Hawks, Doves, and Diplomats; Reputation and Communication in a Modi¯ed Hawk-Dove Game ¤ Anne E. Sartori Princeton University April 13, 2001 Abstract This paper shows the possibility of communication when players interact repeatedly playing a Hawk-Dove game that is modi¯ed to include incomplete information. Players are randomly matched and have no persistent types. Building on previous work (Sartori 2001), I show that players in this situation may obtain reputations for honesty that allow them to communicate their types. Going beyond that work, I prove that players obtain gains from trade over time through communication; realization of such gains provides a motive for communication. I also show that this behavior is not just possible but \likely" by two formal criteria: it is Pareto optimal and collectively stable. Finally, I suggest that communication is more likely when there are greater gains from trade and that it remains possible when both players prefer war to acquiescence. ¤ This paper is very much in-progress. Please do not cite without the permission of the author. Comments are welcome and can be given to the author at [email protected] or (609) 258-4748. I thank Kelly Chang, Joanne Gowa, Jim Fearon, and Howard Rosenthal for helpful conversations and comments. Any errors are, of course, my own.
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Hawks, Doves, and Diplomats;
Reputation and Communication in a Modi¯ed
Hawk-Dove Game¤
Anne E. Sartori
Princeton University
April 13, 2001
Abstract
This paper shows the possibility of communication when players interact repeatedly
playing a Hawk-Dove game that is modi¯ed to include incomplete information. Players
are randomly matched and have no persistent types. Building on previous work (Sartori
2001), I show that players in this situation may obtain reputations for honesty that
allow them to communicate their types. Going beyond that work, I prove that players
obtain gains from trade over time through communication; realization of such gains
provides a motive for communication. I also show that this behavior is not just possible
but \likely" by two formal criteria: it is Pareto optimal and collectively stable. Finally,
I suggest that communication is more likely when there are greater gains from trade
and that it remains possible when both players prefer war to acquiescence.
¤This paper is very much in-progress. Please do not cite without the permission of the author. Comments
are welcome and can be given to the author at [email protected] or (609) 258-4748. I thank Kelly
Chang, Joanne Gowa, Jim Fearon, and Howard Rosenthal for helpful conversations and comments. Any
errors are, of course, my own.
1. Introduction
Our aim is to show you what sort of a city you will have to ¯ght
against, if you make the wrong decision (Thucydides 1972, 78).
Communication plays a paramount role in many areas of human relations. People ¯nd
ways to communicate even when they have strong incentives to lie. In the speech above,
representatives of Athens speak in the Spartan Assembly in an attempt to avoid war. The
Spartans know that the Athenians wish to appear strong, whether or not they are. In this
case, the Athenians threats succeed at least in postponing war, though not in preventing it
forever. However, in other cases, listeners simply disbelieve the speech that they hear.
Under what circumstances is speech an e®ective means of communication? The crisis
bargaining literature in political science is pessimistic about the general e®ectiveness of
diplomacy. One school of thought, the \audience cost" school, focuses upon the importance
of domestic audiences in making signals costly; it suggests that diplomacy is primarily a tool
for democracies (Fearon 1994). Another school, deterrence theory, suggests that credibility
is attained by ¯ghting to show resolve (Schelling 1966).
This paper models two parties communicating as a version of the repeated Hawk-Dove
game that is popular among evolutionary biologists (Maynard Smith 1982), modi¯ed to
include incomplete information and talk. I use this model to expand our understanding
of international relations in several ways. First, I prove that two-sided communication is
a logical outcome of this game and that it allows players to obtain a mutually bene¯cial
trade of issues over time.1 Second, I suggest that communication is not just possible; it
also is likely by two formal criteria. Third, I show that communication and trade are more
likely when there are greater gains from trade. Finally, I suggest that communication and
trade also are possible when both parties prefer war to acquiescence. The paper builds upon
Sartori (2001), but it goes beyond that work in the four ways just mentioned.
The paper thus provides further theoretical basis for the claim that international-relations
theory is too pessimistic about the prospects for diplomacy. Contrary to the implications1This proof follows a conjecture in Sartori (2001).
1
of the audience-cost literature, domestic audiences are unnecessary to communication. Con-
trary to those of deterrence theory, acquiescence can be as a great a boon to credibility as
is ¯ghting.
The paper is organized as follows. The next section reviews the Hawk-Dove model, dis-
cusses my modi¯ed version, and presents the equilibrium in which players use communication
to realize gains from trade over time. The third section shows that this communicative equi-
librium is likely to obtain. The fourth discusses when communication is more or less likely.
The fourth section of the paper concludes with the signi¯cance of the model, its weaknesses,
and directions for future research.
2. The Realization of Gains from Trade Over Time
When the Soviets threatened to invade Czechoslovakia in 1968, the United States did not
try to deter the imminent attack by threatening to ¯ght over Czechoslovakia.2 The U.S.
inaction was not an isolated event; states often acquiesce to other's demands in the fact
of threats (Bueno de Mesquita and Lalman 1992). This seemingly uncontentious fact is
surprising in the context of deterrence theory. According to the logic of the theory, a state
that acquiesces to a challenger's demands is likely to obtain a reputation for lacking resolve;
hence, according to Schelling (1966), the importance to the U.S. of ¯ghting in Vietnam.
However, if an irresolute state threatens to ¯ght, its adversaries may never know that it was
unwilling to do so; the blu® may succeed. Thus, if a state does not consider an issue worth
a ¯ght, the logic of deterrence theory suggests that it should blu®: a blu® may succeed, and
if it fails, the state will be no worse o® than if the state had failed to ¯ght without blu±ng.
Leaders do not act in the way that deterrence theory prescribes. Why do states often
fail to blu®? This section explains why states { and other parties { may often tell the truth
in order to preserve their reputations for honesty. Doing so maintains their credibility for
when they really need diplomacy most: when they are engaged in a dispute over issues that
they consider extremely important. Of course, states sometimes do blu®. However, they
also engage in a \trade" of issues over time, honestly acquiescing when they care little about2The U.S. did threaten to halt the SALT I negotiations, but did not threaten to intervene militarily.
2
the issues, and talking tough when they care a lot.
