Lect012_Slides

# Lect012_Slides - Repeated Games Repeated Games This week we...

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epeated Games Repeated Games • This week we examine the effect of repetition on strategic behavior in games with perfect information. • If a game is played repeatedly, with the same players, the players may behave very differently than if the game is layed just once (a ne- ot ame) e g repeatedly borrow a played just once (a one shot game), e.g., repeatedly borrow a friend’s car versus renting a car. • Two types of repeated games: Finitely repeated : the game is played for a finite and known number of rounds, for example, 2 rounds/repetitions. Infinitely or Indefinitely repeated : the game has no predetermined length; players act as though it will be played indefinitely, or it ends only with some probability.

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Finitely Repeated Games • Writing down the strategy space for repeated games is difficult, even if the game is repeated just 2 rounds. For l id th f ii tl td t t i f example, consider the finitely repeated game strategies for the following 2x2 game played just twice. L R U • For a row player: D –U 1 or D 1 Two possible moves in round 1 (subscript 1). – For each first round history pick whether to go U 2 or D 2 h hi t i The histories are: (U 1 ,L 1 )(U 1 ,R 1 )(D 1 ,L 1 1 ,R 1 ) 2 x 2 x 2 x 2 – 16 possible strategies!
Strategic Form of a 2-Round Finitely Repeated Game his quickly gets messy! This quickly gets messy! L 2 R 2 L 2 R 2 L 1 R 1 U 2 D 2 U 2 D 2 U 1 L 2 R 2 L 2 R 2 D 1 U 2 U 2 D 2 D 2

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Finite Repetition of a Game with a Unique Equilibrium Fortunately, we may be able to determine how to play a finitely repeated game by looking at the equilibrium or equilibria in the one-shot or “stage game” version of the game. For example, consider a 2x2 game with a unique equilibrium, e.g. the risoner’s ilemma: higher numbers=years in prison are worse Prisoner s Dilemma: higher numbers=years in prison, are worse. Does the equilibrium change if this game is played just 2 rounds?
A Game with a Unique Equilibrium Played Finitely Many Times Always Has the Same Subgame Perfect Equilibrium Outcome • To see this, apply backward induction to the finitely repeated game to obtain the subgame perfect Nash equilibrium (spne). • In the last round, round 2, both players know that the game will not continue further. They will therefore both play their dominant strategy of Confess. gy • Knowing the results of round 2 are Confess, Confess, there is no benefit to playing Don’t Confess in round 1. Hence, both layers play Confess in round 1 as well players play Confess in round 1 as well. • As long as there is a known, finite end , there will be no change in the equilibrium outcome of a game with a unique ilib i hi i l f t t equilibrium. This is also true for zero or constant sum games.

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Finite Repetition of a Sequential Move Game • Recall the incumbent-rival game: In the one-shot, sequential move game there is a unique subgame perfect equilibrium where the rival enters and the incumbent accommodates.
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Lect012_Slides - Repeated Games Repeated Games This week we...

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