Download When Fully Informed States Make Good the Threat of War: Rational

B.J.Pol.S. 36, 645–669 Copyright © 2006 Cambridge University Press
doi:10.1017/S0007123406000342 Printed in the United Kingdom
When Fully Informed States Make Good the Threat
of War: Rational Escalation and the Failure
of Bargaining
CATHERINE C. LANGLOIS
AND
JEAN-PIERRE P. LANGLOIS*
Why would fully informed, rational actors fight over possession of a valued asset when they could negotiate
a settlement in peace? Our explanation of the decision to fight highlights the incentives that are present when
the defender holds a valued asset coveted by the challenger. The defender receives utility from possession of
the contested asset and sees any compromise as a loss that is lower if postponed. The challenger, instead, sees
any compromise as a gain that is more valuable if reached earlier. Faced with the defender’s vested interest
in the status quo, the challenger needs to threaten war and may have no choice but to implement the threat
to force a settlement. For the defender, the threat of war is a deterrent that might incite the challenger to back
down. In the perfect equilibria that we describe, the players’ ability to threaten each other credibly allows them
to maintain incompatible bargaining positions instead of helping them narrow their differences. But the very
credibility of these threats leads our rivals to engage in what can become lengthy protracted wars.
Why would fully informed, rational actors fight over possession of a valued asset when
they could negotiate a settlement in peace? We offer an explanation of the decision to fight
that focuses on an interplay between bargaining and deterrence that has been overlooked
in the formal literature on the causes of war. Our rivals can bargain in search of a settlement,
and they can wield traditional deterrent threats of escalation to war. This is made possible
by associating a simple escalation ladder, which assumes that the defender holds the asset
coveted by the challenger, to a bargaining model.1 Our approach therefore highlights the
incentive of incumbent ownership to delay settlement, and the rivals’ use of reciprocal
threats of escalation to incite the other side to accept less favourable terms.
In a standard argument, fully informed rivals should agree to share without fighting
because war reduces the size of the pie. The argument may satisfy all sides if the asset to
be shared belongs to neither party. But the logic is unlikely to convince a defender, who
holds and enjoys the prize, to simply hand it over on demand. Since the defender receives
utility from possession of the contested asset he sees any compromise as a loss that is lower
if postponed. He will therefore, in case of challenge, consider the tradeoff between a
peaceful sharing now and a holding of the asset for longer, despite the risk of escalation
posed by an aggressive challenger. The challenger, instead, sees any compromise as a gain
that is more valuable if reached earlier. Not only do the two sides have opposing interests
regarding the terms of the bargain, their interests also diverge when it comes to the timing
of the settlement. If the defender stands firm, refusing to hand over any part of the prize,
* McDonough School of Business, Georgetown University; Department of Mathematics, San Francisco State
University, respectively. The authors, considering their contribution to be equal, have listed their names
alphabetically. We thank James Morrow, Keith Ord and Dennis Quinn for their comments and two anonymous
reviewers for their constructive feedback.
1
We therefore look at an asymmetric deterrence situation which allows us to incorporate the simplest possible
escalation ladder into our model. Moreover, as Zagare and Kilgour point out, such ‘one-sided deterrence
relationships have obvious empirical and theoretical import’: Frank Zagare and Marc Kilgour, Perfect Deterrence
(Cambridge: Cambridge Studies in International Relations, Cambridge University Press, 2000), p. 135.
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the challenger may withdraw, deterred by the risk of escalation to war. But faced with the
defender’s vested interest in the status quo, the challenger may have no choice but to
implement the threat of war to force a settlement. Viewed in this light, the defender’s
incentive to drag his feet, in full cognizance of the challenger’s priorities and capabilities,
becomes an important ingredient in a rationalist explanation for war.
If the ongoing benefits of possession should make the defender reluctant to turn over
the contested asset on demand, can the challenger threaten the defender enough to persuade
him to relinquish at least some of it? Bargaining under threat has been studied extensively,
but the exact interplay between the terms of the bargain and the rivals’ deterrent threats
deserves closer scrutiny. Fearon and Powell, for example, argue formally that, in the
absence of bargaining indivisibilities or commitment problems, fully informed rivals
should always settle their differences peacefully.2 But their conclusion relies on the
comparative statics of alternative outcomes. The rivals can fight and then share, or they
can share without fighting. Given a redistribution of the contested asset, fully informed
rivals would be better off agreeing to it before fighting, thus avoiding the costs of conflict.
But how is the particular redistribution of the contested asset determined? In a dynamic
approach to bargaining, any redistribution must be formally offered by one of the two sides.
In choosing an offer the defender can indeed reason that there will be more of the pie left
to be shared if there is no fighting, but he may secure a larger share of a possibly diminished
pie by threatening to fight for it. Indeed, both parties can use the threat of war strategically
in an attempt to incite the other side to accept less favourable terms.
This article develops a game theoretic model of bargaining, escalation and war in which
a defender, who enjoys ongoing possession of the contested asset, faces a challenger’s
demand that he relinquish all or part of it. Both parties are fully informed, can bargain and
fight repeatedly, and support their respective positions on the sharing of the asset with
calibrated threats of escalation. We find that the rivals’ ability to threaten each other is a
critical determinant of the outcome of bargaining. Instead of promoting a peaceful sharing
of the contested asset, the rivals’ reciprocal threats allow them to make and maintain
incompatible claims rationally. And the very credibility of these threats leads them to
engage in what can become lengthy protracted wars, driven as much by the need to impose
costs as the prospect of battlefield victory.
COSTLY CONFLICT BETWEEN FULLY INFORMED RIVALS
In a growing body of literature it is shown that equilibria in games between fully informed
rivals can be inefficient, meaning that both parties could have increased their payoff by
adopting different strategies. Typically, costly conflict is at the root of the inefficiency.
Much of this literature is dedicated to stochastic games in which external circumstances
can change from one period to the next. For example, Powell shows that changes in the
relative power of rivals can lead the declining power to fight rather than accept a sharing
2
James D. Fearon, ‘Rationalist Explanations for War,’ International Organization, 49 (1995), 379–414; Robert
Powell, Nuclear Deterrence Theory: The Search for Credibility (New York:Cambridge University Press, 1990).
As we discuss below, these authors do explain costly conflict in equilibrium despite full information if the payers
are engaged in a stochastic game. The inefficient outcome is then attributed to commitment problems. See, for
example, Robert Powell, ‘The Inefficient Use of Power: Costly Conflict with Complete Information’, American
Political Science Review, 98 (2004), 231–41; James D. Fearon, ‘Why Do Some Civil Wars Last So Long?’ Journal
of Peace Research, 41 (2004), 275–302.
When Fully Informed States Make Good the Threat of War
647
of territory that could be contested when its rising rival becomes stronger.3 In Fearon, a
government and a rebel group negotiate regional control.4 But the government can be
strong or weak with some exogenous likelihood. Thus when the government is weak, the
rebels have an incentive to fight rather than accept a more generous level of control because
the government may be strong in the next period and renege on the agreement. In Acemoglu
and Robinson changes in the cost of challenging the elite in power creates an incentive
for the elite to renege on its promises when the cost of challenge is high.5 Revolutionary
movements therefore have an incentive to challenge the elite when the cost of doing so
is low. In a more recent analysis, Powell shows that if changes in actor relative power are
large and rapid enough, then all equilibria of the stochastic game are inefficient.6 In all these
cases, inefficiency comes from the inability of parties to commit credibly to an efficient
outcome if circumstances change.7 In our model, by contrast, circumstances remain stable
ruling out this possible source of inefficiency.
Following Clausewitz’s famous description of ‘war as simply a continuation of political
intercourse, with the addition of other means’,8 the intimate link between war and
bargaining has been developed at length by Pillar and discussed in formal terms by
Wagner.9 Both authors develop the idea that wars are fought to sharpen the terms of an
ultimate bargain by providing each side with information about the other’s capabilities and
resolve. For Wagner, the key to the relationship between war and bargaining is the fact
that the outcome of a fight is expected to modify the terms of a post-war bargain.10 A
decision to fight rather than settle is a gamble on the possible bargaining outcomes that
can emerge once the battle has been fought. It is the ‘inconsistent expectations of the
consequences of fighting’ and the resulting value of war to inform that prevent the parties
from accepting a negotiated compromise instead of escalating the conflict. It follows that,
as Pillar observes, ‘negotiations are more likely to open once the parties arrive at a common
view of the future course of the war and both become aware that they hold such a common
view’.11 A growing body of recent literature, following Wagner, focuses on ‘private
information as a key source of conflict and the revelation of that information as a central
component of the process of war termination’ (Filson and Werner).12
3
Robert Powell, In the Shadow of Power (Princeton, N.J.: Princeton University Press, 1999).
Fearon, ‘Why Do Some Civil Wars Last So Long?’
5
Daren Acemoglu and James Robinson, ‘A Theory of Political Transitions’, American Economic Review, 91
(2001), 938–63.
6
Powell, ‘The Inefficient Use of Power’.
7
Related work examines the incentive to run up inefficient levels of public debt (Alberto Alesina and Guido
Tabellini, ‘A Positive Theory of Fiscal Deficits and Government Debt’, Review of Economic Studies, 57 (1990),
403–14; Thorsten Persson and Lars Svensson, ‘Why Stubborn Conservatives Would Run a Deficit: Policy with
Time-Inconsistent Preferences’, Quarterly Journal of Economics, 104 (1989), 325–45) or to create inefficient
administrative procedures (Rui de Figueiredo, ‘Electoral Competition, Political Uncertainty, and Policy
Insulation’, American Political Science Review, 96 (2002), 321–35).
8
Carl von Clausewitz, On War, edited and translated by Michael Howard and Peter Paret (Princeton, N.J.:
Princeton University Press, 1984), p. 605.
