Economics 650

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Economics 650

• Game Theory

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Repeated Play

• All of our examples so far have been one-off.
• Game theorists suspected that repeated play could
  make a difference, especially in social dilemmas.
  
• The "folk theorem" was supposed to say that when
  social dilemmas were played repeatedly,
  cooperative outcomes would be rather common.

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The Campers' Dilemma
Amanda and Buffy are camp counselors for the
summer and they are sharing a room with a TV and
DVD player. DVD's can be rented from the camp
store for the week-end for 5. Amanda and Buffy
would each get 4.00 worth of enjoyment from a
week-end movie DVD, so if each of them rents a
DVD on a particular week-end they can each get
8.00 worth of enjoyment at a cost of a 5 rental.
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Payoffs
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The Twist

• They play exactly 10 times during the summer --
  once per week for 10 weeks.
• If Amanda chooses to "rent" this week, Buffy can
  reward her by continuing to choose "rent" next
  week,
• But are those rational "best response"
  strategies?
• use the theory of games in extensive form.

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If They Played Just Twice
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NO JOY!

• The subgame perfect equilibrium is to choose the
  noncooperative "don't rent" strategy at both
  repetitions.
• We can extend this reasoning to the whole ten
  weeks of camp.

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The Chain Store Paradox
A large chain store we shall call Chainco has
stores in twenty American communities, with local
companies preparing to enter those twenty
markets. Thus, Chainco expects to play twenty
market-entry games over the next few years.
Intuition suggests that Chainco should retaliate
in the early games, in order to deter future
entrants. But is this a subgame perfect strategy?
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A Single Repetition
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Backward Induction 1.

• Consider the last in the series of 20 repetitions
  of the game.
• Clearly there is nothing to be gained in this
  case by retaliating.
• (enter, don't retaliate) is the subgame perfect
  solution for the single play.

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Backward Induction 2.

• Now consider the 19th repetition.
• Any threat to retaliate will be incredible.
• There is no purpose in retaliating in the 19th
  repetition either.
• The same reasoning applies to the 18th, the 17th,
  and so on.

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Paradox

• The conclusion is that (in a subgame perfect
  equilibrium) the chain store never retaliates.
• Thus, entry takes place and its profits drop.

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Conclusion 1

• Repeated play can be analyzed as we analyze
  sequential games.
• Although intuition suggests that repeated play
  would lead to different results than one-off
  play, this does not always follow.

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Conclusion 2

• This seems a deep puzzle, since phenomena like
  cooperation in hopes of future reciprocation,
  retaliation, and reputation do seem important in
  human affairs.

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The Paradox
We have seen that repeated play can be
paradoxical. The common factor is that the
interactions have an end point. In the Camper's
Dilemma, camp is over after ten weeks. The motive
for cooperative behavior unravels from that last
week to the first. Similarly, in the Chain Store
example, there is a last entry threat when all
other markets have been entered.
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An Answer?
If there were no end point if the game were to
continue to infinity this particular problem
would not arise. But can that really make sense?
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To Make Indefinitely Repeated Games Make Sense,
We Need

• Discounting
• Uncertainty

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Discounting
Since infinitely repeated games are played
over infinite future time, the payoffs must be
discounted to present values. Suppose the payoff
each period is 3, discounted at factor
? 3?3?23 31??2 ,
since the sum of the sequence of powers of ??is
1/(1-?).
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Rate and Factor
Definitions discounting The discount rate is
the rate at which a dollar in hand this year
could grow with compound interest, and hence also
the rate at which a dollar received one period in
the future needs to be discounted to make it
equivalent to a dollar today. The discount factor
is the value of a dollar next period as a
fraction of the dollar in hand now.
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Uncertainty
The point is that there is no definite end to the
repetition. If there is a definite end, then
retaliation and reward strategies will unravel.
But suppose that there is no definite ending
time, but there is some probability that the play
will end on any particular round.
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No Time Discounting
Let the probability of a new round of play be
plt1, Y the payoff if there is a next round, and V
the expected value of the payoff on the next
round. Vp(Y)(1-p)(0) pY Therefore, d p
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Probability 1
Let the rate of time discount from one play to
the next be r. Then Therefore,
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Putting them together
Let plt1 be the probability of another round of
play and r be the rate of time discount. Then the
expected discounted value of the payoff
is Therefore, in general
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From now on,
The discount factor, ?, is the product of two
components, one a pure time discount and the
other the probability that this is the last play
of the game.
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Recall the Effort Dilemma
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A More Complicated Rule
Definition Tit-for-Tat In game theory,
Tit-for-Tat refers to a rule for choosing
strategies of play in a repeated social dilemma.
The rule is to play "cooperate" until the
opponent plays "defect," and then to retaliate by
playing "defect" on the following round.
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Trigger Strategies 1
Definition Trigger Strategy A rule for
choosing strategies in individual repetitions in
an indefinitely repeated game is called a
"Trigger Strategy" if the rule is that
noncooperative play triggers one or more rounds
of noncooperative play by the victim in
retaliation.
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Trigger Strategies

• The Grim Trigger, which means that a single
  noncooperative strategy from one player triggers
  a switch to never cooperate.
• Forgiving Triggers -- examples
• Tit-for-Tat
• Tit-for-2-Tats
• 2 Tits-for-a-Tat

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Assume p0.9, r0.01
We will first try the Tit-for-Tat strategy rule.
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What Happens?
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Deterrence
Thus, Andy's expected value payoff if he shirks
is V1 15 2? 10?2 10?3 However, if
he works it is V2 10 10? 10?2 10?3
Andy is deterred if V1 lt V2, that is, 15 2? lt
10 10?, that is, 5 lt 8?, or ?gt 5/8. OK!
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A Bigger Problem
But what if the probability of a next round of
play is p0.6? Then Tit-for-Tat won't bring about
cooperation. Stronger measures are called for.
Will the Grim Trigger work?
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The Grim Trigger
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Grim Trigger Payoffs
Andys shirking payoff is V3 15 5? 5?2
5?3 15 2.895 29.53
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Grim Trigger Deterrence
Andy is deterred if V3 lt V2, that is, that is,
15(1-?)5? lt 10 15 15? 5? lt
10 15-10lt10? ?gt5/10 1/2 OK
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Interim Summary

• We see that
• Tit for Tat will induce cooperation if dgt5/8
• The Grim Trigger will induce cooperation if dgt1/2
• There will be no cooperation if dlt1/2

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Collusive Pricing
The most important economic applications of
trigger strategy reasoning are probably to
collusive pricing.
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Pricing Dilemma
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Trigger Strategies for the Duopolists

• A Tit-for-Tat strategy will work if 5?gt8?, that
  is, ?gt1/2.
• The Grim Trigger strategy will work if ?gt3/7.

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Summary
When social dilemmas are played "one-off,"
noncooperative play leads to bad results all
around. If the repetition continues only for a
definite number of rounds, repeated play does not
lead to cooperation. When we allow for the fact
that there is uncertainty with some probability
of continuing the relationship for another play,
we can introduce "trigger strategies,which can
sometimes support cooperative play.