On conflict over natural resources

July 5, 2017 | Autor: John Maxwell | Categoria: Ecological Economics, Applied Economics, Dynamics, Dynamic Simulation, Natural Resource, ENVIRONMENTAL SCIENCE AND MANAGEMENT

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Ecological Economics 70 (2011) 698–712

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Ecological Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e c o n

Analysis

On conﬂict over natural resources Rafael Reuveny a,⁎, John W. Maxwell b, Jefferson Davis c a b c

School of Public and Environmental Affairs, Indiana University, Bloomington, IN 47405, United States Kelley School of Business, Indiana University, United States Stat-Math, Indiana University, United States

a r t i c l e

i n f o

Article history: Received 27 November 2008 Received in revised form 22 September 2010 Accepted 9 November 2010 Available online 11 January 2011 Keywords: Game theoretic model Dynamics Simulations Policy

a b s t r a c t This paper considers a game theoretic framework of repeated conﬂict over natural resource extraction, in which the victory in each engagement is probabilistic and the winner takes all the extracted resource. Every period, each contesting group allocates its capabilities, or power, between resource extraction and ﬁghting over the extracted amount. The probability of victory rises with ﬁghting effort, but a weaker group can still win an encounter. The victorious group wins all of the extracted resources and converts them to power, and the game repeats. In one model, groups openly access the resource. In a variant of the model, the stronger group can access a larger part of the resource than its rival, while in a second variant of the model the advantage of the dominant group is made more decisive than in the ﬁrst two models. Our models generate outcomes that mimic several aspects of real-world conﬂict, including full military mobilization, defeats in one or repeated battles, victories following defeats, changes in relative dominance, and surrender. We examine comparative dynamics with respect to changes in the resource attributes, resource extraction, initial power allocation, ﬁghting capabilities, and power accumulation. The policy implications are evaluated, and future research avenues are discussed. © 2010 Elsevier B.V. All rights reserved.

1. Introduction Conﬂicts over natural resource extraction may go on for many years against a background of resource dependence, a weak state, and underdeveloped property rights. This generalization is not in dispute, but some studies argue that actors ﬁght over resources when they are scarce, while others argue they ﬁght over resources when they are abundant. We develop models of conﬂict that apply for both situations. Our models build on the game theoretic approach developed by Hirshleifer (1988, 1991). The rival actors allocate their effort to production and ﬁghting and seek to maximize their gain by taking over the output of their rival. The ﬁghting takes place against a background of anarchy, deﬁned as a situation lacking an accepted authority, social norms, and property rights. The Hirshleifer approach is useful for our purpose, as ﬁghting implies that actors decide to take matters into their own hands, rejecting existing systems of law, order, and norms of peace. However, it has a limitation in that the actors clash only once and the game ends. The one-shot game cannot address questions involving repeated ﬁghting over resources. For example, does a rise in the resource stock over time lessen the conﬂict? How does the conﬂict affect, or how is the conﬂict affected by, changes in the allocated efforts over time?

⁎ Corresponding author. Tel.: +1 812 855 6112; fax: +1 812 855 7802. E-mail address: [email protected] (R. Reuveny). 0921-8009/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2010.11.004

How are these dynamic issues affected by resource and group features? We model repeated conﬂict over resource extraction among two groups of agents: two states, rebels and state forces, or two communities. Every period, each group allocates its power, deﬁned as a composite indicator of available capabilities or efforts, to resource extraction and ﬁghting in order to take over the resource extracted by their rival.1 Victories are stochastic, though not entirely random. The probability of victory rises with the ﬁghting effort, but a weaker group can still win. The victor's conﬂict spoils amount to all of the extracted resource in the given period. The groups see a decline in their power due to depreciation and ﬁghting damage, but the winner converts the conﬂict spoils into power, which the loser cannot do. Having more power, the victor is in a better position at the start of the next engagement since it can allocate more effort to ﬁghting. We apply this framework in three contexts. In the ﬁrst, the groups are assumed to have open access to the resource. In the second, a more powerful group has access to a larger part of the resource. In the third model, the stronger group's relative power is more decisive, though the victory is still stochastic. Given their mathematical complexity, our models can only be solved or simulated numerically. Summarizing our simulation results, a larger resource stock intensiﬁes the conﬂict since it raises the extraction, or the spoils. A group with greater ﬁghting efﬁciency

1

The terms capability, effort, and power are used interchangeably.