9
Paul R. Pillar, Negotiating Peace: War Termination as a Bargaining Process, (Princeton, N.J.: Princeton
University Press, 1983); R. Harrison Wagner, ‘Bargaining and War’, American Journal of Political Science, 44
(2000), 469–84.
10
Wagner, ‘Bargaining and War’, p. 480.
11
Pillar, Negotiating Peace, p. 59.
12
Darren Filson and Suzanne Werner, ‘A Bargaining Model of War and Peace: Anticipating the Onset, Duration
and Outcome of War’, American Journal of Political Science, 46 (2002), 819–38, p. 819. There is a vast literature
on war and bargaining that appeals to uncertainty and we do not review it here. For example, Powell models rivals
4
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But if wars are fought to inform, we argue, the conclusion of the information-gathering
phase should mark the onset of peace. Yet enduring rivals show us that conflicts sometimes
fester for a very long time: the sixty-three enduring rivalries occurring between 1816 and
1992 examined by Bennett have an average duration of twenty-five years.13 Of course, such
rivals do not spend all of their time waging war against each other, but neither do they reach
the settlement that their increasingly intimate experience with each other’s capabilities and
resolve should predict. Why then would well-informed rivals fail to find a settlement that
both prefer to war? Fearon, in his discussion of what he terms the ‘central puzzle of war’,
writes that no rationalist approach ‘adequately resolves the central puzzle namely that war
is costly and risky, so rational states should have incentives to locate negotiated settlements
that all would prefer to the gamble of war’.14 In fact, if the rivals are fully informed about
each other’s capabilities and resolve, both Fearon and Powell argue formally that, in the
absence of bargaining indivisibilities or commitment problems, they should always settle
their differences peacefully in equilibrium.15 It is a matter of efficiency.
There are few attempts to explain war as the rational decision of fully informed players
who can also negotiate. In related work, Fearon models bargaining over co-operative
solutions that are preferred by both sides to the status quo, but differ in the relative
advantage to one or the other party.16 He models bargaining as a war of attrition where
it is costly to both sides to hold out, but where holding out may lead the other to back down
and accept the less favourable bargain. With full information about the enforceability of
the possible agreements, Fearon shows that, in equilibrium, there will be an expected
non-zero delay before one of the rivals concedes the prize. But Fearon’s model is about
the interaction between bargained agreements and their enforcement in Prisoner’s
Dilemma type situations; it is not designed to explore the rationality of war.17
Slantchev examines bargaining between fully informed rivals who enter a so-called
‘conflict game’ if they fail to settle on a proposed sharing of the pie.18 The model embeds
a disagreement game in a Rubinstein bargaining game as in Busch and Wen.19 One of the
(F’note continued)
that have asymmetric information on each other’s preferences (Robert Powell, ‘Bargaining in the Shadow of
Power’, Games and Economic Behavior, 15 (1996), 255–89; and Powell, In the Shadow of Power). The more recent
literature examines the learning that occurs as a result of fighting and bargaining. In Filson and Werner one of
the rivals is uncertain about the other’s type and bargains accordingly. The response to negotiations as well as
the outcome of battle informs about the likelihood that the rival is strong or weak (see Filson and Werner, ‘A
Bargaining Model of War and Peace’, and Darren Filson and Suzanne Werner, ‘Bargaining and Fighting: The
Impact of Regime Type on War Onset, Duration and Outcome’, American Journal of Political Science, 48 (2004),
296–313). In a related vein, Smith and Stam model states that have inconsistent beliefs about the likelihood of
winning the next battle (or fort) and these beliefs converge as the rivals fight battle after battle (Alastair Smith
and Allan Stam, ‘Bargaining and the Nature of War’, Journal of Conflict Resolution, 48 (2004), 783–813).
13
Scott D. Bennett, ‘Integrating and Testing Models of Rivalry Duration’, American Journal of Political
Science, 42 (1998), 1200–32
14
Fearon, ‘Rationalist Explanations for War’, p. 380.
15
Fearon, ‘Rationalist Explanations for War’; Powell, Nuclear Deterrence Theory.
16
James D. Fearon, ‘Bargaining, Enforcement and International Cooperation’, International Organization, 52
(1998), 269–305.
17
Fearon, ‘Bargaining, Enforcement and International Cooperation’.
18
Branislav L. Slantchev, ‘The Power to Hurt: Costly Conflict with Completely Informed States’, American
Political Science Review, 97 (2003), 123–35.
19
Ariel Rubinstein, ‘Perfect Equilibrium in a Bargaining Game’, Econometrica, 50 (1982), 97–110;
Lutz-Alexander Busch and Quan Wen, ‘Perfect Equilibria in a Negotiation Model’, Econometrica, 63 (1995),
545–65.
When Fully Informed States Make Good the Threat of War
649
disagreement outcomes is war. Peace is assumed to be an equilibrium of the disagreement
game, and war is the worst outcome for both parties. Slantchev identifies subgame perfect
equilibria in which the rivals can agree to a settlement that is delayed or interrupted for
a few turns while they fight.20 Such equilibria are supported by the threat of reversion to
the extremal equilibria that pick out the worst such outcome for each of the players. Thus
Slantchev shows that fully informed rivals may choose to fight in equilibrium despite the
bargaining possibilities. However, Slantchev’s rivals know with certainty which
settlement will prevail in the chosen war equilibrium, could have agreed to implement it
at the date chosen for fighting, and would have been better off doing that than by agreeing
to fight. Thus Slantchev’s rivals do identify settlements that both prefer to war. The fact
that they could pick an inefficient war equilibrium does not explain why they would
do so.
If perfectly informed rivals map out deterministic futures, explaining war requires us
to explain why rivals would choose to fight rather than adopt those futures that keep peace
while implementing the settlements that could be obtained by war. But the decisions made
by perfectly informed rivals need not lead to deterministic futures. In contrast to Slantchev,
our rivals disagree on the terms of the bargain. Each side maintains a bargaining position
and accepts a fight in the hope that the other side will give up. In Slantchev’s model, the
rivals fight to make sure that the terms of the agreed bargain will be upheld. Our rivals
fight because they disagree on the terms of the bargain and want to get their own way. As
long as the game lasts, neither party knows, for sure, which bargaining position will prevail.
Indeed, we find that, in equilibrium, the rivals can create strategic uncertainty by
threatening probabilistic rather than deterministic escalation in the face of unacceptable
demands by a challenger or unsatisfactory counteroffers by the defender. Moreover, the
calibration of this uncertainty makes the rivals indifferent between a peaceful settlement
now and a future resolution that can involve fighting along the way.
A M O D E L O F B A R G A I N I N G, E S C A L A T I O N A N D W A R
The Model Structure and Payoffs
To capture the interplay between bargaining and deterrence, we attach a bargaining model
and a ‘costly lottery’ model of war to the standard asymmetric deterrence model discussed
by Zagare and Kilgour.21 Bridges built between the bargaining, war and deterrence models
create a repeated game structure that allows the players to decide how much fighting takes
place and when the game ends. The rivals, a defender and a challenger, are perfectly
informed about each other’s priorities and capabilities, and war imposes the worst
per-period payoff on each party. A crisis develops if the challenger seeks to gain possession
of all or part of an asset held by the defender. The asset, which we can think of as territory,
yields a per-period utility that is enjoyed by the defender throughout the crisis. This gives
him an incentive to resist the challenger’s demand, to forgo a quick bargained settlement,
and to threaten war in the hope that the challenger will back down. As the game evolves,
the players interact in three possible modes: bargaining, crisis and war. Each player gets
to make a decision in each of these modes. Each decision node is labelled with two letters:
the first refers to the mode and the second to the player. Thus at decision point BD the
20
21
Slantchev, ‘The Power to Hurt’.
Zagare and Kilgour, Perfect Deterrence.
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LANGLOIS AND LANGLOIS
Fig. 1. The structure of the game
*Indicates per period payoffs that repeat indefinitely.
defender makes a decision while the players are in bargaining mode, at CC it is
the challenger’s turn to make a decision while the players are in crisis mode, and at WD
the defender makes a decision in war mode. Our construct ensures that each player controls
one strategic variable in each of the three modes, a necessary feature if both players are
to get a chance to bargain, deter and, if necessary, decide to fight. The model is reproduced
in Figure 1.
The game begins at the Start node when the challenger demands a positive share x of
the defender’s asset.22 At node BD, the defender can either submit to this request, ending
the game in State II in which he concedes share x to the challenger and keeps share (1 ⫺ x)
for himself, or seek a bargain by making a counteroffer y. At node BC, the challenger can
finalize a bargain by accepting offer y, ending the game in State I in which she gets
22
Demanding x ⫽ 0 is substantially equivalent to not challenging in the first place and indeed results in an offer
y ⫽ 0 and an immediate exit by the challenger in the equilibria we obtain.
When Fully Informed States Make Good the Threat of War
651
share y and the defender keeps (1 ⫺ y). But should offer y not give her satisfaction, the
challenger can press on with her demand, or make a new one, thus entering the crisis phase.
At CD the defender can submit to this renewed demand and end the game in State II as
above, or resist the continuing challenge to the status quo. At CC, the challenger can
escalate or backdown, moves that would end the classic deterrence game. But in this
repeated framework the challenger’s moves at CC merely conclude a turn of the game by
leading the players to one of two states:
(1) Backdown leads to State III in which the defender continues to enjoy the full benefits
of holding the entire prize while the challenger has yet to obtain anything.
(2) Escalate leads to State IV in which the rivals fight for one turn incurring the costs of
war. The battle can lead to a decisive victory for one or the other party, who then takes
over the entire asset. With probability v the defender wins continued, and uncontested,
possession of the asset ending the game in State I⬘. With probability u, the challenger
wins possession of the whole asset in battle and the game ends in State II⬘. But with
probability t ⫽ 1 ⫺ u ⫺ v, the battle is not decisive and player decisions determine the
future course of the game.
