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Who Wins The Peace? Elites, Citizens, and theDecision for War∗

Colin Krainin†and Kristopher W. Ramsay‡

September 2, 2016

Abstract

It is often argued that different groups in society faces different costs and benefitsfrom war. In this paper we consider a theory of bargaining and war where the objectof dispute has multiple characteristics, some public good in nature, others private. Webuild a model of crisis bargaining over mixed goods to explore how regime type and“crisis type” interact to influence the probability of war and the distribution of benefitsin peace. We show that, in general, democracies get less private benefits but morepublic goods in peace, a state where the elite capture the median is more conflict pronethan a liberalized state, and that taxation of the elite by the citizens as a form executivecontrol can increase the probability of war. If citizens can set different tax rates overpeaceful bargains and war spoils, they can endogenously determine the “political bias”of the decision-making class over war and peace.

∗We thank Bella Wang for excellent research assistance.†Princeton University‡Princeton University

1

Introduction

Issues at stake in international bargaining rarely have uniform implications to all groups

within the bargaining states. Furthermore negotiations often contain multiple elements.

Trade deals may increase the overall pie, but negatively impact workers in a specific indus-

try. Arms reduction agreements, territorial transfers, and government to government cash

transfers all lead to a different distribution of benefits in a society. In order to begin to

address these issues, we develop a model of international bargaining with two new elements.

First we model states bargaining over both public goods that benefit all of society and pri-

vate goods that only elites capture. Second, we consider different regimes by varying the

key decision maker in the bargaining model. Here we very the decision making regime in the

home country in two ways. First we consider a decision making process where elites, who

share private goods and consume public goods decide whether there is war or peace. Next

we consider a regime that expands the ruling elite, possibly to be very large. We can think

of this as a regime that co-opts large groups of citizens and then shares smaller fractions of

the private good. Finally we consider a model of liberalization where the majority citizens,

who do not share in the benefits of private spoils, decide whether to accept a peace or go

to war. Our main finding is that while fully liberalized democracies have advantages in

obtaining peaceful outcomes in international disputes, transitional or half-measures such as

expanding the size of the elite class or redistributive taxation can often increase the potential

for conflict.

Formally, instead of bargaining over a single object (Fearon 1995), states simultaneously

bargain over both public goods, x, and private goods, y. One state, Foreign (F ), with known

strength makes a take-it-or-leave-it offer to Home (H). Home is modeled as having one of

two “ types,” is either strong or weak with a known prior probability. Home is composed of

two groups, elites E and ordinary citizens (or out group) O. We consider two institutional

forms: a liberalized democracy, where the median voter (a member of O) decides whether

or not to accept Foreign’s offer and an oligarchy, where only members of the elite decide

2

whether or not to accept a peace proposal. Though, in an oligarchy, the ruling group can be

large, possibly including the median, a member of O.

The key determinant of conflict in this setting is the attractiveness to Foreign of a risk

free (pooling) offer that both types of Home will accept versus a risky (separating) offer

that only weak types of Home will accept. In our setup, who gets to make this decision

within Home is critical in determining the relative attractiveness of these two types of offers.

When Home is an oligarchy, we find that the following institutional changes all increase the

potential for conflict: (1) decreasing the elite’s cost of war; (2) expanding the size of the elite;

and (3) increasing the ability of citizens to capture private goods from the elite (possibly

through redistributive taxation).

Collectively, these results lead to our conclusion that partially transitioning from a small

oligarchy to a more inclusive but non-democratic regime frequently increases the potential

for conflict. The intuition behind this family of results is that efforts to include more citizens

in the decision making process, or to transfer some of the benefits of private goods from elite

decision makers to all citizens, frequently encourages the elite decision makers to increase

their bargaining demands. This effect is driven by a need to compensate a smaller proportion

of the bargaining pie domestically with a larger overall share of the international bargaining

pie. Generally, these demands are increasing by a larger proportion for strong types than

weak types, increasing the incentive for risky (separating) offers.

We can also obtain three additional results conditional on increasing the amount of

private goods. When Home is a full democracy, then increasing the amount of private goods

available decreases the potential for conflict. Alternatively, if Home is an oligarchy, then

increasing the amount of private goods available increases the potential for conflict. When

there are sufficiently high levels of private goods, liberalization (switching from an oligarchy

to a democracy) decreases the potential for conflict. These results imply that altering the

level of private goods has differing impacts on the potential for conflict for different pairs

of countries in conflict. Increasing the level of private goods (1) increases the potential for

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conflict for two oligarchies, (2) decreases the potential for conflict in mixed dyads, and (3)

does not affect the potential for conflict between democracies.

The intuition behind these results on increasing the level of private goods is that the

decision makers in democracies cannot obtain them and therefore do not care to bargain

for them. Hence, the elites in democracies lose out on access to private goods relative to

oligarchies. If one of the states is a democracy, but the other is not, then increasing the

level of private goods allows the democracy to compensate an oligarchy with private goods

without decreasing the utility of the the median voter. Liberalizing from an oligarchy to a

democracy can cause Home to increase its demands in public goods, but when the level of

private goods is sufficiently high, there always exists enough private goods to compensate a

foreign oligarchy. However, if both states are oligarchies, the potential for conflict increases

as the level of private goods increases because for each additional unit of total private goods,

Foreign keeps a greater proportion of a bargain when facing a known weak type rather than

a known strong type. Hence, under private information, the spread between size of a bargain

necessary to satisfy both type versus only one type increases, increasing the incentive for a

risky offer.

A final implication of our model is that decreasing the ability of citizens to capture

private goods specifically in war time (possibly through redistributive taxation). Hence, if

citizens can set differing tax rates on peaceful bargains and war spoils, than the citizens can

directly determine the decision making group’s incentives toward or against war. Hence, this

type of taxation policy provide a microfoundation for the concept of political bias used in

some agency models of conflict. Political bias lead to a qualitatively different effect than

simply altering the decision maker’s cost internalization of war since it can lead to war

even in the case of complete information (Jackson and Morelli 2007; Krainin and Slinkman

2016). Different taxation structures will imply different bias levels for the elites. Increasing

(decreasing) the effective bias of the elites will increase (decrease) the proportion of the

international bargaining pie captured by Home, but also increase (decrease) the potential

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for conflict in crisis bargaining.

As a model that takes direct account of the impact of domestic politics on international

bargaining and war outcomes, this paper adds to a literature that began with the introduction

of the two-level game in Putnam (1988). Several strands of the literature have taken different

approaches to connecting the domestic level to the international level. This paper connects

most closely with agency models of war (Jackson and Morelli 2007; Fearon 2008; Krainin and

Slinkman 2016). Other approaches include the selectorate model (Bueno de Mesquita et al

1999) where a leader makes international decisions in order to maintain her ability to remain

in power and signaling models that investigate the informational role of internal opposition

(Schultz 1998).

