How a War Ends: A Rational Model Approach
Author(s): Donald Wittman
Source: The Journal of Conflict Resolution, Vol. 23, No. 4 (Dec., 1979), pp. 743-763
Published by: Sage Publications, Inc.
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How a War Ends
A RATIONAL
DONALD
MODEL APPROACH
WITTMAN
Department of Economics
University of California, Santa Cruz
This article discusses the necessary conditions for two countries at war to come to a
settlement and explores how domestic and military costs, time preferences and attitudes
toward risk affect the timing and the outcome of the peace. It views the termination of
war as a process of rational calculations by the participants; unless both sides believe
that they can be made better off by a settlement, the war will continue. An important
result of this approach is that a reduction of hostilities may reduce the probability of a
settlement taking place and thus prolong the war. It is also shown that increasing the
probability of winning may not increase the probability of a settlement and that a country
which only values the present need not be at a disadvantage in the negotiation.
A. INTRODUCTION
This article discusses the necessary conditions for two countries at
war to come to a peaceful settlement and explores how domestic and
military costs, time preferences, and attitudes toward risk affect the
timing and the outcome of the peace. In the process some popular
explanations usually discussed solely in political terms are instead
analyzed in terms of military costs, and a number of generally accepted
views are shown to be incorrect. Because there is a great amount of
symmetry between how a war ends and how a war begins (or peace
ends), the theoretical structure can be applied equally as well to investi-
AUTHOR'S NOTE: I would like to thank Mitchell Page who is responsible for the
original conception of this paper. Only his extreme modesty prevented him from being
a coauthor. Earlier versions were presented at the 1977 annual meeting of the Public
Choice Society, New Orleans, and at the 1977 annual meeting of the American Political
Science Association, Washington, DC.
JOURNAL OF CONFLICT RESOLUTION, Vol. 23 No. 4, December 1979 743-763
( 1979 Sage Publications, Inc.
743
744
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gating the initiation of war. However, to keep the analysis tractable
and empirically more verifiable (it is hard enough to measure the costs
of a war, let alone the costs of a war which was not fought), most of the
analysis will be devoted to how a war ends.
The purpose of the paper is to create a-theoretical framework for
analyzing the termination of a war. While empirical illustrations are
presented, they are not meant to be the complete description of particular wars; rather they are meant to provide concrete understanding
of the abstract theory, to demonstrate how the theory might be applied
in analyzing empirical data, and to yield insights into a variety of
conflicts. A detailed historical or empirical investigation would encompass the body of another paper.
B. NECESSARY
CONDITIONS
FOR ENDING
THE WAR
An agreement (either explicit or implicit) to end a war cannot be
reached unless the agreement makes both sides better off; for each
country the expected utility of continuing the war must be less than
the expected utility of the settlement. I Even an unconditional surrender
is an agreement by both sides to end the war, and means that both
countries prefer peace to continued fighting. For the loser, in particular,
it means that the country prefers unconditional surrender to total
annihilation.2
In order to discuss the circumstances under which these conditions
for a peaceful settlement are met, we will first analyze each country's
expected utility if it continues the war. Strictly speaking this is the
utility function of the country's leader(s). Of course, there may be
some disagreement among the ruling elite, but typically this should
be insignificant compared to the disagreement between the belligerent
1. Many historians of warfare have failed to see that both sides must agree to end
the war. For example, Calahan (1944) claims that peace is made by the vanquished. With
regard to how a war begins, there are various theories (scapegoat, imperialist, psychoanalytic) which treat one country as being the cause of a war. There is, however, a body
of work which treats both countries as being the cause of a war (for example, see the work
of Blainey, 1973, and Carroll, 1969).
2. It should be noted that in only a handful out of hundreds of wars between nationstates has the victor shown overwhelming superiority and demonstrated the ability to
completely overrun the enemy's homeland. Thus World War 11 was something of an
anomaly (see Kecskemeti, 1958).
Wittman / HOW A WAR ENDS
745
countries. The assumption of a single decision-maker is necessary for
logical clarity.
Country X's expected utility from continuing the war depends on
the costs of the war, and the probabilities and utilities of its winning
and losing. The more utility that X derives from winning, (and) the
greater the probability that X does in fact win, and the less the costs to
X of conducting the war, the greater X's expected utility from continuing it. The expected utility of continuing the war will be denoted
by UX (w). The superscript t is present to emphasize that expected
utility calculations will change over time as information about the
outcome changes.3 These variables will now be considered in more
detail.