If neither party wins the asset in battle, the game continues with a decision by the
defender at WD. At WD, the defender can surrender to the challenger’s demand x, ending
the game in State II, or he can pursue violent action initiating a new round of war if the
challenger also decides to fight at WC. But at WC, the challenger can also initiate a return
to bargaining and a temporary end to the violence by backing down, letting the defender
enjoy possession for one turn in State III.23
The rivals collect payoffs when they visit the states of the game, all of which are denoted
by roman numerals. States III and IV mark the end of a turn and the beginning of a new
one, and can recur in any order as long as neither side prevails on the battlefield or chooses
to end the game on the opponent’s terms. These two states yield per-turn payoffs for each
side and mark the discounting of the future. By contrast, final states I, I⬘, II and II⬘ yield
a discounted sum of payoffs corresponding to the agreed terms or those imposed by
victory.
We assume that share z of the prize yields per-period utility Ui(z) ⫽ z to player i.24 In
particular, upon reaching State III, the defender receives utility z ⫽ 1 since he still holds
the entire prize while the challenger, who has nothing, receives z ⫽ 0. For simplicity, we
assume that both sides discount the future at a same rate (0 ⬍ ⬍ 1). Consequently,
enjoying a share z of the prize for the foreseeable future yields utility
(1 ⫹ ⫹ 2 ⫹ … ⫹ n ⫹ … )z ⫽ z/(1 ⫺ ). For example, if the defender accepts the
challenger’s demand of x at BC, he receives 1 ⫺ x per period, or the discounted sum
(1 ⫺ x)/(1 ⫺ ), while the challenger gets x/(1 ⫺ ), corresponding to per period payoff
23
Backdown does not mean that the challenger drops her demand altogether. It merely means that she shies
away from immediate military action while her current demand stands. Whatever y the defender may offer at the
next visit to BD, the challenger can repeat or modify her demand at her next visit to BC. The model can undoubtedly
be complicated to allow more options and decision turns for the two sides. Our modelling choices were driven
by a need for simplicity while allowing for repeated choices in the three main phases of the game (bargaining,
crisis and war). The model variations that the authors investigated did not affect the qualitative results.
24
We could also assume that the rivals are risk averse. Player utilities for share z would then read Ui(z) ⫽ z
where 苸 [0,1] represents the rivals’ degree of risk aversion. The lower , the more risk averse the players. We
briefly explore the impact of risk aversion on model outcomes when we develop our numerical examples (see fn.
40) and provide formulae for strategic probabilities in equilibrium in fn. 35.
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LANGLOIS AND LANGLOIS
x. Likewise, a decisive victory at node IV is worth 1/(1 ⫺ ) to the winner and 0 to the
loser.25
We assume that escalation to war in State IV yields negative net expected payoffs
⫺ c ⬍ 0 to the challenger and ⫺ d ⬍ 0 to the defender. These payoffs account for the
defender’s rent and the probability of decisive victory by one or the other party. Precisely,
if cC and cD, are the costs of engaging in battle for challenger and defender:
⫺ c ⫽ ⫺ cC ⫹ ⫻ u ⫻
1
1⫺
and
⫺ d ⫽ 1 ⫺ cD ⫹ ⫻ v ⫻
1
.
1⫺
(1)
For the defender, the actual costs of fighting are mitigated by the payoff he receives from
continuing possession. And, for both sides, costs are mitigated by the possible benefits of
battlefield victory through the respective probabilities u and v of winning the whole asset
in battle. But neither side can have a positive payoff from visiting State IV. If either did,
standard arguments from classical deterrence theory would apply since one or the other
would prefer war to accepting the other side’s terms in the one-shot escalation game. The
theoretical challenge that we address arises when war is undesirable for both sides.
Corollary 1 in the Appendix verifies logical consistency of our formulation.26
Challenger and defender are standard expected discounted utility maximizers and
choose strategy accordingly. To express the players’ objectives consider a sequence of
player moves through the graph of Figure 1. Given any current decision node of the graph,
such a sequence is valued according to the future payoff states visited. If S denotes the
payoff state visited at turn then a payoff path is a sequence ⫽ {S1, S2, … , S …}. Such
a sequence could cycle within the graph of Figure 1 visiting States III and IV indefinitely,
or it could end with the players exiting the game in one of the four exit states I, I⬘, II or
II⬘. Each sequence of decisions made by the players results in a particular payoff path. For
example, the sequence of choices from decision node BC: ‘demand, resist, backdown,
offer, demand, resist, escalate, fight, backdown, offer, demand, submit’, results in the
payoff path ⫽ {III, IV, II} with the players exiting the game in State II.
Player i’s discounted value for a payoff path is defined as:
V i ⫽ Ui (S1) ⫹ Ui (S2) ⫹ … ⫹ ⫺ 1Ui (S) ⫹ … ⫽
冘
⬁
⫽1
⫺1
Ui (S).
(2)
For example, payoff path ⫽ {III, IV, II} has discounted value for the defender:
V D ⫽ 1 ⫹ (1 ⫺ cD) ⫹ 2
1⫺x
.
1⫺
The standard formulation of player i’s expected utility, viewed from decision node N is:
Ei(N) ⫽
25
冘 P()V
i
(3)
Payoffs indicated in Figure 1 are per period payoffs. The challenger’s payoff is listed first.
We could also reduce the value of the pie when the players escalate the conflict to war by assigning different
discount factors III and IV at the two payoff nodes III and IV. Since IV ⱕ III this is formally equivalent
to III ⫽ and IV ⫽ t with a reinterpretation of parameter t. In the current interpretation we assume t ⫹ u ⫹ v ⫽ 1
but we can instead let t ⫹ u ⫹ v ⬍ 1 with the difference being lost in the fighting. The future would then be more
discounted after any war turn. This would discount costs more after a war episode, meaning that each side’s
capacity to inflict war costs in the future is reduced by fighting. Such a modification would leave formulae for
probabilities in 6a–e unchanged, although a lower t would affect the probability values and the necessary conditions
(s and r decrease, and condition c ⫹ d ⬎ t/(1 ⫺ ) is less stringent).
26
When Fully Informed States Make Good the Threat of War
653
where the sum is taken over all possible paths following N, and P() is the probability
that path will be travelled.27
Bargaining, Challenge and War in Perfect Equilibrium
The standard solution concept used in repeated game models is that of subgame perfect
equilibrium (SPE). The players adopt strategies that specify how they plan their present
and future moves in reaction to any possible past. When a decision node N for player i
is reached after some prior history of play, i’s strategy expresses his intended move as well
as demands or offers when needed. Given his opponent’s strategy, player i should rationally
maximize his expected utility Ei(N) by his own choice of strategy. When this situation
holds for both sides and for any possible prior history, the strategies form a SPE.28 It is
rarely possible to describe all SPEs in a repeated game structure such as ours. But it is
possible to characterize whole classes of SPEs, and our objective, here, is to describe two
classes that lead the players to fight, rather than settle peacefully in case of challenge,
although they are fully informed.
A class of equilibria constructed using extremal equilibria. A class of war equilibria is
constructed using extremal equilibria that yield the worst rational outcomes for each of
the players. As shown in Theorem 1 in the Appendix, extremal SPEs of our game are as
follows:
␧C (for the challenger): At Start the challenger demands any x, at BD the defender offers y ⫽ 0,
and at BC the challenger accepts. Should any of these moves fail, at CD the defender plans
to resist and at WD to fight. And at WC the challenger plans to back down. These moves are
in perfect equilibrium and result in the challenger’s worst outcome of 0 at each of her decision
nodes.
␧D (for the defender): At Start the challenger demands x ⫽ 1 and at BD the defender submits.
Should either of these moves fail, the challenger plans to demand x ⫽ 1 at BC, to escalate at
CC, and to fight at WC. The defender also plans to submit at CD and to surrender at WD. Should
the challenger backdown at CC, play switches to ␧C. These moves result in the defender’s worst
outcome of 0 at each of his decision nodes. Note that the reversion to ␧C after an unintended
backdown is designed to make escalation optimal for the challenger should the defender fail
to submit.
Extremal equilibria can be used to construct a variety of SPEs in similar fashion. The
example that follows is instructive. The rivals consider the following plan P : ‘The
challenger first demands everything (x ⫽ 1) and the defender offers nothing (y ⫽ 0). At BD,
the challenger repeats her demand. Then the rivals engage in a single round of war by
“resist, escalate, fight and backdown”, and at the next visit to BD, the defender offers z
that the challenger immediately accepts.” If the condition
(1 ⫺ )c ⱕ t2z ⱕ t ⫺ (1 ⫺ )d
(4)
holds, the plan is part of a SPE built on extremal equilibria (see Proposition 1 in the
Appendix for details). The equilibrium is simply supported by an expectation of immediate
27
See Drew Fudenberg and Jean Tirole, Game Theory (Cambridge, Mass.: The MIT Press, 1995), chap. 5.
Technically, that history must lead to a subgame that is self-contained in terms of decision and information
structure. This is always the case in our game structure. Also note that the prior history need not result from
equilibrium play in a SPE.
28
654
LANGLOIS AND LANGLOIS
reversion to his or her extremal equilibrium for the deviating player. And because an
extremal equilibrium is a SPE, the deviating player will not be able to deviate further
profitably and will suffer from failing to comply with the initial plan.
The war episode that is built into the type of equilibrium we have just described is not
a retaliatory move implemented to make good on a deterrent threat. Instead, the rivals fight
preventively to evade the danger of reversion to a self-defeating extremal equilibrium that
involves no further fighting. A possible interpretation of plan P proposed by Slantchev (for
his own model) is that the two sides must demonstrate the resolve to fight in order to reach
a fair bargain.29 In Slantchev’s vision, reversion to an extremal equilibrium is simply a
change of player expectations on how the game is going to be played that is contingent
on the demonstration of resolve. Nevertheless, players behaving according to extremal
equilibria engage in rounds of costly war to reach a mutually acceptable bargain that is
already in plain sight. They both expect to agree to its terms, and it is only the threat of
reversion to extremal equilibria that prevents either side from offering and agreeing to this
expected bargain in the very first round.
