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REPUTATION IN THE SIMULTANEOUS PLAY
OF MULTIPLE OPPONENTS
Drew Fudenberg
David M. Kreps
Number 466
March 1987
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
REPUTATION IN THE SIMULTANEOUS PLAY
OF MULTIPLE OPPONENTS
Drew Fudenberg
David M. Kreps
Number 466
March 1987
Reputation
in the
Simultaneous Play of Multiple Opponents*
Drew Fudenberg and David M. Kreps
March 1987
Abstract. Imagine that one player, the "incumbent," competes with several "entrants."
Each entrant competes only with the incumbent, but observes play
contests.
in all
Previous work shows that, as more and more entrants are added, the
may dominate
incumbent's reputation
in sequence.
We
identify conditions
play of the game, if the entrants are faced
under which similar results obtain when the
we
entrants are faced simultaneously, and
find specifications in
simultaneous entrants has a dramatically different
We
effect.
which adding more
also
show
that, with
either sequential or simultaneous play, incumbents need not prefer the situation in
which their reputations can and do dominate play to the "informationally isolated"
case in which each entrant observes only play in
1.
its
own
contest.
Introduction
This paper continues the study of reputation effects in games in which one player
engaged in "predatory
conflict"
with several opponents, each of
the one. In line with the recent literature,
its
several opponents the entrants.
and one of
its
We
we
will call
refer to the
previous play by
its
is
engaged only with
the single player the incumbent and
competition between the incumbent
unsure about how
its
opponent(s)
must consider how
this will affect
but also the predictions, and hence future play, of
for those predictions,
the reputation that
will play,
and each uses the
opponent(s) to help predict future actions. Accordingly, in selecting
current actions, each player
synonym
is
opponents as the basic contest. The notion of reputation enters into these
games because each player
loose
whom
is
will
its
not only immediate rewards
opponent(s). Using reputation as a
each player chooses current actions based in part on
be built or maintained. Previous work (Kreps and Wilson (19S2),
Milgrom and Roberts (1982)) showed that,
in
some
situations, the incumbent's reputation
would, as more and more entrants were added, come to dominate play of the game,
if
the
Financial support from the following sources is gratefully acknowledged: National Science Foundation Grants SES84-08586 and SES85-09697, the Sloan Foundation, and the
Institute for Advanced Studies at the Hebrew University.
2009752
entrants were faced in sequence.
1
Our aim
is
when
to see whether similar results obtain
the entrants are faced simultaneously. Along the way,
we
will also
sharpen what
known
is
about models in which opponents are faced sequentially.
The models that we examine
yield
two basic
insights.
First, the
maintenance of a
reputation in equilibrium depends on a tradeoff between the short run costs of maintaining
that reputation (including the opportunity costs of not milking
lead to
its
demise) and the longer run benefits that accrue from
models, the incumbent's short run cost
run benefits accrue from the
is
most
it
it.
fully, if
that would
In the sequential contests
the cost of fighting a single entrant, while
many opponents
its
long
to be faced in future contests. If the reputation
has a positive value against each opponent, then as the number of opponents grows, the
short run cost
is
eventually outweighed by the long run benefits. There
on the incentives of the entrants to invest
in reputation,
no similar
effect
and so the reputation of the
single
is
incumbent dominates.
The same
ambiguous
basic tradeoff arises in simultaneous contest models, but with generally
run costs
results because the short
run benefits
rise
as well, but only
short run costs will
we
when
find that, as
rise
with the number of opponents.
the long run benefits rises
.
more
quickly than
with sequential contests, the reputation of the one
incumbent dominates. This happens in some formulations of simultaneous play, but
not in others that are equally natural.
it
does
The simple addition of opponents does not ensure
that the reputation of the one will dominate, even
particular reputation are positive.
Long
when
the net benefits per opponent of a
The reputation of the one only dominates
of competition causes the long term benefits to outweigh short run costs.
if
We
the structure
will
show how
the ability of the entrants to reenter influences the direction of this comparison.
Second, the early literature might be interpreted as suggesting that incumbents generally favor conditions
under which their reputations can (and do) dominate
play.
Compare
the informationally linked situation above, in which each entrant observes earlier play in
contests, to a situation of informational isolation, in
in its
N
own
contest.
linked contests
1
By "dominate
is
and
its
is
which each entrant observes only play
between the weak incumbent's expected value from
expected value in TV informationally isolated contests; not
N
play of the game," we mean that the incumbent will, in equilibrium,
reputation regardless of how the entracts act, and so the entrants' equilibrium
given by an easily solved optimization problem.
maintain
play
(The comparison
all
its
linked vs. one isolated.)
One might have thought
that whenever the incumbent's reputation
dominates linked play, the incumbent prefers linkage to
isolation. Indeed, this
is
the case in
the basic model of Kreps and Wilson (1982). But a small variation on their model (which
model of Milgrom and Roberts (1982)) shows that
in the spirit of the basic
is
the case in general. Under linkage, the incumbent
short run costs of keeping
would sustain
and yet not
were
if it
suffer long
The paper
bit less
lost.
are outweighed by the long term opportunity losses that he
Under
isolation, he
term opportunity
We present
complex than
in
conclude section 2 with our
not
is
defend his reputation because the
may be
able to avoid the short run costs,
losses, the best of all possible worlds.
begins, in section 2, with a very brief recapitulation of the basic
the early literature.
and a
it
may
this
a formulation a
bit
more complex than
in
model of
Kreps and Wilson
Milgrom and Roberts. This added complexity allows us to
first
point:
An incumbent
might prefer informational isolation
to informational linkage, even in cases in which the incumbent would fight to maintain
its
reputation.
In the early literature (and in section 2), the individual contest has a very simple tem-
poral structure: First the entrant moves, and then the incumbent responds. Simultaneous
contests of this sort would be uninteresting:
incumbent had to
act, the
individual contest, the concession
roughly a
(If
game
of
entrants
made
their (only)
moves before the
incumbent's play could not influence the entrants' future actions,
and so there would be no reputation
is
If
war of
For this reason, we use a richer model of the
effect.
game found
attrition, in
in
the back half of Kreps and Wilson. This
which a player wins
if its
opponent concedes
first.
neither concedes, then both are losers.) Section 3 begins with a brief recapitulation of
the equilibrium of the single contest
these contests, and
if
we confirm that the
(Analysis of this latter point
is
played in isolation.
results
consigned
Then we look
at sequences of
from the early literature continue to hold.
an appendix.)
to"
Sections 4 and 5 concern simultaneous play. In section 4
we analyze a formulation
in
which the reputation of the incumbent does not dominate play as the number of entrants
increases. Instead, the
number
of entrants
key to the formulation in section 4
retains the winnings
it
may have
is
is
that
irrelevant to the equilibrium outcomes.
if
The
and when the incumbent does concede,
it
secured from earlier concessions by some of the entrants.
In this case, the short run cost of maintaining a reputation
is
proportional to the
number
of entrants there are remaining, as are the reputation's long term benefits. Thus, changing
the
number
of entrants has
no
In contrast, in section 5
effect
on the nature of the equilibrium.
we formulate simultaneous play
does ever concede, accrues no further benefits from contests
basic tradeoff, costs are proportional to the
number
while the gain from maintaining the reputation
entrants.
Thus there
is
is
so that the incumbent,
has already won.
it
of entrants
who
if it
In the
haven't yet conceded,
proportional to the total
number
of
all
the potential that benefits will exceed costs, and the incumbent's
reputation will dominate. This does happen, for (at least) one specification of the model.
In subsection 5.1,
we present the
for a simple case with
basic model.
In subsection 5.2,
we
find the equilibrium
two entrants. Even with two entrants, we are able to observe subtle
brought about by linkage. In 5.3 we show how to compute the equilibrium for any
effects
number
of entrants. Asymptotic results are obtained in subsection 5.4; these are in direct
accord with those obtained when the individual contests are played in sequence.
Our use of terminology such
is
meant
as incumbent, entrant, simultaneous entry
and reentry
to suggest applications to industrial organization, but the reader should not
misled by these terms into imagining that
In particular,
when
there
is
we intend the most obvious
be
interpretions of them.
simultaneous entry, one naturally imagines a single market into
which a number of entrants enter at once.
One would expect
in that situation that the
actions of any one entrant would have a direct effect on the payoffs of other entrants. This
would be an interesting game to investigate, but
game
it is
not the one that we examine. In the
considered here, each entrant's payoffs depend only on
the incumbent; other entrants effect
incumbent. The incumbent
actions
it
is
it
own
actions and those of
only insofar as their actions change the actions of the
concerned with
its
many opponents
takes in other contests; entrants do not care
except to their single opponent. This
its
is
learning about
it
through
what they reveal about themselves
not to say that the entrants are not worried about
their reputation, but that they are only concerned
about their reputation with the single
incumbent.
The models studied
nonmenon.
many
here only continue to scratch the surface of the reputation phe-
In particular, a priori asymmetries in the characteristics of the entrants lead to
interesting effects.
We
hope to report on some of those
effects in
a sequel.
.
Sequential play with simple contest - a recapitulation.
2.
Kreps and Wilson (1982) and Milgrom and Roberts (1982) deal with models
An
the individual contest has the following basic form.
stay out of a particular market.
If
enters, the
It is
incumbent must react, either by fighting or acquiescing. With prior probability
if
If
and the incumbent
fights, the
— p°
1
the entrant stays out, the incumbent nets a
the entrant enters and the incumbent acquiesces, the incumbent nets
enters
which
entrant decides whether to enter or
the entrant stays out, the contest ends.
incumbent's payoffs in the contest are:
in
incumbent nets —1
And
.
the
>
;
the entrant
if
;
,
with prior probability p°
,
the incumbent has as strictly dominant strategy to fight any entry. For the entrant, with
prior probability
nets
b
>
,
1
—
q°
staying out nets
,
and being fought
—1
yields payoff
entering and being
,
.
And
met with acquiescence
with prior probability q°
We
does better to enter than to stay out, regardless of the incumbent's reaction.
the incumbent as being weak
otherwise.
And
the entrant
for
The
if
first
indifferent
if
there
is
p°
b/[b
equality.
or second payoff structure,
strategies,
we
+
1)
,
stays out
if
(1
—
is
p°)b
entry, the
+ p°(— 1)
is
faced
first; after
and so on. Each entrant
incumbent
is
weak
is
the strict inequality
is
N
the contest against entrant
weak with probability
1
in all contests or strong in all of
The weak incumbent's payoff
where payoffs
incumbent
enters.
and
reversed,
is
3
strong. For a strong incumbent, fighting in
dominant.
will usually
if it
imagine that the incumbent faces a sequence of entrants, indexed by n
Entrant TV
strong
is
2
only.
<
refer to
and equilibrium expected payoffs, giving these
weak, so a weak entrant nets expected payoff
It enters, therefore, if
it is
has the
contest above has a unique sequential equilibrium. If there
acquiesces
Now
if it
and incumbents have dominant
in specifying equilibrium strategies
weak types of players
payoff structure; the incumbent
first
weaA' or strong,
is
respectively. Since strong entrants
be lax
has the
if it
the entrant
,
total
all
is
-
q°
,
is
completed,
= N, ..., 1
N—1
is
faced,
independent of the others. The
them, with prior probability p° that
markets where there
the
sum
is
entry
of payoffs in the
N
is
strictly
contests,
in each contest are as above.
2
Kreps and Wilson (1982) consider the case where q° =
Milgrom and Roberts (1982)
have a somewhat richer structure of payoffs, but the ones given here will suffice to make
our points.
For the rest of the paper, we will ignore all such knife-edge cases.
.
.
,
If
N
the
contests were conducted under conditions of informational isolation, where
entrant n does not observe the actions of the incumbent in earlier contests, then entrant n
would begin
its
contest with prior p° that the incumbent
would acquiesce
if
In this case, the
there
Hence weak entrants enter
entry.
is
strong,
is
weak incumbent nets zero
weak incumbent nets an expected
—
(1
Things are much more complex
if
N
the
if
and only
>
in every contest; if p°
N(l —
q°)a per contest, or
and a weak incumbent
if
6/(6
p°
<
+
1)
6/(6
,
+
1)
then the
q°)a overall.
contests are played under conditions of
informational linkage, where entrant n observes the past behavior of the incumbent. Then,
the incumbent acquiesces in contest
if
guarantees that
all
n
subsequent entrants
it
,
reveals that
it
is
weak.