The costs of the war to country X (Cx) could be either military or
political; however, we will tend to concentrate on military costs (e.g.,
destruction of resources, number of soldiers killed).
If it wins, the benefits to country X in the postwar years may include
economic spoils and a sympathetic puppet government. The utility of
country X in the postwar years is clearly less if it loses. The phrases
"X wins" or "X loses" are used only as easy verbal connotations for two
states of the world. For X, winning is a preferred outcome to losing,
but "winning" and "losing" need not even be defined for the analysis.
Thus no interpersonal comparisons of utility are required. Furthermore, we could devise a whole series of possible outcomes instead of
just two. However, this would just make the verbal argument more
complex without changing the essential logic.
Pxwis country X's subjective probability of winning the war. Because
each side may have different information available to it, Pxwneed not
be identical to [I-Pyw] where Pywis country Y's subjective probability
of winning the war. Px, and Pywmay change over the course of the war
as objective circumstances affect the probability of one side winning.
Thus Px, and [1-Py,] are positively correlated with the objective
probability of winning the war. For example, after a battle in which X's
army was unanticipatedly routed, we would expect both Px, and
[I-Pyw] to decrease but not necessarily by the same amount. Thus the
factors which affect the unknown objective probability are the same
3. Porsholt (1966, 1971) has a utility maximization model, but his approach does
not take into account the costs of losing a war nor does he discuss war termination.
Rational models have been used to explain a wide variety of political, economic, and
biological behavior. For some recent applications of rational behavior to international
behavior, see Zinnes and Gillespie (1976).
746
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factors which affect the country's subjective probability. It should be
noted that C,, and Pw need not be related, for if the hostilities increase
this may not change the probabilities of either country winning the war,
but it will increase the military costs to both countries.
Because war and postwar events are not instantaneous, expected
utility from continuing the war depends on the present value of future
outcomes. In turn, present value depends on country X's discount rate.
If country X has a high discount rate, it does not weigh outcomes in
the distant future very heavily in its calculations. Thus only events in
the near future will be very critical. A low discount rate means that
outcomes in the distant future weigh almost as heavily as outcomes
in the near future.
The analysis of country Y's expected utility would proceed in the
same way.
Having analyzed the expected utility from continuing the war, we
now turn our attention to the expected utility from a negotiated settlement. The utility that each country receives from a settlement depends
on what kind of agreement has been made. Country X will obviously
derive greater satisfaction from an unconditional surrender by Y than
an unconditional surrender by X. Since the reverse is true for Y, it is not
unreasonable to assume that the better one country does in the settlement the worse the other one will do. The range of possibilities would
then run from an unconditional surrender by X to an unconditional
surrender by Y. Of course, an unconditional surrender may be totally
inconceivable, e.g., the U.S. surrendering unconditionally to North
Vietnam. In such cases we might want to redefine surrendering unconditionally to apply to what happened in Vietnam. But for our
analysis, it does not make any difference whether the ultimate goal
can be obtained, just as it makes no difference in utility theory that the
consumer can never be satiated. Points in between from right to left
might be X giving up its prewar control over colony Z, the prewar
status quo, and X taking away additional land from country Y. This
can be seen as a single continuum (see Figure 1).4 Sy is an unconditional
surrender by Y and represents the greatest utility that X can receive in
any agreement. The farther right a negotiated settlement takes place
the less utility that X receives and the more that Y receives. This can
be put into mathematical symbols. Let U' (s) be the present discounted
4. One need not even have just a simple continuum. A multidimensional negotiated
settlement is consistent without analysis as long as the settlements are on the pareto
optimal frontier for the two countries.
Wittman / HOWA
WAR ENDS
Ut-()
UT I L IOTY
UTILITYUt
(5)_
S
SETTLEMENT
I2
UTILITY
Y
UTILITY
x
UNCONDITIONAL
SURRENDER
BY X
UNCONDITIONAL
SURRENDER
BY Y
Figure 1: Utility from a Negotiated
S
747
Settlement
takes place, the greater utility for Y and the
NOTE:
The farther right the agreement
surrender
for X. S is the settlement
measure.
lesser utility
by
SX is an unconditional
of the utilities
is
is an unconditional
surrender
magnitude
X. S
by Y. The relative
affecting
up or down without
unimportant.
L.e., one or both curves could be shifted
regarding
the analysis.
Furthermore
most of the propositions
make no assumptions
Therefore
the utility curves could be caved in instead
concavity
(only monotonicity).
of bowed out.
utility that X receives from settlement s made in year t, and let U' (s)
be Y's present discounted utility. Then U' (s) is a decreasing function
of s and Uy (s) is an increasing function of s5.