While such equilibria explain war between perfectly informed rivals, they require that
war costs remain contained. Indeed, as shown in Proposition 1 in the Appendix, if:
c ⫹ d ⱖ t/(1 ⫺ ),
(5)
there are no SPEs for our game in which war can occur unless it has been threatened
probabilistically. In particular, the deterministic war episodes that occur in the equilibria
based on reversion to extremal equilibria no longer exist. And even if Condition 5 is not
satisfied, the nature of the pure equilibria we have discussed becomes increasingly
constrained as c ⫹ d approaches limit t/(1 ⫺ ).30 Condition 5, if t ⫽ 1, states that the sum
of the war costs to challenger and defender exceeds the discounted value of the prize. If
t ⬍ 1, the condition is made less stringent by the prospect of an outright victory by one or
the other party.
War equilibria based on probabilistic threats. While the war equilibria based on the
reversion to extremal equilibria have been discussed by Slantchev to illustrate the
possibility of war between fully informed rivals,31 the equilibria that we are about to
describe have been overlooked. Yet their behavioural implications are of interest in an
attempt to explain war between rational fully informed rivals. In the class of SPEs we
exhibit, if the players maximize expectations with reference to their position in the game
only, but without regard for the history that brought them there, the resulting SPE is called
Markov perfect (MPE).32 For simplicity, we assume that the two sides adopt bargaining
29
Slantchev, ‘The Power to Hurt’.
Indeed, as c ⫹ d approaches t/(1 ⫺ ), the number of turns of war that can be built into these pure equilibria
decreases. More precisely, up to n war turns can be built into the pure strategy equilibria if c ⫹ d ⱕ tn/(1 ⫺ n).
This is because, given an agreed bargain (x, 1 ⫺ x), each rival must find it in his or her interest to follow the plan,
and incur the costs associated with the war episodes, in order to get the agreed share of the prize. For the same
reason, pure equilibria cannot generate outcomes in which the rivals fight wars that end up costing more than the
coveted asset is worth.
31
Slantchev, ‘The Power to Hurt’.
32
See Fudenberg and Tirole, Game Theory, chap. 13, for an analysis of Markov Perfect equilibria. Although
a MPE has a simple structure it retains all properties of SPEs, in particular that no deviation of any kind can be
profitable. An elementary proof of this claim for our particular game structure is available from the authors upon
request.
30
When Fully Informed States Make Good the Threat of War
655
positions at the start of the game, and stick to them as the game unfolds. But our results
can be extended to repeated adjustment of demands and offers.33
Given a demand x by the challenger, and an offer y by the defender, within an appropriate
range to be specified, we show in Theorem 2 in the Appendix that the following choices
by the rivals are part of a MPE:34
At BC the challenger chooses:
demand x with probability s ⫽
tx ⫺ y ⫺ , accept y with probability 1 ⫺ s.
x⫺y
(6a)
At CD the defender chooses:
resist with probability p ⫽
x⫺y
, submit with probability 1 ⫺ p.
x ⫺ y
(6b)
At CC/WC the challenger chooses:
escalate/fight with probability q ⫽
(1 ⫺ )c ⫹ (1 ⫺ t)x
,
(1 ⫺ )(c ⫹ d) ⫹ 1 ⫺ t
backdown with probability 1 ⫺ q.
(6c)
At WD the defender chooses:
fight with probability r ⫽
tx ⫺ y ⫺ , surrender with probability 1 ⫺ r.
t(x ⫺ y)
At BD the defender chooses to reiterate offer y with probabilty z ⫽ 1.
(6d)
(6e)
Parameter, ⫽ (1 ⫺ )c/. The above values are probabilities when the following
conditions are met:
(i) Parameters c and d must be positive ensuring that war is costly for both sides.
(ii) Parameter cannot exceed t. Thus c must be bounded by t/(1 ⫺ ). If t ⫽ 1 this
means that a single turn of war for the challenger must not cost more than the value
of holding the prize for the foreseeable future. With t ⬍ 1, the condition is net of the
chances of a decisive battle win by one party or the other.
(iii) Demand x must be at least /t and offer y must satisfy y ⱕ tx ⫺ .
Condition iii turns out to be the sole condition on x and y for a MPE to exist. Thus, any
pair (x, y) within the range illustrated in Figure 2 can provide a rational demand and a
rational offer for the two sides that will be part of a MPE.
Given a demand x the choice of a particular offer y within the range of possibilities for
the defender is consequential. If the defender offers share y ⫽ (tx ⫺ ), the challenger will
accept the offer with certainty and there will be no conflict since s ⫽ 0 in 6a.35 Offer (tx ⫺ )
33
Details of this generalization are available from the authors upon request.
Theorem 2 in the Appendix specifies out of equilibrium behaviour as required for a SPE (Reinhart Selten,
‘Re-examination of the Perfectness Concept for Equilibrium Points in Extensive Games’, International Journal
of Game Theory, 4 (1975), 25–55).
35
If Ui(z) ⫽ z for both parties, then probabilities 6a–d read: p ⫽ (x ⫺ y)/(x ⫺ y), r ⫽ (tx ⫺ y ⫺ )/
s ⫽ {(1 ⫺ y) ⫺ (1 ⫺ (tx ⫺ )1/)}
and
q ⫽ {(1 ⫺ ) ⫺ (1 ⫺ x) ⫹ (1 ⫺ (tx ⫺ )1/)}/
t(x ⫺ y),
1/ {(1 ⫺ )(1 ⫹ d) ⫺ t(1 ⫺ x) ⫹ (1 ⫺ (tx ⫺ ) ) }. Probability s ⫽ 0 now obtains when y ⫽ tx ⫺ . Overall,
when the rivals are risk averse, it takes a lower counter-offer from the defender for the challenger to accept it for
certain. The risk-averse challenger tends to be more cautious and this shows up in the impact of risk aversion on
model outcomes (see fn. 40).
34
656
LANGLOIS AND LANGLOIS
Fig. 2. Equilibrium demands and offers
is the best that the challenger can hope for, despite the fact that the defender’s offer falls
short of her demand x. But will the defender offer this much? If the defender chooses to
make an offer y ⬍ tx ⫺ in response to the challenger’s demand of x, then with strictly
positive probabilities following 6a–e, the challenger will reiterate her demand x, the
defender will resist that demand, and the challenger will escalate the conflict to war.
Moreover, once at war, defender and challenger will decide to continue fighting with
strictly positive probability. When the defender responds to demand x with an offer y that
falls short of (tx ⫺ ), costly fighting becomes a possible outcome, but the challenger could
also accept the defender’s offer. So what counteroffer will the defender make when faced
with the challenger’s demand x?
The discounted expected payoff to the defender of making any offer within the range
[0,tx ⫺ ], at decision point BD is the same, and the defender has no incentive to avoid
war. In fact the defender’s discounted utility is entirely determined by the challenger’s
demand, and he could just as well decide to counter any demand x by offering the status
quo outcome of y ⫽ 0 rather than any positive share of the contested asset. To be sure, this
increases the probability of war, but the possibility that the challenger could eventually
submit, accepting to remain at the status quo, balances out the negative utility consequences
of war. There is no settlement that the defender prefers strictly to war, because war is
intimately linked to the possibility of giving up less than he would need to give up to avoid
it.
In contrast to the war equilibria based on the threat of reversion to extremal equilibria,
the rivals do not agree on what the bargain should be in the MPEs described in 6a–e. A
deterministic outcome is not in plain sight. In fact the MPEs of 6a–e are based on the
When Fully Informed States Make Good the Threat of War
657
players’ expectation of the extent of their disagreement, and they calibrate the reciprocal
threat of protracted fighting to incite the other side to accept unfavourable terms. For sure,
outcomes will critically depend on the defender’s choice of strategy given a challenge x.
A tough defender will credibly threaten to resist and fight with high probability, while a
determined and greedy challenger will credibly threaten to escalate the conflict to war. By
adopting tough bargaining positions and behaving according to the probabilities set forth
in 6a–e, our rivals also implicitly threaten that the outcome of disagreement will be
catastrophic with some probability. By opening up the possibility of escalation to war, the
rivals threaten possibly protracted fighting and war costs that accumulate beyond the value
of the prize. Such an outcome could not be credibly threatened in a pure strategy
equilibrium. But, the very fact that the rivals are willing to fight costly battles to get what
they want cuts both ways: the high probability of costly conflict can perhaps deter the
challenger from pressing the issue further, or the defender from resisting rather than giving
in to the challenger’s demands. In either case one rival gets his or her way, an outcome
that the other is unwilling to agree to at the outset.
The equilibria we have just described are based on rational but stubborn bargaining
positions but they can be modified to accommodate adjustments in demands and offers.
For instance, the methods of Theorem 2 can be generalized to a sequence of demands xn
and offers yn within the region of Figure 2. The probabilities of 6a–e then become sequences
sn, pn, qn and rn.36 However, unless the rivals pick a starting position (x0, y0) such that
counter-offer y0 is immediately acceptable to the challenger (y0 ⫽ tx0 ⫺ ), the crisis will
escalate to war with positive probability. The calculated probabilities of various events may
be different from those obtained in the simpler case that we focus on, but the substantive
result remains: the fully informed rivals can fight to get what they want. Moreover, because
the defender’s motivation to drag his feet remains whole, there is no reason to assume that
any bargaining position in the sequence would have the defender offer enough for the
challenger to accept with certainty. And unless further behavioural assumptions are
introduced, equilibrium conditions alone will not determine the sequence that the rivals
could choose. The introduction of behavioural assumptions that would lend structure to
the possible sequence of rival bargaining positions in equilibrium would take the next
modelling step. In particular, rivals could respond to prior developments in various ways.
