Long-Term Conflict: How to Signal a Winner? Georgios Katsenos Department of Economics,

Long-Term Conflict:
How to Signal a Winner?
Georgios Katsenos∗
Department of Economics,
University of Hannover
January 28, 2009
Abstract
We study a two-player Tullock contest with asymmetric information, in which
each player can submit a costly signal to his opponent prior to exerting effort.
Separation is possible if and only if the probability of facing a strong opponent is
sufficiently small. In this case, weak types are better-off in a pooling equilibrium;
whereas strong types expect to benefit from signaling if and only if their potential
strength superiority is sufficiently high. As this strength differential increases, the
weak type’s extra costs under pooling outweigh the strong type’s extra costs under
separation; thus, signaling becomes ex ante welfare enhancing.
JEL classification: D44, D72, D74, D82.
Keywords: Tullock contest, asymmetric information, signaling, conflict.
∗
I wish to thank Heidrun Hoppe, Jack Ochs and the seminar audience at the University of Hannover
for helpful discussions and comments. Of course, any remaining errors are my own.
Email address: [email protected].
1
1
Introduction
During the earlier stages of a long-term conflict, the engaging parties often expend levels
of effort that appear disproportionate to the value of the rewards available at the time.
In such cases, the parties typically aim at displaying their power, especially in the eyes
of their opponents, so as to establish a reputation that will hopefully serve them when
it really matters.
In this paper, we study the role of signaling in conflict. In particular, we examine
the incentive to signal strength, in contrast to some recent literature investigating the
advantages of feigning weakness.1 We determine the conditions under which strong parties may benefit from revealing their power; while we show that weak parties are always
better-off in the absence of information exchange. Finally, we examine when the total
gains from signaling exceed the costs, that is, when signaling is welfare enhancing.
Most characteristically, during the Cold War, the United States devoted to the Apollo
program enormous resources,2 so as to land on the Moon ahead of the Soviet Union.3
They did so despite knowing that a victory in the race to the Moon would provide
no military benefit. Rather, they thought of the first lunar landing as an opportunity to demonstrate to the world, in particular, to their adversary, their industrial
and technological superiority.4 In retrospect, the investment appears to have paid off.
The Soviet Union recognized the weakness of its position; and once it failed to redress
the balance of power,5 rather than risk an all-destructive nuclear war, it conceded victory peacefully, by initiating the reform of its economic and political system.
In a similar manner, R&D laboratories occasionally engage in technological challenges
that bear no direct commercial returns. These endeavors usually aim at the development of showcase products that can highlight a firm’s leadership in its field; that is,
at establishing or maintaining a reputation which, among other benefits, can discourage the future efforts of the firm’s competitors. For example, between 1989 and 1997,
IBM financed the development of a chess computer capable of winning a match against
the reigning world champion. This highly publicized project resulted in a success when
“Deep Blue” beat Gary Kasparov, arguably one of the strongest chess players ever.
1
For example, see M¨
unster [32] or Slantchev [38].
According to the NASA History Division [33], the cost of the Apollo project was approximately $25
billion (or $150 billion in 2008 currency). For comparison, the Manhattan Project cost approximately
$2 billion (or $22 billion in 2008 currency).
3
J.F. Kennedy was quite explicit about the objectives of the Apollo program. As he told the NASA
Administrator J.E. Webb, “[e]verything that we do should be tied into getting on to the Moon ahead
of the Russians. [...] Otherwise we shouldn’t be spending this kind of money, because I am not that
interested in space.” (Webb [41], as quoted in Launius [27].)
4
As argued in Logsdon [30], “[t]he Moon Project was chosen to symbolize U.S. strength in the headto-head global competition with the Soviet Union.” For the political motives behind J.F. Kennedy’s
decision to intensify the American space effort, see also Logsdon [29] and NASA History Division [33].
5
In fact, during the 1970’s, the Soviet Union intensified its nuclear program, in an attempt to
force quick resolution. However, the military advantage that it obtained did not suffice to achieve this
objective; while the associated costs would prove to be beyond the capacity of the Soviet economy.
2
2
Having achieved its goal, IBM did not pursue its chess computing activities any further,
either scientifically or commercially.6
We model long-term bilateral conflict as a two-player, two-period game, consisting of a
signaling and a fighting phase. At the beginning of the game, each player is uninformed
about the technological capabilities, i.e., the effort costs, low or high, of his opponent.
During the first phase of the conflict, we assume that there is no prize to be contested,
so, as in Spence [39], the players’ efforts aim purely at signaling. Subsequently, during
the second phase, the parties compete for a prize of common value by exerting efforts
that depend, first, on their actual strength, second, on the strength they have signalled
to their opponent and third, on the strength of their opponent as they perceive it.7
Competition takes the form of a Tullock contest with parameter r = 1, so that each
player’s winning probability equals the share of his effort in the total effort.8 The
use of a stochastic success rule aims at capturing the element of uncertainty that
appears in various competitive situations, such as R&D races, promotion contests,
sports and war. In a complete information environment, this rule has been shown,
by Skaperdas [37] and Clark and Riis [9], to be the consequence of some intuitive axioms, most importantly, of invariance of the contest structure with respect to the number
of the participants and homogeneity of the winning probabilities in the players’ efforts.
In addition, as shown in Baye and Hoppe [5], Fullerton and McAfee [17] and Hirshleifer
and Riley [21], such a rule can be the outcome of a contest with a deterministic success
rule (an all-pay auction) in which the contestants’ quality of output is a random variable
of the effort levels they choose.
In this setting, since there are no rewards in the first period, there is always a pooling
equilibrium in which the players exert no signaling efforts and exchange no information.
