JOURNAL

OF ECONOMIC

THEORY

Reputation27, 253-279(1982)and Imperfect

Information

DAVID

M. KREPS AND ROBERT WILSON

Graduate

School of Business, Stanford
Stanford,
California
9430.5

University.

Received June 18, 1980: revised June 22. 1981

A common observation in the informal literature of economics (and elsewhere) isthat in multistage “games,” players may seek early in the game to acquire a
reputation for being “tough” or “benevolent” or something else. But thisphenomenon is not observed in some formal game-theoretic analyses of finite
games, such as Selten’s finitely repeated chain-store game or in the finitely repeatedprisoners’ dilemma. We reexamine Selten’s model, adding to it a “small” amount ofimperfect (or incomplete) information about players’ payoffs, and we find that thisaddition is sufficient to give rise to the “reputation effect” that one intuitively
expects. Journal of Economic Literature,
Classification Numbers: 026. 2 13, 6 11.

1. INTRODUCTION
The purpose of this paper is to present some game-theoretic models that
illustrate the role of a firm’s reputation. Allusions to reputational effects
recur in the industrial organization literature on imperfect competition, butformal models and analyses have been lacking. Scherer [ 211, for example,
points to
the demonstration effect that sharp price cutting in one market can have on the
behavior of actual or would-be rivals in other markets. If rivals come to fear from amultimarket seller’s actions in Market A that entry or expansion in Markets B andC will be met by sharp price cuts or other rapacious responses, they may be
deterred from taking agressive actions there. Then the conglomerate’s expected
benefit from predation in Market A will be supplemented by the discounted present
value of the competition-inhibiting effects its example has in Markets B and C.
(page

338)

The intuitive appeal of this line of reasoning has, however, been called the“chain-store paradox” by Selten [24], who demonstrates that it is notsupported in a straightforward game-theoretic model. We shall elaborate
Selten’s argument later, but the crux is that, in a very simple environment,
there is no means by which thoroughly rational strategies in one market
could be influenced by behavior in a second, essentially independent market.
2530022-053
Alll/82/040253-27SO2.00/0

Copyright
0 1982 by Academic
Press, Inc.
rights of reproduction
in any form reserved.

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What is lacking, apparently, is a plausible mechanism that connects behavior
in otherwise independent markets.
We show that imperfect information is one such mechanism. Moreover,the effects of imperfect information can be quite dramatic. If rivals perceive
the slightest chance that an incumbent firm might enjoy “rapacious
responses,” then the incumbent’s optimal strategy is to employ such behavior
against its rivals in all, except possibly the last few, in a long string ofencounters. For the incumbent, the immediate cost of predation is aworthwhile
investment to sustain or enhance its reputation, thereby deterring
subsequent challenges.
The two models we present here are variants of the game studied by Selten[24]; several other variations are discussed in Kreps and Wilson 181. The
first model can be interpreted in the context envisioned by Scherer: A
multimarket monopolist faces a succession of potential entrants (though in
our model the analysis is unchanged if there is a single rival with repeated
opportunities to enter). We treat this as a finitely repeated game with theadded feature that the entrants are unsure about the monopolist’s payoffs,
and we show that there is a unique “sensible” equilibrium where, no matter
how small the chance that the monopolist actually benefits from predation,
the entrants nearly always avoid challenging the monopolist for fear of the
predatory response. The second model enriches this formulation by allowing,in the case of a single entrant with multiple entry opportunities, that also theincumbent is uncertain about the entrant’s payoffs. The equilibrium in thismodel is analogous to a price war: Since the entrant also has a reputation toprotect, both firms may engage in battle. Each employs its aggressive tactic
in a classic game of “chicken,” persisting in its attempt to force the other toacquiesce before it would itself give up the fight, even if it is virtually certain(at the outset) that each side will thereby incur short-run losses.
After reviewing Selten’s model in Section 2, we analyze these two models
in Sections 3 and 4, respectively. In Section 5 we discuss our results andrelate them to some of the relevant literature. In particular, this issue of the
Journal includes a companion article by Milgrom and Roberts ] 131 that
explores many of the issues studied here in models that are richer ininstitutional detail. Their paper is highly recommended to the reader.

2. THE CHAIN-STORE PARADOX
The models we analyze are variations on the chain-store game studied by
Selten [24]. Consider a sequential game with two players called the entrant(or potential entrant) and the monopolist. The entrant moves first, electing
either to enter or to stay out. Following entry, the monopolist chooses either
to acquiesce or to fight. If the entrant stays out, the incumbent is not calledREPUTATIONAND

IMPERFECT255

INFORMATIONPAYOFFSENTRANTSTAYSENTRANT’SMONOPOLIST’S

0

II

b

0

b -1

-1

OUT

FIG.

MONOPOLISTACQUIESCESMONOPOLISTFIGHTS

1. S&en’s chain-store game.

upon to move. Payoffs to the players, depending on the moves selected, aregiven in Fig. 1. We consider the case that a > 1 and 0 < b < 1.
How will this game be played? If the entrant enters, the monopolist
choosesbetween the payoffs 0 if it acquiescesand -1 if it fights, so surely it
will acquiesce. Anticipating this response,the entrant choosesbetween 0 if it
stays out and b if it enters, and so it will enter. This is one Nash equilibriumof the game, but there is another: If the entrant were to anticipate that themonopolist would fight entry, then the entrant would want to stay out. Notethat it costs the monopolist nothing to adopt the strategy “fight if entry” if
no entry occurs. So this is a second Nash equilibrium. But this second
equilibrium is not so plausible as the first. It depends on an expectation by
the entrant of the monopolist’s behavior that, faced with thefuit accompli ofentry, would be irrational behavior for the monopolist. In the parlance ofgame theory, the second equilibrium is imperfect. We suppose that theentrant adopts the “rational expectation” that the monopolist will acquiesce
to entry, and we expect the first equilibrium to ensue.
Consider next the case that the game in Fig. 1 is played a finite number oftimes. A single monopolist plays a successionof N different entrants, where
the monopolist’s total payoff is the sum of its payoffs in the N stage games.
Allow the later entrants to observe the moves in all earlier stages of thegame. Scherer’s reasoning predicts that in this case the “reputation” effectmight come to life: The monopolist, by fighting any early entry, might
convince later opponents that it will fight, thus deterring later entries. Indeed,
if this were the case, then also the early round opponents would not enter,not wishing to be abused for demonstration purposes. However, as Selten
argues,this does not withstand scrutiny. In the last stage the monopolist will
not fight becausethere are no later entrants to demonstrate for. So in the laststage, entry will surely occur. But then in the penultimate stage, themonopolist again has no reason to fight-it is costly in the short run and has
no effect on the last stage. The next-to-last entrant, realizing this, will surely
enter. This logic can be repeated, unraveling from the back: In each stage
entry and acquiescencewill occur. To be precise, this is the unique perfectNash equilibrium of the game; cf. Selten [22, 23, 241. Apparently, this modelis inadequate to justify Scherer’s prediction that reputational effects will playa role.