2.1. The Modi¯ed Hawk-Dove Model
To show the possibilities for communication and trade, I use a modi¯ed version of the Hawk-
Dove game. The classic Hawk-Dove game (Figure 2.1) represents an interaction between
two animals ¯ghting over prey. Each animal can act like a hawk or like a dove. If both are
dovish, they split the prey. If both are hawkish, they ¯ght; the value of prey is decreased in
the ¯ght, and they split what is left. Finally, if one animal is hawkish and the other dovish,
the hawkish animal enjoys the prey and the dovish one gets nothing. In the Hawk-Dove
game, each player would like to be the hawk, but only if the other player is acting like a
dove; a < c < d < b.3
Hawk-Dove is a simpli¯ed version of many political situations involving bargaining. For
example, if two states act dovish in negotiations, they will remain at peace and split the pie.
If both act hawkish, they will engage in a painful war, expending valuable lives and materiel
in the process and perhaps damaging whatever goods are being disputed. If one state bullies
and the other acquiesces, then one state wins and the other loses. The loser, however, may
be better o® with acquiescence than it would with war because war involves deaths and lost
resources.
The game has an element of coordination: there is no pure-strategy equilibrium without
it. Nevertheless, it also has an element of con°ict: if the game is played once, the two pure-
strategy equilibria are fHawk, Doveg and fDove, Hawkg, and there is always one player
who would prefer the other equilibrium. The only \fair" (symmetric) outcome occurs when
each plays the mixed-strategy equilibrium. Without communication, this mixed-strategy
equilibrium also intuitively seems the most likely to obtain, since players have no way within
the game of generating expectations that would lead to coordination.
The Hawk-Dove game is relevant to politics, but three elements extant in many political3Authors di®er slightly in their speci¯cation of a Hawk-Dove game. Here, I use the payo® ordering given
by Berstrom and Godfrey-Smith (1998) . Maynard Smith (1982) is less restrictive, and in particular seems
to allow c < a (war is better than acquiescence). Later in the paper, I consider a situation in which one
player for whom c < a plays another for whom c > a.
3
Figure 2.1: The Hawk-Dove Game
situations are lacking. First, most political situations involve players who can talk to each
other before they take any actions. Second, players may consider issues or goods of di®er-
ing importance. For example, when a militarily weak party wins a war against a military
superpower, as the North Vietnamese did against the United States, observers often have
the sense that it achieved victory because it cared more about the outcome. Finally, players
may have private information about the importance that they ascribe to the good.
The following, modi¯ed Hawk-Dove game (Figure 2.2) includes these features. In the
work that follows, I consider a situation in which players with discount factor ± play this
game inde¯nitely (0 < ± < 1). In the modi¯ed game, each player is one of two types, high
(h) or low (l), determined by nature at the beginning of each period t, with h > l; the
probability that a player is of type \h" (or \l") at time t is 12 . Each player learns its own
type, but this type is private information. The types represent the utility that the player
receives from getting the goods that are under dispute { that is, the player's utility when
it plays \Hawk" and its opponent plays \Dove." After learning its type, each player must
send a message, \high" or \low," before play; messages are sent simultaneously. The players
hear each other's messages, and then simultaneously choose actions: each player's choice of
action is between \Hawk" and \Dove."4 At the end of the stage game, players observe each
other's actions and each other's types before proceeding to the next iteration. Note that4A player's message (\high" or \low") can { but need not { reveal which action (\Hawk" or \Dove") it
intends to take.
4
Figure 2.2: A Modi¯ed Hawk-Dove Game
regardless of the players' types, this game is Hawk-Dove; a < c < d < h, l. Here, I add the
assumption that l+c¸2d. I also assume that in the ¯rst period, t=1, one player is randomly
chosen to be Player 1.
Most assumptions of this game are innocuous. I brie°y consider ¯ve here. First, the
assumption of an in¯nitely-repeated game does not ¯t all situations, but is a good approx-
imation for international relations, in which states interact without the prospect of a ¯nal
interaction. Second, the assumption that players are randomly assigned types corresponds
to a situation in which players interact over time and over di®erent issues. In the real world,
values for the issues are not truly random, but players do value some issues more and others
less. Third, the assumption that players' types are revealed at the end of the stage game
is a simpli¯cation; in many real-world situations, including international disputes, players
learn something but not everything about each other's types in the course of an interac-
tion. Fourth, the assumption that players prefer acquiescence to war is not innocuous but
is relaxed later in the paper.
The assumption that l+c¸2d is not necessary for existence of the equilibrium that I
discuss below, but assures Pareto optimality. If this inequality does not hold, the sum of the
parts available through compromise (\Dove," \Dove") is greater than the sum of the parts
obtained by the two players if one player primarily gets its way and the other acquiesces.
In practice, it is most likely that a pie divided in either of two ways remains the same total
size; then, l+c=2d and the condition holds. Similarly, the assumption that one player is
randomly chosen to be player 1 in the ¯rst period is unnecessary for existence, but simply
5
makes the equilibrium symmetric.
If the payo® from mutual dovishness were higher than that from unilateral triumph, this
game would no longer be Hawk-Dove and the players always would compromise. Even with
the present payo®s, there is likely to be an alternative equilibrium in which states compromise
in the present period. This paper ignores the possibilities for such compromises in order to
focus on the possibilities for states to communicate honestly and to trade over time, and to
avoid war.
Since I use the word \communication" often in the work that follows, I de¯ne it here.
In this paper, players \communicate" if each learns about the other's type from the other's
message: if Player k's type is \h," Player j6=k revises its beliefs about Player k's type after
hearing Player k's message, and believes that it is more likely that Player k is of type h than
it did before receipt of the message. Similarly, if player k's type is \l," Player j believes that
it is more likely that Player k is of type l after hearing k's message than it did before receipt
of the message. When I refer to \the communicative equilibrium," I mean the equilibrium
presented in Proposition 1 below.
This model and the equilibrium that I discuss next can be extended easily to a situation
in which a population of players interact. If so, each player must observe the outcome of all
others' interactions.
2.2. The Trade in Equilibrium
When players have di®erent values for the issues and interact inde¯nitely, they can obtain
a mutually-bene¯cial \trade" of issues over time by honestly communicating their types. In
this trade, each player has the issues decided in its favor (obtains the outcome it prefers,
associated with pro¯le in which it plays \Hawk" and its opponent plays \Dove"), when it
has a high value for the issues and the other has a low value, but gives up when the it has
a low value for the issues and the other has a high value. This section of the paper shows
in detail how this trade occurs. First, I present strategies and beliefs and discuss the crucial
constraint for existence of the equilibrium. Next, I discuss the substantive signi¯cance of the
equilibrium. Readers who wish to avoid the mathematical details should skip to section 2.3.
