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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

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.



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