256

KREPS

AND

3. ONE-SIDEDWILSONUNCERTAINTY

Our contention is that this inadequacy arises becausethe model does not
capture a salient feature of realistic situations. (This contention was madefirst by Rosenthal [ 171, whose work we shall discussin Section 5.) In practical situations, the entrants cannot be certain about the payoffs to themonopolist. They may be unsure about the monopolist’s costs, or they may
be uncertain about nonpecuniary benefits that the monopolist reaps-thismay be a monopolist who enjoys being tough. The latter might be morecolorfully stated by saying that the monopolist plays tough “irrationally”;
according to Scherer [21, p. 247 1, “... fear of irrational or avowedly
rapacious action, then, rather than the expectation of rational pricing
responses,may be what deters the potential new entrant from entering on alarge scale.” For whatever reason, the entrants may initially assesssome
positive probability p that the monopolist’s payoffs are not as in Fig. 1 but
rather (in the simplest case) as in Fig. 2, reflecting a short-term benefit froma fighting response.In this case, later entrants, observing earlier moves, willrevise their assessmentp on the basis of what they see. Perhaps in this casethe reputation effect will come alive.
We model this formally as follows. There are N + 1 players, for N a
positive integer. One of the players is the monopolist; the others are called
entrant N, entrant N - l,..., entrant 1. The monopolist plays the game in
Fig. 1 against each entrant in turn: First it plays against entrant N, thenN - 1, etc. (We always index time backwards, and we refer to stage n as thatpart of the game that involves entrant n.) The payoffs for each entrant are
given in Fig. 1.
The monopolist’s payoffs are more complex: Its total payoff is the sum(undiscounted for now) of its payoffs in each stage, where the stage payoffs
are either all as in Fig. 1 or all as in Fig. 2. The monopolist knows which
payoff structure obtains. The entrants, on the other hand, initially assess
probability 6 that the monopolist’s payoff structure is the secondone. As thegame progresses,each entrant (and the monopolist) observesall prior moves.
Consequently, the history of moves prior to stage n may enable entrant n torevise this assessmentif the history reveals some information about therelative likelihoods of the monopolist’s two possible payoff structures.

MONOPOLIST’SENTRANT

STAYS

FIG.

OUT

2.

0
MONOPOLISTACQUIESCESMONOPOLISTFIGHTS

Payoffs

for a tough-10

monopolist.

PAYOFFSREPUTATIONAND

IMPERFECTINFORMATION

257

This model conforms to Harsanyi’s [7] formulation of a game with incomplete information. Alternatively,
it is a game with imperfect information(among the entrants)
and perfect recall, in which “nature”
initially
determines the monopolist’s payoff structure, and nature’s move is observedby the monopolist but not the entrants. In line with the first interpretation,
we refer to the weak monopolist or the strong monopolist, meaning themonopolist if its payoffs are as in Fig. 1 or Fig. 2, respectively.
.
Since the players have perfect recall, there is no loss of generality in
restricting attention to behavior strategies (Kuhn [ 10)). We wish to identify
a Nash equilibrium of this game and, moreover, we wish the equilibrium
identified to be perfect. That is, we wish to exclude equilibria that are basedon expectations by one player of another’s behavior that would not be
rational for the latter to carry out if called upon to do so. Because our games
have incomplete information, Selten’s [22] concept of subgame perfection isinadequate. His concept of “trembling-hand”
perfection (Selten [23]), on theother hand, is difficult to employ in games with strategy spaces as complex
as those present here. So we use an analogous equilibrium concept called asequential equilibrium. This is a refinement for extensive games of the usual
Nash equilibrium that captures the spirit of Selten’s perfectness criterion butthat is much easier to apply. General definitions and properties of sequential
equilibria are given in Kreps and Wilson [9], which we summarize here.
There are three basic parts to the definition of a sequential equilibrium:(a) Whenever a player must choose an action, that player has someprobability assessment over the nodes in its information set, reflecting whatthat player believes has happened so far. (b) These assessments are
consistent with the hypothesized equilibrium strategy. For example, theysatisfy Bayes’ rule whenever it applies. (c) Starting from every information
set, the player whose turn it is to move is using a strategy that is optimal forthe remainder of the game against the hypothesized future moves of its
opponent (given by the strategies) and the assessment of past moves by otherplayers and by “nature” (given by the assessment over nodes in the information set). The difference between this and the standard concept of a Nashequilibrium is that (c) is required for every information set, including thosethat will not be reached if the equilibrium strategies are followed. So eachplayer will be willing to carry out its strategy at every point in the game, if
ever it is called upon to do so. The properties are: Sequential equilibria exist
for all finite extensive games. They are subgame perfect Nash equilibria. Fora fixed extensive form and probabilities of nature’s moves, as we vary the
payoffs it is generic that all strict sequential equilibria are trembling-handperfect, and the equilibrium path of each sequential equilibrium is anequilibrium
path for some trembling-handperfect equilibrium.
Every
trembling-hand perfect equilibrium is sequential.
In the context of the game analyzed here, the definition of sequential

258

KREPS

AND

WILSONequilibrium specializes as follows. An equilibrium comprises a (behavior)
strategy for each player and, for each stage n = N,..., 1, a function p, taking
histories of moves up to stage n into numbers in [0, l] such that: (a) Startingfrom any point in the game where it is the monopolist’s
move, themonopolist’s strategy is a best response to the entrants’ strategies. (b) For
each n, entrant n’s strategy (contingent on a history h, of prior play) is abest response to the monopolist’s strategy given that the monopolist is strong
with probability p,(h,). (c) The game begins with pN = 6. (d) Each p,, iscomputed from pn+ , and the monopolist’s
strategy using Bayes’ rule
whenever possible. (We will not write (d) precisely-it
will be transparent
when we give the equilibrium below. That (d) implies “consistency
ofbeliefs” in the sense of Kreps and Wilson [9] may not be apparent, but it
does follow from the simple structure of the game being considered here.)
The interpretation is that p, gives the probability assessed by entrant n thatthe monopolist is strong as a function of how the game has been played up
to stage n. Note that in (a) the monopolist’s assessment over nodes in its
information set is omitted, because all of its information sets are singletons.
We now give a sequential equilibrium for this game. This particularsequential equilibrium has the fortuitous property that, in terms of play fromstage IZ on, p, is a sufficient statistic for the history of play up to date n.
That is, the choices of the players at stage n depend only on p, and (for themonopolist) the move of entrant n; and p, is a function of pn+, and themoves at stage n + 1. We are lucky to be able to find a sequential
equilibrium with this simple structure: it is not generally the case that one
can find sequential equilibria for which the players’ assessmentsaresufficient statistics for past play. (See remark (A) below.)
We begin by giving the functions p,. Set pN = 6. For n < N, if the historyof play up to stage n includes any instance that entry was met byacquiescence, set p,, = 0. If every entry so far has been met by fighting, andif k is the smallest index (> n) such that there was entry at stage k, then set
pn = max(bk-‘,
S). If there has been no entry, set p, = 6.
This corresponds to the following recursive definition:(a)

If there is no entry at stage n + 1, then p,, =p,,+ , .

(b) If there is entry at stage n + 1, this entry is fought, and p,,+ , > 0,then pn = max(b”, p,+ ,),(c) If there is entry at stage n + 1 and either this entry is met byacquiescence or p,,+ , = 0, then p,, = 0.
Now that we have described how pn is computed at every node in thegame tree, we can give the strategies of the players in terms ofp,.
Strategy of the Monopolist(a)If the monopolist

is strong, it always

fights entry.

