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Publicly Available Published by De Gruyter February 9, 2018

Rules for Dividing a Disputed Resource in the Context of the Classical Liberal Argument for Peace

  • Michelle R. Garfinkel EMAIL logo and Constantinos Syropoulos

Abstract

In this paper, we study alternative forms of conflict resolution, both peaceful and non-peaceful, between two countries that compete for claims to a resource used to produce potentially traded goods. Consistent with the classical liberal argument, peace supports mutually beneficial trade, whereas war preempts it. War always induces countries to allocate resources into non-contractible arming (“guns”) for superiority in conflict. Under peaceful settlement, countries might choose to arm as well for gaining leverage in negotiations, but arming is typically less than what it is under war. Building on the observation that arming itself affects the countries’ bargaining sets, we compare the efficiency properties of division rules generated by three prominent bargaining solutions – namely, splitting the surplus, equal sacrifice, and Nash bargaining – and show how they depend on the gains from trade.

JEL Classification: C72; C78; D30; D74; F51

1 Introduction

The longstanding classical liberal view in the international relations literature holds that globalization and/or trade liberalization promotes global peace (e.g. Polachek, 1980). Specifically, to the extent that war between nations disrupts trade between them as Glick and Taylor’s (2010) evidence suggests, greater economic interdependence increases the costs of war and thus should decrease the likelihood of its onset. This view finds some support empirically; overall, however, the evidence is mixed.[1] Moreover, the particular mechanisms underlying this link are not fully understood. Existing theoretical work shows that whether the presence of trading opportunities makes conflict more or less severe (as measured by resources diverted to the military) depends on a number of factors that drive trade and its possible gains, including the source of comparative advantage and the countries’ relative sizes.[2] Even less understood are how different methods of conflict resolution between nations, who possibly trade with each other, affect the potential gains they could realize by settling their differences peacefully.

An obvious advantage of not resorting to war, in addition to leaving open the possibility of trade, is that it allows the two sides to avoid war’s destructive effects on productive resources. A related but perhaps less obvious advantage derives from the anticipation that the dispute will be resolved peacefully – namely, to free up resources that otherwise would have to be devoted to building one’s military, thereby leaving more for production and trade. In the context of international relations where contracting between countries is incomplete, the magnitude of this benefit would be moderated by the countries’ perception that building one’s military force in advance of their negotiations, even if not deployed, is necessary to improve one’s bargaining position and, thus, obtain a better “deal.” If building military force is seen as being more important this way, the possible net gains from a peaceful settlement are smaller; and, conversely, if building such force is seen as being relatively unimportant, the possible net gains are larger. Hence, the conventions or norms of negotiation that spell out how the two sides of a dispute reach an agreement matter for determining the gains that can be realized under a peaceful settlement over war.[3]

In this paper, we explore how such norms interact with the gains from trade possible under peaceful settlement. More specifically, based on a simplified version of the model presented in Garfinkel and Syropoulos (2017b) of resource disputes and trade between two countries, we compare the efficiency properties of three different rules of division under settlement. These rules correspond to three prominent bargaining solutions – namely, splitting the surplus, equal sacrifice and Nash bargaining.[4] They differ in terms of the role assigned to the threat points or payoffs under war, as defined by the quantity “arms” or “guns” brought by each side to the negotiation table. Specifically, the split-the-surplus solution involves an equal division of the difference between the sum of the two sides’ payoffs under the agreement and the sum of their threat-point payoffs; the equal sacrifice solution requires that each side’s sacrifice in payoffs, defined as the difference between one’s maximum feasible payoff (or ideal payoff) and the payoff under the agreed upon division, be equalized; and, the Nash bargaining solution involves maximizing the product the two sides’ surpluses realized under the agreement relative to their respective threat points. The different roles assigned to the threat points mean that the rules induce different incentives for arming in anticipation of settlement, which imply different quantities of the resource remaining for production and trade. Therefore, like Anbarci, Skaperdas, and Syropoulos (2002), we find the relative efficiency of the rules depends, at least in part, on their relative sensitivity to the threat points or payoffs under war. As in that paper, variation in such sensitivity across rules can be interpreted as variation in the norms against threats, with greater sensitivity implying weaker norms against threats, greater equilibrium arming and thus greater inefficiency.

Moving beyond Anbarci, Skaperdas, and Syropoulos (2002), however, we find that the shape and size of the bargaining set also depend on the potential gains from trade.[5] If the gains from trade are large, then at least one of the division rules considered (either that which follows from the equal sacrifice solution or that which follows from split the surplus solution) can support peace with no arming at all; and, in this case, Nash bargaining, which always induces positive arming, is dominated by the other bargaining protocols. However, if the gains from trade are sufficiently small, then all of the bargaining solutions considered involve strictly positive arming; in this case, equal sacrifice – the solution concept that relies least on the disagreement payoffs – is Pareto dominant.

How do the payoffs under peaceful settlement compare with the payoffs under conflict? Under the premise of the classical liberal hypothesis that war disrupts trade, each of the rules considered under peaceful settlement delivers a higher payoff than the payoff under war. What’s more, whether the gains from trade are large or small, each of the rules considered supports peaceful settlement as the unique equilibrium outcome. Put differently, provided that there are some gains from trade, war is not a possible equilibrium outcome.[6]

The remainder of the paper proceeds as follows. In the next section, we present a simple one-period model of conflict and trade, along with a preliminary analysis of outcomes and payoffs under war and under peaceful settlement for any division rule considered and given the gun choices made by each side of conflict. In Section 3, we turn to study each of the rules of division implied by the three bargaining solutions and derive our central findings. Section 4 offers concluding remarks, including possible extensions of the analysis. Technical details are relegated to Appendix A.