We now investigate the necessary conditions for ending a war.
A necessary condition for a settlement to take place in year t is that
there exists a settlement s* such that Ut (s*) > Ut (w) and Ut (s*) >
Ut (w). I.e., the settlement must make both countries better off, than
by continuing the war.6 This can be seen in Figure 2.
5. A settlement may be tacit instead of explicitly negotiated. It is conceivable that
the utility of a settlement changes as the war progresses. E.g., the values of controlling a
country's mineral deposits are reduced if all the mining equipment has been destroyed in
the war. However, the variability of the utility from a settlement is a lot less than the
variability of the costs of the war. Furthermore, many of the changes in the utility of a
settlement can be attributed to the costs of the war. Therefore we will assume that the
utility of any settlement is constant over time.
6. Each country actually compares the expected utility of a settlement today, with
the expected utility of continuing the war for k more years (k = 1, 2 ... ) and then having a
negotiated settlement. However, U' (w) = A, is still the lower bound and thus still a
necessary condition.
Our article discusses the necessary but not the sufficient conditions for a settlement
to a war. It is entirely possible that, even if both countries can benefit by coming to a
settlement, the war will not end for each side may be trying to gain more favorable terms.
A discussion of sufficient conditions would involve unresolved issues in game theory
including bargaining strategy and the solution to the bargaining set.
JOURNAL OF CONFLICT RESOLUTION
748
U (S)
y
UTILITY
Ut
y
(W)
t
U(w)
x
=Bt
t
a
A
X\--
-
_
t
y
A_
S
Figure 2: Feasibility
of a Negotiated
~~~~~~~~~~~~~~~~~~~
S
Settlement
the war, X will
At from continuing
utility
expected
If X receives
Ut (w)
NOTE:
settleas At. Those
which gives X at least as much utility
a settlement
only accept
ment points to the left of Al give X more utility than U t (w). Y will only agree to a
to be
is to the right of Bt. Thus in order for a settlement
if the settlement
settlement
is to the right of BI and to the left of Al. In this
a point must exist which
feasible,
is feasible.
settlement
case a negotiated
X can expect utility U' (w) equal to At by continuing the war so X
will not agree to any settlement which offers country X less utility
than A, (i.e., any point to the right of A'). Y can expect utility U' (w)
equal to Bt by continuing the war so Y will not agree to any settlement
which offers Y less utility than Bt (i.e., any point to the left of B0).At is
X's minimal acceptable settlement and Bt is Y's minimal acceptable
settlement. In Figure 2 a settlement in year t is possible since an agreement can make both countries better off.7 In Figure 3 a peaceful settlement is impossible since X will agree only to those points to the left of
At and Y will agree only to those points to the right of Bt. Thus, there
is no overlap. (See the Appendix for a game theoretic formulation.) The
possibility of an agreement will now be investigated.
7. We are interested in whether a negotiated settlement is feasible and how various
circumstances affect the likelihood of a negotiated settlement being feasible; we are not
interested in what the settlement will be if it is feasible. Thus we do not provide point
estimates of the negotiated settlement nor an analysis of the bargaining strategy (the
path to the settlement within the feasible set). For this kind of analysis of the bargaining
game see the work by Nash (1950), Harsanyi (1956), Raiffa(1953), Schelling(197 1), Cross
(1969), Coddington (1968) and by Ikle and Leites (1962).
Wititman
/ 110W A WAR ENDS
749
t~~~~~~~~~~~~~~~~~~~~~
(w)
Bt
Ut (w)
A
Uy
t
x
t
Sb
A't
I
!
B'
s
Figure 3: A Peaceful Settlement Is Not Feasible
NOTE: In this case there is no point to the left of At and to the right of B1. Therefore a settlement is impossible.
C. THE EFFECT OF CHANGING P8
During the course of a war each country is constantly reassessing
its probability of winning in response to new information regarding
the progress of the war. It is assumed that an event which results in
country X increasing its estimated probability of winning will also
result in country Y decreasing its estimated probability of winning
(however, the amount of increase in Pxwneed not equal the amount of
decrease in Pyw).
What happens if in year t the probability of country Y winning the
war is reduced? Does this make a negotiated settlement in year t any
more likely? Country Y's expected utility from continuing the war is
decreased; therefore country Y is willing to accept less in the negotiations. It may seem that a settlement in year t is more likely; but this
is not true, because an increase in the probability of country Y losing
means an increase in the probability of country X winning and thus
country X's expected utility from continuing the war increases. This can
be seen in Figure 4. When the probability of X winning the war is
increased, X's minimal demand is increased while Y's minimal demand
8. In this section we isolate the effect of a change in P on expected utility. It is possible that a change in P is accompanied by a change in the length of the war which in turn
will effect the discounted utility. We do not consider this possibility in this section. In
section H we consider the effect of a change in the length of the war.
