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TOPIC 6
REPEATED GAMES
• The same players play the same game G period after
period. Before playing in one period they perfectly
observe the actions chosen in the previous periods.
(perfect observability).
• G is the stage-game (or constituent game).
• G(T) is the repeated game with horizon T (that is,
played T times).
• The strategic behaviour in G(T) can be very different
from the behaviour in G (played once).
• We will focus on a repeated Prisoners´ Dilemma
(PD): under what conditions can cooperation be
obtained between selfish players?
REPEATED GAMES: SOME PREVIOUS ISSUES.
• How do players evaluate a flow or stream of payoffs?
• By the sum of their present or discounted value, that is, its
value in t = 1 euros (present euros).
• The discount factor of a player, d, is the present value (t = 1
euros) of one euro obtained tomorrow (t = 2).
• With an interest rate r > 0, d =1/(1+r).
• Therefore, 0 < d < 1. It measures the relative importance of the
present versus the future. If d is close to 1 means that the
future is very important. If it is close to 0, future payoffs are
then relatively unimportant.
• Besides the interest rate, some other factors influence d, such
as the tastes, the probability of continuing active in the game….
REPEATED GAMES: SOME PREVIOUS ISSUES.
• The horizon of a repeated game.
• - finite horizon: there is a last period (a deadline)
which is common knowledge among the players.
• - infinite horizon: when the relationship does not
have a predetermined lenght.
• For instance, in each period there is a positive
probability p of playing again in the next period and
a probability (1 – p) of ending the game. This
situation is equivalent to an infinite horizon repeated
game, where the discount factor of the players is d´
= p.d, where d is the real discount factor.
REPEATED GAMES: SOME PREVIOUS ISSUES.
• A strategy of a player in G(T) is a complete plan that
specifies in every period what action should be
chosen as a function of each possible previous
history of the game.
• The history of the game in period t is just the
sequence of vectors of actions observed until period
t – 1.
• Unconditional or uncontigent strategies: for
instance in a repeated PD,
• “NC after any history (no matter what your rival has
done in the past)” or “C after any history”.
REPEATED GAMES
• Strategies of a conditional cooperator (reciprocity): play
cooperatively as long as your rival does so, but any defection
on his part triggers a period of punishment.
• TRIGGER STRATEGY:
• “Begin by cooperating (C) in t = 1, then C if everybody has
cooperated at every previous period. But, if any player does not
cooperate, then switch to the strategy NC after any history”
• This is an unforgiven strategy. The lenght of the punishment
does not depend on the behaviour of your opponent during the
punishment.
• TIT-FOR-TAT STRATEGY:
• “Cooperate at the start and then, play in each period as your
opponent did in the previous period.”
COLLUSION
• Two firms set prices simultaneously. The good
produced by every firm is homogeneous and its
aggregate market demand is given by a function
D(p). Both firms have an identical constant average
(and marginal) cost c >0. They play repeatedly this
game and have identical discount factor d > 0.
• Find the conditions for collusion (in the monopoly
price) to be sustainable in the repeated game.
• What happens if the number of firms increases?
• (does collusion depend on market concentration?)
COLLUSION
• Two firms with identical discount factor d = 0,65,
compete in prices in two identical and independent
markets. In market 1, it takes one period to observe
the opponent´s price, but in market 2, it takes two
periods to observe the opponent´s price and then
react to it (there is an information lag).
• Discuss if collusion is sustainable in each market
separatedly.
• Asume now that both firms link collusion to its
maintenance in both markets (multimarket contact).
COLLUSION
• Two firms with identical discount factor, compete in
prices. The demand function at date t is μt-1D(pt),
where μ.d < 1. Derive the set of discount factors
such that full collusion is sustainable in the repeated
game. What would this model predict about the
relative ease of sustaining collusion in expanding
and declining industries?
QUALITY AND REPUTATION.
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A consumer chooses whether or not to purchase a service from a firm.
If the consumer does not purchase, then both players receive a payoff
of 0. If the consumer decides to purchase, then the firm must decide
whether to produce high or low quality. In the former case, both
players have a payoff of 1. In the latter case, the firm´s payoff is 2 and
the consumer´s payoff is -1.
A) Suppose this is the stage-game of an infinitely repeated game. Find
an equilibrium in which high quality is provided every period.
B) Suppose now that a long-lived firm (B) plays against a sequence of
short-lived consumers (At) who only live one period but perfectly
observe how B behaved in the past.
C) What problems might appear if the consumers have to incur a cost
of checking B´s history?
What would happen if there is imperfect observation? For instance,
suppose that even if B intends to provide high quality, there is a small
probability of making a mistake and providing low quality.
THE FIRM AS A DEPOSITORY OF REPUTATION.
• Suppose now that firms (Bt) are also short-lived. For
example, they only live for one period. What would
be the equilibrium in this case?
• Suppose that each Bt lives two periods. In period t,
Bt plays the quality game and in period t + 1 retires
and lives of his savings.
• B1 creates the firm B-Honest Co. (BH) with a good
reputation of providing high quality. Describe an
equilibrium of the repeated game in which each Bt
acquires BH and maintains its reputation. Assume
for example, that the market price of BH (with
reputation) is 10 (and it would be 0 without
reputation).
THE FIRM AS A DEPOSITORY OF REPUTATION.
• The consumers´ strategy: “At trusts Bt if Bt
owns BH and no previous owner of BH has
provided low quality in the past. In other
case, At does not purchase.”
• Given this strategy and if Bt can sell BH in
period t + 1 by 10 monetary units, his optimal
strategy is to acquire BH and keep its good
reputation providing high quality.
• Each owner of BH is willing to maintain its
reputation in order to recover its acquisition
cost (his consumption in the second period
depends on this fact).
FINITELY REPEATED GAMES
• The only perfect equilibrium outcome in a finitely repeated
prisoners´dilemma is “always do not cooperate”, that is, the
repetition every period of the equilibrium of the stage game.
• If the stage game has a multiplicity of inefficient Nash equilibria
(NE), then the previous result does not hold. That is, it might
exist a perfect equilibrium of the finitely repeated game in
which there is cooperation in all except for the last period.
• Intuition: it is possible to punish a player for deviating in the
next-to-last period by specifying that if he does not deviate the
NE he prefers will occur in the last period, and that deviations
lead to the NE he likes less.
A GAME WITH OVERLAPPING GENERATIONS OF
PLAYERS.
• Consider a repeated game in which overlapping generations of
players live for 10 periods, so that at each date t there is one
player of age 10 who is playing his last round, one player of age
9 who has two rounds still to play, and so on down to the new
player who will play 10 times.
• Each period, the 10 players simultaneously choose whether to
work hard (at a cost 1) or to shirk (at a cost zero), and their
choices are revealed at the end of the period; players share
equally in the resulting output, which is twice the number who
choose to work. Payoffs in the repeated game are the sum of
the per-period utilities.
• Find a Nash equilibrium of the repeated game in which
everybody, except the player of age 10 cooperates (works
hard).
A GAME WITH OVERLAPPING GENERATIONS OF
PLAYERS.
• Strategy: “Age-10 players always shirk. So long as
no player has ever shirked when his age is less than
10, all players of age less than 10 work. If a player
has ever shirked when his age is less than 10, then
all players shirk.”
• Notice that the “cooperative” equilibrium we have
derived remains an equilibrium if we suppose that
workers observe only the total number of shirkers
but not their identities.