6
In the game, a player is \caught" deviating from equilibrium if it is observed taking an
action that is inconsistent with its equilibrium strategy. A player could deviate in a given
period and be unobserved doing so. For example, a player might deviate by planning to
play \Hawk" with probability one if its opponent were to say \high;" this deviation would
be unobserved if the opponent were to say \low." In the equilibrium in Proposition 1, each
player always tells the truth about its type. If it were to lie, it would always be caught
deviating since types are revealed at the end of the period.
The equilibrium strategies lead to two di®erent relevant futures for each player (continu-
ation values), one if the player in question is caught deviating, and one if it does not deviates
or deviates and is not caught. The worse future occurs if a player is caught deviating; then
the players do not communicate and the player that deviated always acquiesces. Substan-
tively, this future represents a \reputation for blu±ng," since one way for a player to obtain
it is to lie about its value for the good and then be caught in its blu®. The better future
occurs when a player plays its equilibrium strategy or blu®s but is not caught. Substantively,
this future represents a \reputation for honesty" since a player obtains this future, in part,
by being seen as honestly revealing its type.
Proposition 1. Consider two actors playing the Hawk-Dove game shown in Figure 2.2 and
in the surrounding text. The following symmetric strategies and beliefs form a Perfect
Bayesian Equilibrium if ± > 12 :
Strategies:
Tell the truth about your type.
If the period is t=1 and your type and your opponent's announced type are the same,
play \Hawk" if you are Player 1.
If the period is t=1 and your type and your opponent's announced type are the same,
play \Dove" if you are Player 2.
If neither you nor your opponent ever has been caught deviating from the equilibrium
in period t>1:
7
If your type and your opponent's announced type are di®erent, play \Hawk" if your
type is h and \Dove" if your type is l;
If your type and your opponent's announced type are the same and you played \Hawk"
the previous time that your types were the same, play \Dove."
If your type and your opponent's announced type are the same and you played \Dove"
the previous time that your types were the same, play \Hawk."
If your opponent has been caught deviating in any way from the equilibrium: say
\high" with probability 12 and \low" with probability 1
2 and play \Hawk."
If you have been caught deviating in any way from the equilibrium: say \high" with
probability 12 and \low" with probability 1
2 and play \Dove."
Beliefs:
At the beginning of each period t, the other player is of type h with probability 12 and
of type l with probability 12 .
If neither you nor your opponent has ever been caught deviating and you hear \h," the
other player is of type h with probability 1.
If neither you nor your opponent has ever been caught deviating and you hear \l," the
other player is of type l with probability 1.
If either you or your opponent has been caught deviating in the past and you hear \h,"
the other player is of type h with probability 12 and l with probability 1
2 .
If either you or your opponent has been caught deviating in the past and you hear \l,"
the other player is of type h with probability 12 and l with probability 1
2 .
Proof. The proof of equilibrium is primarily in the Appendix. Here, I discuss only the
low type's temptation to lie, which is the crucial incentive compatibility constraint for the
existence of the equilibrium. Since the game and the strategies of the proposed equilibrium
are symmetric, it su±ces to check the incentives and beliefs of Player 1, assuming that Player
2 abides by its equilibrium strategy.
8
In equilibrium, the high type of player more frequently gets its way, and thus the low
type has an incentive to pretend to be of the high type. If it does so, it is more likely to
obtain a more-favorable outcome today, but then is unable to communicate in the future.
The conditions under which the low type prefers to tell the truth limit the range of the
equilibrium.
There are two relevant histories: one in which Player 1 played \Hawk" the last time that
the players' types were the same (\History 1") and one in which it played \Dove" the last
time that they were the same (\History 2"). Here, I consider the incentives of Player 1 after
History 1. In the appendix, I show that the incentives of Player 1 after History 2 lead to the
same constraint.
If the low type of Player 1 plays according to its equilibrium strategy, it honestly reveals
its type. After History 1, it plays \Dove" regardless of its opponent's statement about type,
since it played \Hawk" the last time that the players announced the same type. Player 2
correctly recognizes that Player 1 has a low value for the disputed good, and plays \Hawk"
regardless of its own type. Player 1's continuation value and future incentives do not depend
upon its present type or the history coming into the present period since the types are not
correlated over time and there is a 50% chance that the players will announce the same types
in the present period, so that the history at the start of the next period will be di®erent. In
each future period, Player 1 expects that there is a 50% chance that it will be of the high
type, and a 50% chance that its opponent will be of the high type. It will receive its highest
payo®, h, whenever it is the only high type and half the time that its opponent's type also is
high (a total of 38 of the time). It will receive the acquiescence payo®, c, whenever it is the
only low type and half the time that the two players' types are the same, either high or low
(12 the time). Finally, it will receive its lower value for the issues, l, half the time that both
players have the low type (18 of the time). Before the exchange of messages, the expected
utility of the low type of Player 1 if it plays its equilibrium strategy is thus,
E[ul(s¤)jHistory 1] = c+±(38h+
12c+
18 l)
1 ¡ ± : (2.1)
where ui(s¤) is the utility to player 1 of type i of playing its equilibrium strategy (i2fh,lg)
9
if Player 2 plays its equilibrium strategy and E[ ui(s¤)] is that utility expected over its
adversary's possible types and the future types of both itself and its adversary in equilibrium.
To see whether or not the low type of Player 1 bene¯ts from lying, consider its expected
utility if it deviates as follows: it says \high" and then follows the strategy of a high type. If
Player 2 hears \high," it will play \Dove" unless it is of the high type itself, in which case it
will play \Hawk" because Player 1 played \Hawk" the previous time that the players were
of the same type. Since players learn their opponent's type at the end of the stage game,
Player 1 will be caught blu±ng and the players will fail to communicate in subsequent
rounds, playing \Dove, Hawk." Thus,
E[ul(lie; play like high type)jHistory 1] =12l+
12c+±c+±2c+ ::: =
12l+
12c+
±c1 ¡ ± : (2.2)
Note that the low type cannot gain the high type's payo® by pretending to be the high type
since it really does not value the issues highly. It can, however, get its way more often, and
thus obtain a payo® of l more often than the lower c.
Equations 2.1 and 2.2 demonstrate the pros and cons of lying. The gains from lying are
the short-run bene¯ts of being thought to be a high type. If the low type of Player 1 lies
today, its current di®erence in payo® is:
12c+
12l ¡ c = 1
2(l ¡ c) > 0: (2.3)
Since this quantity is positive, lying results in an immediate bene¯t.