2 A simple model of conflict and trade

Consider an environment in which each of two agents/countries, i = 1, 2, is endowed with Ri units of a productive resource such as land, oil or water. Once held securely, this resource can be used to produce (possibly tradable) commodities. However, initial claims are not secure. Each agent invests Gi units of Ri in a non-contractible and irreversible influence activity called “arming” (or “guns” for short) to contest the remaining resource jointly held by both: X¯i(RiGi).

Their competing claims over X¯, given their prior arming choices, can be resolved in one of two ways: through a winner-take-all contest (“war”) or through bargaining and negotiation (“peaceful settlement”). Under war which precludes trade, each agent’s probability of winning the contest depends on the agents’ relative arming.[7] Under peaceful settlement, the agents divide X¯ on the basis of some predetermined bargaining solution concept. Once X¯ is divided and the agents produce their goods, they engage in free trade with each other.

A distinguishing feature of peaceful settlement in this setting is that it is conducted under the threat of war. That is to say, if the two agents reach an impasse in their negotiations, war breaks out, with each agent’s probability of winning all of X¯ depending on his/her relative guns. As will become evident, arming provides, through its impact on the disagreement or war payoffs, agents with leverage in their negotiations. But, by diverting resources away from useful production, arming also affects (i) the size of the “prize” and (ii) the trade-off in payoffs along the payoff possibility frontier. To the extent that division rules differ in their reliance on the just noted dimensions of the bargaining problem, their efficiency properties could differ as well. The question is: How?

2.1 Production and trade for given effective endowments

We now turn to production and trade that are important for characterizing the bargaining set and thus for answering the question above. Let the share of X¯ obtained by agent i under peaceful settlement based on the solution concept h ∈ {ss, es, nb} be denoted by λhi, where “ss” is identified with splitting the surplus, “es” with equal sacrifice, and “nb” with Nash bargaining. Furthermore, let ZiλiX¯ be agent i’s effective endowment.[8] We assume for simplicity that Zi can be used to produce, on a one-to-one basis, a nationally differentiated and potentially tradable commodity. We interpret this commodity as an intermediate input to the production of a final consumption good, say “butter.”[9]

To capture the gains from trade in a compact and intuitive way, we assume the production function for butter takes the constant elasticity of substitution (CES) form. Production by agent i is given by

(1)Fi=F(D1i,D2i)=[(D1i)σ1σ+(D2i)σ1σ]σσ1,

where Dji denotes the quantity of intermediate input j ∈ {1, 2} demanded by agent i ∈ {1, 2}, and σ > 1 denotes the elasticity of substitution between the two intermediate inputs. The specification in (1), which is symmetric, increasing, linearly homogeneous, and (provided σ < ∞) strictly concave in its arguments, reflects the benefit of employing a greater variety of distinct inputs in production, analogous to the “love of variety” exhibited by Dixit-Stiglitz preferences; and, this benefit, which can be interpreted as the gains from trade, increases as the two intermediate goods become more distinct (i.e. as σ (>1) falls).[10]

Whether we consider trade or autarky, agent i chooses the allocation of his effective endowment Zi to maximize his output and thus consumption of butter Fi shown in (1). Of course, the constraints relevant for that choice depend on whether the two agents trade or not. Under autarky, each agent i is forced to rely solely on his own productive capabilities; thus, Dji=0 for ij and Fi = F(⋅) = Zi in this case.

The case of free trade is a bit more involved. Let Yi, pji and γji respectively denote agent i’s income, price and expenditure share on good j. One can verify that γji(pji/Pi)1σ, where Pi[j(pji)1σ]11σ, and furthermore that Dji=γjiYi/pji, where Yi=piiZi. Now let the “world” price of good j be denoted by πj. Our assumption of free trade implies pji=πj for each agent i = 1, 2 and both goods j = 1, 2. Naturally, relative prices are endogenously determined. By Walras’ Law, these prices follow from the world market-clearing condition πjDji=πiDij (ij), and satisfy

(2)πTπj/πi=(Zi/Zj)1/σ=(λi/λj)1/σ,

where the far right expression was obtained from the definition ZiλiX¯.

2.2 Payoffs under peaceful settlement

To derive an expression for agent i’s payoff function under trade presuming peaceful negotiations are successful (Vi = Fi), we combine (2) and the relations noted above with (1) to obtain, after simplification,

(3)ViVi(λi,Gi,Gj)=viX¯,

where

(4)vi(λi)λi[1+(λi/λj)1σσ]1σ1=[(λi)σ1σ(λi)σ1σ+(λj)σ1σ]F(λi,λj).

The far right expression in (4) reveals that the payoff to agent i, per unit of the contested resource X¯, coincides with his competitive share of the output that would be produced in an integrated competitive economy.[11] That is to say, each agent is rewarded with the value of his marginal product times the quantity of the resource he controls.

Since as one can easily confirm argmaxλiF(λi,λj)=12 and vi + vj = F(λi, λj), each agent would be willing to sacrifice some efficiency (i.e. a reduction in F(λi, λj)) in return for a larger share λi of the contested resource. It is straightforward to verify that, provided the two intermediate inputs are distinct (i.e. σ < ∞), vi is strictly concave in λi, that λTiargmaxλivi(12,1) and that limσ1λTi=12, while limσλTi=1.[12] Clearly, the elasticity of substitution σ plays a key role in the characterization of the critical values of λi for payoffs under trade. Indeed, as we will see shortly, the value of σ figures prominently in the determination of the bargaining set and the ranking of payoffs that the division rules h ∈ {ss, es, nb} deliver.

2.3 Payoffs under war

Turning to war, let ϕi(Gi, Gj) be the probability that agent i (≠j) wins. For simplicity, we assume the ratio form of the conflict technology:

(5)ϕi(Gi,Gj)={GiGi+Gj, if Gi+Gj>012, if Gi+Gj=0.

