making peace on the cheap - robert j. carroll...i call this making peace on the cheap, as the third...
TRANSCRIPT
Making Peace on the Cheap
Robert J. Carroll
March 26, 2019
Abstract
I study a class of peacemaking strategies relating the bargaining model ofwar to the neoclassical theory of exchange and, in so doing, develop a generalequilibrium model of aid, trade, and war. In the full-blown version of themodel, a third party revises states’ economic and military resources, thoughthese revisions are costly. After the adjustment, the states make a trade anddetermine whether to fight for the right to consume all of both commodities.I study a very general version of the model from the differentiable point ofview, which provides a deeper geometric understanding of the constituentbargaining and trading models. My main result ensures the existence of acost-minimizing pacifying aid schedule, so that peace truly can be made onthe cheap.
Department of Political Science, Florida State University. Email: [email protected] to Phil Arena, Navin Bapat, Kyle Beardsley, Ryan Brutger, Mark Crescenzi, AmandaDriscoll, Mark Fey, Mike Gibilisco, Hein Goemans, Tasos Kalandrakis, Brenton Kenkel, BethanyLacina, Jeff Marshall, Brian Pollins, Amy Pond, Curt Signorino, Mark Souva, and Randy Stonefor helpful comments. The paper has benefited from presentations at Rochester, SUNY-Buffalo,and at the 2013 annual meeting of the Peace Science Society (International). John Duggan’s onlinenotes on optimization and Pareto efficiency have also been quite helpful. Thanks also to the W.Allen Wallis Institute of Political Economy, the star lab, the Kroc Institute for Interna-tionalPeace Studies, and the Institute for Humane Studies. All errors are my own.
C O N T E N T S
1 The Leading Example 41.1 The States’ Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 The Terms of Trade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.3 War as a Costly Lottery . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.4 Visualizing the States’ Setting . . . . . . . . . . . . . . . . . . . . . . . . 71.5 The Peacemaker’s Adjustment . . . . . . . . . . . . . . . . . . . . . . 91.6 Pacifying Transfer Schedules, Three Styles . . . . . . . . . . . . . . . . 111.7 Profit! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 Proof of Concept 18
3 The General Case 253.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1.1 The Disputants’ Parameter . . . . . . . . . . . . . . . . . . . 253.1.2 The Consumption and Allocation Spaces . . . . . . . . . . 253.1.3 Exchange via the Naïve Walrasian Protocol . . . . . . . . . 263.1.4 War as a Costly Lottery . . . . . . . . . . . . . . . . . . . . . 283.1.5 The Third Party’s Aid Schedule . . . . . . . . . . . . . . . . 28
3.2 The Main Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2.1 Can Peace Be Made...? . . . . . . . . . . . . . . . . . . . . . . 303.2.2 ...On The Cheap? . . . . . . . . . . . . . . . . . . . . . . . . . 30
4 Conclusion 32
A Appendix 34A.1 Preliminaries for the General Case . . . . . . . . . . . . . . . . . . . . . 34A.2 Peace-Point Aid Schedules . . . . . . . . . . . . . . . . . . . . . . . . . 35A.3 Finding the Cheapest Pacifying Aid Schedule . . . . . . . . . . . . . 39
A.3.1 Compactness of the Trade-to-Peace Set . . . . . . . . . . . . 39A.3.2 Continuity of the Trade-to-Peace Set . . . . . . . . . . . . . 45A.3.3 Aid-Local Cost Minimization . . . . . . . . . . . . . . . . . . 47
Most of us agree war is awful, so it makes sense that we spend so much timelearning ways to prevent it. In this paper, I will study a class of peacemakingstrategies based on the neoclassical theory of exchange. Under this mechanism, athird party revises the initial conditions of a familiar bargaining problem througheconomic and military aid. The former alters the disputants’ distribution ofresources, while the latter shifts the bargaining range by adjusting their relativemilitary capabilities. In doing so, the aid schedule updates the terms of trade andthe potential for peace after the exchange, and in some cases, this can make peacewhere once there was not. I call this making peace on the cheap, as the third partyneed not create a peaceful distribution of wealth and power herself; instead, sheonly has to shift the status quo enough to let trade do the rest of the job.
The formal model studied here is a multi-dimensional variant of the canonicalbargaining model of war, wherein two states first try to negotiate a peacefulsettlement and then decide whether to fight a costly war lottery. The model ismulti-dimensional because trade requires at least two commodities. In bargainingmodels like this one, the analyst must choose some protocol dictating how thepotential belligerents strike a deal. To impose the onus of peacekeeping ontothe third party, I compel the disputants to trade via a simple Walrasian protocolignorant of the plausible violence. The resulting theory marries the bargainingmodel of war and the general equilibrium theory of value in a novel way. Thoughthe model’s explanations for war lie outside the traditional set, they provide botha useful baseline and an ideal hard case for the proposed mechanism.
By modeling exchange via perfect competition, the analyst reduces the com-modity space to a single equation equilibrating supply and demand. Under stan-dard assumptions common to both the bargaining and Walrasian models, thisequilibrium equation possesses many remarkable properties; these allow for athoroughly geometric analysis. Using some deceptively simple tools, I provideresults on the fundamental structure of the bargaining model of war in higher-dimensional spaces and on the way this model relates to the smooth general equi-librium exchange model. For example, Lemma A.13 demonstrates that the tra-ditional bargaining range retains its topological properties once embedded intomore complicated environments, which in turn allows us to map back to the setof pacifying trades and the set of peaceable initial allocations. Once this moreabstract geometric analysis is complete, it is easy to evaluate the possibility ofpeacemaking on the cheap: the main substantive result (Theorem 3.6) guaranteesthat a third party can find a cost-minimizing pacifying aid schedule. This resultprovides some reason for optimism, then, since it seems intuitive that peacemak-ing should be more likely if it is relatively inexpensive.
1
These methodological decisions have been made for substantive, rather thantechnical, reasons. To learn how third parties can make peace—a matter of nor-mative importance—we must first determine how their actions change bargain-ing problems. This task is not all that straightforward. In a tangible sense, thethird party’s aid schedule stretches the set of potential distributional outcomes.In doing so, the aid schedule morphs the respective sets of Pareto-optimal andpeaceful allocations, the intersection of which is precisely the bargaining range.These latter quantities move with both economic and military aid. After that, thebargaining range’s transformation influences the sets of pacifying Walrasian equi-libria and the initial allocations that yield peace through trade. In other words,the model layers concepts atop one another, and it is crucial to understand howthese layers relate (i.e., the properties of the maps between them) and the substan-tive consequences of those relationships (i.e., the topological properties of therelevant sets of resource distributions, relative capabilities, equilibria, and out-comes). Regardless of the protocol, we theorists of international relations mustcontinue to think hard about the underlying structure of bargaining and howoutside actors, conflict processes, or exogenous shocks influence that structure.
The literature on third-party peacemaking is predictably robust. Many schol-ars focus on how to resolve the usual bargaining problems (Fearon 1995). Infor-mational accounts, especially about mediation, have received the most attention,with a veritable “revolution” in the literature emerging in recent years (Gartner2013). These mechanisms fall into two genera: facilitation and information shar-ing. The latter has been formalized more (Kydd 2003, 2006; Smith and Stam 2003;Rauchhaus 2006) despite mixed evidence (Bercovitch 1986; Bercovitch and Hous-ton 1996; Quinn et al. 2006; Beardsley 2008; Savun 2008). Indeed, the empiricssuggest such tactics aren’t effective relative to other kinds of peacemaking or evento unmediated conflict. Fey and Ramsay (2010) proffer an explanation: just asdisputants have incentives to lie to one another, so too do they have incentivesto lie to the mediator, rendering the mediator a useless go-between.
It comes as no surprise, then, that many have turned their attention to moreforceful kinds of peacekeeping efforts. Third parties may engage in more activeforms of arbitration, even going so far as to act as agenda setter (Camiña andPorteiro 2009). They might change the incentive structure under which the dis-putants operate by the provision of carrots or—much more commonly—sticks(Goltsman et al. 2009). Of more direct interest to scholars of conflict, strong thirdparties can use their might to influence outcomes, be it through direct peacekeep-ing missions (Fortna 2004, 2008) or by securing agreements by enforcing theirterms (Favretto 2009). These mechanisms represent a more muscular approach to
2
making peace, but recent scholarship on informational mediation has indicatedthat such measures may be necessary for peacemaking to have bite.
The model also provides a simple theory relating conflict and trade, so somelinkages are in order. On the liberal side, many scholars (Keohane and Nye 1989;Bliss and Russett 1998; Russett and Oneal 2001; Oneal et al. 1996; Oneal andRussett 1997, 1999) have found a persistent, positive effect of trade on peace, evenif the model accounts for obvious concerns of endogeneity (Hegre, Oneal andRussett 2010). Of special note, the work of Solomon Polachek (Polachek 1980,1997; Polachek, Robst and Chang 1999) provides the only formal microfounda-tions for why trade would have a pacifying effect. In the model most similar tothis one, Böhmelt (2010) takes a similar approach oriented around opportunitycosts forgone, but he concerns himself with trade ties outside the belligerent dyad.On the realist side, those influenced by Hirschman (1945) have observed that thegains from trade are not always distributed evenly, thus bringing about new rela-tive gains calculations that do not always favor peace (Gilpin 1981; Mearsheimer1990). This highlights the idea that not all trades are the same, which raises con-cerns of saliency across contexts (Barbieri 1996, 2002). Less ideologically, manyempirically-minded scholars have subjected the trade-war relationship to a bat-tery of statistical tests, positing that trade follows the flag (Pollins 1989a,b; Gowaand Mansfield 1993; Gowa 1994), that the relationship is not necessarily mono-tonic (Mansfield 1994), or that the question suffers from measurement problems(Mansfield and Pollins 2001; Gartzke and Li 2003).
A word on technique is also appropriate. The re-introduction of the calculusin the study of general economic equilibrium is due to the pathbreaking paperby Debreu (1970), who used Sard’s lemma to prove the set of economies withan infinite set of equilibria is, in a precise sense, pathological. This gave rise tothe study of so-called regular economies. Debreu’s breakthrough, in turn, wasinfluenced by the classic Topology from the Differentiable Point of View by Milnor(1965). The rapid application of differential topology to economic equilibrium(particulary at Berkeley under Debreu, Stephen Smale, and Andreu Mas-Colell)gave rise to remarkably clear published lecture notes on the subject by Dierker(1974) and Mas-Colell (1985). My most obvious debts here are to Yves Balasko(Balasko 1988, 2009, 2011), whose postmodern approach based on the naturalprojection from the set of equilibria back to the parameter space is most appropri-ate for the present problem. There is much more to be done in theoretical IR inthis spirit, both by focusing on other important substantive quantities from ourtheories and by repeating the exercise on bargaining protocols that aren’t quiteas well-behaved as naïve Walrasian exchange.
3
The paper will proceed in four parts. To develop intuitions, I describe a verysimplified leading example of the model in Section 1. Section 2 provides proofof concept by re-interpreting the Camp David Accords (which everybody in thisliterature seems to have an opinion on). Finally, I develop and analyze the generalmodel of aid, trade, and war in Section 3. I then conclude.
1 T H E L E A D I N G E X A M P L E
Before launching into the full-blown model, it will be useful to introduce in-tuitions in a simple parametric setting. (It is hoped this also jogs the reader’smemory from micro classes long since forgotten—or in the author’s case, nevertaken.) The presentation will therefore be quite informal, with a rigorous treat-ment postponed until the general case. Many of the assumptions made here willbe relaxed in the more general setting; the reader’s kind patience is appreciated.
We study an interaction between three actors: a peacemaker (indexed i “ 0)and two states (i P t1,2u). The third party acts more like a social planner thanan actor with a fully-integrated set of economic interests, so you can animateher as an organization or a state much larger than the potential disputants. Thebasic idea here is that the peacemaker may adjust the initial conditions prior tothe interaction among the states, where the terms of that adjustment will be assimple as possible. Accordingly, we begin by describing that interaction; after,we will discuss the peacemaker’s options.
1.1 T H E ST AT E S ’ S E T T I N G
The states are each concerned with consumption of two goods (indexed ` P t1,2u).For the mechanism to be “on the cheap,” we need at least two goods, for reasonsthat will become obvious later on. We write State i ’s consumption of good ` asx`i , and we write her overall consumption vector as xi “
`
x1i , x2
i
˘
. We assumethese live in the consumption set Xi “ R2
``. State i has preferences over theseconsumption vectors, namely
uipxiq “`
x1i
˘αi`
x2i
˘1´αi .
where αi P p0,1q tells us how much State i prefers Good 1 to Good 2.Each state enters into the world with an initial endowment of resources. Call
theseωi “`
ω1i ,ω2
i
˘
PR2``. This latter assumption means that each state begins
with at least a little of both goods. Throughout, we will have reason to refer to
4
ω “ pω1,ω2q, which is the overall distribution of resources among the states.Define also τ` “ω`
i `ω`2 , which is the total amount of Good `. Finally, we have
τ “`
τ1,τ2˘
. Note that ω P Ω :“ R4`` is a distribution of resources, whereas
τ PR2`` is an admissable consumption vector.