The drawbacks of lying are the future reputational costs. Its future di®erence in payo®
is:±c
1 ¡ ± ¡ ±(38h+
12c+
18 l)
1 ¡ ± =4c¡ 3h¡ l
8±
1 ¡ ± :
Since h; l < c; 4c¡ 3h¡ l < 0:
To analyze the costs of lying, it is helpful to reparameterize the higher payo® of getting one's
way by considering its value in relation to the lower value. Let h=l+x. Then the future
di®erence in payo® to a low type who deviates is
4c¡ 3(l + x) ¡ l8
±1 ¡ ± =
12(c¡ l) ±
1 ¡ ± ¡ 38x±
1 ¡ ± : (2.4)
10
This reparameterization allows us to compare the future costs and present bene¯ts of
deviating from the equilibrium strategy. Multiplying both sides of the equation by two, the
bene¯ts, from equation 2.3, are (l ¡ c). The ¯rst term of the costs, -(l ¡ c) ±1¡± (equation
2.4), represents the utility lost when the the player does not attain its future goals. The
second term of the costs, ¡34x
±1¡± , represents the \extra" costs that come because the player
is failing to attain goals when it considers the issues most important.
The total future costs of deviating outweigh the present bene¯ts when:
l ¡ c · (l ¡ c) ±1 ¡ ± +
34x±
1 ¡ ± : (2.5)
When the parameters have values such that inequality 2.5 is satis¯ed, the strategies and
beliefs form a Perfect Bayesian Equilibrium. Note that this inequality always is satis¯ed
when ± ¸ 12 , since in this case the ¯rst term on the right-hand side on its own is greater than
or equal to the term on the left-hand side; the second term on the right-hand side always is
positive. If ± < 12 , the inequality may nevertheless be satis¯ed depending upon the values of
±, l, c, and x.
Proposition 2. There exists a Perfect Bayesian Equilibrium of the in¯nitely-repeated game
shown in Figure 2.2 which is identical to the equilibrium described above except that the
\punishment" for deviating from the equilibrium strategies lasts for only a ¯nite number of
periods.
Such an equilibrium is interesting substantively: in international relations, states do not
appear to gain permanent reputations for blu±ng, nor do they appear to lose the ability
to communicate forever. In addition, this variant of the equilibrium requires less heroic
assumptions about players' memories. If the \punishment" lasts for one or two periods,
players must only remember one or two periods of previous play.
Remark 1. The proof of this equilibrium follows directly from the proof above. Again, we
must pay particular attention to incentives for the low type to lie today. As before, the
continuation value is lower for a player that lies than for one that tells the truth. However,
if the punishment is restricted to a ¯nite number of periods, the two continuation values
11
will be closer together. Thus, the equilibrium with a ¯nite punishment exists but for fewer
values of the parameters ±, c, l, and x.
2.3. Discussion
Since the strategies and beliefs shown above form an equilibrium, communication is logically
possible in this modi¯ed Hawk-Dove game. Communication results in a \trade" of issues
over time; players are more likely to obtain the issues when they value them highly, and less
likely when they consider them less important. Moreover, communication is possible because
player obtain reputations for blu±ng if they lie. A player that uses its talk honestly today
is able to obtain a trade of issues over time; if, tomorrow, it ¯nds itself in an interaction in
which it considers the issues important, it will be able to communicate to its opponent that
that is the case; with a high probability, the opponent will back down and let that player
have what it wants. A player that uses its talk dishonestly, however, obtains a reputation
for blu±ng; it loses the ability to communicate and it loses the future issues entirely. Of
course, this model contains many simpli¯cations. In the real world, players sometimes do
blu® and a player with a reputation for blu±ng sometimes does attain its goals, as in Sartori
(2001). Even so, however, a player that blu®s may be caught, and a player with a reputation
for blu±ng for blu±ng is less likely to get its way.5
This simple model helps to explain President's Johnson's failure to blu® over Czechoslo-
vakia. Remember that Johnson did not threaten to use force to try to deter the threatened
Soviet invasion of Czechoslovakia in 1968. In the 1968 Czechoslovakian crisis, President
Johnson seems to have realized the bene¯ts of being honest about where the U.S. was {
and was not { committed enough to be prepared to ¯ght. The President knew that if
he threatened to get involved in Czechoslovakia and the Soviets called his blu®, the U.S.
would back down and lose credibility. When Soviet Ambassador Dobrynin told Johnson
that the United States' interests were not a®ected by the Soviet action in Czechoslovakia,
\[i]n response he was told that U.S. interests are involved in Berlin where we are committed5The game could be modi¯ed so that a randomly drawn pair of states or other players was matched to
interact in each time period t. In this case, the equilibrium holds as long as one assumes that each player
has full information about all others' previous play.
12
to prevent the city being overrun by the Russians" (National Security Council 1996, 274).
The trade in this case was fairly explicit: the Soviets could take Czechoslovakia. However,
if, in the future, they threatened Berlin, U.S. leaders would threaten to ¯ght and they would
mean what they said.
The analysis also con¯rms the implications of Sartori (2001) about the conditions under
which states are more and less likely to succeed in deterring attacks. States are more likely
to blu® when they begin disputes with reputations for blu±ng{ that is, when they recently
have been caught blu±ng in other disputes. They are more likely to acquiesce when they
ascribe little import to the subject of a dispute, more likely to try deterrence when they
consider the issues more important. Finally, since the possession of a reputation for blu±ng
by either state interferes with communication, a defender's threats are less likely to succeed
in deterring attacks when either the defender or the challenger recently has been caught
blu±ng.
3. Is Communication Likely?
One frequent criticism of repeated games and of games of incomplete information is that
the outcomes are indeterminate. The model presented above has many equilibria, including
equilibria in which players do not communicate; their words do not convey information about
their types because they \babble." Communication may be possible, but is it likely?
This section of the paper argues that communication is indeed likely by two formal
criteria. The communicative equilibrium described above is Pareto optimal under reasonable
restrictions on the payo®s and the strategy that players use in the equilibrium is collectively
stable.
Some scholars have argued that players should coordinate on a Pareto-optimal equilib-
rium if they are able to communicate.6 If players can communicate, why would they \choose"
an equilibrium if there were another that would make one better o® without making the other
worse o®? In the in¯nitely repeated game, since a strategy is a plan for how to play the en-6Fudenberg and Tirole (1992) [21] and Morrow (1994) [95] both review this idea, though both argue
against it.
