Battle of the sexes · Battle of the sexes. 1 0 1 5 opera with her mom with in-laws 0 2 4 0 2 0 0 4...

Strategic Characterization of the Index of an Equilibrium

Arndt von Schemde

Bernhard von Stengel

Department of MathematicsLondon School of Economics

0 2

04

02

0 4

opera

football

opera

he

she football

Battle of the sexes

1

0

1

5

opera withher mom

with in−laws

0 2

04

02

0 4

opera

football

opera

he

she football

Battle of the sexes

1

0

1

5

opera withher mom

with in−laws

0 2

04

02

0 4

opera

football

opera

he

she football

Battle of the sexes

1

0

1

5

opera withher mom

with in−laws

0 2

04

02

0 4

opera

football

opera

he

she football

Battle of the sexes

1

0

1

5

opera withher mom

with in−laws

0 2

04

02

0 4

opera

football

opera

he

she football

Battle of the sexes

Make NE unique by adding strategies

Given: nondegenerate (A,B), Nash equilibrium (NE) (x,y).

Question:Is there a game G extending (A,B) by adding strategies so that (x,y) is the unique NE of G?

e.g.: G obtained from (A,B) by adding columns,(x,y) becoming (x,[y, 0,0,...,0]) for G.

Make NE unique by adding strategies

Given: nondegenerate (A,B), Nash equilibrium (NE) (x,y).

Question:Is there a game G extending (A,B) by adding strategies so that (x,y) is the unique NE of G?

e.g.: G obtained from (A,B) by adding columns,(x,y) becoming (x,[y, 0,0,...,0]) for G.

• can be done for pure-strategy NE

• but not for mixed NE of “battle of the sexes”

Strategic characterization of the index

We will show a conjecture by Josef Hofbauer:

Theorem:

For nondegenerate (A,B), Nash equilibrium (x,y):

index (x,y) = +1

game G extending (A,B) so that (x,y) is the unique equilibrium of G.

for m x n game: G obtained from (A,B) by adding 3m columns.

Sub-matrices of equilibrium supports

Given: nondegenerate (A,B), A>0, B>0,Nash equilibrium (x,y).

A = (aij), B = (bij)

Axy = (aij) isupp(x), jsupp(y) Bxy = (bij) isupp(x), jsupp(y)

Axy , Bxy have full rank |supp(x)|,nonzero determinants.

Index of an equilibrium (Shapley 1974)

Given: nondegenerate game (A,B), A>0, B>0,

Nash equilibrium (x,y).

index (x,y) = (1)|supp(x)|+1 sign det(Axy Bxy)

{ +1, 1 }

Properties of the index

independent of- positive constant added to all payoffs- order of pure strategies- pure strategy payoffs outside equilibrium support

pure-strategy equilibria have index +1

sum of indices over all equilibria is +1

the two endpoints of any Lemke-Howson path areequilibria of opposite index.

3 0 0

0

2 2

0

2

3

1 3

1

2

3

2

Symmetric NE of symmetric games

12

3

1 3

2

3 0 0

0

2 2

0

2

3

1 3

1

2

3

2

Symmetric NE of symmetric games

12

3

1 3

2

3 0 0

0

2 2

0

2

3

1 3

1

2

3

2

Symmetric NE of symmetric games

12

3

1 3

2

3 0 0

0

2 2

0

2

3

1 3

1

2

3

2

Symmetric NE of symmetric games

12

3

1 3

2

−+

−

+

3 0 0

0

2 2

0

2

3

1 3

1

2

3

2

Symmetric NE of symmetric games

12

3

1 3

2

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions1 3 43 2 1B =

Points instead of best−reply regions

New "dual" construction

Given: nondegenerate m × n game (A,B), m ≤ n .

• X subdivided into best response regions

• dualize X: best response regions for j → points j∆

"unplayed strategy" facets of X → large unit vectors

technical construction: "dual polytopes"

• vertices x of regions become simplices x∆

∆Construction of X

3

65

47

12

∆Construction of X

3

65

47

12 x

∆Construction of X

3

65

47

12

∆Construction of X

∆

∆

∆

∆

∆

∆

∆3

65

47

12

∆Construction of X

∆

∆

∆

∆

∆

∆

∆3

65

47

12

∆Construction of X

∆

∆

∆

∆

∆

∆

∆3

65

47

12

∆Construction of X

x∆

∆

∆

∆

∆

∆

∆

∆3

65

47

12 x

Incorporating the other player

So far: X∆ subdivided, dualized according to player II's payoffs B

Now: for each vertex x of a best response region,with labels k > m : best response of player II

or k ≤ m : unplayed strategy of player I

• subdivide x∆ into regions of player I's best responseswhere − if k > m : use column k of A

− if k ≤ m : player I "as if" playing k (artificial unit vector payoff),

⇒ picture X∆ with labels 1...m only, equilibria: all labels.

2∆

4∆

6∆5∆

1∆

7∆

3∆

231

subdivide x∆ via player I’s best responses

2∆

4∆

6∆

4∆ 7∆

5∆

1∆

4/5 1/5 0

1∆

7∆

3∆

21

3

equilibrium iff

subdivide x∆

all labels 1...m

0 10

10 0

8 8

2

3

4 7

8

8

21

1

0

0

40/35

40/35

40/35

1

30/351/354/35

2∆

4∆

6∆

4∆ 7∆

5∆

1∆

4/5 1/5 0

1∆

7∆

3∆

21

3

equilibrium iff

subdivide x∆

all labels 1...m

0 10

10 0

8 8

2

3

4 7

8

8

21

The full dual construction

1

2 1

3

Equilibria have all m labels

1

2 1

3

Index = orientation

+1

−1+1

1

2 1

3

Lemke−Howson paths

1

2 1

3

Opposite index of endpoints

+1

13

12−1

+1

−1+1

1

2 1

3

Completely mixed NE of 3x3 coordination game made unique

1 0 0 0 0 1A = 0 1 0 1 0 0

0 0 1 0 1 0

11 10 10 12 8.9 10B = 10 11 10 10 12 8.9

10 10 11 8.9 10 12

3

1

23

1

2

−

−

−+

++

+

3

1

23

1

2

+213

+213

+213

3

12

+213

3

12

+213

3

12

+213

Summary

Dual construction

Points = payoff columns (replies) of player II

Facet subdivision = player I’s best replies

Visualization and definition of index

Illustration of L-H algorithm

Low dimension m1, for any n

Applications Strategic characterization of index

Components of equilibria, hyperstability