Single-Deviation Principle and Bargaining Mihai Manea MIT - PowerPoint PPT Presentation

Single-Deviation Principle and Bargaining Mihai Manea MIT

Multi-stage games with observable actions ◮ finite set of players N ◮ stages t = 0 , 1 , 2 , . . . ◮ H : set of terminal histories (sequences of action profiles of possibly different lengths) ◮ at stage t , after having observed a non-terminal history of play t = ( a , . . . , a − ) � H , each player i simultaneously chooses an 0 t 1 h action a t i ∈ A i ( h t ) ◮ u i ( h ) : payoff of i ∈ N for terminal history h ∈ H ◮ σ i : behavior strategy for i ∈ N specifies σ i ( h ) ∈ ∆( A i ( h )) for h � H Often natural to identify “stages” with time periods. Examples ◮ repeated games ◮ alternating bargaining game Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 2 / 23

Unimprovable Strategies To verify that a strategy profile σ constitutes a subgame perfect equilibrium (SPE) in a multi-stage game with observed actions, it suffices to check whether there are any histories h t where some player i can gain by deviating from playing σ i ( h t ) at t and conforming to σ i elsewhere. u i ( σ | h t ) : expected payoff of player i in the subgame starting at h t and played according to σ thereafter Definition 1 A strategy σ i is unimprovable given σ − i if u i ( σ i , σ − i | h t ) ≥ u i ( σ ′ , σ h t ) for − i | i every h t and σ ′ i with σ ′ i ( h ) = σ i ( h ) for all h � h t . Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 3 / 23

Continuity at Infinity If σ is an SPE then σ i is unimprovable given σ − i . For the converse. . . Definition 2 A game is continuous at infinity if | u i ( h ) − u i (˜ lim sup h ) | = 0 , ∀ i ∈ N . t →∞ { ( h , ˜ h ) | h t =˜ h t } Events in the distant future are relatively unimportant. Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 4 / 23

Single (or One-Shot) Deviation Principle Theorem 1 Consider a multi-stage game with observed actions that is continuous at infinity. If σ i is unimprovable given σ − i for all i ∈ N, then σ constitutes an SPE. Proof allows for infinite action spaces at some stages. There exist versions for games with unobserved actions. Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 5 / 23

Proof Suppose that σ i is unimprovable given σ − i , but σ i is not a best response 1 to σ − i following some history h t . Let σ i be a strictly better response and define 1 ε = u i ( σ , σ − i | h t ) − u i ( σ i , σ − i | h t ) > 0 . i Since the game is continuous at infinity , there exists t ′ > t and 2 2 σ i s.t. σ i is 1 at all information sets up to (and including) stage t ′ , 2 identical to σ σ i i coincides with σ i across all longer histories and 2 1 i ( σ , σ − i | h t ) − u i ( σ − i | h t ) | < ε/ 2 . | u , σ i i Then 2 i ( σ , σ − i | h t ) > u i ( σ i , σ − i | h t ) . u i Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 6 / 23

Proof σ i by replacing the stage t ′ actions following 3 2 σ i : strategy obtained from any history h t ′ with the corresponding actions under σ i 3 Conditional on any h t ′ , σ i and σ i coincide, hence 3 i ( σ , σ − i | t ′ ) = u i ( σ i , − i | h t ′ ) . u h σ i As σ i is unimprovable given σ − i , and conditional on h t ′ the subsequent σ i differs only at stage t ′ , 2 play in strategies σ i and 2 u i ( σ i , σ − i | h t ′ ) ≥ u i ( σ , σ − i | h t ′ ) . i Then 3 2 i ( σ , σ − i | h t ′ ) ≥ u i ( σ , σ − i | h t ′ ) u i i 2 σ i coincide before reaching stage t ′ , 3 for all histories h t ′ . Since σ i and 3 2 u i ( σ , σ − i | h t ) ≥ u i ( σ , σ − i | h t ) . i i Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 7 / 23

Proof i by replacing the stage t ′ − 1 actions σ 4 i : strategy obtained from σ 3 following any history h t ′ − 1 with the corresponding actions under σ i Similarly, u i ( σ 4 i , σ − i | h t ) ≥ u i ( σ 3 i , σ − i | h t ) . . . The final strategy σ t ′ − t + 3 is identical to σ i conditional on h t and i u i ( σ i , σ − i | h t ) = u i ( σ t ′ − t + 3 , σ − i | h t ) ≥ . . . i ≥ u i ( σ 3 i , σ − i | h t ) ≥ u i ( σ 2 i , σ − i | h t ) > u i ( σ i , σ − i | h t ) , a contradiction. Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 8 / 23

