On the commitment value and commitment optimal strategies in bimatrix games Stefanos Leonardos 1 and Costis Melolidakis National and Kapodistrian University of Athens Department of Mathematics, Division of Statistics & Operations Research January 9, 2018 1Supported by the Alexander S. Onassis Public Benefit Foundation. 1/24
Overview Motivation 1 Definitions – Existing results 2 Results 3 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria Future Research 4 2/24
Outline Motivation 1 Definitions – Existing results 2 Results 3 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria Future Research 4 3/24
Motivation I � J. von Neumann & O. Morgenstern (1944): used 2 auxiliary games to present the solution of 2-person, 0-sum games. Minorant game: player I in disadvantage. I chooses his mixed strategy x first, and then II , in full knowledge of x (but not of its realization) chooses his strategy y . Majorant game: player I in advantage (order reversed). � Payoffs of player I Minorant game: α L = max x ∈ X min y ∈ Y α ( x , y ) Majorant game: α F = min y ∈ Y max x ∈ X α ( x , y ) Simultaneous game: any solution v A must satisfy α L ≤ v A ≤ α F . � Minimax theorem: α L = α F in mixed strategies = ⇒ v A unique solution. 4/24
Motivation II � Generalize to 2-person, non 0-sum games: not straightforward. � Reason: 3 different points of view Matrix values v A : optimize against the worst possible Equilibrium α N : optimize simultaneously Optimization α L : optimize sequentially � In 0-sum games: v A = α N = α L . In non 0-sum? � Our aim is to study the relation between the three notions matrix values & max-min strategies Nash equilibria payoffs & strategies commitment values & optimal strategies in the associated sequential games. 5/24
Outline Motivation 1 Definitions – Existing results 2 Results 3 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria Future Research 4 6/24
Existing results I � Von Stengel & Zamir (2010). Theorem (1) In a (degenerate) bimatrix game, the subgame perfect equilibria payoffs of the leader form an interval [ α L , α H ]. The lowest leader equilibrium payoff α L is given by α L = max j ∈ D max x ∈ X ( j ) min k ∈E ( j ) α ( x , k ) and the highest leader equilibrium payoff α H is given by α H = max j ∈ BR II ( x ) α ( x , j ) = max max j ∈ N max x ∈ X ( j ) α ( x , j ) x ∈ X If the game is non-degenerate, then α L = α H is the unique commitment value of the leader. 7/24
Existing results II � Von Stengel & Zamir (2010). Example � a b c d e � a b c d e � � 2 6 9 1 7 4 4 0 2 4 T T A = B = 8 0 3 1 0 4 4 6 5 0 B B 7/24
Existing results II � Von Stengel & Zamir (2010). Example � a b c d e � a b c d e � � 2 6 9 1 7 4 4 0 2 4 T T A = B = 8 0 3 1 0 4 4 6 5 0 B B Nash equilibria: (many) Nash payoffs: Player I from 1 to 7, Player II 4 7/24
Existing results II � Von Stengel & Zamir (2010). Example � a b c d e � a b c d e � � 2 6 9 1 7 4 4 0 2 4 T T A = B = 8 0 3 1 0 4 4 6 5 0 B B Equivalent strategies: b ∈ E ( a ) 7/24
Existing results II � Von Stengel & Zamir (2010). Example � a b c d e � a b c d e � � 2 6 9 1 7 4 4 0 2 4 T T A = B = 8 0 3 1 0 4 4 6 5 0 B B Equivalent strategies: b ∈ E ( a ) Weakly dominated strategy: e by a (or b ). 7/24
Existing results II � Von Stengel & Zamir (2010). Example a b c d e a b c d e 2 6 9 1 7 4 4 0 2 4 T T ∗ 5 ∗ 4 A = 3 T + 2 1 6 2 1 2 . 3 B = 3 T + 2 1 4 4 4 1 . 3 3 B 3 B 8 0 3 1 0 4 4 6 5 0 B B Equivalent strategies: b ∈ E ( a ) Weakly dominated strategy: e by a (or b ). � x L , j F � �� � � with α L = 5 , β F = 4. 3 , 2 1 = , c 3 7/24
Existing results II � Von Stengel & Zamir (2010). Example � a b c d e � a b c d e � � 2 6 9 1 7 4 4 0 2 4 T T A = B = 8 0 3 1 0 4 4 6 5 0 B B Equivalent strategies: b ∈ E ( a ) Weakly dominated strategy: e by a (or b ). � x L , j F � �� � � with α L = 5 , β F = 4. 1 3 , 2 = , c 3 � x H , j F � = ( T , e ) with α H = 7. 7/24