Moreover, when each player’s probability of facing a strong opponent is sufficiently low,
there are also separating equilibria, in which strong types exert positive levels of signaling
effort. In this case, a weak type is (interim) better-off under pooling. On the contrary,
a strong type would be better-off in a full-information contest. When we account for the
signaling costs, however, we find that a strong type benefits from efficient separation if
and only if his potential strength superiority is sufficiently large. Finally, a strong type’s
total effort costs are higher under separation, whereas those of the weak type are higher
under pooling. The weak type’s extra spending dominates that of the strong type if
and only if the strength differential is sufficiently large. In this case, signaling is ex ante
welfare enhancing.9
Our work relates to two fields of literature, Tullock contests and conflict resolution.
6
For the history of the “Deep Blue” chess computer, narrated by one of its developers, see Hsu [22].
Thus, the first-period effort affects the second-period outcome only indirectly, though signaling.
8
For obvious reasons, this contest is often called a “lottery”. Compared to the general case of r > 0,
it allows for analytical tractability, an issue especially relevant in the welfare comparison.
9
A similar conclusion would hold, albeit with less stringent demands upon the players’ potential
strength differential, if credible information exchange could be costless.
7
3
Tullock contests have been used to address various economic questions, most famously,
in the rent seeking literature, as in Tullock [40].10 With complete information, when
the players’ effort costs are asymmetric, Dixit [11] showed that the stronger (weaker)
player’s effort is a strategic complement (substitute) to the effort of the weaker (stronger)
player, in the terminology introduced by Bulow et al. [7]. Thus, in our setting, without
considering signaling costs, the stronger of the two contestants would be better off if his
strength were perceived in an exaggerated manner; and vice versa for the weaker player.
In the presence of one-sided incomplete information, Hurley and Shogren [24] analyze
the relation between the players’ expected efforts (and probabilities of success) and the
nature (mean and variance) of the uncertainty about their relative strength. For twopoint discrete distributions of the unknown type (“low” or “high” values), Schoonbeek
and Winkel [36] determine the conditions under which the privately informed player may
exert zero effort. The case of two-sided incomplete information, for two-point distributions, is examined numerically by Hurley and Shogren [23] and analytically by Malueg
and Yates [31]. Finally, Fey [16], using a fixed-point argument, proves the existence of
a Bayesian equilibrium for the case of uniformly distributed types.
The effects of signaling, as this may occur in a sequence of two Tullock contests for prizes
of equal value, are examined by M¨
unster [32]. In this setting, there are two players, each
incompletely informed about his opponent’s valuation of the prizes, which can be either
positive or zero. It is shown that a high-valuation player can gain in the second contest
as well as overall by “sandbagging” in the first contest, that is, by exerting no effort,
so as to signal low valuation. Consequently, in equilibrium, the players’ aggregate effort
decreases.11 Unlike the setting we study, the low-valuation type is unwilling to exert
any positive level of effort, so, signaling is effective only against the high-valuation type;
therefore, there are no incentives to signal strength.
Long-term conflict has been analyzed in the peace economics literature12 , especially,
within the context of arms races, the study of which was pioneered by Richardson [34].
In the earlier dynamic models, information, in the form of the belligerents’ capabilities,
was assumed to be complete; thus, no signaling could occur. Incomplete information
was introduced by Brito and Intriligator [6], with the aim of explaining why rational
agents may engage in costly war rather than reach a mutually beneficial agreement, a
problem discussed in Fearon [14, 15].13
10
For a survey of the literature on Tullock contests, see Lockard and Tullock [28] or C´orchon [10];
for more general contest formats, see Konrad [26].
11
The effects of signaling have been also examined in the literature on interim performance evaluations and feedback policies in multi-stage tournaments; see Aoyagi [2], Ederer [13], Gershkov and Perry
[19], Goltsman and Mukherjee [20] and Yildirim [42]. Our work differs from this literature in assuming
that the first-stage effort does not directly contribute in a player’s probability of success.
12
This line of literature typically examines whether the optimal intertemporal choice of weapons
(“guns”) and consumption goods (“butter”) leads to war, which occurs when power becomes sufficiently
imbalanced. For a survey, see Anderton and Carter [1] or Isard and Anderton [25]. It is closely related to
the literature on the economics of conflict, which studies topics such as the emergence and enforcement
of property rights; for a survey, see Garfinkel and Skaperdas [18].
13
The occurrence of costly disputes, such as wars, strikes, legal trials or time-consuming negotiations,
4
Closer to our work, S`anchez-Pag´es [35] examines whether the information generated in
small-scale battles can prevent larger wars. In particular, during a bargaining process,
a player may decide to postpone resolution (in the form of either a negotiated agreement or a costly full-scale war) so as to demonstrate his power in a limited-scope battle.
However, the players do not choose how much effort to expend during their encounters;
thus, they do not face the trade-off between their signaling and war efforts that we study.
The choice of war effort is considered by Slantchev [38], who shows that the desire to
reduce the costs of winning a potential war (which is modeled as a Tullock contest)
may propel a strong type to feign weakness (by offering prior to the fight a settlement
that is too generous, given his strength). In the equilibrium constructed, it is only the
strong type of the responder that may reject the proposed terms and engage in war;
therefore, against such a potential war opponent, the player offering the resolution can
gain by appearing weak, even at the possible cost of a less favorable peaceful settlement.
Our paper shows that war costs, alone, do not suffice to induce an incentive to feign
weakness. When both the weak and the strong types of one’s opponent may engage in
war,14 a player can increase his gains from war by signaling strength.