750
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is reduced, i.e., both minimal demands shift to the left.9 It cannot be
said whether country X's minimal demand moves more left (or less)
than country Y's minimal demand.
Thus, such phrases as, "We are bombing them in order to bring them
to the negotiating table";10 "all we want and have ever wanted in a
negotiated settlement is . . . "; "the better we do the more the enemy will
come to terms and the sooner we will reach a settlement," should be
taken with a grain of salt, for the better "we" do the more we will demand. For example, the U.N. forces changed their minimal acceptable
demands from repulsion of the North Koreans to a Unified Democratic
Korea after the U.N. forces were victorious in battle (see Halperin,
1963).
Propaganda statements are not the only source of mistakes regarding
the effect of changes in the probability of winning. Similar errors exist
in much of the academic literature. Because most of the research on
war has been concerned with its beginning rather than its termination,
most of the errors are centered on the question of why wars begin.
However, as noted earlier, there is a great symmetry between how
wars end and how wars begin. Therefore, the same underlying logic
should hold in both cases. One of the great controversies in the literature is whether a preponderance of power vis-a-vis a balance of power
situation makes an outbreak of war more or less likely.
A preponderance of power means that one side has a high probability and the other side has a low probability of winning. A balance
of power means that the probability of winning is 50/50. On one side
of this debate stand people such as Quincy Wright (1965), Inis Claude
(1962) and Wayne Ferris (1973: 26), who believe that if one side is very
likely to win, there is a higher probability of war. Ferris even marshalls
a large quantitiative study to support this view. But the other side
believes that "a balance of power increases the chance of war" (Organski, 1958: 292) and has countered with its data (e.g., Garnham, 1976)."
9. The analysis holds only for noncorner solutions, i.e., the minimal agreement
acceptable to X was not already at immediate unconditional surrender by Y. In this case
X's minimal demand could not increase (because of satiation). However, even if a nation
expected to achieve a "complete and total victory" without cost to itself if it continued
the war, this would not be a corner solution as there would still be room to negotiate the
timing of the unconditional surrender.
10. There may be some truth in this phrase-see
Section E.
11. Empirical measures of balance of power vis-a-vis preponderance of power may
encompass other aspects besides relative probability of winning. Most importantly,
balance of power may indicate a lack of information as to which side is more powerful.
This can result in differing subjective probability measures by the potential belligerents.
Wittmn!n
HOW A
VAR ENDS
751
Uy (w) =Bok-AUy(
U
)
y~~~~~~~~~~~~~~~~
w) =B
U' (w)=A] _
Uy (w)=A
_
t
A;I
A
1I
y
Sy
-'
S
BjB
l AO,
SX
IL
Figure 4: The Effect of an Increase in Pxw and a Decrease in Pyw
NOTE: When X's subjective probability of winning (Pxw) increases, X's expected
utility fron continuing the war increases from AO to Al. Therefore X's minimal
demand increases (i.e., moves to the left) from Al to At. When Y's subjective probability of winning decreases, its expected utility decreases and its minimal demand
decreases (i.e., moves to the left) from Bt to B1 . Since both minimal demands move
to the left, a settlement need not be more likely.
The analysis in this article suggests that this debate is irrelevant.
There is no relationship between the probability of winning and the
probability of war and therefore we will find no consistent empirical
relationship between the two. War and peace are substitute methods
of achieving an end. If one side is more likely to win at war, its peaceful
demands increase; but at the same time the other side's peaceful demands decrease. Thus we do not know whether an overlap is more or
less likely. Furthermore, if one side's increase in subjective probability
of winning is equal to the other side's decrease in subjective probability
of winning, there will be no change in the probability of war.
D. REDUCTION IN INTENSITY OF FIGHTING
A reduction in the intensity of the fighting is often hailed as a step
toward ending the war. For example, in the United States every time
In turn, divergent subjective probability estimates can have an effect on the probability
of war (see Section F). In this article the analysis is formulated in terms of two sides. The
concept of balance of power is more complex when there are many sides. For a discussion
of balance and equilibrium see Kaplan (1957) and Bueno de Mesquita (1978).
