State Support for Rebels and Interstate Bargaining

Xiaoyan Qiu∗

April 18, 2021

Abstract

Since the end of World War II, leaders have frequently supported rebel groups in other countries as a coercive strategy in international disputes. However, the strategic rationale by which rebel groups gain international support is non-obvious. Many recip- ient groups are too weak to viably win and are hostile to the sponsoring state’s goals. Using a formal model, I explain that the fundamental objective of transnational rebel support is to gain bargaining leverage against a rival state by depleting its resources to counter internal and external challenges. This subversive effect provides a sufﬁcient incentive for sponsoring the rebels even when favorable conditions suggested by previous studies are absent. Sponsoring rebels is attractive even if conventional warfare is not comparatively costly and even if rebel and sponsor preferences diverge. More- over, given the goal of destabilizing rival regimes, potential sponsors prefer to support weaker rebel groups and provide more support to them.

Word Count: 9,870

∗Department of Political Science, University of Rochester, [email protected]. Since the end of World War II, leaders have frequently used support for rebel groups as a coercive strategy in international disputes. For example, Muammar Gaddafi funded, armed, and trained the rebel group National Liberation Front of Chad and used it as a pawn in the Libya-Chad dispute over the Aouzou Strip in the early 1970s. Ethiopia has backed several Somali factions since 1996 to gain leverage in its dispute with Somalia over Ogaden. Iran is a longtime sponsor of Kurdish rebellions against its rival Iraq. Such cases occur very frequently, and scholars have increasingly recognized that antagonistic interstate relations are an important reason for state support of rebel groups (Maoz and San-Akca, 2012). Early research documents various motivations behind external support for insurgent and terrorist groups, such as destabilizing or weakening neighbors, projecting power, regime change, exporting political ideology, and aiding kin (Byman et al., 2001; Byman, 2005). Recent studies raise a fundamental question of why states sponsor rebels rather than using their own military to achieve these policy goals (Salehyan, Gleditsch and Cunningham, 2011). This shifts the literature to primarily conceptualize such support as a delegation of conflict to non-state allies. State sponsors, facing formidable costs of conventional warfare and coup threat from a powerful military, use support for insurgent groups as a cheaper and safer substitute for direct military confrontation (Salehyan, 2010; Tamm, 2016). In particular, scholars argue that ideological and ethnic affinities between the sponsor and the group facilitate support because these groups are more likely to share the sponsor’s preferences (Salehyan, Gleditsch and Cunningham, 2011; San-Akca, 2016). Since rebels conduct conflict on the sponsor’s behalf, the sponsor wants the rebels to win. This suggests that sponsors should support well-organized rebel groups that demonstrate a certain level of competence (Salehyan, Gleditsch and Cunningham, 2011). However, these factors do not capture the fundamental force driving external support. In a large number of cases, states voluntarily sponsor insurgent groups for reasons unaccounted for by existing theories. States frequently support rebel groups in other countries with which they share no ideological or ethnic ties. In over half (1,192/2,276 52%) of triad-years —a triad-year is a ≈

1 rebel group, its target, and a foreign sponsor in a given year—of active external support identified by San-Akca(2016), the sponsor and the group share no ethnic, religious or ideological affiliations. More surprisingly, some sponsors even support hostile or threatening groups. For example, Iran supported leftist Kurdish insurgents during the Cold War, although Iran was a close U.S. ally and Iran itself was plagued by domestic Kurdish rebellion. This is more than a rare exception. A sponsor actively supports rebels in 352 triad-years despite the group sharing ethnic ties with a minority in its own country (San-Akca, 2016), which might pose a threat to the sponsor if their co-ethnics gain power or independence (Cederman, Girardin and Gleditsch, 2009). Sponsorship of extremely weak groups that cannot win is also puzzling. Out of 403 insurgent groups identified by the Expanded Armed Conflict Data (EACD), 159 receive overt troop support from an external state, and more than one-third (59) of them are much weaker than the targets they compete with (Salehyan, Gleditsch and Cunningham, 2011).1 These groups almost have no chance of prevailing against their target governments due to a lack of resources and effective leadership. Overall, more than 42% (102/242) of all actively supported groups share no ideological or ethnic ties with the sponsor and are weaker or much weaker than the target. These empirically observed patterns force us to ask the following questions. Why do states voluntarily support incompetent rebel groups with clearly unaligned preferences? More specifically, why do states support rebel groups with which they share no ideological or ethnic ties and that are much weaker than their target governments? The existing literature cannot answer this question because it has overlooked the bargaining advantage that sponsoring even a weak or hostile rebel group can bring. In particular, constant domestic fighting with an empowered insurgent group weakens state consolidation and hinders resource extraction (Lee, 2018). External support increases rebels’ capacity to compete for control of territory, population, and lootable resources. This decreases the target government’s ability to gather revenues and extract resources from the society, reducing its total available budget to cope with both internal and external challenges. By supporting the rebels, the sponsor indirectly weakens the target and gains bargaining leverage against it. 1 Rebel strength is measured at the time of support.

2 This bargaining advantage comes at a cost. Supporting rebels consumes resources that could otherwise be spent on building the conventional military to combat the opponent directly. In an anarchic world where war is the last resort, a state’s own military power forms the basis for its international standing. Excessive support for the insurgents might drain the resources needed to build a competent military capable of dealing with the external enemy. This puts the sponsor in a less favorable bargaining position. In deciding whether or not to aid the rebels and how much support to provide, the potential sponsor must carefully balance these competing considerations. This article develops a model to formalize this fundamental trade-off faced by a potential sponsor. Two states, a potential sponsor and a target, engage in an international dispute and the target also faces domestic insurgency. The potential sponsor allocates its limited resources between arming the military and supporting a rebel group fighting the target. Similarly, the target splits its military budget between counterinsurgency and fighting the sponsor. The more support the potential sponsor provides to the rebels, the smaller the total military budget available to the target. After resource allocation, the potential sponsor and the target negotiate over their international dispute. The analysis yields three main results. First, the fundamental reason that states support rebel groups is to weaken the international opponents and gain bargaining leverage against rival regimes. The bargaining advantage comes from the fact that such support significantly reduces the target’s total available budget to deal with internal and external threats. The baseline model focuses on a potential sponsor’s fundamental trade-off and is stripped of any confounding factors that might obscure the core mechanism. Model extensions explore how a sponsor’s other considerations interact with the fundamental trade-off to affect support decisions. Although numerous articles have examined empirical correlates, we have little understanding of the core strategic trade-offs that animate state support for rebel groups. This article shows that the main benefit that sponsors derive from supporting rebel groups is to deplete the target governments’ resources. At first glance, this might not seem like a particularly striking observation —after all, it is one of the many motivations suggested in the empirical literature (Byman et al., 2001; Byman, 2005).

3 Instead, the striking finding is that weakening rival governments is the core motivation for supporting rebel groups, and other prominent factors from the literature are neither necessary nor sufficient. Consistent with existing theories, states are indeed more likely to support rebel groups that share their ideological or ethnic preferences (Salehyan, Gleditsch and Cunningham, 2011; San-Akca, 2016) or are more effective at combating the target governments (Salehyan, 2010). However, states are also willing to support groups with whom they share no ideological ties, whose goals they actively oppose, or who are less effective at combating the target government than their own armies. The reason is that supporting such groups can still weaken the rival government. Ideological and ethnic concerns aside, the enemy of my enemy is my friend. Second, I find that the cost-efficiency of supporting the rebels is neither necessary nor sufficient for explaining state support for rebel groups. On the one hand, support for rebels is desirable when it effectively drains the target’s resources regardless of the cost of arming the sponsor’s military relative to supporting the rebels. Increasing the cost of arming the sponsor’s own military does not unambiguously motivate it to finance the rebels more. On the other hand, the low cost of rebel support does not guarantee the sponsor’s willingness to support them if the return —depleting the target’s resources —is unpromising. This result directly contrasts with an important theoretical argument in the existing literature that treats support for rebels as a cheap alternative to costly conventional warfare (Salehyan, 2010). The third result explains why sponsors actively support weak rebel groups. If the sponsor cares primarily about overthrowing the target government, then supporting weak rebel groups wastes precious resources. However, given the goal of depleting the enemy’s resources as much as possible, the sponsors prefer to support weaker groups and provide more support to them. This happens because external support for rebels exhibits decreasing marginal returns. While undercutting the sponsor’s own military strength by the same degree, support for weaker groups shrinks the target’s budget by a larger amount, motivating the sponsor to assist weaker groups. Extremely weak groups are hired not to win the war but to exhaust the target. External support delivers the most bang for the buck precisely when the rebels are weak. This novel result explains why states vol-

4 untarily support extremely weak rebel groups incapable of winning even when the sponsor is not concerned about control over the rebels or target retaliation. Besides, the expectation of receiving support from a rival regime emboldens small groups to form and compete for domestic power by increasing their chances of survival and success. This article carries broader implications for the study of international and civil conflict. First, the existing literature on third parties in international conflict primarily focuses on state actors, including state allies and mediators, and international organizations (Fang and Ramsay, 2010; Meirowitz et al., 2019). Many scholars recognize support for rebels as an important state tactic to destabilize the enemy and extort concessions.2 However, theories of conflict have not adequately addressed how external support for rebel groups undermines a rival regime and how such support influences the international dispute that contributes to it in the first place. This article illustrates a clear mechanism, thereby contributing to a growing literature on coercive bargaining that starts to appreciate non-state militant groups’ role in shaping international negotiation (Schultz, 2010; Bapat, 2012; Carter, 2015). Second, antagonistic interstate relations incentivize states to unsettle rival regimes by med- dling in their domestic conflict. While the existing literature argues that external threat may be beneficial for state building (Tilly, 1992; Thies, 2004, 2005), this article joins recent scholarship to suggest that international rivalry is detrimental to domestic stability (Maoz and San-Akca, 2012; Lee, 2018). The negative effect of outside involvement on civil war is primarily driven by a subset of cases where states intervened to pursue agendas separate from domestic belligerents’ goals (Cunningham, 2010). Studying precisely this scenario, the article is particularly relevant for understanding deleterious international influence on domestic conflict. The article also speaks to the theoretical literature on alliance formation. One strand of literature studies the efficiency puzzle of alliance formation using multi-player contests (Skaperdas, 1998; Garfinkel, 2004; Konrad and Kovenock, 2009). Others focus on the information problem 2 Previous studies loosely speak of a similar logic without specifying how the weakening effect is

achieved. See, for example, Byman et al.(2001); Hughes(2012).

5 pervasive in interstate alliance and its inﬂuence on international bargaining (Morrow, 1994; Ben- son, Meirowitz and Ramsay, 2014; Fang, Johnson and Leeds, 2014; Smith, Forthcoming). This article addresses the nature of transnational alliances in the unique civil war environment. While states form alliances to compete for a common prize or achieve a common policy goal, I model a situation where the sponsor has an agenda orthogonal to its rebel allies’ aspiration to overthrow the target government. Second, while states ally to deter aggressors and share defense burdens, in this model, the decision to ally with a rebel group is made in the background of an ongoing civil war that depletes resources. Support for insurgents allows the sponsor to weaken its enemy before getting directly involved. This distinctive feature proves to be a critical condition for state-rebel alliance formation.

THE MODEL

Two states, the Sponsor and the Target, bargain over a pie of size one. It could be a disputed territory or some policy disagreement over which the two states have opposed preferences. Be- fore bargaining, states simultaneously make allocation choices subject to budget constraints. The

Sponsor divides an exogenously given budget b1 between building a conventional military, m1, and supporting rebels in the Target, s with θ m1 + s = b1. Here θ > 0 captures the cost of military- building relative to supporting rebels. Arming the military is less expensive than sponsoring the rebels if θ < 1. When θ = 1, they have the same unit cost. Figure 1a visualizes the Sponsor’s budget constraint. Simultaneously, the Target allocates its resources proportionally between ﬁghting the Sponsor internationally f2 and ﬁghting the civil war fc with f2 + fc = 1. Here f2 and fc indicate the fraction of resources allocated to each front, respectively. The actual amount of resources is the fraction times the Target’s total budget after the rebels receive support from the Sponsor.

Given the rebel group’s original strength mr and the support it receives s, the total budget available to the Target is b2(mr +s). To capture the idea that heated domestic ﬁghting weakens the Target’s domestic control and resource extraction, I make several assumptions about the Target’s budget. Figure 1b plots an example of the budget function satisfying all the assumptions.

6 Figure 1: An illustration of players’ budgets

(a) Sponsor’s budget constraint (b) Target’s budget function

m1 b2(mr + s)

b1 θ b2 function of s function of mr + s

θ m1 + s = b1

s s 0 b1 0

Notes: The left panel illustrates the Sponsor’s budget constraint θ m1 + s = b1 with θ = 1.25 and b 15. The right panel plots the Target’s total budget after support b m s b a with b 12, 1 = 2( r + ) = 2 a+mr +s 2 = a = 3, and mr = 2. The solid line plots a function of s with rebel strength mr = 2 ﬁxed. The dashed line plots a function of mr + s. Here b2 is the Target’s budget independent of the rebel group, and a represents the Target’s state capability to withstand domestic challenge.

Assumption 1. Assumptions on b2():

1. b is twice continuously differentiable. 2 : R+ R++ →

2. b20 ( ) < 0 and b200( ) > 0 so that the total budget is a decreasing convex function of post- · · support rebel strength.

3. b2(0) = b2 so that the Target obtains the maximum b2 when there is no rebellion.

4. b2 y > 0 for all y with lim b2 y 0 so that the Target always has some resources. ( ) y ( ) = →∞ After the allocation choices, two states engage in Ultimatum bargaining. The Sponsor makes

3 the offer (1 x, x), where x [0, 1] stands for the Target’s share. The Target chooses to accept or − ∈ reject. Acceptance leads to a settlement giving each state the proposed share, and rejection results 3 Equilibrium level of support and other key implications do not depend on which state is the

proposer.

7 in a costly war. The war outcome depends on the distribution of power endogenously determined by both actors’ resources and allocation decisions. In case of a war, the probability of the Target prevailing against the Sponsor is described by the widely used ratio contest success function:

  1 if m f b m s 0,  2 1 = 2 2( r + ) = p f2 b2(mr + s), m1 = f b m s  2 2( r + ) otherwise . m1+f2 b2(mr +s)

Fighting is costly and destroys 1 φ (0, 1) fraction of the pie. The victor enjoys the whole pie − ∈ minus the part destroyed by ﬁghting. The Sponsor’s expected war payoff is φp m1, f2 b2(mr +s) , and that of the Target is φp f2 b2(mr + s), m1 + αp fc b2(mr + s), mr + s , where α > 0 is the Target’s value for domestic control. The Target values both the international dispute and domestic stability, and a larger α implies a higher weight of the domestic front.

