On the commitment value and commitment optimal strategies in - PowerPoint PPT Presentation

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. 1/42

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 Appendix 5 Traveler’s dilemma A simple game, or not? 2/42

Outline - section 1 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 Appendix 5 Traveler’s dilemma A simple game, or not? 3/42

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. 4/42

Motivation II � Generalize to 2-person, non 0-sum games: not straightforward. � Reason: 3 different points of view Matrix values v A : players optimize against the worst possible Equilibrium α N : players optimize simultaneously Optimization α L : players 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 strategies & payoffs commitment values & optimal strategies in minorant/majorant games 5/42

Objective � Minorant/majorant games in non zero-sum games: factors to determine order of play for the players: open? optimizing against non-unique best responses: SPNE ( ε -argument) � Question: Suppose we do not really know whether a game is going to be played sequentially or simultaneously. When would a prediction be never- theless Nash? � Main goal: Examine a solution of non 0-sum games originating from the optimization – vs the equilibrium – point of view. 6/42

Outline - section 2 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 Appendix 5 Traveler’s dilemma A simple game, or not? 7/42

Notation � We consider the mixed extension Γ of an m × n bimatrix game ( A , B ) played by players I and II 1 Pure strategy sets: ( M , N ). Mixed strategy sets ( X = P m , Y = P n ). 2 Payoff functions: α ( x , y ) := x T Ay , β ( x , y ) := x T By . 3 Value of matrix A for I : v A = max x ∈ X min j ∈ N α ( x , j ). 4 Best reply regions: X ( j ) := { x ∈ X : β ( x , j ) ≥ β ( x , j ′ ) } . Full-dimensional: D := { X ( j ) : X o ( j ) � = ∅} . 5 Nash equilibria: x N ∈ NE ( X ) , y N ∈ NE ( Y ) with payoffs � α N , β N � . � A bimatrix game is 1 non-degenerate for player i: if no mixed strategy of i has more pure best replies among the strategies of player j than the size of its support. 2 non-degenerate: condition holds for both i = 1 , 2. 8/42

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. 9/42

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 9/42

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 ) 9/42

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 ). 9/42

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 (1 / 3) T (1 / 3) T A = B = 8 0 3 1 0 4 4 6 5 0 (2 / 3) B (2 / 3) B Equivalent strategies: b ∈ E ( a ) Weakly dominated strategy: e by a (or b ). � x L , j F � �� � � with α L = 5. 3 , 2 1 = , c 3 9/42

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. 1 3 , 2 = , c 3 � x H , j F � = ( T , e ) with α H = 7. 9/42

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) 9/42

Outline - section 3 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 Appendix 5 Traveler’s dilemma A simple game, or not? 10/42

Outline - subsection 3.1 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 Appendix 5 Traveler’s dilemma A simple game, or not? 11/42

2 × 2 bimatrix games � Study small games Proposition In every 2 × 2 bimatrix game � at least one player commits to a Nash equilibrium strategy in the game at which this player moves first. � there exists an equilibrium payoff β F of the follower that is lower than his unique Nash equilibrium payoff in Γ iff ( ℓ 1) the leader has a strongly dominated strategy and ( ℓ 2) an equalizing strategy x d = (1 − d , d ) over the follower’s payoffs, � � ≥ α N for some j ∈ N . x d , j such that α If ( ℓ 2) holds with strict inequality, then this β F is the unique equilibrium payoff of the follower. � Necessity of conditions: extension to higher dimensions was not possible. 12/42

Outline - subsection 3.2 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 Appendix 5 Traveler’s dilemma A simple game, or not? 13/42

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 ) ⇒ β ( x 1 , y ) = β ( x 2 , y ) α ( x 1 , y ) = α ( 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. 14/42

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. � Implication: 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. 14/42

Outline - subsection 3.3 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 Appendix 5 Traveler’s dilemma A simple game, or not? 15/42