This,
on
its
(
—1
)
must weigh the
,
against the long term affect that acquiescence will have
From the
reputation, and hence on the entry decisions of subsequent entrants.
literature,
we obtain the
Proposition
contest,
When
1.
N
the incumbent faces
The nature
early
following result.
and the contests are informationally
outcome.
turns out,
one sequential equilibrium of
will enter in at least
the game. Hence the weak incumbent, in choosing actions in contest n
short term costs of fighting
it
of the equilibrium
entrants in sequential play of the simple
linked, there
outcome
is
a unique sequential equilibrium
N
for large
depends on the sign of
S°-o/(o4l).
(a) If q°
>
a/'(a
+
1)
then the incumbent,
,
occurs "early" in the game.
4
Hence
as
N—
if
oo
,
weak, acquiesces at the
first entry,
which
the weak incumbent's average payoff per
contest approaches zero.
(b)
H
q°
<
there are n(p°)
a/ (a
or
+
1)
more
,
then for every p° >
contests
left
there
is
a number n(p°) such that,
to go, the incumbent
is
if
certain to fight any entry.
Accordingly, weak entrants with label n(p°) or higher choose not to enter, and the weak
incumbent's average payoff per contest approaches
The
results, as
the
If
q°)a
—
q°
as TV
A
simpler argument
is
—
<
oo
.
available for obtaining the asymptotic
developed in Fudenberg and Levine (1987).
A
synopsis of the simpler argument
< 6/(6 + 1) the first entry occurs in the first round of the game.
entry occurs at the earliest date that the entrant is strong.
p°
first
-
early literature derived the unique equilibrium outcome, obtaining the asymptotic
payoff functions as a corollary.
4
(1
,
If
p°
> 6/(6+1)
.
can be given here: Imagine that the incumbent follows the strategy of never conceding.
Since the
weak entrants
small, every time that a
weak entrant enters and
amount that
strong must rise by an
is
incumbent
called
is
upon
probability), doing so
out unless the probability of being founght
will stay
is
N
such that
if
bounded away from
must have a nonneglible
there
fought the probability that the incumbent
to invest in his reputation by a
posterior probability that the incumbent
of
is
is
is
That
is,
a(l
—
must
a(l
q°)
—
weak opponent (with
Thus there
positive probability of entry
is
If this
.
by k (or more) weak entrants,
is
bound
positive, then the
is
is
all
weak
future
bounded below by
tight, since the
incumbent
strong entrants to have a positive average value asymptotically. In the case
fight the
— q°) — q° <
negative.
expression
positive
a number k independent
shows that the incumbent's asymptotic average payoff
q°
each time the
on that reputation (measured by the
effect
strong).
zero.
then fighting by the incumbent leads to a reputation strong enough to deter
entrants. This
sufficiently
is
then even
,
if all
Hence the incumbent
weak entrants
(if
weak)
are deterred, the average value of fighting
will acquiesce early on, for
is
an asymptotic average
value of zero.
The
<
case q°
a/(a
+
1)
has received the most attention in the literature. Here the
incumbent can credibly play the strategy "always
of acquiescence (entry by subsequent
To emphasize the
same
weak entrants) outweigh the short-run
different sizes: Payoffs in the last contest (contest
as before. Payoffs in constest
is
a(a+l)
so doing deters
5
all
With
these payoffs,
future entry. Thus,
it
if
is
a-f-1
,
and
his gain
are
)
1
n are c + 1 times those in contest n —
example, in contest 2 the incumbent's cost of fighting
entry
costs of fighting.
crucial nature of the basic cost/benefit tradeoff, consider a formulation
where the various contests have
exactly the
because the long-run consequences
fight"
1
;
for
from deterring
does not pay the incumbent to fight today, even
p°
< b/(b+
1)
,
for
any
N
if
the unique equilibrium
future
weak
entrants by fighting, but the cost of doing so outweighs any conceivable future gain.
It is
is
for all of the entrants to enter. This
the cost/benefit tradeoff,
which
is
that the incumbent can deter
all
and not the frequency of play or the number of opponents per
se,
the key to understanding reputation effects.
When moving
in reputation,
5
means
One can
we
to a situation of informational linkage favors the incumbent's investment
will say that
also scale
the incumbent has gained in strategic backbone. Naturally,
up the payoffs to the entrants, but
7
this is clearly irrelevant.
this stiffened
backbone comes
The incumbent must
at a cost.
scared off which, in our simple model, amounts to
strategic
backbone brings with
a loss of strategic
it
>
This occurs when q°
play.
may be
+
a /(a
backbone outweigh the costs of
incumbent
flexibility.
1)
have a cost that outweighs the
lost flexibility given that
if
markets are linked, the (weak)
weak, prefer that the contests take place under conditions of
> a/(a+
and p° < b/(b
1)
+
linkage and isolation. All the entrants enter
costly to maintain a stiff
When
q°
need not dominate
Moreover, even when the benefits of a stiffened
.
informational linkage or informational isolation? Refer to figure
q°
< a/(a+
backbone with
if
the weak incumbent
,
there
is
is
+
informational isolation, and
1), informational linkage
and the gain from a
are informationally isolated,
When
p°
>
b/(b
+
weak incumbent must
reputation.
in contrast, the
q°
(if
the strong entrants
The
too
is
preferred.
if
Here
the contests
backbone outweighs the cost of
weak incumbent
is
(if
> b/(b+
is
< a/(a+
< b/(b+
q°
natural
1)
),
1)
if
N
,
weak
or
it
come
entrants,
backbone)
means, however, that the
along,
if it
wants to keep
up the reputation altogether,
its
costs too
if it
maintains the reputation at the cost of fighting
).
where p° and
q°
are each small. This
"strong" types are interpreted as "irrational" firms. But the
and p° > b/(b+
ensures that, for large
sufficient to deter the
loss of strategic flexibility
either gives
1)
strictly prefers informational
isolation to linkage (obtaining a stiffened
early literature focuses on the second case,
particular focus
to choose
stiffened
fight all the strong entrants that
The weak incumbent
to maintain
case q°
,
The accompanying
provides no gains.
much
1)
moving from informational
so that
it is
it.
The incumbent's ex ante reputation
isolation.
between
indifferent
linkage.
and p° < b/(b
1)
1)
1.
the stiffened backbone deters some entrants, whereas none are deterred
maintaining
stiffened
worse off for having to make the choice.
Would the incumbent,
When
not
Milgrom and Roberts make
stiffened backbone, so that the incumbent's reputation
from the
A
the strong entrants.
may sometimes
clear that this loss of strategic flexibility
benefits
all
fight all those that are
1)
should also be borne in mind.
The
first
inequality
the incumbent's reputation dominates play of the game.
between having a reputation
for relative strength or
the weak incumbent prefers strength. But the
8
If
forced
having one for weakness, then
weak incumbent
prefers
most of
all
that
it
not have to prove
equilibrium, but
who would be
3.
because
itself,
it
began a priori with a good reputation.
it
does so more to avoid losing
its
It will fight, in
good reputation and keep scared those
who would not otherwise
scared without linkage than to scare those
be.
The Basic Concession Game
The
early literature
makes use
of a very simple contest,
To obtain
and then the incumbent responded.
against
many opponents, we need a
to the incumbent's play.
and Wilson (1982, section
and
its
We
where the entrant moved once,
interesting analyses of simultaneous play
contest in which the entrants have nontrivial responses
game
use, for the rest of this paper, the concession
4) as the basic contest,
and
in this section
we
of Kreps
describe the
game
equilibrium. (Notation will be changed slightly from Kreps and Wilson (1982).)
There are two players, an incumbent and an entrant. The game
is
played in continuous
Time runs backwards. At each time
time, over the interval from time one to zero.
neither player has yet conceded to the other, either can choose to do so.
side or the other concedes, the
game
As soon
t
as
ends. If neither side concedes by time zero, the
,
if
one
game
ends.
As
in section 2,
each player
that the incumbent [entrant]
player.
A
player's type
is
is
is
one of two types, strong or weak, with the prior probability
strong being p° [q°], independent of the type of the other
private information.
Payoffs to a player depend on which
occurs,
and the player's type.
either) party concedes first,
when
the concession
For strong players (both incumbent and entrant),
dominant strategy never to concede. As
drop the modifier "weak"
(if
in section 2, then,
we
will
it is
is
a
be sloppy and sometimes
in discussing the equilibrium strategies of the
and entrant. For weak types of player,
it
(weak) incumbent
best to have one's opponent concede, second
best to concede oneself, and worst to have an opponent that does not concede while not
conceding oneself. Rewards and losses are proportional to the time periods over which they
are received/incurred. Specifically, there are constants a
>
and
b
>
with the following
assignment of rewards:
Suppose the entrant concedes
1
—
t
,
first,
and the incumbent "wins"
at
time
t
.
Thus the two
for a period of length
t
.
fight for
a period of length
The incumbent
(if
weak) nets
reward
ai-(l-i),
or a per unit time that
the entrant
it
wins, with a loss of
-
weak) nets -(1
(if
unit time during the fight. If the
—
nets bt
—
(1
t)
,
incumbent concedes
to the side that concedes,
first. If
first
t)
time
That
-1 per
concedes, and
t
is,
the (weak) entrant
,
the cost of fighting
b per unit
time
if
game, the two (weak) players engage
in a
is
game
the incumbent
—1
neither side concedes before time zero, then each loses a total of
game
6
.
of "chicken."
neither concedes at the outset, each thereafter concedes at a "rate" until such time as the
one or the other has concedes or each
that at time
is
certain that the other
neither side has yet conceded.
t
probability that the incumbent
We
is
tough, and
t )
will
concede over the period
that are o(h)
time
t
—
must be
indifferent
and
is
(1
—
qt)ph
,
denote the
is
tough.
has not conceded by
if it
h) with probability 7r(t,p t q t )h
,
,
+
o(h)
.
up to terms
(That
(supposed to be) playing this mixed strategy
The former
benefits.
is,
7r
in equi-
and waiting
t
nets zero for the remainder of the game.
With
concede over this time period, and the costs
namely
weak, and
between immediate concession at time
h and then conceding.
latter has potential costs
will
for the concession probabilities.)
Since the (weak) incumbent
it
—
(if
and the (weak) entrant with probability p(t,p t ,qt)h
,
and p give hazard rates
librium,
(t,t
two players - p t
the probability that the entrant
hypothesize an equilibrium where the incumbent
time
,
q-t
tough. Precisely, suppose
is
Play to this time has led to (Bayesian)
posterior reassessments concerning the toughness of the
h
.
it
nets zero for the remainder of the
and the entrant gets a reward of
In the equilibrium to this
If
first at
and the (weak) incumbent nets —(1 —
t)
And
unit time during the fight.
or zero per unit time after
minus one per unit time as before, conceding
concedes
-1 per
until
The
probability close to one, the entrant will not
will total
—h But
.
there
that the entrant does concede before time
6
t
—
is
h
.
a chance of order
This involves the
It is important that rewards be proportional to the amount of time left. The analysis
would change substantially if there were a fixed reward if the other side concedes first,
regardless of when the concession takes place. Also, throughout this paper we will be. less
than perfectly formal. The reader may be troubled by our use of a continuous time game,
for example. But note that this game can be viewed as a one-shot selection by each side
of a "concession time" - a time at which one will concede if the other side hasn't done so
first. This formulation is perfectly legitimate, and the analysis that follows describes an
equilibrium for
it.
10
marginal probability that the entrant
that the entrant will concede
at
.
if
weak (ph).