752
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there was a troop withdrawal from Vietnam, people became more
optimistic about the war ending sooner. In this section we will show
that there is good reason to believe that the opposite will resulta reduction in hostilities (either unilaterally or bilaterally) in year
t may actually prolong the war and make a settlement in year t+ 1 less
likely.
A country will unilaterally reduce its war effort only if this action
increases its expected utility. Often the increase in expected utility is
explained in terms of reducing political costs, but a reduction of military costs may be just as valid an explanation; e.g., the decision by the
United States to stop bombing North Vietnam is often seen as the
United States yielding to the pressure of world opinion; and the withdrawal of U.S. troops from Vietnam is seen as a result of the voters'
demand for a disengagement. There is, however, considerable evidence
to suggest that these changes in policy were due to military considerations. E.g., The Pentagon Papers claim that the decision to stop bombing North Vietnam was based on a desire to reduce the American pilot
and aircraft losses over heavily defended Hanoi and Haiphong, and
on the realization that the bombing had little impact on reducing the
North Vietnam war effort. Similarly, the U.S. decision to switch from
a ground to an air war in South Vietnam can be viewed in military
terms. Changing to an air war reduced the military costs yet did not
significantly reduce the probability of her winning, so the United
States' expected utility increased.12
This is a dynamic explanation for changes in strategy. A nation may
be heavily committed to fighting a war, but if it becomes apparent that
a reduction in its efforts would result in only a slight decrease in the
probability of its winning, the country may decide to withdraw some
of its forces-in this way its expected utility increases even though the
probability of its winning decreases. Thus, a unilateral reduction in
military effort may be carried out for purely military reasons. Furthermore, this reduction in effort will result in lower casualties for, and a
greater probability of winning by, the opposition, thereby increasing
its expected utility from continuing. Therefore, the possibility of a
settlement to end the war is reduced as the cost of continuing the war
is decreased for both sides.
The same holds true for a bilateral agreement to decrease the intensity of the war. This will only take place if both countries are better
off by the agreement, i.e., the cost of continuing the war is decreased.
12. For a complex game theoretic analysis of Vietnam, see Zagare (1977).
Wittman / HOW A WAR ENDS
753
But this may reduce the possibility of a complete settlement and thus
the war may be prolonged. In Figure 5 the countries have reduced their
war effort because this increases their expected utility of continuing
the war; which in turn raises both of their minimal acceptable settlements-X's
minimal demand moving left and Y's minimal demand
moving right. This may explain the continuing war between Israel and
the Arab countries.13 The low-level fighting does not force the belligerents to reach a peace settlement. The long wars in the eighteenth
century (seven European wars lasted for seven years or longer) can
also be explained as partially a consequence of the low level of fighting.
These wars tended not to be fought during winter or at night. More
importantly, defensive war was at a peak, meaning a slowly fought
war via a siege and blockade.14
A more formal test also supports the hypothesis that reduced intensity of fighting increases the length of the war. Using data from
Singer and Small (1972) for 93 wars between 1816 and 1965, we obtained the following regression results:
Months =2.5888-..0124BD
(1.290)
+
1.9022N + 18.4771D
(2.677)
(3.785)
+
0.2141T
(1.225)
F =5.7
Numbers in parentheses are number of standard deviations away from
zero. Months is the number of months the war lasted. BD is battle
deaths per capita per month. The hypothesis is that the more battle
deaths per capita per month the shorter the war. This hypothesis is
confirmed at the 10% level of significance (a one-tailed test). N is the
number of nations involved in the war. For extrasystemic wars which
involve a war between the country and one of its "colonies" the number
of nations involved was considered to be two (where Singer and Small
counted only one nation). Since part of the nation was fighting with
another part, it is not appropriate to treat the nation as a single entity
for the purposes of this investigation. The hypothesis is that the more
nations involved, the longer the war will be. This is because even if the
13. It is significant that Egypt was the first country to come to a peace settlement
with Israel for the costs of being at war with Israel were greater for Egypt than for the
other Arab countries. Of course, a full discussion of the Mideast war would include
the role of the superpowers.
14. See Blainey ( 1973) for a fuller description of these wars. It is especially important
to remember that theoretical analysis holds other things equal. Thus while we may find
a very severe war to be long lasting, other things being equal, we would have expected the
war to have lasted even longer if the war had been less severe.
754
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RESOLUTION
Vt (S)
y
U (w) =B1
Uy w)A0
Cx
--
(w) =A0
i
|
, I
.
S
.I
,
U
,.
A01
ABlA
I
-,
Figure 5: A Partial Settlement
I,
s)
..