With an agreement (1 x, x), each party receives a payoff equal to its settlement share. Sim- − ilarly, the Target cares about domestic stability and beneﬁts from successful counterinsurgency. The Target’s total expected utility from a settlement is x + αp fc b2(mr + s), mr + s .

COMMENTS ON THE MODEL

This section brieﬂy discusses several key assumptions and provides appropriate scope conditions for the model. The Target’s Budget The model assumes that the Target’s defense budget strictly decreases in post-support rebel strength mr + s, as illustrated by the downward slope in Figure 1b. Outside involvement on the rebel side prolongs civil wars, increases fatalities, and renders peaceful settlement more difﬁcult (Heger and Salehyan, 2007; Cunningham, 2010; Moore, 2012). Both theory and empirical evidence suggest that civil wars negatively impact economic growth and investment (Abadie and Gardeazabal, 2003; Cerra and Saxena, 2008). Also, stronger groups (post-support) hurt the target more. First, stronger groups can more effectively undermine the government’s control over territory and population, undercutting its tax base (Martinez, 2017). For example, the

8 Liberation Tigers of Tamil Eelam (LTTE) established monopolistic control and set up quasi-state institutions in part of Sri Lanka for a long time, crowding out the government’s coercive appara- tus and tax institutions. Second, stronger groups are better at competing with the government over control of lootable resources, such as oil deposits and diamond mines. These natural resources con- stitute an important source of government revenue, especially in underdeveloped countries where the threat of civil war is more prominent (Collier and Hoefﬂer, 2004; Humphreys, 2005). The re- bellious União Nacional para Independencia Total de Angola (UNITA) controlled Angola’s most

productive diamond mines and raised $3.7 billion through diamond sales between 1992 and 1997 (Malaquias, 2001).

4 Also, b2( ) is convex, meaning that the marginal damage to the Target’s budget decreases as · post-support rebel strength increases, illustrated by the steeper slope for lower values of mr + s in Figure 1b. One hundred dollars’ worth of food, manpower and weapons might be crucial for a nascent rebel group organizing its ﬁrst attack but offers little help to a strong group in a head- on confrontation with government forces. Additional support at the margin is more effective in

2 reducing the target’s budget for weaker groups ( ∂ b2( ) > 0). A small amount of external assistance ∂ s∂ m·r can help startup insurgent groups evade state repression longer than they should have, dragging the government into a costly counterinsurgency campaign. Fursan al-Joulan, a small Syrian rebel

group with about 400 local ﬁghters, receives approximately $5, 000 per month from Israel—a fairly small amount vital for the group’s sustained existence (Jones, Raydan and Ma’ayeh, 2017). The same amount would be less effective in helping a group like the Anyanya (1955-1972) ﬁghting

to separate from Sudan with 5, 000 10, 000 soldiers. − External support for rebels might also increase the salience of the civil war so that civilians in the target states are willing to bear higher taxes out of security concerns. The government may shift non-military resources to military usage to better contain the increased threat. External support for rebels might trigger new outside assistance for the target government (Salehyan, Gleditsch and 4 This is a natural derivation of the government-rebel contest over resources. Online Appendix A.1

provides details.

9 Cunningham, 2011). All of these scenarios would change the maximum resources the target can access. The model’s assumptions are more likely to hold in cases where outside support contributes to sustained conflict so that the destructive effect on the target’s budget dominates. However, even in those cases, having to deal with an empowered rebel group consumes latent resources that could have been mobilized to fight external enemies. Overt vs. Covert Support In the model, actors make allocation decisions simultaneously. This captures the covert nature of most support. Some states openly support rebels to satisfy their domestic audience. However, even in those cases, the precise level of support is not revealed at the time of resource allocation. More frequently, sponsors deny any ties with rebels when questioned (Schultz, 2010). Financial and material support can go through back channels. Intelligence agen- cies take part in protecting secret transfers. Even in the most detectable case of military support, the sponsors’ soldiers can carry out tasks disguised as rebels (Hoekstra, 2018). Covert support carries little reputation cost. Many sponsors successfully avoid international criticism by careful plan- ning, preparation, and disguise. When exposed, sponsors can gloss their actions as humanitarian aid (Lee, 2018). As such, the model omits reputation concerns. Rebels’ Post-Settlement Fate The article focuses on the effect of support on interstate bargaining and does not model rebels’ post-settlement fate. In the real world, sponsors often abandon their former agents with no hesitation after fulfilling international goals. Abandonment can have a disastrous effect on rebel groups. Sponsors have incentives to provide information to the target about the groups once they become inconvenient. Pakistan, a longtime sponsor of the Taliban, co- operated with the U.S. government to capture a top Taliban commander in 2010. Despite the risk, rebel groups understand the vital importance of external support and seldom refuse the essential assistance (Carter, 2012).

10 EQUILIBRIUM ANALYSIS

This is a sequential game of complete information, and the solution concept is Subgame perfect equilibrium (SPE, or simply equilibrium hereinafter). I begin the analysis with the Target’s decision to accept or reject an offer.

Lemma 1. Given any allocation choices, the Sponsor offers x ∗ = φp f2 b2(mr + s), m1 in equi-

librium, and the Target accepts any x x ∗ and rejects any smaller amount. ≥ As in typical bargaining models, the Sponsor offers the Target’s reservation value, the amount that makes the latter indifferent between acceptance and rejection, and is always accepted.5 The deal depends on the distribution of power endogenously determined by both actors’ allocation decisions. The Sponsor chooses the best way to split its available resources between building a conventional army and sponsoring the rebels. The Target optimally allocates resources between counterinsurgency and international warfare. The following lemma characterizes the Sponsor’s best response to the Target’s allocation choice.

Lemma 2. Given any (f2, fc) such that f2 + fc = 1, the Sponsor’s best response is

 [0, b1) if f2 = 0 (1a)   s(f2) = 0 if f2 = 0 and b2(mr ) + b20 (mr )b1 0 (1b)  6 ≥  ˆs if f2 = 0 and b2(mr ) + b20 (mr )b1 < 0, (1c) 6 where ˆs (0, b1) solves b2(mr + s) + (b1 s)b20 (mr + s) = 0. ∈ −

When f2 = 0, any s < b1 is a best response because it guarantees the Sponsor a victory. When

f2 = 0, if the size of the marginal reduction in the Target’s budget at s = 0 is larger than the two 6 5 It is well-known that uncertainty can lead to war (Fearon, 1995). War is possible in equilibrium

with uncertainty over the destructiveness of ﬁghting (φ). This article omits it to focus on the

potential sponsor’s strategic trade-off rather than the prospect of war.

11 states’ initial resource ratio ( b20 (mr ) > b2(mr )/b1), the Sponsor provides support but also trains − its own military. Otherwise the Sponsor only invests in the military and does not ﬁnance the rebels. The next lemma describes the Target’s best response to the Sponsor’s allocation choice.

Lemma 3. Fix any (m1, s) such that θ m1 + s = b1, the Target’s best response is

 if s = b1 (2a)  ;   if s b and φ m1(mr +s)  0 = 1 α < b m s m s 2 (2b)  6 [ 2( r + )+ r + ]  2 φ [b2(mr +s)+m1] f2 s 1 if s b1 and > (2c) ( ) = = α m1(mr +s)  Ç 6  φm1  b mr s mr s m if s b and  α(mr +s) [ 2( + ) + + ] 1 = 1  2  − φ m1(mr +6s) [b2(mr +s)+m1] (2d)  Ç φm 2 , ,  1 α [b2(mr +s)+mr +s] m1(mr +s)  1 b mr s α(mr +s) + 2( + ) ∈

where m1 = (b1 s)/θ. − For the Target to not give up on either front, the Sponsor’s strength relative to the group after support cannot be too unbalanced. If the Sponsor is extremely powerful, as in condition (2b), the last fraction of resources spent on fc is more proﬁtable than the ﬁrst fraction spent on f2, and the Target gives up the international front. By contrast, if the group becomes extremely powerful after

support, as in condition (2c), the last fraction spent on f2 is more beneﬁcial than the ﬁrst fraction

spent on fc. The Target spends all resources ﬁghting the external enemy. Building on both actors’ best responses, Proposition1 states that an equilibrium exists for all parameter conﬁgurations.

Proposition 1. A subgame perfect equilibrium exists.6

Figure2 graphically illustrates the logic. The solid line depicts the Sponsor’s best response

to f2, and the dashed line represents the Target’s best response to s. An equilibrium exists if the lines cross at least once. If the Target gives up entirely on the international front, any feasible positive investment in the conventional military is a best response as it guarantees the Sponsor a total victory. If the Target spends a positive effort on the international front, the Sponsor’s military

investment ˆs falls in the range of [0, b1), represented by the vertical solid line. Since the Target’s 6 Online Appendix Proposition A.1 provides a complete characterization.

12 best response continuously changes in s, the dashed line crosses the vertical solid line exactly once, indicating that there exists exactly one interior equilibrium. If condition (1a) of Lemma2

and condition (2b) of Lemma3 are simultaneously satisﬁed for some (s, f2), we also have multiple equilibria where the Sponsor engages in both military building and rebel support while the Target only ﬁghts the insurgents, as illustrated by the overlapping part on the x-axis.

Figure 2: Equilibria in the baseline model

Notes: The ﬁgure uses b m s b a with a m 1 2( r + ) = 2 a+mr +s = r = and b2 = 11. Other parameters are b1 = 10, φ = 0.8, θ = 0.9, and α = 1.

BUDGET REDUCTION MECHANISM

The fundamental force driving the external support is its subversive effect on a rival regime and the bargaining leverage it brings. Remark1 below details the budget reduction mechanism through which the subversive effect is achieved.

Remark 1 (Budget Reduction). (a) The Sponsor supports the rebel group if the size of a marginal reduction in the Target’s budget is larger than the initial resource ratio of two states without sup-

13 port. (b) The Sponsor supports the rebel group to the extent that the size of budget reduction equals the resource ratio after support. (c) The Sponsor does not support the rebel group if there is no budget reduction effect.

Remark1(a) tells that the Sponsor supports the rebels if the size of a marginal reduction in the Target’s budget for the ﬁrst dollar spent on external support is larger than the sponsor-target resource ratio conditional on rebel strength, as characterized by the following inequality:

Size of budget Resource ratio reduction at s = 0 without support z }| { z }| { b20 (mr ) > b2(mr )/b1 . (3) −

Otherwise, the potential sponsor does not support the insurgent group. Remark1(b) further shows that the size of the budget reduction effect determines the support level. In equilibrium, the interior allocation decision must satisfy:

Resource ratio Size of budget after support s∗ reduction at s ∗ z }| { z }| { b m s b m s 2( r + ∗) . (4) 20 ( r + ∗) = b s − 1 ∗ − The Sponsor supports the rebels until the marginal damage it causes to the Target’s budget is equal to the sponsor-target resource ratio after support. Notice that the support decision and the support level do not depend on the cost of arming the military θ. Remark1(c) highlights the importance of the budget reduction mechanism by showing that if sponsoring rebels does not hurt the Target’s total ﬁnancial stock, there will be no support. In- tuitively, if the support has no effect on the Target’s available resources but reduces the amount left for the Sponsor’s own armed forces and leads to a weaker conventional military, the resulting balance of power will shift in favor of the Target. This only puts the potential sponsor in a more disadvantageous bargaining position, preventing it from providing any support. This mechanism highlights the underlying incentive driving the decision to support rebels. Sponsors often support rebels simply because it offers an excellent opportunity to destabilize rival

14 regimes. For example, Gordon and Lehren(2010) explain that by supporting Iraqi militias after Saddam Hussein’s fall, “the Iranians understand that they are not going to dominate Iraq, . . . they are going to do their best to weaken it —to have a weak central government that is constantly off balance, that is going to have to be beseeching Iran to stop doing bad things without having the capability to compel them to stop doing bad things. And that is an Iraq that will never again threaten Iran.” The goal of support is political—subversion, containment, and extortion.

SUPPORT WEAK REBELS

Why do extremely weak rebel groups with no hope of victory receive support? Because they can wear down the target regimes and bring the sponsors bargaining leverage. Due to the decreasing marginal return of external support, weaker groups hurt the target more for the same amount of support. As a result, it is more desirable for the sponsors to support weaker insurgent groups and provide more support to them in the absence of other concerns, as summarized by Proposition2.

Proposition 2. There exists a threshold level of rebel strength that potential sponsors only support

rebel groups weaker than the threshold. Moreover, the level of support s∗ decreases in mr .

The above result requires the budget function and the size of its derivative to be log-convex.

These are not very restrictive assumptions, and the example used throughout this article b2(mr + s b a with a > 0 satisfies both. A direct consequence of the assumption is that additional ) = 2 a+mr +s support at the margin more effectively hurts the target’s budget if the rebel group is weaker. The lower the rebels’ initial strength, the more efficient an increase in the level of support is in generating bargaining advantages for the sponsor. Thus the optimal level of support is higher for weaker groups. While undercutting the sponsor’s own military strength by the same degree, support reduces the target’s total budget by a larger amount when rebel groups are weaker, motivating the sponsor to provide more support. When the rebels are sufficiently strong, additional support only does negligible damage to the target’s budget but hurts the sponsor’s military strength, discourag- ing the potential sponsor from providing any support.

