Game Theory
For Microeconomics at MPP
Chapter 1: Static Games of Complete
Information
Normal-form Games
The Prisoner’s Dilemma game
Prisoner 2
Cooperation
Deviation
Cooperation
-1 ,-1
-9 ,0
Deviation
0 , -9
-6 ,-6
Prisoner 1
Definition The normal-form representation of an
n-player game specifies the player’s strategy spaces,
S1,…,Sn and their payoff functions u1,…,un. We denote
this game by G={ S1, …,Sn ; u1,…,un }.
Strictly Dominated Strategies
Definition Strategy s’i is strictly dominated by
strategy s’’i if for each combination of other player’s
strategies, i’s payoff from playing s’i is strictly less
than i’s payoff from playing s’’i .
Iterated Elimination of Strictly
Dominated Strategies
Player 2
Up
Player 1
Down
Left
Middle
Right
1,0
1,2
0,1
0,3
0,1
2,0
T
L
0 ,4
M
4 ,0
B
3 ,5
M
R
4 ,0
5 ,3
0 ,4
3 ,5
5 ,3
6 ,6
Nash Equilibrium
Definition In the normal-form game G={ S1, …,Sn ;
u1,…,un }, the strategies (s*1,s*2,…,s*n) are Nash
equilibrium if for each player i, s*i is player i’s best
response to the strategies specified for the n-1
players, (s*1,…,s*i-1,s*i+1,…,s*n) .
T
L
0 ,4
M
4 ,0
R
5 ,3
M
4 ,0
0 ,4
5 ,3
B
3 ,5
3 ,5
6 ,6
Pat
Opera
Fight
Opera
2 ,1
0 ,0
Fight
0 ,0
1 ,2
Chris
The Battle of the Sexes
Mixed Strategies and
Existence of Equilibrium
Player 2
Heads
Tails
Heads
-1 ,1
1 ,-1
Tails
1 ,-1
-1 ,1
Matching Pennies
Player 2
Hawkish
Dovelike
Hawkish
-1 ,-1
2 ,0
Dovelike
0 ,2
1 ,1
Player 1
Hawk- Dove game
Definition In the normal-form game G={ S1, …,Sn ;
u1,…,un }, suppose Si={si1,…,siK}. Then a mixed strategy
for player i is a probability distribution pi=(pi1,…piK),
where 0≤piK≤1 for k=1,…,K and pi1+・・・+piK=1.
Player 2
Player 2
L
L q
M
T
3 ,-
0 ,-
Player 1 M
0 ,-
3 ,-
B
1 ,-
1 ,-
A mixed strategy strictly dominates B.
T
Player M
1
B
M 1-q
3 ,-
0 ,-
0 ,-
3 ,-
2 ,-
2 ,-
B is best response for player 1 to some
mixed strategy of 2, (q,1-q).
Existence of Nash Equilibrium
Theorem (Nash (1950)) In the n-player normal-form
game G={ S1, …,Sn ; u1,…,un }, if n is finite and Si is
finite for every i, then there exists at least one Nash
equilibrium, possibly involving mixed strategies.
For any strategic (or social) situation, there is at
least one equilibrium.
However, multiple equilibria are probable.
A useful property of mixedstrategy Nash equilibria
Given a mixed-strategy pi, the support of pi is the set
{sij ∈ Si | pij>0}, i.e., the set of strategies assigned with
positive probability.
Each strategy in the support of a mixed Nash
equilibrium strategy earns the same payoff for
the other players’ mixed Nash equilibrium strategy.