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dynamic games with complete information. SPNE definition of subgame perfect Nash equilibrium. A subgame perfect Nash equilibrium is a vector of strategies that, when confined to any subgame of the original game, have the players playing a Nash equilibrium within that subgame.

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dynamic games

with complete information


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SPNEdefinition of subgame perfect Nash equilibrium

A subgame perfect Nash equilibrium is a vector of strategies that, when confined to any subgame of the original game, have the players playing a Nash equilibrium within that subgame.

In a game of perfect information, the SPNE coincides with the backward induction solution.


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SPNE: example dynamic game with imperfect information

-2,-1

T

Coke

T

Pepsi

A

-3, 1

E

A

T

0,-3

Coke

O

Coke

0, 5

A

1, 2


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SPNE: example dynamic game with imperfect information

Nash equilibria


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SPNE: example dynamic game with imperfect information

q

1-q

Post-entry subgame

p

1-p


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SPNE: example dynamic game with imperfect information

Nash equilibria of the post-entry subgame:

(p=1, q=1)

(p=0, q=0)

(p=1/3, q=1/2)

SPNE:

[(Enter, p=0), q=0]

[(Out, p=1), q=1]

[(Out, p=1/3), q=1/2]


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SPNE: example repeated prisoner’s dilemma

Stage game


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SPNE: repeated prisoner’s dilemma

-6,-6

c

2

c

-3,-9

1

nc

c

-9,-3

nc

-4,-4

nc

c

c

-3,-9

2

c

2

1

0,-12

nc

nc

c

-6,-6

nc

c

-1,-7

nc

1

c

-9,-3

2

c

nc

-6,-6

c

1

nc

c

-12,0

nc

-7,-1

nc

nc

c

-4,-4

2

c

-1,-7

nc

c

1

-7,-1

nc

-2,-2

nc


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SPNE: example repeated prisoner’s dilemma

1st period


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SPNEinfinitely repeated games

  • Discount factor δ:

  • discounting the future (ex: impatience)

  • uncertainty about the future: positive probability that the game will be repeated in the future

  • If payoff of the stage game is Π, discounted total payoff is Π+δΠ+ δ2Π+…+ δtΠ+…= Π/(1-δ)


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SPNEinfinitely repeated games: trigger strategies

  • the (grim) trigger strategy comprises two parts: a desired behavior on the part of the players and a punishment regime that is triggered whenever either player violates the desired behavior

  • ex: infinitely repeated prisoner’s dilemma:

  • «start by playing “don’t confess”; continue playing “don’t confess” if nobody has confessed in the past; if someone has deviated in the past, play “confess” forever»


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SPNEinfinitely repeated games: trigger strategies

  • Check for all possible histories: 2 types of subgames

    • no deviation until then:

      • Payoff of conforming: -1/(1-δ)

      • Payoff of deviating: 0+δ.[-3/(1-δ)]

    • a deviation has occurred in the past:

      • Payoff of conforming: -3/(1-δ)

      • Payoff of deviating: -6+δ.[-3/(1-δ)]


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SPNEinfinitely repeated games: trigger strategies

  • the (grim) trigger punishment can sustain the “nice” behavior provided the players have a sufficiently high discount factor.

  • ex: infinitely repeated prisoner’s dilemma:

    • δ>1/3


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