1.2 T H E T E R M S O F T R A D E
Throughout this model and its generalized cousin, we focus on naïve Walrasianexchange. This is one of many “bargaining protocols” that could have been usedto model the transition from initial endowments to post-exchange outcomes;others might include take-it-or-leave-it offers issued by either party, or Rubensteinbargaining, or Nash bargaining, or whatever.1 (Walrasian exchange happens topossess many desirable properties to be exploited in the general case.) In choosingthis protocol, we impose the burden of peacemaking fully onto the (appropriately-named) peacemaker’s shoulders. We study a price vector, q “ pq1, q2q P R2
``,where q` tells us how much one unit of Good ` is worth (in dollars).
Each state acts to maximize consumption utility given initial resources andthe value environment; that is, they solve the constrained maximization problem
maxxi :q¨xiďq¨ωi
`
x1i
˘αi`
x2i
˘1´αi .
This yields solutions—i.e., demand functions fi :R3``ÑR2
``—of the form
fipq , q ¨ωiq “
ˆ
αi q ¨ωi
q1,p1´αiqq ¨ωi
q2
˙
.
1The perfect competition undergirding Walrasian equilibrium may seem implausible withonly two states, but it turns out that Walrasian equilibria are equivalent to a common variant ofbargaining equilibria under reasonable conditions (Yildiz 2003; Dávila and Eeckhout 2008; Penta2011), even in the two-state, two-commodity case where bargaining would seem to be the onlychoice.
5
A Walrasian equilibrium is a price vector‹q so that supply equals demand:
total demand for Good 1hkkkkkkkkkkikkkkkkkkkkj
α1‹q ¨ω1‹q
1 `α2‹q ¨ω2‹q
1 “ τ1,
p1´α1qq ¨ωi‹q
2 `p1´α2qq ¨ωi
‹q
2
loooooooooooooooooomoooooooooooooooooon
total demand for Good 2
“ τ2.
Solving, we find the unique equilibrium price ratio
‹q
1
‹q
2 “α1ω
21 `α2ω
22
p1´α1qω11 `p1´α2qω
12
.
We therefore fix‹q
2“ 1;
‹q may refer to the relative price of Good 1 or the vector
of prices (the second entry being one) depending on the context. This shouldintroduce no problems.
1.3 WA R A S A C O S T LY LO T T E RY
In keeping with the standard literature, we conceive of war as a costly lotteryafter the Walrasian exchange just discussed. State 1 wins the war with probabilityp P p0,1q, and State 2 wins with probability 1´ p. The resulting expected utilitiesfor war are
U1pwarq “ p u1pτq´ c1,U2pwarq “ p1´ pqu2pτq´ c2,
where c1, c2 ą 0 are the costs of war measured in utils. This is a simple multidi-mensional variant of the standard expected utilities. If
Uipwarq ď ui
´
fi
´
‹q ,‹q ¨ωi
¯¯
for both states, then we have peace. Otherwise, we have war.
6
State 1’s Good 1
State 2’s Good 1St
ate
2’s G
ood
2 State 1’s Good 2
Figure 1. The Edgeworth box.
1.4 V I S UA L I Z I N G T H E ST AT E S ’ S E T T I N G
Before we get into the peacemaker’s task, we can first visualize the states’ setting;the key tool will be the classic Edgeworth box. An example is shown in Figure 1.The Edgeworth box features two origins: at the southwest corner, State 1 hasno resources, while State 2 has all resources; at the northeast corner, State 2 hasno resources and State 1 has all resources. Any given allocation x is ω1
1 east ofthe western boundary and ω1
2 west of the eastern boundary; the same goes fornorth-south with Good 2. The box is therefore of dimension τ1 by τ2; in thisexample, the box is drawn to scale with τ1 “ p1`
?5qτ22 to maximize beauty.
The line running from the southwest corner to the northeast corner depictsthe set of all Pareto-efficient allocations. We can offer a more complete charac-terization; as the figure shows, a Pareto-efficient point is one where the marginal
7
State 1’s Good 1
State 2’s Good 1St
ate
2’s G
ood
2 State 1’s Good 2
Figure 2. The peace lens and bargaining range in the Edgeworth box.
rates of substutition are equal.2
We can also imagine how the war decision plays out in the Edgeworth box.In Figure 2, the lower contour sets for the expected utilities of war are depictedin gray. In the southwestern gray region, State 1 prefers fighting to the bundlesin question; in the northeastern gray region, the same goes for State 2. Thethick black line depicts the familiar bargaining range: the allocations that areboth efficient and peaceful. (As a preview of coming attractions, the basic topo-logical properties of the bargaining range—in particular, its compactness andconnectedness—will play a key role in the general analysis.) Contra unidimen-
2We can fully characterize the Pareto efficient points under our functional form impositions:
Bu1Bx21
Bu1Bx21
pxq “Bu2Bx2
2
Bu2Bx22
pτ´ xqðñp1´α1qx
1
α1x2“p1´α2qpτ
1´ x1q
α2pτ2´ x2q,
ðñ x2“
p1´α1qα2τ2x1
pα2´α1qx1`α1p1´α2qτ1.
If α1 “ α2, the homotheticity of the ui s yields the linear x2 “τ2
τ1 x1.
8
sional bargaining models, an uncountable bargaining range exists in two dimen-sions even if c1 “ c2 “ 0, so long as α1 ‰ α2—that is, once we’ve got a rich enoughconsumption space (which might even be the same physical good in two locations,or at two times, or under two states of the world), heterogenous preferences arealone sufficient for a non-singleton bargaining range; the costs of war are notstrictly necessary. (This interesting truism also depends on the standard assump-tion that the utility functions are concave.) As State 1’s probability of victory( p) increases, the entire apparatus—lens, bargaining range, &c.—moves northeast,and likewise for the southwest as p decreases.
1 .5 T H E P E AC E M A K E R ’ S A D J U S T M E N T
We now turn our attention to how the peacemaker can adjust the underlyingbargaining problem. She does so by “adjusting” the underlying economic andmilitary situation between the two states. In this leading example, the terms ofthat adjustment will be exceedingly simple: lump-sum economic and militarytransfers. Though these may appear unrealistic—which peacemaker takes fromthe rich and gives to the poor, or vice versa?—they do describe the outcomes ofeconomic and military interventions, albeit in reduced form. And, with concaveutility functions like the one used in the leading example, aid and transfers maynot work at appropriately similar scales. These disclaimers aside, the lump-sumcharacterization to follow ought to help the reader understand the underlyinggeometry of the situation; in the general case, we study true aid—which stretchesthe setting—rather than working within a fixed box.
Suppose the peacemaker has at her disposal three policy instruments: a lump-sum transfer on Good 1, A1 P r´ω1
1,ω21s; a lump-sump transfer on Good 2, A2 P
r´ω21,ω2
2s; and a lump-sum military transfer, Am P r´p, 1´ ps. Collect theseinto a transfer schedule, A :“ pA1,A2,Amq. After the transfer, we have endowments
rω1 :“`
ω11 `A1,ω2
1 `A2˘ ,
rω2 :“`
ω12 ´A1,ω2
2 ´A2˘ .
Observe that this preserves τ. Likewise for military transfers:
rp :“ p `AmP r0,1s.
Thus, the entire situation has been adjusted.Again, the peacemaker’s adjustments can easily be visualized in the context of
our Edgeworth box. Consider Figure 3. Begin from the black dot depicting the ini-
9
State 1’s Good 1
State 2’s Good 1St
ate
2’s G
ood
2 State 1’s Good 2
Figure 3. The peacemaker’s adjustment in the Edgeworth box.
tial endowment,ω. Through economic instruments A1 and A2, the peacemakercan slide this dot across the Edgeworth box; because the transfer is lump-sumin this example, any adjustment remains inside the box. Slides northeast benefitState 1, whereas slides southwest benefit State 2. Of course, the peacemaker mightslide to the northwest or southeast, which gives one state one good at the expenseof the other. And, the peacemaker might only slide in one dimension, which youcan interpret as a monetary transfer. As for the military adjustment Am: Figure 3demonstrates that it shifts the entire peace lens, including the bargaining range.If Am ą 0, the peace lens shifts northeast; if Am ă 0, it shifts southwest.
In this simple setting, consider the following payoffs for the peacemaker:
U0pAq “
$
&
%
R´ κ1
2
“
A1‰2´
κ2
2
“
A2‰2´
κm
2 rAms
2 , peace
´κ1
2
“
A1‰2´
κ2
2
“
A2‰2´
κm
2 rAms
2 , war.
Here R ě 0 is a (for now exogenous) reward for peace and the respective κ ą 0terms scale marginal costs for adjustment. So for now, adjustment is just a matterof cost minimization; we will focus on cost-minimizing peaceful allocations.
10
1.6 PAC I F Y I N G T R A N S F E R S C H E D U L E S , T H R E E ST Y L E S
We work quite slowly here and develop no new results; instead, this will helpprovide geometric intuitions and introduce the key technical tools. Our primaryobservation is straightforward.
1.1 ObservationIn the leading example, there exists a pacifying transfer schedule.
In other words, the peacemaker can indeed make peace. The pacifying transferschedules lie along a continuum: on the one end lies pure economic transfer, andon the other lies pure military transfer. These rely on different, albeit related,properties of the underlying trade protocol. We consider each in turn beforeturning attention to hybrid transfers.
In the case of pure economic transfer, we rely on the Second FundamentalTheorem of Welfare Economics (which will be a workhorse in the general case,too). To state the theorem:
1.2 Theorem (Second Welfare Theorem)Any Pareto-efficient allocation is the outcome of some Walrasian equilibrium withappropriate lump-sum transfers.
(The proof may be found in any introductory microeconomics textbook (e.g.Mas-Colell, Whinston and Green 1995, Proposition 16.D.1).) In particular, The-orem 1.2 tells us that every allocation in the (pre-modified) bargaining range issupportable if the peacemaker chooses appropriate economic transfers A1 and A2.
The Second Welfare Theorem (and thus the case of pure economic transfer)are easy to visualize; see Figure 4. In this example, State 1 is relatively powerful(hence the peace lens lies more toward the northeast), relatively rich in Good 2(hence the endowment lies far north in the box), and relatively poor in Good 1(hence the endowment lies far west in the box). Indeed, State 1’s relative povertyin Good 1 is sufficient for her to be dissatisfied not just with the initial endowment,but also with the mutually beneficial Walrasian equilibrium that would obtainfromω. In other words, if the peacemaker does not act, trade will fail to pacify,and war will occur.
The Second Welfare Theorem tells us that, for any element of the bargainingrange (call it x), there exists a line of endowments that yields x as an equilib-rium outcome. Thus, for any endowment lying along one of these lines, tradebrings peace. In the figure, the endowments satisfying this property have beenhighlighted in green; in the sequel, we will call this the trade-to-peace set. Quite
11
State 1’s Good 1
State 2’s Good 1St
ate
2’s G
ood
2 State 1’s Good 2
Figure 4. Pacifying transfer schedules: the case of pure economic transfer. Holding thepeace lens fixed, the peacemaker chooses A1,A2 ą 0 to shift ω northeast to rω; this, inturn, yields the peaceful Walrasian equilibrium allocation f . Any lump-sum transfer toan allocation in the green zone would have brought about peace; this is the union of alllevel-set-separating hyperplanes (i.e., linear fibers) running through the bargaining range.
simply, then, the peacemaker’s task is just to find some lump-sum transfer thatmoves the initial endowmentω to some modified endowment rω that lies in thistrade-to-peace set. To do so, the peacemaker must make State 1 relatively richerin keeping with her relative strength. Thus, sinceω lies southwest of the trade-to-peace set, we have A1,A2 ą 0. Hadω fallen northeast of the trade-to-peace set,then State 1 would have been too rich relative to her power, and so the lump-sumtransfers would have been negative.
Let us now consider the other extreme case, where A1 “A2 “ 0—so that thepeacemaker lives up to her name through military transfers rather than economictransfers. Here there are no lump-sum transfers to work with, so the SecondWelfare Theorem cannot help us. We instead use the canonical existence resultfor Walrasian equilibria:
12
1.3 Theorem (Walrasian Existence)A Walrasian equilibrium exists for all ω.
(Consult Mas-Colell, Whinston and Green (1995, Proposition 17.C.1).) In partic-ular, there is a Walrasian equilibrium from the initial endowment. We also knowmuch more about this outcome, per the First Fundamental Theorem of WelfareEconomics.
1.4 Theorem (First Welfare Theorem)The outcome of any Walrasian equilibrium is Pareto-efficient.
(Again, consult Mas-Colell, Whinston and Green (1995, Proposition 16.C.1).)Thus, the outcome of Walrasian exchange from the initial endowment is part ofsome bargaining range, though that bargaining range might not comport withthe status quo balance of power.