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

allocates less effort to conﬂict and is less likely to surrender since it has a larger marginal return to ﬁghting. A group more efﬁcient at extraction allocates more effort to extraction since it has a larger marginal return to extraction, but as a result it is more likely to surrender. Groups worse at converting spoils to power and groups having less power initially surrender more often. These results hold across our models, but power-based access reduces the spoils, which dampens the ﬁghting, and a more decisive conﬂict increases the marginal return to conﬂict, intensifying the ﬁghting. Real-world resource conﬂict is obviously more complex and multifaceted than in our models, but features of our models are often observed in reality. Our approach may apply to cases in which groups ﬁght repeatedly and the situation is quite anarchic. Our assumed all-or-nothing victory may not ﬁt all cases, though we believe it captures the reality of many conﬂicts in which the winner clearly takes the vast majority of the conﬂict spoils. This assumption seems to better match many conﬂicts than the popular alternative assumption in which the combatants share the spoils according to their ﬁghting efforts. We can thus think about our models as providing an upper limit for a continuum of resource extraction splitting ratios. Our models' outcomes tend to mimic aspects of real-world resource conﬂict, including concurrent extraction and ﬁghting, a decline in the resource stock, full mobilization to ﬁghting, defeat in one or repeated battles, victory following defeat, changes in relative dominance, and surrender. It may therefore be useful to cautiously examine the policy implications of our ﬁnding. The remainder of the paper proceeds as follows. Section 2 provides background. Sections 3–5 present models and simulations. Section 6 applies the models to real world resource conﬂicts and examines policy implications, and Section 7 summarizes and suggests future research. 2. Background Our paper brings together elements from the literature on conﬂict over resource extraction, the predator–prey literature in ecology, the economic literature on conﬂict, and the ecological economic literature on conﬂict over resources. These literatures are too large to fully review here. We discuss a number of studies that provide the background for our models. In the social sciences, the idea that actors ﬁght over scarce resources dates back to Malthus (1798). Elaborating on this logic, contemporary studies expect conﬂict when demand exceeds supply, and when actors block access to scarce resources based on factors such as race, ethnicity, or religion.2 For example, ﬁsh scarcity leads to piracy and violence among ﬁshermen (UN, 1998), and clashes between Britain–Iceland (Jóhannesson, 2004), Canada–Spain, Malaysia–Thailand, and Japan–Russia (Renner, 1996; Reuveny, 2002). Arable land scarcity plays a role in the El Salvador–Honduras 1969 War (Durham, 1979), the Somalia–Ethiopia 1977–78 War (Myers, 1993), and the ongoing Darfur War (Jeffrey, 2005).3 Water scarcity contributes to the ongoing Arab-Israeli conﬂict and other cases.4 Food scarcity fuels long conﬂicts in Peru (McClintock, 1984) and Sub-Saharan Africa (Holst, 1989). 2 On resource conﬂict within states, see, e.g., Myers (1993), Dasgupta (1995), Lietzmann and Vest (1999), Homer-Dixon (1999), Baechler (1999), Kahl (2006), and Reuveny (2002, 2007, 2008). 3 Other examples include conﬂicts in the Philippines (Hawes, 1990), Haiti (HomerDixon, 1999), Sudan (UNEP, 2007), South Africa (Percival and Homer-Dixon, 2001), New Guinea (Hirshleifer, 1995), Rwanda (Renner, 1996; Lietzmann and Vest, 1999), Mexico (Homer-Dixon, 1999; Brown, et al., 1999), Bangladesh, India (Swain, 1996), and Nigeria (The Economist, 2001). 4 Examples include disputes between Brazil–Paraguay; Ethiopia–Somalia; Egypt– Sudan–Ethiopia–Tanzania; Syria–Turkey–Iraq; South Africa–Lesotho; India–Bangladesh; Senegal–Mauritania; and internal conﬂicts in Yemen, Darfur, China, Ethiopia, and Somalia (e.g., Myers, 1993; Renner, 1996; Pomfret, 1998; Libiszewski, 1999; Beach et al., 2000; Klare 2002; Reuters, 2006; Gleick, 2008; Jeffrey, 2005; Kasinof, 2009; Zahran, 2010).