REPUTATIONAND

IMPERFECT259

INFORMATION(b) If the monopolist is weak and entry occurs at stage n, themonopolist’s
response depends on n and pn: If n = 1, the monopolist
acquiesces. If n > 1 and pn > b”- ‘, the monopolist fights. If n > 1 andthe monopolist
tightswithprobability((1 - b”-‘)p,)/
pn < b”-l,(( 1 -p,) b’-‘)and acquiesces with the complementary probability. (Notethat when pn = 0, the probability of fighting is zero, and when p, = b”- ‘, theprobability of fighting is one.)
Strategies of the Entrants
If p, > b”, entrant n stays out, If p, < b”, entrant n enters. If p, = b”,
entrant n randomizes, staying out with probability l/a.
PROPOSITION
1. The strategies
sequential equilibrium.

and beliefs given

above constitute

a

ProoJ
We only sketch the proof, leaving details to the reader. In thecontext of this game, there are two things to verify: First, the beliefs of theentrants must be consistent with the strategy of the monopolist, in the sensethat Bayes’ rule holds whenever it applies. Second, starting from any information set in the game, no player has the incentive (in terms of the payoff
for the remainder of the game) to change its selection of move at that information set. For entrants, this verification is made using the beliefs given
above. (Once this is verified, the Bellman optimality principle together withthe fact that beliefs are Bayesian consistent ensures that no player canunilaterally change its strategy and benefit starting from any point in thegame tree.)
The verification of Bayesian consistency is easy. If no entry takes place atstage n, nothing is learned about the monopolist, and we have pnp, =p, insuch instances. If p, > b”-I, then the monopolist is supposed to fight entry.
If p, = 0. then the monopolist is supposed to acquiesce. So in these cases,
Bayes’ rule implies that pnpl =p, (as long as the monopolist follows its
strategy). In each case, this is what we have. Finally, for p, E (0, b”- ‘), there
are positive probabilities that the monopolist will acquiesce and that it willlight entry. It only acquiesces if it is weak, and, indeed, in this case we havep,- 1 = 0. If it fights, Bayes’ rule requires thatpn-, = Prob(monopolist

strong 1monopolist fights)= Prob(monopolist

strong and lights)/Prob(fights)= Prob(fights

Prob(fights 1strong) . Prob(strong)
1strong) . Prob(strong) + Prob(fights 1weak)
1 .Pn= 1 .P,+
642/U/2-2[(Cl -b”-‘)p,,)/((l-P,>b”-‘)I[1 -P,I

= b”-‘,. P(weak)

260

KREPS

AND

WILSON

which is what we have posited. Thus beliefs and strategies are Bayesian consistent.
Note that there are two instances in which Bayes’ rule does not apply:p,, > b”-’
and the monopolist acquiesces to entry; pn = 0 and the monopolist
tights. In each case we set pnP1 = 0. In words, we assume that any
acquiescence is viewed by the entrants as “proof’
that the monopolist is
weak, and the entrants are unshakeable in this conviction once it is formed.
This assignmentof beliefs off the equilibrium
path is somewhat
arbitrary-there
are other assessments that work as well. But this assignment
is not wholly capricious-there
are assessments that would not give an
equilibrium. (This will be discussed more fully below.)(Repeating an earlier contention, this set of assessments is consistent in
the sense of Kreps and Wilson [9]. A direct proof is not difficult.)
Verification that the entrants are playing optimally is straightforward.
If
pn > b”-‘,
entrant n expects entry to be fought, and so it stays out. Ifp,, E (b”, b”-*), acquiescence will occur with positive probability, but withprobability
less than 1 - 6. Again it is better to stay out. If pn = b”,
acquiescence follows entry with probability 1 - 6, and the entrant is indifferent. If p,, < b”, the probability of acquiescence exceeds 1 - 6, and theentrant enters.
To see that the strong monopolist is playing optimally, note that if theentrants follow the strategy above, acquiescence at any point results in more
future entries than does fighting. In the short run fighting is better for thestrong monopolist, and in the long run fewer entries are better, so the strongmonopolist will always fight.
Finally, for the weak monopolist, one can verify inductively that giventhat these strategies are followed from stage n to stage 1, the expected payoff
to the weak monopolist from stages n to 1 is given by the following functionof pn:u,(P,) = 4t - &3J + 1) + 1
= a(t

=1

-

k(p,)

+

1)

if

6” < p, = bk(Pn’-‘,if6” <p,, < bkcpn’-‘,if

pn = b”, and

if

p,

<

b”,

where k(p) = inf(n: b” <p} for p > 0, and k(O) = 00. Now suppose thatentry occurs at stage n. By acquiescing, the monopolist receives zero both in
this stage and in the rest of the game (since pn-, will be set equal to zero).
By fighting, the monopolist receives -1 in this stage and future expected
payoffs of 0 ifp, = 0, 1 ifp, E (0, b”-‘1, and more than 1 ifp, > b”-‘. Thus
the weak monopolist is happy to follow the strategy given above. 1

REPUTATIONAND

IMPERFECT261

INFORMATION

It is easiest to understand the nature of this equilibrium by tracing throughthe play of the game for “typical”
values of 6 and b, say 6 = l/10 andb = l/2. Note that in this case, k(6) = 4. Refer to Fig. 3. At stage N(presumed to be greater than 4) the game begins with pN = 6. At this stage,
the monopolist would fight entry regardless of its payoffs, so entry isforestalled. The game evolves along arrow (a) to the point pN- , =pN = 6.
Note that if there is entry, the monopolist is willing ex post to fight-toacquiescemoves the game along arrow (b) to pN-, = 0, from which point themonopolist nets zero. Fighting costs 1 immediately, but acquiescing costs
much more in the future. (Note that all that is necessaryis that acquiescence
cost at least one-as long as acquiescenceresulted in pNm, < l/16 this would

STAGE N-l

I
PN.,’

’

I

STAGE

N-2

STAGE

4

I

STAGE

3

P2 = I

STAGE

2

STAGE

1

p,-;

P&=0)

(0

1

iP‘$‘l
r3-

--I

I

-.-I

p,-0

P, = 1
FIG.

3.

Temporalevolutionof the

game.

262

KREPS

AND

WILSONbe so in this case. So here is one place where the non-Bayesian reassessments
need not be precisely as specified above to have an equilibrium. But note that
pN-, < l/16 is necessary-otherwise
the weak monopolist would rather
acquiesce than fight at this stage.) The game continues in this fashion(arrows (c), (e), and (f)) until date 4 (=/c(6)). At this date the monopolist
might acquiesce if it is weak and if it is challenged-thestrategy of the weakmonopolist is chosen so that following entry, acquiescence leads to p3 = 0(arrow (g)) and fighting leads to p3 = b3 (arrow (h)). But this does not give
entrant 4 enough incentive to enter-thegame actually evolves along arrow(i) to p3 = 6. At date 3 the weak monopolist again will randomize ifchallenged (so that arrows (j) and (k) give the posteriors), and now there ishigh enough probabilityof acquiescence for entrant 3 to enter. If themonopolist acquiesces, the game moves along arrow (j) to pz = 0. At thispoint the monopolist is known to be weak, and entrants 2 and 1 both enterwith the monopolist acquiescing each time. The monopolist thereafter is
supposed to acquiesce; if it tights instead at, say, date 2. entrant 1 disregards
this and continues to believe that the monopolist is weak. That is, p, = 0 if
either the monopolist acquiesces or if it fights. (Note that we could have
p1 < l/2 if the monopolist fights, and still we would have an equilibrium. But
if pI > l/2, then the weak monopolist would prefer to fight, upsetting theequilibrium. Again there is some freedom in defining beliefs off the equilibrium path, but not complete freedom.) Back at stage 3, if the monopolist
fights entry, the game moves along arrow (k) to pz = l/4. At this point
entrant 2 is indifferent between entering and staying out, and chooses
between the two randomly.
If entrant 2 enters, the weak monopolist
randomizes between acquiescence (arrow (n)) and fighting (arrow (p)). If
entrant 2 stays out, the game moves along arrow (q) to p, = l/4 (and entrant
1 surely enters).
The remarkable fact about this equilibrium is that even for very small 6.
the “reputation”
effect soon predominates. Even if the entrants assess a onein-one-thousand chance that the monopolist would prefer (in the short run)to fight, if there are more than ten stages to go the entrant stays out because
the monopolist will surely fight to preserve its reputation. Note the “discontinuity” that this causes as the number of stages in the game goes to infinity:
lim v,(6)/N

= a

if

6 > 0, and

=o

if

6=0.