Thus, military transfer is a matter of moving the trade-to-peace set toward theinitial endowment, rather tham moving the initial endowment toward the trade-to-peace set. See Figure 5. In this example, State 1 is powerful but relatively poor.In contrast with the previous example, where the peacemaker used economic aidto make State 1 as rich as she was powerful, here the peacemaker uses militaryaid to make State 2 as powerful as she is rich.
There might also be hybrid pacifying transfer schedules (and these are to bethought of as primary); these include some mix of economic and military aid togenerate peace. We will consider a particular hybrid pacifying transfer under thefollowing (homeric) simplifying assumptions designed to ease intuitions:
• the states share the same preferences over goods, so that α1 “ α2 :“ α; and
• war is costless, so that c1 “ c2 “ 0.
Since the states have the same preferences, the equilibrium price simplifies to
‹q
1“
ˆ
α
1´α
˙ˆ
τ2
τ1
˙
.
In this simple setting, the peacemaker’s task is to find some A that solves theconstrained cost-minimization problem
minκ1
2
“
A1‰2`κ2
2
“
A2‰2`κm
2rAm
s2
13
State 1’s Good 1
State 2’s Good 1St
ate
2’s G
ood
2 State 1’s Good 2
Figure 5. Pacifying transfer schedules: the case of pure military transfer. Holding theinitial endowment and its subsequent equilibrium fixed, the peacemaker chooses Am ă
0 to shift the trade-to-peace set southwest; this, in turn, yields the peaceful Walrasianequilibrium allocation f .
such that both states weakly prefer peace after they trade from the modified en-dowments:
f 11
´
‹q ,‹q ¨ rω1
¯αf 21
´
‹q ,‹q ¨ rω1
¯1´α“ rp
`
τ1˘α `τ2˘1´α ,
f 12
´
‹q ,‹q ¨ rω2
¯αf 22
´
‹q ,‹q ¨ rω2
¯1´α“ p1´ rpq
`
τ1˘α `τ2˘1´α .
These must both hold with equality because of our simplfying assumptions: iden-tical preferences and zero war costs close the peace lens to a single point atprpτ, p1´ rpqτq. At this point, both states are indifferent between war and peace.(This is therefore a rather extreme test of our peacemaking mechanism!)
Despite our simplifying assumptions, the actual solutions for an optimal inte-rior hybrid pacifying transfer schedule A are cumbersome and not of much directsubstantive use. Instead, we focus on a few simple properties of this transferschedule under our simplifying assumptions.
14
State 1’s Good 1
State 2’s Good 1St
ate
2’s G
ood
2 State 1’s Good 2
Figure 6. Pacifying transfer schedules: the hybrid case. Under our strong simplifyingassumptions, peace obtains only at rpτ, which is Pareto-efficient. Thus, by combiningAm ă 0 and A1,A2 ą 0, the peacemaker shifts both the trade-to-peace line and the modi-fied endowment toward one another.
1.5 RemarkAt an optimal transfer, the slope of the line from ω to rω is given by
A2
A1“
ˆ
1´αα
˙ˆ
τ1
τ2
˙ˆ
κ1
κ2
˙
ą 0ùñ sgnpA1q “ sgnpA2
q.
In other words, on the economic side, the peacemaker leans more on Good 2 if (1)Good 2 offers more consumption utility; (2) Good 2 is more rare; and (3) Good2 is easier to transfer. This makes good sense, especially under the assumptionthat κ1 “ κ2. If this is the case, then the slope of the line is just p1´ααq
`
τ1τ2
˘
,which happens to be 1
‹q
1. The slope of the line between endowments and Wal-rasian equilibria is ´
‹q
1, meaning that, if κ1 “ κ2, the optimal transfer schedule
‹
A chooses the line perpendicular to the trade line. This is, of course, the shortestdistance betweenω and the the trade line. As κ1 and κ2 diverge, the adjustmentline shifts accordingly. In the limit, as κ1 grows large (or κ2 grows small), the
15
adjustment line grows close to vertical; as κ2 grows large (or κ1 grows small), theadjustment line grows close to horizontal.
Indeed, given our functional form impositions, we can get a precise character-ization of when the peacemaker gives wealth or power to a state.
1.6 RemarkAt an optimal interior pacifying transfer schedule,
A1,A2ě 0ðñ p ě
αω11
ω11 `ω
12
`p1´αqω2
1
ω21 `ω
22
ðñAmď 0.
In other words, the economic transfer favors State 1 if and only if State 1 is pow-erful relative to her wealth. Likewise, the military transfer favors State 1 if andonly if State 1 is weak relative to her wealth. Thus, the depiction in Figure 6depicts a solution to broad class of peacemaking problems—were the shoe on theother foot, the economic transfer would hinder State 1 and the military transferwould help her. This condition offers simple comparative statics on our small setof parameters, which we now record.
1.7 RemarkCeteris paribus:
1. the optimal A1 and A2 are both increasing in p, while the optimal Am is de-creasing in p;
2. the optimal A1 and A2 are increasing in State 2’s endowments and decreasingin State 1’s endowments;
3. the optimal Am is decreasing in State 1’s endowments and increasing in State1’s endowments;
4. the optimal A1 and A2 are increasing in α if and only if ω11τ1 ď ω2
1τ2; and
5. the optimal Am is increasing in α if and only if ω11τ1 ě ω2
1τ2.
Again, these are all commonsensical.
16
1.7 P R O F I T !
The title of the paper hints at some savings on the peacemaker’s side; these savings,in turn, ought to help us deduce conditions under which peacemaking is morelikely. To get at the question, we now consider a peacemaker that is unaware of thepotential for trade—henceforth the ignorant peacemaker. We will study the tragicways of the ignorant peacemaker in the context of the simplifying assumptionsimposed in the previous subsection—the peacemaker is unbiased; the states sharethe same preferences over the goods; and war is costless. These assumptionssharpen the explication but remain unnecessary for the broader points.
Under these assumptions, peace obtains only at the allocation prpτ, p1´ rpqτq.Because of the ignorant peacemaker’s ignorance, she does not realize that tradefrom any allocation along the green hyperplane in Figure 6 will create peace;instead, she believes that she must use the transfer schedule to compel
`
ω11 `A1,ω2
1 `A2˘
loooooooooomoooooooooon
post-transfer allocation
“`
pp `Amqpω1
1 `ω12q, pp `Am
qpω21 `ω
22q˘
loooooooooooooooooooooooooomoooooooooooooooooooooooooon
unique peaceful allocation
.
Take note that this requirement does not depend at all on α—so, the ignorantpeacemaker does not take preferences into account.
Visually, the simplifying assumptions make the ignorant peacemaker’s igno-rance more clear. Consider Figure 7, which depicts a potential interior ignorantpeacemaking transfer schedule. Because this transfer schedule fails to take advan-tage of exchange, the military transfer is comparatively larger than it was in theprevious example. Likewise, the economic transfer—which now must move theendowment all the way to the peace point—is also comparatively larger.
It is interesting to wonder what factors influence the peacemaker’s savingsfor anticipating trade. Sadly, the difference in the two costs does not have a verybeautiful expression. To help understand the origins of savings, we consider aspecial numerical example; the details are relegated to a footnote.3 We plot thesavings as a function of α with p P t110, 12, 910u in Figure 8. In general, themost savings arise when wealth and power and balanced, indicating that only asmall adjustment to the status quo is required. Conversely, if wealth and powerare unaligned—so that a large change is required regardless of the peacemaker’scleverness—then the savings are relatively more modest.
3Suppose the two goods are equally costly for economic transfer: κ1 “ κ2 “ 1. Suppose themilitary transfer is much more costly: κm “ 5. Suppose the two goods are equally abundant, thatState 1 is rich in Good 1, and State 2 is rich in Good 2: ω1
1 “ω22 “ 2 andω2
1 “ω12 “ 1.
17
State 1’s Good 1
State 2’s Good 1St
ate
2’s G
ood
2 State 1’s Good 2
Figure 7. The ignorant peacemaker’s pacifying transfer. Anywhere along the dashedgreen line would have made peace more cheaply conditional on the (inflated) Am .
In other words, Figure 8 provides evidence of a subtle triple interaction: theeffect of preferences on the potential for peace is itself contingent on the balanceof power and the distribution of resources among the potential belligerents. Byletting trade do some of the work, the peacemaker explicitly incorporates prefer-ences but reduces the information otherwise required to navigate the remainderof the interaction.
2 P R O O F O F C O N C E P T
The model described above is unrealistic in that pricing mechanisms are never asnaïve as the one used above. But the proposed mechanism generalizes to contextseasier than the model’s, and so it is prudent to consider what the things mightlook like in the empirical reality we have been given. To that end, this sectionprovides evidence of the mechanism’s workings in an illustrative case: the UnitedStates’ mediation in the Egyptian-Israeli conflict, which ultimately culminatedin the signing of the Camp David Accords in 1978 and the Egypt-Israel Peace
18
Peac
emak
er’s
Savi
ngs
Figure 8. The peacemaker’s savings for anticipating trade. In this numerical example,State 1 is rich in Good 1 and thus moves from comparatively poor to rich as we movefrom left to right.
Treaty in 1979.4 In his study on the subject of bargaining at Camp David, Telhami(1992–3) observes that “the strategic shift toward Israel-Egyptian peace...is bestexplained by key realist variables—the distribution of economic and militarypower...” (630). Here we do not consider how wealth and capability map intopower, but the general idea carries through.
We must consider the context in which the interaction transpired. Econom-ically, the stage was set by a 1946 Arab League-wide boycott of goods from theYishuv: the loose organization of Jewish settlers in Palestine that predated theIsraeli state proper. Save for what could be traded clandestinely, no goods flowedacross Israel’s borders to (or from) Arab neighbors, Egypt included. Politically,after the Six Day War of 1967, Israel found herself in possession of a valuable
4This case is as interesting as any, with its shuttling, its pitched arguments at Camp David,its occasional viewing of heavyweight title fights by heads of state. I do not aim to do the casedramatic justice but only to show how the ultimate provision of peace required the elementsdiscussed above.
19
bargaining chip in forging lasting peace with Egypt: the Sinai Peninsula. De-sch (2008) observes that the Sinai was critical for three of the four pillars of theEgyptian economy. First, most of Egypt’s oil reserves were located in or near theSinai. Second, the blockage of the Suez Canal meant that Egypt could not chargelucrative tolls to use it. Finally, tourism—which leaned heavily on the Sinai’shistorical import in both the Abrahmic and pharaonic traditions—dropped pre-cipitously after the 1967 war.5 And so it is unsurprising that, upon ascending tothe Egyptian presidency in 1970, Anwar Sadat made reacquisition of the Sinaihis top priority; to demonstrate his commitment, in July 1972, Sadat expelledthousands of Soviet advisors to maximize his chances of achieving negotiatedsuccess under an American umbrella. Though the gesture did not go unnoticed,it did not compel the US, who did not want to provoke the Soviets any further,to make enough concessions to nudge Israel toward talks.
The failure to achieve diplomatic success ultimately led to Egyptian involve-ment and leadership on the Arab side in the 1973 Arab-Israeli war. Much to hisSyrian counterpart Hafez al-Assad’s chagrin, Sadat secretly went into the conflictwith very limited objectives: the war was not one of military reacquisition, whichwas surely a fool’s errand, but rather one of compelling the Americans to set thenegotiating table. After achieving modest and surprising territorial successes inthe 1973 war’s early going, Sadat found his position weakening with the secondthrust of the Israeli counteroffensive finding its rhythm. An OPEC embargo ofany nation supplying Israel—a thinly-veiled accusation of American meddling—shook leaders in Washington from their suddenly isolationalist slumber, and USSecretary of State Henry Kissinger quickly began his famous campaign of shut-tle diplomacy before forging a cease-fire at Kilometer 101. The Kilometer 101agreement was the first between Israel and an Arab state since 1949.
And then began the set of negotiations that would necessitate Camp David.More comprehensive, multilateral peace talks in Geneva failed in December 1973,and so Kissinger, no longer encumbered by the Soviets or diplomats from theUnited Nations, intensified his shuttle diplomacy, scuttling from Cairo to TelAviv and back again to craft a more lasting peace that built upon the Kilometer 101agreement. The resulting Sinai I agreement, signed in January 1974, stipulatedthat the Israelis would withdraw from the west bank of the Suez Canal and twentymiles inland from the east bank into the Sinai. This gave Egypt an economic pillarback, but it did not completely pacify her. The rest of the benefits came in theform of promised American economic aid, which began flowing in mid-1974. But
5The fourth pillar, remittances from Egyptians working abroad, remained unchanged.
