Infinitely Repeated Games

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Infinitely Repeated Games
In an infinitely repeated game, the application
of subgame perfection is different - after any
possible history, the continuation must be a NE
- but after any history each subgame looks like
the original game We cannot use generalized
backward induction, because there is no last
period - the trick is to recognize that each
subgame is identical to the whole game - this
simplifies things since only need to consider the
initial game A technical issue we must
discount future payoffs using a discount factor (
0ltdlt1) - without discounting the payoffs are not
finite
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Repeating the one-shot NE is always an SPNE
2 1 C D
C 3, 3 1, 4
D 4, 1 2, 2
If (2) plays D always - then a NE can come from
(1) playing D - everyone playing D always is a
SPNE In the finite repeated prisoners dilemma
game a unique SPNE is (D, D), (D, D)(D, D) In
the infinitely repeated version, there are
multiple SPNE as long as players are sufficiently
patient - one SPNE is always (D, D), (D, D)(D,
D) - the existence of other SPNE implies that
cooperation is possible - Cooperation can
yield a higher long-run payoff
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Infinite prisoners dilemma
2 1 C D
C 3, 3 1, 4
D 4, 1 2, 2
One-shot unique NE is D, D Finite repeated
unique SPNE (D, D), , (D, D) Infinitely
repeated Claim if players are sufficiently
patient (if the discount factor is sufficiently
high), then cooperation can be sustained in
a SPNE in the infinitely repeated game. Nash
Reversion - consider the following trigger
strategy Play C if you have always seen C,
otherwise play D. - if both players follow
this strategy, we always observe (C, C)
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trigger strategy
Under what conditions is there a SPNE in trigger
strategies Is the trigger strategy a best
response for (1) if (2) uses the trigger
strategy? For the trigger strategy to be a BR
for (1)
2 1 C D
C 3, 3 1, 4
D 4, 1 2, 2
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Patience matters in infinitely repeated games If
players do not value the future at all, the
analysis of repeated games is the analysis of
repeated one-shot games
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The Folk Theorem
Using the trigger strategy we can get many
different paths Path of cooperation in
even periods and non-cooperation in odd
periods - - then a trigger that goes to D if
they ever observe anything different from this
and we would get DD forever Path
of alternating Or any other path you can
think of Even if the stage game has an unique
equilibrium, there may be SPNE in the infinitely
repeated game in which no stages outcome
is a NE of a stage game The Folk Theorem
take a game, play it infinitely often -if
players are patient enough you can get a wide
variety of subgame paths -some paths might
require a very high discount rate a lot of
patience -you need further refinements in order
to predict behavior
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Infinitely repeated Bertrand

• 2 firms they play Bertrand each period for an
  infinite of periods
• the firms discount future payoffs
• Recall that in the one-shot game with two firms
  P1 P2 C is the unique NE.
• - Any NE in the one shot game is a NE in the
  repeated game
• If firm (2) plays P2 C every period
• Then firm (1) gets 0 profits no matter what,
  so P1 C is a BR1
• P1 P2 C is also a SPNE
• In the infinitely repeated Bertrand game, many
  other price paths are possible
• - there are other SPNEs ( a lot of them)
• - we will focus on the one in which both
  players choose the monopoly price PM

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Dynamic Bertrand with tacit collusion
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Collusion as N rises
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Too many possibilities exist
This is the folk theorem A reasonable
focal point too many possibilities -
firms choose symmetric strategies -
on the frontier so each gets Anything
can happen Because there are so many equilibria
Using SPNE alone we cannot predict what is to
happen in infinitely repeated games Any price
between the marginal cost and the monopoly price
can be sustained Any payoffs in the triangle can
be the average per period payoffs in a SPNE - if
players are sufficiently patient they can use a
Nash reversion strategy - anything beats getting
zero forever - However, with Nash reversion,
players cannot do worse than the NE
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Cournot with Nash reversion
In Cournot, there are also SPNE with trigger
strategies - firms can tacitly collude
- what keeps it up is the prospect of future
gains
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Minimax payoffs
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Cournot with Non-Nash reversion
What does non-Nash reversion strategy look like?
The Folk Theorem states that the only lower bound
on payoffs in an infinitely repeated game when
players are sufficiently patient is given by the
minimax payoffs (rather than the Nash eqm.
Payoffs) One player can force the other to
the minimax payoffs In Cournot, the minimax
payoff is zero
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Infinitely repeated games pose problems for
analysis - because of the infinite of
equilibria it is difficult to predict the path of
play - it is difficult to perform comparative
statics - we can explain anything the analysis
does not add much value
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Using infinitely repeated games
One response is to ignore repeated game
considerations (1) focus on simple dynamic
games (2) assume players repeat a one-shot NE
when it has intuitive properties Another
response is introduces a state space - assumes
players strategies are a function of the current
state, not of the history - value functions with
Markov perfect equilibria - this removes
history dependence A third response is to use
the insights to explain conditions under which
cooperation can occur -.make standard
assumptions such as players are rational - find
restriction on parameters that make cooperation
possible
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Tit for Tat
This repeated PD strategy has only one period of
memory players cannot carry a grudge
Strategy (TFT) t1 Cooperate tgt1
Cooperate if opponent played C in period t
1 Defect if opponent played D in period t
1 This is not equilibrium analysis There are
some good properties of TFT 1) nice
starts out cooperating, never initiates
defection 2) simple easy to follow, easy
for opponent to understand 3) forgiving
after D, it is willing to cooperate again if
opponent does 4) provocable never lets
cheating go unpunished Axelrods experiments
he collected strategies for computerized Prisoner
Dilemma games - In a round-robin tournament
each strategy played every other strategy. -
TFT was the winner. - Nevertheless, a simple
defect strategy will always beat TFT, but
provides a low payoff
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