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

5

Appendix Traveler’s dilemma A simple game, or not?

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Outline - section 1

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

5

Appendix Traveler’s dilemma A simple game, or not?

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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: players optimize against the worst possible Equilibrium αN: players optimize simultaneously Optimization αL: players 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 strategies & payoffs commitment values & optimal strategies in minorant/majorant games

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Objective

Minorant/majorant games in non zero-sum games: factors to determine order of play for the players: open?

• ptimizing 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

• ptimization – vs the equilibrium – point of view.

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Outline - section 2

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

5

Appendix Traveler’s dilemma A simple game, or not?

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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 = Pm, Y = Pn). 2 Payoff functions: α (x, y) := xTAy, β (x, y) := xTBy. 3 Value of matrix A for I: vA = maxx∈X minj∈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: xN ∈ NE (X) , yN ∈ NE (Y ) with payoffs

• αN, βN

. 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.

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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

• 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 (1/3)T

2 6 9 1 7

(2/3)B

8 3 1

• B =

a

b c d e (1/3)T

4 4 2 4

(2/3)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.

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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.
• 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 - section 3

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

5

Appendix Traveler’s dilemma A simple game, or not?

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Outline - subsection 3.1

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

5

Appendix Traveler’s dilemma A simple game, or not?

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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 βF of the follower that is lower than his unique Nash equilibrium payoff in Γ iff (ℓ1) the leader has a strongly dominated strategy and (ℓ2) an equalizing strategy xd = (1 − d, d) over the follower’s payoffs, such that α

• xd, j
• ≥ αN for some j ∈ N.

If (ℓ2) holds with strict inequality, then this βF is the unique equilibrium payoff of the follower. Necessity of conditions: extension to higher dimensions was not possible.

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Outline - subsection 3.2

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

5

Appendix Traveler’s dilemma A simple game, or not?

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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. 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: vA is different.

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Outline - subsection 3.3

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

5

Appendix Traveler’s dilemma A simple game, or not?

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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. Proof: geometric using xL

ǫ,i := (1 − ǫ) · iL + ǫ · i.

Implication: Necessary condition for the leader to improve over all his Nash equilibria payoffs: he chooses to commit optimally only to mixed

• strategies. In line with concealment interpretation, von Neumann & Mor-

genstern (1953) and Reny & Robson (2004). Necessity of conditions: Not true for degenerate.

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Example: Theorem I

Sketch of the proof: Example (Non-degenerate 3 × 2) A =

  

t1 t2 s1

2

s2

3

s3

3

  

B =

  

t1 t2 s1

0.9 1

s2

3

s3

3

  

Nash equilibrium:

• xN, yN

=

• 0, 1

2, 1 2

• ,
• 1

2, 1 2

• with αN = βN = 1.5.
• xL, jF

=? Theorem I implies that xL must be mixed.

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Example: Theorem I

Sketch of the proof: Example (Non-degenerate 3 × 2) A =

  

t1 t2 s1

2

s2

3

s3

3

  

B =

  

t1 t2 s1

0.9 1

s2

3

s3

3

  

Nash equilibrium:

• xN, yN

=

• 0, 1

2, 1 2

• ,
• 1

2, 1 2

• with αN = βN = 1.5.
• xL, jF

=? Theorem I implies that xL must be mixed.

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Example: Theorem I

Sketch of the proof: Example (Non-degenerate 3 × 2) A =

  

t1 t2 s1

2

s2

3

s3

3

  

B =

  

t1 t2 s1

0.9 1

s2

3

s3

3

  

Nash equilibrium:

• xN, yN

=

• 0, 1

2, 1 2

• ,
• 1

2, 1 2

• with αN = βN = 1.5.
• xL, jF

=? Theorem I implies that xL must be mixed.

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Example: Theorem I

Sketch of the proof: Example (Non-degenerate 3 × 2) A =

  

t1 t2 s1

2

s2

3

s3

3

  

B =

  

t1 t2 s1

0.9 1

s2

3

s3

3

  

Nash equilibrium:

• xN, yN

=

• 0, 1

2, 1 2

• ,
• 1

2, 1 2

• with αN = βN = 1.5.
• xL, jF

=? Theorem I implies that xL must be mixed.