Our work contributes to the literature by analyzing the incentive to signal strength prior
to a Tullock contest, in the presence of two-sided incomplete information and explicit
signaling costs. The use of a Tullock contest allows for the probability of winning the
prize and the costs of competing for it to be determined endogenously, by the players’
effort decisions, thus relaxing some of the assumptions usually made in the literature
on conflict resolution. Finally, we allow signaling to be effective against both a strong
and a weak opponent, so, we can study the possibility of signaling strength, rather than
that of pretending weakness.
In the next section, we present the model describing our problem. In section 3, we
construct a pooling equilibrium; while in section 4, we construct separating equilibria.
We compare the players’ payoffs in the two equilibria in section 5. Finally, we conclude
in section 6.
when an immediate agreement can benefit all parties, known as the “Hicks paradox”, has been studied
thoroughly in the bargaining literature; for a survey, see Ausubel et al. [4].
14
History presents quite a few examples of weak parties that preferred the risk of military conflict
over peaceful resolution. Notably, following its defeat in Sicily, Athens rejected the Spartan peace offers
two times, in 410 BC and 405 BC, and continued the Peloponnesian War against unfavorable odds.
(For details, see Diodorus Siculus [12], book XIII: 52-53, and Aristotle [3], 34.) More recently, in 1940,
Great Britain was unwilling to discuss the possibility of peace, following its defeat in France.
5
2
Model
Two players i, j = 1, 2 compete against one another in a Tullock contest, taking place at
time t = 2, preceded by a round of signaling, at time t = 1. The winner of the contest
gains a prize of value v > 0, common to the two players.
If player i exerts effort e2i ≥ 0 against his opponent’s effort e2j ≥ 0, he wins the contest
with probability
 2
 2ei 2 , if e2i + e2j > 0;
ei +ej
p(e2i , e2j ) =
 1
,
if e2i = e2j = 0.
2
In particular, each player’s effort in the signaling round, e1i , does not directly affect his
probability of winning the contest.
For each player i, the cost of exerting signaling or contest effort eti is
C(eti | ci ) = ci eti ,
where ci ∈ {cL , cH }, for 0 < cL < cH , is player i’s marginal cost of effort.15 To simplify
the exposition, in particular, to avoid corner solutions off the equilibrium path, we will
assume that cH < 4cL .
Each marginal cost ci is private information of player i. The two marginal costs are
drawn independently, at the beginning of the game, from a probability distribution
(
ρ,
if ci = cL ;
p(ci ) =
1 − ρ, if ci = cH ,
where ρ ∈ (0, 1) reflects the players’ (common) prior beliefs.
Following the completion of the signaling round, the two players observe the efforts e11 ,
e12 exerted in it. Therefore, each player’s strategy consists of the two effort functions
s1i : {cL , cH } −→ R+
and
s2i : {cL , cH } × R2+ −→ R+ .
That is, the signaling effort e1i depends only on the marginal cost ci ; while the contest
effort e2i depends on both the marginal cost ci and the history (e11 , e12 ).
15
Our results would not change if we allowed for different (constant) marginal costs of signaling and
contest efforts, c1i ∈ {c1L , c1H } and c2i ∈ {c2L , c2H }, where 0 < ctL < ctH , for t = 1, 2, as long as each
player’s strength persists over time, that is, c1i = c1L ⇔ c2i = c2L .
6
After observing the effort e1j , each player i can use his (equilibrium) knowledge of the
signaling strategy s1j so as to update his beliefs about the cost cj . When the players use
pure strategies, if s1j (cL ) 6= s2j (cH ), player i can perfectly infer player j’s marginal cost
(separation); otherwise, if s1j (cL ) = s2j (cH ), no belief update occurs (pooling).
The two players are assumed to be risk neutral. Thus, given efforts eti , etj , for t = 1, 2,
player i’s game payoff will be
Π(e1i , e2i , e1j , e2j | ci ) = p(e2i , e2j ) v − ci (e1i + e2i ).
The solution concept will be that of perfect Bayesian equilibrium. At each time t = 1, 2,
each player must behave optimally, given the other player’s strategy and his beliefs,
with the players’ beliefs being Bayesian on the equilibrium path (and arbitrary off it).
Finally, we will restrict attention to symmetric pure-strategy equilibria. Thus, whenever
convenient, we will drop the subscripts i, j from the players’ strategies.
3
Pooling
First, we show that a pooling equilibrium always exists. In this equilibrium, the two
players exert no signaling effort; and they choose their contest efforts according to their
own marginal costs and the probability ρ that their opponent is strong.
Lemma 1.
In a two-player Tullock contest for a prize of value v, in which each player believes that
his opponent’s marginal cost of effort is cL , with probability ρ, and cH , with probability
1 − ρ, where 0 < cL < cH , there is a unique symmetric Bayesian equilibrium (eL , eH ).
In addition, the equilibrium efforts eL and eH are respectively increasing and decreasing
in the probability ρ.
The equilibrium efforts, eL = s2i (cL ) and eH = s2i (cH ), for i = 1, 2, solve the equations
ρ
eH
1
+ (1 − ρ)
4 eL
( eL + eH )2
=
cL
v
(1)
ρ
eL
1
+ (1 − ρ)
2
( eL + eH )
4 eH
=
cH
v
(2)
The proof, which is in the Appendix, shows that such a solution exists.16 In addition,
v
ρv
it implies that the equilibrium efforts are eL ∈ ( 4c
, 4cvL ) and eH ∈ ( (1−ρ)
, 4cvH ).
4cH
L
16
A special case of this problem, ρ = 1/2, is analyzed in Malueg and Yates [31] and Fey [16].
7
Proposition 2.
There is a pooling equilibrium in which the players exert no signaling effort.