SX
I
I
May Decrease the Probability
of a Full Settlement
NOTE: When both countries' expected utilities from continuing the war increases
(A< to A1, B0 to B1), their minimal demands also increase (A, to A' and B' to
Bi) and the likelihood of an overlap decreases. In this example there is no longer an
overlap in the second case.
necessary conditions for two of the belligerents to end the war exist, the
necessary conditions for another pair of belligerents to end the war may
not exist, and therefore the war will continue (possibly with fewer
participants). Again the hypothesis is confirmed.
The last two variables are control variables. D is a dummy variable
for an extrasystemic war. The dummy variable is meant to capture
the differences between interstate wars and extrasystemic wars which
were usually colonial. T stands for year and is a proxy for increased
technological change over time. As will be shown in Section F, technological change creates greater uncertainty about the outcome of a
war which in turn makes peace less likely (the effect of technological
change on the death rate is captured in variable BD), thus we would
expect the coefficient of T to be positive, which it is. Finally the F
statistic shows that the equation as a whole is significantly different
from zero.
Wittman / HOW A WAR ENDS
E. NEGOTIATION
755
STRATEGY
A country may undertake action which increases the cost of its
pursuing the war (e.g., heavy bombing of nonmilitary areas defended
by antiaircraft batteries) even if it does not increase the probability
of its winning, if this action inflicts heavy costs on the other side. Here
the object is to win at the negotiation table, for if a country faces greatly
increased costs, its minimal acceptable agreement will be reduced. Since
the cost of continuing the war will also increase for the country doing the
bombing, its minimal acceptable agreement will likewise be reduced
(but not by as much as that of the bombed country). Since both countries' minimal acceptable demands are reduced, a settlement to the war
is more likely, and because the country being bombed has a much
greater reduction in its minimal acceptable agreement than the country
doing the bombing, the country doing the bombing is likely to do better
in the negotiations than if it had refrained from bombing. The German
bombing of London in World War II and U.S. bombing of North
Vietnam in the war in Southeast Asia are examples where the main
purpose was to increase the cost of the war to the other side. From a
military standpoint these bombings were not strategic and did little
to increase the probability of the bombing country winning although
they did increase the likelihood of a settlement to the war and on more
favorable terms to the country doing the bombing. This can be seen
in Figure 6.
F. SUBJECTIVE
PROBABILITIES
War is often due to optimistic estimates regarding the probability
of winning and the cost of war. This can readily be demonstrated by
posing the following question. Given the hindsight of the actual cost
of the war and the actual peace settlement, how many countries would
have gone to war rather than accepted the actual peace settlement in
the first place? Clearly, the United States when it initially entered into
the Vietnam conflict greatly overestimated its ability to wage a successful fight with minimal cost.
As noted before, Pxw need not equal [I-Pyw] as the probability
estimates are based on different sources of information. Looking over
all years, both war and peace, we would not expect rational actors to
consistently over- or underestimate their true probability of winning.
756
JOURNAL
U0 (w) =Ao
OF CONFLICT
RESOLUTION
0y(
U0 (w) =B0
(W) =B1
UY
)
UX(W =Ae ~~~~~~~~~~1II
!~~
Y
!
~
(S)
~~~~~~~~~~~
{
L
X
condtiosfoa
heneesar
Coseuetl,
afte th bobin
thrsAnoelp
Figure 6: Nonstrategic Bombing May Increase the Probability of Settlement
NOTE: If Y's bombing of X causes much damage to X but has little effect on X's
probability of winning, the utility from continuing the war decreases greatly for
Y (Bo to
The result is that negotiations are more
and slightly for B1).
X (AO toAm)
bombing there is no overlap but
likely to take place. In this example beforare
i
there is an overlap. Consequently, the necessary conditions for a
after
ng a
negotiated
settlement
are fulfilled.
ILe., on average we should expect that Px,, + Py,, = 1. However, at times
countries will misjudge. Sometimes
Px,, + Pyw will be greater than one
less than 1.
and sometimes
+ PYW
> 1, then the countries are jointly optimistic
If PXW
(note once
again that it makes no difference whether Pxw + Pyw = .60 + .60 or Px +
Pyw= .95 + .25). The morejointly optimistic the countries are, the greater
the probability of war and the less the chance of a peaceful settlement
(in time t). Because of the objective negative sum nature of war, the
necessary conditions for peace based on subjective probabilities are
less likely to be met under divergent subjective probability estimates
than when the probability estimates are consistent. When the subjective
probability estimates are consistent (Pxw + Pyw= 1), then war will appear
as a negative sum game and the necessary conditions for peace will be
met. When the subjective probability estimates diverge (Pxw + Pyw - 1),
then the necessary conditions for peace will be met if the countries are
jointly pessimistic (Pxw + Pyw< 1) but not necessarily if they are jointly
optimistic. With this latter possibility the necessary conditions for
peace may no longer be satisfied. Times of rapid technological change
Wittnan / HOW A WAR ENDS
757
in weaponry during peace time or rapid changes on the battlefield
during a war are likely to create divergent subjective probabilities.