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Example: Theorem I

Set of mixed strategies of player I: simplex X = P3

(1, 0, 0) (0, 1, 0) (0, 0, 1)

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Example: Theorem I

Best reply regions in the simplex X = P3

(1, 0, 0) (0, 1, 0) (0, 0, 1)

• 0, 1

2, 1 2

• X (2)

X (1)

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Example: Theorem I

Best reply regions in the simplex X = P3

(1, 0, 0) (0, 1, 0) (0, 0, 1)

• 0, 1

2, 1 2

• X (2)

X (1) 1.5 1.5 2 +

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Example: Theorem I

Best reply regions in the simplex X = P3

(1, 0, 0) (0, 1, 0) (0, 0, 1)

• 0, 1

2, 1 2

• X (2)

X (1) xL =

• 30

31, 0, 1 31

• vs t2

1.5 1.5 2 +

• xL, jF

=

• 30

31, 0, 1 31

• , t2
• with αL = 2 1

31.

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Example: Theorem I

Sketch of the proof: Example (Non-degenerate 3 × 2) A =

  

t1 t2

30 31 s1

2

s2

3

1 31 s3

3

  

B =

  

t1 t2

30 31 s1

0.9 1

s2

3

1 31 s3

3

  

• xL, jF

=

• 30

31, 0, 1 31

• , t2
• with αL = 2 1

31.

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Counterexample: degenerate bimatrix game I

Theorem I fails in degenerate bimatrix games, even if all best reply regions are full-dimensional. Example (Degenerate 3 × 2) A =

  

t1 t2 s1

2

s2

3

s3

3

  

B =

  

t1 t2 s1

1 1

s2

3

s3

3

  

BRII (s1) = conv{t1, t2}: degenerate for player I. Nash equilibrium:

• xN, yN

=

• 0, 1

2, 1 2

• ,
• 1

2, 1 2

• with αN = βN = 1.5.
• xL, jF

= (s1, t2) with αL = 2: not a pure strategy equilibrium of Γ.

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Counterexample: degenerate bimatrix game II

Best reply regions in the simplex X = P3

(0, 1, 0) (0, 0, 1) (1, 0, 0)

• 0, 1

2, 1 2

• X (2)

X (1) xL vs

• jF = t2
• 1.5

1.5 2 0

Theorem I fails, because: xL

ǫ,3 = (1 − ǫ) s1 + ǫs3 /

∈ X (5)

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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 his Nash equilibrium payoff, as a leader. i.e. αL > αN. Proof: constructive using a unilateral deviation

• xN, j1
• .

Implication: the strategy profile

• xN, j1
• Pareto-dominates the com-

pletely mixed Nash equilibrium: α

• xN, j1
• > aN and β
• xN, j1
• = bN.

Hence: commitment ensures coordination in equilibrium at no cost. Necessity of conditions: for xN maximin, improvement not possible.

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Outline - section 4

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

5

Appendix Traveler’s dilemma A simple game, or not?

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Future Research

Some useful intuitions, still: initial questions remain open. Directions for future research may be The leader-follower games are the means to place each player in an advantageous and a disadvantageous – compared to the simultaneous move game – situtation in 0-sum games. Can one find other auxiliary games that work in the same manner – i.e. provide bounds for the solution – in other classes of games? Difficulty: results without restricted applicability. Back to our initial question: we do not really know whether a game is going to be played sequentially or simultaneously. When would a pre- diction be/not be Nash? Approach with epistemic models: determine the order of play by the beliefs of the players. Difficulty: formalize the proper epistemic framework.

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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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Outline - section 5

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

5

Appendix Traveler’s dilemma A simple game, or not?

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Outline - subsection 5.1

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

5

Appendix Traveler’s dilemma A simple game, or not?

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Traveler’s Dilemma

• K. Basu (1994): Airline damages 2 identical antiques of 2 travelers.

Manager devises a scheme: each traveler writes down the price of the antique as dollar between 2 and 100 without conferring together

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Traveler’s Dilemma

• K. Basu (1994): Airline damages 2 identical antiques of 2 travelers.