20
the resolution was not an ultimate one: indeed, the agreement concludes withthe following provision: “This agreement is not regarded by Egypt and Israel asa final peace agreement. It constitutes a first step toward a final, just and durablepeace according to the provisions of Security Council Resolution 338 and withinthe framework of the Geneva Conference.”
With such a glowing endorsement of the peace they just forged, one mightthink that the US would set to work on more “final, just and durable” peace.The issue was not simple: other Arab states were whispering about the correctcourse of action in the wake of Egypt’s apparent independence from their in-fluence, and American inability to make hay with Syria only exacerbated theproblem. Unsurprisingly, the US, Egypt, and Israel began work on another lim-ited disengagement agreement (quite fittingly named Sinai II) pushing the Israelifront to about halfway across the upper Sinai, giving Egypt access to strategicallyand economically important fields in the peninsula’s central expanse. Again, theAmericans did not get to make progress for free, as they upped existing economicand military aid to Israel and economic aid to Egypt. The agreement, signedin September 1975, included a much more optimistic statement of inconclusive-ness: “The Government of the Arab Republic of Egypt and the Government ofIsrael...are determined to reach a final and just peace settlement by means of ne-gotiations called for by Security Council Resolution 338, this Agreement beinga significant step towards that end.”
To get a sense of how the events discussed above affected American subsidiza-tion of the two disputants, see Figure 9. Here American economic aid to bothEgypt and Israel is depicted across time in the top row.6 The vertical axis is loga-rithmically scaled and reported in constant terms of millions of 2014 US dollars.We see that American economic support for Israel was relatively steady from thelatter’s inception onward. However, we also see a significant increase in supportfrom 1974 (around $200 million) to 1975 (over $1.2 billion). (The graphs’ graybands demarcate the years 1974–1978, when active peacemaking was transpiring.)That level remains relatively constant from 1975 through the 1990s. Meanwhile,the increase in funds to Egypt was even more dramatic. It began from a lowerstarting point: Egypt found herself completely cut off from American fundingin the wake of the 1967 war. Sadat’s rejection of Soviet help paid some paltrydividends: in 1972, the US provided her with a $6 million grant, followed by a$3.3 million grant in 1973. By 1974, however, the funds jumped to $82 million,
6The data are from the United States Agency for International Development report. For moredetails, see https://www.usaid.gov/results-and-data/data-resources.
21
Egypt IsraelE
conomic
Military
Trade
1960 1980 2000 1960 1980 2000
1e+05
1e+07
1e+09
1e+05
1e+07
1e+09
1e+05
1e+07
1e+09
Year
Con
stan
t Dol
lars
Figure 9. Economic aid, military aid, and trade flows for the case. Economic and militaryaid are from the United States. Trade data reflects imports. Gray rectangle depicts timebetween Sinai I (1974) and the Egypt-Israel Peace Treaty (1979). All figures reflect constant2014 US dollars.
and by 1975 the levels reached their relatively steady state of $1.3 billion.We would think peace likely—what with all the side payments—but all the
cash in the world could not resolve the fundamental tension between the states:the matter of the Sinai still was not fully resolved, nor was the question of trooplocation given that Egypt had intentionally weakened herself by rejecting Sovietsupport. Neither did the states have fully functioning relations: trade still was notestablished, and roads between the states’ major cities were nonexistent. Indeed,given that the amount of economic support between the two states was roughlythe same during the financial escalation from 1973–1975, one might argue thatthe Americans had done little to change the states’ location in a grand Edgeworthbox. Without military readjustment and trade, the peace was undercooked.
22
This was the argument new National Security Advisor Zbigniew Brzezinskiadvanced while arguing about the correct course of action with new Secretary ofState Cyrus Vance when the two participated in the first meetings of the PolicyReview Committee in February 1977 (Quandt 1986). Brzezinski argued that“Formal recognition of Israel and an end to the state of war would not be enough.More tangible actions, such as diplomatic relations and a willingness to engagein normal peaceful relations, like trade and tourism, would have to be part ofthe package” (43–44). Subsequently, the normalization of relations, particularlytrade relations, became a staple of American proposals throughout the famousnegotiations at Camp David in 1978. So too was trade important to Sadat; asthe Camp David negotiations began, Sadat presented Jimmy Carter with twodocuments. The first was his initial peace proposal, which, like any openingsalvo in bargaining, was a low-ball offer. The second was a list of those issueson which Sadat was most prepared to budge, which might be interpreted as anattempt to set the agenda for Carter’s own agenda-setting. At the top of thelist were consular relations, free movement across borders, and enhanced traderelations (Carter 1982). Carter was especially pleased to see these at the top ofSadat’s list (340–341), particularly because the issue of trade was already known tobe important to Israel. What’s more, trade was second on Carter’s own wishlist,second only to a shared Jerusalem (325), and the “free movement of goods andpeople” was an associated principle of the American proposal (367). This set thestage for trade relations’ service as the mutually understood keystone to peace.
This is particularly interesting, as Egypt had publicly, and sometimes pri-vately, maintained that it would not budge on trade, even though Sadat privatelytold President Carter that there might exist wiggle room regarding the embargoon April 4, 1977, conditional on reaching a peace agreement (Carter 1982, 282–283). Yet, Sadat’s speech to the Israeli Knesset on November 20, 1977 includedan impassioned plea for peace covering five main points: an ending of the occu-pation of Arab territories; achievement of self-determination for the Palestinianpeople; the right of all states in the area to live in peace; commitment of all statesin the region to agree to resolve differences through peaceful means; and endingthe state of belligerence in the region. Economic factors were notably absent.In a conversation with Carter on February 4, 1978, Sadat again maintained that“never in our lifetime would any Arab meet [Carter’s] definition of peace andestablish normal trade...” (307). So, Sadat’s announcement of a strong, positiveposition on trade was a dramatic spark late in the peacemaking process.
The military scenario, of course, was more difficult to manipulate, particu-larly given that Israel was in the strongest bargaining position of the three. In
23
particular, Israel found herself able to demand American subsidization of twonew airfields in the Negev desert in exchange for her airfields in the Sinai, whichended up requiring a $3 billion investment in military aid to Israel. For everypercent increase in military support for Israel, however, the Americans seeminglypromised an equal amount to Egypt, though the support was targeted more to-ward rebuilding the Egyptian armies and bringing them up to speed after a fewyears without external assistance. By the time the final agreement was struck,each state enjoyed massive military aid. Consider the middle row of Figure 9: wesee that Egypt gets her first military aid from the US in 1979, when the aid wasa staggering $3.9 billion. This massive shock to the system surely represented achange in the military status quo, though aid to Israel also experienced a bumpas airfields were built. Again, the military aid to both states eventually reached akind of steady state, with Egypt receiving roughly $1.75 billion per year in mili-tary aid through the rest of the series and Israel receiving roughly $2.5 billion. So,American support merely brought the Egyptians up to respectability withoutshifting the balance of power in any meaningful way.
So did the normalized relations yield meaningful levels of trade? See thebottom row of Figure 9. These trade data come from the Correlates of Warinternational trade data set (Barbieri, Keshk and Pollins 2009; Barbieri and Keshk2012). The vertical axis captures trade flows and is again logarithmically scaled.7
Here the effect of normalized relations is at its most dramatic: trade increasesfrom nothing in 1979 to figures well into the hundreds of millions of dollarsin 1980. While flows from Israel to Egypt dance around quite a bit, they aregenerally on the rise throughout the available time series. Flows from Egypt toIsrael, on the other hand, are quite a bit steadier and maintain a relatively highlevel in the tens of millions of dollars. These figures generally represent a sizeableproportion of the economic aid given to each of the states. Consider 1980, whenEgypt received roughly $2.8 billion in economic aid from the US. Egypt alsopurchased around $200 million in goods from Israel that year, or around 7% ofthe aid. So while aid does much of the explanatory work here, it remains thattrade shifted dramatically and significantly in the wake of the Accords.
The case clarifies a subtle point: peace was not possible prior to the completeadherence to balance power, wealth, and trade in a single, unified way. In themid-1970s, the US found itself investing heavily in Israel—both economically andmilitarily—and making non-trivial economic side payments to Egypt. But thiswas not enough for lasting peace: the states did not trade, and Egypt’s military
7The trade data use constant 2009 dollars, so I adjusted them for inflation to 2014 dollars.
24
position was not strong enough to make the trading outcome a peaceful one,anyway. And so we see that the final maneuver in the Camp David Accords—tobalance the militaries through even heavier investment in both states’ militariesand to normalize trade relations—was the one that squared the circle of peace.This demonstrates the importance of an integrated approach to forging peace.
3 T H E G E N E R A L C A S E
Intuitions developed, we now introduce the general model of aid, trade, and war.This removes the odd feature that the peacemaker plays Robin Hood by issuinglump-sum transfers. We will see that the aid-based approach introduces severalinteresting subtleties; indeed, it is not all that straightforward to write the modeldown in the first place.
3 .1 T H E M O D E L
In terms of actors, we have a peacemaker (i “ 0) and a set of potential disputantstates indexed i “ 1, . . . , I . The disputant states are concerned with consumptionof L goods indexed ` “ 1, . . . , L. When I is small, the model works well forinterstate violence; when I is large, it is more useful for intrastate violence.
3.1 .1 T H E D I S P U T A N T S ’ PA R A M E T E R
The source of exogeneity,8 and the focus of the peacemaker’s interest, is the list
θ“ xω,µ, cy PΩˆM ˆC “RI L``ˆR
I``ˆR
I`` “:Θ.
The conveys the initial economic allocationω, the initial military allocation µ,and the initial costs of war c . We refer to this as the parameter, and to Θ as theparameter space. We begin by describing the status quo at some inital θ and thendiscuss the terms of parametric adjustment through aid.
3 .1 .2 T H E C O N S U M P T I O N A N D A L L O C AT I O N S PAC E S
A vector xi “ px1i , . . . , xL
i q is called a bundle, and the set of all consumable bundlesis Xi :“RL
``. (So, consumption has an unattainable lower bound; this openness
8Other potential sources of exogeneity (e.g., the states’ preferences over commodities or po-tential consumption sets) seem outside a peacemaker’s control—I could be wrong.
25
at the bottom is a very useful simplfying manœvre, but it also delivers somestronger results.) Call an I -tuple of bundles x “ px1, . . . , xI q PRI L an allocation.
Each state begins with an endowment of goods,ωi “ pω`i q PXi . The endow-
ment allocation is ω “ pωiq. Endowment allocations are impotant enough tolend their name to the set of allocations: ω P Ω “ RI L
``. The total amount ofgood ` is τ` “
ř
i ω`i , and the vector of these is τ “ pτ`q PXi .
An allocation x is feasible ifř
i x`i ďř
i ω`i for all `. In the other direction,
given a total resource vector τ PXi , we can restrict attention to feasible bundles:
Ωθ :“!
x PΩˇ
ˇ
ˇ
ÿ
x`i ďÿ
ω`i for all `“ 1, . . . , L
)
ĹΩ.
We denote this Ωθ, but it really only depends on τ (hence only theω part of θ).Each state i has a utility function ui : Xi ÑR. We assume:
3.1 AssumptionThe utility function ui PRXi satisfies:
1. smoothness: ui is partially differentiable up to any order;
2. smooth monotonicity: the gradient D uipxiq ą 0;
3. smooth strict concavity: the Hessian D2uipxiq is negative definite;
4. necessity: for any v PR``, the level set txi PXi | uipxiq “ vu is closed inRL;
The set of all such functions isU ĹRXi .
Most of these are slightly stronger than is absolutely necessary, and they arein keeping with the usual conventions of formal IR theory. The concavity ofui implies that interpersonal comparisons are somewhat well-founded, so thesecond bargaining convention introduces no special problems. That said, sincethe total endowment will soon be enhanced (potentially) by the third party, wewill usually write out uipτq for completeness’ sake.
3.1 .3 E XC H A N G E V I A T H E NA Ï V E WA L R A S I A N P R O T O C O L
We again load the full burden of peacemaking onto the third party’s shoulders,we have the states engage in a naïve Walrasian protocol for exchange.
26
Each good is assigned a price, p` ą 0, and we collect these into the price vectorp “ pp`q. We normalize the prices like so:
p P S “!
s “´
s`¯
“RL``
ˇ
ˇ
ˇs L“ 1
)
,
so that good L is numeraire. As a variable, then, p P S has dimension L´ 1. Asbefore, states maximize utility given a set of prices and their wealth level,
maxxi : p¨xiďp¨ωi
uipxiq.
We call the solution to this maximization problem the demand, fi , and record:
3.2 LemmaFor any ui PU , we have a single-valued demand fi : S ˆR`ÑXi ,
fipp, p ¨ωiq “ argmaxxi : p¨xiďp¨ωi
uipxiq.
Further, fi is a diffeomorphism ( i.e., smooth and bijective with a bijective inverse).