15 At first glance, the result seems inconsistent with existing empirical findings that extremely weak groups are less likely to receive external military support (Gent, 2008; Salehyan, Gleditsch and Cunningham, 2011; Szentkirályi and Burch, 2018). However, these studies use samples that are not relevant for assessing the model’s prediction. More specifically, they exclude some relevant cases and include some unrelated cases. First, they focus on military and troop support, which accounts for only 16% of all supported groups (Högbladh, Pettersson and Themnér, 2011) and less than 6% of triad-years of support (San-Akca, 2016). Second, the tests are performed for all rebel groups or all sponsor-rebel dyads. The model suggests that if sponsors primarily care about destabilizing the rival regimes, they prefer to support weaker groups. This relationship might not hold for all rebel groups, especially when the sponsors have strong ideological/ethnic affiliations with the groups and benefit from a rebel takeover. Although I could not find the most appropriate sample to test the claim, I perform empirical tests on a more relevant subsample by examining military support for rebels fighting targets that were engaged in international rivalry. The results are presented in Online Appendix Table D.1, which replicates Salehyan, Gleditsch and Cunningham (2011) and shows that rebel strength loses significance in the subsample of rebels fighting targets with international rivals. This novel modeling result explains why extremely weak rebel groups with no hope of victory receive external support. If potential sponsors care primarily about overthrowing the target governments, incompetent groups known to have unaligned preferences should not receive support at all. However, given the goal of depleting the target’s resources as much as possible, sponsors prefer to support weak rebels. With diminishing marginal returns, external support delivers the most bang for the buck precisely when the rebels are weak. For example, the 19th of April Movement

(M-19) had no more than 1, 500 combatants and was much weaker than the Colombian regime by the time it received Cuban and Nicaraguan support. External support was effective in keeping the guerrilla movement active and embroiling Colombia in greater quagmire. Weakness can be an advantage. Felter and Fishman(2008, p. 35) observe that in sponsoring a plethora of Iraqi local militias after Saddam Hussein’s fall, Iran was fully aware that “the groups were often little more

16 than neighborhood militias; many lacked a cohesive ideology.” Nonetheless, Iran provided support because “from Iran’s perspective, however, these prospective weaknesses may have been attractive. Iran was not looking for a militia capable of taking over the Iraqi government, but rather an ally capable of knocking the government, and its American suitor, off its bearings. . . . The new JAM splinter movements offered Iran the ability to disrupt Iraq with fewer political complications.” The literature suggests various other reasons why sponsors support weak groups. First, weak groups and those without strong centralized command are easier to manipulate. Staniland(2012, p. 170) notes that Pakistan’s Inter Services Intelligence (ISI) agency supported local armed groups in Kashmir to “keep these other groups alive as part of a fractured movement more amenable to its control” for fear that the Jammu Kashmir Liberation Front (JKLF) might become strong enough to threat Pakistan. Second, sponsors might refrain from supporting powerful insurgent groups to avoid uncontrolled escalation and target retaliation. For example, Pakistan covertly armed Islamic extremists during the Soviet-Afghan war. General Zia-ul-Haq of Pakistan reminded the then head of the ISI that “the water in Afghanistan must boil at the right temperature.” (Coll, 2005, p. 63) He feared that pushing the secret war against the Soviet Union too much would backfire. These are all reasonable explanations for state support of weak rebels. This article, however, highlights an additional explanation directly linked to the core mechanism that activates state support for insurgents. The result holds even when sponsors are unconcerned about controlling the rebels or avoiding target retaliation. The result also sheds light on the abundance of small non-militant groups in persistent interstate rivalries like the one between India and Pakistan. Suppose there is a fixed startup cost for all groups, and a rebel leader only organizes the group if the expected future return justifies the cost. Online Appendix C.3 shows that the expectation of receiving external support encourages small rebel groups to form and enter the domestic competition for power because they now have a higher chance of survival and success.

17 COST-EFFICIENCY IS IRRELEVANT

The existence of an interior equilibrium is substantively meaningful and highlights a result directly contrasting a common understanding in the existing literature. The literature on state support of insurgent groups emphasizes that (1) insurgents can ﬁght the target more effectively than the sponsor’s army, (2) support for rebels is a cheap substitute for costly conventional warfare, and (3) rebel groups having ethnic and ideological ties with the sponsor are better agents because they are more likely to share the sponsor’s preferences and less likely to shirk (Salehyan, 2010). This model purposefully omits these components, yet the sponsor still offers support in equilibrium, implying that these conditions suggested by the literature are not necessary. More specifically, in the model, the Sponsor and the rebel group share no common policy goals and have no ideological or ethnic associations. The sponsor’s army is not militarily less effective than the rebels.7 Furthermore, the Sponsor’s support decision does not depend on θ, indicating that there exists an equilibrium where the Sponsor provides support regardless of the cost of rebel support relative to arming the military. This is true even when θ 1. Importantly, when all favorable con- ≤ ditions for rebel support suggested by previous studies are absent, the subversive effect of support provides a sufﬁcient incentive for the sponsor to sabotage its enemy by assisting insurgent groups.

Proposition 3. The Sponsor’s equilibrium conventional military strength m1∗ decreases in θ, and the equilibrium level of rebel support s∗ is constant in θ.

β 7 One way to formalize military effectiveness is to specify p x, y x with β > 0. The ( ) = xβ +y

player with input x is militarily more effective than the player with input y if β > 1 (Grossman,

1991). The baseline model is equivalent with β = 1, where players are equally effective. Online

Appendix C.2 formally considers how military effectiveness affects the decision to support rebels.

18 Proposition3 further shows that increasing the cost of arming the military does not unambiguously motivate the sponsor to ﬁnance the rebels more.8 As military-building becomes more

costly (θ increases), the Sponsor’s conventional military strength m1∗ decreases, yet its support for

rebels s∗ remains unchanged in equilibrium. While an increased cost of arming driving the military investment down is intuitive, the logic behind unchanged rebel support is subtle. In equilibrium, the Sponsor’s (interior) allocation decision must satisfy:

Marginal cost of Marginal beneﬁt of s relative to m1 s relative to m1 z}|{ z }| { 1 m1 b20 (mr + s) = , (5) θ − b2(mr + s)

which simpliﬁes to =θ m1 z }| { b2(mr + s) = (b1 s) b20 (mr + s). (6) − − An increase in θ has two countervailing effects. First, it directly decreases the marginal cost

of supporting rebels relative to military building, which is 1/θ on the left side of equation (5). Second, it decreases the military strength one can build with same amount of budget. As a result, it indirectly decreases the marginal beneﬁt of supporting rebels relative to military building, the

right-hand side of equation (5). Since the marginal beneﬁt of s relative to m1 positively depends on one’s own military strength and thus inversely relates to θ, the right-hand side can be rewritten as 1/θ multiplied by a constant. These two effects cancel out, and the resulting equation (6) is independent of θ. As θ increases, support for rebels becomes relatively cheaper. However, the increased price of arming also lessens the sponsor’s own military strength and therefore decreases the marginal beneﬁt of supporting the rebels. The net effect motivates the sponsor to sustain the 8 Proposition3 focuses on the interior equilibrium because it is more realistic. Online Appendix

Proposition A.2 shows that as θ increases, the range of multiple equilibria shrinks and eventually

disappears. Since a large θ better describes the real world where supporting rebels is much

cheaper, the main text focuses on the interior equilibrium.

19 same level of support. The existing literature only considers the ﬁrst effect but overlooks the second. Therefore, it does not fully capture the inﬂuence of the cost of arming on external support for rebels. This result is not driven by the linear cost of arming the military. Online Appendix A.3 shows that even with a convex cost function widely used to model increasing marginal cost, a higher cost of arming the military does not always lead to more support.

OTHER CONSIDERATIONS OF THE SPONSOR

This section extends the baseline to incorporate three prominent motivations behind state support for rebels suggested by the literature: ideological and ethnic concerns, support to satisfy the domestic audience, and support as a substitute for an ineffective army. I highlight substantively interesting results below and defer all formal statements to the Online Appendix.

Ideological and Ethnic Concerns

Online AppendixB considers the sponsor’s ideological and ethnic concerns. In addition to international considerations, the sponsor suffers/beneﬁts from a rebel victory: V p mr + s, fc b2(mr + s) . Here V R is the ideological congruence between the sponsor and the group, and p mr + ∈ s, fc b2(mr +s) is the probability of a rebel victory. This is a reduced-form way of incorporating the agency issue in supporting rebel groups. One interpretation is that the sponsor, the target, and the rebel group have respective ideal points. The target and the group implement their ideal policies while in power. The sponsor’s utility from ideological and ethnic concerns depends on the distance

from its ideal point to the implemented policy and its probability of implementation. Thus, V parameterizes how much the sponsor likes/dislikes the group compared to the target. Cases in which the sponsor seeks a regime change in the target and supports rebels to install an ideologically and ethnically more friendly government fall under this scenario (Byman et al., 2001). The extension recovers the conventional wisdom that sponsors prefer to support rebel groups with which they get along ideologically or share ethnic ties. The sponsor provides more support to rebels with higher ideological/ethnic congruence (V ). This is unsurprising since now the sponsor

20 directly benefits from a rebel victory. As the potential sponsor and the group’s preferences become increasingly aligned, the sponsor is willing to provide more support to help the rebel group take over the target regime. The result is consistent with existing empirical findings. For example, San- Akca(2016) finds that the existence of ideological and ethnic ties between a potential sponsor and a rebel group increases the likelihood of providing any support. She also finds that these affiliations are associated with a higher level of support. Salehyan, Gleditsch and Cunningham(2011) find that the existence of transnational ethnic ties increases the probability that a group receives troop support from at least one foreign state. However, I also find that ideological and ethnic concerns do not completely override the core mechanism driving the support decision. Suppose the sponsor supports the insurgent group if it

only cares about weakening the target (V = 0). It continues the support, albeit at a lower level, when its ideological/ethnic divergence from the rebels is not too large (small V < 0). Because assisting the rebels significantly weakens the target’s resource extraction, the overall subversive effect overshadows the danger of a victory by the rebel group. This explains why we observe in- stances of patronage relations where the sponsors finance insurgent groups they genuinely dislike. For example, the right-wing government of Iran provided safe havens to the Patriotic Union of Kur- distan (PUK) targeting Iraq during the Cold War (1976-1983), despite the fact that the PUK is a leftist group and has close connections with Kurdish secessionist movements in Iran. Islamic Pak- istan provided training and financial aid to the National Socialist Council of Nagaland-Isak-Muivah (NSCN-IM), a Christian separatist group fighting for the independence of Nagas in Northeast In- dia. Support for these groups can still deplete the target’s resources even though their victory is unpalatable.

Support to Satisfy the Domestic Audience

Some sponsors support rebels to satisfy their domestic audience and beneﬁt from the act of support per se. For example, Chadian President Idriss Déby secured his political survival by submitting to coup plotters’ request to support co-ethnic Zaghawas rebelling in Darfur in 2005 (Tamm, 2016).

21 Leaders can also aid ethnic or religious kin to bolster their political position at home (Byman et al., 2001). India, for example, supported the LTTE to gain votes from the large Tamil population in

India. Online Appendix C.1 formalizes this idea. The sponsor directly receives B s for providing · support s [0, b1], where B > 0 is the unit benefit of support. ∈ The extension finds that the sponsor provides a higher level of support if the unit benefit of support (B) increases. A related implication is that the sponsor provides more support if it brings direct benefit (B > 0) compared to no direct benefit (B = 0). This result is not surprising because the sponsor directly benefits from the act of support. This is consistent with existing empirical findings that the level of support positively associates with the existence of ethnic ties between a potential sponsor and a rebel group (San-Akca, 2016). Supporting transnational ethnic kin not only brings a more friendly regime if the rebels win (V > 0) but also directly benefits the leader of the sponsoring state (B > 0). The extensions suggest that the positive association between shared ethnic ties and support for rebels can work through two different channels.

The level of support also increases in B. Empirically, B should be larger in states where the relevant domestic issue is more salient. In the context of ethnic/religious politics, the issue should be more salient if the rebel group fights for the sponsor’s discriminated or repressed ethnic/religious kin. While I am unaware of any direct large-N test of the hypothesis, Byman et al.(2001) and Byman(2005) identify aiding kin as an important motivation behind state support for rebel groups and provide many examples of states supporting groups fighting on behalf of discriminated or suppressed kin. A surprising finding is that, under certain conditions, even when the sponsor offers no support in the baseline (B = 0), it will support rebels if the act of support brings a direct benefit, regardless of how small it is (any B > 0). The condition is that the target’s military allocation is sensitive to the support level, which happens when the target puts roughly balanced weights on the international and domestic fronts (moderate φ/α). Thus, the result helps explain the robust positive correlation between sponsor-rebel ethnic ties and external support found in the empirical literature. However,

22 it is worth emphasizing that it works by affecting the target’s allocation decision, which in turn shapes the sponsor’s support decision.

Military Effectiveness

States might support rebels that can ﬁght the target more effectively than the state’s own military, which lacks local knowledge and population support (Salehyan, 2010). Online Appendix C.2

formalizes the military effectiveness of the sponsor. Let β > 0 denote the military effectiveness

of the sponsor’s army against the target compared to that of the rebels, with higher β indicating a more effective state military. The probability that the sponsor prevails against the target is

  1 if m f b m s 0,  2 1 = 2 2( r + ) = p m1, f2 b2(mr + s) = mβ  1  β otherwise. m1 +f2 b2(mr +s)

If β > 1, the sponsor’s army is more effective than the rebel group at combating the target. If β <

1, the opposite is true. β = 1 recovers the baseline, where they are equally effective. Everything else remains the same. The extension produces two findings. First, consistent with the literature, as the sponsor’s army becomes less effective at fighting the target, it is more likely to support the rebels and offers more support. When the subversive effect of the support is sufficiently large (equation (3) holds), the sponsor supports the rebels for sufficiently low military effectiveness (β < β˜). Moreover, the level

of support decreases in β. Second, however, contrary to the existing theory, rebels are better at ﬁghting the target than the sponsor is not necessary for rebel support. When the subversive effect of external support —the model’s core mechanism —is large enough, the sponsor voluntarily aids the

rebels even when the sponsor is equally good or better at ﬁghting the target than the rebels (β 1). ≥

23 IRANIAN SUPPORT FOR THE KDP

The model’s core mechanism is that sponsors voluntarily support weak rebel groups to deplete the target government’s resources and gain bargaining leverage in the international dispute against the target. Moreover, a hostile ideological or ethnic relationship between the sponsor and the rebel group does not override the core mechanism. A sponsor will support incompetent groups with unaligned preferences if such sponsorship can drain the target state’s resources and extort concessions. This section applies the mechanism to comprehend Iranian support for the Kurdistan Democratic Party (KDP) of Iraq. Iranian support for the KDP in the 1960s and 1970s deﬁes common explanations of state support for rebel groups.9 The KDP was not a good candidate as a proxy force for Iran. First, Iran and

KDP had opposed ideological and ethnic preferences (V < 0). The KDP was a Kurdish militant group with pro-Soviet inclinations ﬁghting for Kurdish independence in northern Iraq. At that time, Iran was one of America’s closest allies in the Middle East. Besides, Iran faced an active Kurdish insurgency in its northwest, directly bordering Iraqi Kurdistan. A successful Kurdish rebellion in Iraq might eventually have spill-over effects on the sizable Kurdish minority in Iran. Second, the

KDP was incompetent (small mr ). It was much weaker than the Iraqi government. Without external support the KDP would have stood no chance against Iraq’s counterinsurgency campaigns.10 Nonetheless, Iran gave massive support to the KDP between 1961 and 1975, consistent with the model presented here. As the model suggests, Iran supported the Kurds to wear down Iraq’s resources and tip the balance of power in favor of itself, thereby gaining bargaining advantages in their boundary dispute over the Shatt al-Arab waterway. The support played a crucial role in their bargaining that led up to the Algiers Accord in 1975, in which Iraq made signiﬁcant territorial concessions in exchange for 9 Online Appendix D.1 discusses alternative explanations in more detail. 10 The KDP suffered catastrophic losses and was forced to take refugee in Iran and Turkey within

three months after Iran withdrew support in 1975 (Chaliand and Black, 1994, p. 65).

