Embed Size (px)
Citation preview
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