It is straightforward to show that fi is well-defined, and smoothness is also easywith a quick appeal to the implicit function theorem; the strengthening of smooth-ness to diffeomorphism (which is critical for our result) is more delicate. Wedenote the set of all demand functions arising from ui PU asF ĹX SˆR`
i .Prices arise to equilibrate supply (here exogenously given by the endowment)
and demand. Define aggregate demand asř
i fipp, p ¨ωiq, and aggregate excessdemand as z : S ˆΩÑRL, with
zpp,ωq “ÿ
i
fipp, p ¨ωiq´ÿ
i
ωi .
Owing to fi ’s smoothness and the smoothness of the linear functional p ¨ωi , z isitself smooth, which will be important. We arrive at the equilibrium equation,
zpp,ωq “ 0.
Many of the results to follow testify to the remarkable properties of this equation.We say pp,ωq P S ˆΩ is a Walrasian equilibrium if zpp,ωq “ 0. Any p P S
satisfying zpp,ωq “ 0 is an equilibrium price vector for allocation ω. Finally,define the equilibrium manifold,
E “ tpp,ωq P S ˆΩ | zpp,ωq “ 0u Ď S ˆΩ.
The equilibrium manifold E will be one of the stars of our show.
27
3.1.4 WA R A S A C O S T LY LO T T E RY
After the exchange, the states may decide whether to start a war. Conflict isunilateral, so that a single dissatisfied state is sufficient, and it is modeled, perusual, as a costly lottery for the right to consume all the resources. We abusenotation slightly and write uip0q “ 0 for both states i ; because ui satisfies thebargaining convention (part 4 of Assumption 3.1), this introduces no problems.
The terms of the war lottery are determined by each state’s military might.State i has military might µi ą 0, and the distribution of military might is theI -tuple µ “ pµ1, . . . ,µI q; the set of all of these is M “ RI
``. The probabilitiesof victory in a winner-take-all lottery are given by qpµq, where each qipµq ě 0and
ř
i qipµq “ 1. We assume q is smooth and that qi is increasing in mi anddecreasing in m j for j ‰ i . War is costly: State i pays cost ci ą 0, measured inutils. Collect these into the cost vector c “ pc1, . . . , cI q PC “RI
``.We therefore have the I expected utilities for war
Uipwar,θq “ qipµquipτq´ ci .
Observe that this expression combines all aspects of θ: endowments (throughτ “
ř
i ωi ); military might (through qipµq), and war costs (directly through ci ).A post-trade allocation x “ px1, . . . , xiq PΩθ is peaceful if
uipxiq ěUipwar,θq
for all states i . That is, all states must be satisfied relative to war, and stateschoose peace if indifferent. We denote the property of “peacefulness at θ” asPpθq. The set of all parameter combinations that yield peace after trade is givenby T :ΘÑΩ. The trade-to-peace set T pθq,
T pθq “ tω1 PΩθ | uip fipp˚pθq, p˚pθq ¨ω1iqq ďUipwar,θq for all iu ,
is our primary object of interest.
3 .1 .5 T H E T H I R D PA R T Y ’ S A I D S C H E D U L E
The third party moves first and with a complete understanding of all the detailsdescribed to this point. She has at her disposal a simple policy instrument, theaid schedule a “ pe , mq PA :“RI L
` ˆM , where M ĹRI is compact. The first partof this is economic aid,
e “ pe1, . . . , eI q “``
e11 , . . . , eL
1
˘
, . . . ,`
e1I , . . . , eL
I
˘˘
.
28
Economic aid shifts the endowments, yielding the revised endowment
pωi “`
ω1i ` e1
i , . . . ,ωIi ` e I
i
˘
.
The revised endowment allocation is pω “ p pω1, . . . , pωI q. The revised amount ofgood ` is pτ` “
ř
i pω`i , and the vector of these is pτ “
`
pτ1, . . . , pτL˘
.The second component of the aid schedule is military aid,
m “ pm1, . . . , mI q .
This shifts the probability that State i wins to qipµ`mq. The aid schedule alsoshifts war costs via gi :R``ˆAÑR``, so that we now have
pci “ gipci ,aq.
We assume gi is smooth, increasing in its inputs (strictly so in ci ), and we convenegipci , 0q “ ci for all ci . (In particular, gi can be set up to increase the costs of warwithout influencing any other factor, so that “enforcement” models are subsumedby this apparatus.)
The result of the third party’s aid schedule is a new value of the parameter,
pθpaq “ x pω, pµ,pcy “ xωi ` ei ,µi `mi , gipci ,aqyi PΘ.
From our definitions, pθ : AÑΘ is smooth.The peacemaker retains simple payoffs. Should the outcome of the states’
interaction be peace, they receive
u0 pa,θq “ R´ kpaq,
where k : RI L` ˆM Ñ R is a cost function for aid schedule pe , mq and R ě 0 is
some exogenous reward of peace. Suppose k is smooth and strictly increasing inall inputs. Conversely, should the outcome be war, we have
U0 pa,θq “ ´kpaq,
This concludes our generalization of the leading case.
3 .2 T H E M A I N R E S U LT S
We now wrestle the bear.
29
3.2.1 C A N P E AC E B E M A D E . . . ?
Before we greedily search for optimal pacifying aid schedules, we should firstensure that we have a well-defined problem. As things sit, the peacemaker canstretch the Edgeworth box (which is now far too complicated to draw, since wehave many states and many commodities) through economic aid. They can alsoinfluence the terms of war through military aid. Consequently, the set of efficientallocations, the bargaining range, the peace lens, and the trade-to-peace set are allmoving targets; there are not yet any guarantees that the aid schedule will be ableto keep pace with those targets.
We quickly see that the problem is well-defined in the most minimal sense.
3.3 PropositionThere exists a pacifying aid schedule.
Happily, the easiest way to prove Proposition 3.3 is by construction; in the Ap-pendix, you may find an algorithm for constructing peace-generating aid sched-ules. These are certainly not cost-minimizing, but they are concrete, easy to find,and conceptually straightforward. They also play an important role in the searchfor cheap pacifying aid schedules.
3 .2 .2 . . .O N T H E C H E A P ?
We now focus on cost-minimizing, peace-generating adjustments. The primaryquestion is: does there exist such an adjustment? To be able to answer a questionlike that, we will need the structure of the underlying bargaining problem to bewell-behaved in two key ways. First, we need to know that the set of allocationsthat generate peace is compact. If compactness failed to obtain, then there mightbe sequences of endowments (and thus of economic aid schedules) that failed toconverse to some attainable cheapest point. It is therefore good news that we canreport:
3.4 PropositionFor any θ PΘ, the trade-to-peace set, T pθq, is compact and connected.
Second, we need to know that the structure of the underlying bargainingproblem varies smoothly in concert with the underlying parameter θ. Were thisnot the case, then the peacemaker’s aid schedule might induce large changes inthe underlying problem, which in turn might hinder the ability to find a cheapestpacifying aid schedule. With this in mind, we report:
30
3.5 PropositionThe maps θ ÞÝÑ T pθq and a ÞÝÑ T ppθpaqq are continuous.
The tedium of the analysis for these results is relegated to Appendix A. How-ever, as the method of proof—in the “postmodern” style of equilibrium analysis—is not standard in IR research (and is quite useful!), it is worth sketching its details.To ensure existence of an optimal aid schedule, we will need compactness and con-tinuity on the set of allocations that yields peaceful equilibrium allocations—thetrade-to-peace set, T pθq. The argument follows the chain
apθÞÝÑ pU
pθpaqrPppθpaqqs
looooooomooooooon
peaceful, optimalutility imputations
ĹRI`
u´1
ÞÝÑΩpθpaqrPp
pθpaqqslooooooomooooooon
peaceful equilibriumallocationsĹRI L
``
f ´1
ÞÝÑ ErPppθpaqqslooooomooooon
peacefulequilibriaĹRLpI`1q
``
πÞÝÑ T ppθpaqq.
looomooon
endowmentsyielding peaceĹRI L
``
We have introduced most of these concepts: we now introduce the final two. Thefirst is a set of peaceful, efficient utility imputations: a vector of outputs from eachstate’s utility function, each higher than the respective expected utilities of war.The second, π : E Ñ Ω, is the natural projection, which inputs equilibria andoutputs underlying parameters. Given this set-up, the analysis is straightforward:
1. we first show show that the set of peaceful, efficient utility imputationspUθrPpθqs is compact and continuous in a;
2. show that if pUθrPpθqs is compact, the set of peaceful equilibrium alloca-tions ΩθrPpθqs is also compact and continuous in a;
3. show that ifΩθrPpθqs is compact, the inverse demand function f ´1 ensuresthat set of peaceful equilibria ErPpθqs is compact and continuous in a; and
4. show that if ErPpθqs is compact, the natural projection π ensures that thetrade-to-peace set T pθq is compact and continuous in a.
Again, interested readers are referred to Appendix A.With these results in hand, it is easy to answer our question. The answer is
affirmative: peace can indeed be made on the cheap.
3.6 TheoremThere exists a cost-minimizing pacifying aid schedule.
31
4 C O N C L U S I O N
In this paper, I have developed a new model of aid, trade, and war and used itto generate a very hard problem for intervention to solve. Amazingly, the thirdparty is able to overcome the structural problems that make war so appealingfor the potential belligerents even in the case where the latter are uncooperative.The peace made via intervention has strong normative appeal in that it resolvesthe problem completely: not only do military and economic endowments matchone another, but the final transfer that causes peace—which is the key element ofpeacemaking here—aligns with preferences. This is a critical difference betweenthis mechanism and others like military intervention or unassisted bargaining.
Of course, I have used this theoretical apparatus only to answer one questionout of many, and it is not difficult to think of interesting extensions. Here arethree. First, we have said nothing about stability over time. The model’s plausi-bility in dynamic settings is a function of the analyst’s willingness to presupposethe existence of all futures markets. Few international relations scholars—thisone included—would be willing to put much faith in such assumptions, so anexplicitly dynamic model could help matters. In particular, if the initial endow-ments were identical in every stage, the third party would have to execute thesame intervention over and over again. However, it could well be that the gainsfrom trade and the act of intervention affect future endowments (one would hopethrough growth), which might make future intervention less expensive or evenunnecessary. On the other hand, unwise intervention might exacerbate existingdistributional problems. It would be interesting to know whether a third partycould quickly induce a peace that itself evolved into something self-enforcing, orwhether third parties must continue to invest in a region to maintain peace.
Second, we have said little about the third party’s incentives beyond cost-minimization. We have begun to make serious headway in understanding theempirics of foreign aid flows (e.g., Alesina and Dollar 2000) and some of the incen-tives behind those choices (e.g., Bueno de Mesquita and Smith 2007, 2009). Themodel given here offered some very simple options for modeling the third party,but it still could do more. Future extensions will benefit from the foundationalresults derived here, but obviously these extensions may introduce their ownspecial problems.
Third, and related, we have considered a solitary third party, and as it happensthese kind of problems often draw the attention of many third parties, be theyother states or international organizations. These third parties each bring theirown set of biases and valuations, and the way that these interact with one another
32
could complicate the model in a very interesting way that aligns well with scholars’game-theoretic conception of reality. If history is any indication, multiple thirdparties occur reasonably often and in those conflicts that seem most important.Analyzing a model of this type—which is straightforward thanks in no small partto the geometric results we worked so hard to derive—might let us know whenthere are “too many cooks in the kitchen,” which might help answer questionsabout regional stability across types of polarity or in the presence of a middlingregional pest. Of course, it could also be that multiple interveners with similarbiases could work together to reduce their individual costs, thus bringing aboutpeace when unilateral intervention is too costly.
These extensions highlight the promise of the enterprise moving forwardwithout undermining the findings reported above. Future scholars should viewthis model as a foundation upon which new theories of intervention can be builtwithout fear of indeterminacy or a lack of peaceful outcomes. In general, weought to think of ways to ensure that peace is not just feasible, but also that it ischeap enough for reasonable construction.
33
A A P P E N D I X
Here we prove some of the results not shown in the text.
A.1 P R E L I M I N A R I E S F O R T H E G E N E R A L C A S E
We now study the general case as described in Section 3.1. To begin with, werecord some very useful results. First, observe that there is always at least onepeaceful point.
A.1 LemmaFor any θ, the allocation qµΩ :“ pq1pµqΩ, . . . , qI pµqΩq is peaceful.
Proof. Because each ui is strictly concave, we have
ui pqi pµqΩq ą qi pµqui pΩq ą qi pµqui pΩq´ ci “Ui pwar,θq,
which is all that is required for the claim. ||
We will refer to qµΩ as the “peace point,” since we always know it is peaceful.Of course, it will never be the only peaceful allocation; it just happens to be theeasiest one to keep track of.
We now re-state some results about Walrasian equilibria in the general case.These are all well-known, so their proof is omitted. First, existence:
A.2 PropositionIn the general case, a Walrasian equilibrium exists.
Next, the First Welfare Theorem.
A.3 PropositionAny Walrasian equilibrium yields an efficient allocation.
Finally, the Second Welfare Theorem.