N-rCC

The obvious question at this point is: To what extent is this equilibrium
unique? It is not the case that it is the unique Nash equilibrium for thisgame, for the following four reasons.
(a)There are other Nash equilibria

that are not sequential equilibria.

REPUTATIONAND

IMPERFECTINFORMATION

263(That is, that are not, roughly speaking, perfect.) For example, it is a Nashequilibrium for the monopolist to light any entry (regardless of its payoffs),
and for the entrants never to enter. But this behavior is not “ex post rational”
for the weak monopolist in stage one. In general, we wish to allow only
sequential equilibria, and we confine attention to those for the remainder ofthis discussion.
(b) There are sequential equilibria where the strong monopolist
acquiesces to entry. For example, if N = 2, b = l/2, and 6 = 213 (very highprobability that the monopolist is strong), it is a sequential equilibrium forentrant 2 to enter, the monopolist to acquiesce to this entry regardless of its
payoffs, and for entrant 1 to adopt the strategy: Stay out if the monopolist
acquiesces in stage 2; enter if the monopolist fights in stage 2. (In stage one,the monopolist responds with its ex post dominant action.) This is sequential
because it is supported by the following beliefs of entrant 1:
Prob(monopolist

strong ) acquiescence in stage 2) = pz = 2/3,

Prob(monopolist

strong 1fight in stage 2) = l/4.

For the given strategies, the first of these reassessments follows from Bayes’rule, and the second is “legitimate”
because Bayes’ rule does not apply:
There is zero prior probability that the monopolist will light in stage 2.
Although this is a sequential equilibrium, we contend that it is not very
sensible. The flaw is in the beliefs of entrant one-if there is lighting in stage
2, entrant 1 revises downward the probability that the monopolist is strong.
Intuitively it seems at least as likely that the strong monopolist would defectand light as that the weak monopolist would do so. Thus it seems intuitivethat entrant one will assess
Prob(strong

1fight) > Prob(strong( acquiesce).

But if we insist on this condition holding, then the equilibrium given
immediately above is excluded.
Putting this formally, we will call the beliefs {p,} of the entrants plausible
if given two histories h, and h; of play up to stage n, if h, and h; are thesame except that some plays of “fight” in h, are “acquiesce” in hi, thenp,(h,) >p,(h;).
We wish to allow only sequential equilibria that aresupported by plausible beliefs. Note that this is not true of the equilibrium
immediately above, but it is true of the equilibrium Proposition 1.
(c) In the sequential equilibrium given in this section, there is some
freedom in describing what happens off the equilibrium path. For example,
we have said that if pn = 0 and the monopolist fights entry, then the entrants
set p, _ I = 0. Thus once p, = 0 in our equilibrium, every subsequent entrant
enters. But we would also have an equilibrium if we set pn-, = b”-’ after

264

KREPS AND WILSON

such a defection from the equilibrium,
and then entrant n - 1 would
randomize between entering and staying out. Note well, this concerns the
behavior of entrant n - 1 only off the equilibrium path, but in terms ofstrategies it is a different equilibrium. We cannot hope to have uniqueness off
the equilibrium path.
(d) Finally, there is a bit of freedom in defining equilibria along theequilibrium path when 6 = b” for some n < N: The behavior of entrant n inthis case need not conform to the strategy above-any randomization will
work.
Except for these four problems, we do get uniqueness:
PROPOSITION 2. If 6 # b” for n < N, then every sequential equilibriumwith plausible beliefs has on-the-equilibrium-path strategies as described
previously. Thus every sequential equilibrium with plausible beliefs has thevalue functions given above.

The proof is by induction and is left to the reader. We simply note that incarrying out the induction one establishes the following:(a) The value function of the strong monopolist (in equilibrium) will bea nondecreasing function of p,, , and the strong monopolist will therefore fightany entry.
(b) The value function of the weak monopolist will be a nondecreasing
function of pn and will be given by the formula in the proof of Proposition 1
for 6 # b”, m < n.
(c) If there is entry at stage n and if the monopolist lights this entry,
then entrant n - 1 must stay out with probability exactly l/a.
By going through this proof, the reader will see the intuition behind thisequilibrium, which we will try to summarize here. As long as beliefs areplausible, the strong monopolist
will always fight entry. Thus any
acquiescence is conclusive proof that the monopolist is weak. Moreover,such evidence once given must result in zero payoff for the monopolist-theargument of Selten that we have given in Section 1 applies (with minor
modifications).
If entrant n is to enter, then it must be that there is
probability 1 - b (at least) that the monopolist will acquiesce, which requiresthat the weak monopolist is randomizing or simply acquiescing. This also
requires that pn & b, and, from Bayes’ rule, that if this entry is met byfighting, then pnPl ap,/b. Thus if we begin with 6 > b”, there can be at
most m entrants who have a positive probability of entering. As N gets large,
then, the value to either monopolist must asymptote to aN, and, for N > 2m,
the weak monopolist would always wish to light entry. (In fact, in theequilibrium in turns out that this is true for N > m.)

REPUTATIONAND

IMPERFECTINFORMATIONWe close this section by listing several extensions and embellishments
the basic model.

265

of

(A) We have dealt above with the case a > 1. If 0 < a < 1, then thesame basic structure for the equilibrium emerges, in that for sufficiently largen, entrants do not enter because the monopolist will fight with probability
one. The play near the end of the game is more complicated however. In
particular, one cannot obtain an equilibrium where entrant n’s strategy
depends only on p,--it depends instead on p, and the history of play in thelast j rounds, where j is the smallest integer such that &I > 1.
(B) If the monopolist discounts its payoffs by a factor p per period, thefollowing results. If p > l/a, then the equilibrium is precisely as above exceptthat the randomizing
probabilitiesof the entrants must change. If
p < l/(u + 1), then the equilibrium is quite different-the
weak monopolist
acquiesces at the first entry, so entrants enter ifp, < b and stay out ifp, > b.
For p such that l/(u + 1) < p < l/u, the basic character of the equilibrium isjust as in the case of p > l/u-forlarge enough n entrants stay out because
the monopolist will fight any entry. But the equilibrium is complicated forsmall n, resembling the equilibrium in the undiscounted case where a < 1.
(C) Suppose that instead of the sequential game depicted in Fig. 1, eachstage consists of a two-by-two simultaneous move game, Table I, where thepayoffs with probability
1 - 6 are shown in (a) and the payoffs withprobability 6 are shown in (b). (We assume 0 < b < 1 and a > 1.) Otherwise
the structure of the game is the same: One of these two bimatrices is chosen
at the outset, according to the probabilities given. One monopolist plays
against N entrants in sequence. The monopolist knows which bimatrix waschosen; the entrants do not.
For 6 = 0, the argument of Selten is easy adapted to show that the uniqueequilibrium (perfect or not) has row 2, column 1 played in each stage. Thisis because row 2 is strongly dominant in the stage game. But for 6 > 0 we
get an equilibrium almost identical to the one discussed above: For stages nsuch that b” < 6, the monopolist plays row 1 regardless of which bimatrix
was selected, and the entrant responds with column 2. (The play of the gameis a bit different near the end of the game.) So we see that a little incomplete

