Resource distribution and stable alliances with endogenous sharing rules

European Journal of Political Economy Vol. 18 (2002) 129 – 151 www.elsevier.com/locate/econbase

Resource distribution and stable alliances with endogenous sharing rules Suk Jae Noh Department of Economics, Hallym University, 1 Okchon-Dong, Chuncheon 200-702, South Korea Received 5 April 2000; received in revised form 12 February 2001; accepted 15 March 2001

Abstract This paper analyses alliance formation in a general equilibrium model of a conflict among three agents over a common pool of income. When alliance formation enhances the relative effectiveness of alliance’s appropriative activities, the two better endowed agents form an alliance with a fully egalitarian sharing rule, by which they maintain the size of the common pool at the maximum but keep the share of the outside adversary at the minimum. In this case, paradoxically, under some conditions, the welfare of the poorest agent is greatest. If the relative effectiveness of alliance’s appropriative activities is only moderately increased by alliance formation due to an incentive problem, an alliance can be formed only between the two less endowed agents. In each case, we derive the conditions on the initial distribution of resource endowments for a stable alliance and consider welfare implications. D 2002 Elsevier Science B.V. All rights reserved. JEL classification: C70; D60; D74 Keywords: Production; Appropriation; Resource distribution; Enforceable sharing rules; Stable alliances

1. Introduction In his analysis of ‘anarchy,’ Jack Hirshleifer (1995) proposed that an anarchic system is always liable to break up into organization. As an intermediate stage in the evolution of anarchy into a particular form of organization, we can naturally think of the development of an anarchy where a war of each against all prevails into a group conflict.1

E-mail address: [email protected] (S.J. Noh). 1 Kropotkin (1972) proposes that group conflict with mutual aid within each group may better represent the state of nature than the Hobbesian war. 0176-2680/02/$ - see front matter D 2002 Elsevier Science B.V. All rights reserved. PII: S 0 1 7 6 - 2 6 8 0 ( 0 1 ) 0 0 0 7 2 - 6

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This paper takes up this issue and considers the endogenous formation of alliances. When there is a conflict among three economic agents over a common pool of income, alliance formation in this paper results from interaction among (1) the distribution of resource endowments, (2) the endogenously determined sharing rule in an alliance, and (3) the change in the effectiveness of agents’ appropriative activities attributable to alliance formation. I consider a three-stage game. In the first stage, agents decide whether to join an alliance. In the next stage, alliance members choose the sharing rule that determines the division of income within an alliance. In the final stage, agents allocate resources between production and appropriation. A mixture of cooperative and noncooperative game elements is used in deriving alliance formation. When sharing rule is chosen, alliance members choose the sharing rule that maximizes alliance income.2 Otherwise, a standard subgame perfect Nash equilibrium concept is employed. Once an alliance is formed, unless the relative effectiveness of alliance members’ appropriative activities is lowered as a result of alliance formation, a fully egalitarian sharing rule is chosen in all types of alliances. Even though a more egalitarian sharing rule reduces the alliance’s share in the common pool, it allows more resources to be devoted to production so that the size of the common pool becomes larger. Given the production and conflict technologies, the latter effect always dominates. Among possible alliances, I focus on a stable alliance that requires three conditions. First, the sharing rule in an alliance should be enforceable in the sense that allocating resources in accordance with the chosen sharing rule is the best policy for each alliance member. Second, the welfare of each alliance member should be at least greater than the welfare that can be achieved in the individual conflict. Third, the alliance should not be blocked by other alliances. The first condition implies that each alliance member’s resource endowment should be less than a certain amount. The ceiling on the resource endowment of each alliance member becomes more restrictive as the relative effectiveness of alliance members’s appropriative activities becomes smaller. Combined with the third condition, it follows that, when the increase in the effectiveness of appropriative activities is small with alliance formation, the richest agent cannot be an alliance member. When however this effect is large, the two better endowed agents form an alliance to take advantage of the fact that they together can make the size of the common pool as large as possible and at the same time keep the adversary’s share as small as possible. The second condition places some restrictions on the magnitude of the poorest agent’s resource endowment, as well as on the relative magnitudes of the two less endowed agents’ resource endowments.

2 This cooperative element is not a readily acceptable assumption in view of the otherwise noncooperative nature of the problem. More work is needed to address this problem. However, given this assumption, instead of just assuming that the chosen sharing rule is binding, we investigate the conditions under which the chosen rule is incentive compatible. This approach is in the spirit of Ray and Vohra (1997), who mix cooperative and noncooperative elements of game theory to derive an equilibrium binding agreement.

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The analysis is conducted within a general equilibrium framework in which resource allocation by each economic agent between production and appropriation is explicitly modeled. This allows us to show how the welfare of each agent is affected by alliance formation.3 The adoption of a fully egalitarian sharing rule changes a three-party conflict into a two-party conflict and more resources are used for productive purposes than in the individual conflict. We also show that when a stable alliance is formed between two better endowed agents, the poorest agent also benefits from the alliance formation. Furthermore, we obtain, in some circumstances, a paradoxical result that the welfare of the poorest agent becomes the greatest among the three. On the other hand, if a stable alliance is formed between the two less endowed agents, the richest agent can be disadvantaged by alliance formation.

2. The model Consider a conflict among three economic agents over a common pool of income. The exogenously given resources of the economy, R, are distributed such that R1zR2zR3 and R = R1 + R2 + R3. Ri represents the resource endowment of agent i. Therefore, the richest agent, agent 1, has at least one third of total resources and the poorest agent, agent 3, has at most one third of total resources. Each agent allocates his resources between production and appropriation, denoted respectively by P and A, Ri ¼ Pi þ Ai ,

i ¼ 1,2,3:

ð1Þ

Productive technology is described by Y ¼ P1 þ P2 þ P3 ,

ð2Þ

where Y represents the common pool of consumable output. The production function is linear and there is no difference in productiveness among the agents. Also, there is no complementarity among agents’ productive activities. Furthermore, the production function is not changed by alliance formation. 3 Recent contributions that analyse conflict in this spirit include Anderton et al. (1999), Hirshleifer (1988, 1991, 1995), Garfinkel (1990), Grossman and Kim (1995, 1996), Noh (1999a,b, 2002), and Skaperdas (1992). However, these contributions focus on a two party conflict. Considering alliance formation, Skaperdas (1998) suggests that, in a game of one cake and three players, there is a tendency for two ‘weak’ players to form a stable alliance against the strongest contender. However, being a parital equilibrium approach, his analysis considers a contest with a fixed prize and in which each player’s probability of success is not a choice variable but is fixed by the exogenously given strategic endowment. Furthermore, due to the nature of elimination tournament, the outside adversary drops out of the picture when the second round contest between alliance members begins. In many situations, it seems more realistic to assume that settling the differences between alliance members is influenced by the presence and identity of the outside adversary. From the standpoint of a general equilibrium approach, a similar criticism applies to the rent-seeking literature that applies an endogenous sharing rule in a group contest and to the political science literature on alliance formation. See, for example, Katz and Tokatlidu (1996) and Niou and Ordeshook (1990), respectively.