13
tire game, one might expect that players will \choose" an equilibrium that is ex ante Pareto
optimal { Pareto optimal before the players know their types.7
The communicative equilibrium above is weakly ex ante Pareto optimal as long as c+l
¸ 2d. Let u1t(s¤; s¤)+u2t(s¤; s¤) be the sum of the players' utilities in time t if they play
according to the equilibrium. If the equilibrium were not Pareto optimal, there would need
to be some time period t in which some equilibrium other than this one led the sum of the
players' utilities to be higher than u1t(s¤; s¤)+u2t(s¤; s¤). However, the equilibrium (s¤; s¤)
maximizes the sum of the players' utilities in every period as long as c+l ¸ 2d, since l, h, c
> a. Thus, it is weakly Pareto optimal before the players know their types.
This game has other ex ante Pareto-optimal equilibria. However, any Pareto-optimal
equilibrium involves coordination through communication. Without communication, each
player believes that its opponent is equally likely to be of either type, so the players cannot
coordinate to play (\Hawk," \Dove") when Player 1 is of the high type and Player 2 is of the
low type and (\Dove," \Hawk" ) when their types are reversed. If they do not coordinate
conditional upon the pair of types (Player 1's, Player 2's), then in some periods the sum
of their utilities will be strictly less than in this equilibrium. In other words, when the
sum of their utilities would have been h+c if they had played (s¤; s¤), it will be something
lower. Moreover, it never will be higher. Thus, any equilibrium without communication
and coordination is Pareto dominated by this one as long as c+l ¸ 2d.8
Other scholars have criticized Pareto optimality as a criterion for equilibrium choice when
players have divergent interests (Morrow 1994, 95). Another way to think about equilibrium
selection is as the outcome of evolution.
To consider the evolutionary properties of this strategy, I modify the game to include
more players. In this modi¯ed version, two players from a population are randomly matched7In the stage game, any equilibrium of the stage game in which players play either \Hawk, Dove" or
\Dove, Hawk" is Pareto-optimal both ex post (after the players know their types) and ex ante (before they
know them).8I discuss above what it means substantively for c+l to be strictly less than 2d: the sum of Player 1's and
Player 2's shares of the disputed good is greater in the case of compromise than when the good is divided
unevenly between the two.
14
to play the game in Figure 2.2 inde¯nitely. One player is randomly assigned the designation
of \Player 1" by the °ip of a coin.
Axelrod (1984, 56) de¯nes the invasion of a strategy as follows: a strategy s' \invades" s¤
if the utility that a player receives playing s' against a player who plays s¤ is higher than the
utility a player would receive playing s¤ against the player who plays s¤. Strategies that are
uninvadable by this de¯nition are \collectively stable." By the criterion of collective stability,
the strategy that is played in the communicative equilibrium is likely, since any strategy that
is a Nash equilibrium with itself is collectively stable (Axelrod 1984, 217, footnote 1).9
Communication is therefore \likely" in several senses of the word. Not only is it equi-
librium behavior, but the equilibrium is ex ante Pareto optimal and the strategy played by
both players in equilibrium is collectively stable.
The next section of the paper takes for granted that parties can communicate, and asks
some questions about when communication is more likely and when it is less likely.
4. Under What Conditions Do Parties Communicate?
When Johnson decided not to deter an attack on Czechoslovakia, he was thinking about
Berlin, an issue of great importance to the U.S. and its allies. While Berlin was also important
to the Soviets, attaining a change in the status quo there was perhaps less important to them
than preserving the communist government of Czechoslovakia. The likelihood of honest
communication also depends upon the magnitude of the potential gains from trade.
Remember from above that the limits on honest speech come from the player's incentive to
lie when it has a low value for the disputed issue or good. The costs of honest communication
for this player are (l ¡ c), the di®erence between its payo® from getting its way on this
unimportant issue by lying and the payo® from acquiescing. The ¯rst term of the bene¯ts
from honesty, (l¡ c) ±1¡± (equation 2.4), represents the utility gained when the player attains
future unimportant goals through communication. The second term, 34x
±1¡± , represents the
9Of course, for this strategy to work, players must aware of each other's past play. In practice, this means
that the population must not be too large.
15
\extra" future bene¯ts of honesty that come because the player is able to use communication
to attain the goals that it considers the most important. As long as players value tomorrow
at least half as much as today, the one-time bene¯ts of deviating will be o®set or more than
o®set by the losing future issues that are unimportant; that is (l ¡ c) · (l ¡ c) ±1¡± .When the players value the future less, the potential gains from trade are crucial in deter-
mining when the communicative equilibrium exists. Remember that x represents the extent
to which \important" issues are much more valuable to the player than the \unimportant"
issues. When x is large { that is, when the gains to trading over time are large { the in-
centives to blu® today are more likely to be overwhelmed by the future loss in ability to
communicate and to attain goals. For example, if players only value tomorrow one quarter
as much as today, then communication is possible when the gains from trade are at least 83 as
great as the di®erence between gaining unimportant issues and acquiescing to an opponent's
demands. Communication is more likely when it is more pro¯table: when the gains from
trade over time are large.
In the Hawk-Dove game that I have been considering (Figure 2.1 and the accompanying
text), the likelihood of communication and the circumstances under which an actor gets its
way are una®ected by the costs of war. As long as the payo®s have the Hawk-Dove ordering,
the low type's temptation to lie is the only constraint on equilibrium (and this constraint
does not depend upon a).
Of course, the modi¯ed Hawk-Dove model simpli¯es from reality. In particular, in an
international dispute, it may be the case that one player has a substantial military advantage
and can take what it wants by force. Though the state will lose some resources by going to
war, it may nevertheless envision an easy enough ¯ght so that the bene¯ts of acquiring the
territory or having some issue decided in its favor outweigh these costs. If so, a>c for this
state. If the militarily disadvantaged party prefers acquiescence to war, a remains less than
c for this state. This game is represented in Figure 4.1.
If only one player prefers war to acquiescence, the communicative equilibrium described
in proposition 1 does not exist. The warmongering player always prefers to play \Hawk,"
irrespective of what the other is doing. Moreover, as long as this player insists upon playing
16
Figure 4.1: A Hawk-Dove Game in Which One Player Prefers War to Acquiescence
\Hawk," the other player, which still prefers acquiescence to war, will acquiesce. Thus, not
only does this player have short-run incentives to play \Hawk," but there is no way to punish
it for deviating from an equilibrium strategy that sometimes includes playing \Dove."