Proof:
Suppose that each player’s posterior beliefs are identical to his prior ones, independently
of the signaling efforts. In the contest, therefore, the players will exert efforts as described in Lemma 1. So, since the players cannot benefit in the continuation game by
signaling, they will exert no effort in the first period.
Corollary 3.
In the pooling equilibrium, the two types’ expected payoffs are homogeneous of degree
zero in the parameters (cL , cH ).
Therefore, in the pooling equilibrium, the two types’ interim payoffs are fully determined
by the types’ relative strength γ = cH /cL . In the sequel, we will use the parameter γ to
state those results which depend on the size of cH relative to cL .
An outcome-equivalent pooling equilibrium also exists in settings in which the players
observe only a (possibly random) function of their signaling efforts, as long as the firstperiod prize is of trivial value. For example, it might describe the players’ behavior, if
the first round took the form of a Tullock contest in which only the winner’s identity
was observed.
4
Separation
Suppose that the two players exert signaling efforts sL = s1 (cL ) 6= s1 (cH ) = sH , so that
their types are separated prior to the contest. In this case, following signaling efforts
(s1 (˜
c1 ), s1 (˜
c2 )), for c˜i ∈ {cL , cH }, each player i chooses his contest effort ei = e(˜
c i , cj | c i )
knowing his type to be ci , his opponent’s type to be c˜j and that his opponent perceives
his type to be c˜i . Since we consider only single-player deviations from (s1 (c1 ), s1 (c2 )),
in examining the incentives of player i we can assume that c˜j = cj .
In the contest, player j will act as if the cost c˜i is player i’s actual marginal cost, so, as
shown in Fullerton and McAfee [17], he will exert effort
ej = e(cj , c˜i | cj ) =
c˜i
v.
(˜
ci + cj )2
Anticipating player j’s behavior, player i will maximize his expected payoff by exerting
contest effort
8
"r
ei = e(˜
ci , cj | ci ) =
c˜i
(˜
ci + cj ) − c˜i
ci
#
v
.
(˜
ci + cj )2
Notice that ei ≥ 0, since cH < 4cL . Thus, player i’s optimal payoff in the contest is
"r
πi = π(˜
ci , cj | ci ) =
c˜i
(˜
ci + cj ) − c˜i
ci
# ·r
ci
(˜
ci + cj ) − ci
c˜i
¸
v
.
(˜
ci + cj )2
(3)
It is easy to check that, for ci ∈ {cL , cH },
(
∆π(cj | ci ) = π(cL , cj | ci ) − π(cH , cj | ci )
> 0, for cj = cH ;
< 0, for cj = cL .
Therefore, in agreement with Dixit [11], player i is better off signaling strength against
a weak opponent and weakness against a strong opponent. In addition,
∆π(cj | cH ) > ∆π(cj | cL ),
(4)
so that the (possibly negative) gains from signaling strength against an opponent of
type cj , for cj ∈ {cL , cH }, decrease in a player’s type ci .17
For separation to be possible, the additional signaling effort of the strong type must be:
a. Sufficiently high, so that misrepresentation is not profitable for the weak type.
b. Not too high, so that signaling is profitable for the strong type.
When the weak type exerts no signaling effort, these conditions translate into the following two inequalities:
ρ π(cH , cL | cH ) + (1 − ρ) π(cH , cH | cH ) ≥
ρ π(cL , cL | cH ) + (1 − ρ) π(cL , cH | cH ) − cH sL
(5)
ρ π(cL , cL | cL ) + (1 − ρ) π(cL , cH | cL ) − cL sL ≥
ρ π(cH , cL | cL ) + (1 − ρ) π(cH , cH | cL )
(6)
Incentive compatibility for the weak type, (5), requires that sL ≥ max{0, sL }, where
¯
17
This is the discrete version of the single-crossing condition,
9
∂2π
ci , cj
∂ci ∂˜
ci (˜
| ci ) < 0.
1
{ ρ [π(cL , cL | cH ) − π(cH , cL | cH )] + (1 − ρ) [π(cL , cH | cH ) − π(cH , cH | cH )] } .
sL =
¯
cH
Incentive compatibility for the strong type, (6), is possible, for some effort level sL > 0,
if and only if
ρ < ρ¯1 =
π(cL , cH | cL ) − π(cH , cH | cL )
.
[π(cL , cH | cL ) − π(cH , cH | cL )] − [π(cL , cL | cL ) − π(cH , cL | cL )]
(7)
That is, for any marginal costs cH > cL > 0, a strong contestant will be willing to signal
his type as long as the probability ρ of confronting another strong contestant is low.
When condition (7) holds, any signaling effort sL ∈ [0, s¯L ], where
s¯L =
1
{ ρ [π(cL , cL | cL ) − π(cH , cL | cL )] + (1 − ρ) [π(cL , cH | cL ) − π(cH , cH | cL )] } ,
cL
is incentive compatible for the strong type.
Assuming condition (7), the incentive compatibility constraints (5) and (6) can be both
satisfied, for signaling efforts sL ∈ [sL , s¯L ], if and only if sL ≤ s¯L , that is, if and only if
¯
¯
1
{ρ [π(cL , cL | cH ) − π(cH , cL | cH )] + (1 − ρ) [π(cL , cH | cH ) − π(cH , cH | cH )]} ≤
cH
1
{ρ [π(cL , cL | cL ) − π(cH , cL | cL )] + (1 − ρ) [π(cL , cH | cL ) − π(cH , cH | cL )]}
cL
(8)
Verbally, the maximal signaling effort that the weak type is willing to exert must not
exceed the maximal signaling effort that the strong type is willing to exert.