Hence (up to terms of order o(h)
strategies
if
and only
weak
is
(
—
1
qt
times the conditional probability
)
the weak incumbent
)
incumbent
In this case, the benefit to the
between the two
indifferent
is
is
if
= -h +
-q
(l
Dropping the h and repeating the argument
t
)phat.
for the
weak entrant
gives the first
two equi-
librium conditions:
= -1 + (1 = -1 +
If
(1
-p
the weak incumbent does use the strategy
ever there
is
and
qt )pat
t
(3.1a)
'
)irbt.
ir
,
(3.16)
then pt will change through time.
a concession, this of course reveals that the incumbent
drop to zero. But as long as there
application of
B ayes' formula
=
Pt
is
no concession (and
it
>
),
-
q t )p.
pt
Pt(l
~ Pt)*
and
qt
=
q t (l
means dp t /d(—t)
,
=
dp
some constant k >
We assert
Pt
will
<
1
.
that k
1*
pa
or
q
(3.2)
we
see that posteriors
= fcpV
.
=
1 is required if this
Then, tracing back through
reach one at a time
when
qt
=
k
is
to give an equilibrium. Suppose, for example,
and
(3.1)
< .1
.
(3.2),
and benefits up to
that the incumbent
is
this
time net out to zero.
tough.
If
one can see that, before time zero,
Imagine that the incumbent
the strategy of no concession until just after this time.
costs
Simple
along a curve
fj2
that k
rise.
since time runs backwards.)
Substituting (3.2) into (3.1) and dividing one equation by the other,
for
should
will
t
yields
(See Kreps and Wilson (1982). Here, p t
will evolve
weak, and p
is
If
And
According to
at this
(if
(3.1), the
time the entrant
weak (which happens with probability
weak)
1
—
k
),
is
tries
expected
convinced
the entrant
drops out immediately. Hence waiting until this time nets the (weak) incumbent a strictly
positive expected value,
is
and
it
could not be an equilibrium for
it
to use
ir
precluded. So, to have an equilibrium of the sort described, posteriors
11
.
Similarly, k
>
1
must evolve along
the curve
q
If
(q°,p°)
falls
labelled Region
T
time
,
I,
(if
What
not? Refer to figure
if
,
2. If
it
is
tough
if it
follow the equilibrium described above.
lie
(q°,p°)
falls in
a
the region
region
II,
make the
sufficient to
doesn't concede equal to
From
this involves
then the game begins as follows: At
(p°)
b^a
Thereafter,
.
the (weak) incumbent concedes
immediately with positive probability so that (once again)
posteriors
(To see
weak) concedes immediately with probability
posterior probability that
we
(3.3)
"small" relative to p°
with q°
the entrant
p»/\
along this curve, we do have an equilibrium.
checking by the reader.)
little
=
if
is
no concession, the
for
example, then the
there
along the curve.
These strategies do give an equilibrium.
(weak) entrant randomizes at the
If
This
start.
is
we
lie in
region
I,
a best response by the entrant: Dropping
out nets zero for the entrant, as does continued fighting (as the posteriors begin to
move
along the curve).
Note that
any positive expected value from the game
for each (weak) player,
right at the start. After the initial randomization (if any),
expected value to both sides
(if
weak)
if
the
game
is
realized
hasn't ended, the
zero.
is
Using the methods of Fudenberg and Tirole (1986), one can show that this
is
the unique
equilibrium of this game.
We
plays
introduce this concession
many
game
entrants simultaneously. But before doing so,
play of the concession
game
yields the
the simple contests of section
of entrants, n
= N,N —
is
faced
Under informational
the
full
is
we wish
when
the incumbent
to record that sequential
results as does sequential play with
imagine that the incumbent faces a sequence
Each entrant
1,...,1.
first, for
same asymptotic
Specifically,
2.
of the others, while the incumbent
Entrant TV
to study reputation effects
is
strong with probability q°
either strong in all contests or in
unit of time, followed by entrant
isolation, the equilibrium
is
simply
N
,
independent
none of them.
N—
1
,
and so on.
copies of the equilibrium just
described.
Equilibrium play of the
N
contests under informational linkage
provide the sequential equilibrium of this
game
in the
quite complex.
We
appendix, and we (only) record here
that the incumbent's asymptotic payoffs are as with simple contests.
12
is
Proposition
2.
N
Imainge that the incumbent faces
entrants in sequential play of the
concession game.
>
(a) If q°
JV
—
oo
+
aj{a
1)
the incumbent's average payoff per contest approaches zero as
,
Moreover, the incumbent
.
<
(b) If q°
+
a/(a
(if
weak) concedes "early" on
in the
1), the incumbent's average payoff approaches
"most" of the game, the incumbent
game.
a(l
-
q°)
-
q°
.
For
not concede whether weak or strong, so that weak
will
entrants capitulate at the outset.
The appendix should be consulted
for exact statements.
average values can be derived more simply than
The comparison
(a) If q°
3.
> a/(a +
1)
since v n
(b) If p°
>
<
weak incumbent under conditions of informational
also as before.
and p° <
isolation;
(For any given n
zero.
the appendix, using the general machinery
(Refer to figure 3.)
between linkage and
But
of the payoff to the
and linkage are
Proposition
note also that the asymptotic
and Levine (1987).
of Fudenberg
isolation
in
We
1
,
since
,
(q°)
a/b
,
then the incumbent
under either regime,
t;"
>
,
its
expected value per contest has limit
the incumbent might favor informational linkage.
the advantage from informational linkage
(a/(a + l)) a / 6 and a/(a + l)
asymptotically indifferent
is
<
q°
<
(p°)
b/a
,
is
slight.)
then the incumbent asymptotically
favors informational isolation. Indeed, under isolation the incumbent has a positive expected
payoff per contest, whereas
its
expected payoff per contest
falls
to zero with informational
linkage.
(c) If
p°
>
{a I {a
+
l))
informational isolation.
a/
fc
and
q°
<
a/(a
+
1)
The incumbent has a
,
then the incumbent asymptotically favors
positive average payoff in either regime; in
particular, its reputation dominates play for large
it
loses
more from the
loss of strategic flexibility
n under conditions of linkage.
than
it
gains from inducing
But
more weak
entrants to stay out for sure.
(d) If p°
<
(a /(a
+
l))°/
informational linkage.
b
and
The
q°
<
a/(a
+
1)
,
then the incumbent asymptotically favors
gains from stiffened strategic backbone outweigh the costs of
lost strategic flexibility.
The proof
is
a simple matter of algebra, given proposition
13
2,
and
it is left
to the reader.
4.
Simultaneous play with captured contests
Now we
N
incumbent faces
is
either
tough
in all contests or
and the entrants are either weak or tough, independent of each other and of the
incumbent. Entrants observe what the incumbent does
assessments accordingly.
We
has not yet conceded.
incumbent. Rather,
it
(This
entrant gets payoffs from
is
its
and update
their
which the incumbent,
in all contests in
if it
which the entrant
not imposed by restricting the strategies available to the
be equilibrium behavior
will
in all the contests
restrict attention to equilibria in
concedes in any contest, must concede simultaneously
in
imagine that one
entrants in TV distinct contests which are played simultaneously, over
the same unit length of time. As before, the incumbent
in none,
We
turn to simultaneous play of the concession game.
own
we examine.) Each
in all the equilibria
contest; the incumbent's payoff
is
sum
the
of
its
payoffs
each contest.
Note well that when facing many entrants
in sequence, the
longer horizon over which to amortize investment in
Thus
entrant.
it
is
its
incumbent has a much
reputation than does any single
"natural" with sequential contests that, in some cases at least, the
incumbent finds that the long-run benefits of a reputation outweigh
its
while the entrants find that reputation's costs outweigh
With simultaneous
contests, the
incumbent and the entrants amortize investments
Thus the
horizon.
its benefits.
effect of increasing the
number
short-run costs,
in reputation over the
of simultaneous entrants
is
same
not obvious a
priori.
At
this point, the specification of the simultaneous contest
happens
if
some, but not
all,
of the entrants have conceded by date
incumbent concedes? In equilibrium, the incumbent concedes
contests.
But what happens
that this question
to get the reward
is
(
in the contests -that the
moot with
Do
is
t
,
incomplete.
at
in aii the
What
which point the
remaining active
incumbent has already won? (Note
sequential contests.) Does the (weak) incumbent continue
a per unit time) from those contests
those flows of rewards for the time remaining?
previously?
game
it
has already won, or does
And what
it
lose
of the entrants that conceded
they continue to receive zero for the remainder of the game, or can they
reenter to get b per unit time for the time that remains?
In this section,
scenario.
we
will
The incumbent
complete the description of the game according to a no reentry
retains the rewards
from contests already won, and entrants who
14
conceded earlier get no benefit from the (subsequent) concession of the incumbent.
Proposition
4.
In the
no reentry scenario, the equilibrium
strategies for the
incumbent and
the entrants are the same whether contests are linked informationally or are isolated. These
strategies are independent of the
N
number
of entrants.
The expected
payoffs for
weak
entrants are precisely as in the one-vs.-one equilibrium of section 3, while the expected
payoff to the incumbent
is
N
simply
times the expected reward from the equilibrium of
section 3.
In saying that the equilibrium strategies are the same,
we mean that
(after initial
randomizations) each entrant uses the hazard rate p defined in (3.1a), and the incumbent
r defined
uses the hazard rate
in (3.1b). Posteriors evolve along the curve q
=
p
b/a
(unless,
of course, a player concedes). Initial randomizations are as in the one-vs-one equilibrium:
In region
I
of figure 2, the (weak) entrants simultaneously and independently randomize at
the start of the game, so that the posterior proability that each remaining entrant
is
(j9
6 /°
,
)
independent of the toughness of the others. In region
undertakes an
initial
All this follows
k of the
N
randomization.
-k
from the equations analogous to
,
is
unchanged from
complex. There are k entrants to
k(l
—
q t )p
— qt)p,
The
fight, so
Suppose we reach a point where
indifference equation for each entrant,
the rate of cost expenditure
if
each
is
is
—k
more
But there
.
conceding (independently) at hazard
(unconditional on type), then the hazard rate for the "next" concession
giving the incumbent a rate of expected gains equal to k(l
the use of the no-reentry scenario assumption that,
is
(3.1).
before. For the incumbent, things are slightly
are k entrants that might concede, and
rate (1
the (weak) incumbent
7
entrants have not yet conceded.
which determines
II,
tough
is
able to "bank" the
amount
at
.)
,
k(l
-
(fixing p
)
(Note here
is
q t )pat,
from continuing to
7
)pat_.
indifference equation
(4.1)
gives us the pair of equations (3.1).
and the expected benefits
t
an entrant concedes, the incumbent
Thus the incumbent's
= -k +
which, cancelling the k
if
—q
is
Both the
cost (per unit time)
fight rise linearly
with k
,
so the
Of course, informational linkage "forces" the incumbent to concede, if at all, at times
which are perfectly correlated among the contests. Under informational isolation, concession
in the different contests could take place at, say, independently distributed times.
15
equilibrating concession rate
continues to obtain.
Without
is
unchanged with changes
in
k
.
Hence the old equilibrium
8
reentry, linking the contests does not stiffen the incumbent's backbone, re-
and
gardless of q°
N
.
Contrast this with the previous section, where for small q° the
incumbent's reputation dominates play for large
N
.
There, increasing
N
increased the
(opportunity) cost of concession, but does not change the (momentary) cost of fighting.
For the incumbent's reputation to dominate
Here, both costs increase at the same rate.
with simultaneous play, the cost of concession must (be able to)
the
number
It is
helpful to interpret this result using the "strategic backbone/strategic flexibility"
of strategic
flexibility.
That
One
is,
gains strategic backbone only through the (potential) loss
strategic flexibility
wish to treat a given opponent differently
would be treated
change
others.
this
more quickly than does
of entrants.
notions introduced earlier.
if this
rise
Of
if
if
is
lost if one, for strategic reasons,
how
others are watching from
would
that opponent
others could not observe. This translates to a gain in strategic backbone
in one's strategic reactions has salubrious effect
on the
initiating actions of
course, to wish to treat one opponent differently from others,
opponent be somehow distinguished from the others. But
in the
it is
necessary that
no reentry scenario
and the equilibrium we have constructed, entrants are distinguished only
have conceded and others have not. All those that have not conceded are
insofar as
some
identical. Since,
without reentry, the incumbent need not be concerned with entrants that have already
conceded, there
is
no
loss in strategic flexibility.
