GAME THEORY

Game Theory

• What is Game Theory?• What is strategic environment?• What are the main assumptions in Game

Theory?• How are games represented?• What are the elements of Games ?

Game representations

Extensive form

Pepsi

1, 2

3, 4

Coca ColaUp

Down

Up

Down

5, 6

7, 8

Coca Cola

Up

Down

Matrix form (aka normal formaka strategic form)

player 1’sstrategy

player 2’s strategy

1, 2Up

Down

Up, Up Up, Down

3, 4

5, 6 7, 8

Down, Up

D, D

3, 41, 2

5, 6 7, 8

Dominated Strategy

• What is a dominated strategy?

Rules of the Game

• The strategic environment– Players– Strategies– Payoffs

• The assumptions– Rationality– Common knowledge

• The rules– Timing of moves– Informational conditions

Formal definition

Definitions

• Let Ai be the set of actions available for player I

• a = (a1, a2, …, ai,…) be an action profile: An action for each player in the game.

• write (ai', a-i) if i chooses ai', other players according to a

• ui (ai

, a-i ): payoff for player i from playing action ai and others playing a-i

Prisoner’s Dilemma

Prisoner’s DilemmaNot Confess Confess

Not Confess -2, -2 -5, -1

Confess -1, -5 -3,-3

Comments

• Simultaneous actions does not imply taking actions at the same time.

• Rationality implies knowing the structure of the game.

• It does not mean that there is coordination on beliefs / cooperation

• Rationality is an assumption not necessary in reality.

Comments

• (Not Confess, Not Confess) is a pareto optimal outcome

• This is common knowledge• Will Not Confess be chosen by both?• Depends upon their beliefs about the actions

of other players

Prisoner’s DilemmaNot Confess Confess

Not Confess -2, -2 -5, -1

Confess -1, -5 -3,-3

NASH EQUILIBRIUM

Nash Equilibrium

Nash Equilibrium:• A set of strategies, one for each player, such

that each player’s strategy is a best response to others’ strategies

Everybody is playing a best response• No incentive to unilaterally change my strategy

• a* =( a* 1 , a* 2 , ……) = =( a* i , a* -i) is a Nash equilibrium if for every player i and every action, ai that is element of Ai: ui(a*) ≥ ui (ai , a-i*)

where ui is the payoff function representing thepreferences of player I

Nash Equilibrium

L C

R

U 8, 3 0, 4 4,4

M 8,5 1,5 5,3

D 3,7 0,1 2,0

L R

L 1,-1 -1, 1

R -1, 1 1,-1

Coordination Games

Opera Movie

Opera 2,1 0,0

Movie 0,0 1,2

SUV Price WarsDiscount No Discount

Discount 3,3 8, 0

No Discount 0, 8 5,5

Games of Chicken

• Entry into small markets

Firm 2Stay Swerve

Firm 1Stay -50 , -50 100 , 0

Swerve 0 , 100 50 , 50

Stag Hunt Game

• Each can individually hunt a stag or hunt a hare together.

• You can hunt a stag alone.• You can only hunt a hare when you hunt it

together.• Hunting a hare alone means no dinner!

Stag Hunt GameHare Stag

Hare 3,3 -1, 0

Stag 0, -1 0,0

Games of Assurance

Joint research ventures– Each firm may invest $50,000 into an R&D project– Project succeeds only if both invest– If successful, each nets $75,000– Stag Hunt Firm 2

$50K $0

Firm 1$50K 75 , 75 -50 , 0

$0 0 , -50 0 , 0

Nash Equilibrium

• May or may not have to exist in pure strategies.

• Can be multiple in a single game.