The following lemma will help us establish the validity of the last condition; its proof
can be found in the Appendix.
Lemma 4.
For all marginal costs cH > cL > 0, we have
1
1
[ π(cL , cL | cH ) − π(cH , cL | cH ) ] >
[ π(cL , cL | cL ) − π(cH , cL | cL ) ]
cH
cL
(9)
1
1
[ π(cL , cH | cH ) − π(cH , cH | cH ) ] <
[ π(cL , cH | cL ) − π(cH , cH | cL ) ]
cH
cL
(10)
10
Inequalities (9) and (10) imply that there is a threshold ρ¯2 ∈ (0, 1), given by
ρ¯2 =
1
cL
1
cL
∆π(cH | cL ) −
[∆π(cH | cL ) − ∆π(cL | cL )] −
1
cH
1
cH
∆π(cH | cH )
[∆π(cH | cH ) − ∆π(cL | cH )]
,
(11)
such that condition (8) holds if and only if ρ ≤ ρ¯2 .
Lemma 5.
For all marginal costs cH > cL > 0, we have ρ¯2 < ρ¯1 .
Therefore, for separation to be incentive compatible, it is both necessary and sufficient
that ρ ≤ ρ¯2 , that is, that the probability of facing a strong opponent is sufficiently small.
In particular, notice that ρ ≤ ρ¯2 < ρ¯1 < 1/2.
Proposition 6.
Given marginal costs cL , cH > 0, suppose that the probability of facing a strong opponent
is ρ ∈ (0, ρ¯2 ], where ρ¯2 is defined by equation (11). Then there is a symmetric separating
equilibrium in which the strong type exerts signaling effort s1 (cL ) = sL , for any level
sL ∈ [sL , s¯L ], while the weak type exerts no signaling effort, s1 (cH ) = 0. Otherwise,
¯
there is no symmetric separating equilibrium.
Clearly, there is a continuum of equilibrium signaling efforts, sL ∈ [sL , s¯L ]. However,
¯
applying Cho and Kreps’ intuitive criterion18 reduces the number of equilibrium signaling efforts to sL = sL , by requiring that player j must believe that player i is strong
¯
following any signaling effort s1i > sL .19 In the sequel, we will focus on this most efficient
¯
separating equilibrium.
Corollary 7.
The threshold ρ¯2 , the signaling costs cL sL , and the two types’ expected payoffs in the
separating equilibrium are homogeneous of degree zero in the parameters (cL , cH ).
Therefore, as in the pooling equilibrium, the two types’ interim payoffs under efficient
separation are fully determined by the types’ relative strength γ = cH /cL .20
Finally, the threshold probability ρ¯2 is decreasing in the relative strength cH /cL . Thus,
there is more scope for signaling when the difference in the two types’ strength is small.
18
See Cho and Kreps [8].
However, there are still multiple equilibria, depending on player j’ beliefs following s1i ∈ (0, sL ).
¯
20
Corollary 7 follows from straightforward calculations; its proof is therefore omitted.
19
11
5
Welfare Comparisons
Even though it is difficult to solve analytically the system of equations (1) and (2), so as
to describe explicitly the firms’ efforts in the pooling equilibrium, we can use elements
from the argument that proved Lemma 1 so as to compare the players’ welfare in the
pooling and the (most efficient) separating equilibria. Of course, this comparison makes
sense only when a separating equilibrium exists, that is, when ρ ≤ ρ¯2 .
Proposition 8.
Given marginal costs cL , cH > 0 such that γ = cH /cL ∈ (1, 4], suppose that the probability
of facing a strong opponent is ρ ∈ (0, ρ¯2 ], where ρ¯2 is defined by equation (11). Then
- The weak type’s expected payoff is higher under pooling than under separation.
- There exists a probability ρˆ1 = ρˆ1 (γ) such that the strong type’s expected payoff is
higher under separation than under pooling if and only if ρ ≤ ρˆ1 .
- There exists a probability ρˆ0 = ρˆ0 (γ) ≤ ρˆ1 (γ) such that the separating equilibrium
is ex ante more efficient than the pooling equilibrium if and only if ρ ≤ ρˆ0 .
Corollary 9.
There is a value γˆ1 ≈ 2.04 such that the probability ρˆ1 (γ) is increasing, for γ ∈ (1, γˆ1 ),
and ρˆ1 (γ) = ρ¯2 (γ), for γ ∈ [ˆ
γ1 , 4]. There are also values γˆ0a ≈ 2.2 and γˆ0b ≈ 3.72
such that the probability ρˆ0 (γ) = 0, for γ ∈ (1, γˆ1a ], is increasing, for γ ∈ (ˆ
γ1a , γˆ1b ), and
ρˆ1 (γ) = ρ¯2 (γ), for γ ∈ [ˆ
γ1b , 4].
The weak type is better-off in an incomplete-information contest. On the other hand,
the strong type’s expected payoff in the contest, net of signaling costs, is higher under
full information.21 Thus, the strong type is better-off in the separating equilibrium if
and only if the signaling costs are sufficiently low. This occurs when the probability
of facing a strong opponent is small; or, equivalently, when the two types’ difference in
strength is large.
Comparing the players’ effort costs, the strong type spends more under separation while
the weak type spends more under pooling. For values of cH close to cL , the extra costs
of the strong type dominate those of the weak type; therefore, in this case, pooling is ex
ante more efficient than separation. As the difference in strength increases, the strong
type’s extra spending under separation decreases to zero while the weak weak type’s
extra spending under pooling increases. Therefore, for values of cH sufficiently larger
than cL , separation is ex ante more efficient than pooling.
21
The argument proving this result is symmetric to the one regarding the welfare of the weak type.
