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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.