Empirically, our results most clearly have implications for the large literature that inves-

tigates conflict proneness in authoritarian regimes versus transitional democracies versus full

democracies (Mansfield and Snyder 1995; Thompson and Tucker 1997; Ward and Gleditsh

1998; Gleditsh and Ward 2000; Goemans 2000; Narang and Nelson 2009; Baliga, Lucca,

and Sjostrom 2011; Colgan 2013). This literature is inconclusive as to the likelihood of

conflict in transitional democracies. Our model indicates a previously unidentified force for

the increased conflict potential of transitional democracies. However, shifting fully from an

oligarchy to a democracy is often peace enhancing in our model. Since our model plays out

in a simple, one-shot setting we see a more fully dynamic extension of our model as the

logical next step. Our expectation is that the foreign power experiences a tension between

the increased static incentive for conflict with transitional democracies and the desire to wait

out the transition period in order to enjoy the more favorable bargaining environment once

democracy full takes root.

5

Baseline Model

We describe our model on two levels: the international level and the domestic level. Our

focus is on how domestic institutions impact international bargaining and the potential for

war. In order to cleanly present these effects, we simplify the international level as much as

possible.

International Level

Two states, H and F , bargain over a public good x ∈ (0, x] and a private good y ∈ (0, y].

The presence of more than one type of good is the only aspect of the international level

that is qualitatively distinctive from previous crisis bargaining models featuring asymmetric

information.

F makes a take-it-or-leave-it offer (x, y) to H. H either peacefully accepts the offer,

or rejects the offer and war results. We assume a simple quasilinear form of utility for all

decision makers over the public and private good. In peace, for a given offer (x, y), F

receives

uF (x, y) =√(x− x) + (y − y) .

The concavity of payoffs from x assumes that at low levels of the public good, the marginal

value of public goods is high relative to the marginal value of private goods. However, the

marginal value of public goods is decreasing relative to private goods. Therefore, at higher

levels of the public good, the marginal value of public goods is low relative to private goods.

This type of utility function is appropriate in this case because we can think of the public

good as a particular good while allowing the private good to be “money” i.e. something that

can be traded for other unmodeled goods.

H has a probability of victory in war dependent on H’s type and incurs costs described in

the next section. With probability q ∈ (0, 1), H is a weak type (l) and with probability 1−q,

H is a strong type (s). When H is type s, then H wins with probability p ∈ (0, 1), while

6

when H is type l, H wins with probability 0 < p′ < p. F then wins a war with probability

1− p against a strong type (1− p′ against a weak type ) and pays cost cF . In a war against

a strong type, F receives

wF = (1− p)[√

x+ y]− cF .

Domestic Level

While F is assumed to behave as a unitary actor, H’s domestic institutions are explicitly

modeled. H contains n actors divided into two groups: E (for elites) and O (for outside

group). There are m ≥ 1 members of E, while the rest of the population (n−m) belongs to

the outside group. We generally assume that m is small relative to n. The only difference

between the groups is that members of E receive private goods from an international bargain,

while members of O do not.

We consider two institutional settings. We refer to the first setting as an oligarchy. In an

oligarchy, the members of E possess the power to agree to a bargain or not. While we use

the term oligarchy, this first setting captures a number of political structures ranging from

a pure dictatorship (m = 1) to various forms of limited democracy (m relatively high, but

m < n/2). We refer to the second setting as a democracy. In a democracy, we assume that

the median voter has the power to agree to a bargain or not. Moreover, we assume that m

is small enough that the median voter is a member of O.

It is possible to imagine cases where m is high enough in the oligarchy setting that the

median voter is needed to agree to a deal.1 In fact, we will think of cases where m is higher

as cases where a state is more democratic than one with a lower m. However, the democracy

setting is distinctive in that we cover the case where the median voter has decision making

power, but may not have the same incentives as the, possibly quite small, elite that receives

private goods in a bargain.

Within a state, public goods are non-rival and non-excludable2, so both groups receive1An example might be a society where a minority group is excluded from voting and private goods.2We use the term public good throughout, but our model also applies to common resource goods (non-

7

the same amount of public good utility, while private goods must be divided among the elite.

Hence, in peace for a given offer (x, y), E receives utility

uE (x, y) =√x+

y

m,

while O receives

uO (x, y) =√x.

In war, all n players divide the cost of war. However, a deciding member’s cost is

modified by γ > 0 to reflect that they may pay a proportionally different amount of costs.3

For instance, if m = 1 and γ = n, then we are back in the unitary actor case. Hence, in war

when H is strong E receives

wE = p

(√x+

y

m

)− γ cH

n,

while O receives

wO = p(√

x)− cH

n.

Bargain Values

Once F makes an offer, H must either accept the bargain or go to war. In the oligarchy

setting, the members of E make this decision while in a democracy the median voter, a

member of O, decides. In order for a decision maker in H to want to accept a bargain, she

excludable, but rivalrous). The only modeling difference, which does not qualitatively impact the analysis,is that an individual’s value of common resource goods are scaled down to reflect the population size.

3Note that costs need not sum to cH if γ 6= 1. Altering the level of γ, alters the total cost faced by H. Analternative assumption would be to set elite cost to γ cH

n and citizen cost to (1− γ) cHn . This change would

not qualitatively impact any of our results in this paper, but may quantitatively impact the results presentedhere as well as substantively impacting the results of certain possible extensions. These alternative ways offormulating costs have different real world implications. Our assumption is that the institutional structureof government affects how elites are impacted by war, but it does not directly affect ordinary citizens. So,for instance, election outcomes may be affected by war decisions. The alternative assumption is that theiris a direct trade-off in the costs paid by elites and those borne by citizens. In this case, costs are the literalpayment for war.

8

must prefer the bargain to her war value. In the case of a strong type democracy, this means

that a peaceful bargain must satisfy

√x ≥ p

(√x)− cH

n.

While in a strong type oligarchy, a peaceful bargain must satisfy

√x+ y

m≥ p

(√x+ y

m

)− γ cH

n.

In the democracy setting, the condition for a peaceful bargain is substantively identical to

the standard model. In the oligarchy setting, there are two important distinctions. The first,

which has historically been emphasized as a reason why democracies may be more peaceful

than other forms of government, is that the cost of war for a member of E need not be the

same as it is for the majority of society. When γ = 1, members of E and O share equally

in the cost of war. When γ > 1, members of E suffer higher costs in war than O, while

the opposite is the case when γ < 1. The argument is that typically, γ < 1 and therefore

oligarchies are less peaceful than democracies. Indeed, in our model it is easy to show that

lower values of γ will increase the risk of war in oligarchies relative to democracies.

The second distinction is that members of E value two different goods. Public goods

that are valued by all members of society and private goods that must be divided among

the m members of E. Since private goods are divvied up amongst the elite, the larger the

size of the elite, the less each member values a given offer y.