24 Iranian cessation of support for the KDP. This case demonstrates that external support’s subversive effect provides sufficient incentive to aid the rebels targeting the sponsor’s international rival even when the conditions for proxy warfare are absent. The Shatt al-Arab is a navigable waterway of approximately 204 kilometers that discharges into the Persian Gulf. Its lower part down to the Gulf constitutes the Iran-Iraq frontier and has been a source of conflict between the two countries dating back to the seventeenth century. The dispute revived after World War II and continued to fester into the 1960s, characterized by intermittent negotiations and military clashes on the frontier. In April 1969, Iran unilaterally abrogated their 1937 Boundary Treaty, starting a period of crisis until the Algiers Accords of 1975. The crux of the disagreement was that Iran wanted to revise the boundary to follow the thalweg (the center of the principal navigable channel), while Iraq claimed the entire Shatt according to the 1937 Boundary Treaty. To pressure Iraq into concessions, Iran started giving massive support to the KDP in 1961. Iran shipped commodities, arms, and equipment into the Kurdish region, helped infiltrate heavy and light arms from other sources, trained the insurgents in Iranian military academies, and provided intelligence assistance (Abdulghani, 1984). The model’s core mechanism states that sponsors support the rebels to deplete the target’s resources as much as possible and gain bargaining leverage against the target. This is consistent with Iran’s intention. By sponsoring the KDP, Iran aimed at draining Iraq’s resources and immobilize its military capabilities. The ultimate goal was to weaken the Iraqi government and extort concessions in their border dispute. Iran’s intention is well-documented in the Congressional Pike Report based on a series of closed hearings conducted by the House Subcommittee on Intelligence.11 According to the Pike Report (1976, p. 91), the United States and Iran had used the Kurds as “a uniquely useful tool for weakening [our ally’s enemy’s] potential for international adventurism (all brackets in the original source, hereinafter the same).” They hoped that “the insurgents simply continue a level of hostilities sufficient to sap the resources of our ally’s neighboring country [Iraq].” The 11 A substantial proportion of the Pike Report was leaked and published in Village Voice.

25 Shah of Iran also told Henry Kissinger in 1973 that “he would keep Iraq occupied by supporting the Kurdish rebellion within Iraq, and maintaining a large army near the frontier.” (Kissinger, 1982, p. 675) Iran was keenly aware that the Kurdish insurgency was mere “a card to play” in the Shatt dispute. U.S. and Iran had secretly agreed that once Iran came to an agreement with Iraq over their common boundaries, Iran would stop sponsoring the Kurds. The Shah had secretly offered to cease aiding the Kurds in exchange for Iraqi concessions regarding the Shatt as early as the summer of 1972 (Bengio, 2012, p. 140). This is confirmed in a CIA memo of October 17, 1972, which stated, “[Our ally] has apparently used [another government’s] Foreign Minister to pass word to [his enemy] that he would be willing to allow peace to prevail [in the area] if [his enemy] would publicly agree to abrogate [a previous treaty concerning their respective borders].” (Pike, 1976, p. 87) Iranian support for the KDP achieved its intended effects. While it is difficult to assess the precise quantities described in Remark1, there is little doubt that the support drained Iraq’s financial and military resources substantially. It helped the KDP create a viable fighting force able to confront the superior Iraqi military. Sustained insurgency drained the Iraqi government financially. It created a practical hurdle for Iraq to profit from the Kurdistan region’s rich oil reserves, a vital financial source for the government. The Kurds destroyed parts of the Kirkuk refinery in March 1969, causing a severe financial blow to the Iraqi government. The KDP was also able to establish control over a non-trivial part of Iraqi territory and population, isolating a significant portion of the Iraqi government’s tax bas. The fight with the Iran-backed KDP also diverted Iraq’s already strained budget away from wrestling with Iran. The estimated cost of combating the Kurds in 1969 alone was £100 million, about 30% of Iraq’s total budget that year. As the domestic conflict escalated, Iraq’s expenditure on fighting the Kurdish rebels skyrocketed. Iraq’s estimated cost of the war against the Kurds was more than $4 billion, accounting for a substantial proportion of its limited financial resources (Abdulghani, 1984, p. 134).

26 Iraq deployed a majority of its armed forces to combat the KDP and consumed almost all its available weapon stocks. Empowered rebels occupied Iraq’s military forces in the north, severely constraining Iraq’s ability to deal with the border dispute if the conflict with Iran were to escalate. Between 1961 and 1966, three of Iraq’s five army divisions and a substantial portion of its air force engaged in fighting the Kurdish insurgency (Chubin and Zabih, 1974, p. 181). In the full-scale civil war of 1974-5, Iraq mobilized 80,000 men supported by 800 tanks and eight squadrons of aircraft against the Kurdish rebels (Abdulghani, 1984, p. 155). Iraq practically had committed all its available forces to fight the Kurds—even its police force joined the war (Vanly, 1980, p. 181). Saddam Hussein later revealed that, before the end of the war in March 1975, the Iraqi air force had only three heavy bombs left, and there was “a great shortage of ammunition” to continue the war (Abdulghani, 1984, p. 156). The war with the KDP weakened Iraq, generating bargaining leverage for Iran. After the Shatt dispute reopened in April 1969, Iran and Iraq went through four rounds of negotiations, in which Iranian support for the Kurds was a central point of contention. The first three rounds failed because Iraq refused to back down on the Shatt. Increasingly exhausted by fighting with the Iran-backed KDP, Iraq eventually made concessions in the Algiers Accord of 1975. In the agreement, Iraq agreed to the thalweg principle demanded by Iran in exchange for Iran withdrawing support for the KDP. Although the agreement was facilitated by a quid pro quo, its timing lends support to the model. Iran had repeatedly made the exact same offer earlier, and Iraq had rejected it every time. It was only after the civil war severely weakened Iraq that it eventually made the concession.

CONCLUSION

Why do states voluntarily support incompetent rebel groups known to have unaligned preferences? This article studies a model of state support for rebels amid an interstate dispute. One state in the dispute sponsors rebels ﬁghting the other state to sap the target state’s resources. The sponsor thereby gains international bargaining advantages. The analysis also yields several results worth reemphasizing.

27 First, shared preferences facilitate but are not necessary for state support of insurgent groups. The subversive effect of support provides a sufficient incentive even in the absence of favorable conditions suggested by previous studies. Ideological or ethnic misalignment does not override this subversive incentive. Second, foreign sponsors prefer to support weak rebel groups in attempts to wear down their opponent. Given the goal of depleting the enemy’s resources, not only do rebels incapable of victory receive external support, but it is also more desirable for the sponsors to support relatively weak groups. Third, in contrast to previous research, the article finds that cost- efficiency of supporting the rebels is neither necessary nor sufficient for explaining state support for insurgent groups. This challenges the conventional wisdom that support for rebels is a cheap substitute for costly interstate conflict. This article contributes to the study of both international and civil conflict in general and state support for insurgent groups more specifically. First, it complements existing theories of external support for rebel groups by illustrating a previously under-appreciated mechanism. The model not only recovers results consistent with existing empirical findings but also explains a significant number of cases unaccounted for by existing theories. Second, the article contributes to the study of third parties in international conflict by exploring non-state armed groups’ influence on international negotiation. Third, while the existing literature argues that external threat may be beneficial for state-building, this article joins recent scholarship suggesting that interstate rivalry is detrimental to domestic stability.

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33 State Support for Rebels and Interstate Bargaining Supplementary materials intended for online publication

CONTENTS

A Supplementary Information for the Baseline Model1

A.1 Proofs of the Baseline Results...... 1 A.2 Convexity of the Budget Function...... 5 A.3 General Function for the Cost of Arming...... 5

B Extension: Ideological and Ethnic Concerns7

C Additional Model Extensions 16

C.1 Support to Satisfy the Domestic Audience...... 16 C.2 Military Effectiveness...... 18 C.3 Rebel Group Entrance...... 19

D Supplementary Information: Empirical Implications 20

D.1 Iranian Support for the KDP: Alternative Explanations...... 20 D.2 Rebel Strength and External Support...... 21 A Supplementary Information for the Baseline Model A.1 Proofs of the Baseline Results Proof of Lemma2. The Sponsor’s problem is max u1(m1, s) = 1 φp f2 b2(mr + s), m1 s,m1 − s.t. θ m1 + s = b1.

Let m1 = (b1 s)/θ and substitute into the objective function. It is then transformed into the following single− variable maximization problem: max u1(s) = 1 φp f2 b2(mr + s), (b1 s)/θ s − − s.t. s [0, b1]. ∈ 1 I start with the case where f2 = 0. u1(s) = 1 for all s [0, b1) and u1(b1) = 1 2 φ < 1. It ∈ − immediately follows that the best response of the Sponsor to f2 = 0 is s(0) = [0, b1). Now consider the case where f2 = 0. The ﬁrst order condition yields 6 ∂ u φ f b2(mr + s) + (b1 s)b0 (mr + s) 1 2 2 . (A.1) = − 2 ∂ s − θ [(b1 s)/θ + f2 b2(mr + s)] − φ f2 Since θ is negative and the denominator is positive, only need to consider the sign of the − ∂ N(s) numerator N(s) b2(mr + s) + (b1 s)b20 (mr + s). ∂ s = (b1 s)b200(mr + s) 0 implies that the numerator weakly≡ increases in s. N −s b m b > 0, and− N s ≥b m b b m , ( ) s=b1 = 2( r + 1) ( ) s=0 = 2( r )+ 1 20 ( r ) the sign of which is indeterminate. | |

If b2(mr ) + b1 b20 (mr ) 0, then there exists a unique interior ˆs that solves b2(mr + s) + (b1 ≤ − s)b20 (mr + s) = 0. The second order condition is also satisﬁed at ˆs since

2 ∂ N(s) ∂ u1 φ f2 ∂ s [(b1 s)/θ + f2 b2(mr + s)] N(s) 2[ 1/θ + f2 b20 (mr + s)] 2 = − − × − 3 ∂ s ˆs − θ [(b1 s)/θ + f2 b2(mr + s)] ˆs − φ f2 (b1 s)b200(mr + s) < 0, = − 2 − θ [(b1 s)/θ + f2 b2(mr + s)] ˆs − where the second equality follows from N(ˆs) = 0. If, however, b2(mr ) + b1 b20 (mr ) > 0, then ∂ u1(s) ∂ s < 0 for all feasible s. The corner solution s = 0 is obtained. Proof of Lemma3. The proof looks at four separate cases. Case 1. s = b1. fc b2(mr +s) Since mr > 0, we have p fc b2 mr s , mr s . For all f2 0, 1 , the Target’s ( + ) + = fc b2(mr +s)+mr +s ( ] expected utility is ∈ (1 f2)b2(mr + b1) u2(f2) = φ + α − , (1 f2)b2(mr + b1) + mr + b1 − which strictly decreases in f2. Thus, for all f2 > 0, there exists f20 (0, f2) that is a better response. ∈ This rules out all f2 = 0 as best responses to m1 = 0. The Target’s expected utility for f2 = 0 is 6

1 φ b2(mr +s) u2 0 α . The difference between the expected utility from any positive f2 and ( ) = 2 + b2(mr +s)+mr +s that from f2 = 0 is

φ (1 f2)b2(mr + s) b2(mr + s) D(f2) = + α − , (A.2) 2 (1 f2)b2(mr + s) + mr + s − b2(mr + s) + mr + s − φ which decreases in f2. Since lim D(f2) = 2 > 0, and D(f2) continuously changes (decreases) in f2 0 → f2, there exists f2 > 0 such that D(f2) > 0, implying f2 = 0 is not a best response to s = b1. Thus, f2(s = b1) = . ; Now consider the three cases where m1 = 0. 2 φ m1(mr +s) [b26 (mr +s)+m1] Case 2. m1 0 and 2 , . = α [b2(mr +s)+mr +s] m1(mr +s) For this6 case, the Target∈ ’s optimization problem is

f2 b2(mr + s) fc b2(mr + s) max u2(f2, fc) = φ + α f2,fc f2 b2(mr + s) + m1 fc b2(mr + s) + mr + s s.t. f2 + fc = 1.

As the Target’s objective function is strictly concave in (f2, fc) and the feasible set is convex, the first-order conditions are both necessary and sufficient for an interior global maximum. The first-order conditions imply that at the maximum

φm α m s 1 ( r + ) . (A.3) 2 = 2 [m1 + f2 b2(mr + s)] [fc b2(mr + s) + mr + s] Combining equation (A.3) with the Target’s budget constraint yields:

Ç φm1 b mr s mr s m α(mr +s) [ 2( + ) + + ] 1 f2 = − , (A.4) Ç φm1 1 b mr s α(mr +s) + 2( + ) which is contained in the interval [0, 1] over the parameter region given in this case. φ m1(mr +s) Case 3. m1 0 and < 2 . = α [b2(mr +s)+mr +s] 6 φ m1(mr +s) φ α(mr +s) Note that < 2 implies < 2 , which is equivalent with α [b2(mr +s)+mr +s] m1 [b2(mr +s)+mr +s]

φm1 α(mr + s) < . m f b m s 2 f b m s m s 2 [ 1 + 2 2( r + )] f2=0 [ c 2( r + ) + r + ] fc =1 It follows that the Target’s marginal benefit from allocating its first unit of resources to fight the international enemy is less than the marginal benefit from the last unit of resources spent on fighting the domestic insurgents. Thus, it is best for the Target to choose f2 = 0. 2 φ [b2(mr +s)+m1] Case 4. m1 0 and > . = α m1(mr +s) 6 φm1 α(mr +s) The same logic applies to this case, where 2 > 2 . The [m1+f2 b2(mr +s)] f2=1 [fc b2(mr +s)+mr +s] fc =0 marginal return from spending the last unit of resources on confronting the Sponsor exceeds the marginal return from the first unit allocated to counter-insurgency. Thus, the Target optimally chooses f2 = 1 and fc = 0.