According to this specification, agent i’s winning probability is increasing in his own guns Gi, decreasing in his rival’s guns Gj and is symmetric so that ϕi=ϕj=12 when Gi = Gj.[13]

Recall that, since there is no trade under war, the expression in (1) implies that production depends linearly on the agent’s own effective endowment: Fi = Zi. Now let ZWX¯ denote the effective endowment enjoyed by the winner of the conflict and ZL = 0 denote the loser’s effective endowment. Then, we can write agent i’s expected payoff under conflict, denoted by Ui, as

(6)UiUi(Gi,Gj)=ϕiZWi+(1ϕi)ZLi=ϕiX¯.

For future reference, define uiϕi, so that ui + uj = 1 and Ui = uiX¯. As one can confirm, by differentiating the far right expression in (6) with respect to Gi, recalling X¯=i(RiGi), using (5) and examining the resulting first-order conditions (FOCs), agent i’s best response to agent j’s choice Gj, Bci(Gj), is given by

(7)Bci(Gj)=max[Ri,Gj(Ri+Rj)Gj],

which implies a symmetric equilibrium under war, Gci=Gc=14(Ri+Rj) for i = 1, 2, when neither agent is resource constrained in his arming choice.[14]

2.4 A comparison of payoffs for given guns

To understand the nature of the bargaining protocols we wish to study – and, in particular, the way in which they condition arming incentives – it helps to fix guns at some level and ask how the per unit of X¯ payoff vectors, (vi, vj) under settlement and (ui,uj) under conflict, compare. As suggested earlier, the elasticity of substitution σ is a crucial parameter here. Figure 1 illustrates the dependence of the above payoffs (per unit of X¯) along the payoff possibility frontier (PPF) for several values of σ. The black, solid-line segment that connects point 1 on the two axes depicts the possible payoff values under conflict, ui. Points along this schedule define the agents’ disagreement payoffs which, of course, depend on their relative arming. The midpoint of this schedule depicts the combination of the agents’ expected payoffs under war that arises when G1 = G2. In the extreme case, where σ → ∞ such that the intermediate tradables are perfect substitutes, the PPF for (vi,vj) coincides with the u1 + u2 = 1 frontier; in this special case, trade and, thus, settlement are pointless.

Figure 1: Bargaining Sets and their Dependence on the Gains from Trade.
Figure 1:

Bargaining Sets and their Dependence on the Gains from Trade.

Now suppose σ is finite, which implies as noted earlier that vi(⋅) is strictly concave in λi. Then, for any feasible pair of guns (Gi, Gj), there exist many combinations of shares, λi and λj = 1 − λi, for which both agents are weakly better off under settlement (viui for i = 1, 2), and at least one agent is strictly better off (vi > ui). The collection of such combinations gives us the bargaining set. Figure 1 depicts this set, in the symmetric case where Gi = Gj, for alternative (finite) values of σ.[15] Importantly, the lower is the value of σ (>1) and thus the greater are the possible gains from trade, the larger are the potential gains from settlement and thus the larger is the size of the bargaining set. The particular rule of division used by the two agents, then, pins down a point on the relevant PPF. For example, a division of the contested resource according to ϕi given Gi = Gj would pick the (vi,vj) combination where the 45 line intersects the relevant PPF (see Figure 1). More generally, the strict concavity of vi(⋅) implies, again for given (Gi, Gj), the two sides always find it appealing to settle their disputes peacefully by dividing the resource X¯ according to ϕi instead of initiating war and winning all of X¯ with probability ϕi.

But, while the division of the resource according to ϕi is one possibility, other division rules, such as the ones noted earlier, could be considered. To prepare the groundwork for this analysis, we start with the definition of an agent’s “ideal” payoff. Let λ~j denote the smallest share of the contested resource that agent j would be willing to accept in a peaceful settlement. This share satisfies

(8)v~jvj(λ~j)=uj(),

with uj = ϕj defined in (5) and vj defined in (4). This share for agent j would leave agent i with a share λ~~i=1λ~j that generates a payoff

(9)v~~ivi(λ~~i).

This payoff combination is illustrated in Figure 1, assuming Gi = Gj and σ = 3, at point E. However, depending on the degree of substitutability between the two inputs (σ), λTiargmaxλivi could be less than λ~~i, in which case agent i would obtain a higher payoff by accepting a smaller share of the contested resource – i.e. vi(λTi)>vi(λ~~i) – which his rival would gladly accept. This reasoning suggests that an agent’s “ideal” payoff in the bargaining set (denoted by wi) can be defined as

(10)wi=max[vi(λ~~i),vi(λTi)].

The next lemma builds on these ideas to characterize some important properties of the payoffs vi and wi as they depend on the potential gains from trade:

Lemma 1

Under settlement, agent i’s payoff vi and ideal payoff wi for i = 1, 2 satisfy the following relationships:

  1. Ifσ ⋚ 2, thenvi(12)vi(1).

  2. Suppose G1 = G2 = G. Then,wi=vi(λTi)forσ ≤ 2 andwi=vi(λ~~i)forσ ≥ 2.

Proof: See Appendix A.

Part (1) suggests a potential trade-off between ownership and trade. In particular, it establishes that, when the gains from trade are sufficiently large (σ < 2), both agents prefer to share the disputed resource equally and then engage in trade over controlling the entire resource in which case trade would not be possible. Otherwise (when σ ≥ 2), both agents prefer to control the entire resource.[16] Part (2) shows that, when both agents produce identical quantities of guns and the gains from trade are relatively large (σ ≤ 2), each agent’s ideal payoff coincides with the payoff he would enjoy if he were free to choose the division of X¯ solely on the basis of self interest, as indicated by λTi. By contrast, when σ ≥ 2, an agent’s ideal payoff depends on the disagreement payoffs, as indicated by λ~~i that gives his rival the lowest acceptable payoff. Figure 1 illustrates these points.

