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

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On the commitment value and commitment optimal strategies in bimatrix games

Stefanos Leonardos1 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.

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Overview

1

Motivation

2

Definitions – Existing results

3

Results 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria

4

Future Research

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Outline

1

Motivation

2

Definitions – Existing results

3

Results 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria

4

Future Research

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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: αL = maxx∈X miny∈Y α (x, y) Majorant game: αF = miny∈Y maxx∈X α (x, y) Simultaneous game: any solution vA must satisfy αL ≤ vA ≤ αF. Minimax theorem: αL = αF in mixed strategies = ⇒ vA unique solution.

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Motivation II

Generalize to 2-person, non 0-sum games: not straightforward. Reason: 3 different points of view Matrix values vA: optimize against the worst possible Equilibrium αN: optimize simultaneously Optimization αL: optimize sequentially In 0-sum games: vA = αN = αL. 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.

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Outline

1

Motivation

2

Definitions – Existing results

3

Results 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria

4

Future Research

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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 [αL, αH]. The lowest leader equilibrium payoff αL is given by αL = max

j∈D max x∈X(j) min k∈E(j) α (x, k)

and the highest leader equilibrium payoff αH is given by αH = max

x∈X

max

j∈BRII(x) α (x, j) = max j∈N max x∈X(j) α (x, j)

If the game is non-degenerate, then αL = αH is the unique commitment value of the leader.

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Existing results II

Von Stengel & Zamir (2010). Example A =

a

b c d e T

2 6 9 1 7

B

8 3 1

• B =

a

b c d e T

4 4 2 4

B

4 4 6 5

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Existing results II

Von Stengel & Zamir (2010). Example A =

a

b c d e T

2 6 9 1 7

B

8 3 1

• B =

a

b c d e T

4 4 2 4

B

4 4 6 5

• Nash equilibria: (many)

Nash payoffs: Player I from 1 to 7, Player II 4

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Existing results II

Von Stengel & Zamir (2010). Example A =

a

b c d e T

2 6 9 1 7

B

8 3 1

• B =

a

b c d e T

4 4 2 4

B

4 4 6 5

• Equivalent strategies: b ∈ E (a)

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Existing results II

Von Stengel & Zamir (2010). Example A =

a

b c d e T

2 6 9 1 7

B

8 3 1

• B =

a

b c d e T

4 4 2 4

B

4 4 6 5

• Equivalent strategies: b ∈ E (a)

Weakly dominated strategy: e by a (or b).

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Existing results II

Von Stengel & Zamir (2010). Example A =

  

a b c d e T

2 6 9 1 7

1 3 T+ 2 3 B

6 2

∗5

1 2.3

B

8 3 1

   B =   

a b c d e T

4 4 2 4

1 3 T+ 2 3 B

4 4

∗4

4 1.3

B

4 4 6 5

  

Equivalent strategies: b ∈ E (a) Weakly dominated strategy: e by a (or b).

• xL, jF

=

• 1

3, 2 3

• , c
• with αL = 5, βF = 4.

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Existing results II

Von Stengel & Zamir (2010). Example A =

a

b c d e T

2 6 9 1 7

B

8 3 1

• B =

a

b c d e T

4 4 2 4

B

4 4 6 5

• Equivalent strategies: b ∈ E (a)

Weakly dominated strategy: e by a (or b).

• xL, jF

=

• 1

3, 2 3

• , c
• with αL = 5, βF = 4.
• xH, jF

= (T, e) with αH = 7.

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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 ≤ αL and h ≤ αH. So, in degenerate games with αL < αH vA ≤ ℓ ≤ αL, vA ≤ ℓ ≤ h ≤ αH which for non-degenerate games simplifies to vA ≤ ℓ ≤ h ≤ αL = αH. Lower and upper bounds for Nash equilibria payoffs. Based on these results, characterize the bimatrix games for which vA = αL = αH: accept Nash (e.g. 0-sum games) vA = h < αL: question Nash (e.g. Traveler’s Dilemma)

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Outline

1

Motivation

2

Definitions – Existing results

3

Results 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria

4

Future Research

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Outline

1

Motivation

2

Definitions – Existing results

3

Results 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria

4

Future Research

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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.