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The productive technology in conjunction with the resource constraint indicates the essential feature that, the more resources are devoted to appropriative activities, the less will remain for consumable output. Agent i’s share in the common pool of output depends on whether or not he belongs to an alliance. In the individual conflict where no alliance is formed, the agent’s share, denoted by pi, is given by the conflict technology, pi = (hiAi) /(h1A1 + h2A2 + h3A3), where hi > 0 represents the effectiveness of agent i’s appropriative activities. Since this paper emphasizes the possible increase in the effectiveness of appropriative efforts as a result of alliance formation, I assume, for simplicity, that hi = 1 for all i in the individual conflict. That is, pi ¼

Ai , A1 þ A2 þ A3

i ¼ 1,2,3:

ð3Þ

This technology implies that, given the appropriative activities of other agents, an agent’s share increases at a decreasing rate as his appropriative activities increase. Final consumption of agent i in the individual conflict, therefore, is given by Ci = piY. When an alliance is formed between agents i and j, denoted by {ij}, each alliance member’s final consumption is determined by two factors, {ij}’s share in total consumable output and each member’s share in the alliance income. Alliance {ij}’s share in total output, denoted by pij, is given by, pij ¼

hAij , hAij þ Ak

h z 1,

i, j,k ¼ 1,2,3,

ð4Þ

where Aij = Ai + Aj4 and i, j, and k represent different agents.5 Of course, we have pk = 1pij. h is assumed to be equal to or greater than 1. It will be shown later that when h < 1, a stable alliance cannot be formed. Even though appropriative activities of each agent are identically effective in the individual conflict, if an alliance is formed, it is assumed that alliance members’ appropriative activities become more 4 That is to say, we assume that alliance’s appropriative activities are perfect substitutes between alliance members. If they are imperfect substitutes, we can represent Aij, in some special cases, as max[Ai, Aj] or min[Ai, Aj]. They are known respectively as ‘best shot’ and ‘weakest link’ technology in aggregating alliance member’s provision of appropriative activities. Sandler (1993) surveys the literature that deals with the public good nature of an alliance’s defense. 5 An alternative way in which we capture the effect of alliance formation is to use a power form of conflict technology where pi = Aim/(A1m + A2m + A3m) and pij = Aijm/(Aijm + Akm). m indicates scale parameter. Using this form, Skaperdas (1998) shows that a stable alliance exists only when m z 1. To distinguish whether alliance formation affects the conflict situation through the scale parameter, m, or effectiveness parameter, h, is a difficult but interesting problem. In Noh (1999b), I show that using m or h would yield qualitatively the same results if agents’ vulnerability to appropriation do not differ much, which I also assume in this paper. Therefore, the results of this paper would go through with the use of a power form. Note that m z 1 corresponds to h z 1 of this paper. Skaperdas (1996) derived this power form under the anonymity axiom that is not adequate when the effectiveness of appropriative activities is different among agents.

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effective relative to those of the outside adversary. This may, for example, reflect the division of labor in the alliance or geographical advantage in a war among nations.6 Let us denote alliance {ij}’s income by Cij = pijY. We assume that Cij is distributed between alliance members according to the following sharing rule: fz ¼

a Az þ ð1  aÞ , 2 Aij

0 V a V 1,

z ¼ i, j,

ð5Þ

where fz indicates agent z’s share in Cij.7 a represents the fraction of alliance income that is distributed on egalitarian grounds and 1a represents the fraction distributed according to each member’s relative contribution to the alliance’s total appropriative activities. Therefore, when a = 1, alliance income is evenly divided between alliance members independently of each member’s contribution to total appropriative activities of the alliance. Consequently, final consumption of alliance member z in {ij}, denoted by Cijz, becomes: Cijz ¼ fz pij Y ,

z ¼ i, j:

ð6Þ

Given these assumptions, we consider the following sequence of actions. In the first stage, each agent decides whether to participate in an alliance. In the second stage, if no alliance is formed, we have the individual conflict. However, if an alliance is formed, alliance members choose the sharing rule to maximize the alliance income. In the third stage, given the sharing rule, the agents solve the problem of resource allocation to maximize their respective final consumption.

3. Individual conflict Supposing that no alliance is formed in the first stage, we can investigate Nash equilibrium implications of the individual conflict for resource allocation. In this case, the conflict technology is given by Eq. (3). 3.1. R3 z 2R /9 Suppose that the resource endowment of the poorest agent is greater than (2 R) /9. Then no agent faces a binding resource constraint for appropriative activities and we obtain an interior solution.8 6 Sandler (1999) considers alliance formation among nations that face a common adversary. Alliances are driven by the savings in defense costs because of a reduction in the size of borders to defend. 7 The same form of sharing rule has been used in group rent-seeking literature. See, for example, Nitzan (1991). We could generalize the sharing rule by adding, with apporopriate weighing, Pz / Pij, relative contribution to alliance’s total productive activities, to Eq. (5). This would only reinforce our result that alliance formation saves resources for production. 8 From now on, when we say agent i faces a binding resource constraint, this implies that his resource constraint is binding for appropriative activities, i.e., Ai z Ri.

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Each agent i maximizes Ci = piY by optimally choosing Ai. With the substitution of resource constraints, production and conflict technologies into C1, the first-order interior condition for agent 1 is: @C1 A2 þ A3 ¼ ½R  A1  A2  A3   A1 ¼ 0: @A1 A1 þ A2 þ A3

ð7Þ

We obtain similar first-order conditions for other agents. By increasing appropriative efforts, each agent increases his share in the common pool, but he decreases the size of the common pool. Solving the first-order conditions of three agents simultaneously, we have: A1 ¼ A2 ¼ A3 ¼

2R , 9

C1 ¼ C2 ¼ C3 ¼

R , 9

A1 þ A2 þ A3 2 ¼ : 3 R

ð8Þ

Final consumption is equalized among three agents regardless of the distribution of resource endowment. This phenomenon has been referred to as the ‘‘paradox of power’’ by Hirshleifer (1991). Note that a relatively less endowed agent allocates a relatively larger fraction of resources to appropriative activities. That is, he has a comparative advantage in appropriation. The resources allocated to appropriative activities as a ratio of total resources in the economy, which will be called the resource wastage ratio, becomes 2/3 in the interior solution. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2R R  4R3 þ R2 þ 8RR3 ˆ 3.2. R3 < and R2 z R ¼ 9 8 Only the poorest agent faces a binding resource constraint. Substituting A3 = R3 and solving the first-order conditions of agents 1 and 2 simultaneously, we obtain: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! A þ A þ A 1 8R3 1 2 3 ˆ 1þ 1þ A1 ¼ A2 ¼ R, ¼ , 4 R R  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Rˆ 3R  R2 þ 8RR3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi , C1 ¼ C2 ¼ R þ R2 þ 8RR3