Divergent subjective probabilities increase the probability of a
substantial joint optimism (vis-a-vis the probability of joint optimism
when the subjective probabilities do not diverge-which
is zero). In
turn, this decreases the likelihood of the necessary conditions for peace
being met.
G. ATTITUDE
TOWARD
RISK
If both countries' utility functions are concave (bowed out), they
will be risk averters (i.e., they tend to prefer a sure thing), while if both
countries' utility functions are convex (caved in), they will be risk
preferrers. In general, the continuation of a war is riskier than a negotiated settlement (the outcome of a war being more uncertain than
the results of a particular settlement). In such cases, other things being
equal, a settlement is more likely to take place when the countries are
risk averters than when they are risk preferrers. (This can be seen in
Figure 7.)
H. TIME PREFERENCE'5
Often the side which is oriented toward the present is portrayed
as being at a serious disadvantage to the country whose time horizon is
in terms of centuries. I.e., the first country is too impatient and therefore will be in a weak bargaining position at the negotiating table,
while the other country will take advantage of this impatience by prolonging the war and demanding more in a negotiated settlement.
Although this argument is plausible, it need not be true. In order to
15. In the analysis presented in this article, the countries discount over time. Therefore they consider the possibility of future wars in their calculations. Because there is
rarely a third party enforcer of contracts (negotiated settlements) in international relations, the belligerents usually look for self-enforcing contracts. In the past, mountains
and wide rivers often served as natural enforcers of the contract. These natural barriers
make offense difficult and defense relatively easy. An arbitrary division across a fertile
plain could more easily be cheated on, once the armies regrouped. This may explain the
history of Poland's complete subjugation in comparison to the more mountainous Balkan
states.
758
JOURNAL
OF CONFLICT
RESOLUTION
CONCAVE UTILITY
FUNCTION
RISK AVERTER
FUNCTION
CONVIEX UTILITY
RISE PREFERER
U (W)
t
2
-U(W)
+
1
Sl
---I
-
-U(L)
Y
I
X
|
Y
I
Uts
Figure 7: Risk Aversion Compared to Risk Preference
if X wins and there is an unconditional
NOTE:
U(W) is X's utility
surrender
by Y.
from Utt (w) which is X's expected
if it continues
This is to be distinguished
utility
if X loses and X surrenders
the war. U(L) is X's utility
to Y. In this
unconditionally
of W and 1/3 chance
of L. If the utility function
X has 2/3 chance
of X is
example,
X's minimal
demand
will be less (more to the right) than if its utility funcconcave,
tion is convex.
understand why this argument need not be true, we will consider
expected utility in greater detail. The present value of expected utility
is the sum of discounted expected utilities over time. The sum can be
broken up into two parts-expected discounted utility (or disutility)
during the war years and expected discounted utility during the postwar years. The more the country discounts the future, the greater
weight the country places on the present (war years) in its calculations.
If utilities are positive, greater discounting of the future will also result
in a lower present value of expected utility as future utility is discounted
so heavily. This, however, does not mean that the country will demand
less in a negotiated settlement because the value of the negotiated
settlement over time is also discounted, therefore future benefits
accruing from a settlement are also heavily discounted.
Whether greater discounting of the future increases or decreases a
country's minimal acceptance point depends on its probability of
winning the war, the costs of fighting, the utility of its winning or
losing, and the stream of utility from a negotiated settlement. If the
level of warfare is very low, the expected utility during the war years
may be greater for the losing country than the expected utility when
the war is over. In this case, a country likely to lose, which cared only
Wittman / HOW A WAR ENDS
759
about the present, would try to prolong the war, if possible via a lower
level of fighting, even if it made losing more likely. A lower level of
fighting would mean that the country's utility in the present would
be higher, and since the country did not care about the future, its sum
of discounted expected utilities would be greater.16 On the other hand,
a country which weighted the present and future equally would not
prolong the war if it reduced the probability of winning as this action
would reduce its expected utility. The costs would be less in any one
year but they would be spread over more years and so there would be
no advantage to prolonging the war. Furthermore, the probability
of its winning would be reduced so it would actually be worse off. In
the polar extreme, if the fighting were very heavy and the disutility of
fighting were greater than the disutility of losing, then the country
which valued the present would have a negligible minimal acceptance
point, while a country which valued the future would have a more
significant minimal acceptance point.