TABLE

(a)

I

(b)

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information can not only make an imperfect equilibrium
perfect (moreaccurately, sequentialFit
can also make as part of a sequential equilibrium
the play of an action that with very high probability is strongly dominated in
the stage game.
(D) Paul Milgrom has pointed out to us that similar equilibria can befound even when every player in the game knows the payoffs of themonopolist, as long as this knowledge is not common knowledge. That is
suppose all the entrants know the monopolist’s payoffs, but they are notcertain whether their fellow entrants have this information. Then (with theproper precise specification) they fear that the weak monopolist will fight(for large n), in order to maintain its reputation among the other entrants.
This being so, the entrant will not enter. And the monopolist, even if it
knows that all the entrants know that it is weak, may be willing to tightentry early on, in order to help “convince” subsequent entrants that it (themonopolist) is not sure that the entrants know this. (Precise arguments ofthis form are found in Milgrom and Roberts [ 131.) Selten’s argument
requires that it is common knowledge that the monopolist is weak. In reallife contexts this is a very strong assumption, and weakening it ever soslightly (more slightly than we have done above) can give life to the“reputation” effect.
(E) We have dealt exclusively with the case of a single monopolist
playing against N different entrants. Another interesting case is where asingle monopolist plays N times against a single entrant. For the game wehave analyzed in this section, this turns out to have no effect on theequilibrium. (We leave this to the reader to verify.) But as we shall see in thenext section, this is due (at least in part) to the fact that there is no uncertainty about the payoffs of the entrants.

4. TWO-SIDED

UNCERTAINTY

In this section we consider what happens when the monopolist is unsure
about the payoffs of the entrants. The most interesting formulation of thisproblem is where a single monopolist plays the stage game of Fig. 1 a totalof N times against a single opponent. The payoff to each player is the sum of
the player’s payoffs in each stage. The monopolist’s payoffs are as in Fig. 1
or Fig. 2, with probabilities 1 - 6 and 6, respectively. The entrant’s payoffs
are as in Fig. 1, for some b such that 0 < b < 1 with probability 1 - y andfor some other b > 1 with probability y. Each player knows its own payoffs
at the start of the game, and each is unsure of the payoffs of its opponent.
The payoffs are statistically independent.
Continuing the terminology of Section 3, we shall refer to the weak entrant

REPUTATIONAND

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INFORMATION

as the entrant if its payoffs satisfy 0 < b < 1, and the strong entrant if its
payoffs satisfy b > 1.
Note that the strong entrant does better to enter than to stay out in any
stage, even if the monopolist is sure to fight. Because it seems plausible thatentry will not decrease the probability that the monopolist will acquiesce
subsequently, we look for equilibria where the strong entrant always enters.
Thus any failure to enter brands the entrant as weak, at which point we areback to the situation of Section 3. (Recall that it did not matter there
whether there was a single entrant or N entrants.) Similarly, we look for an
equilibrium where the strong monopolist always fights. Thus any failure totight brands the monopolist as weak, following which the entrant always
enters and the monopolist always acquiesces. We search, then, for an
equilibrium of the following sort: The strong entrant always enters. The
strong monopolist always fights. The weak entrant chooses a strategy that isa mixture of “stopping rules”: A stopping rule gives the date at which theentrant will “give in” and not enter if the monopolist has not acquiesced yet.
(The entrant may later re-enter, as we will then follow the equilibrium of
Section 3.) The weak monopolist will also mix among stopping rules: A
stopping rule for the monopolist gives the date at which the monopolist will
first acquiesce if the entrant has not retreated first. If one side or the othergives in, we move to either the situation of Section 3 or to where entryacquiescence follows until the game ends.
Giving a complete specification of the equilibrium that is obtained is
extraordinarily tedious, because it is based on some very involved recursions.
Still, we can give a rough description of what happens. At any stage n theprevious play of the game is summarized into two statistics: p,, theprobability assessed by the entrant that the monopolist is strong; qn, theprobability assessed by the monopolist that the entrant is strong. (The game
begins with pN = 6 and qN = y.) Thus the “state space” of the game at stage nis the unit square, as depicted in Fig. 4. The edge q, = 0 is the subject of

FIG.

4.

State space of thegame

at stage

n.

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Section 3. The edge p,, = 0 can be analyzed using the argument of Seltenwith the conclusion: The entrant always enters, and the weak monopolist
always acquiesces.
The square is divided into two regions by a curve, as shown. If (p,, qn)
lies in region I, say at the point x, then the entrant enters regardless of its
payoffs, and the weak monopolist
randomizes. If the weak monopolist
acquiesces, the game evolves to the pointy (actually, to this point in the next
square-thesquare for stage n - 1). If it lights, or if it is strong and hencetights, the entrant uses Bayes’ rule to compute p, _ r, landing at the point z
on the curve, and just beyond the curve in region II of the next square. If(p,, qn) lies in region II, say at x’, then the weak entrant randomizes. If it
stays out, the equilibrium of Section 3 ensues-andthe next stage begins a
the point y’. If it enters (or if it is strong and therefore enters), themonopolist recomputes the probability that it is strong, landing at the pointx” along the curve. Then the monopolist randomizes (if weak), and the gameevolves to y” or to z”. Both z and z” are in region II of the next square, so
the next round begins with randomization by the weak entrant, and so on.
Except for the very start of the game, when pN = 6 and qn = y, most of theplay takes place along the curve. (Actually, the curve shifts slightly as nchanges.) So we see an initial jump to the curve (or to one of the two edges),followed by a slow climb up the curve with ever present the chance that ajump to one of the edges will occur. With probability one, a weak player will
eventually give in, so we either jump to an edge eventually or, if both players
are strong, we reach the point p. = q. = 1. This is a game of “chicken,”where once begun, each side (if weak) randomizes
between a small
probability of giving in and a large probability of daring the other side forone more round. The relative size of these probabilities is required by theconditions of an equilibrium: Daring once more costs something this round,but giving in is costly for the rest of the game. So it must be that daring once
more does not give either player a substantial chance of immediate gain; theopponent must be about to “dare” once again with large probability.
While it is tedious to give the exact equilibrium in the discrete time
formulation, it is relatively simple to do so in a continuous time version of
the game. So we shall now develop that continuous-time version. (We should
forewarn the reader: We will be somewhat sloppy in what follows. But
everything we say can be made exact.)
To begin, consider the game of Section 3 played against a single entrant
over the time period N to 0. Instead of playing at times N, N - I,..., I, for the
stakes (per stage) given in Section 3, we imagine that an integer K is given,and that the game is played “more frequently, for reduced stakes,” with playat times N, (KN - 1)/K, (KN - 2)/K ,..., l/K, for stakes l/K times the stakes
given. It is the number of times that the monopolist has left to demonstrate
its “toughness”
that is decisive in Section 3, so we find that if k(6) = n, theREPUTATIONAND