12
ρ
0.5
0.4
A
B
0.3
D
C
0.2
0.1
1.0
1.5
2.0
2.5
3.0
3.5
4.0
cH c L
Figure 1: Separation versus pooling.
In Figure 1, we compare the two equilibria.22 Separation is possible only outside region
A. In region B, both types prefer pooling; whereas in regions C and D, the weak type
prefers pooling and the strong type prefers separation. Finally, separation is ex ante
more efficient than pooling if and only if the parameters (ρ, cH /cL ) are in region D.
Ex post, two players of the same type, weak or strong, are better-off under pooling
than under separation, since this results in lower effort levels without affecting any of
the players’ winning probabilities. On the other hand, a player facing an opponent of
different type is better-off under separation, since his opponent’s effort is lower.
6
Conclusions - Extensions
This study offers insights on the manner in which long-term conflicts evolve, in particular, on the signaling activities of the belligerents prior to their decisive confrontation.
We have shown that strong parties benefit from treating the earlier, non-decisive stages
of a conflict as opportunities to demonstrate their power, so as to discourage their opponents from exerting higher effort in the later stages. Thus, we have provided a rationale
for efforts that appear to be disproportionate to the value of the rewards in question.
Our work can be extended in various ways:
22
The value v of the prize is not relevant for welfare comparison.
13
First, one might allow for different types of signaling opportunities, that reveal information other than each player’s exerted effort. Especially, it would be interesting to know
whether the players would be better-off if the signaling round took the form of a Tullock
contest (for zero prize), in which only the winner’s identity is revealed.
In addition, the effects of signaling can be amplified when participation in the secondperiod contest entails fixed costs. In this case, following the conclusion of the signaling
phase, it is likely that the weaker party will concede victory by not proceeding into the
fighting round, in line with the ending of the Cold War.
Finally, one might allow for a larger set of types, so that the distinction between strength
and weakness can be determined endogenously. In this case, semi-separating equilibria
may occur, so that sufficiently strong types are fully separated, by exerting efforts that
are increasing in their strength, while weak types are pooled.
These extensions are the subject of current research.
14
Appendix: Proof of Results
Proof of Lemma 1:
The payoff of player i from exerting contest effort ei = ei (ci ) ≥ 0 against player j’s effort
strategy ej = ej (cj ) ≥ 0 is
Π(ei , ej | ci ) = ρ
ei (ci )
ei (ci )
v + (1 − ρ)
v − ci ei (ci ).
ei (ci ) + ej (cL )
ei (ci ) + ej (cH )
The first-order condition for optimality is
ρ
ej (cH )
ej (cL )
v
+
(1
−
ρ)
v = ci ,
( ei (ci ) + ej (cL ) )2
( ei (ci ) + ej (cH ) )2
for ci ∈ {cL , cH }, which, after imposing the symmetry condition ei (cL ) = ej (cL ) = eL
and ei (cH ) = ej (cH ) = eH , results in equation (1) or (2), depending on the value of ci .
After rearranging the terms in the equations (1) and (2), we get
4 (1 − ρ) eL eH = (4
4 ρ eL eH = [4
cL eL
− ρ) ( eL + eH )2
v
(12)
cH eH
− (1 − ρ)] ( eL + eH )2
v
(13)
Since the left-hand side is positive, we must have eL ≥ e¯L =
ρv
4 cL
and eH ≥ e¯H =
(1−ρ) v
.
4 cH
Combining equations (12) and (13) results in
eL =
1 − ρ cH
2ρ − 1 v
eH +
ρ cL
4 ρ cL
(14)
After substituting the last equation in equation (13), we get
(4ρ)2 cL eH [4 (1 − ρ) cH eH + (2ρ − 1) v] v =
[4 cH eH − (1 − ρ) v] [4 (1 − ρ) cH eH + (2ρ − 1) v + 4ρ cL eH ]2
(15)
To show that for any probability ρ0 ∈ (0, 1), there is an effort e0H = eH (ρ0 ) such that
the last equation holds, let the function φ : R+ × (0, 1) −→ R be defined by
φ(eH , ρ) = [4 cH eH − (1 − ρ) v] [4 (1 − ρ) cH eH + (2ρ − 1) v + 4ρ cL eH ]2
− (4ρ)2 cL eH [4 (1 − ρ) cH eH + (2ρ − 1) v] v
15
0) v
Given a probability ρ0 ∈ (0, 1), it is easy to check that φ(¯
eH , ρ0 ) < 0, where e¯H = (1−ρ
.
4 cH
∂φ
In addition, there exists a value e˜H > e¯H such that ∂eH (eH , ρ0 ) ≥ 0, for all eH ≥ e˜H .
2
Finally, (∂e∂ Hφ)2 (eH , ρ0 ) > 0, for all eH ≥ e¯H . From the last two observations, we can
infer that φ(eH , ρ0 ) > 0, for all eH ∈ [˜
eH , ∞) sufficiently large. Therefore, by continuity,
there is a unique effort level e0H = eH (ρ0 ) ∈ (¯
eH , ∞) such that φ(eH , ρ0 ) R 0, for any
0
eH R eH such that eH ≥ e¯H . In particular, φ(e0H , ρ0 ) = 0, as required.
To show that the equilibrium effort e0H is decreasing in the probability ρ0 , notice that
∂φ
(e0H , ρ0 ) > 0. In addition, by the Implicit Function Theorem,
∂eH
deH
φρ [eH (ρ), ρ]
(ρ) = −
,
dρ
φeH [eH (ρ), ρ]
for all ρ in a neighborhood of ρ0 . Therefore, it suffices to show that
∂φ 0
(e , ρ0 )
∂ρ H
> 0.