However, if both players prefer war to acquiescence, I conjecture that a slightly modi¯ed
version of the equilibrium exists:
Proposition 3. If a>c>d>h,l for both players in Figure 2.2, the strategies and beliefs in
Proposition 1 form an equilibrium with one modi¯cation. If a player is caught deviating in
any way from the equilibrium, its strategy is: \If you are caught deviating in any way from the
equilibrium, in all future periods say `high' with probability 12 and `low' with probability 1
2 and
play `Hawk.' " Though this equilibrium exists, there are additional constraints on existence
of the equilibrium beyond the constraint in Proposition 1.
Proofs of this proposition will follow in a later version of the paper. When both parties
prefer war to acquiescence, the existence of communication and coordination depend upon
the payo® from war, as well as upon the discount factor, the gains from trade, and the payo®s
from having an unimportant issue decided in your favor, from acquiescing.
17
5. Conclusion
Therefore, the coastal islands, while useful, are by no means indispensable to the
successful prosecution of the broad policy above outlined...
It was these considerations which led the United States not to include the coastal
positions in the `treaty area' de¯ned by our Mutual Defense Treaty (Dulles, 1955,
8).
States are reluctant to make promises that they do not intend to keep. A part (though
not all) of the U.S. policy of strategic ambiguity in regard to the defense of Taiwan can
be explained by this fact. In the quotation above, John Foster Dulles notes that the U.S.
deliberately excluded the o®shore islands from the Mutual Defense Treaty with Taiwan
because it did not consider them important.
When they do make promises, states often abide by the agreements that they do make
despite substantial incentives to reneg. For example, writing about the United States'
commitment to decontrol domestic oil prices, made at the Bonn summit of 1980, Robert
Putnam says:
...the Germans and the Japanese irretrievably enacted their parts of the bargain
more than six months before the president's action on oil price decontrol and
nearly two years before that decision was implemented. Once they had done so,
the temptation to the president to reneg should have been overpowering, but
in fact no one on either side of the decontrol debate within the administration
dismissed the Bonn pledge as irrelevant. In short, the Bonn `promise' had polit-
ical weight, because reneging would have had high political and diplomatic costs
(Putnam 1988, 438).
Why are states so careful with their promises? What are these \political and diplomatic
costs?" Pro®ering advice to diplomats, de Calliµeres writes, \Even were deceit not in itself
repugnant to every right-minded person, the negotiator should recollect that he is likely for
the rest of his life to be engaged in diplomatic business, and that it is essential for him to
18
establish a reputation for straight and honest dealing so that thereafter men may be ready
to trust his word" (in Nicolson 1963).
Like the diplomats who speak for them, states interact repeatedly in the international
system. If they blu® and are caught, they know that others will not trust their future
diplomacy. Thus, like President Johnson in the Czechoslovakian crisis of 1968, leaders are
cognizant of the bene¯ts of honesty. As the Czech case illustrates, states often acquiesce
to each other's demands so that others will not accuse them of crying wolf when they state
a willingness to ¯ght over truly important issues arise in the future. They abide by their
promises in part to avoid the reputational costs of reneging. By acquiescing to others'
demands on less important issues and by following through on threats and promises, states
maintain the ability to communicate, and are able to use communication to attain a mutually
bene¯cial \trade" of issues over time.
This paper analyzes an in¯nitely repeated Hawk-Dove game, modi¯ed to include in-
complete information. Hawk-Dove is suggestive of many political situations that involve
bargaining, including relations between states. The analysis shows that coordination and
communication are not only possible in this game; they are likely. The paper thus provides
a rationale for diplomacy. It also suggests that diplomacy is more likely to work among
states that care more about the future and in situations in which it is most needed { when
there are greater gains from trade over time.
The work that I present here suggests that communication and coordination are not
possible if only one party prefers war to acquiescence, though they are possible if both parties
have this preference. This result is troubling, since many international disputes are ones in
which one party is substantially favored by the balance of forces. Nevertheless, it probably
is an artifact of a simpli¯cation in the present model. In actual international disputes, a
state's expected value from war is likely to be highly correlated with the importance that
it ascribes to the issue; a state that values the issue highly has greater resolve, leading to a
higher value of the parameter \a" in the model. In a future version of the model, I intend to
explore a situation in which the relationship between a and c depends upon a state's value
for the issues: when the state is the high type, it prefers war to acquiescence, but when it
19
is the low type, it prefers acquiescence to war. In such a model, I expect that there will
be an equilibrium in which states communicate the importance they ascribe to issues and
coordinate a \trade" of issues over time.
The applications in this paper are to international relations, the subject that the author
knows best. Nevertheless, the work shows that communication and coordination are pos-
sible whenever parties interact inde¯nitely, as long as their preferences can be represented
reasonably as Hawk-Dove, they care su±ciently about the future, and they have di®erent
\favorites" among goods that will be contested. One interesting area for exploration is the
possibility for such communication among animals, though Maynard Smith (1982) reports
mixed evidence about such communication. The communicative strategies in this work do
require some thought: players must recognize opponents and keep track of their deviations.
Perhaps this behavior is possible only for diplomats, and not for hawks and doves.
20
References
Axelrod, Robert, The Evolution of Cooperation, New York: Basic Books, 1984.
Bergstrom, Carl T. and Peter Godfrey-Smith, \On the Evolution of Behavioral
Heterogeneity in Individuals and Populations," Biology and Philosophy, 1998, 13,
205{231.
Bueno de Mesquita, Bruce and David Lalman, War and Reason, New Haven: Yale
University Press, 1992.
Dulles, John Foster, \Preliminary Draft of Possible Statement of Position for
Communication to the Republic of China," 1955. 4, April 7.
Fearon, James Dana, \Domestic Political Audiences and the Escalation of International
Disputes," American Political Science Review, September 1994, 88 (3), 577{591.
Fudenberg, Drew and Jean Tirole, Game Theory, Cambridge, Massachusetts: The
MIT Press, 1992.
Morrow, James D., Game Theory for Political Scientists, Princeton: Princeton
University Press, 1994.
National Security Council, \Summary Notes of the 590th Meeting of the National
Security Council," in \Foreign Relations of the United States, 1964-1968; Volume XVII;
Eastern Europe," United States Government Printing O±ce, 1996.
Nicolson, Harold, Diplomacy, New York: Oxford University Press, 1963.
Putnam, Robert D., \Diplomacy and Domestic Politics: The Logic of Two-Level
Games," International Organization, summer 1988, 42 (3), 427{460.
Sartori, Anne E., \The Might of the Pen; A Reputational Theory of Communication in
International Disputes," 2001. Forthcoming in International Organization.