2 The following Proposition A.1 provides a complete equilibrium characterization and formally ˜ states the exposition of Proposition1 in the main text. Deﬁne f2(s) as

Ç φm1 m s [b2(mr + s) + mr + s] m1 ˜ α( r + ) f2(s) − , Ç φm1 ≡ 1 b mr s α(mr +s) + 2( + )

b1 s where m1 = θ− . Remember that ˆs solves b2(mr + s) + (b1 s)b20 (mr + s) = 0. − Proposition A.1. A subgame perfect equilibrium exists. More speciﬁcally, the following holds in equilibrium. ˜ • If b2(mr ) + b1 b20 (mr ) 0, then s∗ = 0, and f2∗ = min max 0, f2(0) , 1 . ≥ { { } } ˜ • If b2(mr ) + b1 b20 (mr ) < 0, then s∗ = ˆs, and f2∗ = min max 0, f2(ˆs) , 1 . { { } } φ (b1 s∗)(mr +s∗) • Any s∗ (0, b1) and f2∗ = 0 is an equilibrium if α < θ b m− s m s 2 . ∈ [ 2( r + ∗)+ r + ∗] Proof. As discussed in the main text, for a mutual best response to exist, I need to show that (1)

under all parameter conﬁgurations, there exists a s0 [0, b1) such that for all f2 [0, 1], s0 is a ∈ ∈ best response; and (2) for all s [0, b1), the Target has a unique best response. The second claim follows directly∈ from Lemma3. The rest of the proof proceeds to show that the ﬁrst claim is true and together they imply equilibrium existence.

Claim 1. There exists a s0 [0, b1) such that for all f2 [0, 1], s0 is a best response. ∈ ∈ Proof of Claim 1. The proof looks at two cases.

Case 1. b2(mr ) + b1 b20 (mr ) < 0 If f2 = 0, condition (1c) of Lemma2 is satisﬁed, thus s0 = ˆs [0, b1). If f2 = 0, we have 6 ∈ s(0) = [0, b1), which contains ˆs. Thus, s0 = ˆs in this case. Case 2. b2(mr ) + b1 b20 (mr ) 0 ≥ If f2 = 0, condition (1b) of Lemma2 implies a corner solution s0 = 0. If f2 = 0, we have 6 s(0) = [0, b1), which contains 0. Thus, s0 = 0 in this case.

The second claim indicates that the Target has a unique best response to s0, denoted f ∗(s0). 2 Claim 1 implies that s0 is also a best response to f2∗. s0, f2∗(s0) constitutes a mutual best response, and thus an equilibrium exists. The characterizations directly follow from Lemma2 and Lemma 3.

Proposition A.2. As θ increases, the range of multiple equilibria shrinks. Proof. The condition for multiple equilibria is

φ b s m s /θ α b s m s < ( 1 )( r + ) θ < ( 1 )( r + ) , − 2 − 2 α [b2(mr + s) + mr + s] ⇒ φ · [b2(mr + s) + mr + s] which is more difﬁcult to satisfy as θ increases, indicating that the range of multiple equilibrium shrinks.

Proposition A.3. As b mr s b mr goes to 0, equilibrium support s also goes to 0. 20 ( + ) s=0 = 20 ( ) ∗

Proof. Proposition A.3 restates Remark1(c) formally. The logic is straightforward. As b20 (mr ) goes to 0, b20 (mr ) < b2(mr )/b1 always holds and the corner solution s∗ = 0 is obtained. An − a example of this is b mr s b , where a > 0 is a parameter. A larger a corresponds to a 2( + ) = 2 a+mr +s ﬂatter b2(mr + s). As a , b20 (mr ) 0. → ∞ →

Proposition A.4. If b2( ) is log-convex, then there exists b1 > b1 > 0 such that for b1 (b1, b1), · ∈ there exists mr > 0 such that s∗ = 0 if mr mr , and s∗ > 0 if mr < mr . If, in addition, b20 ( ) is ≥ − · log-convex, then s∗ strictly decreases in mr for all s∗ > 0. Proof. Proposition A.4 formally restates Proposition2 in the main text. The proof is comprised of two parts. I will look at them sequentially.

Part (1). There exists b1 > b1 > 0 such that for b1 (b1, b1), there exists mr > 0 such that s∗ = 0 ∈ if mr mr , and s∗ > 0 if mr < mr . ≥ f m f m s b s b m s b m s b b m First, I show that ( r ) ( r , ) s=0 = ( 1 ) 20 ( r + ) + 2( r + ) s=0 = 1 20 ( r ) + ≡ | − b20 (mr ) | b2 mr 0 has at most one solution. Since b2 is log-convex, is strictly increasing, which ( ) = ( ) b2(mr ) b1 b20 (mr ) · b1 b20 (mr ) means that − is strictly decreasing. Let − d mr , where d mr > 0 is a strictly b2(mr ) b2(mr ) ( ) ( ) decreasing function of mr . Rearrange and we have that for≡ all mr ,

f (mr ) = b1 b20 (mr ) + b2(mr ) = [1 d(mr )]b2(mr ). (A.5) − The RHS (and therefore f (mr )) is positive if d(mr ) > 1 and negative if d(mr ) < 1. Since d(mr ) is a strictly decreasing function of mr , d(mr ) = 1 has at most one solution. b20 (mr ) Second, I show that for b1 b , b1 , there exists exactly one solution. Since − is positive ( 1 ) b2(mr ) ∈ b20 (mr ) b20 (0) and strictly decreasing, it must converge as m . Let lim − r 0 and − r b2 mr b2 0 mr ( ) ( ) 1 1 → ∞ →∞ ≡ ≥ ≡ r0 > r. Deﬁne b1 r , and b1 lim γ . Thus for all b1 (b1, b1), we have d(mr = 0) > 1 and 0 γ r ≡ ≡ → ∈ lim d(mr ) < 1, and as a result exactly one solution exists, denoted mr . The statement follows mr →∞ directly from the monotonicity of d(mr ). ∂ s Part (2). ∗ < 0 for all s > 0. ∂ mr ∗ An interior solution is characterized by (b1 s∗)b20 (mr + s∗) + b2(mr + s∗) = 0. Using the Implicit function theorem, we have −

b s b m s b m s ∂ s∗ ( 1 ∗) 200( r + ∗) + 20 ( r + ∗) − . ∂ m = b s b m s r − ( 1 ∗) 200( r + ∗) − The denominator is positive. Let N mr , s∗(mr ) (b1 s∗)b200(mr + s∗) + b20 (mr + s∗). Now ≡ − I show N mr , s∗(mr ) > 0 for all mr < mr . From Part (1), we know s∗(mr ) = 0, implying ∂ f (mr ) N mr , s mr b1 b mr b mr . Note that this is , which is positive as shown in ∗( ) = 200( ) + 20 ( ) ∂ mr mr

4 Part (1). Take the derivative of N mr , s∗(mr ) with respect to mr and we have

∂ Nm , s m r ∗( r ) ∂ s∗ ∂ s∗ ∂ s∗ = (b1 s∗)b2000(mr + s∗)(1 + ) b200(mr + s∗) + b200(mr + s∗)(1 + ) ∂ mr − ∂ mr − ∂ mr ∂ mr ∂ s∗ = (b1 s∗)b2000(mr + s∗)(1 + ) + b200(mr + s∗) − ∂ mr ∂ s b0 (mr + s∗) Plug in ∗ . b s b m s 2 b m s ∂ m = ( 1 ∗) 2000( r + ∗)( b s b m s ) + 200( r + ∗) r − −( 1 ∗) 200( r + ∗) 2 − b200(mr + s∗) b2000(mr + s∗)b20 (mr + s∗) − . = b m s 200( r + ∗)

The denominator is positive. Log-convexity of b20 ( ) implies − · 2 2 ∂ log b20 (y) b000(y)b0 (y) [b00(y)] 2 2 2 > 0, −2 = − 2 ∂ y [b20 (y)] ∂ N mr ,s (mr ) which means b y b y b y 2 > 0 for all y. Therefore ∗ < 0 for all m < m . 2000( ) 20 ( ) [ 200( )] ∂ mr r r − ∂ s Together with N m , s m > 0, we have N m , s m > 0 for all m < m . Therefore ∗ < 0 r ∗( r ) r ∗( r ) r r ∂ mr for all mr < mr . A.2 Convexity of the Budget Function I show that the convexity of the budget function can be derived from the Target-rebel contest over

resource control. Suppose they compete over total resource b2, and the proportion earned by each φ(Mi ) party is pi = φ M φ M if φ(Mi) + φ(Mj) > 0, and pi = 1/2 otherwise, with i, j G, R . Mi ( i )+ ( j ) ∈ { } is i’s military strength, and φ(Mi) measures the impact of Mi in the contest. If φ() is increasing and concave —a common assumption in most contest models, then the target’s obtains resource φ(MG ) b , which is decreasing and convex in MR. In the context of rebel support, MR can be 2 φ(MG )+φ(MR) naturally interpreted as post-support rebel strength mr + s. A.3 General Function for the Cost of Arming This part shows that with a general functional form for the cost of arming the military, increasing cost of building the military does not always drive the potential sponsor to provide higher level of support. Now assume the budget constraint of the Sponsor is θκ(m1) + s = b1, where κ(m1) : m R+ R+ is an increasing, convex function of 1 common to all potential sponsors, and θ > 0 parameterizes→ the cost of arming the military. So in line with the description, the assumptions on

κ(m1) are κ0( ) > 0 and κ00( ) 0. Let m1 satisfy θκ(m1) = b1. The FOC condition characterizing · · ≥ the interior solution then becomes f (m1∗, θ) = b2(mr + b1 θκ(m1∗)) + θ m1∗κ0(m1∗)b20 (mr + b1 − − θκ(m1∗)) = 0. One can also verify that the SOC is satisﬁed. Apply the Implicit Function Theorem

5 and we have ∂ f m , θ /∂ θ ∂ s∗ ∂ ( 1∗ ) = [b1 θκ(m1∗)] = κ(m1∗) + θκ0(m1∗) ∂ θ ∂ θ − − ∂ f (m∗1, θ)/∂ m∗1 + z }| { = b20 (mr + b1 θκ(m1∗)) − − × +/ − z }| 2 { m1∗κ(m1∗)κ00(m1∗) m1∗[κ0(m1∗)] + κ0(m1∗)κ(m1∗) − 2 , m1∗κ00(m1∗)b20 (mr + b1 θκ(m1∗)) θ m1∗[κ0(m1∗)] b200(mr + b1 θκ(m1∗)) | − −{z − } − ρ the sign of which is indeterminate. For a general class of cost functions θκ(m1) = θ m1 with ∂ s∗ ρ ρ 2 ρ 1 2 ρ > 1, ∂ θ = 0 because the numerator is m1∗θ(m1∗) θρ(ρ 1)(m1∗) − m1∗[θρ(m1∗) − ] + ρ 1 ρ − − θρ(m1∗) − θ(m1∗) = 0. The condition for no support is f (m1, θ) = b2(mr ) + θ m1κ0(m1)b20 (mr ) < 0. Note that m1 = 1 κ− (b1/θ), so ∂ m 1 b b 1 1 1 . = 1 2 = 2 ∂ θ κ κ b /θ θ θ κ m1 0 − ( 1 ) · − − 0( ) And ∂ f (m1, θ) ∂ m1 ∂ m1 = b20 (mr ) m1κ0(m1) + θ κ0(m1) + θ m1κ00(m1) ∂ θ · ∂ θ ∂ θ m κ m b m m κ m κ m κ m 1 00( 1) = 20 ( r ) 1 0( 1) ( 1) ( 1) κ m · − − 0( 1) b0 (mr ) 2 m κ m κ m m κ m 2 κ m κ m , = κ m 1 ( 1) 00( 1) 1[ 0( 1)] + ( 1) 0( 1) − 0( 1) · | − {z } | {z } +/ + − the sign of which is indeterminate. Note that the ambiguous part is the same numerator evaluated ∂ f (m1,θ) at m1. Thus for a general class of cost functions discussed above, ∂ θ = 0, indicating that the condition separating positive level of support and no support is independent of θ. Similarly, the condition for multiple equilibria is

α κ 1 b s m s θ < − ( 1 )( r + ) , − 2 φ · [b2(mr + s) + mr + s] which is more difﬁcult to satisfy as θ increases, indicating that the range of multiple equilibrium shrinks.

6 B Extension: Ideological and Ethnic Concerns

Lemma B.1. Suppose V > 0. Given any (f2, fc) such that f2 + fc = 1, the Sponsor’s best response is

 if f2 = 0 (B.1a)  ;  ˜s if f2 = 0 and h(s) 0 h(s) (B.1b) s(f2) = 6 ≥ ≥  0 if f2 = 0 and h(s) < 0 (B.1c)  6 b1 if f2 = 0 and h(s) > 0 (B.1d) 6 where

φ f b2(mr + s) + (b1 s)b0 (mr + s) b2(mr + s) (mr + s)b0 (mr + s) h s 2 2 V f 2 , (B.2) ( ) = − 2 + c − 2 − θ [(b1 s)/θ + f2 b2(mr + s)] [fc b2(mr + s) + mr + s] − h(s) max h(s), h(s) min h(s), and ˜s solves h(s) = 0. s [0,b1] s [0,b1] ≡ ∈ ≡ ∈ Proof. The Sponsor’s problem is max u1(s) = 1 φp f2 b2(mr + s), (b1 s)/θ + V p mr + s, fc b2(mr + s) . (B.3) s [0,b1] ∈ − −

Case 1. f2 = 0. For all s [0, b1), the Sponsor’s expected utility is ∈ mr + s u1(s) = 1 + V . b2(mr + s) + mr + s The third part strict increases in s, and hence there is always a better response. The Sponsor’s φ mr +b1 expected utility from s = b1 is u1(b1) = 1 2 + V b m b m b . The difference between the − 2( r + 1)+ r + 1 expected utility from any s slightly smaller than b1 and that from s = b1 is

φ mr + s mr + b1 D(s) = + V , 2 b2(mr + s) + mr + s − b2(mr + b1) + mr + b1

φ which strictly increases in s. Since lim D(s) = 2 > 0, and D(s) continuously changes in s, there s b1 → exists s close enough to b1 such that D(s) > 0, implying s = b1 is not a best response. Thus s(f2 = 0) = . ; Case 2. f2 = 0. The objective function u1(s) is continuous in s when f2 = 0. Since [0, b1] is a closed and6 bounded interval in R, the Extreme Value Theorem guarantees6 the existence of a solution. Taking the ﬁrst partial derivative with respect to s yields that

