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1
G
AME
T
HEORY IN THE
S
OCIAL
S
CIENCES
Political Science 135/Economics 110
Hand out on Repeated Games
Why study repeated games?
Many forms of social interaction involve repeated interaction.
Housemates interact repeatedly until the lease is up.
High technology firms interact repeatedly
in an oligopolistic market.
Members of Congress interact repeatedly at least until the next
election.
States interact repeatedly when they decide how much to spend on arms and for
defense.
Repeated games are the simplest place to study the effects of repeated interaction.
In a
repeated game, the players meet every period and play the same “game” over and over again.
More precisely, the players repeatedly play the same
constituent game, oneshot game,
or
stage
game
.
In a repeated game, the outcome of previous interactions never changes the situation the
players will confront in the next period, i.e. the stage game never changes.
In effect, then,
repeated games strip out or abstract away from all of the possible other effects that repeated
interaction might have in order to focus solely on the effects that repeated interaction can have
on the behavior of the actors.
What are the effects of repetition?
As soon as we allow for repetition, then we are introducing
time and in particular the future into the analysis.
This means that actors can begin to make
threats or promises.
Once a future exists, a player can threaten to punish another player in the
future if that player does not do what the first player wants in the current period.
So, asking
what the effects of repetition are is equivalent to asking how the ability to make threats or
promises affects the possible outcomes of an interaction.
In more technical terms, how do the
equilibria of the stage game compare to the Nash equilibria or subgame perfect equilibria of the
repeated game.
We saw that if the players simply play a Nash equilibrium of the stage game over and
over, then this will also be a Nash equilibrium of the repeated game.
(Can you see why it is also
a subgame perfect equilibrium of the repeated game?)
So, asking how repetition affects the
possible outcomes really reduces to asking how many more outcomes might we expect to see in
a repeated game?
The Nash Folk Theorem for Infinitely Repeated Games:
From the perspective of wanting to
make
precise and focused predictions about the outcomes of a strategic interactions, the news
from our analysis of repeated games is not very good.
The result is:
Any individually rational
outcome of the stage game can be supported as a Nash equilibrium outcome of the repeated
game if the
players
are
sufficiently
patient,
i.e.,
if
the
players
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 Fall '10
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