A.4 PropositionAny efficient allocation is the outcome of a Walrasian equilibrium from some alloca-tion.
We now combine these with Lemma A.1.
A.5 LemmaThere exists at least one peaceful Walrasian equilibrium.
34
Proof. The peace point is itself peaceful. The Walrasian equilibrium taking thepeace point as its endowment must be peaceful, too. ||
Predictably, we also have the following.
A.6 LemmaThere exists at least one peaceful efficient allocation.
Proof. The Walrasian equilibrium we just studied in Lemma A.5 yields anefficient outcome, per Proposition A.3. ||
In other words, the bargaining range is nonempty.
A.2 P E AC E -P O I N T A I D S C H E D U L E S
For reasons that will become obvious (or may already be), we study one com-modity at a time and focus in particular on aid schedules satisfying
p1´ qipµ`mqq´
ω`i ` e`i
¯
´ qipµ`mq
˜
ÿ
j‰i
ω`i `
ÿ
j‰i
e`i
¸
“ 0
for all states i “ 1, . . . , I . We will assume that the aid schedule a “ 0 does not sat-isfy this requirement; if it does, then there is nothing to do, as we are already at thepeace point. We will also begin from the m “ 0 case (no military intervention);to save on notation, let qipµ`mq “ qi for now.
This construction has two nuances at work: first, that there is not a well-behaved system of I answers; and second, that there are no immediate guaranteesthat whatever solutions we do find are positive (as required by our aid motiva-tions). Our construction takes both of these into account.
We first turn our attention to the Jacobian of the system of equations thatdefine peace-point economic aid schedules. In particular, we have
»
—
—
—
—
—
–
1´ q1 ´q1 ¨ ¨ ¨ ´q1 ´q1´q2 1´ q2 ¨ ¨ ¨ ´q2 ´q2
...... . . . ...
...´qI´1 ´qI´1 ¨ ¨ ¨ 1´ qI´1 ´qI´1
1´ř
jăI q j 1´ř
jăI q j ¨ ¨ ¨ 1´ř
jăI q jř
jăI q j
fi
ffi
ffi
ffi
ffi
ffi
fl
.
35
For all i “ 1, . . . , I ´ 1, multiply columm i ` 1 by ´1 and add to column i toobtain the matrix
»
—
—
—
—
—
—
—
—
—
—
—
—
—
–
1 0 0 0 ¨ ¨ ¨ 0 0 0 ´q1´1 1 0 0 ¨ ¨ ¨ 0 0 0 ´q20 ´1 1 0 ¨ ¨ ¨ 0 0 0 ´q30 0 ´1 1 ¨ ¨ ¨ 0 0 0 ´q4...
......
... . . . ......
......
0 0 0 0 ¨ ¨ ¨ 1 0 0 ´qI´30 0 0 0 ¨ ¨ ¨ ´1 1 0 ´qI´20 0 0 0 ¨ ¨ ¨ 0 ´1 1 ´qI´10 0 0 0 ¨ ¨ ¨ 0 0 ´1
ř
jăI q j
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
.
We first show that the entire matrix is not invertible. For all i “ 1, . . . , I ´ 1,multiply column i by qi and add it to column I to obtain
»
—
—
—
—
—
—
—
—
—
—
—
—
—
–
1 0 0 0 ¨ ¨ ¨ 0 0 0 0´1 1 0 0 ¨ ¨ ¨ 0 0 0 ´q10 ´1 1 0 ¨ ¨ ¨ 0 0 0 ´q20 0 ´1 1 ¨ ¨ ¨ 0 0 0 ´q3...
......
... . . . ......
......
0 0 0 0 ¨ ¨ ¨ 1 0 0 ´qI´40 0 0 0 ¨ ¨ ¨ ´1 1 0 ´qI´30 0 0 0 ¨ ¨ ¨ 0 ´1 1 ´qI´20 0 0 0 ¨ ¨ ¨ 0 0 ´1
ř
jăI´1 q j
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
.
For all i “ 2, . . . , I ´ 1, multiply column i by qi and add to column i to obtain»
—
—
—
—
—
—
—
—
—
—
—
—
—
–
1 0 0 0 ¨ ¨ ¨ 0 0 0 0´1 1 0 0 ¨ ¨ ¨ 0 0 0 00 ´1 1 0 ¨ ¨ ¨ 0 0 0 ´q10 0 ´1 1 ¨ ¨ ¨ 0 0 0 ´q2...
......
... . . . ......
......
0 0 0 0 ¨ ¨ ¨ 1 0 0 ´qI´50 0 0 0 ¨ ¨ ¨ ´1 1 0 ´qI´40 0 0 0 ¨ ¨ ¨ 0 ´1 1 ´qI´30 0 0 0 ¨ ¨ ¨ 0 0 ´1
ř
jăI´2 q j
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
.
36
Repeating this procedure a total of I´1 times (each time moving the first columnto the right by one), we arrive at the degenerate matrix
»
—
—
—
—
—
—
—
—
—
—
—
—
—
–
1 0 0 0 ¨ ¨ ¨ 0 0 0 0´1 1 0 0 ¨ ¨ ¨ 0 0 0 00 ´1 1 0 ¨ ¨ ¨ 0 0 0 00 0 ´1 1 ¨ ¨ ¨ 0 0 0 0...
......
... . . . ......
......
0 0 0 0 ¨ ¨ ¨ 1 0 0 00 0 0 0 ¨ ¨ ¨ ´1 1 0 00 0 0 0 ¨ ¨ ¨ 0 ´1 1 00 0 0 0 ¨ ¨ ¨ 0 0 ´1 0
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
.
We conclude two things from this analysis: first, that the peace-point system willbe badly behaved if we study all I equations; and second, that it will be well-behaved if we study any I ´ 1 equations or fewer. (We wouldn’t have been ableto kill column I without each other victory probability)
We now re-index the states by their relative happiness on commodity `. Wedo so with the function i` : t1, . . . , I uÑ t1, . . . , I u defined as
i`p j q “
ˇ
ˇ
ˇ
ˇ
ˇ
#
k P t1, . . . , I u
ˇ
ˇ
ˇ
ˇ
ˇ
ω`k
τ`´ qk ă
ω`j
τ`´ q j
+ˇ
ˇ
ˇ
ˇ
ˇ
` 1
In the knife-edge case of ties, resolve them by ranking with the original index.We therefore have a well-formed subset of N as our index; throughout the rest ofthis subsection, we simply let i “ i`piq. (This will be done for each good `.)
We therefore have the situation that
ω`1
τ`´ q1 ď ¨ ¨ ¨ ď
ω`I
τ`´ qI .
Call the number of strictly-negative terms in this chain 1 ă n ă I . We known ă I because at least one state must be strictly dissatisfied relative to the peacepoint. Likewise, we know n ą 1 because at least one state must be strictly satisfiedrelative to the peace point. We fix e`I “ 0.
We are now in position to find a solution. We drop the superscript for com-modities to use it for iterations. Set the counter at c “ n´ 1, and partition thestates as A“ t1, . . . , nu and B “ tn`1, . . . , I u; these are the “dissatisfied” and “sat-isfied” states, respectively. The general idea is to offer aid to render the dissatisfiedstates (just) satisfied without over-perturbing the satisfied states.
37
We proceed as follows while că I :
1. Increase the counter by one: c“ c` 1.
2. Find the´
e pcqi
¯
iPAthat solves
p1´ qiqei ´ qi
ÿ
jPAztiu
e j “ qiτ´ωi
for all i PA. (From our previous analyses of the Jacobian, such a solutionexists and is unique so long as |A| ă I .)
3. Check to see if this aid has perturbed any previously-satisfied states. Inparticular, calculate
ρ j “ω j ´ q j
˜
τ`ÿ
kPA
e pcqk
¸
for all j P B . If ρ j ě 0 for all j P B , then STOP.
4. Otherwise, let b P B be the state with the lowest value of ρ j . Now define
A“AYtbu,B “ B z tbu,
and repeat from Step 1.
The algorithm clearly generates peace, and the reader can confirm that each re-sulting aid schedule is positive. We conclude:
3.3 PropositionThere exists a pacifying aid schedule.
A quick appeal to the Intermediate Value Theorem (which applies because of theJacobians derived above) offers us the following:
A.7 CorollaryThe peace-point construction is continuous in qi (hence in military aid m).
We define eP pmq as the peace-point construction at qpmq. Observe that the costfunction kpeP pmq, mq is continuous in m. When combined with the compactnessof M , we conclude:
A.8 CorollaryThere exists a cost-minimizing aid schedule in the peace point class.
38
A.3 F I N D I N G T H E C H E A P E S T PAC I F Y I N G A I D S C H E D U L E
We proceed in a number of steps. The first job is to learn about the structure ofthe set of peaceful, efficient allocations; by Proposition A.4, this also happens tobe the set of peaceful, efficient Walrasian equilibrium outcomes. It turns out thatthis is most easily done by studying the set of utility imputations.
We aim to show two things. Both of these pertain to the trade-to-peace set,
T pθq “ tω1 PΩθ | uip fipp˚pθq, p˚pθq ¨ω1iqq ďUipwar,θq for all iu .
First, we will show that T pθq is compact and connected for any θ P Θ. Second,we will show that T pθqmoves continuously with θ. We will show these in turn.
A.3.1 C O M PAC T N E S S O F T H E T R A D E -T O -P E AC E S E T
Recall that Ω“RI L`` and that
Ωθ :“!
x PΩˇ
ˇ
ˇ
ÿ
x`i ďÿ
ω`i for all `“ 1, . . . , L
)
ĹΩ.
To allow for zero consumption now and again, we also have
cl pΩθq :“!
x PRI L`
ˇ
ˇ
ˇ
ÿ
x`i ďÿ
ω`i for all `“ 1, . . . , L
)
.
(For reasons that will probably be clear later on, we did not have to close up theupper boundary.) We have the following.
A.9 LemmaThe closure of the set of feasible allocations, cl pΩθq, is convex and compact.
Proof. We begin with convexity. We can write
cl pΩθq “RI L` X
!
x PRI Lˇ
ˇ
ˇ
ÿ
x`i ďÿ
ω`i for all `“ 1, . . . , L
)
.
The former is a half-space, and the latter is the (L-times) Cartesian product ofa series of half-spaces. Both are therefore convex; thus, so too is their intersec-tion.
For boundedness, observe that the point p´1, . . . ,´1q (repeated I L times) isstrictly less than or equal to any point in cl pΩθq. Likewise, the point
˜˜
ÿ
i
ω1i ` 1, . . . ,
ÿ
i
ωLi ` 1
¸
, . . . ,
˜
ÿ
i
ω1i ` 1, . . . ,
ÿ
i
ωLi ` 1
¸¸
39
is strictly greater than or equal to any point in cl pΩθq. So, cl pΩθq is boundedabove and below, and hence is bounded.
For closedness, return again to the intersection used to establish convexity.Both of the constituent sets contain their boundary points; hence, so too doestheir intersection. We conclude that cl pΩθq is closed, and we are done. ||
We need to extend our utility function to work in this setting. Let each ui P
ĂU , where this set includes all the assumptions enumerated in Assumption 3.1.However, define ui : clpXiqÑR, and for any
x 1i P!
y P clpXiq
ˇ
ˇ
ˇy` “ 0 for at least one `“ 1, . . . , L
)
,
let
ruipx1iq “ lim
zÑx1iuipzq
where each element of the sequence z lives in Xi “ RL``. We also convene that
ruip0q “ 0 for all i , which doesn’t influence generality all that much. (It onlyinfluences generality inasmuch as we view ui as measurable thanks to its concavity,whereas concavity has been assumed mostly for convenience.) The codomain foreach rui P
ĂU is therefore restricted to R`.Now define
Uθ “
pv1, . . . , vI q PRI`
ˇ
ˇ
ruipxiq “ vi for some xi P clpΩθq(
.
This is the set of utility imputations. Thanks to the continuity of each rui , Uθ hasa free disposability property: if v P Uθ and 0ď v 1 ď v , we must also have v 1 P Uθ.We will now strengthen this to convexity.
A.10 LemmaFor any θ PΘ, Uθ is compact and convex.
Proof. Compactness is easy. Observe that Uθ is the image of the compact setclpΩθq by the continuous map
px1, . . . , xI q ÞÝÑ pru1px1q, . . . , ruI pxI qq ,
and so it is compact.
Now convexity. Choose any v, v 1 P Uθ and a α P p0,1q. By definition, thereexists at least one attainable allocation x P clpΩθq such that rui pxi q “ vi for all
40
states i ; the same goes for an x 1 and v 1. Since clpΩθq is convex, the allocationx2 “ αx`p1´αqx 1 is also in clpΩθq. From concavity of each utility function„
v2 “`
ru1px21 q, . . . , ruI px
2I q˘
ą αv `p1´αqv 1.