Manager devises a scheme: each traveler writes down the price of the antique as dollar between 2 and 100 without conferring together if both write the same price: true price, both get that amount. But

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Traveler’s Dilemma

• K. Basu (1994): Airline damages 2 identical antiques of 2 travelers.

Manager devises a scheme: each traveler writes down the price of the antique as dollar between 2 and 100 without conferring together if both write the same price: true price, both get that amount. But if they write different prices: both get paid the lower price, and the person with lower number will get +$2 as a reward, the one with the higher number will get −$2 as a punishment.

28/42

Traveler’s Dilemma

• K. Basu (1994): Airline damages 2 identical antiques of 2 travelers.

Manager devises a scheme: each traveler writes down the price of the antique as dollar between 2 and 100 without conferring together if both write the same price: true price, both get that amount. But if they write different prices: both get paid the lower price, and the person with lower number will get +$2 as a reward, the one with the higher number will get −$2 as a punishment. E.g., if Lucy writes $46 and Pete writes $100, Lucy will get $48 and Pete will get $44.

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Traveler’s Dilemma

• K. Basu (1994): Airline damages 2 identical antiques of 2 travelers.

Manager devises a scheme: each traveler writes down the price of the antique as dollar between 2 and 100 without conferring together if both write the same price: true price, both get that amount. But if they write different prices: both get paid the lower price, and the person with lower number will get +$2 as a reward, the one with the higher number will get −$2 as a punishment. E.g., if Lucy writes $46 and Pete writes $100, Lucy will get $48 and Pete will get $44. Resembles Bertrand duopoly.

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Traveler’s Dilemma

• K. Basu (1994): Airline damages 2 identical antiques of 2 travelers.

Manager devises a scheme: each traveler writes down the price of the antique as dollar between 2 and 100 without conferring together if both write the same price: true price, both get that amount. But if they write different prices: both get paid the lower price, and the person with lower number will get +$2 as a reward, the one with the higher number will get −$2 as a punishment. E.g., if Lucy writes $46 and Pete writes $100, Lucy will get $48 and Pete will get $44. Resembles Bertrand duopoly. Rationality leads to a difficult to accept Nash equilibrium solution.

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Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

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Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

Can be solved with iterated deletion of strongly dominated strategies.

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Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

Can be solved with iterated deletion of strongly dominated strategies. First step: strategy (5) is strongly dominated for player I (and II).

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Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

Can be solved with iterated deletion of strongly dominated strategies. First step: strategy (5) is strongly dominated for player I (and II).

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Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

30/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

30/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

30/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

30/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

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Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

But then: strategy (4) is strongly dominated for both players.

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Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

31/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

31/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

Now strategy (3) is strongly dominated for both players.

32/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

32/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

32/42

Traveler’s Dilemma

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

33/42

Traveler’s Dilemma: Nash Equilibrium

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

33/42

Traveler’s Dilemma: Nash Equilibrium

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

Strategy profile (2, 2) is the only Nash equilibrium.

33/42

Traveler’s Dilemma: Nash Equilibrium

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (4 × 4 Traveler’s Dilemma) A =

    

(2) (3) (4) (5) (2) 2 4 4 4 (3) 3 5 5 (4) 1 4 6 (5) 1 2 5

    

B =

    

(2) (3) (4) (5) (2) 2 (3) 4 3 1 1 (4) 4 5 4 2 (5) 4 5 6 5

    

Strategy profile (2, 2) is the only Nash equilibrium. Equilibrium payoffs are equal to safety levels, vA = h: not convincing as a solution.