N1(s) N2(s) z ≡}| { z ≡}| { ∂ u s φ f b2(mr + s) + (b1 s)b0 (mr + s) b2(mr + s) (mr + s)b0 (mr + s) h s 1( ) 2 2 V f 2 . ( ) = − 2 + c − 2 ≡ ∂ s − θ [(b1 s)/θ + f2 b2(mr + s)] [fc b2(mr + s) + mr + s] | − {z } | {z } D1(s) D2(s) ≡ ≡ (B.4)

7 With f2 = 0, h(s) is continuous in s, and since [0, b1] is a closed interval, h(s) has both a maximum 6 and minimum value by the Extreme Value Theorem. Define them as h(s) and h(s) respectively. If h(s) < 0, u1(s) strictly decreases in s, and s(f2) = 0. If h(s) > 0, u1(s) strictly increases in s, and s(f2) = b1. If h(s) 0 h(s), the solution ˜s is interior and can be characterized by the first order condition h(s) = 0.≥ The≥ second order condition is satisfied at ˜s. To simplify notations, I use terms defined in equation (B.4). Evaluating the SOC at ˜s yields

2 ∂ u1(s) φ f2 N10(s) φ f2 2N1(s)D10 (s) N20(s) 2N2(s)D20 (s) V f V f . (B.5) 2 ˜s = 2 + 3 + c 2 c 3 ∂ s − θ [D1(s)] θ [D1(s)] [D2(s)] − [D2(s)] ˜s | {z } | {z } | {z } | {z } 1 2 3 4

To sign equation (B.5), ﬁrst notice that N2(s), D1(s) and D2(s) are all strictly positive. In addition, N10(s) = (b1 s)b200(mr +s) 0 with equality obtained only at s = b1 and N20(s) = (mr +s)b200(mr + − ≥φ f2 N1(s) N2(s) − s < 0. The FOC tells that 2 V fc 2 , and hence N1 ˜s has the same sign as V for ) θ [D1(s)] ˜s = [D2(s)] ˜s ( ) all fc = 0. For6 V > 0, 1 and 3 in equation (B.5) are negative. After substituting in the FOC, 2 4 becomes −

φ f2 N1 s 2 ( ) D ˜s D ˜s D ˜s D ˜s 3 [ 10 ( ) 2( ) 20 ( ) 1( )] θ [D1(s)] D2(s) ˜s × −

φ f2 N1 s fc b1 mr 2 ( ) N ˜s + f b m ˜s m ˜s f b m ˜s = 3 [ 1( ) 2 2( r + ) + ( r + ) 2 20 ( r + )] θ [D1(s)] D2(s) ˜s × − θ − θ − 0, ≤ where the equality holds only at ˜s = ˆs0, the solution for f2 = 1. This implies that equation (B.5) is negative and the SOC holds at ˜s. This also implies that the solution is unique for V > 0. h(s) continuously changes in s, and the SOC holds for any interior ˜s. If h(s) crosses 0 more than once, there cannot exist two consecutive maximizers without a minimizer in between, at which the SOC would fail.

The case of V < 0 is more complicated and equation (B.5) cannot be signed without further a assumptions. In fact, with b mr s b , a > 0, for a range of parameters, equation (B.5) 2( + ) = 2 a+mr +s has more than one solution, and multiple local maximizers exist. Under some degeneracy, there are two global maximizers. Simulations reveal two possible scenarios. First, one maximizer is 0 and the other can be characterized by the FOC. Second, both maximizers are interior. Under both scenarios, although the best response correspondence is upper hemicontinuous, it is not convex- valued at the point where two maximizers exist. As a result, Kakutani’s ﬁxed-point theorem does not apply and equilibrium existence might fail. In Figure B.1, the blue dots correspond to Sponsor’s

best response to f2, and one can clearly see a discontinuous jump. An equilibrium exists for the set

of parameters presented in Figure B.1, as marked by (s∗, f2∗) next to the intersection of two lines. However, if the discontinuous jump happens around the point they are supposed to intersect, an equilibrium might not exist. The next lemma establishes bounds of s(f2) for V > 0.

8 Figure B.1: Equilibrium in the extension with V < 0

Notes: The ﬁgure uses b m s b a with a 1, m 0.2 and 2( r + ) = 2 a+mr +s = r = b2 = 20. Other parameter values are b1 = 9, φ = 0.8, θ = 1.5, α = 1, and V = 0.2. −

Lemma B.2. Let ˆs0 be the maximizer when V = 0, and let s(f2, V ) be the best response to f2 for V = 0. For all V > 0 and f2 = 0, s(f2, V ) is continuous in V and f2, s(1, V ) = ˆs0, and 6 6 lims(f2, V ) = b1. f2 0 →

Proof. Continuity of s(f2, V ) is readily ascertainable from Lemma B.1. If f2 = 1, fc = 0, and the FOC would be equivalent to the baseline, where the best response is ˆs0. If f2 approaches 0, first observe that for f2 close enough to 0, s(f2, V ) = b1. Suppose towards a contradiction ˜ 6 ˜ that s(f2, V ) = b1 for some f2 sufficiently close to 0. However, for f2 sufficiently close to 0, φ 1 N2(s) lim h s lim V fc 2 < 0, implying that the best response is not b1. Now ˜ ( ) = f m˜ θ f2 b2(mr +s) + [D2(s)] f2 f2 2 2 − → → ˜ φ f2 N1(˜s) N2(˜s) suppose lims f2, V b1 < b1. Then limh s lim limh s lim 2 V fc 2 f 0 ( ) = f 0 ( ) = f 0 ˜ ( ) = f 0 θ [D1(˜s)] + [D2(˜s)] = 2 2 2 s b1 2 − → → → → → N2(˜s) ˜ V 2 > 0, implying that the best response is not b1. Thus lims f2, V b1. [b2(mr +˜s)+mr +˜s] ( ) = f2 0 → Lemma B.3 characterizes the bound of f2(s) as s approaches b1. Since V does not enter the Target’s utility, this result holds for the baseline and the extension.

Lemma B.3. lim f2(s) = 0. s b1 → 2 m1(mr +s) [b2(mr +s)+m1] Proof. As s b1, m1 0, 2 0 < φ/α, and > φ/α. So f2 s [b2(mr +s)+mr +s] m1(mr +s) ( ) must be interior→ and characterized→ by equation→ (A.4). It is readily ascertainable→ ∞ that the RHS of

equation (A.4) goes to 0 as s b1. → The next lemma characterizes the slopes of both players’ best response at the boundary.

Lemma B.4. Assume V > 0. lim ∂ s(f2,V ) , and lim ∂ f2(s) . ∂ f2 = ∂ s = f2 0 s b1 → −∞ → −∞

9 Proof. Lemma B.2 has shown that as f2 0, s(f2, V ) is interior and continuous in f2. Deﬁne → h(s, f2) = h(s) and apply the Implicit Function Theorem:

∂ s(f2, V ) ∂ ˜s ∂ h(˜s, f2)/∂ f2 = = ∂ f2 ∂ f2 − ∂ h(˜s, f2)/∂ s which has the same sign as ∂ h(s,f2) because the denominator is negative by the SOC. ∂ f2 Evaluating the numerator yields

∂ h ˜s, f2 φ N1 s φ 2f2N1 s b2 mr s N2 s 2fc N2 s b2 mr s ( ) ( ) ( ) ( + ) V ( ) V ( ) ( + ) = 2 + 3 2 + 3 ∂ f2 − θ [D1(f2, s)] θ [D1(f2, s)] − [D2(fc, s)] [D2(fc, s)] ˜s φ f2 N1(s) D1(f2, s)D2(fc, s) = 3 − θ [D1(f2, s)] D2(fc, s) × f2 D1(f2, s)D2(fc, s) + 2b2(mr + s)[D1(f2, s) + D2(fc, s)] . fc − ˜s Evaluating the denominator yields

N s D f , s D f , s ∂ h(˜s, f2) φ f2 N1(s) 10( ) 1( 2 ) 2( c ) = 3 ∂ s − θ [D1(f2, s)] D2(fc, s) × N1(s) N20(s)D1(f2, s)D2(fc, s) 2D10 (f2, s)D2(fc, s) 2D20 (fc, s)D1(f2, s) . − N2(s) − − ˜s

Plug them back and rearrange. Since lim D1(f2, ˜s) = 0, and lim D2(fc, ˜s), lim N2(˜s) and lim N1(˜s) f2 0 f2 0 f2 0 f2 0 are positive and ﬁnite, we have → → → →

b1 ˜s lim − b m ˜s [ θ f2 2( r + )] ∂ ˜s f2 0 − b1 ˜s lim = → = lim − + limθ b2(mr + ˜s) f2 0 ∂ f2 − lim1/θ −f2 0 f2 f2 0 → f2 0 → → → ∂ ˜s = lim + θ b2(mr + b1), f2 0 ∂ f2 →

where the third equality is obtained by applying L’Hopital’s Rule. Since θ b2(mr + b1) = 0, lim ∂ ˜s or . We can easily rule out . Therefore, lim ∂ ˜s . 6 ∂ f2 = ∂ f2 = f2 0 f2 0 → ∞ −∞ ∞ → −∞ Now turn to the Target’s best response. First, if s b1, f2(s) is interior and characterized by → equation (A.4). Denote the numerator N3(s) and the denominator D3(s). Observe that lim N3(s) = s b1 → 0, and lim D3(s) = b2(mr + b1) > 0. Therefore we have s b1 →

∂ N3(s) ∂ D3(s) ∂ f2 s s D3(s) N3(s) s 1 ∂ N2 s lim ( ) lim ∂ ∂ lim ( ) = × − 2 × = s b1 ∂ s s b1 [D3(s)] b2(mr + b1) s b1 ∂ s → → § → ª 1 Æ 1/2 = 1/θ + φ/αθ lim [(b1 s)(mr + s)]− [b2(mr + s) + mr + s] b2 mr b1 s b1 ( + ) − → − = , −∞ 10 1/2 where the last equality follows from lim[(b1 s)(mr + s)]− = . s b1 → − −∞ Proposition B.1. A subgame perfect equilibrium exists for all V > 0.

Proof. Previous lemmas establish the Target’s best response function f2 : (0, 1] [0, b1) and → the Sponsor’s best response function s : [0, b1) (0, 1] for V > 0. Define T = (0, 1] [0, b1), → × t = (f2, s) T, and f (t) = f (f2, s) = f2(s) s(f2). A strategy profile t∗ T is an equilibrium if ∈ × ∈ and only if f : T T admits a fixed point such that t∗ = f (t∗). Since the domain is not compact, Brower’s fixed-point→ theorem is not directly applicable. With additional mathematical structure, equilibrium existence can be established. With the norm topology, T is a non-empty subset in a locally convex Hausdorff topological vector space E. Since s(f2) and f2(s) are both continuous, f (t) is continuous. I now claim that there exists ε > 0 and C [ε, 1] [0, b1 ε] T such that f (t) C for all t C. Suppose not towards a contradiction.≡ Then for× all ε >−0, we⊂ have either ⊂ ∈ s(ε) > b1 ε or f2(b1 ε) < ε. If the former is true, then − − ∂ s f , V s ε δ s ε lim ( 2 ) lim lim ( + ) ( ) = 0 − f2 0 ∂ f2 ε δ 0 δ → → → s(ε + δ) s(ε) Continuity = lim lim δ 0 ε 0 δ − → → s(δ) b1 b1 δ b1 Lemma B.2 = lim > lim = 1, δ 0 δ− δ 0 − δ − → → − which contradicts Lemma B.4. Similarly, if f2(b1 ε) < ε is true, then − ∂ f2(s) f2(b1 ε) f2(b1 ε δ) lim = lim lim − − − − s b1 ∂ s ε 0 δ 0 δ → → → f2(b1 ε) f2(b1 ε δ) Continuity = lim lim δ 0 ε 0 − − δ − − → → 0 f2(b1 δ) 0 δ Lemma B.3 = lim > lim = 1, δ 0 − δ − δ 0 −δ → → − which contradicts Lemma B.4. C T is obviously compact and convex. Corollary 1 of Tian (1991) applies and a fixed point exists.⊂ Figure B.2 elaborates it graphically. The solid blue line plots the Sponsor’s best response s(f2) and the dashed red line plots the Target’ best response f2(s). The hollow circle at (b1, 0) represents that s(0) = and f2(b1) = due to the discontinuity in each state’s utility at this ; ; point. As a consequence the closed intervals [0, b1] and [0, 1] cannot be proper domains for the best response correspondences so that Brower’s fixed-point theorem is not directly applicable. However, with uniqueness of best response for any given strategy of the other player, continuity of the best response functions, and the slopes of the functions at the boundary, we know the solid blue line must approach (b1, 0) from the left and the dashed red line must approach the same point from right above, so there must exist a region where the solid blue line is below the dashed red line. In addition, the other end point of the dashed red line lies between 0 and 1 on the y axis, and s(f2) ˆs0 as f2 1, implying that the other end point of the solid blue line is above or exactly on the dashed→ red line.→ As a result, the two lines must cross each other at some point, and the point of crossing is an equilibrium.

11 Figure B.2: Equilibria in the extension with V > 0

Notes: The ﬁgure uses b m s b a with a m 1 and 2( r + ) = 2 a+mr +s = r = b2 = 12. Other parameter values are b1 = 10, φ = 0.8, θ = 1, and α = 1.

Denote the equilibrium (sV∗ , m2,∗ V ). It is readily ascertainable that sV∗ [ˆs0, b1). ∈ Due to problems discussed early, equilibrium existence might become a problem for V < 0. a For the purpose of this article, I show that when b mr s b , for V sufﬁciently close 2( + ) = 2 a+mr +s to 0, an interior Nash equilibrium exists. The more general case is left for future research. Note a that for the rest of Section ??, I assume b mr s b , with a > 0. If ˆs 0, it is 2( + ) = 2 a+mr +s 0 = straightforward to see that s(f2) = 0 for all V < 0. The following results only consider the case where ˆs0 > 0. It is also easy to see that for V < 0, s(f2) [0, ˆs0]. ∈ Fix f2 (0, 1), s, and other parameters, deﬁne V 0 V h(s) = 0 . The explicit expression for ∈ ≡ { | } V 0 is φ f N s D s 2 V 2 1( )[ 2( )] , (B.6) 0 = 2 θ fc N2(s)[D1(s)]

which is negative because N1(s) < 0 for s [0, ˆs0]. Since ∂ h(s)/∂ V > 0, h(s) < 0 for all V < V 0, ∈ and h(s) > 0 for all V > V 0. In the latter case, s is ruled out for a maximizer. As f2 0, V 0 0, → → and as f2 1, V 0 . → → −∞ Fix f2 (0, 1), s, and other parameters with , deﬁne V 00 as ∈  φ f A s Á f B s  2 ( ) c ( ) if B s < 0 (B.7a) 3 3 ( ) V 00 −θ[D1(s)] [D2(s)] ≡  if B(s) 0, (B.7b) −∞ ≥

where A(s) = N10(s)D1(s) 2N1(s)D10 (s), and B(s) = N20(s)D2(s) 2N2(s)D20 (s). V 00 is the value − − of V such that ∂ h(s)/∂ s = 0 whenever feasible. The following results make sure that V 00 is well deﬁned.