Hence there can be no gain
in strategic
backbone.
Imagine, however, that the
game began with
probabilities. In particular, consider the case of
be strong, and the other
is
entrants that were tough with different
two entrants, one of
whom
is
almost sure to
almost certainly weak. Without linkage, the incumbent would
net almost a against the "weak" entrant and zero against the "strong" entrant, or a in
total. It
should not be surprising that the incumbent does not do as well under conditions
Since we did not restrict the strategy space of the incumbent so that concession in all
contests must take place simultaneously, we must say what happens should the incumbent
concede in some, but not all, contests. At this point, entrants all assess probability one that
the incumbent (i) is weak and (ii) will concede everywhere immediately, and the incumbent
(if weak) does concede everywhere. The careful reader can check that this out-of-equilibrium
behavior supports the equilibrium we have given.
16
of informational linkage, because
cannot afford to concede to the "strong" entrant until
it
On
has obtained concession from the weak.
it
shown, that
if
the parameters a and b
incumbent could gain by linkage.
the other hand,
we
believe, but
have not
as well as the prior, vary across contests, then the
,
9
Simultaneous contests with reentry
5.
Two models
5.1.
In the sequential contests
model we observed a
priors were equal, because in the course of play the
current opponent was
its
simultaneous play,
viz.,
it
is
more
loss of stategic flexibility,
incumbent could come to believe that
be tough than are
likely to
its
likewise the case that identical priors
some entrants concede while the others
fight.
even though
future opponents.
become unequal
Without reentry
With
posteriors;
this doesn't matter,
because the incumbent needn't be concerned with entrants that have already conceded.
But
if
entrants that have conceded can reenter,
may be
flexibility
we can
anticipate that the loss of strategic
may
therefore be a gain from stiffened
consequential, and that there
strategic backbone.
To investigate
In both,
if
this,
we consider here two
variations on simultaneous play "with reentry."
the (weak) incumbent ever concedes, then
remainder of the game. That
is,
it
nets a zero flow of benefits for the
the incumbent loses benefits accrued from contests already
won. The variations are distinguished
in
the incumbent's concession. In the
first
entrants receive reward at a rate of
b
what happens to entrants that conceded
variation, called (hereafter) reentry scenerio A, all
per unit time after the incumbent concedes, regardless
of whether those entrants conceded earlier or not. In reentry scenario
have not already conceded receive
therefore,
B
will
more
become
this positive flow of reward.
descriptive of scenario
clear as
The remainder
prior to
A
then
it is
The
J5,
only entrants that
description reentry
is,
of B; the reason for introducing scenario
we proceed.
of this section
is
organized as follows. In the next subsection, we analyze
the equilibrium for two entrants and for a
>
1
,
both to provide a warm-up
for the general
case and to illustrate the tension between gain in strategic backbone and loss of strategic
flexibility.
9
In subsection 5.3,
we
present the equilibrium for general values of
Cases of asymmetries of these and other sorts
17
may be
treated in a sequel.
N
,
a and
,
b
.
More
precisely,
we show how the equilibrium can be computed
not obtained closed form solutions.
Finally, in subsection 5.4,
N
limiting form of the equilibria in 5.3, for large
summary
recursively; but
we present
(if
analysis of the
under the assumptions of scenario B.
,
of the results of subsection 5.4 are that for any p° £ (0, 1)
then the incumbent
we have
,
if
q°
> a/(a +
A
1)
weak) concedes at the outset with probability approaching one
(proposition 4), while for q°
< a/ (a +
1)
weak incumbents concede
all
,
at the outset with
probability approaching one (proposition 5). These results, once established,
(weak) incumbent's expected payoff per contest under scenario B
of opponents faced sequentially,
is
show that the
precisely as in the case
and proposition 3 applies to simultaneous contests under
scenario B, precisely as stated before for sequential contests.
5.2.
The case
N
=
2
and a >
1
>
Consider the case of two entrants and a
then the (weak) incumbent's flow of payoffs
Hence the concession of the
concede.
1
a
is
.
—
If either
1
>
of the
if
two entrants concedes,
the second entrant does not
entrant ensures that the incumbent will never
first
subsequently concede; fighting to the end dominates concession, no matter what the second
entrant does.
entrant
is
This in turn implies that
one entrant does concede and the remaining
if
weak, the remaining entrant, knowing that the incumbent
will
now
fight to the
end, concedes immediately thereafter.
Accordingly,
entrant,
we
derive an equilibrium of the following form.
we have a curve
q
=
f(p) that runs between (0, 0) and (1, 1)
been no concession yet, the incumbent
strong with probability q t
is
incumbent, then
zero.
,
=
f(pt)
,
is
,
such that
strong with some probability pt
,
there has
if
and each entrant
independent of each other and of the strength of the
game
of
Each adopts a strategy of continuous randomization, the (weak) incumbent dropping
,
and each entrant
(if
weak) dropping independently with hazard rate
such that posteriors computed by Bayes' rule
time zero
it is
a single
in the case of
the (weak) players have expected payoffs for the rest of the
all
with hazard rate t
p
As
is
known
If,
move up along the curve
(p,
f(p))
.
Before
reached, either some player has conceded (which effectively ends the game), or
that
all
players are strong; that
at the start of the
game, (p°,q°)
is,
is
the posteriors (1,1) are reached.
above the curve, then the (weak) incumbent
randomizes between immediate concession and beginning to
fight,
such that no immediate
concession causes the entrants to reassess the probability that the incumbent
18
is
strong to
be /
-1
(g°)
And
.
if
(p°,q°)
lies
randomizations by the entrants
below the curve, then the game starts with (independent)
(if
weak) between concession and fighting such that,
they choose to fight, the probability that they are strong
this second, case,
initial
possible that each entrant
it is
rises to
Note that,
f(p°)
if
in
weak, and that after conducting these
is
randomizations, one chooses to fight and the other chooses to concede. Then, since
one entrant has conceded, the incumbent
(because
it is
is
sure never to concede, and the second entrant
weak) concedes immediately. That
is,
a concession by one entrant at the start
increases the chances of a concession by the second, although their strategies are chosen
This
independently.
is
a manifestation of continuous time, where we are imagining that
one player can react instantaneously to the actions of second; the phenomenon
Fudenberg and Tirole (1985).
To
=
f(p)
.
it
remains to give
Analysis just as in section 4
a continuous randomization strategy p
(at time
units
is
t )
and concession
after h
,
is
must be
more
7r
and p
indifferent
— qt)ph
will
which
—h. There
is
means a continuation value
equation, which yields
7r
,
will
(1
—
pt)fth
there
is
probability
probability
bt
.
And
first
entrant
never concede, concession either at
of zero (in either scenario
A
h
(if
that
weak)
Since immediate concession by
concede immediately, for a zero continuation value.
means that the incumbent
the
cost of waiting the extra
that the other entrant will concede; should this occur, the
this entrant
will also identify
between immediate concession
The
units of time.
(with probability very close to one)
,
employed. Each entrant, while employing
the incumbent will concede, which will confer a benefit of
(1
just as in
10
specify the equilibrium,
curve q
is
t
or
t
—
h
or B), and the entrant's indifference
is
= -1 +
(1
-p
t
(5.1a)
)irbt.
For the incumbent's indifference relation, note
first
that
conceding at rate p independently of the other, and each
is
if
each entrant
(if
weak)
weak with probability
1
—
is
qt
10
The reader may be concerned with our less than completely formal style, especially
games in continuous time is underdeveloped. A completely tight
formalization of all the analysis we give in this section is possible, using one of two game
since the general theory of
forms.
minima
In the first, periods of "real time", the lengths of which are determined by the
of stopping times chosen independently by players, are interspersed with periods of
which players react to each others' actions. In the second, players
choose sets of stopping times, where, in particular, each entrant chooses a time at which it
— 1
will concede if k or more of its fellows entrants have conceded, for each k = 0, ...,
We leave all these formal details to the interested reader.
discrete, fictitious time, in
N
19
.
.
independently of the other, then the rate of the "next" concession by an entrant
of the individual rates, or 2(1
—
q t )p.
= -2 +
The terms on
if
-
2(1
q t )p(a
-f
a(l
-
—
sum
the
indifference equation
qt
)
-
(5.16)
q t )t.
the right hand side are the cost of fighting (divided by
the rate of the next concession 2(1
incumbent
Thus we have the
is
h
-2
)
and then
,
qt)p by an entrant times the expected gain to the
one of the two concedes, (a
+
a(l
—
qt )
—
qt)t
.
This last term
from the one entrant who does concede, plus the expected gain (a(l
—
qt )
—
is
the gain at
from the
qt)t
second entrant.
Equations (5.1) are substituted into Bayes' rule (3.2) and integrated with the bound-
—
ary condition that the curve q
written as a function of q
f(p) passes through (1,1). This gives the curve, with p
,
p=
In figure 4
we graph
this
?
2o/6 (l+a)(l-g)/6
e
curve together with the curve for a single entrant, p
reader can easily verify that the picture
is
as
we have shown, with the curve
lying below the curve for one for high values of q
We
>
,
and lying above
for
for
=
q
a/b
The
.
two entrants
low values of q
can see in this picture the tension between the two sides of the informational linkage
of the two contests, the beneficial stiffened strategic backbone
flexibility.
For low values of q
the loss of strategic
flexibility,
,
and the
costly loss in strategic
gains from stiffened strategic backbone exceed losses from
because
if
one entrant can be
made
to concede,
likely
it is
that the stiffened strategic backbone will cause the other to concede. For high values of q
if
one entrant concedes there are good prospects that the other
will
,
have to be fought to
protect the gains already won, so the losses from lost strategic flexibility outweigh the gains
from
stiffened backbone.
We
should note that a comparison of informational linkage vs. informational isolation
in this case
is
not entirely
trivial.
(weak) incumbent nets zero
If
(p°,q°)
in either case.
lies
If
above both curves
the initial data
lie
in figure 4,
then the
in the vertically cross-
hatched region, linkage gives the incumbent a positive expected payoff, while isolation gives
expected payoff zero in each contest, so that linkage
that isolation
is
is
clearly preferred. Similarly,
preferred in the horizontally cross-hatched region.
But
it is
in the region
clear
below
both curves, where the (weak) incumbent has a positive expected payoff in either regime,
20
.
We will
the exact computations must be carried out.
make
spare the reader the details and simply
the assertion that at points (p°,5°) below both curves, and where p°
the level at which the two curves intersect, informational linkage
above the intersection
isolation
5.3.
is
)
below
But
for p°
below both at which informational
Construction of an equilibrium for the general case
quite difficult to
and then we
Let
That
.
there are points (p ,?
preferred.
is
at or
preferred.
The equilibrium
is
level,
is
K
is,
possible.
will
of this
compute
show how,
game
for the general case has a relatively simple
We
precisely.
be the smallest integer such that
K
= [Na/(a +
To keep matters
1)J
where
,
simple,
we
[-J
will
would compute
it.
(N-K)a-K >
{N-K-l)a-(K+l) <
and
denotes "integer part of. Note that
assume that (N
K
— K)a —
>
K
=
is
.
Description of the equilibrium. In either scenario
A
or B, the
game has an equilibrium
the following form. (Refer to figure
=
K+
N
For each k
5.)
curve in the unit square state space, passing through
k lying above the curve for k
(i)
At every time
in the
+
it
begin by giving the form of the equilibrium,
will
inductively, one
form, but
1,
(0, 0)
...,
and
there
,
(1, 1)
,
is
of
a corresponding
with the curve for
1
game, there
is
identical probability that
any entrant
is
strong,
and
the strength of different entrants are independent and independent of the strength of the
incumbent.
We
denote the (common) probability that an entrant
probability that the incumbent
(ii) If
the initial
datum
(p°,q°)
is
strong by p
lies
is
strong by q t
,
and the
.
t
above and to the
T
left
of the
A
curve, then the
game
begins with an immediate randomization by the weak incumbent between concession and
beginning to play, such that, conditional on no concession, the probability that the incumbent
(iii)
is
strong rises to take us to the
If at
any time
in the
game
N
curve.