In any peaceful bargain, F seeks to maximize its utility given the constraints described

above. Lemma 1 characterizes these values when H is known to be strong.

Lemma 1. Assume H is known to be strong. In the unique subgame perfect equilibrium,

bargains are set as follows

9

(1) If H is a democracy, then

(x∗, y∗) =

0, 0 x ≤

(cHnp

)2[p√x− cH

n

]2, 0 x >

(cHnp

)2(2) If H is an oligarchy, then

(x∗, y∗) =

0, 0 y ≤ m[γcHnp−√x]

[p(√

x+ ym

)− γ cH

n

]2, 0 m

[γcHnp−√x]< y < m

p

[γ cHn−(p− 1√

2

)√x]

[p√x+ (p−1)

my − γ cH

n

]2, y y ≥ m

p

[γ cHn−(p− 1√

2

)√x]and

y < m1−p

[(p− 1√

2

)√x− γ cH

n

]x2, m

(p− 1√

2

)√x+ py − γm cH

notherwise

A similar characterization follows if H is known to be weak with p′s instead of ps.

In the democracy setting, H only has a positive war threat when x is high relative to

the population weighted cost of war, modified by the probability of victory. Therefore, H

only receives positive amounts of the public good when this threshold is reached. As seems

natural, when H’s bargain value is positive, it is increasing in H’s probability of victory and

the total amount of public goods available, while declining in war costs. Democracies never

receive any private goods since the median voter does not value them, hence F optimizes by

setting y = 0.

In the oligarchy setting, bargain values are more complex. Similarly to the democracy

case, if the total amount of public and private goods per recipient is not high enough relative

to a modification of war costs, then H does not receive any portion of the international

bargain. There are two further corner cases. In both cases, the total amount of public and

private goods are high enough that H receives a positive portion of the international bargain,

but y is relatively scarce. In the first case, F maximizes by keeping all of the private goods

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for itself and pays H off entirely in public goods. In the second case, F would prefer, on the

margin, to substitute more private goods for less public goods in the bargain offer F makes

to H. However, F is constrained from making this substitution by the resource constraint

on y. One might imagine two further corner cases symmetric to these cases where x∗ = 0 or

x∗ = x, but these cases turn out to be impossible.

The last bargain scenario in the oligarchy setting is the interior solution. At the interior

solution, the total amount of private goods is high enough that F can choose an optimal

mix of public and private goods for its offer. When this is the case, x∗ is constant and only

depends on the total level public goods, while y∗ is then set to just satisfy H’s war threat

given the value of x∗.

In order to see when an interior solution holds, we first define the value

y (p) ≡ max

{m

p

[γcHn−(p− 1√

2

)√x

],

m

1− p

[(p− 1√

2

)√x− γ cH

n

]}.

Then Assumption A describes the condition for the interior case.

Assumption. (A): y ≥ y (p)

We will make use of this assumption at some points in order to simplify the analysis. For

our purposes, this assumption is often quite reasonable. We are the interested in a two good

model in order to identify how domestic institutions impact bargaining over these different

types of goods and consequently the potential for conflict. Hence, we are interested in cases

where private goods are not extremely scarce relative to public goods. One concern might be

that when p is large, y (p) gets large as m gets large. However, in most cases of interest, m

is relatively small. A more pressing concern is that as p→ 0 or p→ 1, y (p) goes to infinity.

However, our results that make use of Assumption A turn out to be robust to cases where

p→ 0 or p→ 1 and Assumption A fails.

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The Potential for Conflict

When H’s strength level is known, F can safely make an offer that H is just willing to

accept. However, H’s strength level is unknown and may be strong (type s) with probability

1− q or weak (type l) with probability q. H cannot credibly communicate its strength level

to F before bargaining since whenever H is weak, it has the incentive to misrepresent its

strength level as strong. F then has three options: (1) make an offer that neither type of H

will accept; (2) make an offer that both strength types will accept; or (3) make an offer that

H will only accept if it is a weak type.

In the (Perfect Bayesian) equilibrium of the baseline model, F never takes option 1 since

the cost of war implies that there exists bargains where both players do better than in war.

Under option 2, F optimizes by offering H just enough that a strong type will accept. We

call this the risk free proposal.4 With this option, F avoids risking war, but possibly pays

too much to a weak type. Under option 3, F optimizes by offering H just enough that a

weak type will accept. We call this a risky proposal, because if H is a strong type, war will

result.5

Let usF be F ’s payoff when it makes a risk free proposal that a strong type will just

accept. Let ulF be F ’s payoff when it makes a risky proposal and the weak type accepts. F

prefers to make a risky proposal when

qulF + (1− q)[(1− p)

[√x+ y

]− cF

]− usF > 0.

The second term on the left hand side is F ’s war value multiplied by the probability of war.

In our simple model, either this inequality is not satisfied and the probability of war is zero

or it is and war occurs with probability 1 − q. Since this inequality separates cases where

war never occurs from cases where it occurs with positive probability, we will refer to it

as the war risk inequality. Therefore, holding all other parameters constant, any change in4Also known as a “pooling” proposal.5Also known as a “separating” proposal.

12

a single parameter that causes the left hand side of the war risk inequality to increase in

value increases the space of parameters for which the war risk inequality is satisfied. When

increasing a parameter increases (decreases) the value of the left hand side of the war risk

inequality, we say that the potential for conflict is increasing (decreasing) in that parameter.

Expanding the Elite

In the oligarchy setting, the elite expands as m increases. This may be viewed as a limited

type of democratization. As m increases, a greater percentage of society is involved in the

decision making process over international bargains and conflict. As a first step toward

understanding the effects of democratization, Lemma 2 demonstrates how F ’s utility is

affected by increases in the number of elites in H.

Lemma 2. Assume H is known to be type s. uF is strictly decreasing in m if and only if

p > γcHn√x+ 1√

2. Otherwise, uF is weakly increasing in m.

A similar condition, replacing p with p′, holds for the case where H is known to be type l.

Lemma 2 states that increasing democratic involvement increases H’s demands in bargaining

when H is sufficiently strong causing F ’s utility to decline when bargains are peaceful. In

order for this level of strength to be achievable in a particular case, it must be that

x >

[γcH

n(1− 1√

2

)]2.

When H is not sufficiently strong, democratization reduces H’s bargaining demands.

The intuition for Lemma 2 is as follows. Increasing the share of the population (m)

receiving private goods decreases the payoff for each member of the elite in peacetime since

private goods received in a bargain are spread over greater numbers of elites. For the same

reason, war payoffs are also lowered when m increases. However, these payoffs are modified

by the probability of victory and modified costs. When the probability of victory is high

13

and modified costs are low, then war payoffs drop more slowly than peacetime payoffs as m

increases. Thus, H requires higher peace values to avoid war leading to lower utility for F .