Lemma B.5. A(s) > 0 for all s [0, ˆs0]. ∈

12 Proof. Expand A(s) and we have

A(s) = N10(s)D1(s) 2N1(s)D10 (s) − = (b1 s)b200[(b1 s)/θ + f2 b2(mr + s)] 2[b2(mr + s) + (b1 s)b20 (mr + s)][ 1/θ + f2 b20 (mr + s)] − 2 − − − − = (b1 s) b200(mr + s) + 2b2(mr + s)/θ + 2(b1 s)b20 (mr + s)/θ 2f2 b20 (mr + s)b2(mr + s) − − 2 − +f2(b1 s)b2(mr + s)b200(mr + s) 2f2(b1 s)[b20 (mr + s)] −2 − − = (b1 s) b200(mr + s) + 2b2(mr + s)/θ + 2(b1 s)b20 (mr + s)/θ 2f2 b20 (mr + s)b2(mr + s) − −b m s − b m s 200( r + ) 2 20 ( r + ) = 2b2(mr + s)/θ 2f2 b20 (mr + s)b2(mr + s) + (b1 s) + 2 (b1 s) − θ − b200(mr + s) − 2 2 b200(mr + s) b20 (mr + s) [b20 (mr + s)] = 2b2(mr + s)/θ 2f2 b20 (mr + s)b2(mr + s) + b1 s + , − θ − b200(mr + s) − θ b200(mr + s)

2 where the fourth equality follows from b2(mr + s)b200(mr + s) = 2[b20 (mr + s)] . The last term is a2 2 b m s 4 [ 20 ( r + )] 1 [a+mr +s] 1 a b2(mr +s) θ b m s = θ 2a = 2θ a m s = − 2θ . Substitute back and we have 200( r + ) 3 + r + − − [a+mr +s] − 2 b200(mr + s) b20 (mr + s) A(s) = 3b2(mr + s)/2θ 2f2 b20 (mr + s)b2(mr + s) + b1 s + > 0. − θ − b200(mr + s)

Lemma B.5 makes sure that when B(s) < 0, V 00 < 0. As B(s) < 0 , V 00 . This, → −∞ → −∞ together with V 00 = if B(s) 0, ensures that V 00 is continuous in s. It is straightforward to see −∞ ≥ that when B(s) < 0, as f2 0, V 00 0, and as f2 1, V 00 . → → †→ → −∞ † Fix f2 (0, 1) and other parameters, deﬁne V V h(0) = 0 . The explicit expression for V is ∈ ≡ { | } 2 † φ f2 N1(s)[D2(s)] V = . (B.8) θ f N s D s 2 c 2( )[ 1( )] s=0 † † N1(s = 0) < 0 ensures that V < 0 is well-deﬁned. For all V > V , h(0) > 0 and therefore the † corner solution s = 0 is ruled out as a maximizer. The next result shows that for all V > V , as f2 0, V 0 > V 00 evaluated at the interior solution. As a result, any V > V 0 will also be larger than → V 00. This means that h(s) > 0, and hence s cannot be a maximizer.

† Lemma B.6. For all V > V , as f2 0, V 0 ˜s > V 00 ˜s. → | | Proof. Since V 00 and V 0 are both negative, this is equivalent with lim V 0/V 00 ˜s < 1. Plug in the f2 0 expressions for both terms and we need to show that →

N1(s)B(s)D1(s) lim < 1 lim N2 s D2 s A s N1 s D1 s B s > 0 f 0 f 0 ( ) ( ) ( ) ( ) ( ) ( ) ˜s 2 N2(s)A(s)D(s) ˜s ⇐⇒ 2 − → → lim N2(s)D2(s)[N10(s)D1(s) 2N1(s)D10 (s)] N1(s)D1(s)[N20(s)D2(s) 2N2(s)D20 (s)] ˜s > 0 ⇐⇒ f2 0 − − − → lim D1(s)D2(s)[N10(s)N2(s) N20(s)N1(s)] 2N1(s)N2(s)[D10 (s)D2(s) D1(s)D20 (s)] ˜s > 0 f2 0 ⇐⇒ → − − −

13 which always holds at an interior ˜s as f2 0 for the following reason. → φ f2 N1(s) N2(s) Rewrite equation (B.5) by substituting in 2 V fc 2 , and we have θ [D1(s)] ˜s = [D2(s)] ˜s

2 ∂ u1(s) φ f2 N10(s) φ f2 N20(s)N1(s) φ f2 2N1(s)D10 (s) φ f2 2N1(s)D20 (s) 2 ˜s = 2 + + 2 + 3 2 ∂ s − θ [D1(s)] θ [D2(s)] N2(s) θ [D1(s)] − θ [D1(s)] D2(s) ˜s | {z } | {z } | {z } | {z } 1 3 2 4

φ f2 2φ f2 N1 s N s N s N s N s ( ) D s D s D s D s = 2 [ 10( ) 2( ) 20( ) 1( )] + 3 [ 10 ( ) 2( ) 1( ) 20 ( )] ˜s −θ[D1(s)] N2(s) − θ [D1(s)] D2(s) − φ f 2 D s D s N s N s N s N s = 3 1( ) 2( )[ 10( ) 2( ) 20( ) 1( )] −θ[D1(s)] D2(s)N2(s) −

2N1(s)N2(s)[D10 (s)D2(s) D1(s)D20 (s)] ˜s. − − † For all V > V , as f2 0, an interior local maximizer ˜s is obtained and the SOC must be satisﬁed. 2 ∂ u1(s)→ This means that ∂ s2 ˜s < 0 for f2 sufﬁciently close to 0, which implies that lim D1(s)D2(s)[N10(s)N2(s) N20(s)N1(s)] 2N1(s)N2(s)[D10 (s)D2(s) D1(s)D20 (s)] ˜s > 0 f2 0 → − − − as required.

Proposition B.2 establishes that for V sufﬁciently close to 0, the Sponsor’s best response is a

continuous function of f2. Brewer’s ﬁxed-point theorem applies and an equilibrium exists. Propo- sition B.2 formally restates Proposition ?? in the main text.

Proposition B.2. Suppose ˆs0 > 0 and f2(s = 0) > 0. There exists V < 0 such that for all 0 > V > V , an equilibrium exists, and the equilibrium level of support s∗∗ > 0.

Proof. Lemma B.6 means that there exists m˜ 2 > 0 such that for all f2 < m˜ 2, V 0 ˜s > V 00 ˜s. Let | | E(f2) f2 V 0 ˜s V 00 ˜s . As a result of Lemma B.6, E(f2) is a closed and bounded set and ≡ { | | ≤ | } † 0 / E(f2). Deﬁne V max V , max V 00 . Since V 00 is continuous in s and f2 when f2 = 0, and ∈ ≡ { f2 E(f2) } 6 s ∈[0,ˆs0] ∈ the domain is compact, V exists by the Extreme Value Theorem. Since 0 / E(f2), and A(s) > 0 for all s, V < 0. ∈

For all V > V , either s = 0 or interior solutions are obtained. By the definition of V 00, any interior solution is a strict local maximizer and therefore the unique maximizer. Thus for all 0 > V > V , given any (f2, fc) such that f2 + fc = 1, the Sponsor’s best response is § 0 if f2 = 0 (B.9a) s(f2) = ˜s if f2 = 0, (B.9b) 6 where ˜s is defined in Lemma B.1. It is straightforward to see that s(f2) is continuous in f2. In addition, f2 0, ˜s 0, and as f2 1, ˜s ˆs0. By Brewer’s fixed-point theorem, an equilibrium exists. Because→ the→ way V → → is constructed, ˜s (0, ˆs0] > 0. The assumption f2(s = 0) > 0 makes sure that (0, 0) is not an ∈ equilibrium, and therefore the equilibrium level of support s∗∗ > 0.

14 Proposition B.3. There exist parameters where the equilibrium level of support increases in V .

Proof. Consider any interior equilibrium. We know that the Sponsor’s interior best response must satisfy equation (B.2), relabeled as h(˜s, V ). Apply the Implicit Function theorem and we have

b2(mr +s) (mr +s)b20 (mr +s) fc − 2 ∂ ˜s ∂ h(˜s, V )/∂ V [fc b2(mr +s)+mr +s] ˜s = = 2 > 0, ∂ V ∂ h ˜s, V /∂ ˜s ∂ u1 − ( ) − ∂ s2 ˜s because the numerator is positive, and the denominator is negative by the SOC. This implies that the Sponsor’s interior best response increases in V . In Figure B.2, a change in V shifts the Spon- sor’s best response to the right but does not alter the Target’s best response, the resulting equilibrium level of support increases. Proposition B.3 is only an existence statement. The precise conditions are mathematically intractable. Because in this extension both the sponsor and the target engage in two-sided conﬂict. This, combined with the nature of contest success functions, renders both players’ best responses non-monotonic, which makes it difﬁcult to pin down the precise direction of the comparative stat- ics. While a mathematical proof is intractable, extensive numerical simulations suggest that for reasonable values of V , the equilibrium support level increases in V , visualized by Online Ap- pendix Figure B.2. The relationship only breaks out for extremely large V (larger than other parameters by two degrees of magnitude). Figure B.3 provides a numerical example of equilibrium support level decreasing in V . As V increases, the Sponsor’s best response shifts to the right but

the Target’s best response remains unchanged, as a result one equilibrium, (s∗, f2∗) labeled in the graph, shifts northeast, indicating a decrease in s∗.

Figure B.3: s∗ decreases in V : a numerical example

Notes: The ﬁgure uses b m s b a with a 1, m 0.2 and 2( r + ) = 2 a+mr +s = r = b2 = 15. Other parameter values are b1 = 10, φ = 0.8, θ = 1, α = 1 and V = 150.

15 C Additional Model Extensions C.1 Support to Satisfy the Domestic Audience Some sponsors support rebels to satisfy their domestic audiences and benefit from the act of support per se. For example, when the rebels’ co-ethnics have an important electoral influence in the sponsoring state, the leader might offer support to gain votes to stay in power. To formalize this idea, besides international considerations, suppose S directly benefits from providing support: Bs for s [0, b1], where B > 0 is the unit benefit of support. To summarize the substantively interesting results,∈ Proposition C.2 show that the equilibrium support level s‡ increases in B. The sponsor provides higher level of support if the unit benefit of support is higher. Proposition C.3 finds that, under certain conditions, even when the sponsor offers no support in the baseline, it will support rebels for any B > 0, regardless of how small B is.

Lemma C.1. Suppose B > 0. Given any (f2, fc) such that f2 + fc = 1, the Sponsor’s best response is

 if f 0 (C.1a)  2 =  ;†  s if f2 = 0 and g(s) 0 g(s) (C.1b) s f 6 ≥ ≥ ( 2) = 0 if f 0 and g s 0 (C.1c)  2 = ( ) <  6  b1 if f2 = 0 and g(s) > 0 (C.1d) 6 where φ f b2(mr + s) + (b1 s)b0 (mr + s) g s 2 2 B, (C.2) ( ) = − 2 + − θ [(b1 s)/θ + f2 b2(mr + s)] †− g(s) max g(s), g(s) min g(s), and s solves g(s) = 0. s [0,b1] s [0,b1] ≡ ∈ ≡ ∈ Proof. The Sponsor’s’s problem is max u1(s) = 1 φp f2 b2(mr + s), (b1 s)/θ + Bs. (C.3) s [0,b1] ∈ − −

Case 1. f2 = 0. The same logic in the proof of Lemma B.1 applies and the Sponsor’s best response is . ; Case 2. f2 = 0. The objective function u1(s) is continuous in s. Since [0, b1] is a closed and bounded interval6 in R, the Extreme Value Theorem guarantees the existence of a solution. Taking the ﬁrst partial derivative with respect to s yields that

N1(s) z ≡}| { ∂ u s φ f b2(mr + s) + (b1 s)b0 (mr + s) g s 1( ) 2 2 B. (C.4) ( ) = − 2 + ≡ ∂ s − θ [(b1 s)/θ + f2 b2(mr + s)] | − {z } D1(s) ≡

With f2 = 0, h(s) is continuous in s, and since [0, b1] is a closed interval, g(s) has both a maximum and a minimum6 value by the Extreme Value Theorem. Define them as g(s) and g(s) respectively. If g(s) < 0, u1(s) strictly decreases in s, and s(f2) = 0. If g(s) > 0, u1(s) strictly increases in s, † and s(f2) = b1. If g(s) 0 g(s), the solution s is interior and can be characterized by the first ≥ ≥ 16 † order condition g(s) = 0. The second order condition is satisfied at s . To simplify notations, I use terms defined in equation (C.4). Evaluating the SOC at s† yields

2 † † † † ∂ u1(s) φ f2 N10(s )D1(s ) 2N1(s )D10 (s ) < 0, 2 = − † 3 ∂ s s† − θ · D1(s )

† † 00 † † where the inequality follows because D1(s ) > 0, N10(s ) = b1 b2(mr + s ) > 0, and D10 (s ) = † † † 1/θ + f2 b20 (mr + s ) < 0. Also, since g(s ) = 0, N1(s ) and B have the same sign, implying − † N1(s ) > 0.