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Outline

1

Motivation

2

Definitions – Existing results

3

Results 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria

4

Future Research

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Back to the drawing board

Study specific classes of games. Definition A bimatrix game Γ is weakly unilaterally competitive, if for all x1, x2 ∈ X and all y ∈ Y α (x1, y) > α (x2, y) = ⇒ β (x1, y) ≤ β (x2, y) α (x1, y) = α (x2, y) = ⇒ β (x1, y) = β (x2, y) and similarly if for all y1, y2 ∈ 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.

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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

• f the commitment game is equal to his commitment value which is equal

to his matrix game value, i.e. vA = αL = αH and vB = βL = βH 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: vA is different.

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Outline

1

Motivation

2

Definitions – Existing results

3

Results 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria

4

Future Research

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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.

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Sketch of the proof

Proof: geometric using xL

ǫ,i := (1 − ǫ) · iL + ǫ · i.

(1, 0, 0) (0, 1, 0) (0, 0, 1) X (2) X (1) (1, 0, 0) (0, 1, 0) (0, 0, 1) X (2) X (1)

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Sketch of the proof

Proof: geometric using xL

ǫ,i := (1 − ǫ) · iL + ǫ · i.

Consider pure strategy iL = (1, 0, 0).

(1, 0, 0) (0, 1, 0) (0, 0, 1) X (2) X (1) (1, 0, 0) (0, 1, 0) (0, 0, 1) X (2) X (1)

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Sketch of the proof

Proof: geometric using xL

ǫ,i := (1 − ǫ) · iL + ǫ · i.

Consider pure strategy iL = (1, 0, 0).

(1, 0, 0) (0, 1, 0) (0, 0, 1) X (2) X (1)

Non-degenerate: xL

ǫ,i ∈ X (2) for every i.

If I chooses iL, then it must be Nash.

(1, 0, 0) (0, 1, 0) (0, 0, 1) X (2) X (1)

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Sketch of the proof

Proof: geometric using xL

ǫ,i := (1 − ǫ) · iL + ǫ · i.

Consider pure strategy iL = (1, 0, 0).

(1, 0, 0) (0, 1, 0) (0, 0, 1) X (2) X (1)

Non-degenerate: xL

ǫ,i ∈ X (2) for every i.

If I chooses iL, then it must be Nash.

(1, 0, 0) (0, 1, 0) (0, 0, 1) X (2) X (1)

Degenerate: moving towards i = (0, 1, 0) and i′ = (0, 0, 1) changes the objective function.

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General bimatrix games II

Equilibria in completely mixed strategies. Theorem (Part II) Let Γ be a non-degenerate bimatrix game with a completely mixed Nash equilibrium (xN, yN), such that xN is not a maxmin strategy. Then player I may strictly improve over this Nash equilibrium payoff as a leader. Proof: constructive using a unilateral deviation

• xN, j1
• .

Upshot: commitment ensures coordination in equilibrium at no cost, since this strategy profile Pareto-dominates the completely mixed Nash equilib- rium: α

• xN, j1
• > aN,

β

• xN, j1
• = bN

Necessity of conditions: not true if xN maximin (not restrictive).

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Outline

1

Motivation

2

Definitions – Existing results

3

Results 2 × 2 bimatrix games Weakly unilaterally competitive games General bimatrix games: pure & completely mixed equilibria

4

Future Research

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Future Research

Some useful intuitions, still initial questions remain open. Are there other auxiliary games that work in the same manner – i.e. provide bounds for the solution – in more general classes of games? Difficulty: results without restricted applicability. Predict the order of play through the players’ beliefs: if every player believes that some player is a leader (e.g. negotiations), then he may act as such without . Difficulty: formalize the proper epistemic framework.

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Future Research

Some useful intuitions, still initial questions remain open. Are there other auxiliary games that work in the same manner – i.e. provide bounds for the solution – in more general classes of games? Difficulty: results without restricted applicability. Predict the order of play through the players’ beliefs: if every player believes that some player is a leader (e.g. negotiations), then he may act as such without . Difficulty: formalize the proper epistemic framework. Many other: n ≥ 2 followers, bimatrix games (again), non bimatrix- games . . .