After an episode of fighting, accommodating leaders might seek to narrow their
differences, but states might also harden their positions in the face of accumulating war
costs in an effort to recoup past losses. The merits of such refinements notwithstanding,
they do not alter the substantive nature of the outcomes predicted by the MPEs of 6a–e.
As long as the rivals adopt incompatible bargaining positions at any point, the crisis can
escalate to war and wars can be fought repeatedly.
The equilibria that we describe involve probabilistic moves. Does this mean that states
in our model must make decisions on the flip of a coin? Such an interpretation is
unnecessary and misleading. Moreover, as we mentioned earlier, if war is costly enough
it can only occur as a result of credible probabilistic threats (see Proposition 1 in the
Appendix). In fact, ours is at least in part a deterrence model for which the use of
probabilistic threats is widely accepted.37 Schelling, in his discussion of the randomization
36
Details of this generalization are available from the authors upon request.
A probability of war is also frequently the equilibrium outcome of incomplete information games that
describe bargaining in the shadow of conflict. For example, Banks describes an asymmetric information game for
which the probability of war in equilibrium increases with the informed player’s payoff from war (Jeffrey S. Banks,
37
658
LANGLOIS AND LANGLOIS
of promises and threats, writes: ‘it is interesting to notice that attaching a probability of
fulfillment to our threat is … substantially equivalent to scaling … the size of the threat.’38
The probabilistic moves chosen by our rivals serve the same purpose. They express
expectations on the size of the threats and the timing of their execution. As a result of
equilibrium play, the rivals can avoid fighting altogether by backing down or submitting
to the other’s demands at critical decision points. But they can also fight so much that the
accumulated costs of war for each side exceed the value of holding the prize in perpetuity.
These and all intermediate events occur with some probability. Seen in this light,
equilibrium play, according to 6a–d, is a statement about the risks and threats associated
with a bargaining position. It is not about going to war on the roll of a die. The numerical
examples of the next section will further illustrate this point.
TO PREVAIL OR TO LOSE BUT AT WHAT COST?
Choosing a Bargaining Position
If the rivals maintain incompatible bargaining positions, they risk costly fighting.
However, compromise at the bargaining table does not reduce the risk of costly war
commensurately, leaving both challenger and defender with strong incentives to adopt
tough bargaining positions. This point finds illustration in the calculations of Table 1,
which illustrate the consequences of equilibrium behaviour as described in 6a–e. We fix
the costs of battle to challenger (cC) and defender (cD) to 3, and the discount rate at
⫽ 0.95.39
Given an expected response y by the defender, the challenger can choose from a range
of strategic options, from the timid to the tough. However, if the defender is expected to
stand firm, choosing y ⫽ 0, the calculations reported in bold in Table 1 illustrate that the
challenger’s chances of obtaining the share x that she demands goes down as she becomes
less aggressive. Given parameter values, a timid demand of 16 per cent of the contested
asset (x ⫽ 0.16) would actually be met by an equilibrium counter-offer of y ⫽ 0 that the
challenger would then simply accept. This is minimum demand /t. But as the challenger
becomes bolder increasing her demand to x ⫽ 0.5, and x ⫽ 1, her chances of actually
obtaining such outcomes increase, but so does the probability of costly war. In all but the
most timid of demands, costly war is possible, but the defender is more likely to submit
to the higher demand of x ⫽ 1 than to anything lower.
As expected, the probability that war is avoided increases as the gap between the rivals’
demand and offer narrows. War is avoided with certainty under two sets of circumstances:
the challenger asks for too little and immediately accepts the defender’s counter-offer to
maintain the status quo, or the defender agrees to offer a large enough share of the contested
asset so that the challenger chooses to accept immediately rather than risk costly war. But
(F’note continued)
‘Equilibrium Behavior in Crisis Bargaining Games’, American Journal of Political Science, 34 (1990), 599–614).
Our mixed equilibria also predict a probability of war, but in contrast to Banks, we specify the escalation process
by which war occurs and it is the escalation process itself that generates the war probability in equilibrium. Banks,
instead, assumes an a priori theoretical relationship between player choices and the probability of war.
38
Thomas C. Schelling, The Strategy of Conflict (Cambridge, Mass.: Harvard University Press, 1960, 18th
printing 2002), p. 182.
39
To be sure the costs of engaging in battle will impact model outcomes. For example, battle costs will be
instrumental in determining the value of the counter-offer that the challenger accepts with certainty. However,
the patterns that we have identified assuming cC ⫽ cD ⫽ 3 cut across the range of battle costs that the model admits.
When Fully Informed States Make Good the Threat of War
TABLE
1
659
The Impact on Model Outcomes of Rival Bargaining Positions
Challenger’s options
(y ⫽ 0)
x⫽1
s: Probability challenger demands
x at BC
p: Probability defender resists
at CD
q: Probability challenger escalates
at CC
r: Probability defender fights
at WD
Is War Avoided?
Probability that war is avoided
x ⫽ 0.5 x ⫽ 0.16
Defender’s options
(x ⫽ 1)
y⫽0
y ⫽ 0.5 y ⫽ 0.84
0.842
0.684
0
0.842
0.684
0
1.000
1.000
1
1.000
0.952
0.790
0.667
0.583
0.526
0.667
0.667
0.667
0.842
0.684
0
0.842
0.652
0
0.220
0.442
1
0.220
0.445
1
0.601
0.399
1
0
0.439
0.561
0.513
0.487
1
0
2.17
0
5.33
1.87
0
1.26
0
3.56
1.25
0
Whose bargaining position prevails?
Probability challenger accepts y 0.439
Probability defender accepts x
0.561
How long will it all last?
Expected length of the crisis
5.33
Expected number of turns of
war
3.56
any counter-offer less generous comes with the risk of costly warfare. The defender’s
choice for a bargaining position has subtle implications. The italicized figures in Table 1
are of particular relevance. Faced with demand x ⫽ 1, the defender accompanies tough
counter-offer y ⫽ 0 with the threat that he will resist for certain if the challenger insists
(p ⫽ 1) and will choose to fight rather than surrender in case of war with 84.2 per cent
probability (r ⫽ 0.842). As a result, the challenger will accept the defender’s counter-offer
of y ⫽ 0 with 43.9 per cent probability. But interestingly, as the defender softens his
position, offering y ⫽ 0.5, he also tones down the deterrent threats embodied in
probabilities p and r so that the challenger’s probability of accepting this far more generous
offer increases less than commensurately with the defender’s generosity. The challenger
accepts an offer of half the contested asset with 51.3 per cent probability only. These results
remain valid if we assume that the rivals are risk averse.40
The defender faces the following dilemma: he can avoid war by giving up most of the
contested asset at the outset, but if he decides to offer less and risk costly warfare, then
Assuming Ui(z) ⫽ z for both parties, we re-computed the statistics of Table 1 for various values of . Risk
aversion essentially deters the challenger from reiterating her demand at BC (probability s decreases) while the
defender is, by and large, just as willing to resist at CC and to fight at WD. For example, given cC ⫽ cD ⫽ 3 and
⫽ 0.95 as in Table 1, if the protagonists pick bargaining position x ⫽ 1, y ⫽ 0, then as decreases to 0.8,
probability s that Challenger reiterates her demand at BC drops to 73.1 per cent while she would have done so
with 84.2 per cent probability had she been risk neutral. Overall, as risk aversion increases, the challenger accepts
the defender’s counter-offer with increasing probability, war is avoided with increasing probability, and the
expected length of the crisis is reduced. A full numerical exploration of the impact of risk aversion, is available
from the authors upon request.
40
660
LANGLOIS AND LANGLOIS
the probability that the challenger will accept that offer does not increase significantly
unless he is willing to give up more than half of the contested asset. And the probability
that war will be avoided exhibits similar behaviour. Such an analysis of the situation could
well encourage the defender to adopt a tough stance, refusing to accommodate a challenger
whose incentive, ex ante, is to be aggressive.41 Do real world rivals fail to compromise,
backing their tough bargaining positions with the credible threat of war? Consider the
Vietnam war: the failure of the 1965 peace initiatives to end the nascent war has been
attributed to ‘Hanoi’s intransigence’ by Defense Secretary Robert McNamara42 and the
rigidity of the Johnson administration’s bargaining position by authors such as Porter and
Kolko.43 Moreover, uncompromising bargaining positions went hand in hand with the
willingness to fight. For example, General Giap’s announced strategic objective, according
to Karnow,44 ‘was to continue to bleed the Americans until they agreed to a settlement that
satisfied the Hanoi regime.’ As far as North Vietnam was concerned, the costs of war would
force the United States to accept peace on Hanoi’s terms.
Similar patterns can be observed in less dramatic struggles for strategic territory.
Argentina relentlessly challenged Chile for possession of islands in the Beagle Channel.
Each side declared its willingness to fight rather than compromise. ‘Sovereignty is not to
be negotiated,’ declared Argentine junta member General Viola in August of 1978, while
foreign minister Cubillo summarized Chile’s point of view in the following terms: ‘We
are well accustomed to the threats that come from the Argentine side, and they will not
change for anything our position on this matter’.45 The two countries escalated the dispute
militarily on ten occasions between 1952 and 1982 before the dispute was finally settled
in Chile’s favour in 1984.46 Overall, Huth notes that challengers do not compromise when
strategic territory is at stake,47 and Singer reports that fewer than thirty cases of
accommodation are to be found among the roughly six hundred militarized interstate
41
Interestingly, allowing for decisive battle wins does not encourage fighting. Numerical simulations, available
from the authors upon request, show that the impact of a decisive win in battle is limited to relatively small changes
in the calibration of outcomes. Nevertheless, the fact that deterrence succeeds with higher probability when wars
can be won outright is noteworthy, as it confirms our central finding: perfectly informed rivals wield deterrent
threats to gain or maintain possession of the contested asset, and make good on these threats by engaging in costly
warfare. The possibility of winning outright emboldens challenger and defender once war is the next possible step,
and this encourages the challenger to accept the defender’s counteroffer before escalation. Powell points out that
a war as ‘costly lottery’ assumption may not be neutral (Robert Powell, ‘Bargaining Theory and International
Conflict’, Annual Review of Political Science, 5 (2002), 1–30). In fact, Wagner argues that the assumption is
distorting and ‘can only lead to misleading conclusions’ (Wagner, ‘Bargaining and War’, p. 469). Our analysis
suggests that the ‘costly lottery’ assumption reinforces incentives that are present when wars merely impose costs.
