The game has two players. The entrant decides whether to Enter the market each period or
to Stay Out instead. The incumbent decides whether to Fight entry or Accommodate, if
entry occurs. This goes on for T repetitions. The entrant only wants to Enter if the
incumbent will Accommodate. What the incumbent wants depends on his type. With
probability delta, the incumbent is Tough, and gets a higher payoff from Fight than from
Accommodate. With probability (1-delta), the incumbent is Soft, and prefers to
Accommodate. Usually we will think of delta as being small (say, 1%), and T as being
large (say, 900 repetitions).
The general idea of the equilibrium is that the soft incumbent will choose Fight if
there are enough periods left, so as to pretend to be tough and thereby deter entry. As
the last period approaches, this strategy is no longer optimal, so the soft entrant
first starts randomizing and then gives up entirely and accommodates if entry occurs.
The entrant, knowing all this, stays out until near the end, and then starts randomizing
or entering with probability one.
In the plausible perfect bayesian equilibrium, the entrant Stays Out for the first K
repetitions, because the incumbent would choose to Fight entry if it occurred. Then
(generically) there is a period in which the entrant chooses Enter, and then
there is a set of periods in which the entrant either randomizes between Enter and
Stay
Out or simply picks Enter. In all these middle periods, the tough incumbent chooses
Fight (as he always does) and the soft incumbent randomizes between Fight and
Accommodate. If, however, the incumbent ever chooses Accommodate as result of this
randomization, his soft type has been revealed, and thereafter he will always play
Accommodate and the entrant will always choose Enter.
Towards the end, it may happen that the entrant gives up and decides to stay out, and
it will certainly happen that in the last period or few periods the soft incumbent will
choose Accommodate.
(a) The possibility of "cooperative behavior" for the *entire* length of a finite- period game.
The game's equilibrium is in mixed strategies, so there are many possible outcomes that might be observed. One possibility is that the incumbent will never be observed to choose Accommodate.
Here is how that would happen. Suppose the entrant, randomizing, has entered 7 times, but each time the random response of the incumbent has been to Fight. Seeing the incumbent choose Fight so often, the entrant's posterior belief that he is tough rises far above delta. Maybe delta=1%, but the posterior ends up being a 90% probability that the entrant is tough. The last period comes. The entrant knows that if the incumbent is soft he will Accommodate in the last period, but that only has probability 10%. Thus, the entrant will stay out, and even in the last period we won't observe Accommodation.
Entry will be observed in every equilibrium, but not necessarily entry in the last period.
This is different from the repeated prisoner's dilemma, which always has a rational, normal, player choosing Fink in the last period.
(b) Implausible off-equilibrium beliefs and use of the Intuitive Criterion.
On page 263, Kreps and Wilson point out that their game can have unreasonable perfect bayesian equilibria. They give an example like this:
Out of equilibrium belief: If the incumbent plays Fight in the first period, then he is
definitely soft.
Suppose delta=.8, so it is very likely that the incumbent is tough, and T=2, so there
are just two periods. Suppose, too, that the tough incumbent, while preferring Fight to
Accommodate, prefers No Entry by far to entry with either of these two responses.
Here is an equilibrium:
Soft incumbent: Accommodate entry in each period.
This is an equilibrium because the tough incumbent wants to deter entry in the second
period, which he can do only by choosing Accommodate in the first period, and he values
deterring entry more than he values responding to entry with Fight instead of
Accommodate.
Tough incumbent: Accommodate entry in the first period. Fight in the second period.
Entrant: Enter in the first period. Stay Out in the second period, unless the incumbent
chose Fight in the first period, in which case choose Enter in the second period.
The out-of-equilibrium beliefs, however, while perfectly in accord with Bayes' Rule, are implausible. They say that if the impossible happens, and the incumbent chooses Fight, the entrant interpret that as a sign that the incumbent is *soft*. More plausibly, he would regard it as just a mistaken move, that ought not to affect his beliefs at all ("passive conjectures") or as a sign that the incumbent is *tough*. Such beliefs will not support the perverse equilibrium, though.
I think the Cho-Kreps Intuitive Criterion would apply here, even though this is not a signalling game, the type of game in which that criterion is usually applied. Here, intuitively, is how it works. Suppose the equilibrium is as described above, but the incumbent deviates and chooses Fight in the first period. The incumbent also makes the following speech:
I know you expected me to choose Accommodate, and that if you follow your "equilibrium"
behavior I will lose by having chosen Fight instead. But you should not follow that
behavior. You should realize that the only conceivable reason I would choose Fight is
if (a) I thought it would change your beliefs from the "equilibrium" ones, and (b) if I
were Tough. If I were Soft, I wouldn't choose Fight even if I thought I could change
your beliefs however I wanted. So you should believe I am tough-- or at least not
increase your belief that I am soft.
The difference from the typical signalling game is interesting, though. In the
signalling game, the deviator makes this speech so that his deviation will change the
other player's beliefs from what they were before the deviation. In the present game,
the deviator would be willing to make the speech even if its effect were to keep
the other player's beliefs the same as they were before the deviation. Here, the
speech is "defensive" rather than "offensive". I find this makes the refinement more
persuasive, actually. But then, I rather like passive conjectures anyway.
Cho, In-Koo, and Kreps, David M. "Signaling Games and Stable Equilibria." Quarterly Journal of Economics 102 (May 1987): 179-221.
Kreps, David, Milgrom, Paul, Roberts, John & Wilson, Robert (1982). Rational Cooperation in the Finitely Repeated Prisoners' Dilemma. Journal of Economic Theory 27: 245 -52
Kreps, David & Robert Wilson (1982) ``Reputation and Imperfect Information'' Journal of Economic Theory. August 1982. 27: 253-79.
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