Applications Apply the single deviation principle to repeated prisoners’ dilemma to implement the following equilibrium paths for high discount factors: ◮ ( C , C ) , ( C , C ) , . . . ◮ ( C , C ) , ( C , C ) , ( D , D ) , ( C , C ) , ( C , C ) , ( D , D ) , . . . ◮ ( C , D ) , ( D , C ) , ( C , D ) , ( D , C ) . . . C D C 1 , 1 − 1 , 2 D 2 , − 1 0 , 0 Cooperation is possible in repeated play. Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 9 / 23

Bargaining with Alternating Offers Rubinstein (1982) ◮ players i = 1 , 2; j = 3 − i ◮ set of feasible utility pairs U = { ( u 1 , u 2 ) ∈ [ 0 , ∞ 2 ) | u 2 ≤ g 2 ( u 1 ) } ◮ g 2 s. decreasing, concave (and hence continuous), g 2 ( 0 ) > 0 ◮ δ i : discount factor of player i ◮ at every time t = 0 , 1 , . . . , player i ( t ) proposes an alternative u = ( u 1 , u 2 ) ∈ U to player j ( t ) = 3 − i ( t )  1 for t even   i ( t ) =   2 for t odd   ◮ if j ( t ) accepts the offer, game ends yielding payoffs ( δ t 1 u 1 , δ t 2 u 2 ) ◮ otherwise, game proceeds to period t + 1 Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 10 / 23

Stationary SPE Define g 1 = g − 1 2 . Graphs of g 2 and g − 1 1 : Pareto-frontier of U Let ( m 1 , m 2 ) be the unique solution to the following system of equations = δ 1 g 1 ( m 2 ) m 1 m 2 = δ 2 g 2 ( m 1 ) . ( m 1 , m 2 ) is the intersection of the graphs of δ 2 g 2 and ( δ 1 g 1 ) − 1 . SPE in “stationary” strategies: in any period where player i has to make an offer to j , he offers u with u j = m j and u i = g i ( m j ) , and j accepts only offers u with u j ≥ m j . Single-deviation principle : constructed strategies form an SPE. Is the SPE unique? Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 11 / 23

Iterated Conditional Dominance Definition 3 In a multi-stage game with observable actions, an action a i is conditionally dominated at stage t given history h t if, in the subgame starting at h t , every strategy for player i that assigns positive probability to a i is strictly dominated. Proposition 1 In any multi-stage game with observable actions, every SPE survives the iterated elimination of conditionally dominated strategies. Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 12 / 23

Equilibrium uniqueness Iterated conditional dominance : stationary equilibrium is essentially the unique SPE. Theorem 2 The SPE of the alternating-offer bargaining game is unique, except for the decision to accept or reject Pareto-inefficient offers. Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 13 / 23

Proof ◮ Following a disagreement at date t , player i cannot obtain a period t expected payoff greater than M 0 = δ i max u i = δ i g i ( 0 ) i u ∈ U ◮ Rejecting an offer u with u 0 i > M i is conditionally dominated by accepting such an offer for i . ◮ Once we eliminate dominated actions, i accepts all offers u with u i > M 0 i from j . M 0 ◮ Making any offer u � with u � i > �� i is dominated for j by an offer ¯ = λ u + ( 1 − λ ) M 0 , i g j M 0 for λ ∈ ( 0 , 1 ) (both offers are accepted u i immediately). Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 14 / 23

Proof Under the surviving strategies ◮ j can reject an offer from i and make � a counteroff er next period that � leaves him with slightly less than g j M 0 i , which i accepts; it is conditionally dominated for j to accept any offer smaller than m 1 = δ � 0 � j g M i j j i cannot expect to receive a continuation pa yoff greater than ◮ M 1 � g m 1 � � 2 0 � i g i m 1 � � = = δ max δ , δ i M i i i i j j after rejecting an offer from j � m 1 � � � M 0 �� � � 0 �� 0 2 0 = δ i g i = δ i M i ≥ δ i M i δ i g i δ j g j ≥ δ i g g j M i i j i Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 15 / 23

Proof Recursively define m k + 1 � M k � = δ j g j i j M k + 1 � m k + 1 � = δ i g i i j for i = 1 , 2 and k ≥ 1. ( m k i ) k ≥ 0 is increasing and ( M k i ) k ≥ 0 is decreasing. Prove by induction on k that, under any strategy that survives iterated conditional dominance, player i = 1 , 2 ◮ never accepts offers with u i < m k i ◮ always accepts offers with u i > M k i , but making such offers is dominated for j . Mihai Manea (MIT) Single-Deviation Principle and Bargaining March 9, 2016 16 / 23