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R3 3R  R2 þ 8RR3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C3 ¼ : R þ R2 þ 8RR3

ð9Þ

Final consumption of the two better endowed agents is equalized. This identical level of consumption becomes larger as R3 becomes smaller. In order to enjoy the same level of consumption as agent 1, the required resource endowment of agent 2, Rˆ, becomes larger as R3 becomes smaller. For example, when R3 is close to 0, R2 should be greater than R/4. We note that resource wastage ratio ranges from 1/2 to 2/3. 3.3. R3 < 2R /9 and R2 < Rˆ Only the richest agent has a nonbinding resource constraint for appropriative activities. Substituting A2 = R2 and A3 = R3 into the first-order condition of agent 1 in Eq. (7), we

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obtain: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A1 þ A2 þ A3 R2 þ R3 ¼ , R R  pffiffiffi pffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 R R  R2 þ R3 , C2 ¼ R2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 , C1 ¼ R2 þ R3 p ffiffiffi  R C3 ¼ R3 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 : R2 þ R3

A1 ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi RðR2 þ R3 Þ  ðR2 þ R3 Þ,

ð10Þ

We note that when R2 is close to R3, C2 becomes close to C3. Now, the resource wastage ratio lies between 0 and 2/3. In the absence of alliance formation, the resource wastage ratio becomes smaller as more agents face binding resource constraints for appropriative activities.

4. Endogenous sharing rules and enforceability In this section, I take the alliance structure as given and investigate the determination of sharing rule, assuming that alliance members choose the sharing rule that maximizes the alliance’s income. I also consider the conditions under which the chosen sharing rule is enforceable. Consider the grand coalition. In the absence of an adversary, alliance income is maximized by choosing the fully egalitarian rule that ensures that no agent allocates resources to appropriative activities.9 However, given that two members of the grand coalition maintain the chosen sharing rule, the conflict technology in Eq. (3) indicates that the remaining member can take all of the consumable output with only a small amount of appropriative activities. Hence, the chosen sharing rule is not enforceable for any resource distribution.10 Consider a conflict between an alliance of two agents and one outside agent. There are five qualitatively distinct cases to consider. This is due to the possibility of corner 9 Given the class of sharing rules assumed in Eq. (5), only the fully egalitarian sharing rule maximizes the total consumable output. Even though other criteria such as the relative resource endowment or relative contribution to an alliance’s total productive activities are used to distribute alliance income among members, given that there is no outside adversary, no sharing rule in the grand alliance is enforceable given the conflict technology in Eq. (3). 10 Ray and Vohra (1997) developed the concept of ‘‘equilibrium binding agreements’’ in which, when one agent deviates from a coalition, the deviator does not take the actions of the remaning members as given, but visualizes that the remaining agents will take actions that are best to their interest in group or individually. Using this concept, Kamada (1997) analyses tax harmonizing alliance formation in a three-country version of a tax Leviathan model. Disparities of population density among three countries plays the same role as distribution of resource endowments in this paper. If we adopted this concept, we could show that the fully egalitarian rule in the grand alliance can be enforceable when the resource endowment of the poorest agent is not too small.

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solutions that may affect the choice of sharing rule and the enforceability conditions. In the following two subsections, we deal with Case 1. The other cases are considered in the Appendix A. 4.1. Determination of sharing rules R Case 1: Inpaffiffiffi conflict between {ij} and agent k, suppose that Ri ; Rj z pffiffiffi 4ð h þ 1Þ hR and Rk z pffiffiffi . In this case, we obtain an interior solution. 2ð h þ 1Þ Consider alliance {12}. We begin with the third stage where, given the sharing rule, each agent maximizes final consumption by choosing the optimal level of appropriative activities. The first-order interior condition for agent 1 becomes: @f1 p12 Y @f1 hA12 ¼ ½R  A12  A3  @A1 @A1 hA12 þ A3 ( ) hA3 hA12 þ f1 ½R  A12  A3   ¼ 0, hA12 þ A3 ðhA12 þ A3 Þ2

ð11Þ

2 ). where @f1/@A1 = (1a)(A2/A12 We obtain a similar first-order condition for agent 2. Given a, by increasing appropriative activities, each alliance member increases both his share in the alliance income and the alliance’s share in the common pool of consumable output. But this action decreases the size of the common pool. These two effects are balanced at the optimum. Observe that, since the only alternative to appropriative activities is productive activities with outcome that is subject to appropriation, even when a = 1 free-riding in supplying appropriative activities in the alliance does not arise. The first-order interior condition of the outside agent, agent 3, is:

@C3 hA12 ¼ ½R  A12  A3   A3 ¼ 0: @A3 hA12 þ A3

ð12Þ

Solving these three first-order conditions simultaneously, we obtain:

A12 ¼

hð1  aÞ  2ð2  aÞ þ

A12 þ A3 ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4hð2  aÞ þ h2 ð1  aÞ2

2ðh  1Þð3  aÞ

R,

2a R: 3a

Also we derive,

C12

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hð1  aÞ  2ð2  aÞ þ 4hð2  aÞ þ h2 ð1  aÞ2 R h qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ¼ h1 3a hð1  aÞ þ 4hð2  aÞ þ h2 ð1  aÞ2

ð13Þ

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To determine the sharing rule in the alliance, we assume that alliance members choose a that maximizes alliance income, C12. It turns out that the sign of @C12/@a becomes identical to that of the numerator in A12 and, @C12 > 0 for all a when hz1: @a Therefore, the fully egalitarian rule is adopted in the alliance. To understand more easily what is involved in the problem, suppose that h = 1. In this case, we obtain A12 = ((2a)2 /(3a)2)R, A3 = ((2a) /(3a)2)R, p12 = (2a)/(3a), Y = R/(3a) and C12 = ((2a) /(3a)2)R.11 By choosing a = 0, the alliance replicates the individual conflict. However, as a more egalitarian rule is adopted in the alliance, each alliance member devotes fewer resources to appropriative activities. The outside agent responds to the decrease in the alliance’s appropriative activities by increasing appropriative activities. Consequently, the alliance’s share in the common pool becomes smaller, but the size of the common pool ends up being larger. Since the latter effect dominates the former, C12 is increasing for all values of a. This happens because we assumed that the production technology is linear while there are diminishing returns to appropriative activities. Even when the production function is concave, we can obtain similar results, in particular when h is strictly greater than 1. Given the fully egalitarian sharing rule, the conflict between alliance {12} and agent 3 results in the following allocation of resources: A1 ¼ A2 ¼