3 Division rules and their relative efficiency

We now study the efficiency properties of three division rules generated by the three bargaining protocols identified earlier – namely, the split-the-surplus (ss), equal sacrifice (es) and Nash bargaining (nb) solutions. After formally defining these rules, we compare their implications for arming and payoffs and ultimately for the relative appeal of peace. Henceforth, we adopt the assumption of symmetry across agents/countries:

(11)Symmetry: Ri=R for i=1,2.

Under this assumption, equilibrium spending under peaceful settlement for any rule h ∈ {ss, es, nb} will be symmetric.

As noted earlier, λhi (for i = 1, 2 and h ∈ {ss, es, nb}) denotes the share of the common pool obtained by agent i under solution h. We similarly index payoffs vhi=vi(λhi) and Vhi=Vi(λhi) (=vhiX¯). Keeping guns (and thus X¯) fixed in the background, define the following functional relationships:

(12)ΦΦ(λi,Gi,Gj)=vjujviui
(13)ΨΨ(λi,Gi,Gj)=wjvjwivi
(14)ΩΩ(λi)=vλjjvλii,

for ij = 1, 2, where we use the convention that vλiidvi/dλidvi/dλj (since λj = 1 − λi). Then, the solution concepts noted above can be defined as follows:

(15)Φ(λi,Gi,Gj)=1vj(λssj)ujvi(λssi)ui=1
(16)Ψ(λi,Gi,Gj)=1wjvj(λesj)wivi(λesi)=1
(17)Φ(λi,Gi,Gj)+Ω(λi)=0vj(λnbj)ujvi(λnbi)ui=vλjj(λnbj)vλii(λnbi).

Since λhj=1λhi, the shares λhi induced by the various solution concepts are implicitly defined by the above equations.

These concepts can be visualized with the help of the two panels in Figure 2, drawn for given arms with G1 > G2.[17] In both panels, the combination of payoffs vh1 and vh2 under each of the solutions considered h ∈ {ss, es, nb} lie on the PPF. The ideal payoff for agent 1 w1 is defined by λT1 in panel (A) where σ < 2 and is defined by λ~~1 in panel (B) where σ > 2. For the rule generated by ss, λssi is implicitly defined (by (15)) as point ss in the panels, because Φ = 1 holds along the 45° line going through the (u1, u2) point of disagreement payoffs. Similarly, the λesi share is defined (by (16)) as point es in both panels, because Ψ = 1 holds along the 45° line going through the point of ideal payoffs (w1, w2). Finally, the Nash bargaining solution satisfies λnbi=argmaxλi[vi(λi)ui)][vj(λj)uj]. The expression in (17) represents the FOC to this problem.

Figure 2: Disagreement Payoffs, Ideal Payoffs and Division Rules.
Figure 2:

Disagreement Payoffs, Ideal Payoffs and Division Rules.

It is important to emphasize that changes in guns do not directly affect payoffs vhi. However, changes in guns can affect shares (and thus the payoffs vhi) through their impact on the disagreement payoffs (ui, uj). In the case of equal sacrifice, the effect of guns on λesi is transmitted through their impact on the ideal payoffs which, as we saw earlier, can depend on guns indirectly (though not always), again through the disagreement payoffs.

To identify the impact of guns on payoffs we partially differentiate Vhi=vhiX¯ with respect to Gi, accounting for the effect on the resource that remains, X¯:

(18)Vhi/Gi=X¯(vhiλhi)(λhi/Gi)vhi, i=1,2 and h{ss,es,nb}.

The first term represents the indirect effect of an additional gun on agent i’s payoff through its effect on the division rule; the second term captures the constant marginal cost as an additional gun implies a smaller quantity of the resource X¯ to divide. It should now be evident that the sensitivity λhi/Gi of an agent’s share to his guns is central to his arming decision.

The next lemma describes this sensitivity across the three solution concepts:

Lemma 2

Assume symmetry (11) holds and suppose that λhi responds to changes in Gi(i = 1, 2 and h ∈ {ss, es, nb}). Then, 0<λesi/Gi<λnbi/Gi<λssi/Gi, where each partial derivative is evaluated at an identical quantity of arms Gi = Gj = G.

Proof: See Appendix A.

Maintaining (momentarily) a focus on situations where agents’ shares λhi respond to changes in arming, Lemma 2 points out that, under conditions of symmetry, equal sacrifice gives rise to a division that is least sensitive to an agent’s arms, whereas splitting the surplus gives rise to a division that is most sensitive to an agent’s arms; the sensitivity of the division to an agent’s arms under Nash bargaining lies in the middle. Although Anbarci, Skaperdas, and Syropoulos (2002) do not consider Nash bargaining, they find a similar ranking for the equal sacrifice and split-the-surplus solutions in a different setting. But, the more important point here is this: the bargaining solution that relies least (most) on the threat-point payoffs corresponds precisely to the division that is least (most) sensitive to changes in arming. Interestingly, the Nash bargaining solution lies in the middle of this ranking because, even though it relies on the disagreement payoffs, this reliance is tempered by its dependence on the slope of the PPF and the sensitivity of that slope to arming.

The above discussion raises the question of whether, under our assumption of symmetry, each of the bargaining solutions h ∈ {ss, es, nb} necessarily implies a rule of division λhi that depends positively on agent i’s arming given the rival’s arming. As we now argue, the presence or absence of such an influence of guns depends on the magnitude of the gains from trade (which depends negatively on σ > 1) as well the bargaining solution considered.

