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 hemicontinuous 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 twoperson, zerosum 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
twoplayer constant sum game are equivalent and any pair of equilibrium
strategies is an equilibrium:
For a two player, constantsum 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
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 Spring '11
 JohnPaddy
 Game Theory, player, Nash, best response, pure strategies