699

Applying this approach to major countries, Hobson (1902) and Lenin (1916) argue the business class pushes states to seize foreign resources, leading to imperialism. The German geopoliticians justify the German expansionism before 1945 as a drive for resources and Lebensraum (Heske, 1987). Choucri and North (1975, 1989) argue more generally that economic development and population growth generate “lateral pressure,” an expansionist drive to seize foreign resources that may cause wars. Demonstrating this logic, studies argue that lateral pressure plays a role in World War I (Choucri and North, 1975), the pre-1945 Japanese expansionism (Choucri et al., 1992), the US foreign policy since the 19th century (Pollins and Schweller, 1999), and the current Iranian aggressiveness (Wickboldt and Choucri, 2006). Observers predict that resource scarcity will lead to more conﬂicts in the future as supply falls short of demand due to development and population growth, and climate change intensiﬁes pressures on water, arable land, and agriculture, assuming a business as usual climate change policy. The less developed countries (LDCs) may exhibit more conﬂict since they depend more on resources, are less able to adapt, and have larger populations, but the violence may spread to the developed countries (DCs).5 Other studies argue that groups tend to ﬁght over abundant resources, not scarce, since the resource revenue can ﬁnance their arming and activities. Resource plenty can lead to a prolonged “Dutch Disease,” a decline in export, investments, and economic growth due to currency appreciation, and can be a “curse,” eliciting corruption and rent seeking, increasing grievances, and ultimately leading to violence over resource extraction. The domestic problems may tempt other countries to attack, or promote leaders to rally the people behind the ﬂag by attacking other countries.6 Westing (1986) ﬁnds that access to abundant resources fueled 12 major wars in 1914–1982, including the two World Wars. Yergin (1992) describes the role of oil in World Wars I and II. Oil is a factor in the 1991 and 2003 Iraq Wars, and fuel tensions in the South China Sea and the Caspian Sea Basin (Klare, 2001; Follath, 2006; Mayr, 2006; Judis, 2007). Abundant arable land fuels a long conﬂict in Borneo; oil in Angola; copper in the Bougainville Island; timber in Liberia, Cambodia, Burma and other states (Klare, 2002; Thomson and Kanaan, 2003; Global Witness, 2002, 2010); minerals, metals and oil in the Congo; diamonds in Sierra Leone and Angola; oil and drugs in Colombia; wood and minerals in Indonesia; and cocoa in Côte d'Ivoire (Renner, 2002; PBS, 2008; Gettleman, 2009; Global Witness, 2002, 2010).7 Arising from the work of Lotka (1924) and Volterra (1931), the predator–prey literature in ecology models the dynamics of competition between animal species that feed on each other and consume resources by using a system of differential equations that codiﬁes the behavior of each element (e.g., Slobodkin, 1980; Clark, 2010). Economists studied analogies between this approach and economic competition over time (e.g., Hirshleifer, 1977; Jacquemin, 1987). Political scientists have used somewhat different systems of differential equations to examine the conﬂict and arms races dynamics (e.g., Richardson, 1960; Zinnes and Gillespie, 1976; Luterbacher and Ward, 1985; Hess, 1995). Unlike animal species and resources, however, people may not necessarily follow codiﬁed rules of deterministic behaviors, but rather choose an action they deem to be optimal, taking account of the constraints they face. 5 On increased resource conﬂict see, e.g., World Bank (1995), Klare (2002, 2005), Forney (2004), Reuters (2006), and Follath (2006). On resource conﬂicts precipitated by climate change see, e.g., Reuveny (2002, 2007), Schwartz and Randall (2003), Gore (2007), CNA (2007), Parthemore and Rogers (2010), and Parsons (2010). 6 For example, see Krebs and Levy (2001), Sachs and Warner (2001), Klare (2002), Renner (2002), Le Billon (2001), and World Bank (2004). 7 Le Billon (2001) lists other examples, including Liberia (iron, rubber); Nicaragua, El Salvador, and Guatemala (coffee); Indonesia (oil, copper, gold); Senegal (land); Mauritania (land); Afghanistan (opium); and the Philippines (wood).

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The economic literature on conﬂict incorporates an element of mathematical optimization to conﬂict decision making. The idea can be traced to Hirshleifer (1988, 1991). He assumes that conﬂict occurs in anarchy, which holds in the international system and may apply to a varying degree within states. The actors produce output and typically ﬁght once over it, assuming they do not trade and there is no external intervention. A group allocating more effort to ﬁghting gets a larger share of the output, while a few models pick a winner by chance and award it all of the spoils. Many studies have since used this framework. Grossman and Kim (1995), for example, distinguish between defensive and offensive activities. Hirshleifer (1995) examines various divisions of the spoils. Anderton et al. (1999) add trade. Garﬁnkel and Skaperdas (2000) model a two-period one-shot game, Maxwell and Reuveny (2005) add dynamics to a simple splitting model of conﬂict, and Reuveny et al. (forthcoming) examine differences between splitting- and chance-based outcomes for homogenous groups. The ecological economic literature models conﬂict over resources, though not based on Hirshleifer's approach. Suzuki and Iwasa (2009) show that social norms of conservation deter conﬂict, conceptualized as non-conformist individual pollution. BenDor et al. (2009) show that property rights reduce conﬂict, conceptualized as the extent of individual overﬁshing in a common access ﬁshery. Welsch (2008) models the choice to work in a production or in a resource sector whose extraction is subject to looting, and represents the probability of conﬂict by the size of the labor force working in the resource sector. In sum, we can say that states or non-state actors often ﬁght repeatedly over resource extraction. The outcome of encounters is not known in advance, though an overall victor may arise, ending the conﬂict. One may model such conﬂict in several ways, though it is reasonable to assume that the actors seek to maximize their gain. We propose a game theoretic framework that embeds this element. 3. Base Model The setup of the model is as follows. Two rival groups extract a resource and seek to take over the resource extracted by each other. The groups live in a state of anarchy. In this world, there is no recognized central authority; no accepted social norms of peace or resource conservation; no enforced property right institutions; and no trade between the actors. Since we seek to study the underlying behavior of the two groups, we continue in the Hirshleifer tradition and assume also that there is no external intervention of any type (e.g., military and economic).8 Nit is group i's power at time t, where i = {1, 2}. Each period, each group allocates its power between resource extraction effort, Eit, and ﬁghting effort, Fit. Both efforts embed all the required human, physical, or military capitals. Nit = Eit + Fit ;