Proposition 1

(Symmetric equilibria.) Assume (11), let Gh denote the representative agent’s guns in the symmetric equilibria under each bargaining solution h ∈ {ss, es, nb}, and defineA21σ1(σ1)σ(>0) and B21σ11σ1(>0). Then,

  1. under the split-the-surplus protocol, equilibrium arming satisfies Gss = 0 and/orGss=R/[1+σσ1A]:

    1. ifσ[112,2], there exist two equilibria one of which exhibits no arming;

    2. ifσ(1,112)(2,), there exists a unique equilibrium with positive arming.

  2. under the equal sacrifice and Nash bargaining protocols, there exist unique equilibria such that

    1. Ges = 0 forσ ∈ (1, 2], while Ges > 0 for σ ∈ (2, ∞);

    2. Gnb=R/[1+σσ1(A+B)]for σ ∈ (1, ∞).

Proof: See Appendix A.

Four ideas stand out in connection with Proposition 1. First, under the split-the-surplus solution and depending on the size of the gains from trade, two types of (symmetric) equilibria can arise: one with and another without arming. As explained in Garfinkel and Syropoulos (2017b), this possibility arises in a more general setting and can be explained as follows. For all σ ∈ (1, ∞], given the rival’s arming is strictly positive, each agent i has a strictly positive incentive to arm as well under the split-the-surplus solution. However, when σ < 2, the split-the-surplus rule allocates a strictly positive share of the resource to the rival even when he does not arm and agent i does. Now suppose that σ is marginally below 2 and Gj = 0. Agent i can choose Gi = ε (an infinitesimal quantity) and obtain a share close to 1 or choose Gi = 0 and obtain half of X¯. By part (1) of Lemma 1, agent i prefers the latter. Nevertheless, as σ falls, the relative appeal of choosing Gi = 0 when Gj = 0 eventually diminishes and approaches zero as σ approaches 32, thereby undermining the stability of Gss = 0 for σ(1,32).[18] Second, the symmetric equilibria under the equal sacrifice and Nash bargaining solutions are unique. The reason is that these solutions, unlike the split-the-surplus solution, do not set in motion a process that leads to no arming. Third, the equilibrium under Nash bargaining always involves a positive quantity of guns. In contrast, the equilibrium under equal sacrifice involves no arming when σ ≤ 2 and positive arming when σ > 2. The reason Gnb > 0 is because the solution is always at a negatively sloped portion of the PPF and its implementation requires each agent to produce some guns. The reason Ges = 0 for σ ≤ 2 is because in this case agents’ ideal payoffs are independent of the disagreement payoffs, as emphasized in part (2) of Lemma 1, and, consequently, are invariant to changes in arms. By contrast, when σ > 2, Ges > 0 because agents’ ideal payoffs depend on (u1, u2) and thus on arming decisions. Finally, in interior equilibria, the quantity of arms under splitting the surplus is larger than that under Nash bargaining. This ranking, which can be deduced from the equilibrium quantities described in the proposition, is driven by the ranking of the sensitivities of shares detailed in Lemma 2.

The next proposition, which follows from Proposition 1 with Lemma 1 and Lemma 2, provides a more precise statement of these ideas.

Proposition 2

(Arming Comparisons.) Under the assumption of symmetry (11), the equilibrium quantities of guns associated with the bargaining solutions h ∈ {ss, es, nb} can be compared as follows:

  1. Ifσ(1,112), then Ges = 0 < Gnb < Gss.

  2. Ifσ(112,2], then Ges = Gss = 0 < Gnbor Ges = 0 < Gnb < Gss.

  3. If σ ∈ (2, ∞), then 0 < Ges < Gnb < Gss.

The horizontal axis in Figure 3 plots the equilibrium level of arming associated with each solution concept under peaceful settlement and that under war. Panel (A) considers a case where σ(112,2] (as in part (2) of the proposition), and panel (B) considers a case where σ > 2 (as in part (3)). One can also verify easily, as shown in the figure, peaceful settlement with any rule h ∈ {ss, es, nb} induces lower equilibrium arming than what emerges under conflict for all σ ∈ (1, ∞) under assumption (11).[19]

Figure 3: Equilibrium Arming and Payoffs under the Split-the-Surplus, Equal Sacrifice
                        and Nash Bargaining Solutions
Figure 3:

Equilibrium Arming and Payoffs under the Split-the-Surplus, Equal Sacrifice and Nash Bargaining Solutions

It is worthwhile pointing out how the results in Proposition 2 compare with the results in Anbarci, Skaperdas, and Syropoulos (2002). As suggested earlier in connection to Lemma 2, our finding, that for all σ ∈ (1, ∞) equilibrium arming under equal sacrifice is no greater than that under split-the-surplus (i.e. GesGss), is similar to their ranking. Nonetheless, there are two important differences. First, in the setting of that paper with no trade, the ideal payoffs are always sensitive to the threat-point payoffs, while in setting of this paper where there is trade, they need not be – namely, under the equal sacrifice protocol when σ < 2. As such, we find in contrast to Anbarci, Skaperdas, and Syropoulos (2002) that, depending on the magnitude of the possible gains from trade, the rule of division that follows from the equal sacrifice bargaining solution can support unarmed peace. Second, depending on the potential gains from trade, the marginal benefit from arming for agent i, under the rule of division that follows from the split-the-surplus solution given Gj = 0, can fall short of the marginal cost. More precisely and in contrast to Anbarci, Skaperdas, and Syropoulos (2002), we find that, when σ(32,2), the split-the-surplus protocol can also support unarmed peace.[20]

Proposition 1 and Proposition 2 show how equilibrium arming compares across the three solution concepts. The next step is to compare the associated payoffs. Let Vh be the representative agent’s payoff when Gi = Gh (i = 1, 2 and h ∈ {ss, es, nb}). It is easy to see that Vh=21σ1(RGh), which is linearly decreasing in Gh, as indicated by the downward sloping line in Figure 3. Then, the next proposition follows readily from Proposition 2.