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Traveler’s Dilemma: Commitment

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (Traveler’s Dilemma (original))

A=

       

(2) (3) . . . (99) (100) (2) 2 4 . . . 4 4 (3) 3 . . . 5 5 . . . . . . . . . ... . . . . . . (99) 1 . . . 99 101 (100) 1 . . . 97 100

       

, B=

       

(2) (3) . . . (99) (100) (2) 2 . . . (3) 4 3 . . . 1 1 . . . . . . . . . ... . . . . . . (99) 4 5 . . . 99 97 (100) 4 5 . . . 101 100

       

34/42

Traveler’s Dilemma: Commitment

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (Traveler’s Dilemma (original))

A=

       

(2) (3) . . . (99) (100) (2) 2 4 . . . 4 4 (3) 3 . . . 5 5 . . . . . . . . . ... . . . . . . (99) 1 . . . 99 101 (100) 1 . . . 97 100

       

, B=

       

(2) (3) . . . (99) (100) (2) 2 . . . (3) 4 3 . . . 1 1 . . . . . . . . . ... . . . . . . (99) 4 5 . . . 99 97 (100) 4 5 . . . 101 100

       

Commitment in pure strategies: (100) vs (99) with payoffs (97, 101).

34/42

Traveler’s Dilemma: Commitment

• K. Basu (1994): symmetric bimatrix game with payoff matrices

Example (Traveler’s Dilemma (original))

A=

       

(2) (3) . . . (99) (100) (2) 2 4 . . . 4 4 (3) 3 . . . 5 5 . . . . . . . . . ... . . . . . . (99) 1 . . . 99 101 (100) 1 . . . 97 100

       

, B=

       

(2) (3) . . . (99) (100) (2) 2 . . . (3) 4 3 . . . 1 1 . . . . . . . . . ... . . . . . . (99) 4 5 . . . 99 97 (100) 4 5 . . . 101 100

       

Commitment in pure strategies: (100) vs (99) with payoffs (97, 101). Actually: commitment optimal strategy and value for the leader xL = 1 3 (100) + 1 3 (99) + 1 3 (97) , jF = (99) with payoffs αL = βF = 981

3.

35/42

Traveler’s Dilemma: Properties

Can be solved with the process of iterated elimination of strongly domi- nated strategies Strategy profile (2, 2) unique:

1 Nash equilibrium with payoffs (2, 2): survives refinements 2 correlated equilibrium 3 rationalizable profile

However, does not match experimental, intuitive or socially optimal be- havior.

• K. Basu: People consistently reject the rational choice: by acting irra-

tionally, they end up with a larger reward – an outcome that demands a new kind of formal reasoning. Leader-follower approach as solution concept works well for TrD.

36/42

Traveler’s Dilemma: Revised I

Commitment optimal strategy of the leader (and optimal response of the follower) xL = 1 3 (100) + 1 3 (99) + 1 3 (97) , jF = (99) with payoffs αL = βF = 981

3.

Leader-follower approach as solution concept works well for TrD

1 symmetry preserved: same payoffs for both players, irrespective of order 2 payoffs close to the Pareto-optimal outcome.

Implication: if the rules of the game permit commitment, one expects the leader-follower equilibrium to prevail over the Nash equilibrium of the simultaneous move game.

37/42

Outline - subsection 5.2

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

5

Appendix Traveler’s dilemma A simple game, or not?

38/42

A non zero-sum game I

Story: Firm A chooses the product and firm B chooses the location. Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Firm A prefers product s2, but first and foremost prefers location t1.

Truthful representation of Firm A’s preferences undermines cooperation. Elemination of strongly dominated strategies does not make sense as a solution here.

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1 2 s2 2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN = α

s2, t2 = 2

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN = α

s2, t2 = 2

Commitment value (pure): αL

p

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN = α

s2, t2 = 2

Commitment value (pure): αL

p

38/42

A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

• t1

t2 s1

8

s2

10 2

• ,

B =

• t1

t2 s1

2

s2

2

• Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2

Nash equilibrium: αN = α

s2, t2 = 2

Commitment value (pure): αL

p = α

s1, t1 = 8.

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A non zero-sum game I

Compare vA, αN and αL in Example (Non-degenerate 2 × 2) A =

  

t1 t2 s1

8

0.5s1+0.5s2

9 1

s2

10 2

  ,

B =

  

t1 t2 s1

2

0.5s1+0.5s2

1 1

s2

2

  

Safety level: vA = maxs mint α (s, t) = maxs {0, 2} = 2 Nash equilibrium: αN = α

s2, t2 = 2

Commitment value (pure): αL

p = α

s1, t1 = 8.

Commitment value (mixed): αL = α

(0.5, 0.5) , t1 = 9.

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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!