R pffiffiffi , 4ð h þ 1Þ

1 2 ¼ C12 ¼ C12

A3 ¼

pffiffiffi hR pffiffiffi , 2ð h þ 1Þ

pffiffiffi R hR pffiffiffi : , C3 ¼ pffiffiffi 2ð hÞ þ 1 4ð h þ 1Þ

ð14Þ

Before we proceed to analyse whether the egalitarian rule is enforceable, we consider the implications of alliance formation based on the adoption of the egalitarian sharing rule for resource allocation. From the expression in A12 + A3 in Eq. (13), we see that, regardless of the values of h, the resource wastage ratio is fixed at 1/2, which is smaller than pffiffiffithe wastage ratio in the h interior solution of the individual conflict. Since p12 ¼ pffiffiffi , alliance formation hþ1 transforms a three-agent conflict into a two-party conflict with equal division of a common pool between the two parties when h = 1. However, when h > 1, even though the alliance allocates a smaller amount of resources to appropriative activities than the outside agent, the effective amount of appropriative activities of the alliance becomes greater so that p12 becomes greater than 1/2. 11 When h = 1, A12 is derived either by applying l’Hopital’s rule to A12 in Eq. (13) or directly from the firstorder conditions.

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It is interesting to observe that the fully egalitarian sharing rule has only an allocative role and has nothing to do with equity consideration between alliance members. This is because whatever values of a are chosen, each alliance member allocates an identical amount of resources to appropriative activities, A1 = A2, whenever both alliance members have sufficient resources, as in Case 1. Hence, each alliance member ends up having an identical amount of income. The highest level of identical income is achieved with the fully egalitarian sharing rule. 4.2. Enforceability of sharing rules Case 1: Suppose that a particular sharing rule is chosen in alliance {ij}. If we denote the Nash equilibrium values associated with the chosen sharing rule by a star, the chosen sharing rule is defined to be enforceable for alliance member j if:

V ðAj Þ ¼

h i Aj  A  A R  A Cijj V 0 for Aj V Rj : j i k Aj þ A i þ A k

ð15Þ

Note that we assume that, if an alliance member allocates resources deviating from the chosen sharing rule, the synergy effect of alliance is lost and the conflict technology is given as in the individual conflict. If agent j can find an Aj such that V(Aj) > 0, the chosen sharing rule is not enforceable for j. Therefore, we require V(Aj) V 0 and V(Ai) V 0 for the enforceability of the chosen sharing rule. 2 Substituting A1, A3, and C12 from Eq. (14) into Eq. (15), the enforceability condition for agent 2, V(A2) V 0, is simplified as:

A22 

pffiffiffi pffiffiffi pffiffiffi 3þ h hð2 h þ 1Þ 2 pffiffiffi pffiffiffi R z 0, RA2 þ 4ð h þ 1Þ 16ð h þ 1Þ2

A2 V R2 :

ð16Þ

By symmetry, we obtain the enforceability condition for agent 1 by replacing A2 with A1. It can be shown that the inequality in Eq. (16) is satisfied for all values of A2 when pffiffiffi 9 pffiffiffi 9 h z . However, when h < , the inequality (Eq. (16)) is satisfied either when 7 7 A2 V R* or A2 z R**.12 If alliance formation greatly increases the relative effectiveness of an alliance’s appropriative activities, the alliance, with the egalitarian sharing rule, takes a lion’s share from the enlarged common pool of consumable output. Therefore, the incentive for deviation is small. However, if the synergy effect on appropriation is only moderate, the

12

R* and R** (>R * ) are two roots that solve Eq. (16) as an equality and are given by pffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 3 þ h F ð9  7 hÞð1 þ hÞ pffiffiffi R. 8ð h þ 1Þ

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Table 1 Enforceable alliance {ij} against agent k Cases

Resource distribution

Case 1

pffiffiffi R hR Ri , Rj z pffiffiffi , Rk z pffiffiffi 4ð h þ 1Þ 2ð h þ 1Þ

Case 2

Case 3 Case 4

Case 5

R R pffiffiffi , Rj z pffiffiffi  Ri , 4ð h þ 1Þ 2ð h þ 1Þ pffiffiffi hR Rk z pffiffiffi 2ð h þ 1Þ pffiffiffi pffiffiffiffiffi R3 hR FðR3 Þ, Rk < pffiffiffi R i , Rj z 2h 2ð h þ 1Þ pffiffiffiffiffi pffiffiffiffiffi R3 R3 FðR3 Þ, Rj > FðR3 Þ  Ri , Ri < 2h h pffiffiffi hR Rk < pffiffiffi 2ð h þ 1Þ R R , Rj < pffiffiffi  Ri Ri < pffiffiffi 4ð h þ 1Þ 2ð h þ 1Þ Ri <

Conditions pffiffiffi 9 pffiffiffi 9 h z or h < and 7 7 Ri, Rj V R pffiffiffi 9 pffiffiffi 9 h z or h < and 7 7 pffiffiffi ð2  hÞR Ri z pffiffiffi , Rj V R 4ð h þ 1Þ pffiffiffi 9 hz 7 not enforceable

indeterminate

cost of losing the synergy effect is small so that the incentive for deviation is relatively large. This incentive is however realized only if an alliance member has enough resources. Consequently, agent 2 can be prevented from deviant behavior only when R2 V R*.13 These results are summarized in Table 1 as Case 1. Table 1 lists the types of enforceable alliances along with the associated conditions on h and the distribution of resources. The definition of F(R3) appears in the Appendix A, in which we show that the fully egalitarian sharing rule is adopted in all cases, except Case 5 where the sharing rule is indeterminate. From Table 1, we see that enforceability of the fully egalitarian sharing rule places a ceiling on the resource endowment of each alliance member and the ceiling becomes more restrictive as the synergy effect becomes smaller. pffiffiffi 9 Suppose that h z . In Cases 1, 2 and 3, the resource endowment of the richer ally is 7 not large enough for deviation to be profitable. Consequently, the sharing rule is enforceable for all resource distributions. In Case 2, only the less endowed alliance member, agent i, has a binding resource constraint for appropriation. Case 3 describes a situation where only the outside adversary, agent k, has a binding resource constraint.

R pffiffiffi in Case 1. Consequently, for the existence of an enforceable egalitarian 4ð h þ 1Þ R . This inequality is always satisfied when h z 1. sharing rule, we require that R z pffiffiffi 4ð h þ 1Þ 13

From Eq. (14), R2 z

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However, in Case 4, where both one alliance member and the outside adversary face binding resource constraints, the richer ally always deviates. pffiffiffi 9 Suppose that h < . In this case, deviant behavior is profitable with only a moderate 7 amount of resource endowment. When the less endowed alliance member faces a binding resource constraint, as in Case 2, the richer ally has a greater incentive for deviation than in Case 1. Therefore, we have a more restrictive condition. That is to say, R* > R. In Cases 3 and 4, the fully egalitarian sharing rule is not enforceable for any distribution of resource endowment. Concerning Case pffiffiffi 1, we make two observations. First, agent 3 cannot be the outside adversary when h > 2. In this case, agent 3 is required to have more than one third of total pffiffiffi hR resources of the economy. Second, since pffiffiffi > R*, agent 2 cannot be the outside 2ð h þ 1Þ pffiffiffi 9 adversary when h < . If he were the outside adversary, then we have R1 zR2 z 7 pffiffiffi hR * pffiffiffi > R . This violates the enforceability condition for agent 1. Similar consid2ð h þ 1Þ erations apply for each case in the Appendix A and they are utilized in the derivation of a stable alliance.