Schelling, Thomas C., Arms and In°uence, New Haven: Yale University Press, 1966.
Smith, John Maynard, Evolution and the Theory of Games, Cambridge: Cambridge
University Press, 1982.
Thucydides, History of the Peloponnesian War, Harmondsworth, Eng: Penguin Books,
1972. trans. Rex Warner.
21
Appendix
A. Proofs of Equilibrium
The appendix shows that the strategies in Proposition 1 form a Perfect Bayesian Equilibrium
(PBE) of the modi¯ed Hawk-Dove game in Figure 2.2 and the accompanying text if (but
not only if) ± ¸ 12 . To ascertain that these strategies form an equilibrium, one must check
nine sets of facts. The ¯rst eight involve strategies: neither the high type of player nor the
low type should wish to deviate from its equilibrium strategy once ex ante or ex post as long
as the other player plays its part of the equilibrium, where ex ante means before hearing
the other player's message and ex post means after hearing it. Moreover, these facts must
hold after all relevant histories, where relevant histories are histories upon which strategies
depend. The ninth fact to check is that beliefs are updated according to Bayes' Rule, but one
can see at a glance that beliefs are updated rationally here since each player receives a new
type at the beginning of the stage game. Since the game and the strategies of the proposed
equilibrium are symmetric, it su±ces to check the incentives and beliefs of Player 1. In the
work that follows, I check that no player wishes to deviate to another pure strategy. As long
as a player receives more utility from its equilibrium strategy than from an alternative pure
strategy s', the player will never wish to deviate by putting positive weight on s'.
There are two relevant histories: those in which Player 1 played \Hawk" the last time
that the players' announced types were the same and those in which Player 2 did so. Player
1's expected continuation value depends upon its history. Once it has heard its opponent's
message, Player 1's expected continuation value also depends upon whether or not the two
players' announced types in the present period are the same. These considerations lead to
four scenarios for continuation values in equilibrium after players announce their types:
1. History 1: Player 1 played \Hawk" the last time that the players' announced types
were the same.
1. The players' present announced types are the same. Thus, Player 1 will play
\Dove" and Player 2 will play \Hawk" today. Then, if both players play the
22
equilibrium strategies (s¤,s¤), Player 1 will play \Hawk" and Player 2 will play
\Dove" in time t+1 if the two players' types are the same. In this scenario, in
period t+1, Player 1 will receive h half of the time, when Player 1 is of type h and
Player 2 is of type l, and when both players' types are h. Player 1 will receive l
one fourth of the time, when both players' types are l. It will receive c one-fourth
of the time, when it is the only player of type l. Since there is a 50% chance
that the players are of the same type in period t+1, they enter period 2 with a
50% probability that Player 1 played \Hawk" the last time that the players were
of the same type and a 50% probability that Player 2 played \Hawk" the last
time that the players were of the same type. In present-valued terms, Player 1's
continuation value v1 is:
v1 = ±µ12h+
14l +
14c¶+ ±2
µ38h+
18l +
12c¶+ ±3
µ38h+
18l +
12c¶+ :::
= ±µ12h+
14l +
14c¶+±2
¡38h+
18 l +
12c
¢
1 ¡ ± :
2. The players' present announced types are di®erent. Then, if both players play the
equilibrium strategies (s¤,s¤), the history that Player 1 played \Hawk" the last
time the players' announced types were the same remains the relevant history in
time t+1. Player 1 will play \Dove" and Player 2 will play \Hawk" in time t+1 if
the two players' types are the same. In this scenario, in period t+1, Player 1 will
receive h one-fourth of the time, when it is the only player of the high type, and
c otherwise, when it is the only player of the low type or when the two players'
types are the same.
v2 = ±µ14h+
34c¶+ ±2
µ38h+
18l +
12c¶+ ±3
µ38h+
18l +
12c¶+ :::
= ±µ14h+
34c¶+±2
¡38h+
18 l +
12c
¢
1 ¡ ± :
2. History 2: Player 1 played \Dove" the last time that the players' announced types
were the same.
1. The players' present announced types are the same. Thus, Player 1 will play
\Hawk" and Player 2 will play \Dove" today. In period t+1, Player 1 will play
23
\Dove" and Player 2 will play \Hawk" in time t+1 if the two players' types are
the same. The continuation value for this scenario is just v2 above.
2. The players' present announced types are di®erent. Thus, in t+1 the relevant
history still will be that Player 1 played \Dove" the last time that the players'
announced types were the same; Player 1 will play \Hawk" if their announced
types are the same and Player 2 will play \Dove." The continuation for this
scenario is just v1 above.
In the work that follows, I check that Player 1 does not wish to deviate from its strategy
after learning its type but before hearing Player 2's message. If neither type of Player
1 wishes to deviate after learning its type, then Player 1 will not wish to deviate before
learning its type. Before Player 1 hears Player 2's message, it believes it equally likely that
Player 2 will announce either type and thus equally likely that the players' types will or will
not be the same. Thus, regardless of the history when it enters time t, it believes that there
is a 50% chance that it will leave that period (and enter t+1) having played \Hawk" the
last time that the announced types were the same and a 50% chance that it will leave that
period having played \Dove" the last time that the announced types were the same. Player
1's continuation value is therefore:
v3 = ±[14h+
14c+
14(12l +
12c) +
14(12h+
12c)] + ±2[
14h+
14c+
14(12l +
12c) +
14(12h+
12c)] + :::
=±(38h+
12c+
18 l)
1 ¡ ± :
I also check that Player 1 does not wish to deviate after both players make their an-
nouncements and each hears its opponent's message. This check turns out to be trivial as
long as a<c (which it is in 2.2) because, after hearing each other's messages, both players
do strictly worse in the present period through unilateral deviations than through the coor-
dinating strategies of the equilibrium. However, these constraints are important when a>c
in the second part of the appendix.
24
A.1. The Communicative Equilibrium of the Modi¯ed-Hawk-Dove Game
Player 1 never wishes to deviate o® the equilibrium path as long as the game has the payo®
ordering a<c<d<h,l. O®-path, players learn nothing about each other's types from their
messages and do not condition their strategies on messages, and the o®-path strategies
restricted to the stage game are Nash equilibria. Below, I check that Player 1 does not
wish to deviate from its strategies on the equilibrium path, assuming that Player 2 plays its
equilibrium strategy.