On the other hand, when the probability of victory is low and/or modified costs are high,

then increasing m lowers the war value more than it lowers the peace value for a member of

E, hence H requires less to avoid war which translates into higher utility for F .

Utilizing Lemma 1 and Lemma 2, Proposition 1 demonstrates two different sufficient

conditions for the potential for conflict to be increasing in m.

Proposition 1. (1) If p > γcHn√x+ 1√

2and p′ ≤ γcH

n√x+ 1√

2, then the potential for conflict is

strictly increasing in m. (2) If Assumption A holds for both p and p′, then the potential for

conflict is strictly increasing in m.

The key implication of Proposition 1 is that, on its own, expanding the elite has an am-

biguous direct effect on the potential for conflict in crisis bargaining. While there may exist

cases where elite expansion alleviates the potential for conflict, Proposition 1 demonstrates

that it is also possible that democratization actually increases the potential for conflict. In

particular, this is the case when p is relatively large and p′ is relatively small. That is, when

the strong type of H is sufficiently strong and the weak type of H is sufficiently weak. It

is also the case when Assumption A is satisfied for both strong and weak types so that the

solution is always interior.

The intuition is that when these conditions are satisfied, increasing the size of the elite

increases H’s demands when H is a strong type while weakly decreasing them when H is

a weak type. Moreover, m does not factor into F ’s utility in the event that F wins a war.

These factors combine to imply that in many case, increasing the size of the elite causes a

decrease in the value of the risk free proposal for F while weakly increasing the value of the

risky proposal.

14

Oligarchy versus Democracy

In this section we compare the potential for conflict in oligarchies versus democracies. We

have already noted that as γ shrinks, the potential for conflict in oligarchies is increasing

while the potential for conflict in democracies is static. Hence, oligarchies with institutions

implying a low γ, will have a potential for conflict that is high relative to a state that is

similar in every way, except that it has democratic institutions. Proposition 2 states this

formally.

Proposition 2. When Assumption A holds for both p and p′, then decreasing γ strictly

increases the potential for conflict in oligarchies. Democracies are unaffected by changes in

γ.

Besides differing costs, oligarchies and democracies also differ in their valuation of public

versus private goods. The major difference being that the median voter values public goods,

but not private goods while decision makers in oligarchies care about both goods. Hence,

the total amount of private goods relative to the total amount of public goods to be deter-

mined by international bargaining is potentially critical in determining whether oligarchies

or democracies are more conflict prone. In order to elucidate this effect, Proposition 3 char-

acterizes the impact of increasing the amount of private goods relative to public goods on

the potential for conflict in democracies and oligarchies.

Proposition 3.

1. Increasing the amount of private goods, y, decreases the potential for conflict in democ-

racies.

2. When Assumption A holds for both p and p′, increasing the amount of private goods,

y, increases the potential for conflict in oligarchies.

Proposition 3 states that as the amount of private goods, y, increases, democracies become

less conflict prone while oligarchies become more prone to conflict. Hence, when private

15

goods are a relatively large proportion of what is being bargained over, democracies are

more peaceful than oligarchies. Alternatively, as private goods decrease relative to public

goods, democracies become relatively more conflict prone.

The intuition behind the result is straightforward. As the amount of private goods

increases, F is more and more willing to pay a democracy off in public goods since F is able

to keep a greater amount of private goods for itself. On the other hand, whenH is a oligarchy,

the risky proposal becomes more attractive as the amount if private goods increases. This is

because for each additional unit of total private goods, F keeps a greater proportion when

facing a weak type than a strong type. Hence, as the amount of private goods increases, F

becomes more and more willing to gamble on the risky proposal.

This result is contingent on F valuing private goods. If F does not value private goods

(perhaps because F is also a democracy), the potential for conflict will be invariant in the

level of private goods when H is a democracy. In this case, our model is qualitatively the

same as a standard crisis bargaining model. When F is a democracy, but H is an oligarchy,

increasing the amount of private goods will actually decrease the potential for conflict. The

general implication being that mixed dyads are relatively peaceful when there are sufficient

private goods to pay off elites within the oligarchy while allowing the democracy to retain

a consequently higher proportion of the public good. When oligarchies bargain with other

oligarchies, increasing the amount of private goods only exacerbates the situation, while the

amount of private goods is irrelevant for two democracies bargaining with one another.

As a final point of comparison between oligarchies and democracies, Proposition 4 con-

siders how shifting from an oligarchy to a democracy (what we call liberalization) affects the

potential for conflict in comparison to how expanding the elite to capture the median affects

the potential for conflict.6

Proposition 4. There exists a y∗, such that for all y > y∗, (1) liberalization decreases the6An example of expanding the elite to capture the median might be as follows. Consider a society with

three ethnic groups that is initially controlled by a single group. In some cases, the elite group may be ableto capture the median by expanding elite status to one group, but not the other.

16

potential for conflict, and (2) expanding the elite to capture the median increases the potential

for conflict.

Intuitively, Proposition 4 demonstrates that while certain types of democratization,

namely the full liberalization of the state from an oligarchy to a democracy, may decrease

the potential for conflict, limited democratization in the form of an expanding the decision

making class, often increases the potential for conflict.

That part (2) of Proposition 4 holds is an immediate consequence of Proposition 1. If

private goods are as sufficient portion of the international bargain, then the the optimal

peaceful bargaining under complete information will be interior solutions for both strong

and weak types. In other words, Assumption A will be satisfied for both p and p′. Therefore

the potential for conflict is increasing in m and so expanding m in order to capture the

median voter only increases the potential for conflict.

Part (1) of Proposition 4 follows from allowing private goods to be sufficiently large.

When liberalization occurs, F gets to keep the private goods that it previously used to

payoff the elite in H. It is possible that this comes at the cost of paying more public goods

to H (whenever(p− 1√

2

)√x ≥ cH

n). However, if the total amount of private goods is

sufficiently large, then the loss of some public goods is outweighed by the gain in private

goods. Again, note that this is contingent on F being an institutional type whose decision

makers value private goods.

Taxation

In this section we expand on the baseline model to allow for redistributive taxation. Consider

the oligarchy case where E has the power to accept or reject bargains. However, members of

the out group, O, have a limited amount of power to demand some percentage of the private

goods offered to H through a redistributive tax rate τ ∈ [0, 1]. This ability to tax is meant

to represent various ways political institutions can allow out groups to constrain the decision

17

making of the elite. The larger the tax rate, the more power O has to impact policy.

In this case, a member of E’s utility in peace for a bargain (x, y) becomes

uE (x, y) =√x+ y(1−τ)

m

while a member of O’s utility for the same bargain is

uO (x, y) =√x+ τy

n−m .

As seems natural, we assume that n−m > m.