5. Stable alliances Now consider the first stage of the conflict. Each agent would join the alliance that gives the highest level of consumption. To consider this problem, we define an alliance {ij} as stable if, given that the chosen sharing rule is enforceable as defined in the previous section, we have: Ciji z Ciki ,

Ciji z Ciki

and

Ciji z Ci , Ciji z Cj :

These conditions require that, in a stable alliance, each alliance member should not obtain a higher level of consumption by forming an alternative alliance with the outside agent. Furthermore, each alliance member should be better off or at least be indifferent compared to what each can achieve under individual conflict.14 The derivation of stable alliance is in Appendix B and we summarize the results in the Table 2. Here we discuss the results.

14 Using the above two conditions, Skaperdas (1998) defined a stable alliance and showed that the definition is equivalent to Strong Nash equilibrium, which is a more restrictive concept than coalition-proof Nash equilibrium. Note, however, that our definition requires further that the chosen sharing rule in the alliance is enforceable.

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Table 2 Stable alliance and the required conditions Types

Poorest agent

I

R3 <

R pffiffiffi 4ð h þ 1Þ

pffiffiffi 9 h< 7

pffiffiffi 9 hz 7 {12}: R2 z

pffiffiffiffiffi R3 FðR3 Þ, 2h

R3 is large or R2R3 is large

R 2 pffiffiffi VR3 < R 9 4ð h þ 1Þ

II

same as above

{23}:

pffiffiffi 2 h pffiffiffi R V R3 4ð h þ 1Þ

R pffiffiffi , 4ð h þ 1Þ R pffiffiffi  R3 < R2 < R, 2ð h þ 1Þ

<

R3 is small or R3 is large but R2  R3 is small R {23}: pffiffiffi V R3 4ð h þ 1Þ < R2 < R , R2R3 is small

III

pffiffiffi 2 h R V R3 < pffiffiffi R 9 2ð h þ 1Þ

{12}

{23}:

IV

pffiffiffi R h pffiffiffi R V R3 < 3 2ð h þ 1Þ

{12}, {23}, {13}

same as above

5.1.

pffiffiffi h is close to 9/7, R2 V R

pffiffiffi 9 hz 7

5.1.1. R3 < 2R /9 Suppose that the poorest agent faces a binding resource constraint in the individual conflict, which includes Types I and II. In this case, a stable alliance can exist between pffiffiffiffiffi R3 FðR3 Þ. As an outside adversary, the poorest agent also faces agents 1 and 2 when R2 z 2h a binding constraint. The interests of the two better endowed agents coincide because they together can maintain the common pool at the maximum but keep the share of outside agent to the minimum. In a stable alliance, agent 2 should not face a binding resource constraint as an alliance member. (See Case pffiffiffiffiffi4 in Table 1). Given that agent 2 has sufficient resources as an alliance R3 member ðR2 z FðR3 ÞÞ, he may or may not face a binding resource constraint in the 2h individual conflict. In the latter case, the alliance is always stable. However, in the former case, agent 1 sometimes wants to play the individual conflict. In this case, for the stability of the alliance we require that either R3 is large (given that R3 < (2R) /9) or the difference between R2 and R3 is large. These conditions tend to make the welfare of the agent 1 small in the individual conflict.

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pffiffiffi 2R hR V R3 < pffiffiffi 9 2ð h þ 1Þ In a Type III alliance, even though the poorest agent confronts the individual conflict without facing a binding resource constraint, he is forced to allocate all of his resources to appropriative activities as a result of the alliance formed by the two better endowed agents. When a Type III alliance is formed, we obtain a paradoxical result that the final income of the poorest agent becomes the largest when the synergy effect on appropriation is large but not too large, and at the same time the resource endowment of the poorest agent is not too small.15 Under these conditions, the poorest agent takes almost half of the consumable output, but the remaining consumable output is divided equally between two better endowed agents. This result is more paradoxical than Hirshleifer’s (1991) ‘‘paradox of power’’. This says that, in a two-party conflict, the disparity in the final incomes of two parties is less than the disparity in initial endowments. But, the absolute amount of final income of the better endowed party is always greater than or equal to that of the less endowed party. Also related to our result is the ‘‘joint bargaining paradox’’ in cooperative game theory. This says that, when some reasonable cooperative solution concepts are applied to alliance formation, alliance members become worse off and an outside player becomes better off, compared to the outcome of individual bargaining. Hence there is no alliance formation. See, for example, Harsanyi (1977). However, in this paper, even though alliance members lose relatively to the outside agent, the welfare of each alliance member is increased from the welfare that can be achieved under the individual conflict. Therefore we observe the formation of a stable alliance. pffiffiffi R hR 5.1.3. pffiffiffi V R3 V 3 2ð h þ 1Þ

5.1.2.

In a Type VI alliance, the poorest agent faces a binding resource constraint neither in the individual conflict nor as the outside adversary. This situation is possible only when hV4. Otherwise, agent 3 is required to have more than one third of the total resources of the economy. We note from Eq. (14) that any alliance of two agents yields the same level of welfare for each alliance member, and this level is greater than that achievable in the individual conflict. Hence, any two-member alliance becomes stable. Furthermore, if an alliance is formed, the welfare of the outside agent exceeds that of each alliance member. This is an interesting situation. Each agent, as an alliance member, improves his welfare relative to an individual conflict. However, each agent becomes the winner of the conflict as an outside adversary. 5.2.

pffiffiffi 9 h< 7

The enforceability of a sharing rule dictates that a stable alliance can be formed only between the two less endowed agents. Suppose that the poorest agent faces a binding

15

We can also observe the paradoxical result in a Type II but never in a Type I alliance.

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resource constraint in the individual conflict. In this case, a stable alliance is possible only when agent 2 also faces a binding resource constraint in the individual conflict. To ensure that an alliance is a better option than individual conflict, we require that either R3 be small or, when R3 is large, that R3 be close to R2 in a Type I alliance.16 In a Type II alliance, for stability we require that R3 be close to R2. In a stable alliance of Type I or II, the richest agent consumes more than the alliance members. However, he is harmed as a result of alliance formation. That is, he prefers the individual conflict.17 When the poorest agent does not face a binding resource constraint pffiffiffi in the individual conflict as in Types III and IV, a stable alliance is possible only if h is close to 9/7 and R2 V R*. In this case, the richest agent also benefits from alliance formation. If h is close to 1, the sharing rule cannot be enforceable even in the alliance {23}.