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269

entrant stays out (and the monopolist would surely fight) at all times
t > n/K. As K goes to infinity, we see that the entrant stays out until “thevery last instant” of time.
With this limiting result as motivation, we now consider the “continuoustime” version of the above game, played over a time interval T to 0. At eachtime t E [T, 01, the entrant chooses whether to enter or to stay out, and themonopolist chooses whether to fight entry or to acquiesce. A realization of
the entrant’s strategy is formalized as a (measurable) function e: [T, 0] -+(0, 1 }, where e(r) = 1 means that the entrant is entering at date t. A
realization of the monopolist’s
strategy is formalized as a function
f: [T, O]-, {O, 11, where f(t) = 1 means that the monopolist is fighting at
date t. (We have a “closed-loop” game, so pure strategies would be a pair offunctions e and f where e(t) is F((e(s), f(s)), s < t)-measurable, and f(t) isF(f(s), s < t; e(s), s < t)-measurable. We shall not try to be more precise
about this here; instead we trust the reader’s ability to see how to formalize
what follows.) Given realizations e and f, payoffs to each side are determinedby measuring the lengths of times during which there is not entry (e(t) = 0),
during which entry is fought (e(t) = 1, f(t) = l), and during which there isacquiescence to entry (e(f) = 1, f(t) = 0), and assigning payoffs accordingly.
For example, if A denotes Lebesgue measure, then the weak monopolist’s
payoff is
A(e(t) = 0} aa - A{e(t) = l,f(t) = l}.
In this game, an equilibrium calls for the entrant to stay out as long as themonopolist does not acquiesce and to always enter after any acquiescence isobserved; for the strong monopolist to fight any entry; and for the weakmonopolist to tight as long as it has not acquiesced yet and to acquiesce
forever after an acquiescence. The reader can easily verify that this is anequilibrium. Moreover, if by some “mistake” the entrant entered before time
0, the weak monopolist would want to fight: By acquiescing it saves an“instantaneous” one unit, but then it invites entry for the remainder of thegame-a substantial loss that outweighs the instantaneous savings.
(The reader is entitled to be somewhat skeptical about this. By moving toa continuous-time formulation, we have obtained some of the features of thesupergame (infinitely repeated) formulation. For example, the equilibrium
above is “perfect” even if 6 = 0, just as in the supergame with 6 = 0. But inthe case 6 = 0, this equilibrium is not the limit of discrete-but-more-rapid
equilibrium play. What justifies this particular equilibrium in the case 6 > 0
is that it is the limit of discrete equilibria. We shall return to this point after
we discuss the case of two-sided uncertainty.)
Now consider the continuous-time game where there is uncertainty on
both sides. The formulation is as above, but now there is uncertainty (at theoutset) about the entrant’s payoffs as a function of the realizations of e andJ

270

KREPS

AND

WILSONWe are looking for an equilibrium with the following characteristics:(1) Thestrong monopolist always tights. (2) The strong entrant always enters. (3) By
virtue of (l), if the monopolist ever declines to fight an entry, it is revealed
as weak. Thereafter, the entrant always enters and the weak monopolist
always acquiesces. (4) By virtue of (2), if the entrant ever fails to enter it isrevealed as weak. Assuming that the monopolist has not previously been
revealed as weak, the game proceeds as above, with the weak entrant staying
out until the end and the monopolist always ready to fight.
Just as in the discrete time formulation, an equilibrium with these features
can be recast as an equilibrium in “stopping rules” for the weak entrant andmonopolist-eachchoosing the date at which it will “give in” if its opponent
has not given in yet. If the entrant gives in first (at date t), then regime (4)above takes effect, with the weak monopolist obtaining at for the rest of thegame, and the weak entrant receiving 0. If the monopolist gives in first at t,
regime (3) ensues, with the weak entrant receiving bt and the weakmonopolist receiving 0. Until one side or the other gives in, the weakmonopolist receives -1 per unit of time, and the weak entrant receives b - 1
per unit of time. The equilibrium condition is that each player’s stopping
time should be optimal given the probability distribution of the other’s, andgiven the assumption that the other player, if strong, will never give in. This
game is very similar to the “war of attrition”
game; cf. Riley [ 161 andMilgromand Weber [ 141. It is formally equivalent to a two-personcompetitive auction, where the stopping times are reinterpreted as bids. Thisobservation will be especially useful later when we discuss the connection
between this continuous-time
formulation and the discrete-time formulationof the game. (We are indebted to Paul Milgrom for acquainting us with theand for pointing out the relevance of his work with“war of attrition”
Weber.)
It is easiest and most illustrative to present the equilibrium using a
diagram similar to Fig. 4. In Fig. 5 we have the “state space” of thisgame-theunit square, interpreted
exactly as in Fig. 4. The bottom
boundary is where the entrant is known to be weak. Along this boundary(excluding the left hand endpoint) the weak monopolist’s payoff function (atdate t) is ul(p, 0) = at and the weak entrant’s is ul(p, 0) = 0. The left handboundary is where the monopolist is known to be weak-here(including thebottom endpoint) v,(O, q) = 0 and ~~(0, q) = bt.
The nature of the equilibrium is just as in the discrete case: The state
space is divided into two regions by a curvef(p,q) = 0 that passes throughthe points (0,O) and (1, 1). If the initial data of the game place us inregion I, then the game begins with the entrant entering for sure and themonopolist(if weak)randomizing
betweenfighting and immediate
capitulation. This randomization is such that if the monopolist does fight attime T, the entrant revises its assessment that the monopolist is strong so as

REPUTATIONAND

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271

Vt (p,O)= at, ut (p ,O):O
p- PROBABILITYMONOPOLISTFIG.

THAT THE
IS STRONG

5. State space of the continuous-time game.

to go to the curve f(p, q) = 0. From region II it is the (weak) entrant that
randomizes between immediate capitulation and entry-if
it does enter, themonopolist revises its assessmentthat the entrant is strong up to a point
where the curve f(p, q) is reached. Thereafter, the weak monopolist andweak entrant randomize “continuously” between keeping up the tight andcapitulating-this is done in a fashion so that as long as they continue tofight, the Bayesian reassessmentsof each side that the other is strong causes(p,, qr) to slide up along the curve toward (1, 1). (Of course, if one side or
the other capitulates, transition is made to the appropriate boundary.) Thereis a time P > 0 such that by this time, one side or the other (if weak) has
given in with probability one-if both sides are strong, at this time the point(1, 1) has been reached, and we remain there until time t = 0.
The difference between this equilibrium and the one for the discrete time
game (and the reason that this one is so much easier to compute) is that the
curvef(p, q) = 0 does not change with t in the continuous-time case. This isso because in the continuous-time version of the game, a game of duration
T/2 is strategically equivalent to a game of duration T, so long as the priors(6, y) are the same. All that changes is that everything takes place twice as
rapidly-we could as well think of the game taking place at the same speed
but for half the stakes.The values are half as large, but nothing else changes.
We now present a heuristic derivation of the equilibrium, assumingthat it
has the form outlined above. Note first that along the curve the valuefunctions for each side must be identically zero. This is so because (byhypothesis) both sides are randomizing continuously, and one outcome ofthese randomizations transfers them to points (the lower boundary for theentrant and the left-hand boundary for the monopolist) where the valuefunction is zero. Let rcI(pI, q,) and pI(pI, qJ be the hazard rate functions
associatedwith the weak monopolist’s and entrant’s randomizations at time t

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with posteriors pI and qt lying along the curve. That is, in the time interval(t, t - h) there is (up to terms of order o(h)) probability(1 -pl) .
n,(p,, qJ . h that the monopolist will give in, and (1 - ql) . pl(pr, qt) . h thatthe entrant will give in. Assuming sufficient continuity, if the value functions
are to be constant (and zero) along the curve, it is necessary that the
expected change in value to the weak monopolist be zero. Up to terms oforder o(h), this is-h + [(I - 41) . Pt(Pt, 4r) . hlbtl= o(h).