In war when H is strong E receives

wE = p

(√x+

y (1− τ)m

)− γ cH

n,

while O receives

wO = p

(√x+

τy

n−m

)− cH

n.

Under this setup we derive the following results

Proposition 5. When Assumption A holds for both p and p′,

1. In peace, y∗ and consequently uO is increasing in τ .

2. The potential for conflict is increasing in τ .

Proposition 5 demonstrates two effects of increasing the tax rate, τ . The first effect

is that y∗ is increasing in τ as the elites are forced to bargain harder in order to get the

same amount of utility. Hence, the out group benefits in two ways from taxation. Increased

taxation directly increases the proportion of private goods the out group receives while also

incentivizing the elites to demand a higher portion of the private good pie.

The second effect demonstrates the darker side of increased taxation power. As τ increases

E may bargain too hard, inducing conflict more frequently as τ increases. Hence, the out

18

group may wind up losing out if their taxation power becomes too high and induces conflict.

When this is the case, the out group would counterintuitively prefer institutions that put

greater limits on their ability to redistribute private goods.

Political Bias and War

Building on the taxation section, we are able to extend our model in a way that provides

a microfoundation for the concept of political bias in Jackson and Morelli (2007). In that

paper, a leader (or more generally here, the elite) makes a decision over whether or not

to accept a bargain or go to war with some bias over how the acquisition of resources are

valued in peace and war. For instance, a leader (or elite member) who is biased toward war

is able to capture a higher percentage of resources after a successful war than from a peaceful

bargain. In our model, it is easy to incorporate this notion as the out group having different

taxation technologies available in peace versus war. In this case, everything is the same as

in the taxation section, except that in war, a different tax rate, τ ′ ∈ [0, 1] is applied so that

war utilities become

wE = p(√

x+ y(1−τ ′)m

)− γ cH

n

and

wO = p(√

x+ τ ′yn−m

)− cH

n.

It follows immediately from Jackson and Morelli (2007) that, under this setup, war is

possible even whenH’s type is known for certain. This war result holds for certain parameters

when τ ′ becomes small. When τ ′ is low, members of E have a strong incentive to go to war

since they can keep a higher percentage of the private goods pie gained through war than

of those gained through peaceful bargaining. When τ ′ is low enough, it is possible that E’s

19

incentive to capture goods through war eliminates the peaceful bargaining space, inducing

war with certainty. An important point to note is that this effect is qualitatively different

than changes in γ. While low γ levels can increase the potential for war when H’s type

is unknown, they never induce war when H’s type is known since there always exists a

non-empty bargaining space.

Going beyond the Jackson and Morelli (2007) result, we can derive results on how τ ′

effects the potential for conflict in the incomplete information game.

Proposition 6. When Assumption A holds for both p and p′,

1. In peace, y∗ and consequently uO is decreasing in τ ′.

2. The potential for conflict is decreasing in τ ′.

In this case, lowering the out groups ability to capture rents in war (which is analogous

to increasing E’s war bias) causes E to bargain harder, increasing O utility. However, as E

begins to bargain too hard as τ ′ becomes small, the potential for conflict increases. Note

that these effects are the opposite of the tax described in the previous section which was not

contingent on whether the tax was collected in peace or after a war.

Taken together, Propositions 5 and 6 are quite striking. Institutions that increase the

level of redistribution in peacetime, but decrease it during a war, increase a state’s bar-

gaining power. However, these kinds of institutions also maximize the potential for conflict.

Propositions 5 and 6 also demonstrate exactly which aspects of redistributive institutions

increase the potential for peace. Namely, allowing elites to retain high levels of private goods

in peacetime, but depriving them of the spoils of war, minimizes the potential for conflict.

Conclusion

In this paper, we modified the standard incomplete information crisis bargaining model in

two ways. We allowed for two types of goods to be bargained over (public and private) and

20

we allowed one of the countries, H, to be made up of two groups (elites and citizens). Both

groups benefited from public goods, while only the elites had access to the private goods.

This led to a series of results on how the internal institutional makeup of H impacted the

potential for conflict. Counterintuitively, while the full liberalization of a oligarchy (limited

democracy) can reduce the potential for conflict, half-measures often exacerbate the potential

for conflict. For instance, expanding the decision making class (the elite) and increasing the

redistributive powers of the citizens (out group) often increases the potential for conflict.

There are three immediate extensions of the model. First, it is easy to imagine that

the two countries are bargaining over not just public and private goods, but a variety of

goods that impact the utility of a country’s internal divisions differently. This is especially

relevant when bargaining over the gains from trade. Second, the internal structure of H

presented here is quite reduced form. A more sophisticated model of internal voting and

taxation might capture subtleties missed here. In particular, when the level of strength is

endogenous to taxation (as in Chapman, McDonald, and Moser 2015), the redistributive

and strength increasing properties of taxation may combine to interesting effect. Third,

a dynamic extension where an oligarchy expanded the elite or liberalized over time would

likely demonstrate that such a process induces conflict through the commitment problem

mechanism.

21

References

[1] Baliga, Sandeep, David O. Lucca, and Tomas Sjostrom. 2011. “Domestic Political Sur-

vival and International Conflict: Is Democracy Good for Peace?” Review of Economic

Studies 78(2): 458-486.

[2] Bueno de Mesquita, Bruce, James D. Morrow, Randolph M. Siverson, and Alastair

Smith. 1999. “An Institutional Explanation of the Democratic Peace.” American Polit-

ical Science Review 93(4): 791-807.

[3] Chapman Terrence, Patrick J. McDonald and Scott Moser. 2015. “The domestic poli-

tics of strategic retrenchment, power shifts, and preventive war.” International Studies

Quarterly 59(1): 133–144.

[4] Colgan, Jeff D. 2013. “Domestic Revolutionary Leaders and International Conflict”

World Politics 65(4): 656-690.

[5] Fearon, James D. 1995. “Rationalist Explanations for War.” International Organization

49 (3): 379-414.

[6] Fearon, James D. 2008. “A Simple Political Economy of Relations among Democracies

and Autocracies.” working paper.

[7] Gleditsch, Kristian S. and Michael D. Ward. 2000. “War and Peace in Space and Time:

The Role of Democratization.” International Studies Quarterly 44(1): 1-29.

[8] Goemans, H.E. 2000. “Fighting for Survival: The Fate of Leaders and the Duration of

War.” Journal of Conflict Resolution 44(5): 555-579.

[9] Jackson Matthew O. and Massimo Morelli. 2007. “Political bias and war.” American

Economic Review 97(4): 1353–1373.

[10] Krainin, Colin and John Slinkman. 2016. “Bargaining with a Biased Autocrat.” Journal

of Theoretical Politics. Forthcoming.

22

[11] Mansfield, Edward D. and Jack Snyder. 1995 “Democratization and the Danger of War.”

International Security 20(1): 5-38.

