Prof. Bryan Caplan
bcaplan@gmu.edu
http://www.bcaplan.com
Econ 812
Week 5: Repeated Games,
Competition, and Cooperation, I
I.
FinitelyRepeated Games
A.
We frequently play with the same people over and over again.
B.
Question: If players condition their behavior in one game on your
behavior in previous games, what happens?
C.
Answer: More equilibria may be sustainable.
D.
There are two main cases to consider: finitelyrepeated games and
infinitelyrepeated games.
1.
Note: Games that probabilistically end, with no fixed upper
bound to number of games, count as infinitelyrepeated.
E.
Suppose two players first play a PD game, then a Coordination game,
using last week's payoffs.
F.
Note: The "independent" equilibria of the two games remain
equilibria.
G.
But a Paretosuperior outcome now becomes possible. Suppose that each player plays Left in
the second game if either player failed to Cooperate in the first game, and
Right otherwise. Then both players
play Cooperate, Right, and this is a NE!
H.
How is this possible? If a
player fails to Cooperate in the first game, he gets 6, but then only earns 3
in the second game, for a total payoff of 9. But equilibrium play has a payoff of 10.
I.
What happens if you reverse the order of the two games?
II.
The Paradox of Backwards Induction
A.
Thus, even in finitelyrepeated games, the set of Nash equilibria
expands. But it expands much less
than you would think.
B.
How so? Suppose two players
play the PD game a hundred times.
Couldn't they sustain Cooperation by threatening retaliation?
C.
No. In the last turn, both
players will defect. Since they
both defect in the last turn no matter what, threatening to defect if
your opponent fails to cooperate in the secondtolast game is no
deterrent at all. So people fail to
cooperate then, too,
D.
Pushing this logic backwards all of the way to the first turn,
cooperative play completely "unravels."
E.
How does this differ from the previous example? That combined a dominancesolvable game
with a game with two Nash equilibria. So even in the last turn, a sort of
"revenge" is possible.
Not so if all of the games in the series are dominance solvable.
F.
Aside: In reality, of course, experiments confirm that people do
cooperate in finitelyrepeated games to a greater extent than 1shot
games. Some attempts have been made
to theoretically model this. Most
are based on the premise that players assign a small probability of
irrationality to their opponent.
III.
InfinitelyRepeated Games
A.
Few games literally last forever, but many games always have a chance
to continue. As long as they have
that chance, game theorists call them "infinitely repeated."
B.
With infinite repetition, the previous unraveling logic no longer
holds, making more equilibria sustainable.
Now, the intuition of retaliation works.
C.
Simple example: Repeated PDs.
Suppose we both make the most extreme possible threat (aka "trigger
strategy"): If you cheat me once, I'll never cooperate with you
again. Suppose further that we both
discount the future by b. (Alternately, that the game continues
each turn with probability b). Is this a NE?
D.
If you cooperate, you get _{}. Recalling the
formulae for infinite sums, this adds up to _{}.
E.
If you defect, you get 6 immediately, but then only 1 forever
afterwards. Mathematically: _{}, which adds up to: _{}.
F.
To check to see whether this is a NE, then, we see whether the Nash
payoff weakly exceeds the defection payoff. Is _{}? It is, so long
as b³1/5.
1.
Note: Without discounting, repeated games are a nobrainer. No finite gain from cheating would ever
be worth infinite punishment.
G.
Are other equilibria sustainable?
Of course. You might not
cooperate at all. You might only
punish for one period, then return to cooperation. Intuition: The weaker the punishment,
the higher b must be to make cooperation sustainable (b³1/4 in the latter case).
H.
The Folk Theorem shows that if cooperation is sustainable at all, there
will normally be an infinite number of equilibria.
I.
So what actually happens out of the endless possibilities? As in Coordination games, focal points
probably matter a great deal, but are hard to formally model.
IV.
Reputation
A.
Economists frequently invoke reputation to explain seemingly
moneylosing behavior. Does this
make sense?
B.
Yes. The logic of repeated
play often works even if there is some oneshot interaction. Suppose, for example, that a store owner
decides to cheat or not cheat a customer, and onetime customers decide whether
to buy or not.

Buy 
Don't 
Cheat 
10,2 
0,0 
Don't 
5,2 
0,0 
C.
Using weak dominance, the store owner always cheats, so the customer
never buys.
D.
But suppose that customers know whether the store has cheated in the
past, so they can play (Buy if no past Cheating, Don't otherwise).
E.
Is this a NE? It is if _{}.
F.
The applications of reputational models are endless. Most obviously, reputation is the market
alternative to regulation of product quality and the like.
1.
Question: How does ease of detection affect reputational incentives?
G.
Reputation probably matters for prices as well as quality. Stores may keep prices below daily
profitmaximizing levels because they want a reputation for low prices.
H.
Intuitively, we usually think that reputational incentives lead to
Pareto superior outcomes. But reputational
incentives could actually lead in the opposite direction. Outlaws might try to develop reputations
for ferocity, or dictators for brutality.
I.
How can the standard intuition be rationalized? Add on free entry and exit. Then people with bad reputations earn no
advantage because they have no one to interact with.
1.
The Tullock PDwithpartnerselection experiment.
V.
Monopoly and Contestability
A.
You have all seen the standard monopoly model. The monopolist maximizes PQTC, and sets
MR=MC.
B.
Does this make sense in game theoretic terms? Sure, this is an
equilibrium. But there is also an
equilibrium where consumers refuse to buy anything if P>MC, so the
monopolist sets P=MC. And of course
there are many other equilibria.
1.
Question: What extra assumptions and/or solution concept underlie the
standard account?
C.