Proposition 3

(Payoff Comparisons.) Under assumption (11), the equal sacrifice solution weakly dominates the split-the-surplus and Nash bargaining solutions. Furthermore, the Nash bargaining solution dominates the split-the-surplus solutions in equilibria that involve positive quantities of guns. However, this ranking is reversed when the no arming equilibrium under the split-the-surplus solution is considered. More specifically, the equilibrium payoffs associated with the bargaining solutions h ∈ {ss, es, nb} can be compared as follows:

  1. Ifσ(1,112), then Ves > Vnb > Vss.

  2. Ifσ(112,2], then Ves = Vss > Vnbfor Gss = 0, while Ves > Vnb > Vssfor Gss > 0.

  3. If σ ∈ (2, ∞), then Ves > Vnb > Vss.

Proposition 3 as well as Proposition 2 are illustrated in Figure 3.

Finally, the reader might wonder whether any or all of the equilibria under peaceful settlement based on these division rules are stable. The next proposition addresses this issue.

Proposition 4

(Stability of equilibrium.) The equilibria under each of the rules of division h ∈ {ss, es, nb} are immune to unilateral deviations.

The argument here is rather intuitive, and starts with the observation that, for any feasible arming by the two agents Gi and Gj (not necessarily equal), the payoff for each agent under war is less than the payoff under settlement (and thus trade): ui(Gi,Gj)<vhi(Gi,Gj) for ij = 1, 2, where we now denote agent i’s payoff under war given Gi and Gj per unit of X¯ by ui(Gi, Gj) = ϕi(Gi, Gj) and agent i’s payoff under settlement with division rule h conditional again on Gi and Gj per unit of X¯ by vhi(Gi,Gj). But, if that inequality holds for any feasible pair (Gi, Gj), then it will surely hold for (Bci(Ghj),Ghj), where Bci(Ghj) indicates agent i’s best reply to the rival’s choice, shown in (7), evaluated at Gj=Ghj: ui(Bci(Ghj),Ghj)<vhi(Bci(Ghj),Ghj). That is to say, even if agent i were to anticipate choosing war and arm accordingly given agent j’s arming choice in anticipation of settlement, agent i would be better off choosing settlement over war. But, if agent i anticipates choosing settlement, he would adjust his guns in anticipation of settlement, implying vhi(Bci(Ghj),Ghj)<vhi(Ghi,Ghj). Thus, unilateral deviations from the settlement outcome under rule h that would trigger a war cannot be profitable. Notice that this argument does not require we impose the assumption of symmetry. Perhaps more importantly, this line of reasoning implies that war cannot emerge as an equilibrium outcome in the one-period setting of this paper. In particular, even if either one or both agents were to arm in anticipation of war, neither would have an incentive to declare war.

4 Concluding remarks

The idea that the presence of trade opportunities and the potential gains they imply for participating countries can help promote global peace is by no means new. Indeed, we have shown that, in the setting of our model despite its abstraction from war’s destructive effects, countries will always prefer a peaceful division of the productive resource followed by mutually beneficial trade over a winner-take-all contest (i.e. war) that precludes trade. But, going beyond that intuitive idea, this paper explores how different rules of division that follow from a variety of bargaining solutions rank in terms of their efficiency. One key result here, as in Anbarci, Skaperdas, and Syropoulos (2002) who do not consider trade, is that the ranking depends on how sensitive the rules of division are to arming through their influence on the disagreement payoffs (or the payoffs obtained under war). The greater is that sensitivity or the weaker are norms against threats, the greater are the incentives to arm under peaceful settlement, leaving fewer resources left over for production.

Building on that idea and extending Anbarci, Skaperdas, and Syropoulos (2002), our analysis shows further that the ranking of the sensitivity of these rules depends on the gains from trade. When the gains are sufficiently large, then either the rule based on the equal sacrifice solution or the rule based on the split-the-surplus solution can support unarmed peace that would dominate the rule based on Nash bargaining that generates strictly positive equilibrium arming. Otherwise, each of the rules considered imply arming in equilibrium, and in this case the rule based on the equal sacrifice solution, which is least sensitive to the disagreement payoffs and thus implies the strongest norms against threats, emerges as the Pareto dominant rule.

Since our analysis is based on a very simple setting, it is natural for one to ask how the efficiency rankings extend to more general settings. Consider, for example, our assumption that initial resources are symmetrically distributed across the two agents. It turns out that this assumption is important only insofar as it allows us to abstract from outcomes where one of the two agents is resource constrained in the production of guns. That is to say, the equilibria in guns under peaceful settlement based on any of the rules of division considered as well as the equilibrium under war remain symmetric provided the resource constraint does not bind for either of the agents in his production of guns. As such, the efficiency ranking of the rules of division under settlement found above remain intact. Even when one agent is resource constrained under war, the finding that incentives to arm under peaceful settlement are lower than under war means that it is possible neither agent is resource constrained under settlement, in which case the efficiency rankings would continue to hold. Of course, if the initial distribution of the resource is sufficiently uneven, one of the agents could be resource constrained under one or more of the rules of division for peaceful settlement as well as under war.[21] The implications for the efficiency ranking of the rules of division under settlement are not immediately obvious.[22] Still, settlement would continue to dominate war for both agents since it involves less arming for the unconstrained agent; and, by our arguments above, war could not emerge in the one-period setting analyzed here.