Hence, v2 must also be positive, and since it is the utility image of an attainableallocation, we know it also lives in Uθ. We are done. ||
We now make a few more definitions. We say an attainble x P clpΩθq is a weakoptimum if and only if rupxq is in the upper boundary of Uθ. In other words, xis a weak optimum if vi ą ruipxiq for all i implies v R Uθ. The associated utilityimputation is a weak utility optimum. We denote the set of all weak utility optimaby pUθ. Since our utility functions are strictly monotonic, any weak optimum isa Pareto optimum in the more traditional sense.
Let us now modify these to allow for war. Define
UθrPpθqs “ tv P Uθ | vi ěUipwar,θq for all i “ 1, . . . , I u ,
pUθrPpθqs “!
v P pUθˇ
ˇ
ˇvi ěUipwar,θq for all i “ 1, . . . , I
)
.
These are the set of all peaceful utility imputations and the set of all peaceful weakutility optima, respectively. The former inherits compactness and convexity fromUθ; but we are particularly interested in the properties of the latter.
We have the following.
A.11 LemmaFor any θ PΘ, the set of peaceful weak utility optima, pUθrPpθqs, is homeomorphicto the I ´ 1-unit simplex.
Proof. First, observe that pUθrPpθqs is not empty, per Lemma A.5.
Because ωi ą 0 for each i , and because of our limiting conditions on rui , weknow that 0 P intpUθq and
uPpθq :“ pmaxt0, U1pwar,θqu , . . . ,maxt0, UI pwar,θquq ,
which lives in intpUθrPpθqsq. Further, if uPpθq ď v 1 ă v P UθrPpθqs, then
v 1 R pUθrPpθqs. Therefore, for each q in the simplex, there is a unique vpqq PpUθrPpθqs such that q “ αvpqq for some αą 0. Because UθrPpθqs is compact,this map q ÞÑ vpqq is continuous. We conclude that v is a continuous, one-to-one map from the simplex onto pUθrPpθqs, as required. We are done. ||
41
In particular, this means that pUθrPpθqs is compact and connected. It turns outthat the whole of pUθ is homeomorphic to the same simplex (Mas-Colell 1985,Proposition 4.6.1, p. 154), so that the peaceful weak utility optima are a compact,connected subset of the set of all weak utility optima, which is also compact andconnected. This will prove useful.
We next define the set of efficient allocations given parameter θ PΘ:
Pθ “ tx P clpΩθq | E x 1 P clpΩθq s.t. ruipx1iq ą ruipxiq for all i “ 1, . . . , I u
We again work with the “weak optimum” notion of Pareto efficiency; the strictmonotonicity of the utility functions ensures that this is equivalent to the moretraditional sense. We also consider the peaceful efficient allocations:
PθrPpθqs “ tx P Pθ | ruipxiq ěUipwar,θq for all i “ 1, . . . , I u .
This is the familiar bargaining range. We now record two results about thesequantities of interest.
A.12 LemmaFor any θ PΘ, the set of efficient allocations, Pθ, is homeomorphic to the pI ´1q-unitsimplex.
Proof. See Mas-Colell (1985, Proposition 4.6.2, pp. 155–6). ||
This provides our more relevant result.
A.13 LemmaFor any θ P Θ, the set of peaceful efficient allocations, PθrPpθqs, is homeomorphicto the pI ´ 1q-unit simplex.
Proof. Per Lemma A.12, there is a homeomorphism between pUθ and Pθ. Ob-serve that PθrPpθqs is the image of pUθrPpθqs under this homeomorphism,which is sufficient for the result per Lemma A.11. ||
To recap: we have shown that the set of peaceful efficient allocations is thesame as a simplex (and thus is compact and connected). It turns out that this givesus an even stronger result.
A.14 LemmaThe set of peaceful equilibrium allocations is homeomorphic to the pI ´ 1q-unit sim-plex.
42
Proof. We show equivalence of the set of peaceful efficient allocations and theset of peaceful equilibrium allocations. In the first direction, observe thatevery peaceful efficient allocation is the outcome of some Walrasian equilib-rium, per Proposition A.4. In the other direction, observe that every peacefulequilibrium allocation is efficient, per Proposition A.3. Thus, the sets areequivalent, and we are done. ||
Our next step is to map the peaceful equilibrium allocations to the set ofpeaceful equilibria themselves. Recall that the set of all equilibria is E :
E “
#
pp,ωq P S ˆRI L``
ˇ
ˇ
ˇ
ˇ
ˇ
ÿ
i
fipp, p ¨ωiq “ÿ
i
ωi
+
,
where each fi : S ˆΩθÑXi is a demand function. In keeping with our conven-tions, we let ErPpθqs denote the set of all peaceful equilibria.
To work backwards from allocations to equilibria, we will obviously need afunction of the form
f ´1pp,ωq “
`
f ´11 pp, p ¨ω1q, . . . , f ´1
I pp, p ¨ωI q˘
.
To get at this, consider another extended utility function: vi : RL Ñ R. Ob-serve that the domain for vi is all of Euclidean space rather than the non-negativeorthant (open or closed); let V be the set of all functions on RL that satisfy As-sumption 3.1. (Given our definitions, Xi “ RL
`` is homemorphic to RL, sowe can continue to work with ui on Xi without any fear of changing matters.)Denote
rfipp, p ¨ωiq “ maxx: p¨xďp¨ωi
vipxq.
Observe that rfi maps from S ˆR to RL. We have:A.15 Lemma
The extended demand function rfi is a single-valued diffeomorphism.
Proof. See Balasko (2011, Proposition 3.9, pp. 25–6), who constructs theinverse demand function rather straightforwardly. ||
This means there exists a smooth, bijective inverse demand function rf ´1i :RL Ñ
S ˆR. We collocate these into an inverse societal demand function, rf ´1 :RI L Ñ
SLˆRL, where in particular
rf ´1i px1, . . . , xnq “
´
rf ´11 px1q, . . . ,
rf ´1I pxI q
¯
,
“ ppp1, w1q, . . . , ppI , wI qq .
43
Here pi is the set of prices that can support a purchase of bundle xi given wealthlevel wi “ pi ¨ωi . There is no guarantee that the inverse societal demand functionreflects an equilibrium, but if we input an efficient allocation, the second welfaretheorem guarantees that the output will reflect equilibrium. That is, we can definea function φ˚ : Pθ Ñ E , where in particular
φ˚pxq “!
pp,ωq P S ˆΩθˇ
ˇ
ˇ
rfi pp, p¨ωi q“ř
i ωirfi pp, p¨ωi q“xi for all i“1,...,I
)
.
φ˚ inherits smoothness from rfi ; the only tricky part here is in mapping wealthlevels, wi P R to endowments, ωi P RL. Because many different endowmentscan support a given wealth level, we need a smooth function mapping wealthto endowments. But clearly, the associated projections will all be affine (hencesmooth). In general, the prices that support a given efficient allocation will beunique, as they must reflect the marginal utilities at that point (which are fixed).Because of this truism and the fact that ErPpθqs is the image of the compact,connected set of peaceful equilibrium outcomes by φ˚, we can conclude:
A.16 LemmaFor any θ PΘ, the set of peaceful equilibria, ErPpθqs, is compact and connected.
We now have a compact, connected set of peaceful equilibria. We must turnthis into a set of endowments that yield peace—this is precisely the trade-to-peaceset, T pθq Ĺ Ωθ. We introduce the natural projection, π : E Ñ RI L
``, which isdefined by the formula
pp,ωq ÞÝÑω.
We know the following about the natural projection.
A.17 LemmaThe natural projection, π : E ÑRI L
``, is smooth.
Proof. See Balasko (2011, Proposition 4.10, p. 44). ||
This yields our desired result as a corollary.
3.4 PropositionFor any θ PΘ, the trade-to-peace set, T pθq, is compact and connected.
This obtains immediately because ErPpθqs is compact and connected and T pθqis the image of it by the continuous function π.
44
A.3.2 C O N T I N U I T Y O F T H E T R A D E -T O -P E AC E S E T
Our second task is to show that the trade-to-peace correspondence, T : A ÑRI L``,
is continuous in θ—i.e., that it is both upper and lower hemicontinuous. We doso along the lines of the same chain used to establish compactness. For our firststep, consider the map h :ΘÑRI
`, where in particular
hpθq “ pUθrPpθqs.
Recall that pUθrPpθqs is the set of peaceful weak utility optima.First, upper hemicontinuity. We cannot make the usual appeal the the closed-
graph property, since the co-domain of h is not compact. However, there is asimple workaround that utilizes local boundedness.
A.18 LemmaThe map h :ΘÑRI
` is locally bounded.9
Proof. Fix a θ, and let U Ă Θ be a neighborhood of θ. For any θ1 P U , weknow that hpθ1q is compact (per Lemma A.10). Define the functions h˚pU qand h˚pU q with the formulae
h˚pU q “
˜
infθ1PU
minvP pUθ1 rPpθ1qs
vi
¸I
i“1
,
h˚pU q “
˜
supθ1PU
maxvP pUθ1 rPpθ1qs
vi
¸I
i“1
.
Owing to the compactness of pUθrPpθqs, these are well-defined. (We have touse inf and sup because O is open.) Let H pU q be the closed rectangle fallingbetween these points. Clearly, H pU q is compact. Furthermore, hpθqmust liesomewhere in hpU q. We are done. ||
Because every correspondence that is closed-valued and locally bounded is upperhemicontinuous, we have the following as a corollary.
A.19 CorollaryThe map θ h
ÞÝÑ pUθrPpθqs is upper hemicontinuous.
9We say a map f : X Ñ Y is locally bounded at x PX if there is a neighborhood U ĂX of xand a compact set K Ď Y such that f pU q ĎK . If f is locally bounded at all x PX , then we say itis locally bounded.
45
We next turn our attention to the lower hemicontinuity of h. It is easier toaddress this directly.
A.20 LemmaThe map θ h
ÞÝÑ pUθrPpθqs is lower hemicontinuous.
Proof. Choose an arbitrary θ PΘ and an arbitrary open set O ĹRI such that
hpθqXO ‰H.
Choose an arbitrary v 1 P hpθqXO. From the analysis above, we know thatv 1 “ zpσ ,θq for some σ P∆I´1, where z maps the simplex to pU . (Note thatwe are parameterizing all weak utility optima for z , not just the weak peacefulutility optima.)
To obtain a contradiction, suppose that for each n PN, there is some
θpnq “ pωpnq,µpnq, cpnqq PΘ
in a 1n-neighborhood of θ such that hpθpnqqXO “H.
Define
υpOq “OX!
v PRIˇ
ˇ
ˇv “ αv 1 for some α PR`
)
.
Because O is open, we have αv 1 P O for α sufficiently close to unity; thus,υpOq ‰H for all open sets O.
But because θpnqÑ θ, the continuity of the utility functions rui , the expectedutilities of war Ui p¨q, and the homeomorphism zp¨q ensures that, if n is suffi-ciently large, then for some open set O 1 P∆I´1 with σ PO 1, we have
ph ˝ zqpO 1,θpnqqX υpOq ‰H,
a contradiction. We are done. ||
Since h is both upper and lower hemicontinuous, we draw the following impor-tant conclusion:
A.21 CorollaryThe map θ h
ÞÝÑ pUθrPpθqs is continuous.
From here, we again use the same chain we used to obtain compactness toobtain the following continuity result.
3.5 PropositionThe maps θ ÞÝÑ T pθq and a ÞÝÑ T ppθpaqq are continuous.
46
Proof. Observe:
1. per Corollary A.21, the map from parameters θ to weak peaceful utilityoptima pUθrPpθqs is continuous;
2. per Lemma A.13, the map from weak peaceful utility optima pUθrPpθqsto peaceful efficient allocations PθrPpθqs is continuous;
3. per Lemma A.15, the map from peaceful efficient allocations PθrPpθqsto peaceful equilibria ErPpθqs is continuous; and
4. per Lemma A.17, the map from peaceful equilibria ErPpθqs to peace-generating endowments T pθq is continuous.
As it is the composition of four continuous correspondences, θ ÞÝÑ T pθq isitself continuous.
Finally, observing that pθ : AÑΘ is continuous, we conclude that the map
a ÞÝÑ T´
pθpaq¯
is continuous. We are done. ||
A.3.3 A I D -LO C A L C O S T M I N I M I Z AT I O N
The set of all economic aid schedules is not compact, as it is not bounded above.We need to obtain an upper bound without rejecting any possible cost-minimizingeconomic aid schedules conditional on m1. To that end, consider
Epm1q “
!
e PRI L`
ˇ
ˇ
ˇe`i ą e`P ipm
1q for all i “ 1, . . . , I and `“ 1, . . . , L
)
.
This is the set of all economic aid schedules (pacifying or otherwise) strictly greaterthan the peace-point construction coordinate-wise. As kp¨q is strictly monotonein all coordinates, these are all more expensive than the peace-point construc-tion. We therefore do not remove any candidate cost-minimizing aid schedulesby focusing only the compact set
Epm1q “
!
e PRI L`
ˇ
ˇ
ˇe`i ď e`i for some e P Epm1
q
)
.
