Dynamic Games of Complete Information

1
Dynamic Games of Complete Information

• Repeated Game

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Repeated game

• A repeated game is a dynamic game of complete
  information in which a (simultaneous-move) game
  is played at least twice, and the previous plays
  are observed before the next play.
• We will find out the behavior of the players in a
  repeated game.

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Two-stage repeated game

• Two-stage prisoners dilemma
• Two players play the following simultaneous move
  game twice
• The outcome of the first play is observed before
  the second play begins
• The payoff for the entire game is simply the sum
  of the payoffs from the two stages.
• Question what is the subgame perfect Nash
  equilibrium?

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Game tree of the two-stage prisoners dilemma
1
R1
L1
2
2
L2
L2
R2
R2
1
1
1
1
L1
L1
L1
L1
R1
R1
R1
R1
2
2
2
2
2
2
2
2
L2
L2
L2
R2
L2
L2
R2
R2
L2
L2
L2
R2
R2
R2
R2
R2
4444
1414
4141
1111
0151
0454
4045
1015
5101
5404
0055
4540
1510
5005
0550
5500
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Informal game tree of the two-stage prisoners
dilemma
1
R1
L1
2
2
L2
L2
R2
R2
1
1
1
1
(1, 1)
(5, 0)
(4, 4)
(0, 5)
L1
L1
L1
L1
R1
R1
R1
R1
2
2
2
2
2
2
2
2
L2
L2
L2
R2
L2
L2
R2
R2
L2
L2
L2
R2
R2
R2
R2
R2
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Informal game tree of the two-stage prisoners
dilemma
1
R1
L1
2
2
L2
L2
R2
R2
1
1
1
1
(2, 2)
(6, 1)
(1, 6)
(5, 5)
L1
L1
L1
L1
R1
R1
R1
R1
2
2
2
2
2
2
2
2
L2
L2
L2
R2
L2
L2
R2
R2
L2
L2
L2
R2
R2
R2
R2
R2
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Two-stage prisoners dilemma

• The subgame-perfect Nash equilibrium(L1
  L1L1L1L1, L2 L2L2L2L2) Player 1 plays L1 at
  stage 1, and plays L1 at stage 2 for any outcome
  of stage 1.Player 2 plays L2 at stage 1, and
  plays L2 at stage 2 for any outcome of stage 1.

The payoff (1, 1) of the 2nd stage has been added
to the first stage game.
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Finitely repeated game

• A finitely repeated game is a dynamic game of
  complete information in which a
  (simultaneous-move) game is played a finite
  number of times, and the previous plays are
  observed before the next play.
• The finitely repeated game has a unique subgame
  perfect Nash equilibrium if the stage game (the
  simultaneous-move game) has a unique Nash
  equilibrium.
• The Nash equilibrium of the stage game is played
  in every stage.

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Infinitely repeated game

• A infinitely repeated game is a dynamic game of
  complete information in which a
  (simultaneous-move) game called the stage game is
  played infinitely, and the outcomes of all
  previous plays are observed before the next play.
  
• Precisely, the simultaneous-move game is played
  at stage 1, 2, 3, ..., t-1, t, t1, ..... The
  outcomes of all previous t-1 stages are observed
  before the play at the tth stage.
• Each player discounts her/his payoff by a factor
  ?, where 0lt ? lt 1.
• A players payoff in the repeated game is the
  present value of the players payoffs from the
  stage games.

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Static (or Simultaneous-Move) Games of Incomplete
Information

• Introduction to Static Bayesian Games
• Bayesian Nash Equilibrium

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Static (or simultaneous-move) games of INCOMPLETE
information

• Payoffs are no longer common knowledge
• Incomplete information means that
• At least one player is uncertain about some other
  players payoff function.
• Static games of incomplete information are also
  called static Bayesian games

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Prisoners dilemma of incomplete information

• Prisoner 1 is always rational (selfish).
• Prisoner 2 can be rational (selfish) or
  altruistic, depending on whether s/he is happy or
  not.
• If s/he is altruistic then s/he prefers to deny
  and s/he thinks that confess is equivalent to
  additional four months in jail.
• Prisoner 1 can not know exactly whether prisoner
  2 is rational or altruistic, but s/he believes
  that prisoner 2 is rational with probability 0.8,
  and altruistic with probability 0.2.

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Prisoners dilemma of incomplete information

• Given prisoner 1s belief on prisoner 2,
• what strategy should prisoner 1 choose?
• What strategy should prisoner 2 choose if s/he is
  rational or altruistic?

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Prisoners dilemma of incomplete information

• Solution
• Prisoner 1 chooses to confess, given her/his
  belief on prisoner 2
• Prisoner 2 chooses to confess if s/he is
  rational, and deny if s/he is altruistic
• This can be written as (Confess, (Confess if
  rational, Deny if altruistic))
• Confess is prisoner 1s best response to prisoner
  2s choice (Confess if rational, Deny if
  altruistic).
• (Confess if rational, Deny if altruistic) is
  prisoner 2s best response to prisoner 1s
  Confess
• A Nash equilibrium called Bayesian Nash
  equilibrium

15
Battle of the sexes with incomplete information

• Now Pats preference depends on whether she is
  happy.
• If she is happy then her preference is the same.
• If she is unhappy then she prefers to spend the
  evening by himself and her preference is shown in
  the following table.
• Chris cannot figure out whether Pat is happy or
  not. But Chris believes that Pat is happy with
  probability 0.5 and unhappy with probability 0.5

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Battle of the sexes with incomplete information

• How to find a solution ?

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Battle of the sexes with incomplete information

• Best response
• If Chris chooses opera then Pats best response
  opera if she is happy, and prize fight if she is
  unhappy
• Suppose that Pat chooses opera if she is happy,
  and prize fight if she is unhappy. What is Chris
  best response?
• If Chris chooses opera then he get a payoff 2 if
  Pat is happy, or 0 if Pat is unhappy.
• His expected payoff is 2?0.5 0?0.51
• If Chris chooses prize fight then he get a payoff
  0 if Pat is happy, or 1 if Pat is unhappy.
• His expected payoff is 0?0.5 1?0.50.5
• Since 1 gt 0.5, Chris best response is opera
• A Bayesian Nash equilibrium (opera, (opera if
  happy and prize fight if unhappy))

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Battle of the sexes with incomplete information

• Best response
• If Chris chooses prize fight then Pats best
  response prize fight if she is happy, and opera
  if she is unhappy
• Suppose that Pat chooses prize fight if she is
  happy, and opera if she is unhappy. What is
  Chris best response?
• If Chris chooses opera then he get a payoff 0 if
  Pat is happy, or 2 if Pat is unhappy.
• His expected payoff is 0?0.5 2?0.51
• If Chris chooses prize fight then he get a payoff
  1 if Pat is happy, or 0 if Pat is unhappy.
• His expected payoff is 1?0.5 0?0.50.5
• Since 1 gt 0.5, Chris best response is opera
• (prize fight, (prize fight if happy and opera if
  unhappy)) is not a Bayesian Nash equilibrium.

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Normal-form representation of static Bayesian
games
20
Normal-form representation of static Bayesian
games Payoffs
21
Normal-form representation of static Bayesian
games Beliefs (probabilities)
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Strategy
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Bayesian Nash Equilibrium 2-player
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Bayesian Nash equilibrium 2-player
In the sense of expectation based on her/his
belief
player 2s best response if her/his type is t2j
player 1s best response if her/his type is t1i
In the sense of expectation based on her/his
belief