(including the start), there are
have not previously conceded, and
if
(pt,qt)
lies
k
>
K
entrants that
below and to the right of the k curve,
then those of the k "remaining" entrants that are weak randomize independently between
immediate concession and continuing to
play, so that (for each) the posterior probability of
strength given no concession moves to the k curve. Note well that
21
if
we begin
with, say,
k entrants, and
randomize, there
all
—
continue to fight and k
is
some number j < k
positive probability that
j will concede immediately.
If
j
>
K
,
then we
will
again be
will
(now the j curve), and a subsequent randomization
at a point below the relevant curve
to fight will be required.
between immediate concession and continuing
This "cascade" of
instantaneous randomizations ends in one of two ways: Either there are some some number
k'
K
>
of entrants remaining, and the posterior probability of their strength
K
(pt,qt) lies along the k' curve; or there are
(iv) If at
any time
strong or weak,
entrants,
all
game
lies
The point
game
K
(1,1)
—
The equilibrium
the k curve.
reached (which
is
which case, as we are below the k
will
1
N—K
contests remaining, even
1
,
is
curve, the prescription of step
K
if
is
contests
none of the
one
won
K
is
in the
One entrant concedes,
over.
(iii)
above
described has as basis the following consideration:
from the
and a >
This behavior ends one of
happen before time runs out
which case the game
of the entrants have conceded, then the
positive flow
weak incumbent
will
If
is
ever
or fewer live entrants,
all
in
followed.
N
—
K
never concede; the
K
enough to cover the cost of fighting the
remaining ever concede. (In terms of the case
— as soon as
the
first
entrant concedes, the incumbent
sure to fight for the rest of the game.) Hence, at any point at which the incumbent faces
It is
K
entrants remaining, and the state of the
move along
to
in
2
or fewer entrants remaining, then the incumbent,
that are weak will concede immediately.
game). The incumbent concedes,
N=
or fewer entrants remaining.
along the k curve, then the entrants and incumbent concede with hazard rates
three ways:
more
such that
the end with probability one. Accordingly, of the remaining
will fight to
that cause the state of the
or
K
there are
any time in the game there are k >
(v) If at
game
in the
is
is
K
the remaining entrants that are weak will concede immediately.
the prospect of such an event that keeps the incumbent fighting, and (in equilibrium)
the incumbent's instantaneous cost of fighting must just be covered by the instantaneous
expected value composed of the chance that such an event
incumbent
will receive for
the remainder of the
game
if
will
occur times the value the
the event occurs.
The concession
rates of live entrants are adjusted in equilibrium so that this equation holds.
concession rate of the incumbent
is
set so that live entrants are just
by the (instantaneous) chance that the incumbent
22
will
compensated
And
the
for fighting
concede. Equations similar in spirit
,
to (3.1) (and (5.1)) govern concession rates, and hence generate the curves that describe
the equilibrium.
The exact equations that
Consider
on two grounds.
k entrants
alive,
replace
the indifference equation of the incumbent,
first
each conceding with hazard rate p
net zero for the remainder of the game, hence the
The instantaneous
(3.1a) remains zero.
concession takes place
(1
is
—
qt)kp
initial
left
But, whereas
.
it
concession,
hand
is
in
if
(3.1)
there are
the incumbent concedes,
If
.
cost of fighting
concession nets at for the incumbent, here
expected value, given this
more complex than
are a good deal
(3.1)
k
side of the
The
.
(3.1a)
will
it
replacement for
rate at which the next
(and
in
(6.1)) the next
does not. The expression for the incumbent's
is
a monster: The k
—
1
remaining entrants
all
randomize immediately, with concession probability that would carry the posterior up to
the k
—
1
curve (whose placement
derived in the previous step of a recursive derivation).
is
Conditional on the number of the k
k
—
—
we now have another immediate randomization
1 ),
lower index, etc. Working through
still
entrants that do not concede (unless
1
these,
all
we
to carry posteriors
(ii)
which case the incumbent nets expectation zero
in
still live,
the probability that
entrants (the others
on j
),
all
it
ends with j
strong with
K—j
concede immediately,
we have
for the
K
live
probability q" that depends in a nontrivial
way
K
live entrants.
If it
less
it
all
j
<
k
,
then this expected gain
seems rather formidable
is
the easier to deal with.
remainder of the game. Hence the
The instantaneous
.
(This
cost of fighting
is
—1
left
,
If
hand
as in
bt
at
for the entrant.
(If
A
certainly
and B becomes
it
side of the analogue to
And
will net zero
(3.16)
zero.
is
the instantaneous expected
some other entrant concedes,
either the
—
pt)x
game continues
along some other j curve, along which the entrant nets zero expected value. Or the
23
If
analytically.
an entrant concedes,
(3.16).
is
gain arises from the chance that the incumbent concedes, which occurs at rate (1
and nets
is
—t from each of the j remaining that turn out to be strong.)
analogue to (3.16), the distinction between scenarios
B
<
ends with j
already dead entrants, plus at from the each of j remaining that
computable, step by step. But
important. Scenario
k with j
and
the placement of the j curves for
for the
<
j
the incumbent nets an expected reward of t[a((K—j)+j(l — q*))—jq*]
from each of the
As
<
the
(i)
in continuation,
<
some
K
all
is
up to a curve of
get a marginal distribution for
probability that this cascade of randomizations ends on a j curve for
entrants
it
game
K
ends with a cascade leading to
mean a
or fewer entrants, which will
(weak) entrant.) Hence, in scenario B, x
payoff of zero for a
given by precisely (3.16).
is
A, things are more complex. Note that a weak entrant who has conceded
In scenario
does not necessarily have zero expected payoff for the remainder of the game.
K
more than
entrants
left,
there
of immediate concession, which goes on the
it is
Hence the value
hand
left
side of the analog to
the solution of an integral equation.
The
(3.16),
hand
right
(k
times the probability that the incumbent concedes, as before.
bt
—
1)(1
—
q t )p
,
some other entrant might concede, and
weak entrant whose
payoffs
we
outcome
concession (although
entrant that
for
it,
now we must
at least,
is
no
1
)
and the
also, at rate
this will effect the value to the
The exact impact
are attempting to balance.
before, on the distribution of the
But
is
side of the
entrant's indifference equation will include instantaneous cost (incurred at rate
gain
there are
a chance the incumbent will concede, and then the
is
entrants that conceded earlier can reenter to accrue a positive reward.
longer zero. Rather,
If
will
depend, as
of the cascade of randomizations let loose by any
condition that distribution on the knowledge of the one
indeed weak) and on the value of continuation to a weak entrant
each of those outcomes. Finally, since this
is
meant
to evaluate the expected value to
an entrant of conceding after h more time units, we must compute that (no longer zero)
expected value, which involves yet another set of probabilities of how the next cascade
resolve.
Once again, the curves are computable
in theory (although
now they
will
are integral
instead of differential equations).
One
sees
above does
from the
exist.
curves for j
<
the k curve
lies
to (3.1a)
,
this description
For k
—
K + 1, ...,N
as the
—
one "computes" the k curve recursively, with the
,
curve.
1
to prove that an equilibrium of the form given
The important
k already in place.
below the k
how
To
step in the recursion
see that this
k curve comes closer to the k
—
1
is
so,
is
to
show that
note that in the analogue
curve, the probability that
all
k
—
1
entrants continue to fight after an initial concession goes to one. Hence the "gain" term for
the weak incumbent goes to zero.
increase,
k
—
1
in the
To achieve
equality in the analogue to (3.1a)
which increases the slope of the k curve, pushing
curve.
(The exact argument
analogue to (3.16), but we
is
will
a
bit
more complex
not go into details.)
it
p must
away from (and below) the
in scenario
An
,
A, owing to changes
equilibrium exists that does
have the form given, and, with a large computer budget, one could solve (to any desired
24
accuracy) for the position of the curves.
level of
5.4.
Asymptotic
results
While we are unable to obtain a closed form solution
B
in scenario
for the equilibrium,
N
at least, to obtain asymptotic results for large
and a/(a
sequential contests, the comparison of q°
-f 1)
As
.
are able,
on
in the literature
the key.
is
we
Let us point out,
however, that here the asymptotics do not follow from the general argument of Fudenberg
and Levine (1987),
entrants have the
as
same
same length of
for the
no matter how many entrants are involved, the incumbent and the
decision horizon.
time.
Of
That
is,
the reputation of each will be of value
course, the incumbent's reputation
of value against
is
opponents, but we know from the no-reentry case that this fact alone
matters
of course, the comparison of costs
is,
Throughout
is
this section,
irrational, so that
Proposition
4.
In scenario B,
The idea
let
K^
1)
is
never
aN/(a +
game by conceding with
Proof.
we
q°
if
>
a/(a
Thus there
is
amount
and we
,
will
assume that a
+ 1)
,
N—
then as
co the incumbent begins the
of the proof
is
that, for q°
>
a/ (a
+
N — Kn
1)
,
N
as
goes to infinity there
is
entrants will ever concede, because
N
—
entrants are in fact weak.
iT/v
vanishingly small chance that the incumbent will ever get a net positive
will
show that the expected reward vanishes with
only fight for a very small period of time.
of time the incumbent
and the equation
initial
1)J
probability approaching one.
flow of reward (in fact, we'll
incumbent
What
itself integer.
vanishingly small probability that more than
is
not decisive.
and benefits of the reputation.
denote [aN/(a-\-
vanishingly small probability that more than
there
is
more
must be
for the evolution of
randomizations)
is,
willing to fight
p
t
is
is
But
N
For any fixed p°
in the limit, one.
,
,
),
so the
since, in scenario B, the
the time
independent of
N
it
it
takes pt to reach one,
must be that pi
(after
this gives the result of the
proposition.
A
bit
more
formally, note that in the equilibrium,
cumbent to refuse to concede
until such time as
the incumbent does not concede
concede).
The time
r/v
at
if,
which p
(pt,qt)
before this time,
t
reaches
1
25
it
is
in-
reaches (1,1) (where, of course,
N — Kn
in the
a best response for the
°r
game with
more entrants do
N
in fact
entrants, depends on
p^
the probability
differential
that the incumbent
equation p
=
p/bt
In particular, this time approaches
.
of time that elapses until (1,1)
to 1
strong after any initial randomizations and the
is
So we aim to show that ts must approach one as
.
In the
N
game with
entrants,
if
or fewer strong entrants initially.
will,
good deal more).
amount. And
If
If
by waiting until
Kn
is
N—
oo
T/v
,
lose at least
K^
Ks +
more than
(a
+
Kx
—
From
goes
,
then the chance
K^
strong entrants, then
1
—
1)(1
aN
or fewer entrants are strong) X
Kn +
strong entrants) x (a
1
(and, probably, a
r/y)
the most the incumbent
is
" strategy,
the ex ante
and the second probability goes to one. (The estimate given
must be that
it
r/v
—
1
aN
+
1)(1
—
t/v).
a rate exceeding exp(-i\7l/4 )/iV 1 / 6 |
Feller (1968, VTI.6), the first probability goes to zero at
expected value to be nonnegative,
p^
bounded above by:
is
Prob(more than
if
.
can hope to win. Thus, by following the "don't concede until t^
Prob(
and only
amount
the
strong entrants, the incumbent will lose some
or fewer strong entrants, then
expected payoff to the incumbent
if
is,
precisely the probability that there are
there are
there are precisely
there are
if
N
(that
the incumbent waits until time r/y
that the incumbent "wins" before time ryv
the incumbent
in
reached approaches zero)
is
1
,
is
fairly crude.)
Hence
for the
which completes the proof.
D
Proposition
5.
In scenario B,
if
q°
< a/(a+l)
,
then as
N
approaches oo
,
N—K'jj
or
more
entrants concede at the outset (and the incumbent "wins") with probability approaching
one.
Proof.
We
will
not give
all
them from the following
the details, although the reader should be able to reconstruct
sketch.