Thus, if the expected utility during the war years is greater than the
expected utility after the war is ended, the more the country values
the present, the greater would be its minimal demand in any settlement;
and if the utility during the war years is less than the utility during the
postwar years, then the more the country values the present, the lesser
would be its minimal demand.
I. SUMMARY
This article views the termination of war as a process of rational
calculations by the participants. It is assumed that, unless both sides
believe that they can be made better off by a settlement, the war will
continue. War and its ending have been treated as rational acts between
nation states-in contrast to mechanistic and psychoanalytic theories
16. E.g., the cost of a U.S. President admitting that the United States has lost a war
may be greater than the disutility of continuing the war. President Nixon said that he did
not want to be the first president to lose a war. During his first term of office he did not
want to lose nor end the war in South Vietnam by coming to a settlement which admitted
the United States had lost the war (which he felt would hurt his chances for reelection).
Thus, his strategy was to prolong it at a very low military cost and thus increase his expected utility. This suggests that President Nixon may have been responsible for the
relatively low intensity of the fighting during his first term as a way to prolong the war
and thus prolong the facing of defeat.
760
JOURNAL
OF CONFLICT
RESOLUTION
as well as with other theories which do not consider that it takes two
sides to make peace.
While not denying that economic variables may alter behavior,
these economic variables work via the equations in the text. Here
war and peace are treated as alternative means to a nation's ends.
Even conquering a country by war is an alternative to peaceful surrender of the country's sovereignty. Therefore, if economic circumstances somehow do motivate a country to overtake another country's
sovereignty, it need not do so via war; and if the economic system
enables a country to be more successful in war and consequently more
intransigent in its demands, then, as already shown, the other country
will be more willing to compromise.
That it takes both sides to end a war is a useful insight. In this article
we have developed this insight into a number of interesting results via
the use of a theoretical model. An important result of this approach is
that a reduction of hostilities may reduce the probability of a settlement
taking place and thus prolong the war. It is also shown that increasing
the probability of winning may not increase the probability of a settlement and that a country which only values the present need not be at
a disadvantage in the negotiation.
APPENDIX
The concepts used in the main body of the text can also be formulated
in terms of a game.
In Figure Al we have a utility possibility frontier (only the positive
quadrant has been drawn in). At, B. is the noncooperative solution
(war). The hatched-in areas are the set of alternatives which are pareto
superior (the set of cooperative solutions). By looking only at the
frontier, we have the same information as drawn in Figure 2 of the
main text.
In Figure A2 there are no possible points of agreement as the set of
pareto superior alternatives to the war (hatched lines) are drawn outside
the feasible set. In this case a cooperative solution is not feasible;
equivalently, the necessary conditions for a peaceful settlement do not
exist. While war is objectively a negative sum game, the subjective
probability of winning estimates can make it look like a positive sum
game as each country's own estimate of expected utility is based on
Witiman / HOWA
WAR ENDS
761
Y'I s
BY
UTILI
n
Bt.
Sy
Figure Al:
NOTE:
of
frontier
utility
=
tinues.
U,
BT
=
out
(bowed
on
curve)
Surrender
=
(w)
Ut
=
by
X's
Country
(w)
UTILITY
Possible
agreement
Unconditional
SY=
AT
Settlement
Points
X's
At
Sx
war
termination
is equivalent
Y.
Y's
the
discounted
line
Surrender
utility
discounted
expected
are
lines)
settlement
= Unconditional
Sx
expected
Country
(hatched
to
in
by
t
year
in
utility
continues.
Bt
Y's
UTILITY
I
I
i
A
S
X
Figure A2: Settlement
NOTE:
Points
of
X'S
UTILITY
Impossible
Agreement
(hatched
lines)
are
not
feasible.
At
feasible.
in Figure
The
1.
X.
if the
year
war
t if the
conwar
762
JO URNA L OF CONFLICT
RESOLUTION
these subjective probability estimates. As noted in the text, wars become more likely when the countries are jointly optimistic (Pxw +
Pyw > 1). Because of imperfect information, countries may have different subjective probability estimates. Only when the countries are
sufficiently jointly optimistic to overcome the negative sum qualities
is war inevitable (Figure A2). Fortunately, strong joint optimism is
itself not inevitable. Otherwise we would be constantly at war.
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