A.1.1. The High Type of Player 1
Note that the continuation value from deviating and being caught, ±c1¡± from 2.2 in the text,
is strictly lower than the expected continuation value from playing the equilibrium strategy
after any history, regardless of whether the expectation is taken before or after the player
receives its type, and regardless of whether the expectation is taken before or after the the
players exchange messages. Since h, l > c, v1, v2, or v3, > ±c1¡± .
Consider ¯rst the expected utility of a high type of Player 1 after History 1 before the
players send their messages if it plays its equilibrium strategy (s¤). Player 1's expected
utility before the players exchange messages will not be greater if Player 1 sends a di®erent
message. If Player 1 lies, Player 2 will be more likely to play \Hawk" after either history,
and Player 1 will be strictly worse o®, getting c or a when it would otherwise have received
h. Lying thus would have short run disadvantages, as well as the long-run disadvantage of
accruing for Player 1 a reputation for blu±ng.
The high type of Player 1 also can do no better by deviating from s¤ after hearing Player
2's message about its type after either history. The two players' equilibrium strategies after
any pair of messages form a strict Nash equilibrium of the stage game without talk, so
that Player 1 would make itself worse o® in the short run by deviating after an exchange
of messages. Moreover, a deviation would result in the lower continuation value associated
with the inability to communicate.
Since the high type of Player 1 cannot do better either by lying or by telling the truth
and then deviating after either of Player 2's possible messages, it can never do better by
25
deviating from the actions prescribed by s¤ on the equilibrium path. Since it cannot do
better by deviating o® or on the equilibrium path, it cannot do better by deviating.
A.1.2. The Low Type of Player 1
Consider now the low type of Player 1. One can quickly see that the low type of player cannot
bene¯t from deviating from its equilibrium strategy after hearing its opponent's message or
o® the equilibrium path. As with the high type, the low type's strategies, restricted to the
stage game without talk, are Nash equilibria.
The crucial incentive compatibility constraint on existence of the equilibrium is the low
type of player's incentive to lie. In the text, I show that the low type weakly prefers its
equilibrium strategy to one in which it lies and then acts as if it were the high type after
History 1 i® 12(l ¡ c) ¸ 1
2(l ¡ c) ±1¡± + 38x
±1¡± .
Another possible deviation, not considered in the text, is for the low type of Player 1 to
lie and then to play the high type's strategy only when Player 1 played \Dove" the last time
that the players were of the same type (after History 2). The advantage of this approach
is that Player 1 gets its way more often when it recently has given up (played \Dove"). If
the low type of Player 1 pursues this strategy, then Player 2 believes that it is the high
type. If Player 2 itself is of the low type, it plays \Dove" because the low type always plays
\Dove" when it believes that its opponent is of the high type. If Player 2 is of the high type,
it plays \Dove" because it played \Hawk" the last time that the players' announced types
were the same. Here, ui(s) is the utility to player 1 of type i of playing strategy s (i2fh,lg), assuming that Player 2 plays its equilibrium strategy. E[ ui(s)] is that utility expected
over its adversary's possible types and the future types of both itself and its adversary in
equilibrium. Thus,
E[ul(lie; play like high type)jHistory 2] = l +±c
1 ¡ ± : (A.1)
If it plays according to its equilibrium strategy,
E[ul(s¤)jHistory 2] =12c+
12l +±(38(l + x) +
12c+
18 l)
1 ¡ ± : (A.2)
26
Thus, the low type of Player 1 wishes to maintain its equilibrium strategy following this
history i® A.1 is less than A.2, or
or i®
l ¡ c · ±(l ¡ c+34x)
1 ¡ ± :
This condition is the same that obtains when considering the low type's ex ante incentives
to lie without conditioning on history (2.5).
The low type of Player 1 also can do no better by deviating from s¤ after hearing Player
2's message about its type after either history because the two players' equilibrium strategies
after any pair of messages form a strict Nash equilibrium of the stage game without talk,
so that Player 1 would make itself worse o® in the short run by deviating after an exchange
of messages. Moreover, a deviation would result in the lower continuation value associated
with the inability to communicate. Since the low type of Player 1 cannot do better either by
lying or by telling the truth and then deviating after either of Player 2's possible messages,
it can never do better by deviating on the equilibrium path.
I have shown that neither the high or the low type of Player 1 can do better by deviating
from the equilibrium strategy given in Proposition 1 on or o® the equilibrium path as long
as Player 2 plays this same strategy. Since beliefs are updated according to Bayes' Rule, the
strategy pro¯le in which both players play the strategies given in Proposition 1 and hold the
corresponding beliefs is a Perfect Bayesian Equilibrium.
A.2. Communication When One or More Players Prefer War to Acquiescence
The communicative equilibrium described in Proposition 1 breaks down when one player has
the Hawk-Dove payo® ordering a<c<d<h,l and the other prefers to war acquiescence. To see
this, consider a situation in which Player 1 only has payo®s c<a<d<h,l; Player 2's payo®s
remain as before. In this case, if Player 1 deviates from equilibrium, it will not be willing
to abide by its punishment. Following a deviation by Player 1, in equilibrium, both players
should babble; Player 1 should play \Dove," and Player 2 should play \Hawk." However,
for Player 1, \Hawk" is now a dominant strategy of the stage game; if Player 2 plays its
part of the equilibrium, Player 1 will be unwilling to play \Dove," as its equilibrium strategy
27
sometimes dictates. Moreover, there is no punishment that Player 2 is willing to provide to
Player 1 if Player 1 deviates from its equilibrium strategy . As long as Player 1 is playing
\Hawk," Player 2 prefers to play \Dove;" this results in no punishment at all for Player 1
since it gives Player 1 its favorite outcome.
On the other hand, a modi¯ed version of the communicative equilibrium exists when
both players prefer war to acquiescence{that is, when c<a<d<h,l for both players. In this
modi¯ed version, players \punish" each other for deviations by babbling and playing \Hawk"
forever as in Proposition 3 above.
The proof of existence is similar to that of the previous section. However, additional
deviations may now be tempting, since either type of Player 1 now has a short-run incentive
to deviate whenever it expects Player 2 to play \Hawk." The proofs will follow in a later
version of the paper.
28
Player 2 Hawk Dove Player 1 Hawk a, a b, c Dove c, b d, d a < c < d < b Figure 2.1 (page 4) Player 2 Hawk Dove Player 1 Hawk a, a h or l, c Dove c, h or l d, d a < c < d < h, l Figure 2.2 (page 5) Player 2 Hawk Dove Player 1 Hawk g, a h or l, c Dove c, h or l d, d a < c < g<d < h, l Figure 4.1 (page 17)