To
begin, establish
some notation and terminology. Let
A-(i±V
+ l)/2.
a
By assumption,
of
no concession
at
some time
t"
A
is less
than one. Imagine that the incumbent,
until such
>
,
if
weak, plays the strategy
time as posteriors (1,1) are reached. Note that
this
happens
which can be bounded below by integrating the equation for the
•
26
.
with the
evolution of pt
B
the assumption that scenario
N
change with
will
exceed p°
Imagine
Note that
.
,
and
that gives the
we
note,
number
is
game
among
his best responses). Let
entrants after any cascade of randomizations at time
to denote the left continuous versions.
That
is,
Let
S(t)
let
K^
are
is
the
The
consider two stopping times.
or fewer entrants remaining - the
first
time that
must occur prior to
<£(2)k(2)/./V
t"
.
The
first
first is
For definiteness, we
1)
number
t
,
We
etc.
number
of remaining
n(t)(f)(t)
and
of strong entrants given
t
the
first
Note that,
.
the
fix
use k(Z~) and 4>(t~)
will
time
(if it
hitting time of {^{t)
first
> Aa/(a +
denote the (common)
<f>{t)
n(t ) gives the
the expected
is
be the stochastic process
k(<)
and S(t~) be equal to
5(i)
the information revealed up to (and including) time
Now
does not
as he plays this particular waiting
of remaining entrants at time t, and
respectively.
t
larger.)
right continuous versions of these stochastic processes:
,
motion of p
we use
begins with a randomization by the incumbent, p T
posterior probability that the remaining entrants are strong.
K(t~)<f>(t~)
point only that
(It is at this
.
pertains; to ensure that the law of
the
be
will
t"
p°
incumbent watching the game evolve
this
strategy (which,
if
=
condition p\
initial
for large
occurs) that there
< Ks)
N
,
The second
one of rx or r2
stopping time marks a time at which the incumbent has
"won". The second marks a time at which the remaining entrants are surprisingly strong.
The meaning
As
of the second description arises from the following result:
N—
oo
,
the probability that
We leave the proof for later, but
that any entrant
is
value.
this event,
number
t
is
).
We
approaches one.
of strong entrants subsequently,
N
Moreover, for large
suppose that at some time
reached (before r2
time
.
number
So the probability that S(t)/N exceeds Aa/(a +
Now
on
q°
to
the intuition should be clear. Given the prior probability q°
strong, the expected
must have expected value
occurs before
T\
Z,
,
there can be
l)
little
claim that the expected
bounded above by S{t~)
.
of strong entrants just before time
t
The
is
t
N
the time
its
-* co
r-y
.
is
S{t~)
is
the expected
probability of enough concessions at
one of the «(i~) did concede), conditional
decreasing in the
27
is,
as
,
of strong entrants, conditional
see this, note that
to cause the "win" (given that at time
on the number of actually strong entrants,
number
To
.
variance in
must approach zero
the incumbent "wins"; that
S(t)/N
number
of strong entrants.
.
Hence the expected number of strong entrants, conditional on a win at time
than the unconditional expectation S(t~)
incumbent, conditional on a win at time
(aTV
-
(a
+
At time
.
and conditional on «(f~) and
t
two stopping times,
in
,
S(t)/N < Aa/(a
—
expected value of aTV(l
before the times T\ and r2
,
Until these
this conditional
t
,
4>{t~)
X)t"
+
is
,
1)
,
less
is
,
Thus the conditional expected value
.
l)S(t~))t*. Since r2 was not hit before time
have a lower bound on
TV
t
t
to the
at least
and we
.
the maximal rate of cost to the incumbent
is
our equilibrium the instantaneous cost to the incum-
bent must just equal the hazard rate at which the incumbent "wins" times the conditional
expected value
cumbent
it
receives
does win. So we see that the hazard rate of a win for the
if it
and r2
at times before rj
is
But then by integrating the hazard
to time
t"
,
bounded above, uniformly
rate,
we
the chance of a "no win" (that
in TV
,
by l/(a(l — A)i*)
see that, over the time interval
delayed until after r2
is
Ti
in-
) is
from time one
bounded below
by
Prob(no win
If
we know the
x
c
probability of "no win before r2
"
of a win at the outset
To complete the
must go
proof,
—
r2 goes to one as TV
=
max(ri, r2 )
n
Fix any
.
p
must go
to zero in TV
,
then probability
to one.
To show
.
stopping time, and that S(t)/N
r
-(i-0/(»(i-A)O
remains to establish that the probability that
it
oo
at outset)
e
>
is
this,
note that the
maximum
tj
of rx
a bounded martingale, so that E(S(t)/N)
,
and
is
less
and r2
=
q°
,
than
is
a
where
let
a
6
= e(-^-a
Select
N
-\-
q °)/4.
1
sufficiently large that
Prob(fraction of strong entrants
(That
sufficiently large
the set Ai where t2
of strong entrants
11
is
>
q°
Ti
—
q°
—
6
)
<
5.
do exist follows from the estimate given
TV
Now decompose E(S(t)/N)
previous proposition.)
<
;
6
the integral over the set
or less;
A 2 where
set
The
integral over
T\
> r2 and
the fraction
A3 where rx > r2 and the
Recall that clocks run backwards in this model, so that the
is a stopping time, etc.
28
proof of the
into three pieces:
and integral over the
times
in the
maximum
of two stopping
,
fraction of strong entrants
is
above
And
nonnegative.
+
+ Prob(A 3 )) -
>q° -
X Prob(A 2 )
q°
the definition of 6
Ai
is less
,
,
The
.
l))/2
(Prob(Ax)
By
e
6
integral over
Ai
bounded below by
is
from the definition of Aj
,
bounded below by
is
+ -4t)/2 + Prob(A 3
a + 1
(q°
>q° -
than
a/(a
—
.
The
—
q°
integral
times
S
its
bounded below by
is
Prob(A!) X
>q°X
q°
the integral over A3
Hence E(S(t)/N)
probability.
+
times (q°
the probability of Ai
over A2
is
-
+ Prob(A 1 )(-^— a + 1
6
+ Prob(A )(-4T a + 1
26
-
x (q°
6)
Prob^X^- - q°)/2
+
Prob(A 3 )<5
)
1
q°)/2
9 °)/2.
then, the original integral can equal q° only
if
the probability of
which completes the proof.
D
Although
more work, the methods used
takes a bit
it
be used to describe the location of the k curves
for
a < a/(a + l) the
,
a/(a(a
+
1))
.
That
q level of the
is,
the picture
we needed
proposition,
any given starting point)
it
would have to begin
know
to
in
N
in the
1
is
game. So
Intuition
it
to
let
that
;
if
p
who
(in
has to reach
t
for large
become very
reached, there
N
Fixing p £
.
is still
1
in
of time
is
in that
to
know
(or above),
when
we needed
from p°
(bounded below
better off in scenario
it is
A than in
likely to survive in scenario
is
29
itself. If
N
)
left
in scenario A.
B. In scenario
sit
on the
the incumbent
the benefits (in scenario A) that accrue
correct, then proposition 5
A.
in
better for any single entrant to
the other entrants fight the incumbent than to fight
all
first
a short period of time, then
slow; that starting
some amount
In the
become very quick (from
seems clear that one of the two propositions must hold
fought. If this intuition
(0, 1)
the TV entrant game) approaches
In the second proposition,
.
does concede, an entrant on the sidelines gets
to one
p
)
assume that scenario B prevailed.
would suggest that the incumbent
and
K^
that the motion of p t does not
A, entrants face a "free rider" problem
sidelines
at
>
and 5 can
as in figure 6.
is
at a value close to 1
that the motion of p t does not
the posterior
[aN] curve
we needed
In both propositions,
k
(
to prove propositions 4
would be the one more
In fact,
we
believe that both propositions
version of this paper (available
do
in fact survive in scenario
upon request) shows that
this
is
so at the "limit"
which the incumbent faces a continuum of entrants. But scenario B
the point that
we wish
at the structure of the
to make,
game
namely that with simultaneous
to see
if
A.
play,
is
A
earlier
game,
in
make
sufficient to
one must look closely
the reputation effect of the incumbent dominates.
We
leave further analysis to the interested reader.
References
Feller,
William,
edition),
An
Introduction to Probability Theory and Its Applications, Vol.
John Wiley and Sons,
New
1
(3rd
York, 1968.
Fudenberg, Drew and David Levine
with a Single Long-lived Player Facing
[1987]
many
Fudenberg, Drew and Jean Tirole [1986]
,
"Equilbirium Selection in Repeated
,
Short-lived Opponents,"
"A Theory
Games
mimeo.
of Exit in Oligopoly," Econometrica,
in press.
[1985]
,
"Preemption and Rent Equalization
in the
Adoption of
New
Tech-
nology," .Review of Economic Studies, Vol. 52, 383-402.
Kreps, David and Robert Wilson [1982]
of Economic Theory, Vol. 27, 253-279.
"Reputation and Imperfect Information," Journal
,
Milgrom, Paul and John Roberts [1982] "Predation, Reputation, and Entry Deterrence,"
Journal of Economic Theory, Vol. 27, 280-312.
,
30
.
Appendix: Sequential play of the concession game
we analyze
In this appendix,
the end of section
sequential play of the concession game, as outlined at
In particular,
3.
a particular equilibrium, which we
this
is
we provide a proof
will construct
of proposition 2.
n,n —
play in contests
incumbent
is
1,
...,
1
and
version of Proposition
Proposition
The
We
will
prove the following strengthened
2'
> a/(a +
game, v n (p)
<
then v n (p)
1)
such that the incumbent,
above by k(p°)
a(l
—
is
nondecreasing
q°-)/q°
<
1
,
N
a/(a + l)
—
,
m
weak,
then there
is
in
m
p.
is
a constant k(p°) (independent
a total length of time that
will fight for
decays exponentially
of index greater than n(p°)
with certainty.
if
in
and the long run expected value per
Hence the probability that the incumbent
.
the end of round
<
(Recall that contests are
.
probability q° enters as a parameter
contest to the (weak) incumbent approaches zero. There
(c) If q°
We
2.
(a) In the equilibrium to this
(b) If q°
1 is
denoted by v n (p)
is
the last.)
held fixed throughout.
is
is.)
it
that
n beings with the entrants assessing that the
contest
if
strong with probability p
to this function
)
know
not
for
the expected value to the weak incumbent from equilibrium
indexed backwards, so that contest
N
(We do
along the way.
done
is
the only sequential equilibrium of this game, although we believe that
use the following notation:
of
This
(if
bounded
weak) has not conceded by
.
a number n(p°) such that in
all
contests against entrants
the incumbent (weak or strong) will fight for the
,
is
unit time
full
Accordingly, weak entrants in those contests concede immediately.
lim^oo vN (p) =
c(l
-
q°)
-
q°
(for all p°
>
And
).
The equilibrium
The
first
step
is
to give the equilibrium construction.
Recall that the incumbent,
strong, will always fight, and strong entrants always enter. Hence
we
will describe only the
equilibrium strategies of the weak incumbent and weak entrants. Throughout, q°
probability (at the beginning of each stage) that the current entrant
a parameter and
to the
is
suppressed in the notation.
weak incumbent
for play of the
We
is
(o)
If,
at
game from stage n
equilibrium takes the following form at stage n
to stage 1
concedes immediately in
(i) If
v n ^i(p)
>
1
,
all
is
is
the prior
treated as
,
inclusive,
is
when
stage
strong.
:
any point in the game, the incumbent concedes, whether
trants revise assessments that the incumbent
strong
,
use v n (p) to denote the expected payoff
n begins with the entrants assessing probability p that the incumbent
The
if
in equilibrium or out, en-
strong to be zero. Thereafter, the incumbent
contests, and entrants never concede.
then the (weak) incumbent
Al
will refuse to
concede
in the current
round,
even
convinced that the current entrant
if
the outset of this round.
v n _i(p)
(ii) If
<
1
,
immediate
It is
then there
is
strong.
is
in this case
Hence weak entrants
will
concede at
— (a+
l)q°
+ v n -i(p)
that v n (p)
=
a
= pb / a
a curve analogous to the curve q
.
along which play
evolves in this stage, after initial randomization by one side or the other which takes us to
The placement
the curve.
and on p
of this curve depends on n
.