6. Conclusion This paper has considered the formation of stable alliances based on resource distribution among agents and with endogenous sharing rules within alliances. We have seen that various outcomes are possible. We have in particular found circumstances where the least well-endowed agent has the highest welfare as the consequences of alliance formation.

Acknowledgements I thank Herschel I. Grossman and anonymous referees for helpful comments. The financial support of Korean Research Foundation 1998 program is also gratefully acknowledged.

Appendix A Case 2: In a conflict between alliance {ij} and agent k, suppose that alliance member i R R faces a binding resource constraint. This happens when Ri < pffiffiffi , Rj z pffiffiffi 4ð h þ 1Þ 2ð h þ 1Þ

16 In deriving these results, I assume that agent 2 as an alliance member does not face a binding resource constraint. For the case when he faces a binding constraint see footnote 20. 17 To prevent alliance formation, the richest agent can transfer part of his resources to agent 2. This is because, once R2 z Rˆ, the alliance becomes unstable. However, he would never make a transfer to the poorest agent. Note that resource redistribution to deter alliance formation reduces production efficiency. In contrast, in a two party conflict, Grossman and Kim (2000) and Noh (2002) consider the role of transfers in reducing appropriative activities.

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pffiffiffi hR Ri , and Rk z pffiffiffi . Let the alliance be {23} and suppose that A3 = R3 with R3 < 2ð h þ 1Þ R pffiffiffi . Then, from Eqs. (11) and (12), we obtain: 4ð h þ 1Þ rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 1a 1 R2 1  a R3 þ RR3 , þ A2 þ R3 ¼ þ 8 2a 2 16 2  a rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R R2 1  a RR3 : A1 þ A2 þ R3 ¼ þ þ 4 16 2  a From these expressions, combined with the first-order conditions, we derive: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi R 1a 1 R2 1  a R3 þ RR3 : þ C23 ¼  8 2a 2 16 2  a Since @C23/@a > 0 for all values of a, the sharing rule produces: pffiffiffi R hR A2 ¼ pffiffiffi  R3 , A1 ¼ pffiffiffi , 2ð h þ 1Þ 2ð h þ 1Þ pffiffiffi R hR 2 3 ¼ C23 ¼ pffiffiffi C23 : , C1 ¼ pffiffiffi 2ð h þ 1Þ 4ð h þ 1Þ

ðA1  1Þ

We still obtain the interior solution resource allocation because the richer ally supplements the insufficient resources of poorer ally for appropriative activities. Substituting A1 and A2 from Eq. (A1-1) into the enforceability condition (Eq. (15)), we find that V(A3) V 0 for 0 V A3 V R3. This implies that resource constrained ally does not behave opportunistically. pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi 3 h þ 4  hþ h pffiffiffi For agent 2, we note that V(A2)V0 for all values of A2 when R3 z . 4ð h þ 1Þ R R Since R3 < pffiffiffi , the sharing rule is enforceable for agent 2 only when pffiffiffi 4ðphffiffiffiffiffiffiffiffiffiffiffiffiffi þ 1Þ 4ð h þ 1Þ pffiffiffi pffiffiffi pffiffiffi 9 3 h þ 4  hþ h pffiffiffi . This inequality is satisfied whenever h z . Therefore, as in z 7 4ð h þ 1Þ pffiffiffi 9 Case 1, the fully egalitarian sharing rule is always enforceable when h z . 7 pffiffiffi 9 Suppose that h < . Then, V(A2) V 0 requires that R2 V R.18 Therefore, the sharing 7 R rule is enforceable only when R z pffiffiffi  R3 . This inequality implies that R3 z pffiffiffi 2ð h þ 1Þ ð2  hÞR pffiffiffi . 4ð h þ 1Þ 18

R¼

" pffiffiffi # qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi pffiffiffi ffi 2  pffiffiffi 1 1 hþ4 pffiffiffi 4ð h þ 1ÞR3  ð3 h þ 4ÞR  16ðh þ hÞR2 . It is the R  R3  pffiffiffi 2 4ð h þ 1Þ 4ð h þ 1Þ

smaller root that solves V ðA2 Þ ¼ 0:

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From these conditions, we deduce that the sharing rule is enforceable only in alliance pffiffiffi 9 {23} if h V . First, agent 3 cannot be the outside adversary. This is because agent 3 is 7 required to have more than (9/32)R but agent 2 is assumed to have less than (7/64)R in Case 2. Next, consider {13}. For the enforceability, we require R1 V R. This implies that R1 + R2 V 2R. But this violates the resource constraint of the economy because R shows that we must have 2R < R  R3. pffiffiffiffiffi R3 Case 3: In a conflict between {ij} against agent k, suppose that Ri ,Rj z FðR3 Þ, but 2h pffiffiffi hR Rk < pffiffiffi so that only the outside agent faces a binding resource constraint. 2ð h þ 1Þ pffiffiffi hR Consider {12} and suppose A3 = R3 with R3 < pffiffiffi . The first-order conditions of 2ð h þ 1Þ agents 1 and 2 in Eq. (11) give us A12 þ R3 ¼

ðh  1Þð3  aÞR3 þ ð1  aÞhR þ L , 2hð2  aÞ

where L = (1a)2h2R2 + 2h[a2 4a + 5  (1a)2h]RR3 + [(1a)2h2 + (3a)2  2h(a2  4a + 5)]R32. When h = 1, we can explicitly prove @C12/@a > 0 for all values of a. When h > 1, computer simulation shows that the fully egalitarian sharing rule is also chosen in this case. Compared to Case 1, as a more egalitarian rule is adopted in the alliance, the outside agent cannot increase appropriative activities because he is resource constrained. With the fully egalitarian sharing rule, we derive pffiffiffiffiffi 1 R3 1 2 ðFðR3 ÞÞ2 , A1 ¼ A2 ¼ FðR3 Þ, C12 ¼ C12 ¼ 2h 2h

pffiffiffiffiffi R3 FðR3 Þ: C3 ¼ h

ðA1  2Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffi where, for notational simplicity, we define FðR3 Þ ¼ R3 þ hðR  R3 Þ  R3. Noting that F(R3) is decreasing in R3, we have the relation F(R2) V F(R3). 2 Substituting A1 and C12 from Eq. (A1-2) into Eq. (15), we find that V(A2) V 0 for all hR . Therefore, the sharing rule is enforceable when values of A2 when R3 z 9ðh  1Þ pffiffiffi pffiffiffi 9 hR hR pffiffiffi . This inequality is satisfied again when h z . z 7 2ð h þ 1Þ 9ðh  1Þ pffiffiffi 9 19 When h < , the enforceability condition requires R2VR. Since agent 1 faces the 7 same enforceability condition, it follows that R1 + R2 V 2R. However, we can show that the resource constraint of the economy requires R1 + R2 > 2R. Therefore, if the synergy effect on appropriation is only moderate, the chosen sharing rule is not enforceable in Case 3. 19