[12] Narang, Vipin and Rebecca M. Nelson. 2009. “Who Are These Belligerent Democratiz-

ers? Reassessing the Impact of Democratization on War.” International Organization

63(2): 357-379.

[13] Putnam, Robert D. 1988. “Diplomacy and Domestic Politics: The Logic of Two-Level

Games.” International Organization 42(3): 427-460.

[14] Schultz, Kenneth A. 1998. “Domestic Opposition and Signaling in International Crises.”

American Political Science Review 92(4): 829-844.

[15] Ward, Michael D. and Kristian S. Gleditsch. 1998. “Democratizing for Peace.” American

Political Science Review 92(1): 51-61.

23

Appendix

Proof of Lemma 1

Proof. First consider the case where H is a democracy. When this is the case, F chooses

(x∗, y∗) as the solution to the following programming problem

maxx,y√

(x− x) + y − y

subject to√x ≥ p

√x− cH

n

x ≥ 0, y ≥ 0

First note that both the objective and constraint functions are continuously differentiable.

Since uF is decreasing in y and the constraint is unaffected by changes in y, y∗ = 0. Since,

uF is strictly decreasing in x, then the constraint binds. Also further note that the objective

function is quasiconcave, the constraint function is quasiconvex, and the gradient of the ob-

jective function is nonzero at the solution. Hence, the necessary conditions for maximization

are also sufficient.

It is then easy to calculate the interior solution as

√x = p

√x− cH

n

x∗ =[p√x− cH

n

]2Clearly, x∗ < x. So x∗ is always feasible when the right hand side of the equation is positive.

However, F cannot transfer a negative amount of x to H, so there is a corner case where

x∗ = 0 whenever

p√x− cH

n≤ 0

x ≤(cHnp

)2

24

Therefore, when H is a democracy, we can summarize equilibrium bargains as

(x∗, y∗) =

0, 0 x ≤

(cHnp

)2[p√x− cH

n

]2, 0 x >

(cHnp

)2

Now consider the case where H is an oligarchy. F chooses (x∗, y∗) as the solution to the

following programming problem

maxx,y√(x− x) + y − y

subject to√x+ y

m≥ p

(√x+ y

m

)− γ cH

n

x ≥ 0, y ≥ 0

Similar to the democracy case, both the objective and constraint functions are continuously

differentiable. Furthermore, the objective function is quasiconcave, the constraint function

is quasiconvex, and the gradient of the objective function is nonzero at the solution. Hence,

the necessary conditions for maximization are also sufficient.

First, we identify the interior solution by taking the first order condition on x.

1

2√

(x−x)− 1

2√x= 0

x = x− x

x∗ = x2

Since F ’s utility is strictly decreasing in x and y, the constraint binds. We can then use the

constraint to find y.

√x2+ y

m= p

(√x+ y

m

)− γ cH

n

y∗ = m(p− 1√

2


n

25

We now check the corner solution cases. First, if the RHS of the constraint is not positive,

or

p(√

x+ ym

)− γ cH

n≤ 0

y ≤ m[γcHnp−√x],

then F never makes a positive offer. In this case we have (x∗, y∗) = (0, 0).

Using the interior solution, we can check the condition that the total amount of private

goods, y, is sufficient to satisfy y∗.

y ≥ m(p− 1√

2


n

y ≥ m1−p

[(p− 1√

2

)√x− γ cH

n

]

When this is not the case, H receives all of the private goods y. The constraint then implies

that H receives the following amount of x

√x+ y

m= p

(√x+ y

m

)− γ cH

n

x∗ =[p√x+ (p−1)

my − γ cH

n

]2.

First, note that the RHS is always less than x. Next note that since x cannot be a com-

plex number, it must be that p√x + (p−1)

my − γ cH

n≥ 0. When this is the case, then

x∗ ≥ 0. When it is not the case, x∗ is set to 0. This happens when in addition to

y < m1−p

[(p− 1√

2

)√x− γ cH

n

], it is also the case that

p√x+ (p−1)

my − γ cH

n< 0

y > m1−p

[p√x− γ cH

n

].

However, since m1−p

[p√x− γ cH

n

]> m

1−p

[(p− 1√

2

)√x− γ cH

n

], this case can never hold.

26

We must also check the corner case where y∗ ≥ 0. Hence, for an interior solution we also

must have

m(p− 1√

2


n≥ 0

y ≥ mp

[γ cHn−(p− 1√

2

)√x].

Alternatively, when this corner case is satisfied, y∗ is set to 0 and we solve of x∗ using the

constraint

√x = p

(√x+ y

m

)− γ cH

n

x∗ =[p(√

x+ ym

)− γ cH

n

]2.

Therefore, when H is a oligarchy, we can summarize equilibrium bargains as

(x∗, y∗) =

0, 0 y ≤ m[γcHnp−√x]

[p(√

x+ ym

)− γ cH

n

]2, 0 m

[γcHnp−√x]< y < m

p

[γ cHn−(p− 1√

2

)√x]

[p√x+ (p−1)

my − γ cH

n

]2, y y ≥ m

p

[γ cHn−(p− 1√

2

)√x]

y < m1−p

[(p− 1√

2

)√x− γ cH

n

]x2, m

(p− 1√

2


notherwise

Finally, since war is costly, we need not worry about violating F ’s constraints at the same

time as H’s constraints.

27

Proof of Lemma 2

Proof. First, y ≤ m[γcHnp−√x]is never possible when p > γcH

n√x+ 1√

2, since

γcH

n[γcHn√x+ 1√

2

] −√x < γcH

n[γcHn√x

] −√x = ...

... = 0

Next, it is clear from inspection that x∗ is strictly decreasing inm when y < mp

[γ cHn−(p− 1√

2

)√x].

In this case, y∗ is always equal to 0. Therefore, uE is strictly decreasing in m whenever the

condition for this corner case is satisfied. Since y > 0, this means that the RHS of the

condition must be positive in order for it to be satisfied. For p ≥ γcHn√x+ 1√

2, the following

inequality holds

mp

[γ cHn−(p− 1√

2

)√x]≤ m

p

[γ cHn−((

γcHn√x+ 1√

2

)− 1√

2

)√x]= ...

... = mp[0] = 0

Therefore, this case never occurs whenever p > γcHn√x+ 1√

2.