42
Robert K. Brigham, ‘Vietnamese–American Peace Negotiations: The Failed 1965 Initiatives’, Journal of
American–East Asian Relations, 4 (1995), 377–94.
43
Gereth Porter, A Peace Denied: The United States, Vietnam and the Paris Agreement (Bloomington: Indiana
University Press, 1985); Gabriel Kolko, Anatomy of a War: Vietnam, the United States and the Modern Historical
Experience (New York: Pantheon Books, 1985).
44
Stanley Karnov, Vietnam, A History (New York: Penguin Books, 1983), p. 549.
45
Thomas Princen, ‘Beagle Channel Negotiations’ (Pew Case 401, Institute for the Study of Diplomacy. School
of Foreign Service, Georgetown University, 1995).
46
Paul Huth, Standing Your Ground: Territorial Dispute and International Conflict (Ann Arbor: University
of Michigan Press, 1996), p. 197.
47
Huth, Standing Your Ground, p. 146.
When Fully Informed States Make Good the Threat of War
661
disputes that occurred between 1816 and 1985.48 Perhaps some elements of the above logic
drive the aggressive bargaining positions noted in the literature.
The Cost of Failing to Agree
Paul Kennedy’s historical account of the rise and fall of the Great Powers makes frequent
reference to the devastating impact of the costs of warfare.49 ‘After each bout of fighting,
most countries desperately needed to draw breath, [and] to recover from economic
exhaustion,’ Kennedy writes of the eighteenth-century European Great Powers.50 How
costly is the failure to agree to the peaceful bargain in which the defender offers y ⬍ tx ⫺ in response to the challenger’s demand x? With strictly positive likelihood, equilibrium
play could lead our rivals to accumulate war costs in excess of the discounted value of
holding the prize forever. An aggressive, but rational, challenge can therefore lead to
catastrophic war for both sides in our model, and by the same logic challenger and defender
can hope to see the other accept his own terms with little if any fighting. But it is this array
of possible futures, and their likelihood, that defines the deterrent threat that the rivals wield
against each other when choosing to maintain incompatible claims on the contested asset.
While the expected number of turns of war keeps expected accumulated battle costs
contained, this expectation says little about the possible futures that a crisis could take as
our rivals implement their equilibrium strategies. A challenger, faced with an uncompromising defender, can end up with a strictly negative payoff if she makes good on the threat
of violence only to finally accept the status quo. For the defender, the situation is more
complex, since he enjoys possession of the contested asset as long as the crisis lasts.
However, the rivals could fight so much that even if the challenger accepts the status quo,
the defender could have accumulated net war costs in excess of the discounted value of
the prize. Yet the defender could end up keeping the contested asset if the challenger
accepts a counter-offer of y ⫽ 0 and end up with a positive payoff, even if some fighting
has taken place. For the challenger, experiencing a positive payoff outcome requires that
the defender submit to his demands without fighting too much. The probabilities that one
rival or both end up with strictly negative (or positive) payoffs cannot be calculated
analytically. We therefore estimate these probabilities using the Monte Carlo method. We
estimated the likelihood of various payoff events excluding path 1 ⫽ {I}, in which the
challenger immediately accepts the defender’s offer. P(1) can be calculated explicitly. By
estimating the likelihood of 1 separately, we were able to compare one of our probability
estimates to a calculated true value. In all cases, our 95 per cent confidence interval for
the likelihood of 1 contained the true value, and this bolstered our confidence in the
interval estimates of the likelihoods of other payoff events of interest.51
48
David J. Singer, ‘The Appeasement Option: Past and Future’, in Melvin Small and Otto Feinstein, eds,
Appeasing Fascism: Articles from the Wayne State University Conference on Munich After Fifty Years (Lanham,
Md.: University Press of America, 1991).
49
Paul Kennedy, The Rise and Fall of the Great Powers (New York: Random House, 1987).
50
Kennedy, The Rise and Fall of the Great Powers, p. 85.
51
Having consulted Chistopher Mooney, Monte Carlo Simulation (Thousand Oaks, Calif.: Quantitative
Applications in the Social Sciences Series, Paper 116, Sage, 1997), and Ilya Sobol, A Primer for the Monte Carlo
Method (London: CRC Press, 1994), we implemented the Monte Carlo method as follows: for each set of parameter
values we generated 20,000 paths by simulating the rivals’ equilibrium play. We calculated the players’ ex ante
utilities for each path. We then separated all instances of path 1. The remaining paths were sorted into four
662
LANGLOIS AND LANGLOIS
Estimates of the likelihood of catastrophic war are reported in bold in Table 2. The
likelihood of catastrophic war reaches a mean of 19.3 per cent if the rivals hold tough
bargaining positions (x ⫽ 1, y ⫽ 0) and war costs are relatively low, the conditions of
Case 1. But what is perhaps more striking about Case 1 is the high likelihood of a strictly
negative expected utility outcome for each of the rivals. Challenger and defender could
end up in this predicament with mean likelihoods of 38.4 per cent and 57.7 per cent
respectively.52 On the other hand, the challenger could immediately accept the defender’s
status quo offer with 15.6 per cent probability P(1), a boon to the defender. And the
challenger is at least as well off for having challenged with 61.6 per cent likelihood.53 The
downside risk associated with a tough bargaining position is significant for both rivals. But
it seems especially high for the defender. While the challenger has a better than even chance
of remaining at least as well off for having challenged, the defender who chooses to respond
harshly has a more than even chance of experiencing a strictly negative outcome. Higher
battle costs for both parties essentially increase the probability of an immediate acceptance
of the status quo offer by the challenger (P(1) ⫽ 0.526) and this works in the defender’s
favour as illustrated in Case 2. But by lowering the downside risk for both parties, high
battle costs cannot necessarily be expected to prevent a challenge in the first place.
With low war costs, the perspective of a strictly negative outcome might encourage the
defender to adopt a softer bargaining position. We illustrate the consequences in Case 3.
If he is willing to offer as much as one half of the contested asset in the face of a challenge
for all of it, the defender reduces the likelihood that he will end up with a strictly negative
outcome to 44.2 per cent from 57.7 per cent. But this possibility is surely mitigated by the
prospect of handing over half the asset on demand with probability P(1) ⫽ 0.316. It is the
challenger who should welcome the defender’s softer stance, since she can now expect to
be at least as well off as a result of a challenge with a 94.3 per cent likelihood. And the
challenger seems to have little incentive to be less greedy. As Case 4 reveals, a challenge
for 75 per cent of the asset can still leave her strictly worse off with 14.6 per cent likelihood,
even if the defender is willing to be somewhat accommodating by offering y ⫽ 0.2. And,
at best, since no decisive battle win is to be expected, the challenger would end up with
only 75 per cent of the asset instead of getting all of it.54
In an essay on the comparative merits of theory and case studies Robert Jervis writes:
‘The choice between the deductive approach and one that builds on case studies involves
a tradeoff between rigor and richness. A deductive theory must miss many facets of any
individual case.’55 Our theoretical approach to war and bargaining is no exception.
Nevertheless, our rivals threaten a range of outcomes that are observed in real world
settings. An aggressive Iraq, contesting the 1975 Algiers Accord that settled sovereignty
(F’note continued)
groups: paths for which both parties receive strictly negative payoffs; paths for which the players received positive
payoffs; paths for which one or the other receives a strictly negative payoff. Frequencies for each of the five events
were calculated. This process was repeated 1,000 times generating a distribution for the frequencies of each event.
In all cases, with 95 per cent confidence, we were not able to reject the null hypothesis of a normal distribution
(Jarque Bera statistic ⬍ 5.99). We were therefore able to construct 95 per cent confidence interval estimates of
the likelihoods of each of the five events by adding and subtracting 1.96 standard deviations to each of the means.
52
We add the means in the ⬍ 0 column for the challenger and the ⬍ 0 row for defender.
53
Adds to the ⱖ 0 column for challenger P(1), which leaves her with utility 0.
54
The possibility of winning the asset in battle does not significantly reduce the downside risk for either party.
Estimates of the likelihood of various payoff events when battles can be won decisively are available from the
authors upon request.
55
Robert Jervis, ‘Rational Deterrence: Theory and Practice’, World Politics, 41 (1989), 183–207, p. 184.
When Fully Informed States Make Good the Threat of War
TABLE
2
663
Means and 95 Per Cent Interval Estimates of the
Likelihood of Catastrophic War and Other Events
No decisive battle win, u ⫽ v ⫽ 0; tough rivals x ⫽ 1, y ⫽ 0; costs vary.
Case 1: cC ⫽ 3, cD ⫽ 3
Challenger utility
ⱖ0
⬍0
ⱖ0
[0.070, 0.078]
Mean 0.074
[0.186, 0.196]
Mean 0.191
⬍0
[0.377, 0.391]
Mean 0.384
[0.187, 0.198]
Mean 0.193
Defender utility
P(1) 苸 [0.153, 0.163] True value 0.158
Case 2: cC ⫽ 10, cD ⫽ 10
Challenger utility
ⱖ0
⬍0
ⱖ0
[0.138, 0.147]
Mean 0.143
[0.060, 0.066]
Mean 0.063
⬍0
[0.169, 0.180]
Mean 0.174
[0.86, 0.094]
Mean 0.090
Defender utility
P(1) 苸 [0.519, 0.533] True value 0.526
No decisive battle win, u ⫽ v ⫽ 0; costs, cC ⫽ 3, cD ⫽ 3; bargaining positions vary.