R¼

pffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi FðR3 Þ ½FðR3 Þ þ 3 R3  hR  9ðh  1ÞR3 . 4h

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pffiffiffi pffiffiffi hR We observe that pffiffiffi is increasing in h and becomes R/3 when h ¼ 2 . 2ð h þ 1Þ Therefore, agent 1 cannot be the outside adversary in Case 3 when h > 4. Case 4: One alliance member as well as the outside agent face binding resource constraints. Consider {12} and suppose that A2 = R2 and A3 = R3. Substituting A2 = R2 and A3 = R3 into Eq. (11) and solving the equation, we obtain vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi !ffi u u 1 2hð1  aÞ R2 : A1 þ R2 ¼ ½K  R3 , K ¼ tðhR  ðh  1ÞR3 Þ R3 þ h 2a From this and using the first-order conditon, we derive C12 = RR3 + (2/h)R3 (1/h)KR3(R((h1)/h)R3)K 1/2. Then @C23/@a > 0 for all values of a. This case is similar to Case 3 except that, as a more egalitarian rule is adopted, one alliance member cannot release resources to production. When the fully egalitarian sharing rule is adopted in the alliance, the conflict results in pffiffiffiffiffi 1 R3 1 2 A1 ¼ ðFðR3 ÞÞ2 : FðR3 Þ  R2 , C12 ¼ C12 ¼ 2h h This case replicates the results in Case 3 because agent 1 supplements the insufficient supply of the poorer ally’s appropriative activities. Considering the enforceability condition for agent 1, V(A1) V 0, we can show that, for all values of h, there exists no distribution of resource endowment that simultaneously satisfies the enforceability condition and the resource constraint of the economy. Case 5: Suppose that both alliance members face binding resource constraints. This R type of alliance can be formed only between agents 2 and 3 and when R3 < pffiffiffi and 4ð h þ 1Þ R  R3. However, given the class of sharing rule in Eq. (5), the sharing rule R2 < pffiffiffi 2ð h þ 1Þ becomes indeterminate. I assume that the individual conflict occurs in this case.20

Appendix B pffiffiffi 9 h z : We know that an enforceable alliance can exist in Cases 1, 2 and 3. 7 R Type I: R3 < pffiffiffi : Agent 3 can form an alliance either with agent 2 or agent 1. 4ð h þ 1Þ

(1)

In this case, Case 2 applies. Alternatively, agents 1 and 2 can form an alliance according to 20 Even though the choice of sharing rule is indeterminate, suppose, for whatever reason, that the fully egalitarian sharing rule is adopted in the alliance. Following the same procedure as in the text, we can derive conditions for the enforceability of the sharing rule. However, these conditions become very complex. In the special case where R2 = R3, I can show that the fully egalitarian sharing rule is always enforceable.

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Case 3. The welfare levels associated with each alliance along with the required resource distribution are summarized as 8 pffiffiffi pffiffiffi > R hR hR > 3 > C23 ¼ pffiffiffi  R3 , R1 z pffiffiffi : R2 z pffiffiffi , > > > 2ð h þ 1Þ 4ð h þ 1Þ 2ð h þ 1Þ > > p ffiffi ffi p ffiffi ffi < hR hR 3 C13 ¼ pffiffiffi : R2 z pffiffiffi , > > 4ð h þ 1Þ 2ð h þ 1Þ > > pffiffiffiffiffi > > 1 R3 > 1 > ðFðR3 ÞÞ2 : R2 z FðR3 Þ: ¼ : C12 2h 2h pffiffiffiffiffi R3 R R Noting that pffiffiffi FðR3 Þ whenever R3 < pffiffiffi  R3 > , we prove that 2h 2ð h þ 1Þ 4ð hÞ þ 1 1 2 1 3 2 3 = C12 > C13 = C13 = C23 = C23 . Therefore, if a stable alliance is ever formed, it is C12 pffiffiffiffiffi R3 FðR3 Þ. between two better endowed agents when R2 z pffiffiffiffiffi 2h R3 FðR3 Þ, the results of Case 4 apply to alliance {12} and Remember that, when R2 < 2h the chosen sharing rule is not enforceable. Now consider the behaviors pffiffiffi of agents 2 and 3 in pffiffiffiffiffi R R3 FðR3 Þ implies that R2VRˆ and R2 þ R3 < pffiffiffi because we can this case. R2 V 2h pffiffiffiffiffi pffiffiffiffiffi2ð h þ 1Þ R3 R3 R FðR3 Þ for all R3 and that Rˆ > FðR3 Þ. That is, both agents show that pffiffiffi z 2h 2h 4ð h þ 1Þ 2 and 3 face binding resource constraints both in the individual conflict and as alliance members if they form an alliance. Since we assumed individual conflict when both alliance members face binding resource constraints, no stable alliance is possible. R 2R Type II: pffiffiffi VR3 < : When agent 3 forms an alliance with agent 2, Case 1 9 4ð h þ 1Þ applies. If agent 3 forms an alliance with agent 1, we observe either Case 1 or Case 3 depending on the values of R2.p Finally, agents 1 and 2 can form an alliance only according ffiffiffi 2R hR to Case 3. This is because pffiffiffi > . Hence, we obtain 9 2ð h þ 1Þ 8 pffiffiffi > > hR > 3 > pffiffiffi C ¼ > 23 > > 4ð h þ 1Þ > > pffiffiffi < hR 3 C13 ¼ pffiffiffi > > 4ð h þ 1Þ > > > > 1 > 2 1 > > : C12 ¼ 2h ðFðR3 ÞÞ

pffiffiffi R hR pffiffiffi , R1 z pffiffiffi , 4ð h þ 1Þ 2ð h þ 1Þ pffiffiffi 1 hR 3 ðFðR2 ÞÞ2 ¼ : R2 z pffiffiffi , C13 2h 2ð h þ 1Þ pffiffiffiffiffi R3 FðR3 Þ: : R2 z 2h

: R2 z

: R2 <

pffiffiffi hR pffiffiffi , 2ð h þ 1Þ

R We have already shown that F(R3 ) z F(R2) and we know that R2 z R3 z pffiffiffi z pffiffiffiffiffi 4ð h þ 1Þ R3 FðR3 Þ. Therefore, both agents 1 and 2 want to form an alliance between themselves 2h R when R2 z pffiffiffi . 4ð h þ 1Þ