Given ρ0 ∈ (0, 1) and e0H > 0 such that φ(e0H , ρ0 ) = 0, let ρ¯ = 1 − 4 cvH e0H . It is easy to
∂2φ
0
verify that φ(e0H , ρ¯) < 0 and that ∂φ
(e0 , ρ¯) < 0. In addition, the function (∂ρ)
2 (eH , · )
∂ρ H
is linear and monotonically increasing, so, there is a unique value ρ˜ ∈ (¯
ρ, 1) such that
∂2φ
0
(e , ρ) R 0, for ρ R ρ˜. Combining these three observations, we can conclude that
(∂ρ)2 H
(e0 , ρ0 ) > 0.
the value ρ0 satisfying φ(e0H , ρ0 ) = 0 is unique and that ∂φ
∂ρ H
Finally, for the monotonicity of the equilibrium effort eL , we can use equation (14) to
show that
deL
−1 cH deH
1 v
=
+
dρ
(ρ)2 cL dρ
4 ρ2 cL
Since
deH
(ρ)
dρ
< 0, it follows that
deL
(ρ)
dρ
> 0, so that the effort eL is increasing in ρ.
Proof of Corollary 3:
The necessary conditions (1) and (2) can be written as
ρ
1
eL eH
v + (1 − ρ)
v
4
( eL + eH )2
= cL e L
ρ
eL eH
1
v + (1 − ρ) v
2
( eL + eH )
4
= cH eH ,
16
so, the two types’ interim payoffs are
Π∗p (cL ) =
ρ
1
e2L
v + (1 − ρ)
v
4
( eL + eH )2
Π∗p (cH ) = (1 − ρ)
1
e2H
v + ρ
v.
4
( eL + eH )2
Therefore, the two types’ interim payoffs depend on the effort ratio ε = eH /eL . Hence,
it suffices to show that this ratio is fully determined by the relative strength γ = cH /cL .
We can use the ratio ε = eH /eL to further transform the necessary conditions
ρ
1
ε
v + (1 − ρ)
v
4
(1 + ε)2
= cL eL
ρ
ε
1
v + (1 − ρ) v
2
(1 + ε)
4
= cH eH ,
By dividing the second equation by the first one, we get
4ρ ε + (1 − ρ) (1 + ε)2
= γ ε.
ρ (1 + ε)2 + 4 (1 − ρ) ε
Therefore, the effort ratio ε is fully determined by the relative strength γ, completing
the argument.
Proof of Lemma 4:
We prove inequality (10); the argument for inequality (9) is identical.
After substituting equation (3), inequality (10) becomes
£
¤
√
1
1
2
(c
+
c
)
+
c
c
−
2
c
c
(c
+
c
)
−
<
L
H
L
H
L
H
L
H
2
cH (cL + cH )
4 cH
¤
√
1 £ 2
c2H
+
c
c
−
4
−
4
c
,
c
c
c
L
H
L
H
H
H
cL (cL + cH )2
4 c2H cL
which is equivalent to
17
c3L + 3 c2H cL + 4 cH c2L < ( c2H + 3 c2L cL + 4 cH cL )
√
cH cL .
For cH ≥ cL > 0, let
χ(cH , cL ) = ( c2H + 3 c2L cL + 4 cH cL )
√
cH cL − ( c3L + 3 c2H cL + 4 cH c2L ).
Then it suffices to show that χ(cH , cL ) > 0, for all cH > cL > 0.
For all cL > 0, we have
χ(cL , cL ) = 0.
In addition, it is straightforward to show that
∂χ
∂ 2χ
(cL , cL ) =
(cL , cL ) = 0
∂cH
(∂cH )2
while
∂ 3χ
(cL , cL ) = 3/2 > 0,
(∂cH )3
which suffices for the result.
Proof of Lemma 5:
A direct argument, using Lemma 4 whenever needed to characterize the signs of the
multiplicative terms, shows that r¯2 < r¯1 if and only if
∆π(cL | cL ) ∆π(cH | cH ) < ∆π(cL | cH ) ∆π(cH | cL ).
From equation (4), we know that
∆π(cH | cH ) > ∆π(cH | cL ) > 0
and
∆π(cL | cL ) < ∆π(cL | cH ) < 0.
Therefore,
(−∆π(cL | cL )) ∆π(cH | cH ) > (−∆π(cL | cH )) ∆π(cH | cL ),
implying the result.
18
Proof of Proposition 8:
First, consider a player of type cH . As shown in the proof of Corollary 3, this type’s
payoff in the pooling equilibrium is
(ε∗ )2
1
v + ρ
v,
4
( 1 + ε∗ )2
Π∗p (cH ) = (1 − ρ)
where ε∗ = eH /eL ∈ (0, 1) is the equilibrium effort ratio.
The effort ratio ε∗ is the root of the function ϕ(· , γ, ρ) : [0, 1] −→ R, given by
ϕ(ε, γ, ρ) = (1 − ρ − ργε) (1 + ε)2 + 4 [ρ − (1 − ρ)γε] ε,
where γ = cH /cL ∈ (1, 4] and ρ ∈ (0, 1/2]. It is easy to calculate that
ϕ(0, γ, ρ) = 1 − ρ
> 0,
ϕ(1, γ, ρ) = 4(1 − γ) < 0
and that
∂ 2ϕ
(ε, γ, ρ) = 2 [(1 − ρ)(1 − 2γ) − 2γ] − 6 ρ γ ε < 0.
∂ε2
Hence, there is a unique value ε∗ = ε∗ (γ, ρ) ∈ (0, 1) such that ϕ(ε, γ, ρ) R 0, for ε Q ε∗ .