In the y∗ = y corner solution, x∗ is strictly increasing in m since the term (p−1)m

y is

always negative since p < 1, but getting smaller in m. Recall, that the term in brackets

must always be positive as shown in Lemma 1. Private goods are once again, invariant

in this case. Therefore, uE is strictly increasing in m whenever the conditions for this

corner case is satisfied. Therefore, we must show that whenever the conditions for this

case are satisfied, then it is also true that p > γcHn√x+ 1√

2. The conditions for this case

are that mp

[γ cHn−(p− 1√

2

)√x]≤ y < m

1−p

[(p− 1√

2

)√x− γ cH

n

]. Since, uE is strictly

increasing in this case, we must show that two conditions defining this case are only si-

multaneously satisfied if p > γcHn√x+ 1√

2. From the previous paragraph, the first inequality

(mp

[γ cHn−(p− 1√

2

)√x]≤ y) is always satisfied when p > γcH

n√x+ 1√

2, since this gives the

LHS a value less than 0. The RHS of the second inequality (y < m1−p

[(p− 1√

2

)√x− γ cH

n

]),

28

however, equals 0 or less for any value p ≤ γcHn√x+ 1√

2since

m1−p

[((γcHn√x+ 1√

2

)− 1√

2

)√x− γ cH

n

]≤ m

1−p [0] = 0.

Therefore, this case is only non-empty when p > γcHn√x+ 1√

2.

Finally, in the case of the interior solution, it is now y∗ that is changing in m, while x∗

is invariant. Therefore, uE is now moving in the direction of y∗. Therefore, uE is strictly

increasing in m whenever y∗ is strictly increasing in m. The conditions for this is

(p− 1√

2

)√x− γ cH

n> 0

p > γcHn√x+ 1√

2

Furthermore, whenever the inequality is reversed, y∗ is strictly decreasing, or unchanging in

the case of equality. With this, we have covered all cases.

Proof of Proposition 1

Proof. We begin by demonstrating the first conditional statement and then show the second.

Since p > γcHn√x+ 1√

2, the proof of Lemma 2 demonstrates that the case where y ≤

m[γcHnp−√x]as well as the case where y < m

p

[γ cHn−(p− 1√

2

)√x]are ruled out. We

can also demonstrate that since p′ ≤ γcHn√x+ 1√

2, we can rule out the case where y <

m1−p′

[(p′ − 1√

2

)√x− γ cH

n

]since

(p′ − 1√

2

)√x− γ cH

n≤((

γcHn√x+ 1√

2

)− 1√

2

)√x− γ cH

n= ...

... = 0.

Since y ≥ 0, this case is never satisfied.

There are then six cases. While our result can be demonstrated without writing out each

case in detail, we do so in order to clearly show how the risk free and risky proposal values

29

are altered depending on the case.

1. y ≤ m[γcHnp−√x], y < m

1−p

[(p− 1√

2

)√x− γ cH

n

]:

A risk free proposal gives F a payoff of

√x−

[p√x+

(p− 1)

my − γ cH

n

]2,

while a risky proposal gives a value of

q[√

x+ y]+ (1− q)

[(1− p)

[√x+ y

]− cF

].


2, Lemma 2 implies that the value of the risk free proposal is decreasing

for F in m. The risky proposal is invariant in m. Therefore, in this case, the value of a risky

proposal relative to a risk free proposal is strictly increasing in m.

2. y ≤ m[γcHnp−√x], y ≥ m

1−p

[(p− 1√

2

)√x− γ cH

n

]:


√x

2+ (1− p) y −m

(p− 1√

2

)√x+ γm

cHn,


q[√

x+ y]+ (1− q)

[(1− p)

[√x+ y

]− cF

].



for F in m. The risky proposal is invariant in m. Therefore, in this case, the value of a risky

proposal relative to a risk free proposal is strictly increasing in m.

30

3. m[γcHnp−√x]≤ y < m

p′

[γ cHn−(p′ − 1√

2

)√x], y < m

1−p

[(p− 1√

2

)√x− γ cH

n

]:


√x−

[p√x+

(p− 1)

my − γ cH

n

]2,


q

√x−[p′(√

x+y

m

)− γ cH

n

]2+ y

+ (1− q)[(1− p)

[√x+ y

]− cF

].



for F in m. Since p′ ≤ γcHn√x+ 1√

2, Lemma 2 implies that the value of the risky proposal is

increasing for F in m. Therefore, in this case, the value of a risky proposal relative to a risk

free proposal is strictly increasing in m.

4. m[γcHnp−√x]≤ y < m

p′

[γ cHn−(p′ − 1√

2

)√x], y ≥ m

1−p

[(p− 1√

2

)√x− γ cH

n

]:


√x

2+ (1− p) y −m

(p− 1√

2

)√x+ γm

cHn,


q

√x−[p′(√

x+y

m

)− γ cH

n

]2+ y

+ (1− q)[(1− p)

[√x+ y

]− cF

].







31

5. y ≥ mp′

[γ cHn−(p′ − 1√

2

)√x], y < m

1−p

[(p− 1√

2

)√x− γ cH

n

]:


√x−

[p√x+

(p− 1)

my − γ cH

n

]2,


q

[√x

2+ (1− p′) y −m

(p′ − 1√

2

)√x+ γm

cHn

]+ (1− q)

[(1− p)

[√x+ y

]− cF

].







6. y ≥ mp′

[γ cHn−(p′ − 1√

2

)√x], y ≥ m

1−p

[(p− 1√

2

)√x− γ cH

n

]:


√x

2+ (1− p) y −m

(p− 1√

2

)√x+ γm

cHn,


q

[√x

2+ (1− p′) y −m

(p′ − 1√

2

)√x+ γm

cHn

]+ (1− q)

[(1− p)

[√x+ y

]− cF

].







An alternative method demonstrates that the potential for conflict is always increasing

32

in the size of the elite for interior solutions (i.e when Assumption A holds for both p and p′).

Let z be the number of additional citizens by which the elite is expanding. In this case, the

risky proposal has a value to F of

q

[√x

2+ (1− p′) y − (m+ z)

(p′ − 1√

2

)√x+ γ (m+ z)

cHn

]+(1− q)

[(1− p)

[√x+ y

]− cF

]

and the risk free proposal has a value to F of

√x

2+ (1− p) y − (m+ z)

(p− 1√

2

)√x+ γ (m+ z)

cHn.

Taking the difference of these two values (the risky proposal value minus the risk free

proposal value) gives the potential for conflict after the elite captures the median. Subtract-

ing this value from the potential for conflict before the elite expands (i.e. the potential for

conflict when the elite is still size m) gives

z(q[(p′ − 1√

2

)√x− γm cH

n

]−[(p− 1√

2

)√x− γm cH

n

]).

Since p > p′ and q < 1, this value is always negative. Therefore, the potential for conflict is

higher after expanding the elite when the optimal peaceful solution is interior for weak and

strong types.


Proof. First note that the potential for conflict in democracies are unaffected by changes in

γ since γ never enters into the decision making process in democracies. Since Assumption A

holds for both p and p′, the interior solution holds for oligarchies. Hence, a risk free proposal

33

gives F a payoff of

√x

2+ (1− p) y −m

(p− 1√

2

)√x+ γm

cHn,


q

[√x

2+ (1− p′) y −m

(p′ − 1√

2

)√x+ γm

cHn

]+ (1− q)

[(1− p)

[√x+ y

]− cF

].