This result would seem to form a strong basis for the classical liberal view. However, we caution the reader that it need not hold generally in a multi-period setting, under the reasonable assumption that commitments to the division of the resource in the future are not feasible. In particular, even if the two sides settle their dispute in the current period peacefully, they would have to negotiate a division of the resource in the future and that possibly requires additional arming.[23] War, by contrast, could confer on the winner a strategic advantage in future disputes over the resource. One extreme possibility here, studied by Garfinkel and Syropoulos (2017b) while focusing on the split-the-surplus rule, is that victory in a war effectively wipes out the rival and thus eliminates the victor’s need to arm in the future. In that dynamic setting, depending on the magnitude of the gains from trade, the destructiveness of war and the salience of the future, among other factors, war could emerge as the perfectly coalition-proof equilibrium. While under other parameterizations of the model peaceful settlement could Pareto dominate war, the arguments of the present paper in a one-period setting that rule out the profitability of unilateral deviations do not easily carry over to the multi-period setting. Nevertheless, results of the present paper suggest that the other bargaining protocols (namely, equal sacrifice and Nash bargaining) could be more conducive to peaceful settlement than the split-the-surplus solution in a multi-period setting.

Acknowledgement

We are indebted to Nejat Anbarci and Stergios Skaperdas, as well as the Associate Editor, Arzu Kibris, and three anonymous referees, for insightful comments and suggestions.

A Appendix

Proof of Lemma 1:

Part (1): Setting λi = 1/2 and λi = 1 in (4) gives vi(12)=2σ2σ1 and vi(1) = 1, respectively. The result then follows by comparing these payoffs. In particular, if σ = 2, then vi(12)=1=vi(1). In contrast, if σ < 2 (resp., σ > 2), then vi(12)>1 (resp., vi(12)<1).

Part (2): First note that differentiation of vi with respect to λi (while, at the same time, allowing λj to change since λj = 1 − λi) gives

(19)vλii1σviλiλj[σλj(λj)σ1σ(λi)σ1σ+(λj)σ1σ].

Now define ρ λi/λj. Since λi + λj = 1, we have λi = ρ /(1 + ρ) and λj = 1/(1 + ρ). Next observe that the (implicit) solutions to vλii=0 and vjuj =0 give λTi and λ~~i, respectively. We may rewrite these conditions so that, in the former case, we have:

(20)11λTi=1+ρ=σ(1+ρσ1σ).

Similarly, setting Gi = Gj = G (which implies uj=12) and using the definition of vj in the vjuj = 0 case we have

(21)11λ~~i=1+ρ=2(1+ρσ1σ)1σ1.

One can readily see now that both (20) and (21) have unique solutions in which ρ > 1. Division of the expressions in the far right of (20) and (21) gives

2σ(1+ρσ1σ)σ2σ1.

Since the left-hand sides of (20) and (21) involve 1 + ρ, the relationship of the above expression to 1 describes the relationship between λTi and λ~~i. It is now easy to verify that σ21λTi1λ~~i or, equivalently, σ2λTiλ~~i. ‖

Proof of Lemma 2:

To find the impact of an agent’s arming on his share of the disputed resource, first we differentiate the equations in (1214) to obtain:

(22)Φλi=Φvλii+vλjjviui and ΦGi=ΦuGiiuGijviui
(23)Ψλi=Ψvλii+vλjjwivi and ΨGi=ΨwGiiwGijwivi
(24)Ωλi=Ωvλiλii+vλjλjjvλii.

Differentiating the equations in (1517) and using the implicit function theorem yield:

(25)λssi/Gi=ΦGiΦλi=ΦuGiiuGijΦvλii+vλjj
(26)λesi/Gi=ΨGiΨλi=ΨwGiiwGijΨvλii+vλjj
(27)λnbi/Gi=ΦGiΦλi+Ωλi=ΦuGiiuGijΦvλii+vλjj(viui)(Ωvλiλii+vλjλjjvλii).

Since the signs and magnitudes of the expressions in (2224) play key roles in the determination of λhi/Gi shown in (2527), it pays to investigate them now. First, recalling that uiϕi (i = 1, 2) and using (5), we have uGii=ϕGii>0 and uGij=ϕGii<0 as well as vi > ui, which from (22) imply ΦGi>0. Second, we have vλii>0 for λi[12,λTi) (i = 1, 2). As such, under assumption (11) which implies λi=λj=12, vλii>0 holds in equilibrium. Then, it follows from (22) that Φλi|λi=1/2<0 and, thus, from (25), that λssi/Gi>0 holds.

The above discussion also sheds light on the sign of λnbi/Gi and its ranking relative to λssi/Gi. As noted in the text and as can be confirmed by differentiating vλii in (19) to find

(28)vλiλii=1vi{(λj)σ+1σ[(λi)σ1σ+(σ1)(λj)σ1σ](λi)2σ2[(λi)σ1σ+(λj)σ1σ]2}<0,

vi is strictly concave in λi for both players provided σ > 1. In addition, from (14) and the definition of λTi, one can then see Ω ≡ Ω(λi) < 0 for λi[12,λTi). Thus, the last term in the denominator of the far right expression in (27) is positive. This point and the analysis in the previous paragraph imply λssi/Gi>λnbi/Gi>0, the last inequality shown in Lemma 2. In words, the concavity of payoff functions in shares (which enters the determination of distributive shares under nb but not under ss) renders λnbi less sensitive to changes in guns than λssi under the conditions of the lemma.

Turning to λesi/Gi, the definition of the ideal payoffs in (10) reveals that changes in arming will influence these payoffs only in cases where wi=vi(λ~~i) (which as discussed earlier depends on the value of σ, an inverse proxy of the gains from trade). Focusing on this case, (8) implies wGii=v~~λii(λ~~i/Gi)=v~~λii(λ~j/Gi) since λ~~i=1λ~j, for ij = 1, 2. Equation (8) with assumption (11) further implies λ~j/Gi=uGij/v~λjj. Repeating this argument for changes in wj gives

(29)wGii=v~~λiiv~λjj(uGij)
(30)wGij=v~~λjjv~λii(uGii).