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Now we check, in the above two cases, whether two better endowed agents can do better under individual conflict. Recall that, since R3 < (2R)/9, agent 3 faces a binding resource constraint in the individual pffiffiffiffifficonflict. R3 However, since Rˆ > FðR3 Þ, we consider two cases. Suppose that R2 < Rˆ so that 2h agent 2 faces a binding resource constraint in the individual conflict. In this case, agent 1 1 wants to play the individual conflict in some cases. Comparing C1 in Eq. (10) and C12 in Eq. (A1-2), the latter is greater than the former either when R3 is large (given that R3 < (2R)/9), or when the difference between R2 and R3 is large.21 Once R2 z Rˆ, the alliance between two better endowed agents becomes stable. In this case, either in the individual conflict or as alliance members, the consumption levels of both agents 1 and 2 are equalized. Since more resources are available for production with 1 2 alliance formation, it is true pffiffithat ffi C12 = C12 > C1 = C2. 2R hR V R3 < pffiffiffi Type III: : The situation is identical to 1.2 except that agent 3 does 9 2ð h þ 1Þ not face a binding resource constraint in the individual conflict. Since (1/(2h))( F(R3))2 > R/9, the alliance between agents 1 and 2 is stable. The welfare of each agent, in this case, is reproduced as: pffiffiffiffiffi R3 1 2 1 2 ðFðR3 ÞÞ , C3 ¼ C12 ¼ C12 ¼ FðR3 Þ: 2h h

pffiffiffi h h 1 V pffiffiffi V C3 when R3 z (h/h + 8)R. We also observe that We derive that C12 h þ 8 2ð h þ 1Þ pffiffiffi 9 pffiffiffi h R, we obtain a paradoxical when h V 2. Consequently, when V h < 2 and R3 > 7 hþ8 22 1 result that C12 V C . pffiffiffi 3 R hR Type IV: pffiffiffi V R3 V : This case is easy to see. 3 2ð h þ 1Þ pffiffiffi 9 (2) h < : Only Cases 1 and 2 are relevant. 7 R Type I: R3 < pffiffiffi : Since agent 3 faces a binding resource constraint either as an 4ð h þ 1Þ alliance member or as the outside adversary, only Case 2 is relevant and we noted earlier that agent 1 cannot be an alliance member in this case. Accordingly, an alliance can be formed only between agents 2 and 3 but enforceability requires that pffiffiffi 2 h R R pffiffiffi ðA2  1Þ R V R3 < pffiffiffi and pffiffiffi  R3 V R2 < R: 4ð h þ 1Þ 4ð h þ 1Þ 2ð h þ 1Þ

21

This comparison becomes very complex but we can prove at least numerically that when both R3 is small 2 and R2 is close to R3, C1 > C112 . On the other hand, we prove that C12 > C2. That is, agent 2 always wants to form an alliance with agent 1. 22 Note that (h/(h + 8))R < (2R)/9 if h < 16/7. This implies that the paradoxical result is also obtained in Type II if h < 16/7 and R3 > (h/(h + 8))R. However, this result never occurs in Type I.

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R pffiffiffi . 4ð h þ 1Þ This implies that, when the sharing rule is enforceable, it must be the case that both agents R 2 and 3 face binding resource constraints in the individual conflict. Note that pffiffiffi 4ð h þ 1Þ pffiffiffi 9 2R when h < . < 9 7 The welfares of agents 2 and 3 under alliance formation and under the individual conflict To compare with the individual conflict, we first find that Rˆ > R when R3 <

are reproduced from Eqs. (A1-1) and (10) as: pffiffiffi h ¼ ¼ pffiffiffi R, 4ð h þ 1Þ  pffiffiffi R C3 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 R3 : R2 þ R3 2 C23

3 C23

 C2 ¼

pffiffiffi R pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  1 R2 , R2 þ R3 ðA2  2Þ

3 2 > C3.23 A comparison between C23 and C2 is complex. First, we show that C23 However, we can prove numerically that if, given that inequalities in Eq. (A2-1) is satisfied, R3 is small or if R3 is large but the difference between R2 and R3 is small, agent 2 wants to form an alliance with agent 3.24 From Eq. (A2-1), we note that R3 cannot be identical to R2. R 2R Type II: pffiffiffi V R3 < : Since we rule out Cases 3 and 4, no agent faces a 9 4ð h þ 1Þ binding resource constraint as an alliance member in this case. Therefore, only Case 1 is relevant and this case is possible only when agent 3 is an alliance member. We already know that agent 2 cannot be the outside agent if the sharing rule is enforceable. Hence, an alliance can be formed between two less endowed agents and the sharing rule is enforceable only if

R pffiffiffi V R3 V R2 V R : 4ð h þ 1Þ Now consider the individual conflict. Since R* can be greater or less than Rˆ, we consider two cases. If R2 < Rˆ,25 both agents 2 and 3 face binding resource constraints in the individual conflict. Therefore, welfare levels in the individual conflict and the welfare of alliance members are given in Eq. (A2-2). In comparing welfare, we again prove 23

C3 has a maximum value when R3 and R2 + R3 are respectively at the maximum and minimum. Substituting R R 3 R3 ¼ pffiffiffi > C3. and R2 þ R3 ¼ pffiffiffi from Eq. (A2-1) into C3, we show that C23 4ð h þ 1Þ 2ð h þ 1Þ 24 Since R in Eq. (A2-1) is an increasing function of R3, among the resource distributions that satisfy Eq. (A2-1), the difference between R2 and R3 could widen as R3 becomes larger. 25 Note that R2 < Rˆ is possible either when R2 < R* < Rˆ or R2 < Rˆ < R*. However, we can prove that R* z R R with equality at R3 ¼ pffiffiffi . 4ð h þ 1Þ

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numerically that, if R3 is close to R2, the alliance becomes stable. Note, in this case, that R3 can be the same as R2.26 Suppose R2 z Rˆ. In this case, agent 2 is in an interior solution in the individual conflict and enjoys the same level of consumption as agent 1, which is given by Eq. (9). We can show that agent 2 obtains a higher level of welfare under individual conflict. Therefore, a stable alliance is not possible in this case.27 2R Types III and IV: R3 z : Only Case 1 is relevant and we know that, in this case, agent 9 2 cannot be the outside agent. For enforceability of the sharing rule, the resource endowment of each alliance member should be less than R*. Since R* has a minimum pffiffiffi 9 value of R/8 when h = 1 and a maximum value of about 0.23R when h ¼ , agent 1 7 cannot be the alliance member for all values of h. Similarly, when h is close to 1, the sharing rule in the alliancepbetween agents 2 and 3 is not enforceable. Once the alliance if ffiffiffi R hR formed, we know that pffiffiffi > . Consequently, the alliance formed between the two 4ð h þ 1Þ pffiffiffi 9 less endowed agents is stable if h is close to 9/7 and R2VR*. In this case, the richest agent also benefits from the formation of alliance.

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