Case 3: x ⫽ 1, y ⫽ 0.5
Challenger utility
ⱖ0
⬍0
ⱖ0
[0.211, 0.222]
Mean 0.217
[0.024, 0.028]
Mean 0.026
⬍0
[0.403, 0.417]
Mean 0.410
[0.29, 0.034]
Mean 0.032
Defender utility
P(1) 苸 [0.309, 0.323] True value 0.316
Case 4: x ⫽ 0.75, y ⫽ 0.2
Challenger utility
ⱖ0
⬍0
ⱖ0
[0.480, 0.494]
Mean 0.487
[0.083, 0.090]
Mean 0.086
⬍0
[0.075, 0.083]
Mean 0.079
[0.057, 0.064]
Mean 0.060
Defender utility
P(1) 苸 [0.281, 0.293] True value 0.287
664
LANGLOIS AND LANGLOIS
over the Shatt Al Arab, could result in eighty-three months of war with Iran.56 The rivals’
aggressive posture in this dispute led to a war whose direct and indirect costs rose to ‘the
astronomical figure of $1,190 billion’.57 Was sovereignty over the Shatt Al Arab waterway
worth this much? For sure, wider issues of Arab nationalism and religious fundamentalism
added much fuel to a raging fire. But surely war costs accumulated beyond the value of
the contested territory. By contrast, the dispute between Mali and Burkino Faso over 500
square miles of territory including the mineral rich Agacher strip lasted for seventeen years
but escalated to war on only two brief occasions. Throughout the period, Burkino Faso held
mineral-rich territory of value. And Mali’s willingness to challenge and escalate the
conflict to war eventually led to an even distribution of the contested territory in 1987.58
It is probable that neither side fought more than the contested asset was worth if exploited,
and the defender (Burkino Faso) was able to keep possession for a very long while.
CONCLUSION
The probabilistic war equilibria that we exhibit illustrate how a defender’s vested interest
in maintaining the status quo forces the challenger to resort to the credible threat of war
in order to secure any share of the prize. Indeed, as long as the challenger decides to demand
a share of the contested asset, backing down in the face of the defender’s refusal to satisfy
her demand achieves nothing. She must therefore credibly threaten to fight with at least
some probability if she is to maintain her claim. When challenged, the defender will attempt
deterrence by also threatening to fight, hoping that the challenger will not press on with
her demands. But each side faces a dilemma: the defender could ensure peace by offering
a high enough share of the prize. But lower offers that involve the risk of war might be
accepted and leave the defender enjoying the prize for as long as the dispute lasts. In an
expected sense, the defender is indifferent between the rational counter-offers that he can
make given the challenger’s demand. Symmetrically, the challenger could demand little
and obtain it with high probability and no fighting. Or she could ask for more and obtain
it, but given the defender’s counter-offer, she must expect to fight for it with higher
probability. In the equilibria that we describe, the players’ ability to threaten each other
credibly does not help them narrow their differences. Instead, credible threats of war
sustain incompatible bargaining positions. It is rational to demand more as challenger or
to offer less as defender, although a more accommodating stance by either party would
reduce the risk of costly conflict. Yet, the rivals’ bargaining choices have predictable
consequences. Model outcomes depend on parameter values and on the possibility of
decisive battle wins, but, importantly, they depend on the rivals’ bargaining positions. The
wider the gap between defender and challenger, the lower the probability that deterrence
will succeed and the higher the probability of war. But as the probability of war increases,
so does the likelihood that the rivals will fight so much that they will each accumulate war
costs in excess of the value of keeping or acquiring the prize. And this happens with
significant likelihood when the two sides are sufficiently far apart. Nevertheless, in
choosing a bargaining position, each party finds that the prospect of a dire outcome is
balanced out by the possibility that the crisis will end swiftly and on his or her own terms.
56
Huth, Standing Your Ground.
Dilip Hiro, The Longest War; The Iran–Iraq Military Conflict (New York: Routledge, Clapman and Hall,
1991), p. 1.
58
Huth, Standing Your Ground, p. 220.
57
When Fully Informed States Make Good the Threat of War
665
APPENDIX
The challenger is denoted C and the defender D. Moves and states are those of Figure 1. Lemma
1 shows how all expected utility calculations can be reduced from sequences of moves to payoff
paths. It is used in Corollary 1 and Theorem 2.
LEMMA 1: For any node N if move mk ⫽ NMk has probability pk expected utility Ei(N) satisfies (using
the notations of Formulae 2 and 3 in the text):
冘 p E (M )
E (N) ⫽ U (N) ⫹ 冘 p E (M )
Ei(N) ⫽
k i
if N is a decision node
k
k
i
i
k i
if N is a payoff node with payoff Ui(N)
k
k
Proof: Any move sequence k following N reads k ⫽ mkk for some remaining move sequence k
and probabilities satisfy P(k) ⫽ pkP(k). If N is a decision node no payoffs are made at N and
expectations at the next nodes Mk are not discounted. Thus, Vi k ⫽ Vi k and expectations read
Ei(N) ⫽
冘 p P( )V ⫽ 冘 p E (M ).
k
k, k
k
k
i
k i
k
k
If N is a payoff node, i receives Ui(N), discounts the future, and Vi k ⫽ Ui(N) ⫹ Vi k. Thus
Ei(N) ⫽
冘 p P( )(U (N) ⫹ V ) ⫽ U (N) ⫹ 冘 p E (M )
k
k
i
k
i
k,k
i
k i
k
k
Q.E.D.
COROLLARY
1: At node IV, expectations can be written with Formula 1:
ED(IV) ⫽ ⫺ d ⫹ tED(WD)
and
EC(IV) ⫽ ⫺ c ⫹ tEC(WD)
Proof: By Lemma 1, ED(IV) ⫽ 1 ⫺ cD ⫹ (tED(WD) ⫹ v/(1 ⫺ ) ⫹ u.0) (similarly for C). Q.E.D.
THEOREM
1: ␧C forms an extremal SPE for the challenger and ␧D forms an extremal SPE for the
defender.
Proof: We can easily verify that the expected utilities at all decision nodes resulting from ␧C are
0 for the challenger and 1/(1 ⫺ ) for the defender. It is then easy to verify that the given moves
are optimal for each player. At CC, for instance, escalating yields ⫺ c ⫹ 0 ⬍ 0, strictly less than
backing down for the challenger. This is extremal for the challenger, since he can always guarantee
himself a non-negative outcome by accepting any y and choosing to backdown at CC and WC. We
can verify that the expected utilities at all decision nodes resulting from ␧D are given by ⬍ 1/(1 ⫺ ),
0 ⬎ as long as no backdown has occurred (since there is a reversion to ␧C and a resulting change
of expectations in that case). It is then easy to verify that the given moves are optimal for each player.
At CC, for instance, escalating yields ⫺ c ⫹ t/(1 ⫺ ) ⱖ 0 for the challenger. And since backdown
would yield the expectation 0 by ␧C, escalation is best. This is extremal for the defender, since she
can always guarantee herself a non-negative outcome by submitting at BD and CD, and surrendering
at WD.
Q.E.D.
PROPOSITION 1: Assume Condition 5 in the text. Then the strategy profile P forms a SPE. Moreover,
if Condition 6 holds there exists no SPE such that war occurs deterministically.
Proof: The expected utility for C of escalation at CC is ⫺ c ⫹ 2tz/(1 ⫺ ) that must be no less than
the expected 0 of backdown resulting from reversion to ␧C. The expected utility for D of resisting
at CD is ⫺ d ⫹ t(1 ⫹ (1 ⫺ z)/(1 ⫺ )) that must be no less than the expected 0 of submit or
reversion to ␧D. These two conditions form Condition 5. We can easily verify that all other moves
in P are optimal given the expectation P and reversion to the extremal equilibria. Moreover, if there
666
LANGLOIS AND LANGLOIS
exists an SPE such that escalate occurs deterministically at CC, then the two sides’ expected payoffs
at CC must satisfy:
⫺ c ⫹ tEC(WD) ⱖ EC(III) ⱖ 0
and
⫺ d ⫹ tED(WD) ⱖ ED(II) ⱖ 0,
since C can ensure at least 0 at III and D can ensure at least 0 at CD in any SPE. Thus:
c ⫹ d ⱕ t(EC(WD) ⫹ ED(WD)) ⱕ t/(1 ⫺ )
since the two sides’ expected payoffs at WD cannot add up to more than the discounted value
1/(1 ⫺ ) of the whole prize. This last condition is incompatible with Condition 6.
Q.E.D.
2: Assume that ⫽ (1 ⫺ )c/ ⬍ 1. Let x and y, both in [0,1], be such that tx ⱖ and
y ⱕ tx ⫺ . Then the following set of moves forms a MPE:
THEOREM
At the Start node, C demands x and D offers y at BD. Thereafter, denoting C’s last demand by x
and D’s last offer by y:59
At BD, D chooses: never accept x but (with probability z ⫽ 1) offer:
⎧⎪y
(x,y) ⫽ ⎨tx ⫺ ⎪⎩0
if y ⱕ tx ⫺ ;
if ⱕ tx and y ⬎ tx ⫺ ;
if tx ⬍ .
At BC, C chooses: (a) accept y if y ⱖ tx ⫺ ; otherwise (b):
(i) demand x with probability s ⫽
tx ⫺ y ⫺ ; (ii) accept y with probability 1 ⫺ s.
x⫺y
At CD, D chooses: (a) resist if tx ⬍ ; otherwise (b):
(i) resist with probability p ⫽
x ⫺ (x,y)
; (ii) submits with probability 1 ⫺ p.
x ⫺ (x,y)
At CC (or WC), C chooses (a) backdown if tx ⬍ ; otherwise (b):
(i) escalate (fight) with probability q ⫽
(1 ⫺ )c ⫹ (1 ⫺ t)x
;
(1 ⫺ )(c ⫹ d) ⫹ 1 ⫺ t
(ii) backdown with probability 1 ⫺ q.