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About

Journal: International Game Theory Review, forthcoming. Available online at: https://arxiv.org/abs/1612.08888. Conferences: 28th International Conference on Game Theory at Stony Brook (2017), USA, contributed talk. 9th Israeli chapter of the Game Theory Society (2017), Technion, Israel. South Dakota State University, (2017), USA, invited talk. National & Kapodistrian University of Athens, (2017), seminar talk.

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Selected References I

[1] Rudolf Avenhaus, Akira Okada and Shmuel Zamir, Inspector Leadership with In- complete Information. Game Equilibrium Models IV: Social and Political Interaction, 319–361, Springer Berlin Heidelberg, 1991. [2] Tamer Basar and Geert Jan Olsder, Dynamic Noncooperative Game Theory, 2nd ed. rev., SIAM Classics in Applied Mathematics 23,. Society for Industrial and Applied Mathematics, Philadelphia, 1999. [3] Kaushik Basu. The traveler’s dilemma: Paradoxes of rationality in game theory. The American Economic Review, 84(2): 391–395, 1994. URL http://www.jstor.org/stable/2117865. [4] Jean-Pierre Beaud. Antagonistic games. Mathmatiques et sciences humaines [En ligne], 40(157):5–26, 2002. doi:10.4000/msh.2850. [5] Vincent Conitzer. On Stackelberg mixed strategies. Synthese, 193(3):689–703, 2016. doi:10.1007/s11229- 015-0927-6. [6] Claude D’Aspremont and L.-A Gérard-Varet. Stackelberg–solvable games and pre-play communication. Journal of Economic Theory, 23(2):201–217, 1980. doi:10.1016/0022-0531(80)90006-X.

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Selected References II

[7] Joseph Y. Halpern and Rafael Pass. Iterated regret minimization: A new solution concept. Games and Economic Behavior, 74(1):184–207, 2012. doi:10.1016/j.geb.2011.05.012. [8] Jonathan H. Hamilton and Steven M. Slutsky. Endogenizing the order of moves in matrix games. Theory and Decision, 34(1):47–62, 1993. doi:10.1007/BF01076104. [9] Amoz Kats and Jacques Francois Thisse. Unilaterally competitive games. Interna- tional Journal of Game Theory, 21(3):291–299, 1992. doi:10.1007/BF01258280. [10] Philip J. Reny and Arthur J. Robson. Reinterpreting mixed strategy equilibria: a unifi- cation of the classical and Bayesian views. Games and Economic Behavior, 48(2):355– 384, 2004. doi:10.1016/j.geb.2003.09.009. [11] Robert W. Rosenthal. A note on robustness of equilibria with respect to com- mitment opportunities. Games and Economic Behavior, 3(2):237–243, 1991. doi:10.1016/0899-8256(91)90024-9. [12] Eric van Damme and Sjaak Hurkens. Commitment robust equilibria and endogenous timing. Games and Economic Behavior, 15(2):290–311, 1996. doi:10.1006/game.1996.0069.

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Selected References III

[13] John von Neumann and Oskar Morgenstern. Theory of Games and Economic Be- havior, 3rd ed. Princeton University Press, Princeton N.J., 1953. [14] Igal Milchtaich, Crowding games are sequentially solvable. International Journal of Game Theory, 27:501–509, 1998. [15] Bernhard von Stengel. Recursive inspection games. Mathematics of Operations Re- search, 41(3):935–952, 2016. doi:10.1287/moor.2015.0762. [16] Bernhard von Stengel and Shmuel Zamir. Leadership with commitment to mixed

• strategies. Research report LSE-CDAM-2004-01, London School of Economics, 2004.

URL http://www.cdam.lse.ac.uk/ Reports/Files/cdam-2004-01.pdf. [17] Bernhard von Stengel and Shmuel Zamir. Leadership games with con- vex strategy sets. Games and Economic Behavior, 69(2):446–457, 2010. doi:10.1016/j.geb.2009.11.008.

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Source

Online at: https://arxiv.org/abs/1612.08888.

Thank you for your attention!