Thus, provided λi[12,λTi) for both players, wGii>0 and wGij<0 hold. To proceed further, we must make more intensive use of our symmetry assumption. Accordingly, we focus on the case of equal arming (i.e. Gi = G for i = 1, 2), which implies, among other things, uGij=uGii, vλjj=vλii, vλjλjj=vλiλii and enables us to rewrite v~~λii/v~λjj=v~~λjj/v~λii as v~~λii/v~λii (since v~λjj=v~λii). But, the strict concavity of vi in λi implies v~~λii/v~λii<1; thus, wGii=wGij<uGii. Since as can easily be confirmed Φ = Ψ = −Ω = 1 under symmetry, a comparison of (25) and (26) shows that λssi/Gi>λesi/Gi>0. Thus, the division under es is less sensitive than division under ss to an agent’s arming, due to the indirect (and weaker) dependence of the ideal payoffs on the threat point.

What we do not know yet is how λesi/Gi compares with λnbi/Gi. Such a comparison requires a considerable amount of additional work. Applying the implications of symmetry to (26) and (27), we can rewrite these expressions, after simplification, as

(31)λesi/Gi=(v~~λii/v~λjj)uGiivλii=uGii/vλiiv~λjj/v~~λii
(32)λnbi/Gi=uGiivλii(viui)(vλiλii/vλii)=uGii/vλii1(1ui/vi)[vivλiλii/(vλii)2].

Note that v~λjj/v~~λii is the slope of the PPF at wi – or, in the context of Figure 2(B), the slope of the PPF at point E (but with G1 = G2) or at point E in Figure 1.

Now, using the transformations λi = ρ/(1 + ρ) and λj = 1/(1 + ρ) in the definitions of v~λjj and v~~λii we may rewrite v~λjj/v~~λii in (31) as

v~λjj/v~~λii=ρ1σ1σ1+ρ1+ρσ1σσ1+ρ1+ρσ1σ>1, since ρ>1 at vjuj=0.

Turning to (32), ui/vi=21σ1 (since ui = 1/2 and vi=2σ2σ1) and vivλiλii/(vλii)2=σ/(σ1)2 in the denominator. Upon substitution of these expressions in (31) and (32), we arrive at

(33)Λ(ρ,σ)λesi/Giλnbi/Gi=[σ1+ρ1+ρσ1σρ1σ1σ1+ρ1+ρσ1σ][1+σ(121σ1)(σ1)2]>0.

Since the expression inside the first set of brackets is smaller than 1 while the expression in the second set of brackets is larger than 1, it is unclear how es and nb compare in the sensitivities of the shares they give rise to. Nevertheless, we can resolve this ambiguity constructively, with the use of numerical methods that exhaust all possibilities.

To this end, note from (21), that the value of ρ that solves vjuj = 0 coincides with the solution to 1+ρ=2(1+ρσ1σ)1σ1. Denote this solution as ρ0(σ). It is straightforward to show that ρ0 is unique and that 0/dσ < 0. We calculated ρ0(σ) for values of σ ∈ (2,1000) using a grid of 0.001. The calculations relied on Newton’s method. Substituting each value of ρ0 into Λ(⋅) and evaluating the resulting expression confirms that, indeed, Λ(ρ0(σ), σ) < 1 for all relevant values of σ. In short, λnbi/Gi>λesi/Gi>0, which completes the proof. ‖

Proof of Proposition 1:

With our maintained focus on symmetric equilibria that imply λhi=1/2 for all h, we derive the following expressions for the relevant terms appearing in (18): vi=2σ2σ1, vλii=21σ1(σ1)σ, and X¯ = 2(RG). Substituting these relationships in (18) and simplifying the resulting expression gives

(34)VhiGi=2σ2σ1[4(σ1)(RG)σ(λhiGi)1].

The above expression makes two things clear. First, a corner solution with Gh = 0 arises if λhi/Gi=0. The intuition for this result is that, if an increase in an agent’s arming does not raise his share of the contested resource, then his marginal benefit will vanish and it would be pointless to incur the cost of producing any guns. Second, if λhi/Gi>0, then the marginal benefit of producing guns will be positive and, as long as this benefit is not dominated by the associated marginal cost, a positive quantity of arms will be observed. What’s more, because an agent’s marginal benefit is increasing in the sensitivity of his share to his arming, the bargaining solutions can be ranked in terms of their relative sensitivities.

Noting that there is no closed-form solution to λesi/Gi when λesi/Gi>0, we apply the implications of symmetry to λhi/Gi in (2527) for h ∈ {ss, nb} using (2224) to obtain

λssi/Gi=14G[21σ1(σ1)σ]1=14GA>0λnbi/Gi=14G[21σ1(σ1)σ+21σ11σ1]1=14G(A+B)>0,

where A21σ1(σ1)σ and B21σ11σ1, as defined in the proposition. Substituting the above expressions in (34) and solving for G gives the expressions for the unique Gss and Gnb in the interior of the strategy space noted in the proposition.

Let us now take a closer look at each solution concept. Starting with ss, for space reasons and as noted in the text, we just point out that technical details on the Gss = 0 case for σ[112,2] are contained Garfinkel and Syropoulos (2017b). The intuition was explained in the text.

Turning to nb, the solution is always in the interior of the strategy space because an agent always has an incentive to operate on the interior (and downward-sloping part) of the PPF, which requires some arming.

Lastly, with regards to the es solution, we note that whether Ges is zero or positive hinges on the possible dependence (or not) of the agents’ ideal payoffs on their disagreement payoffs. But, as noted in part (2) of Lemma 1, this dependence (or, its absence) hinges on whether σ < 2 (or not). ‖

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Published Online: 2018-2-9

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