Existing results III � Von Stengel & Zamir (2010). Theorem (2) If ℓ denotes the lowest and h the highest Nash equilibrium payoff of player I in Γ, then ℓ ≤ α L and h ≤ α H . So, in degenerate games with α L < α H v A ≤ ℓ ≤ α L , v A ≤ ℓ ≤ h ≤ α H which for non-degenerate games simplifies to v A ≤ ℓ ≤ h ≤ α L = α H . � Lower and upper bounds for Nash equilibria payoffs. � Based on these results, characterize the bimatrix games for which v A = α L = α H : accept Nash (e.g. 0-sum games) v A = h < α L : question Nash (e.g. Traveler’s Dilemma) 7/24
Outline Motivation 1 Definitions – Existing results 2 Results 3 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria Future Research 4 8/24
Outline Motivation 1 Definitions – Existing results 2 Results 3 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria Future Research 4 9/24
2 × 2 bimatrix games � Study small games. Proposition Let Γ be an arbitrary 2 × 2 bimatrix game. Then 1 at least one player commits to a Nash equilibrium strategy in the game at which he moves first. 2 each follower’s payoff is at least as good as some of his Nash equilibrium payoffs in the simultaneous move game (in all but a technical case). � Upshot: incentive to play the game sequentially with the specified order, not simultaneously. � Necessity of conditions: extension to higher dimensions was not possible. 10/24
Outline Motivation 1 Definitions – Existing results 2 Results 3 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria Future Research 4 11/24
Back to the drawing board � Study specific classes of games. Definition A bimatrix game Γ is weakly unilaterally competitive, if for all x 1 , x 2 ∈ X and all y ∈ Y α ( x 1 , y ) > α ( x 2 , y ) = ⇒ β ( x 1 , y ) ≤ β ( x 2 , y ) α ( x 1 , y ) = α ( x 2 , y ) = ⇒ β ( x 1 , y ) = β ( x 2 , y ) and similarly if for all y 1 , y 2 ∈ Y and all x ∈ X . � Introduced by Kats and Thisse (1992). � Retain the flavor of pure antagonism: any unilateral (cf. pure conflict) deviation, that improves a player’s payoff, incurs a weak loss to opponent’s payoff. 12/24
Weakly unilaterally competitive games � For wuc games, the three concepts coincide. Proposition In a (wuc) game Γ, the leader’s payoff at any subgame perfect equilibrium of the commitment game is equal to his commitment value which is equal to his matrix game value, i.e. v A = α L = α H v B = β L = β H and � Proof: Consequence of the definitions. � Upshot: no controversies on optimal behavior or solution in (wuc) games. Naturally generalize 0-sum games. � Necessity of conditions: property fails in other generalizations. � Property extends to N -player wuc games: v A is different. 13/24
Outline Motivation 1 Definitions – Existing results 2 Results 3 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria Future Research 4 14/24
General bimatrix games I � Equilibria in pure strategies. Theorem (Part I) If the bimatrix game Γ is non-degenerate for player I , and I as a leader commits optimally to a pure strategy, then the resulting strategy profile coincides with a pure Nash equilibrium of Γ. � Upshot: to improve over all his Nash equilibria payoffs, the leader opti- mally commits to a mixed strategy. In line with concealment interpretation, von Neumann & Morgenstern (1953) and Reny & Robson (2004). � Necessity of conditions: Not true for degenerate. 15/24
Sketch of the proof ǫ, i := (1 − ǫ ) · i L + ǫ · i . � Proof: geometric using x L (1 , 0 , 0) (1 , 0 , 0) X (2) X (1) X (2) X (1) (0 , 1 , 0) (0 , 0 , 1) (0 , 1 , 0) (0 , 0 , 1) 16/24
Sketch of the proof ǫ, i := (1 − ǫ ) · i L + ǫ · i . � Proof: geometric using x L � Consider pure strategy i L = (1 , 0 , 0). (1 , 0 , 0) (1 , 0 , 0) X (2) X (1) X (2) X (1) (0 , 1 , 0) (0 , 0 , 1) (0 , 1 , 0) (0 , 0 , 1) 16/24
Sketch of the proof ǫ, i := (1 − ǫ ) · i L + ǫ · i . � Proof: geometric using x L � Consider pure strategy i L = (1 , 0 , 0). (1 , 0 , 0) (1 , 0 , 0) X (2) X (1) X (2) X (1) (0 , 1 , 0) (0 , 0 , 1) (0 , 1 , 0) (0 , 0 , 1) Non-degenerate: x L ǫ, i ∈ X (2) for every i . If I chooses i L , then it must be Nash. 16/24
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