That is, the weak monopolist’s immediate cost 4 of keeping up the bluff
should be offset by the small chance (1 - qJ . pI(pI, q,) +h that the entrant
will give in times the large gain ut that will accrue in this event. The
analogous argument for the entrant gives(b - 1)h + (1 -pJ
Dividing* 7T,(pt, qJ * hbt = o(h).

by h and passing to the limit, we getPt(Pv 4r) = l/Ml

and- 41))%(P,Y 4t) = (1 - b)lW(l-P,>)*

Consider next the evolution of the posteriors pt and qt. The probability
table that the monopolist would construct at date t for the joint probability
distribution that the entrant is weak or not and will give in or not in theinterval (t, t - h) is given in Table II (up to terms of order o(h)). Thus theconditional probability that the entrant is strong, conditional on not giving inover the interval (t, t - h), is
qt-h = 4dll

- (1 - 4t)Pt(Pt,

4Jhl.

Thus, ignoring terms of order O(h),(91 - qt-/J/h= 4r(l - 4t)Pt(Pt,St>*

Passing to the limit, this gives 4, = (ql - q:) pI(pt, qJ = qJ(at).
p, =p,(l - b)/(b). Thus along the curve we must have&t/d’,= (q,b)l((l
TABLE

qive
not

m

- b) UP,>.
II

weak

s+rO”g(1 - s+)p,~~+,o+)h0(1 -q,)(l-p,(p,,q+)h)(1 - 9, 141
41(1 - 0, )p, (P, ,o,lh
1 -(1 - q,)p,(~,,q+)hSimilarly

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This is independent of t, and it is easily integrated to give qr = k(pJC where
k is a constant of integration and c = b/((l - b)a). To ensure that (1, 1) ison the curve, we must have k = 1. Therefore the curve is given byf(p,q) s q”-b’lb

-p

= 0.

(Note well the normalization of k so that (1, 1) is on the curve. This will be
important later on.)We can solve similarly for q1 and pt. Integrating 4, = q,/(at) yields
qf = k’t-‘I”.
Analogously, pI = k”t- (l-b”b . The constants k’ and k” aredetermined by the initial conditions. Suppose, for example, that we initiallyhave a prior (6, y) that lies in region II. Then the initial randomization is by
the weak entrant and yields posterior qT = 6’ if the entrant does enter.
Solving for k’ yields k’ = ~5’Tl’~. Solving for k” yields k” = BT(l-bf’b. Thus
pr = B(T/t)(l-b’fb

andqI = ,c(T/t)“?

Note that these yield pt = 1 and q1 = 1 for t = TP = T66”1-b’. (Of course,
both p1 and qt hit one simultaneously as the curve has been normalized topass through (1, l).) The point to note here is that in this equilibrium, theposterior (1, 1) will be reached at a time T” strictly between T and 0 (unless
6 = 0 or 1) so long as neither player gives in previously. But of course, theposterior (1, 1) can only be reached if with probability one both of the weak
players would have given in. So, according to this equilibrium, if the twoplayers are both strong, they will learn this before the game terminates. Putanother way, the date To previously referred to is 86/(1-b)T (for (6, r) inregion II). The formulae change somewhat for (6, y) in region I, but thequalitative conclusions are the same.
Does this heuristic derivation hold up? That is, do we really have an
equilibrium? There are two things to worry about. First, in several places theheuristic arguments that we give depend somewhat on sufficient regularity of
the functions rr, and pt. The reader can make these arguments rigorous forthe functions that we derived. Second (and more substantially), the necessary
conditions that were developed for rr, and pI were necessary for the valuefunctions to be constant along the curve. To have an equilibrium we require
somewhat more: The value functions must be identically zero along thiscurve. This is where the normalization of the curve comes in: At the point(1, l), the value functions are clearly zero for each weak player, as each iscertain that the other is strong. Put another way, suppose (once the curve isreached) that one side or the other is weak and decides not to randomize but
simply to wait out its opponent. The conditions that gave us rrl and p1 ensurethat the change in expected value is zero as this goes on. (For the technicallyminded, apply Dynkin’s
formula to the appropriate generalized Poissonprocess.) And at date To, if nothing has happened the player that is waiting

274

KREPS

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WILSONknows that its opponent is strong-it
should immediately give in for a valueof zero. Thus the value all along the curve is zero. This, with a little careful
argument, sufftces to show that we really do have an equilibrium.
Two final comments about this equilibrium seem in order. First, the valuefunctions to each (weak) player are easily computed. In region II, they are
z+(c?,~) = 0 and ~~(6, y) = [(SC - y)/6’] UT; in region I, they are ~~(6, y) =[(y”’ -6)/y”‘]
bT and ~~(6, JJ) = 0. That is, they are simple linear interpolates of the value of zero along the curve and the values along the bottom
boundary in region II and the left-hand boundary in region I.
Second, we noted earlier that the continuous-time
formulationcanintroduce equilibria that are not limits of the equilibria for discrete-timemodels. We should like to know that the continuous-timeequilibrium justpresented is indeed the limit of the discrete-time equilibrium with which webegan this section. We have not checked all the details, but we are quite surethat this is so. To see this, recall that the discrete time game can be posed as
an optimal stopping problem where the entrant is limited to stopping at, say,discrete times T, (TK - 1)/K,..., l/K and the monopolist
is limited to
stopping
at times(say)
T - 1/(2K), (TK - 3/2)/K,..., 1/(2K).
The
continuous-time
problem is one where stopping at any time t E [T, 0] is
possible. It is easy to move from the discrete-to-continuous-timeversions of
the problem when there is one-sided uncertainty, so we know that we have
convergence of the value of “stopping”
at particular times. As K goes toinfinity, the sets of available strategies also converge, and the methods ofMilgrom and Weber [ 141 apply to show convergence of the equilibria of thediscrete games to the continuous-timeversion. (Indeed, Paul Milgrom has
shown us how, by viewing the continuous-timegame as a game in
distributional strategies, it is simple to derive the equilibrium given above.)
Before concluding this section, we also note that this gives just one sort offormulation of the problem with two-sided uncertainty. We might consider
what happens when a single monopolist plays against a succession ofdifferent entrants (each of whom plays the game once), where the monopolist
is uncertain of the entrants’ payoffs. In such a game we would have to
specify the way in which the entrants’ payoffs are related-theymight all beidentical (perfectly correlated, from the point of view of the monopolist), or
they might be independently and identically distributed,
or something
between these two extremes. Both of these extreme cases are analyzed indetail in Kreps and Wilson [8]. The case of identical entrants gives the mostinteresting comparison
with the model analyzed above: With identicalentrants who only play the game once, the first entrant nearly always “tells
the truth” by refusing to enter if weak, and the weak monopolist will with
substantial probability
fight the first few entries, just to keep the weakentrants “honest.” What this illustrates is that the game of “chicken” that we
see above requires both two-sided uncertainty and that each side has a stakeREPUTATIONAND

IMPERFECTINFORMATION275

in maintaining its reputation. When it is only one side that will participate inmany stages, the other has little motivation to dissemble and will not light
too hard to attain/maintain its reputation. (Another interesting formulationof the problem is where there is a population of entrants and a population ofmonopolists,
and in each round there is a random assignment of one
monopolist to one entrant, in the manner of Rosenthal [ 181 and Rosenthaland Landau [ 191. We have done no analysis of this formulation.)