In particular, we must have ϕε (ε∗ , γ, ρ) < 0.
On the other hand, the cH -type’s payoff in the separating equilibrium is
1
γ2
v + ρ
v.
4
( 1 + γ)2
Π∗S (cH ) = (1 − ρ)
Thus, the weak type’s gain from pooling is
Π∗P (cH ) R Π∗S (cH ) ⇐⇒ ε∗ R 1/γ
Since ρ ∈ (0, 1/2), we have
ϕ(1/γ, γ, ρ) = (1 − 2ρ) (1 − 1/γ)2 > 0,
so that 1/γ < ε∗ , implying that Π∗P (cH ) > Π∗S (cH ), as required.
Second, consider a player of type cL . This type’s payoff in the pooling equilibrium is
Π∗p (cL ) = (1 − ρ)
1
1
v
+
ρ
v.
( 1 + ε∗ )2
4
19
On the other hand, his payoff in the separating equilibrium is
Π∗S (cL ) = (1 − ρ)
γ2
1
v + ρ v − cL sL ,
2
( 1 + γ)
4
with the signaling costs being, in terms of the ratio γ = cH /cL , equal to
√
γ 2 1
1
1√ 2
1 2
[ ρ [ (1 −
) ] + (1 − ρ) [ (1 −
) − ] ] v.
cL sL =
γ) − (
γ
2
1+γ
1+γ
4
To compare the strong type’s payoffs from separation and pooling, define the function
ψ(γ, ρ) =
γ2
1
−
2
( 1 + γ)
(1 + ε(γ, ρ))2
√
γ 2 1
1
ρ
1√ 2
1 2
−
[
[ (1 −
γ) − (
) ] + [ (1 −
) − ] ],
γ 1−ρ
2
1+γ
1+γ
4
for γ ∈ [1, 4] and ρ ∈ [0, 1/2], where ε∗ = ε∗ (γ, ρ) ∈ (0, 1) was defined above as the
unique root of the function ϕ(· , γ, ρ) : [0, 1] −→ R. Clearly, Π∗S (cL ) > Π∗P (cL ) if and
only if ψ(γ, ρ) > 0. So, it suffices to explore the sign of ψ(γ, ρ).
For any γ ∈ [1, 4], it is easy to check that
ψ(γ, 0) ≥ 0,
ψ(γ, 1/2) < 0.
In addition, by the Implicit Function Theorem,
∂ε∗
ϕρ (ε∗ , γ, ρ)
(γ, ρ) = −
∂ρ
ϕε (ε∗ , γ, ρ)
< 0,
since both ϕε (ε∗ , γ, ρ) < 0 and ϕρ (ε∗ , γ, ρ) = − (1 + γε∗ ) (1 − ε∗ )2 < 0. Therefore,
∂ψ
2
∂ε
1
1√ 2
1 2
1
(γ, ρ) =
(γ, ρ) −
[ (1 −
γ) − (
) ] < 0.
3
2
∂ρ
(1 + ε(γ, ρ)) ∂ρ
γ (1 − ρ)
2
1+γ
In conclusion, there is a unique probability ρˆ1 = ρˆ1 (γ) such that ψ(γ, ρ) R 0, for ρ Q ρˆ1 .
By using the Implicit Function Theorem once more, one can show that the function
ρˆ1 = ρˆ1 (γ) is increasing in γ ∈ [0, γˆ1 ]. Finally, one can check that ρˆ1 < ρ¯2 , for γ ' 1,
and that ρˆ1 > ρ¯2 , for γ = 4. Since the function ρˆ1 = ρˆ1 (γ) is increasing while the
function ρ¯2 = ρ¯2 (γ) is decreasing, it follows, by continuity, that there is a unique value
γˆ1 ∈ (1, 4) such that ρˆ1 (γ) Q ρ¯2 (γ), for γ Q γˆ1 .
20
Coming to the ex ante welfare comparison, the difference in each player’s expected effort
cost between the separating and the pooling equilibrium is
∆W = ρ (1 − ρ) [
2ε
2γ
−
] − ρ c L sL .
2
(1 + ε)
(1 + γ)2
To determine the sign od ∆W , define the function ω =
∆W
ρ(1−ρ) v
by
2 ε(γ, ρ)
2γ
−
2
(1 + ε(γ, ρ))
( 1 + γ)2
ω(γ, ρ) =
√
γ 2 1
1
ρ
1√ 2
1 2
−
[
[ (1 −
γ) − (
) ] + [ (1 −
) − ] ],
γ 1−ρ
2
1+γ
1+γ
4
for γ ∈ [1, 4] and ρ ∈ [0, 1/2].
For ρ = 0, we have
ϕ(ε, γ, 0) = 0 =⇒ ε =
so that
1
2 γ −1
√
√
( γ − 1)2
ω(γ, 0) =
(4 γ 3/2 − 5 γ + 2 γ 1/2 − 5).
2
4 γ (1 + γ)
Therefore, there is a value γ0a ≈ 2.19584 such that ω(γ, 0) Q 0, for γ Q γ0a .
For ρ = 1/2, we have
ϕ(ε, γ, 1/2) = 0 =⇒ ε = 1/γ
so that
Finally,
√
( γ − 1) (γ + 3)
ω(γ, 1/2) =
< 0.
4 γ (1 + γ)2
∂ω
(γ, ρ) < 0.
∂ρ
Therefore, for all γ < γ0a , pooling is preferable to separation; while, for all γ ≥ γ0a ,
there exists a unique probability ρˆ0 = ρˆ0 (γ) such that separation is preferable to pooling
if and only if ρ Q ρˆ0 .
21
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