For ease of the next proof, introduce
ψpm1,θq “!
e P Epm1q
ˇ
ˇ
ˇω` e P T
´
pθpe , m1q
¯)
.
47
This is the candidate set of economic allocations that generate peace given somemilitary aid level. We know this set is not empty because peP pm
1q, m1q is in it. Asit is the intersection of the compact sets Epm1q and T ppθpe , m1qq´ω, ψpm1,θq isalso compact. As it is closed-valued with a compact-valued co-domain, the mappm1,θq ÞÝÑψpm1,θq is upper hemicontinuous.
We now define the set
Kpθq “ď
m1PM
ψpm1,θq.
This is the set of all candidate pacifying aid schedules given initial parameter θ.From Proposition 3.3, we know that this set is not empty, as the peace-point aidschedule peP pmq, mq is an element of it for all m PM .
We have the following.
A.22 LemmaFor any θ PΘ, the set of candidate pacifying aid schedules, Kpθq, is compact.
Proof. We must show that Kpθq is closed and bounded. We address bounded-ness first. Consider
K “
»
–
˜
maxm1PM
supePψpm1,θq
e`i
¸L
`“1
fi
fl
I
i“1
.
Because ψpm1,θq is a subset of a compact set, it is itself bounded above andbelow for every m1, so the supremum is a well-defined real number. BecauseM is compact, we write “max” instead of “sup” on it. Clearly, we have
Kpθq Ď!
e PRI L`
ˇ
ˇ
ˇ0ď e`i ďK
`i for all i “ 1, . . . , I and all `“ 1, . . . , L
)
.
As a subset of a bounded set, Kpθq is bounded.
Now closedness. Consider a sequence zn such that zn P Kpθq for all n P Nwith zn Ñ z. We need to see z P Kpθq. Note that each zn is an element ofψpm;θq for at least one m PM . In fact, define
ψ´1pz;θq “ tm PM | z Pψpm;θqu .
Observe that ψ´1pz ;θq “ψ` rtzus, where ψ` is the lower inverse (borrowingthe notation from Aliprantis and Border (2006, Chapter 17)). Because ψ isupper hemicontinuous, this it follows that ψ´1 is closed for any closed sub-set of E—in particular, for the closed subset tzu (Aliprantis and Border 2006,
48
Lemma 17.4, p. 559). Because it is has closed values and a compact-valued(hence bounded) co-domain, ψ´1 is itself compact-valued and upper hemicon-tinuous (Aliprantis and Border 2006, Theorem 17.10, p. 561). This entailsthe existence of a sequence mn with mn P ψ
´1pz;θq for all n P N such thatmn Ñ m˚ and m˚ Pψ´1pz;θq (Ok 2007, Proposition 2, p. 290–1). Thus, forany zn Ñ z, there exists an m˚ such that z Pψpm˚;θq. We are done. ||
Finally, we arrive at the cost minimization problem
minaPKpθq
kpaq.
Observe that K is continuous on its domain,RI L` ˆM , and that Kpθq is a compact
subset of that domain. From the Weierstrass Theorem, we infer our main result.
3.6 TheoremThere exists a cost-minimizing pacifying aid schedule.
49
R E F E R E N C E S
Alesina, Alberto and David Dollar. 2000. “Who Gives Foreign Aid to Whomand Why?” Journal of Economic Growth 5(1):33–63.
Aliprantis, Charalambos D. and Kim C. Border. 2006. Infinite DimensionalAnalysis: A Hitchhiker’s Guide. Third ed. Berlin: Springer.
Balasko, Yves. 1988. Foundations of the Theory of General Equilibrium. Boston,MA: Academic Press.
Balasko, Yves. 2009. The Equilibrium Manifold: Postmodern Developments in theTheory of General Economic Equilibrium. Cambridge, MA: The MIT Press.
Balasko, Yves. 2011. General Equilibrium Theory of Value. Princeton, NJ:Princeton University Press.
Barbieri, Katherine. 1996. “Economic Interdependence: A Path to Peace or aSource of Interstate Conflict?” Journal of Peace Research 33(1):29–49.
Barbieri, Katherine. 2002. The Liberal Illusion: Does Trade Promote Peace? AnnArbor, MI: University of Michigan Press.
Barbieri, Katherine and Omar Keshk. 2012. “Correlates of War Project TradeData Set Codebook, Version 3.0.” Correlates of War.URL: http://www.correlatesofwar.org
Barbieri, Katherine, Omar M. G. Keshk and Brian Pollins. 2009. “TRADINGDATA: Evaluating our Assumptions and Coding Rules.” ConflictManagement and Peace Science 26(5):471–491.
Beardsley, Kyle. 2008. “Agreement without Peace? International Mediation andTime Inconsistency Problems.” American Journal of Political Science52(4):723–740.
Bercovitch, Jacob. 1986. “International Mediation: A Study of the Incidence,Strategies and Conditions of Successful Outcomes.” Cooperation and Conflict21(3):155–168.
Bercovitch, Jacob and Allison Houston. 1996. The Study of InternationalMediation: Theoretical Issues and Empirical Evidence. In Resolving
50
International Conflicts: The Theory and Practice of Mediation, ed. JacobBercovitch. Boulder, CO: Lynne Rienner.
Bliss, Harry and Bruce Russett. 1998. “Democratic Trading Partners: TheLiberal Connection.” Journal of Politics 60(4):1122–1147.
Böhmelt, Tobias. 2010. “The Impact of Trade on International Mediation.”Journal of Conflict Resolution 54(4):566–592.
Bueno de Mesquita, Bruce and Alastair Smith. 2007. “Foreign Aid and PolicyConcessions.” Journal of Conflict Resolution 51(2):251–284.
Bueno de Mesquita, Bruce and Alastair Smith. 2009. “A Political Economy ofAid.” International Organization 63(2):309–340.
Camiña, Ester and Nicolá Porteiro. 2009. “The Role of Mediation inPeacemaking and Peacekeeping Negotiations.” European Economic Review53(1):73–92.
Carter, Jimmy. 1982. Keeping Faith: Memoirs of a President. Fayetteville, AR:University of Arkansas Press.
Dávila, Julio and Jan Eeckhout. 2008. “Competitive Bargaining Equilibrium.”Journal of Economic Theory 139(1):269–294.
Debreu, Gerard. 1970. “Economies with a Finite Set of Equilibria.”Econometrica 38(3):387–392.
Desch, David W. 2008. The Arab-Israeli Conflict: A History. New York: OxfordUniversity Press.
Dierker, Egbert. 1974. Topological Methods in Walrasian Economics. Vol. 92 ofLecture Notes on Economics and Mathematical Systems. Berlin: Springer-Verlag.
Favretto, Katja. 2009. “Should Peacemakers Take Sides? Major PowerMediation, Coercion, and Bias.” American Political Science Review103(2):248–263.
Fearon, James D. 1995. “Rationalist Explanations for War.” InternationalOrganization 49(3):379–414.
51
Fey, Mark and Kristopher W. Ramsay. 2010. “When is Shuttle DiplomacyWorth the Commute? Information Sharing through Mediation.” WorldPolitics 62(4):529–260.
Fortna, Virginia Page. 2004. “Does Peacekeeping Keep Peace? InternationalIntervention and the Duration of Peace after Civil War.” International StudiesQuarterly 48(2):269–292.
Fortna, Virginia Page. 2008. Does Peacekeeping Work? Shaping Belligerents’Choices after Civil War. Princeton, NJ: Princeton University Press.
Gartner, Scott Sigmund. 2013. “Introduction. Symposium: Innovations in theStudy of Mediation and Peacemaking.” Conflict Management and Peace Science30(4):349–353.
Gartzke, Erik and Quan Li. 2003. “Measure for Measure: ConceptOperationalization and the Trade Interdependence-Conflict Debate.” Journalof Peace Research 40(5):553–572.
Gilpin, Robert. 1981. War and Change in World Politics. New York:Cambridge University Press.
Goltsman, Maria, Johanes Hörner, Gregory Pavlov and Francesco Squintani.2009. “Mediation, Arbitration, and Negotiation.” Journal of Economic Theory144(4):1397–1420.
Gowa, Joanne. 1994. Allies, Adversaries, and International Trade. Princeton, NJ:Princeton University Press.
Gowa, Joanne and Edward D. Mansfield. 1993. “Power Politics andInternational Trade.” American Political Science Review 87(3):408–420.
Hegre, Håvard, John R. Oneal and Bruce Russett. 2010. “Trade Does PromotePeace: New Simultaneous Estimates of the Reciprocal Effects of Trade andConflict.” Journal of Peace Research 47(6):763–774.
Hirschman, Albert O. 1945. National Power and the Structure of Foreign Trade.Berkeley, CA: University of California Press.
Keohane, Robert O. and Joseph S. Nye. 1989. Power and Interdependence. NewYork: Harper Collins.
52
Kydd, Andrew. 2003. “Which Side Are You On? Bias, Credibility, andMediation.” American Journal of Political Science 47(4):597–611.
Kydd, Andrew. 2006. “When Can Mediators Build Trust?” American PoliticalScience Review 100(3):449–462.
Mansfield, Edward D. 1994. Power, Trade, and War. Princeton, NJ: PrincetonUniversity Press.
Mansfield, Edward D. and Brian M. Pollins. 2001. “The Study ofInterdependence and Conflict: Recent Advances, Open Questions, andDirections for Future Research.” Journal of Conflict Resolution 45(6):834–859.
Mas-Colell, Andreu. 1985. The Theory of General Economic Equilibrium: ADifferentiable Approach. Cambridge, UK: Cambridge University Press.
Mas-Colell, Andreu, Michael D. Whinston and Jerry R. Green. 1995.Microeconomic Theory. New York: Oxford University Press.
Mearsheimer, John J. 1990. “Back to the Future: Instability in Europe after theCold War.” International Security 15(1):5–56.
Milnor, John. 1965. Topology from the Differentiable Point of View.Charlottesville, VA: The University Press of Virginia.
Ok, Efe A. 2007. Real Analysis with Economic Applications. Princeton, NJ:Princeton University Press.
Oneal, John R. and Bruce Russett. 1997. “The Classical Liberals Were Right:Democracy, Interdependence, and Conflict, 1950–1985.” International StudiesQuarterly 41(2):267–293.
Oneal, John R. and Bruce Russett. 1999. “Assessing the Liberal Peace withAlternative Specifications: Trade Still Reduces Conflict.” Journal of PeaceResearch 36(4):423–442.
Oneal, John R., Frances H. Oneal, Zeev Maoz and Bruce Russett. 1996. “TheLiberal Peace: Interdependence, Democracy, and International Conflict,1950–1985.” Journal of Peace Research 33(1):11–28.
Penta, Antonio. 2011. “Multilateral Bargaining and Walrasian Equilibrium.”Journal of Mathematical Economics 47(4–5):417–424.
53
Polachek, Solomon W. 1980. “Conflict and Trade.” Journal of ConflictResolution 24(1):55–78.
Polachek, Solomon W. 1997. “Why Democracies Cooperate More and FightLess: The Relationship between International Trade and Cooperation.”Review of International Economics 5(3):295–309.
Polachek, Solomon W., John Robst and Yuan-Ching Chang. 1999. “Liberalismand Interdependence: Extending the Trade-Conflict Model.” Journal of PeaceResearch 36(4):405–422.
Pollins, Brian M. 1989a. “Conflict, Cooperation, and Commerce.” AmericanJournal of Political Science 33(4):737–761.
Pollins, Brian M. 1989b. “Does Trade Still Follow the Flag?” American PoliticalScience Review 83(3):465–480.
Quandt, William B. 1986. Camp David: Peacemaking and Politics. Washington,DC: The Brookings Institution.
Quinn, David, Jonathan Wilkenfeld, Kathleen Smarick and Victor Asal. 2006.“Power Play: Mediation in Symmetric and Asymmetric International Crises.”International Interactions 32(4):441–470.
Rauchhaus, Robert. 2006. “Asymmetric Information, Mediation, and ConflictManagement.” World Politics 58(2):207–241.
Russett, Bruce and John R. Oneal. 2001. Triangulating Peace: Democracy,Interdependence, and International Organizations. New York: Norton.
Savun, Burcu. 2008. “Information, Bias, and Mediation Success.” InternationalStudies Quarterly 52(1):25–47.
Smith, Alastair and Allan Stam. 2003. “Mediation and Peacekeeping in aRandom Walk Model of Civil and Interstate War.” International StudiesReview 5(4):115–135.
Telhami, Shibley. 1992–3. “Evaluating Bargaining Performance: The Case ofCamp David.” Political Science Quarterly 107(4):629–653.
Yildiz, Muhamet. 2003. “Walrasian Bargaining.” Games and Economic Behavior45(2):465–487.
54