Game Theory

1
Game Theory

• The study of rational behavior among
  interdependent agents
• Common interest to make the pie as large as
  possible, but
• Competing interests to maximize their own share
  of the pie.
• An agents rational decisions require
  anticipating rivals responses
• Uncertainty is a necessary feature of games

2
Prisoners Dilemma

• During the Stalinist Era
• Russian Conductor studying Tchaikovsky score on
  the train to Minsk
• Arrested by the KGB
• Thrown into prison
• For three days, he is told nothing
• .Then

3
Prisoners Dilemma

• We have your friend Tchaikovsky and he is
  starting to talk
• Should the conductor confess?

4
Prisoners Dilemma Payoff Matrix (C, T)
Tchaikovsky
Confess Dont Confess
Conductor
( -8, -8)
( 0, -15)
Confess
( -15, 0)
Dont Confess
( -1, -1)
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Prisoners Dilemma Suppose Tchaikovsky confesses
Tchaikovsky
Confess Dont Confess
Conductor
( -8, -8)
( 0, -15)
Confess
( -15, 0)
Dont Confess
( -1, -1)
6
Prisoners Dilemma Suppose Tchaikovsky doesnt
confess
Tchaikovsky
Confess Dont Confess
Conductor
( -8, -8)
( 0, -15)
Confess
( -15, 0)
Dont Confess
( -1, -1)
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Prisoners Dilemma

• Conclusion
• The Conductor will confess
• And Tchaikovsky?

8
Prisoners Dilemma Suppose Conductor confesses
Tchaikovsky
Confess Dont Confess
Conductor
( -8, -8)
( 0, -15)
Confess
( -15, 0)
Dont Confess
( -1, -1)
9
Prisoners Dilemma Suppose Conductor doesnt
confess
Tchaikovsky
Confess Dont Confess
Conductor
( -8, -8)
( 0, -15)
Confess
( -15, 0)
Dont Confess
( -1, -1)
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Prisoners Dilemma

• Conclusion
• Tchaikovsky confesses also
• Both get 8 years
• Could get off with one year each if they
  cooperated
• For both, confession is a dominant strategy a
  strategy that yields a better outcome regardless
  of the opponents choice

11
Equilibrium

• Nash Equilibrium Neither player has an
  incentive to change strategy, given the other
  players choice
• Both confess is a Nash Equilibrium
• Both dont confess is not a Nash Equilibrium,
  rival will always want to renege
• Noncooperative equilibrium in the Prisoners
  dilemma results in a solution that is not the
  best possible outcome for the parties.

12
Go Back to Prisoners Dilemma What would the
Conductor and Tchaikovsky decide if they could
negotiate?

• They could both become better off if they reached
  the cooperative solution.
• which is why police interrogate suspects in
  separate rooms.

13
Go back to Prisoners DilemmaIs there a way to
generate the cooperative solution?
Tchaikovsky
Confess Dont Confess
Conductor
( -8, -8)
( 0, -15)
Confess
( -15, 0)
Dont Confess
( -1, -1)
14
Go back to Prisoners DilemmaIs there a way to
generate the cooperative solution?

• Not a Nash Equilibrium
• If Conductor commits to Dont Confess,
  Tchaikovsky has an incentive to confess
• If Tchaikovsky commits to Dont Confess,
  Conductor has an incentive to confess
• Role of a contractto commit parties to actions
  they would not undertake voluntarily

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Economic Application Duopoly Game(?A, ?B)
Firm B
Cheat Collude
Firm A
( 100, 100)
( 250, 50)
Cheat
( 50, 250)
Collude
( 200, 200)
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Duopoly Game

• Nash equilibrium Both Cheat
• Both Cheating means we get closer to the social
  optimum

17
Dominant Firm Game

• Two firms, one large and one small
• Either firm can announce an output level (lead)
  or else wait to see what the rival does and then
  produce an amount that does not saturate the
  market.
• Numbers are firm profit

18
Dominant Firm Game
Dominant
Lead Follow
Subordinate
( 0.5, 4)
( 3, 2)
Lead
( 1, 8)
Follow
( 0.5, 1)
19
Dominant Firm Game Subordinate Leads
Dominant
Lead Follow
Subordinate
( 0.5, 4)
( 3, 2)
Lead
( 1, 8)
Follow
( 0.5, 1)
20
Dominant Firm Game Subordinate Follows
Dominant
Lead Follow
Subordinate
( 0.5, 4)
( 3, 2)
Lead
( 1, 8)
Follow
( 0.5, 1)
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Dominant Firm Game

• Conclusion
• Dominant Firm will always lead..
• But what about the Subordinate firm?

22
Dominant Firm Game Dominant Follows
Dominant
Lead Follow
Subordinate
( 0.5, 4)
( 3, 2)
Lead
( 1, 8)
Follow
( 0.5, 1)
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Dominant Firm Game Dominant Leads
Dominant
Lead Follow
Subordinate
( 0.5, 4)
( 3, 2)
Lead
( 1, 8)
Follow
( 0.5, 1)
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Dominant Firm Game

• Conclusion
• No dominant strategy for the Subordinate firm.
• Does this mean we cannot predict what they will
  do?

25
Dominant Firm Always Leads
Dominant
Lead Follow
Subordinate
( 0.5, 4)
( 3, 2)
Lead
( 1, 8)
Follow
( 0.5, 1)
26
Dominant Firm Game

• Conclusion
• Subordinate firm will always follow, because
  dominant firm will always lead.

27
Equilibrium

• Nash Equilibrium Neither player has an
  incentive to change strategy, given the other
  players choice
• Dominant Lead Subordinate Follow is a Nash
  Equilibrium
• A players best option may be dictated by
  anticipating the rivals best option