At WD, D chooses: (a) fight if tx ⬍ ; otherwise (b):
(i) fights with probability r ⫽
tx ⫺ (x,y) ⫺ ; (ii) surrenders with probability 1 ⫺ r.
t(x ⫺ (x,y))
Proof: To prove equilibrium, we verify that all demands, offers, moves and probabilities stipulated
above are best at each decision node when used as expectations of future play. We can easily verify
that all formulas for {s, p, q, r} are probabilities.60 We next verify that the specified moves and
probabilities are best given expectations and we finally check the optimality of the demands and
offers.
We begin with the case where y ⱕ tx ⫺ . Since the very same x and y are expected at the next
bargaining turn, we first obtain the expected utilities associated to x, y and the probabilities s, p, q
59
These might be different from the initial ones. To substantiate a claim of perfect equilibrium one must allow
any prior development, not just prior moves conforming to equilibrium behaviour. So, whether optimal or not,
prior offers x and y may have been made.
60
s and r are 0 when y ⫽ tx ⫺ and p reduces to 1 when ⫽ 0. Otherwise, all are in (0, 1).
When Fully Informed States Make Good the Threat of War
667
and r. We observe that these probabilities define a Markov chain on the states I, II, III and IV in
Figure 1. The transition between these states is described by the matrix:61
⎛ 1
⎜ 0
T⫽ ⎜
⎜1 ⫺ s
⎝ 0
0
0
1
0
s(1 ⫺ p)
sp(1 ⫺ q)
t(1 ⫺ r)
tr(1 ⫺ q)
⎞
0 ⎟
⎟
spq
⎟
trq ⎠
0
.
It is thus convenient to obtain the expected payoffs at all (payoff) states using linear algebra
techniques. The payoff vectors at these states read:
⎛
⎜
UC ⫽ ⎜
⎜
⎝
⎞
x ⎟
⎟
0
⎟
⫺c ⎠
⎛ 1⫺y ⎞
⎜ 1⫺x ⎟
UD ⫽ ⎜
⎟
⎜ 1 ⎟
⎝ ⫺d ⎠
y
,
.
According to Lemma 1, the expected payoffs satisfy Ei ⫽ Ui ⫹ TEi (for i ⫽ C and D) when p, q,
r and s are replaced (in T ) by the expressions given in the statement of the theorem. It is easy to
check that these expectations must then read:
1
EC ⫽
1⫺
⎛y⎞
⎜x⎟
⎜ ⎟
⎜ y ⎟
⎝ y ⎠
and
1
ED ⫽
1⫺
⎛
⎞
1⫺y
⎜
⎟
1⫺x
⎜
⎟
1 ⫺ tx ⫹ (1 ⫺ )c
⎜
⎟
⎝ t(1 ⫺ x) ⫺ (1 ⫺ )d ⎠
.
The given probabilities will indeed be optimal at each decision node if the deciding player is
indifferent in the expected payoff sense between the moves that are weighed by these probabilities.
This results from replacing p and q by their expressions in terms of x and y and arguing as follows:
At BC, C can use s optimally since EC(I) ⫽
y
⫽ (1 ⫺ p)EC(II) ⫹ pEC(III);
1⫺
At CD and WD, D can use p and r (respectively) optimally since
ED(II) ⫽
1⫺x
⫽ (1 ⫺ q)ED(III) ⫹ qED(IV);
1⫺
At CC and WC, C can use q optimally since EC(III) ⫽ EC(IV) ⫽
y;
1⫺
Moreover, D’s expected payoff at BC satisfies (up to the factor (1 ⫺ )):
(1 ⫺ s)(1 ⫺ y) ⫹ s(1 ⫺ x) ⫽ 1 ⫺ y ⫹ s(y ⫺ x) ⫽ 1 ⫺ tx ⫹ ⬎ 1 ⫺ x
by definition of s. It is therefore best for D never to accept x at BD.
The case where y ⬎ tx ⫺ yields two subcases:
(a) If tx ⱖ , D is expected to offer ⫽ tx ⫺ at BD. The resulting probabilities s, p, q and r
are such that s ⫽ r ⫽ 0, but they satisfy all the optimality conditions of the case y ⱕ tx ⫺ .
61
The Markov chain actually includes states I⬘ and II⬘, and its full transition matrix should formally involve
two additional rows and columns. But I⬘ and II⬘ are absorbing states with fixed probabilities u and v of being
reached and fixed outcomes. All calculations relevant to the decisions x and y and the move probabilities can
therefore be limited to this 4 ⫻ 4 matrix for simplicity.
668
LANGLOIS AND LANGLOIS
In particular, accepting the offer y ⫽ tx ⫺ at BC and not accepting x at BD are optimal for
C and D respectively. All stipulated moves and probabilities in this subcase are therefore also
optimal.
(b) If tx ⬍ , C is expected to accept any offer at BC (since any y ⱖ 0 ⬎ tx ⫺ for such x’s. So,
D’s offer of 0 is indeed best for D at BD. Since D is expected to fight with certainty at WD,
it is clearly best for C to backdown at WC. Accordingly, D’s expected payoff from fight at
WD is 1 ⫹ /(1 ⫺ ) ⫽ 1/(1 ⫺ ) ⱖ (1 ⫺ x)/(1 ⫺ ) for any x. It is therefore best for D to fight
at WD. As a result, it is best for C to backdown, for an expected payoff of 0, rather than
escalate for an expected payoff of ⫺ c ⫹ 0. It is therefore best for D to resist at CD (since
1 ⱖ 1 ⫺ x). Consequently, any demand tx ⬍ at BC yields an expected payoff of 0 for C which
is no better than accepting any offer y from D. All stipulated moves in this subcase are
therefore also optimal.
We finally verify that the stipulated demands and offers are best given expectations.
Given offer y and demand x at BC C can make three kinds of demands :
(a) demand such that y ⱕ t ⫺ . This is expected to yield probabilistic play and an expected
payoff of y at BC;
(b) demand such that t ⬍ . This is expected to yield resist, backdown offer (, y) ⫽ 0 and
accept (by C at BC). This demand therefore yields an expected payoff of 0, no more than
any offered y;
(c) demand such that t ⱖ , but such that y ⬎ t ⫺ .62 This is expected to yield probabilistic
play with offer (, y) ⫽ t ⫺ by D at BD followed by accept at BC. The expected payoff
of such a demand is therefore (t ⫺ k)/(1 ⫺ ) ⬍ y/(1 ⫺ ).
The best expected payoff for C at BC is therefore y and it is achieved by repeating the prior demand
( ⫽ x) when y ⱕ tx ⫺ and by accepting y when y ⬎ tx ⫺ .
Given demand x and prior offer y D at BD can make two kinds of offers :
(a) offer ⱕ tx ⫺ when tx ⱖ . C is expected to reiterate demand x at BC and to accept offer
with probability (1 ⫺ s) which, by definition of s, is such that:
(1 ⫺ s)(1 ⫺ ) ⫹ s(1 ⫺ x) ⫽ 1 ⫺ ⫹ ( ⫺ x)s ⫽ 1 ⫺ tx ⫹ So, D’s expected payoff from any offer ⱕ tx ⫺ at BD (when x ⱖ ) identically equals
1 ⫺ tx ⫹ .
(b) offer ⬎ tx ⫺ . C is then expected to accept it at BC, yielding an expected payoff to D of
1 ⫺ ⬍ 1 ⫺ tx ⫹ regardless of the sign of tx ⫺ .
The best expected payoff for D at BD is therefore achieved by repeating prior offer ( ⫽ y) if
y ⱕ tx ⫺ and, whenever y ⬎ tx ⫺ , to offer (tx ⫺ ) if tx ⱖ and to offer 0 when tx ⬍ , as specified
by Q.E.D.
PROPOSITION
2: The MPE of Theorem 1 results in the following probabilities and expectations:
h1 ⫽
pqsu
1 ⫺ (1 ⫺ q)ps ⫺ qrt
is the probability that challenger wins;
h2 ⫽
(1 ⫺ s)(1 ⫺ qrt)
1 ⫺ (1 ⫺ q)ps ⫺ qrt
is the probability that challenger submits or surrenders;
h3 ⫽
1⫺s
1 ⫺ (1 ⫺ q)ps
is the probability that deterrence succeeds;
62
This includes the case where y ⬎ t ⫺ .
When Fully Informed States Make Good the Threat of War
e1 ⫽
ps
1 ⫺ (1 ⫺ q)ps ⫺ qrt
is the expected number of turns of the game;
e2 ⫽
pqs
1 ⫺ (1 ⫺ q)ps ⫺ qrt
is the expected number of war turns.
669
Proof: In Markov chain terms, h1 is the probability of hitting I⬘ from the start, the same as from node
III. Let k1 be the probability of hitting I⬘ from node IV. One can write:
h1 ⫽ psqk1 ⫹ ps(1 ⫺ q)h1
and
k1 ⫽ u ⫹ trqk1 ⫹ tr(1 ⫺ q)h1.
Solving for h1 yields the given formula; h2 and h3 are obtained in similar fashion. The expected
number of turns from III is g1 ⫽ e1 ⫹ 1 and let f1 be that expectation from IV.
g1 ⫽ 1 ⫹ psqf1 ⫹ ps(1 ⫺ q)g1
and
f1 ⫽ 1 ⫹ trqf1 ⫹ tr(1 ⫺ q)g1.
Solving yields e1. Expectation e2 is obtained similarly.63
63
Q.E.D.
These arguments are standard in Markov chains and can be found in James Norris, Markov Chains
(Cambridge: Cambridge University Press, 1997), p. 12).
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.