It is
given by indifference
equations analogous to (4.1), which in this case are
1
where
=
(1
-p
t
and
)irbt
=
1
(1
-
q t )p(at
+
t;
(A.l)
n _i(p t )),
in these equations refers to the time remaining in the current stage.
t
These
indif-
ference equations, together with Bayes's rule (4.2), give differential equations
p
=
f
bt
and
=
q
—
^
at
for the evolution of the posteriors in the current stage.
curve (analogous to the condition in the single contest
(1,1))
is:
At the time
t*
(in
(A.2)
y
+ Vn-^p)
'
The boundary condition
game
the current stage) at which q
for this
that the curve passes through
=
t
1
(the entrant
certain to
is
be tough), the value to the incumbent beginning the next stage must satisfy the inequality
(A.3)
v n -i(pt-)<f.
If
(A. 3) holds as a strict inequality, then the boundary condition
weak incumbent's strategy has
If
called for
<
(A.3) holds as an equality, then pt-
calls for
We
him
vn
when
is
possible,
(ii) is
it is
,
and the
we
We
will
,
the
number
of
show, as we proceed, that the boundary condition
uniquely specified as we compute v n recursively. As part of the induction
establish that each v n has the following properties:
nonnegative, has value zero at p
is
1
and the weak incumbent's strategy
establish that an equilibrium of this form exists by induction on n
hypothesis,
(iii)
Pf =
to refuse to concede for the remainder of the current stage.
stages remaining in the game.
given in
that
to concede with probability one by this point.
him
1
is
nonzero, and
it is
—
,
is
nondecreasing and
strictly increasing
continuous.
Initiating the induction.
As the
first
stage in our inductive proof, we note that vi
,
the value function associated
with the equilibrium for a single stage of the game given in section
given in
(iii).
has
all
the properties
Thinking of vq as being identically zero, we see as well that the equilibrium
for this last stage
If the
3,
is
described by
(ii).
This initiates the induction.
incumbent ever concedes
A2
.
We note next that
(o)
above
is
indeed compatible with (sequential) equilibrium. As long
weak incumbent wishes
as entrants will never again concede, the
(and being unshakeable in this belief)
is
to concede immediately in
incumbent to concede immediately with probability one
cases. Entrants, expecting the
all
.
never concede. Note that this shows that » n (0)
will
indeed zero. Also, since a strong incumbent
will
never concede, the beliefs entailed in (o)
are consistent with any equilibrium strategies for this game.
Beginning with v n -i(p)
>
1
beginning at the start of stage n
If
u„_i(p
n
)
incumbent
If
>
1
,
then play as described
not concede at
will
with
,
all in this
in step
(i)
is
If
the
weak entrants concede immediately.
stage, then
will
be certain,
if
there
is
strong, and hence the current stage will cost
no
1
.
weak entrant does concede, then p will move to zero, and the incumbent
then net the continuation value u n _i(0) =
Hence it is an equilibrium for the weak
Given
will
is
.
certainly equilibrium play.
weak entrants concede immediately, then the incumbent
concession, that the current entrant
—
<
Consider play
for all j
n
1
n
initial probability p
that the incumbent is strong.
Suppose the properties above have been established
(o), if the
.
incumbent to pay the cost of
(Indeed, this
is
1
and get
in return the continuation value
t'„_i(j)")
the only possible path of equilibrium play at this stage, given that u„_i
gives the continuation value.
If in
some equilibrium the weak incumbent were
with positive probability, then Bayes' rule would force
to concede
beliefs given concession to zero, giving
zero continuation value next period. Since fighting will lead to a p
n~l
beginning the next
no lower than than the current value p n and since t> n _i is (by the induction
hypothesis) strictly increasing above p n (and, hence, is strictly greater than one), it cannot
stage which
is
,
be an equilibrium for the weak incumbent to concede with positive probability.)
This implies that v n (p n )
of v n ^i
The harder
case
<
is
)
.
entrants play as in
v n inherits
1
p above the smallest value such that
(i),
3,
an
(i)
is
.
Note
not equilibrium behavior.
If
then unless the entrant concedes immediately, the incumbent
if
he follows the
is
insufficient to
compensate him
for the cost of
end of the current period.
construct instead an equilibrium of the form outlined in
initial
1
— preliminaries
equilibrium strategy and does not concede
We
the properties
where we begin stage n with p n such that v n _i(p n ) <
1
for
concludes that he faces a strong entrant, but the value of continuation
fighting to the
all
>
(iii)
that, in such circumstances, the behavior described in
the"
Thus
v n {p)
that are outlined in
Beginning with v n (p)
= a-q°(a+l)+v n -i(p n
randomization by one side or the other, such that
then both sides
(if
(ii).
There
as in section
no
initial concession,
rest of the
game. Each then
if there is
weak) have continuation value zero for the
is,
concedes continuously, until such time as either one or the other concedes, or until an
appropriate boundary condition
is
reached.
We
A3
use, as before,
n to denote the concession
.
.
(hazard) rate of the weak incumbent, and p to denote the concession rate of the weak
entrant.
the same sort of logic as in the one stage game, x and p must satisfy the
By
The equation
indifference equations (A.l) given above.
r
for
— the weak
just as before
is
entrant must just be compensated for the cost of fighting by the expected rate at which the
incumbent
won
concede, times the prize
will
if
The equation
the weak incumbent
the incumbent does concede.
p has an additional term, since concession by the entrant gives
at in the current contest and v n -i(p ) beginning the next. Bayes' rule
for
as always, so
is
t
regime of continuous randomizations, posteriors evolve according to the differential
in this
equations (A. 2)
v n (p)
<
— solutions of the
1
The next
step
n
(p ,q°)
and the r function
to consider the solution to the differential equations (A. 2) as a function
is
,
We
Refer to figure Al.
of initial conditions.
some point
differential equations
suppose that we start the current round at
and that there may be an
randomization by one (weak) side or
initial
the other, so the differential equations will be initiated at
P\
>P
n
an d
?i
=
q°
or p\
=p
n
and
>
q\
q°
That
.
is,
some point
we begin the
(
p\ q\
,
with either
)
differential equations
n
running at some point along one of the two rays emanating from (p ,q°) that are shown
Al. Given the assumptions about u n _i in the induction hypothesis, the solutions
in figure
to the differential equations are well behaved in their initial conditions:
(iv)
The
solutions to the differential equations (A. 2)
,
as functions of their initial conditions
along the rays shown in figure Al, are continuous in those initial conditions. Paths from
two
different initial conditions
The equation
Pt
—
1
is
do not
for pt integrates to
intersect.
show
that, since p\
> pN
,
as long as
p
N >
conditions along the two rays, there
continuous bijection between
{(p, 1)
the level
reached by some time prior to time zero. Hence from each of the possible
is
a time
depending on the
t*
n
p > p ]
1J{(1 q)
q
'
5
Accordingly, there
is
initial
>
initial
conditions at
initial
which the path of (pt,qt) exits from the unit square. Moreover, because of
:
,
(iv),
there
is
a
conditions along the two rays and "terminal positions"
q } at which
some particular
(p t q t ) departs the unit square.
,
condition (p, q) along one of the two rays
initial
such that the solution of (A. 2) starting from that point passes out of the unit square at
(1,1).
Figure
Al
is
drawn
so that
(p, q)
the case in general. However the following
is
a unique starting position
(p(p'), q(p'))
lies
is
n
to the right of (p ,q°); this need not
generally true. For every p' € [p°,
lying to the right and/or above
1]
,
be
there
(p.q), along
one of the two rays, such that with (p(p'),q(p')) as starting position, the solution to the
differential equations exits the unit
square at
(p'
,
1)
.
These
initial
conditions are continuous
in p'
Let
t(p')
{p(p')i q{p'))
,
be the time remaining when the curve that exits at
hits
(p',1).
Being excessively formal, r(p')
A4
is
(p', 1)
,
starting at
such that, in the solution
.
to
with
(A. 2)
p(p'),q(p')
condition
initial
and r(p')
,
all
(p(p'),q(p'))
depend on the
the two rays that form the set of possible
(v) For
any
point (p
initial
p'
Moreover, for fixed
P
n
,
q°)
r(p')
,
,
r(p')
is
initial
=
and qT (p>)
p'
(Note that
1-
n
n
point (p ,q°), since (p ,q°) determines
Now we
initial conditions.)
can show:
continuous and strictly decreasing in
nonincreasing in p
is
=
Pt( p ')
,
n
,
and
p'
strictly decreasing
it is
n
€
[p
,
1]
p(p')
if
=
n
.
We
leave to the reader the task of establishing (v), using the differential equations
and the assumed monotonicity
<
v n -i(p)
We
1
p)
(in
of
t>
n _i
Note that r(p
.
n
)
=
and t(1) > 0.
1
— terminal conditions and the equilibrium
can now derive the terminal condition that determines the equilibrium beginning
at stage n with
n
p
We
Refer to figures A2(a) and A2(b).
.
n
v n -i(p') for p' €
[p
,
1]
have graphed there r(p') and
Note that, by assumption, u n _i(p n ) <
.
1
=
r(p n )
either have a picture as in A2(a), in which the two curves intersect once
at some p' G (p n ,l], or we have the picture in A2(b), in. which u n _i(l)
uniqueness and existence of an intersection point
facts that
both u n _i and r are continuous, r
increasing (where
it is
is
(if
(l)
<
r(l)
.
Then we use
until this posterior
value zero. This
is
for the next stage to begin (since the
is
is
reached,
its
optimal continuation
is
r(l)
all
along the curve through (1,1), hence t
incumbent to
its
opponents' strategies.
and u n _i
strictly
is
At
less
this point,
is
consider the case where the curves r„_i
the point of intersection of the two curves.
than the cost the incumbent
incumbent now knows that the
if
the weak incumbent waits
to concede immediately, netting
is
and r do
a best response
intersect.
By
construction,
reached, the value of continuation beginning in the next stage
weak incumbent must incur to get to that
terminus
is
zero.
stage.
We
for the
important that,
in the
r(p*) with positive probability
is
now
is
all
weak
use in this case
when the time
,
where
r(p")
is
just equal to the cost the
Hence the value to the incumbent
Once again, the incumbent's value
be identically zero, and the incumbent
It is
n the
by assumption,
the terminal condition that the curve for stage n passes through the point (p", 1)
p"
The
.
show that the weak incumbent's expected
zero
Now
<
the right terminal condition; as in the single contest analysis, we can
integrate back from this terminal condition to
is
.
strong with probability one). Accordingly,
is
and once only
as terminal condition for stage
the value to the weak incumbent starting in the next stage
current entrant
Hence we
t(1)) follows from the
strictly decreasing,
condition that the curve must pass through the point (1, 1)
would pay to wait
>
j_i(1)
t; T
.
nonzero).
Imagine that v n _ 1
payoff
(A. 2)
at this
along the curve integrates back to
using a best response to
its
opponents' strategies.
second case, the incumbent does not concede at the time
—
it
continues on to the next stage with probability one.
A5
.
Of
r(p*) the incumbent wishes strictly to remain, as the value
course, after the instant
beginning the next stage
.
undiminished, while the cost remaining to be paid to get to
is
that stage decreases as time passes. But at r(p*)
the incumbent
would, by waiting until (just after) r{jp*)
,
<
We
that v n inherits
straightforward:
for each p'
,
If
p
n
and
,
moves a
when
bit, so
do the associated
The
.
obvious.
is
It
The argument was
.
suppose that v n _i(p
n
<
)
1
q
>
,
Since v n (p
.
determines v n
n
a
is
+ ^_i(p
n
)
the v n -i(p
-
(a
n
<
)
+
1
First, if v n _i (p
u n _! (p n ))g°
(1
=
so that v n (p
,
bit,
)
>
,
strictly increasing in
is
and consider v n (p n )
the case v n -i(p n ) > 1. So we may
given for
n
n
>
)
,
we know from the form
we show that
q
)
>
n
)
>
1
,
then
v n (p
n
v n (p )
n
is
n
)
=
(1
=
a
—
—
of the equilibrium
is
n
some
for
(p ,q)
+ v n -i(p n ))
+ v n ^i(p n ) >
q° /q)(a
+
(a
l)q°
n
- q°)(a + r„_i(p )) > v n (p ) The other possibility
= (1 - q°/q)(a + v n -i(p n )) where q is the posterior
By the monotonicity of u„-i we have the desired
n
.
