Chapter 1

Game Theory
For Microeconomics at MPP

Chapter 1: Static Games of Complete Information


Normal-form Games

The Prisoner’s Dilemma game
Prisoner 2
Cooperation

Cooperation

－ 1 －－ 1

Deviation

－ 9 － 0

Prisoner 1
Deviation

0 －－ 9

－ 6 －－ 6


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
Lef

Middle

Up
1 － 0

Right

1 － 2

0 － 1

0 － 1

2 － 0

Player 1
Down

0 － 3

R

L

M

T

0 － 4

4 － 0

5 － 3

M

4 － 0

0 － 4

5 － 3

B

3 － 5

3 － 5

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) .


L

R

M

T

0 － 4

4 － 0

5 － 3

M

4 － 0

0 － 4

5 － 3

B

3 － 5

3 － 5

6 － 6

Pat
Opera
Opera

2 － 1

Fight

0 － 0

Chris
Fight

0 － 0

The Battle of the Sexes

1 － 2


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

Hawkish

Dovelike

-1 － -1

2 － 0

0 － 2

1 － 1

Player 1
Dovelike

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

T

Player 1

M

B

3 －－

0 －－

0 －－

3 －－

1 －－

1 －－

A mixed strategy strictly dominates B.

L

M

Player

q

M

T

3 －－

0 －－

M

0 －－

3 －－

B

2 －－

2 －－

1-q

1

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 mixed-strategy 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.