Subtracting the risk free proposal from the risky proposal and taking the derivative with

respect to γ gives

m cHn(q − 1) ,

which is always negative since q < 1. Therefore, the potential for conflict is decreasing in

γ.


Proof. When H is a democracy, in the interior case, a risky proposal gives the following

payoff

q

[√x−

[p′√x− cH

n

]2+ y

]+ (1− q)

[(1− p)

[√x+ y

]− cF

]and a payoff of

q[√

x+ y]+ (1− q)

[(1− p)

[√x+ y

]− cF

]in the corner case. Clearly, if p induces a corner case, p′ does as well since p′ < p.

Regardless of the case, subtracting the risk free proposal from the risky proposal and

taking the derivative with respect to y gives a value of

q + (1− q) (1− p)− 1 = p (q − 1) .

34

Since q < 1, this means that the risky proposal is always decreasing in value over the risk

free proposal in y.

Now consider how the relative value of the risky proposal and risk free proposal changes

in y in the oligarchy case. By our assumption on y, both p and p′ lead to interior solutions.

Subtracting the risk free value from the risky value and taking the derivative of F ’s with

respect to y, gives

q (1− p′) + (1− q) (1− p)− 1 + p = q (p− p′) .

Since p > p′, the relative value of the risky proposal over the risk free proposal is increasing

in y hence, the size of the conflict space for oligarchies is increasing as y increases.


Proof. First, we show (1). Consider an oligarchy. y is high, so Assumption A holds for both

p and p′. The potential for conflict is the value to F of the risky proposal,

q

[√x

2+ (1− p′) y −m

(p′ − 1√

2

)√x+ γm

cHn

]+ (1− q)

[(1− p)

[√x+ y

]− cF

],

minus the value to F of the risk free proposal

√x

2+ (1− p) y −m

(p− 1√

2

)√x+ γm

cHn.

In a democracy, the potential for conflict is the value to F of the risky proposal,

q

[√x−

[p′√x− cH

n

]2+ y

]+ (1− q)

[(1− p)

[√x+ y

]− cF

],

35

minus the value to F of the risk free proposal

√x−

[p√x− cH

n

]2+ y.

Taking the former difference minus the latter difference gives

q

[√x2+ (1− p′) y −m

(p′ − 1√

2

)√x+ γm cH

n−√x−

[p′√x− cH

n

]2 − y]− ......−

√x2− (1− p) y +m

(p− 1√

2

)√x− γm cH

n+

√x−

[p√x− cH

n

]2+ y.

If this term is positive, than the potential for conflict under oligarchy is greater than under

democracy when all parameters are otherwise identical.

Taking the derivative of this term with respect to y gives

−qp′ + p

which is positive since p > p′ and q < 1. For a given set of parameters, the term in question

is bounded below. Since the domain of y is unbounded above, one can find a y∗ such that

the term in question is positive.

Now we turn to (2). The proof is quite similar to that used to show increases in m always

increase the potential for conflict when the optimal bargain is interior for both strong and

weak types. Consider the case where m enlarges to capture the median. Let z be the number

of additional citizens to which the elite must expand in order to capture the median. In this

case, the risky proposal has a value to F of

q

[√x

2+ (1− p′) y − (m+ z)

(p′ − 1√

2

)√x+ γ (m+ z)

cHn

]+(1− q)

[(1− p)

[√x+ y

]− cF

]

36

and the risk free proposal has a value to F of

√x

2+ (1− p) y − (m+ z)

(p− 1√

2

)√x+ γ (m+ z)

cHn.

Taking the difference of these two values gives the potential for conflict after the elite

captures the median. Subtracting this value from the potential for conflict before the elite

expands gives

z(q[(p′ − 1√

2

)√x− γm cH

n

]−[(p− 1√

2

)√x− γm cH

n

]).

Since p > p′ and q < 1, this value is always negative. Therefore, when y∗ is high enough that

Assumption A holds for both p and p′, expanding the elite to capture the median always

increases the potential for war.


Proof. F chooses (x∗, y∗) as the solution to the following programming problem

maxx,y√

(x− x) + y − y

subject to√x+ y(1−τ)

m≥ p

(√x+ y(1−τ)

m

)− γ cH

n


1

2√

(x−x)− 1

2√x= 0

x = x− x

x∗ = x2

37



√x2+ y(1−τ)

m= p

(√x+ y(1−τ)

m

)− γ cH

n

y∗ = m1−τ

[(p− 1√

2

)√x− γ cH

n

]+ py

Hence, y∗ is increasing in τ . Therefore, uO is increasing in τ .


√x

2+ (1− p) y − m

1− τ

[(p− 1√

2

)√x− γ cH

n

],


q

[√x

2+ (1− p′) y − m

1− τ

[(p′ − 1√

2

)√x− γ cH

n

]]+ (1− q)

[(1− p)

[√x+ y

]− cF

].

Taking the difference and then the derivative with respect to τ , we find that the potential

for conflict is increasing when

− qm

(1−τ)2

[(p′ − 1√

2

)√x− γ cH

n

]+ m

(1−τ)2

[(p− 1√

2

)√x− γ cH

n

]> 0(

p− 1√2

)√x− γ cH

n− q

[(p′ − 1√

2

)√x− γ cH

n

]> 0

which is always true since p > p′ and q < 1. Therefore, the potential for conflict is always

increasing in τ .

38


Proof. F chooses (x∗, y∗) as the solution to the following programming problem

maxx,y√

(x− x) + y − y

subject to√x+ y(1−τ)

m≥ p

(√x+ y(1−τ ′)

m

)− γ cH

n


1

2√

(x−x)− 1

2√x= 0

x = x− x

x∗ = x2



√x2+ y(1−τ)

m= p

(√x+ y(1−τ ′)

m

)− γ cH

n

y∗ = m1−τ

[(p− 1√

2

)√x− γ cH

n

]+ py 1−τ ′

1−τ

Hence, y∗ is decreasing in τ ′. Therefore, uO is decreasing in τ ′.


√x

2+ y − py1− τ

′

1− τ− m

1− τ

[(p− 1√

2

)√x− γ cH

n

],


q

[√x

2+ y − p′y1− τ

′

1− τ− m

1− τ

[(p′ − 1√

2

)√x− γ cH

n

]]+(1− q)

[(1− p)

[√x+ y

]− cF

].

Taking the difference and then the derivative with respect to τ ′, we find that the potential

39

for conflict is decreasing when

p′y 11−τ − py

11−τ < 0

p′ − p < 0

Which is always true since p′ < p. Therefore, the potential for conflict is always decreasing

in τ ′.

40

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Who Wins The Peace? Elites, Citizens, and the Decision … · Elites, Citizens, and the Decision for War ... Forinstance,ifm= 1 and ... F be F’s payoﬀ when it makes a risk free