where the seller goes first Let m be the infimum of buyer payoffs over all PEs in the game where the buyer goes first Observe, for every n, σ n (PE) s.t. Π s ) M /n (game where seller goes first) The following is easily seen to be an PE of the game where the buyer goes first: σ n ' The buyer offers a division (δπ s ), - δπ s )). The seller accepts (in period 0) Given any deviation, play σ n beginning in period. Observe that: Π ') = - δπ ) - δm + δ(/n), for all n Hence m - δm ut if the buyer offers the seller a division (δm + ε, - δm - ε), the seller must accept for any ε > 0 since δm + ε today is better than M tomorrow. Hence m - δm - ε, for all ε > 0. Conclude m = - δm. Now observe that the following is an equilibrium of the game where the seller goes first. The seller offers [ - δ + δ 2 Π ), δ - δ 2 Π )]. The buyer accepts in period 0. σ n '' In case of any deviation, the buyer offers [δπ ), - δπ )]. The seller accepts in period. In case of further deviation, the parties play σ n ' in all subsequent periods. Observe that: Π '') = - δ + δ 2 Π ) - δ + δ 2 M - δ 2 (/n) for all n 0 Hence M - δ + δ 2 M.

13 ut suppose the seller ever offered the buyer a share smaller than δm. urely the buyer would reject. [imilarly, the buyer never makes an offer (which is accepted) in which his own share is less than m.] Consequently: M - δm = - δ( - δm) = - δ + δ 2 M. This demonstrates M = - δ + δ 2 M, and hence M = ( - δ)/( - δ 2 ) = /( + δ). Then m = - δ/( + δ) = /( + δ). Reversing the roles, this show that the seller s payoff in any PE is uniquely /( + δ). Furthermore this can only be realized if the offer is /( + δ) in the first period. Nash bargaining solution Disagreement point Axiom : The solution should not depend on linear transformations of players utility functions. Axiom 2: The solution should be individually-rational and Pareto-optimal. Axiom 3: Independence of Irrelevant Alternatives. Axiom 4: ymmetry. Feasible set Theorem: If the feasible set is convex, closed and bounded above, there is a unique solution satisfying Axioms -4, and it is given by: max (x d ) (x 2 d 2 ) x d where d denotes the disagreement point and x is an element of the feasible set.

14 From Coase (960), The Problem of ocial Cost : # of steers in herd Total crop loss Marginal crop loss Value of crop = $ per unit Cost of fence = $9 Definition: Let G be a static game. Then the T-period repeated game, denoted G(T, δ), consists of game G repeated T times. At each period t, the moves from periods,... t are known to every player. Payoffs are computed by: T u i = δ t u it t = (u it denotes the payoff to player i in period t) If T =, then G(T, δ) is referred to as the infinitely-repeated game. The average payoff to player i is then given by: ( δ) δ t u it t =

15 Finitely Repeated Games tage game with multiple NE 2 x 2 y 2 z 2 x, 5, 0 0, 0 0, 5 4, 4 0, 0 y z 0, 0 0, 0 3, 3 Two pure strategy NE: (x,x 2 ) and (z,z 2 ) Trigger trategy Equilibria First, define a main equilibrium path to be an action suggested for every player i (i =,,n) and for every period: s = ( s,..., sn),( s2,..., sn2 ),( s3,..., sn3),... period period 2 period 3 econd, define (s *,, s n *) to be a Nash equilibrium of the static game G. A trigger strategy for player i in the repeated game G(T, δ) is given by: sit, if every player has played according to s in all σ previous periods,..., (or if =), it = t t s *, if there has been any prior deviation by any player. i

16 Example: Infinitely-Repeated Cournot Game Description of trigger strategies: q it = a c a c 4 s 4 2s a c 3 Payoff along equilibrium path: t= 0 δ Payoff from optimally deviating:, if q = = q, for all s=,..., t,, otherwise. 2 ( )( a c) = ( ) a c t a c a c a c δ 8 ( ) t= t a c ( ) δ ( )( ) ( a c) a ( a c) c + a ( a c) c ( ) = ( ) + ( ) a c δ δ 9 a c Example: Infinitely-Repeated Cournot Game This is a trigger-strategy equilibrium if and only if: Payoff i (equilibrium path) Payoff i (optimally deviating) 2 ( ) a c ( ) δ 8 ( δ ) + δ [ ] 9 δ δ [ ] 7 δ δ a c 2 2 δ ( ) a c δ 64 ( ) + ( ) 9.

17 The Folk Theorem Definition: The n-tuple (x,, x n ) of payoffs to the n players is called feasible if it arises from the play of pure strategies or if it is a convex combination of payoffs from pure strategies. Theorem: Let (e,, e n ) be the payoffs from a Nash equilibrium of G and let (x,, x n ) be any feasible payoffs from G. If x i > e i for every player i, then there exists a subgame-perfect equilibrium of G(, δ) that attains (x,, x n ) as the average payoff, provided that δ is sufficiently close to. Example: Prisoners Dilemma Prisoner I Remain ilent Confess Remain ilent (, ) ( 0, 5) Prisoner II Confess ( 5, 0) ( 4, 4)

18 Example: attle of the exes F oxing allet M oxing allet (2, ) (0, 0) (0, 0) (, 2) Maximally-Collusive Equilibria Example: Two-Phase trategies in Repeated Cournot. If q,t = (a c)/4 and q 2,t = (a c)/4 tart Collusive Phase : Produce (a c)/4 If q, t (a c)/4 or q 2, t (a c)/4 If q, t = (a c)/2 and q 2, t = (a c)/2 Punishment Phase Produce (a c)/2 If q, t (a c)/2 or q 2, t (a c)/2