Lemma C.2. Let ˆs0 be the maximizer when B = 0, and let s(f2, B) be the best response to f2 for B = 0. For all B > 0 and f2 = 0, s(f2, B) is continuous in B and f2, lims(f2, B) = b1 and f2 0 6 6 → lim ∂ s(f2,B) . If ˆs > 0, then s f , B > ˆs for all f 0, 1 . ∂ f2 = 0 ( 2 ) 0 2 ( ] f2 0 → −∞ ∈ Proof. The continuity of s(f2, B) is readily ascertainable from Lemma C.1. Since g(˜s) = 0 at the † φ f2 N1(s ) interior solution, we have B † 2 . We know N1 ˆs0 0 and N1 s < 0 for all s < ˆs0. = θ [D1(s )] ( ) = ( ) † Suppose s(1, B) = s < ˆs0, then the right-hand side of the equation is negative while B > 0, a contradiction. Also, lim g s B 0, implying that lims f , B b . Deﬁne g s, f g s f2 0 ( ) = > ( 2 ) = 1 ( 2) ( ) → f2 0 ≡ and apply the Implicit Function Theorem: →

N s D f ,s † † 1( ) 1( 2 ) 2N1 s b2 mr s ∂ s f2, B ∂ s ∂ g s , f2 /∂ f2 f2 ( ) ( + ) ( ) ( ) . = = † = − ∂ f2 ∂ f2 − ∂ g(s , f2)/∂ s −N10(s)D1(f2, s) 2N1(s)D10 (f2, s) s† − Note that since lim s f , B b , lim s† b . Thus, lim N s† lim b f2 0 ( 2 ) = 1 f2 0 = 1 f2 0 10( ) = f2 0( 1 s† b m b 0, lim→ N s† lim →b m s† b s† b→ m s† b m→ b −, ) 200( r + 1) = f2 0 1( ) = f2 0 2( r + ) + ( 1 ) 20 ( r + ) = 2( r + 1) lim D f , s† 1/θ→, and lim D1(f2→,s) lim b s† −/ f θ b m s† b m f2 0 10 ( 2 ) = f2 0 f2 = f2 0( 1 ) ( 2 ) + 2( r + ) = 2( r + → † − → → − b 1 lim ∂ s , where the last equality follows from the L’Hopital’s Rule. Thus, 1) θ f2 0 ∂ f2 − · → † 1 lim ∂ s b m b † θ ∂ f2 2( r + 1) † ∂ s − f2 0 − 1 ∂ s lim = → = lim + θ b2(mr + b1)/2. f2 0 ∂ f2 − lim2/θ 2 f2 0 ∂ f2 → f2 0 → → † † † There are three solutions. lim ∂ s θ b m b > 0, or lim ∂ s , or lim ∂ s . We ∂ f2 = 2( r + 1) ∂ f2 = ∂ f2 = f2 0 f2 0 f2 0 → † → ∞ → −∞ can easily rule out the ﬁrst two solutions. Therefore, lim ∂ s . ∂ f2 = f2 0 → −∞ Proposition C.1. A subgame perfect equilibrium exists for all B > 0. Proof. Using Lemmas C.1, C.2, B.3 and B.4, the proof is almost identical to that of Proposition B.1 and is therefore omitted.

‡ ‡ Proposition C.2. If ˆs0 > 0, then the equilibrium support level s > ˆs0, and s increases in B.

Proof. Lemma C.2 tells that s(f2, B) > ˆs0 for all B > 0 and all f2 (0, 1]. Proposition C.1 ‡ establishes equilibrium existence. Combing both implies that in equilibrium∈ s > ˆs0. Since the

17 † interior solution satisﬁes g(s ) = 0, apply the Implicit Function Theorem and we have ∂ s† ∂ g s† /∂ B 1 ( ) > 0, = † † = † † ∂ B −∂ g(s )/∂ s −∂ g(s )/∂ s where the inequality follows because the denominator is negative according to the SOC.

φ m1(mr +s) Proposition C.3. Suppose ˆs0 < 0. If there exists s 0, b1 such that < 2 and there ( ) α [b2(mr +s)+mr +s] 2 φ [b2(mr +s)+m1] ∈ ‡ also exists s 0, b1 such that > , then there exists an equilibrium in which s > 0 ( ) α m1(mr +s) for all B > 0∈.

Proof. There are two cases. Case 1. s(f2, B) > 0 for all B and f2 (0, 1]. In this case, it is readily ascertainable that in equilibrium s‡ > 0. ∈ φ m1(mr +s) Case 2. s f2, B 0 for some f2 0, 1 . If there exists s 0, b1 such that < 2 , ( ) = ( ] ( ) α [b2(mr +s)+mr +s] ∈ ∈ max Lemma3 implies that there exists s such that f2(s) = 0. Let the s0 be the largest s that satisfies 2 φ [b2(mr +s)+m1] this condition. If there also exists s 0, b1 such that > , Lemma3 implies that ( ) α m1(mr +s) ∈ min there also exists s such that f2(s) = 1. Let the s1 be the smallest s that satisfies this second 2 m1(mr +s) [b2(mr +s)+m1] max min condition. The proof of Lemma3 tells that 2 < , implying s < s . [b2(mr +s)+mr +s] m1(mr +s) 0 1 max min Since f2(s) is also continuous in s, there exists an interval [s0 , s1 ] (0, b1) such that the range ⊂ max of f2(s) over the interval continuously spans the entire [0, 1] interval. Since s(f2, B) = 0 < s0 for some f 0, 1 , lim s f , B b > smin, and s f , B is continuous in s and B for all 2 ( ] f2 0 ( 2 ) = 1 1 ( 2 ) ∈ → B > 0 and all f2 (0, 1], the two continuous lines must cross at least once. That is, there exists an ∈ † max min equilibrium in which s (s0 , s1 ) > 0. ∈ C.2 Military Effectiveness This subsection formalizes how military effectiveness of the sponsor’s army affects the support decision. Use β > 0 to denote military effectiveness of the sponsor’s army, with higher β indicating a more effective army. If β > 1, the sponsor’s army is more effective than the target’s army and the rebel group. If β < 1, the opposite is true. β = 1 recovers the baseline, where all military forces are equally effective. The probability that the Target prevails against the Sponsor is ( 1 if m f b m s 0, 2 1 = 2 2( r + ) = p f2 b2(mr + s), m1 = f2 b2(mr +s) β otherwise. m1 +f2 b2(mr +s) Everything else from the baseline model remains the same. The following Proposition C.4 sum- marizes how the sponsor’s military effectiveness affects its support for rebels. I find that if the subversive effect of the support is sufficiently large (equation (3) holds), then the sponsor support the rebels for sufficiently low military effectiveness (β < β˜). Moreover, the interior equilibrium

level of support s∗ decreases β, implying that as the sponsor’s army becomes more effective at fighting the target, it offers less support to the rebels. Also note that β˜ > 1. This implies that the condition rebels are more effective than the sponsor in fighting the target regime is not necessary for support. When the support’s subversive effect is large enough, the sponsor voluntarily aids the rebels even when the sponsor is better at fighting the target than the rebels (β 1). ≥

18 Proposition C.4. Let ˆs0 be the equilibrium level of support in the interior equilibrium for β = 1. ˜ ˜ Assume that ˆs0 > 0. There exists β > 1 such that for all β < β, s∗ > 0, and s∗ decreases in β. Proof. The ﬁrst order condition for the interior solution must satisfy

∂ u1 φ f2 b1 sβ 1 β b2(mr + s) + (b1 s)b20 (mr + s) = − − − 2 . (C.5) ∂ s θ θ β − · · [(b1 s)/θ] + f2 b2(mr + s) − β 1 φ f2 b1 s Since θ θ− − is negative and the denominator is positive, we only need to consider the − · sign of the numerator N(β, s) β b2(mr + s) + (b1 s)b20 (mr + s). Observe that N(β, b1) = ≡ − β b2(mr + b1) > 0 and N(β, 0) = β b2(mr ) + b1 b20 (mr ), the sign of which is indeterminate. We ∂ N(β,0) know that ∂ β = b2(mr ) > 0, N(1, 0) = b2(mr ) + b1 b20 (mr ) < 0 due to the assumption that ˜ ˜ ˆs0 > 0, and limβ N(β, 0) > 0. Thus, there exists β > 1 such that N(β, 0) = 0, and N(β, 0) < 0 ˜ →∞ ˜ for all β < β. Combined with N(β, b1) > 0, we have that for all β < β, there exists s∗ (0, b1) N s N s s ∂ N(β,s) ∈ such that (β, ∗) = 0, and (β, ) increases at ∗, implying ∂ s s=s > 0. β | ∗ Let D(β, s) [(b1 s)/θ] + f2 b2(mr + s). The second order condition is also satisﬁed at s∗ because ≡ −

2 N(β,s) ∂ D(β,s) ∂ u1 φ f2(β 1) b1 sβ 2 N(β, s) φ f2 b1 sβ 1 ∂ s D(β, s) N(β, s) ∂ s − − 2 = − − 2 − − 3 ∂ s s s θ θ D(β, s) θ θ D(β, s) s s = ∗ − · · − · · = ∗ ∂ N(β,s) φ f2 b1 sβ 1 ∂ s − < 0, = − 2 θ θ D(β, s) s s − · · = ∗

where the last equality follows from N(β, s∗) = 0. Applying the Implicit Function Theorem yields

N(β,s∗) ∂ s∗ ∂ β b2(mr + s∗) = = < 0, ∂ β ∂ N(β,s) ∂ N(β,s) − ∂ s s=s − ∂ s s=s | ∗ | ∗

indicating that s∗ decreases in β. C.3 Rebel Group Entrance Suppose a rebel leader pays a ﬁxed cost F > 0 to organize an organization from scratch and the prize of the regime is Π. The leader organizes the rebel group if the expected return is larger than

the ﬁxed cost. With support s∗, the rebel group enters if m s Π r + ∗ > F. (C.6) m s f s b m s r + ∗ + c( ∗) 2( r + ∗) Proposition C.5 states that external support increases the probability of victory for small groups and encourages them to enter. As a result the entrance condition given by equation (C.6) is more likely to hold, and the groups enters under a wider set of parameter conditions. Some small groups that would not have formed without support now entered the domestic competition for power.

mr +s∗ mr Proposition C.5. There exists m0r (0, mr ] such that for all mr < m0r , m s f b m s > m f 0 b m . ∈ r + ∗+ c∗ 2( r + ∗) r + c ( ) 2( r )

19 Proof. The difference in probability of rebel victory is

m s m Pr R wins s Pr R wins s 0 r + ∗ r ( ∗) ( = ) = m s f b m s m f 0 b m | − | r + ∗ + c∗ 2( r + ∗) − r + c( ) 2( r )

A(mr ,s∗) z ≡ }| { (mr + s∗)fc(0)b2(mr ) mr fc∗ b2(mr + s∗) − , = m s f b m s m f 0 b m [ r + ∗ + c∗ 2( r + ∗)] [ r + c( ) 2( r )] ·

where the denominator is positive and the sign depends on the numerator A(mr , s∗). lim A(mr , s∗) > mr 0 → 0, and lim A(mr , s∗) = 0. Thus there exists m0r (0, mr ] such that A(m0r , s∗) = 0, and for all mr mr → ∈ mr < m0r , A(m0r , s∗) > 0, implying that the probability of rebel victory with support increases for these groups.

D Supplementary Information: Empirical Implications D.1 Iranian Support for the KDP: Alternative Explanations One explanation for Iranian support of the KDP is that Iran delegated the conflict with Iraq over their common boundaries to the rebel organization. This explanation is inconsistent for two reasons. First, the goal of a war is to win. If the KDP were a proxy, Iran would have genuinely desired its victory. Archival evidence suggests the opposite. According to a CIA memo of March 22, 1974: “We would think that [our ally] would not look with favor on the establishment of a formalized autonomous government. [Our ally] like ourselves, has seen benefit in a stalemate situation...in which [our ally’s enemy] is intrinsically weakened by [the ethnic group’s] refusal to relinquish its semi-autonomy. Neither [our ally] nor ourselves wish to see the matter resolved one way or the other.” (Pike, 1976, p. 87) In addition, a hypothetical victory of the KDP is unlikely to benefit Iran in the boundary dispute. First, an independent Kurdistan in the north cannot grant Iran control over the Shatt al-Arab waterway in the south. Second, armed opposition that intervenes in politics has an incentive to fend off external challenges after coming to power.12 Even if, instead of independence, the Kurds shared power in the Iraqi central government and had a say in Iraqi foreign policy, they were unlikely to prefer concession over the waterway, Iraq’s only outlet to the Persian Gulf. Another explanation is that shared ethnic ties between the Kurds and Iranian Kurdish minority motivated the support. However, the common Kurdish minority hindered rather than prompted Iran’s decision to sponsor the KDP. There was active Kurdish insurgency in Iran as well. In the early 1960s, the Shah was wary of aiding the Kurdish rebellion in Iraq, for fear that it might eventually have spill-over effects on the sizable Kurdish minority in Iran. Iran demanded the KPD break all ties with Kurdish movements in Iran and execute two leaders attempting to organize a guerrilla front in Iran from a safe haven in Iraq (Chaliand and Black, 1994, p. 60). Despite such concern, Iran sponsored the KDP from 1961 onward. 12 McMahon and Slantchev(2015) expresses a similar logic in characterizing coup plotters’ incen-

tive.

20 D.2 Rebel Strength and External Support

Table D.1: Logistic regression estimates of support for rebel groups

Model 1 Model 2 Model 3 Support I Support II Support III Variables (including alleged) (Only acknowledged) (alleged set to missing)

Rebels Strong 1.244 1.199 1.092 −(1.196)− (1.212)− (1.216) Rebels Weak 0.046 0.186 0.223 (0.434) (0.440) (0.455) Strong Center Command 0.717 0.737 0.834† −(0.457)− (0.467)− (0.479)

Government Support 1.538∗∗ 1.402∗∗ 1.599∗∗ (0.420) (0.419) (0.441) Transnational Constituency 0.625 0.826† 0.813† (0.438) (0.446) (0.456)

Territorial Control 0.405 0.712∗ 0.601 (0.423) (0.428) (0.440) More Than One Actor 0.159 0.176 0.074 (0.387)− (0.408)− (0.423) † Ln GNP per capita 0.421 0.506∗ 0.510∗ (0.235) (0.237) (0.242) † CINC 31.481∗ 13.546 18.098 (12.369) (9.138) (10.402)

Democracy 1.752∗∗ 2.094∗∗ 2.111∗∗ −(0.597)− (0.583)− (0.606)

Constant 3.893∗ 4.570∗ 4.632∗ −(1.815)− (1.832)− (1.876)

N 162 162 152 Log Likelihood 90.342 86.674 80.964 AIC− 202.684− 195.349− 183.928 LR test, strength terms 1.560 1.839 1.695 LR test, p-values 0.458 0.399 0.429

Notes: The table replicates Salehyan, Gleditsch and Cunningham(2011) Table 2 for RIVAL = 1. Variable coefficients are reported, with standard errors in parentheses. None of the strength terms is significant. A † joint LR test indicates they are not jointly significant. ∗∗p < .01; ∗p < .05; p < .1.