,
appropriate to starting at (p n ,q°).
result as soon as
moves a
.
"such that v n (p
and the weak incumbent's expected payoff
Consider two cases.
tedious but
is
intersection point p* therefore
that the appropriate starting condition for the corresponding curve
q°
this form),
conditions p(p') and q(p')
initial
remains to show that v n
nonzero. Fix, therefore, some p
some p n > p n
for
the required properties. Continuity of v n
all
this initial condition continuously
it is
an equilibrium of
is
condition at the intersection point q(p") moves continuously
initial
Nonnegativity of v n
n
proof that there
finish the
and (therefore) so does r(p')
which means that the
p
.
have now established that the equilibrium in stage n has the form we outlined.
we must show
in
then the weak entrant
,
— completing the induction step
1
To complete the induction (and
n
Still,
obtain a positive expected value, contradicting
the optimality of the strategy specified by p
v n -i(p)
just indifferent.
is
the incumbent were to concede with positive probability at r(p")
if
p
,
,
q
This follows from the derivation of terminal conditions in these cases: As noted before,
the r curve for the starting value p
— pn
is
(as a function of the terminating p'
greater than the r curve for the starting value p
amount
of time left r
for (p ,q)
,
We
q°
(vi)
(vi) If q°
>
1)
,
for
a/(a
p
for large
we have an equilibrium
n
than
equilibrating p' and the
in
will
of the
everywhere
)
p
be
n
.
But
less for
if
q
<
q
n
(p
,
q)
,
the
than
form described.
N
next characterize the equilibrium for large
> a/(a +
in
Thus the
above can be used to show that r
a contradiction. Thus
The equilibrium
.
must both be greater starting
argument used to prove
n
—p
n
N
.
It is easiest
to begin with the case
which the following holds:
+ 1)
,
then v n (p)
the equilibrium behavior
is
<
a{\
— q°)/q° <
1
,
uniformly in p and n
never of the form described in
A6
(i)
.
Accordingly,
but always has the form
(ii).
There
an upper bound k(p°) on the amount of time the incumbent
is
and (therefore) the probability that the incumbent
N—m
of round
We
decays exponentially
bound on
establish the uniform
which
uniformly in p
has the form
.
Then we know that
(ii),
(1
-
q°)(a
to use the monotonicity of v n in
(Indeed, one can
a(l
— q°)/q°
incumbent,
.
if
show that
.
< °(l — 9
has been shown that u n -i
weak incumbent
to the
+
a(l
—
q°)/q°)
p and compute
for
any p <
1
=
a(l
directly
this
bound
first
contest
is
,
q°)/q°
.
is
)/9°
q°)
<
1
regardless of p n
(An
easier proof
)
,
,
,
is
.)
N
,
<
lim/v_oo v^(p)
not tight:
is
—
a(l
n
(1 — q° / q)(a + v n _i(p
is
w n (l)
This in turn implies that no matter how large
weak, concedes in the
—
<
v\{p)
last stage,
the form of the equilibrium in stage n
and the expected payoff
bounded above by
is
it
will fight,
weak) has not conceded by the end
by induction. In the
vn
than our bound. Suppose that
is less
which
m
in
(if
weak)
(if
the probability that the
bounded away from
zero.
We
leave this
to the reader.)
To obtain the uniform bound on the
reason as follows.
will fight,
If
total
amount
of time that the
weak incumbent
ever p reaches one, by that time the incumbent,
weak,
if
must have conceded with probability one. Hence we wish to show that p reaches one after
the incumbent has fought for an amount of time that is bounded above (depending on p° ).
We
can obtain such a bound from the law of motion of p
p > p/b
The proof
.
=
in
1
,
since
any time p
is
complicated slightly by the fact that the weak incumbent
when p
fighting at times
q
is
,
moving,
may
be
not moving. (Recall that when the state reaches. the boundary
is
the incumbent does not concede for the remainder of the contest.) But in any contest
which the incumbent does
units of time;
if
fight at all,
p must be moving
for the first
1
—
a(l
-
q°)/q°
the boundary was struck before that time, then the time remaining would
exceed the continuation value, in contradiction to the nature of the equilibrium (unless p
has already reached one). So for every one unit of time that the incumbent fights, p moves
for
1
-
before
a(l
—
q°)/q° or more. This, together with a
strikes one, gives
it
Finally, the
the
full
an upper bound on how long the (weak) incumbent
incumbent,
unit of time (until
if
it
weak,
will
concedes).
have to fight every strong entrant
bounded above by the chance that
is
will fight.
it
meets, for
Given an upper bound k(p°) on the amount of
time the weak incumbent will fight, the chance that
contests
bound on how long p can be moving
in
it
those
will live
first
beyond the
k(jp)
+ rn
first
contests,
k(p°)
m
or
+
rn
more
of the entrants faced are weak. This clearly decays exponentially.
Large
N
and
When
(viii) If
q°
q°
q°
< a/(a+
< a/(a +
< a/(a +
greater than n(p°)
,
1)
1)
,
1)
,
a very different picture emerges.
then there
is
a number n(p°) such that in
the incumbent (weak or strong) will fight for the
A7
all
contests of index
full
unit of time with
+
.
certainty.
That
is,
the equilibrium behavior
And
concede immediately.
in those contests
as in
is
lim^r_ O0 v^{p)/N
from the previous step that we can
First recall
above. Accordingly, weak entrants
(i)
=
a
—
bound the
easily
(a
+
l)q°.
amount
total
of time
N
that the weak incumbent randomizes between fighting and conceding, irrespective of
(depending on p°
is
),
since
whenever such randomization
governed by the relationship p
=
p/bt
Suppose that we have n contests
taking place, the evolution of p t
is
> p/b
to play in the game.
left
on the expected length of time that there
be fighting,
will
We
develop a bound
will
the (weak) incumbent adopts
if
the (nonequilibrium) strategy of never conceding, and the entrants follow their equilibrium
strategies.
•
n
E [total
time fighting]
= YJ E
[time fighting in contest k
]
=
k=l
2_]
k=\
E [time
fighting in contest k
E [time
fighting in contest k
5°
/J E
+
k=i
probability
is
weak]
tough]
=
is
weak]
<
k=i
bounded by
+
K.
from the previous paragraph - the total time fighting weak
last inequality follows
opponents
is
[time fighing in contest k and k
nq°
The
and k
is
n
n
2J
and entrant k
K
,
because whenever the opponent
is
weak and
is
fighting, the
p must be moving.
Hence by following the strategy of fighting
all
the
way
to the end, starting with
stages left and against the equilibrium strategies of the entrants, the
weak incumbent
n
will
net an expected value of
a
E [time
not fighting]
— E
which, by the previous estimate,
[time fighting]
is
no
less
an —
=
sufficiently large,
under the hypothesis that q°
the optimal response by the
sufficiently large
n
,
v n (p°)
weak incumbent must do
>
1
,
which
is
+
1)
E [time
fighting]
than
n(a-(a+ l)q°)-(a+
For n
(a
1)K.
<
a/(a
+
1)
,
this exceeds one. Since
at least this well,
we know that
what we wanted. The remainder
trivially.
A8
of
(viii)
for all
follows
vn
is
nondecreasing in n
To complete the
then v n (p
<
that 5°
)
=
a/ (a
v n -\(p
If
v n -i(p
n
n
)
<
+
n
)
1-
+
(a
1)
l)q°
we
+
establish that v n
vn _i(p
n
)
.
Hence we have the
.
is
nondecreasing in n
.
If u n _ a (p
Moreover, the condition v„_i(p
n
)
>
1
n
)
>
1
,
requires
result.
= 0, then the result is
Now assume inductively
trivial.
that
Hence we have
u n _j
>
r n _2
.
left
<
the case where
Looking at the
differential
any termination point (p 1) the curve for stage n — 1
be more steeply sloped than the curve for stage n Moreover, looking at figure A2, the
equations (A.2)
will
—
a
analysis,
,
we
1
see that for
,
,
.
r function for stages n and n —
1
are the same, at least in the neighborhood
and to the
left
n
) we know that the time left upon hitting
of the solution for n — 1 since (as u„_i(jD ) >
n
(p',1) from (p ,q) depends on the evolution of p, the differential equation for which is
,
independent of n
.
Combined with the induction hypothesis that u n -i ^ ^n-2 this means
is moved leftward from the terminal
s
that the equilibrating terminal condition in stage n
condition for stage n
and n —
1
stage n
then, there
if
,
,
—
1
.
Accordingly, for the same starting point (p n ,9°) in stages n
the relevant curve in stage n
is
lies
everywhere above the curve for n
higher probability that the entrant
will
—
1
concede immediately.
.
At
And
the entrant does concede immediately, the incumbent receives a larger reward (from the
value of continuation).
Thus v n >
i>„_i
.
A9
o
-9
«
O
q)
c
o O
t* *—
D- W
i-
(a)
(b)
(d)
(c)
a
a+1
b/(b+1)
p°
— prior probability of
strong incumbent
Figure
1.
Isolation vs. linkage with the simple contest.
With informational
weak incumbent's asymptotic payoff per contest is zero in regions
and a — (a + l)q° in regions (c) and (d). With isolation, the
asymptotic payoffs are zero in regions (a) and (d) and a(l — q°) in regions
(b) and (c). Hence the weak incumbent prefers linkage in region (d), isolation
in (b) and (c), and is indifferent between isolation and linkage in region (a).
linkage, the
(a)
and
(b),
o
>.
03
c
O CO
Q.
,
O
h_
(!)
to
Q.
k_
c
cu
m
c
o
W
cr
p —posterior probability of
strong incumbent
Figure
2.
State space for the concession
game
contest.
.
.a
CD
c
CO
I—
.Q
O
CD
D.
D)
c
o o
D. w
U-
(a/Ca+l))
875
of strong
p°_prior probability
Figure
3. Isolation vs.
In region (a), the
incumbent
linkage with the concession
weak incumbent
is
game
as the single contest.
asymptotically indifferent, as
that with informational linkage, the
in region (c)
and gives
it
up
in (b).
it
nets zero
is prefered. Note
weak incumbent maintains the reputation
In region (d), the weak incumbent prefers
per contest in either case. In regions (b) and (c) isolation
linkage. See Proposition 3 for further details.
CO
JZ
-O 0>
CD
C
o2
»- .<2
.2
—
i
o c
jg
Q.
0>
pt
— posterior probability that
is
Figure
4.
incumbent
strong
The equilibrium
for the case
N
=
2
and a >
1
IK+1 curve
o
^ K+2 curve
>%
.ti
s
IU
rfl
•
o c
m
.Q
D.
/
*
ir
/
f
^f
,0)
O m
c
l.
o
ki—
***
|N curve
(11
«/i
o
to
Q.
io
pt
— posterior probability
istrong
Figure
5.
State space of the
of
incumbent
game with simultaneous
contests
and no
reentry.
c
ro
k_
c
0)
05
c
o
k_
curvet aNJ
CO
o
CO
.Q
o
curve
N
o
Q.
pt
— posterior probability that incumbent
is
Figure
6.
strong
Position of the
[aN] curve
for large
TV
and a £ [a/(a +1),1]
c
c
CD
D)
C
o
w
o
15
o
o
CL
n
1
p
p
— posterior
probability of strong
incumbent
t
Figure Al. Solutions to the
and ending points.
differential equations
(A.2) for various starting
Figure A2. Finding the correct terminal condition for stage n
P
-
P"
pi— boundary condition
(a) In this case, the correct
of r(p)
=
ending point
is
path
of
(p«, 1)
,
where p«
is
the solution
v„_i(p).
o
>.
CD
CL
TJ
03
O
CD
Q.
X
a>
T3
c
CD
CD
pi— boundary condition
(b) In this case,
where r(l) > v n _!(l)
,
of
path
the correct ending point
is
(1, 1)
361
I
095
Date Due
m ^ «*
3
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