Still, the standard account intuitively seems right as far as it
goes. The main problem is that it
neglects potential competition.
D.
Contestability models offer one of the most appealing ways to analyze
potential competition. Basic setup:
An incumbent firm sets its price.
Then a potential entrant decides whether to enter and, if so, at what
price. Consumers buy from the
lowerpriced firm.
E.
Suppose TC=bQ. Then if P_{i}>b,
the entrant enters and charges P_{e}= P_{i} e,
leaving the incumbent with 0 profits.
The only NE is where the incumbent charges P_{i}=b and the
entrant stays out.
F.
What if the entrant has higher costs than the incumbent? Then the incumbent prices just below the
entrant's costs.
G.
What if there are fixed costs, so TC=a+bQ? Then P=b is no longer an equilibrium,
because that implies profits of a<0.
In that case, the incumbent prices at AC instead of MC.
H.
What if there is a sunk cost of a, followed by pricing
decisions? Then the firstmover
acts like a monopolist, since if entry occurs, both firms will compete price
down to b, and both lose money.
I.
What about simultaneous decisions to incur sunk costs? Analyze the following normal form.

In 
Out 
In 
a,
a 
P^{m},0 
Out 
0,
P^{m} 
0,0 
VI.
Allocative versus Productive Inefficiency
A.
Most micro texts focus on the allocative inefficiency of monopoly.
A.
Main intuition: Landsburg on "Why Taxes Are Bad." Units consumers buy anyway involve only
a transfer; units that are no longer bought involve a deadweight loss.
B.
Allocative inefficiencies are normally quite tiny, however, because
they arise only on the marginal units, or DW loss "triangle."
C.
Far less discussed: productive inefficiency. A situation is productively inefficient
iff the AC of producing a given quantity is above the minimum AC.
D.
Productive inefficiencies can easily be large, because they exist on
ALL units produced, yielding a whole DW loss trapezoid.
E.
With contestable monopoly and unequal costs, some allocative
inefficiency persists, but no productive inefficiency.
F.
In contrast, imagine an inefficient monopoly with a price cap at
P=MC. There is no allocative
inefficiency, but still productive inefficiency.
G.
Governmentcreated monopolies versus market monopolies: Both allow for
allocative inefficiency, but the former have a strong potential for productive
inefficiency as well.
VII.
Predation, Entry Deterrence, and Mixed Strategies
A.
"Predation" means many things to many people. What insight can game theory shed here?
B.
Simplest model of predation: limit pricing. There are many potential producers with
varying costs. The lowestcost
producer prices just below the costs of the secondlowestcost producer,
winning the whole market.
1.
This probably happens frequently, with or without "predatory
intent."
C.
More interesting model: Incumbent prices high if no entry, low if
entry; entrant decides whether to enter.
D.
As discussed earlier, there are two NE: (Out, Fight) and (In,
Accommodate). But (Out, Fight) is
not subgame perfect. Once the entry
happens, the incumbent is better off accommodating. The threat to predate is not credible;
the incumbent would be "cutting off his nose to spite his face."
1.
Less formal literature emphasizes that predation is especially costly
to the incumbent; the game theoretic point is simply that even if predation is
cheap, it is more expensive than accommodation.
E.
What if predation game is infinitely repeated? Then predation is potentially
sustainable. It all depends on the
shortterm cost of predation versus the longrun monopoly profits. (Here the standard arguments come into
their own).
F.
Big question about predation: Why can't "two play at that
game"? In other words, why
can't entrants predate against incumbents just as well as incumbents predate
against entrants?
G.
Natural solution: Mixed strategy.
Returning to the previous normal form, note that in addition to the two
PSNE, there is also a MSNE.
Potential monopoly profits balance out potential losses of
"destructive competition."
H.
I maintain that the MS solution makes a lot more sense. There is no way to credibly commit to be
In no matter what. The bigger the
conditional benefit of being a monopoly, the more willing firms will be to try
to win monopoly status.
VIII.
Bertrand and Cournot Competition
A.
The previous arguments rely heavily on what is known as Bertrand
competition (and, to some degree, constant MC). Firms propose prices; all customers buy
from the firm that offers the lowest price, and randomize between equal prices.
B.
In equilibrium, the most (productively) efficient firm takes the whole
market, and charges just below the price of the secondmost efficient
firm. P=MC if at least two firms
can produce in the most productively efficient way.
C.
Bertrand competition strongly undermines the perfectly competitive
benchmark. It shows that you can
get perfectly competitive outcomes with just TWO firms.
D.
Perhaps because of this result, many economists prefer the Cournot
model of oligopoly. Cournot assumed
that firms set quantities rather than prices. The price then independently adjusts to
clear the market.
E.
Formally, define Q as the sum of all N firms' q's, suppose P=abQ, and
firms' MC=0. Bertrand competition
predicts a price of 0 for all N.
What does Cournot predict?
F.
Each firm maximizes Pq_{i}MCq_{i}=_{}. So they set: _{}, which gives the optimal response of firm i given the
behavior of all the other firms.
G.
Natural solution: Look for the symmetric NE, where all firms
produce the same q. Then _{}, so _{}, and _{}.
H.
Now Q goes to the perfectly competition level a/b as N goes to
infinity. Q falls as N falls even
though each firm thinks only of itself and makes no effort to collude.
I.
Big weakness of Cournot: Firms would want to split! Under these assumptions, an infinite N
would arise endogenously.
J.
If you add a fixed cost for each firm, it can also be proven that
Cournot competition with free entry is not even secondbest. Imposing a zeroprofit condition implies
an inefficiently large number of firms.
K.
Once again, though, if one firm could credibly commit to expand its
output and take over the whole market, you would reach the secondbest (P=AC)
outcome.