41 pareto ranked coordination g b g 22 00 b 00 11 3

41

Pareto Ranked Coordination
G B
G 2,2 0,0
B 0,0 1,1
3 equilibria: ( B,B ) , ( O,O ) , and (( 1 3 , 2 3 ) , ( 1 3 , 2 3 ))
42

A Special Game
G B
G 1,0 -1,0
B 0,0 0,0
How many equilibria?
43

A Special Game
G B
G 1,0 -1,0
B 0,0 0,0
Infinite equilibria: (( a, 1 - a ) , ( 1 2 , 1 2 )) for any a ∈ [ 0 , 1 ]
44

45

46

48

49

Fact 2 For any game, the set of mixed strategy profiles is convex. For a
finite game, the set of mixed strategy profiles is compact.
Fact 3 The best response correspondence is convex valued. For a finite
game, the best response correspondence is upper hemi-continuous and
compact valued.
Thus, in a nutshell, allowing for mixed strategies guarantees the existence
of Nash equilibria in finite games.
50

Minmax and Interchangeability
Theorem 4 (Minmax Theorem) For every two-person, zero-sum game with
finite pure strategies, there exists a value V and a mixed strategy for each
player, such that (a) Given player 2’s strategy, the best payoff possible for
player 1 is V , and (b) Given player 1’s strategy, the best payoff possible for
player 2 is - V .
Interchangeability. All equilibrium strategies for either given player in a
two-player constant sum game are equivalent and any pair of equilibrium
strategies is an equilibrium:
For a two player, constant-sum game, if ( s
*
1
,s
*
2
) is an equilibrium and
( s
**
1
,s
**
2
) is an equilibrium, then ( s
*
1
,s
**
2
) and ( s
**
1
,s
*
2
) are also equilib-
ria.
51

How Many Equilibria?
The Oddness Theorem
Theorem 5 (Wilson (1971)) Almost all finite games have a finite and odd
number of equilibria.
(“Almost all” means it is not the case for a set of games possessing Lebesqgue
measure zero in R
∏
i ∈ N
Ai
.)
52

The Generic Oddness of Fixed Points
53

The Generic Oddness of Fixed Points
54

Equilibrium Refinements
● Pareto dominance/efficiency
● Symmetry
Consider Battle of the Sexes: symmetric equilibrium is Pareto dominated.
55

Iterated Dominance
p -Beauty Contest Suppose that 3 individuals each simultaneously an-
nounce a number x
i
∈ [ 0 , 100 ] . Player i ’s payoff is
u
i
( a
1
,a
2
,a
3
) =
1 if a
i
-
p
3
∑
3
i = 1
a
i
< min
j ≠ i
a
j
-
p
3
∑
3
i = 1
a
i
0 if a
i
-
p
3
∑
3
i = 1
a
i
> min
j ≠ i
a
j
-
p
3
∑
3
i = 1
a
i
(and the prize is split evenly if more than one individual wins.)
56