5.

DISCUSSION

We have presented these simple examples to illustrate formally the powerof “reputation”
in finitely repeated games. That reputation is powerful inreality is very well appreciated: In the context of Industrial Organization,
recall the quotation from Scherer in Section 1. Consider the importance ofreputation in contract and labor negotiations;in a firm’s employment
practices; in a firm’s “good name” for its product; in the maintenance of acartel (or in the prisoners’ dilemma game); in international diplomacy. To
each of these contexts, our analytical structure can be applied to yield theconclusions: If the situation is repeated, so that it is worthwhile to maintainor acquire a reputation, and if there is some uncertainty
about themotivations of one or more of the players, then that uncertainty can substantially affect the play of the game. There need not be much uncertainty forthis to happen. The power of the reputation effect depends on the nature of
one’s opponents; notably on whether they also seek to acquire a reputation.
Phenomena that bear the interpretation of “reputation”
are not entirely
new to the literature of formal game theory. They are implicit in much of theliterature on super-games, where the stage game is repeated infinitely often,or where there is always high enough probability of at least one more
repetition (Rubinstein [20] is a representative citation). Indeed, Dybvig andSpatt [6] make explicit use of the reputation interpretation in a super-gamecontext. What is new in this paper (and in Milgrom and Roberts [ 131) is theobservation that with a very little imperfect information, these effects comealive in finitely repeated games. Comparing the two approaches is difficult,but it is worth noting that in the models reported here, the problem ofmultiplicity of equilibria that plagues the super-game literature is substantially alleviated. Also, we believe that we have interesting models of the sorts
of “wars”
that might go on between players to see which equilibrium willensue. But we are far from ready to make a very informed comparison of thetwo approaches-at
this point, we can only claim that this seems to be aninteresting alternative way to produce reputation effects.
A point made briefly in Section 3 is worth repeating here. To keep matters
simple, we posited the simplest type of uncertainty: Players are uncertain
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about the payoffs of their fellows. But it does nearly as well if there is no
uncertainty about players’ payoffs, but there is uncertainty about whether
this is so. In the parlance of game theory, for these effects to disappear,
payoffs must be common knowledge. (Milgrom and Roberts [ 131 present
formal models to back up this contention.) This is a very strong assumption
for any real-life application.
The reader may object that in order to obtain the reputation effect, wehave loaded the deck. That is, we have a model where reputation is easily
shattered, making it all the more valuable; there are at most two types ofeach player; and each player has only two possible actions. To the first ofthese criticisms
we plead guilty: The power of reputation seems to bepositively related to its fragility. As for the second, the models of Milgrom
and Roberts [ 131 have continua of types of monopolists, so this does notseem crucial to our conclusions. And to the third, we do admit that this has
made it easy for us to get a “pooling equilibrium” (to borrow a term fromthe insurance literature), where one type successfully mimics another. The
analysis of Milgrom and Roberts [ 121 shows that with a continuum ofactions, one can also get screening equilibria in these sorts of models. But
this is not necessary: Crawford and Sobel [3] investigate a class of models
with a continua of actions where some pooling is necessary in any
equilibrium. The assumption of only two actions makes things easier for us.
but we doubt that it is crucial.
What is evident from our simple examples is that a very little uncertainty“destabilizes”
game-theoretic analysis in games with a fairly large number ofstages. The reader may suspect that something more is true: By cleverly
choosing the nature of that small uncertainty (precisely-its
support), one
can get out of a game-theoretic analysis whatever one wishes. We have noformal proposition of this sort to present at this time, but we certainly sharethese suspicions. If this is so, then the game-theoretic analysis of this type ofgame comes down eventually to how one picks the initial incomplete information. And nothing in the theory of games will help one to do this.
This reinforces a point made by Rosenthal [ 171. Rosenthal investigates theoriginal chain-store game and makes the point with which we began: The
paradoxical result in Selten’s analysis is due to the complete and perfect
information formulation that Selten uses. In a more realistic formulation of
the game, the intuitive outcome will be predicted by the game-theoretic
analysis. Rosenthal does not provide this analysis, despairing of the analyst’s
ability to solve an adequate formulation. Instead, he suggests an analysis
using the paradigm of Decision Analysis, where one tries to assess directly
how the entrants will respond to early round fighting by the monopolist.
Such an analysis can certainly lead to the intuitive outcome, as shown by
Macgregor [ 111. But, as Rosenthal notes, the weakness in this approach isthe ad hoc assessment of entrants’ behavior. We have carried out a game-REPUTATIONAND

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theoretic analysis of one very simple incomplete information formulation.
We therefore have avoided ad hoc assumptions about the entrants’ behavior.
But we have made ad hoc assumptions about their information, and we havefound that small changes in those assumptions greatly influence the play of
the game. So at some level, analysis of this sort of situation may require ad
hoc assumptions.
We have presented models in this paper that demonstrate the reputation
effect as simply and as powerfully as possible. In order to do this, we havenot tried to model realistic settings from Industrial Organization or someother economic context. (Milgrom and Roberts [ 13) rectify this deficiency:
They concentrate somewhat more on the application of these ideas.) To
illustrate how these ideas might be applied, we close with two examples.
The first concerns the problem of entry deterrence, especially the papers of
Spence [26] and Dixit [4,5]. These papers take the basic framework of Bain
[ 1 ] and Sylos [27] and ask: What can the monopolist do prior to theentrant’s decision point to make predation optimal in the short run? (Theanswers they give include such things as expanded capacity, sales networks,etc.) The relevance of this question is that the threat of predation is only
credible if predation is ex post the optimal response, so the monopolist must
make it so in order to forestall entry. What our model suggests (and what
can be demonstrated formally) is that in repeated play situations, the actions
taken by the monopolist
need not make predation actually ex post
optimal-what
they must do is to make predation possible and, perhaps,
increase the probability assessed by the entrants that it is expost optimal. If
deterrence is the objective, the appearance and not the reality of ex post
optimal predation may be what is important.
The second context is that of a monopolist producer of a durable capital
good where, for whatever reason, the monopolist is unable to maintain arental market but must sell outright his product. In a multiperiod setting,where the monopolist is assumed to be sequentially optimizing, this canseverely diminish the monopolist’s market power. (See Bulow [2] and Stokey[25].) Supposing the monopolist produces subject to a capacity constraint,
the monopolist is often berter off with a tighter constraint. This is because
the constraint prevents the monopolist from “over-producing.”
Then, if that
constraint
is the matter of private information
for the monopolist, amonopolist with a loose constraint can successfully(in an equilibrium)masquerade as having a more stringent constraint, thereby recouping someof his lost market power. In essence, as the number of periods goes toinfinity (as one comes closer to a continuous-time
formulation),themonopolist can successfully
attain the reputation of a “low capacity”
producer even if his capacity is (with probability approaching one) high.
Moorthy [ 151 presents an example along these lines.

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KREPS AND WILSONACKNOWLEDGMENTS

The authors express their gratitude to John Harsanyi, Mary Macgregor, Paul Milgrom,
John Roberts, Robert Rosenthal, Reinhart Selten, Sylvain Sorin and many other colleagues
for their helpful comments and criticisms. This research has been supported in part by
National Science Foundation Grants SOC77-0774 l-AOl, SOC75-21820.AOl, SOC77-0600.
A01 and SES80-06407 to the Institute for Mathematical Studies in the Social Sciences andthe Graduate School of Business, Stanford University.

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