i = f1; 2g:

ð1Þ

To keep the math tractable, the model includes one natural resource, representing a composite of renewable and nonrenewable resources. The resource extracted by group i, Git, is given by Git = βi Sit Eit ;

i = f1; 2g;

ð2Þ

The total resource extraction in period t, ð3Þ

Gt = G1t + G2t

is contested. A group gains control over Gt if it wins the ﬁght; if it loses, it gets nothing. The probability of victory for group i in period t is Pit. The expected payoff of group i, E(Yit), is given by: EðYit Þ = Pit Gt + ð1−Pit Þ⋅0

ð4Þ

i = f1; 2g;

The probability of victory is modeled by the widely used Tullock (1980) contest success function: Pit = Prð group i winsÞ =

αi Fit α1 F1t + α2 F2t

i = f1; 2g;

ð5Þ

where a larger α1 or α2 represents better ﬁghting capabilities, including better ﬁghters, leaders, or arms. Pit is used in generating a random draw of 1 or 0 in period t, as discussed shortly. If this draw equals 1 for i = 1, group 1 wins all of the conﬂict spoil; otherwise, group 2 wins. The probability of victory rises with the conﬂict effort, though as implied by Eq. (5) a group allocating less effort to conﬂict can also win (e.g., P1t N 0, if F1t b F2t, as long as F1t N 0). Each group chooses its ﬁghting allocation in order to maximize its expected payoff in the current encounter, subject to (s. t.) its power. We assume that they have strong leaders or otherwise they face no collective action problems.10 max EðYit Þ s:t:Fit ≥ 0; Nit −Fit ≥ 0 Fit

i = f1; 2g:

ð6Þ

Substituting the expressions for the total good produced and contest success function into Eq. (6), the optimization problem facing actor i is max Fi

αi Fi ½ β SðN1 −F1 Þ + β2 SðN2 −F2 Þ α1 F1 + α2 F2 1

s:t: Fi ≥ 0; Ni −Fi ≥ 0

ð7Þ

i = f1; 2g;

where from here on time subscripts are not shown to simplify the notation, but are understood. The solution of Eq. (7) involves the Lagrangian £i, where λi is the Lagrange coefﬁcient and i = {1, 2}:

£i =

αi Fi ½β SðN1 −F1 Þ + β2 SðN2 −F2 Þ + λi ½Ni −Fi : α1 F1 + α2 F2 1

ð8Þ

αi Fi ½β1 SðN1 −F1 Þ + β2 SðN2 −F2 Þ, the α1 F1 + α2 F2 Kuhn–Tucker conditions deﬁning the optimal allocations are: Recalling EðYi Þ =

£Fi = E YiF −λi ≤0; i

Fi £Fi = 0£λi = Ni −Fi ≥0; λi £λi = 0:

ð9Þ

where Sit is the stock of the resource composite available for group i, and βi is i's extraction efﬁciency.9 The base model assumes that Sit = St, i = {1, 2}, or St is a common pool resource composite.

The conditions in Eq. (9) imply that whenever Fi b Ni, £λi N 0 and therefore λi = 0. Thus, player i's reaction function is given by the solution to the unconstrained optimization problem, denoted Zi below, where i = {1, 2}. When Fi N Ni, the constraint binds and λi N 0. In this case, £λi = 0, and the reaction function becomes Fi = Ni.

8 While we do not explicitly model the mechanics of external intervention, we may examine its effects in the model by introducing changes in parameters and initial conditions. See Sections 6 and 7. 9 This functional form is widely used (e.g., Brander and Taylor, 1998; Maxwell and Reuveny, 2005).

10 Considering the current and all the future extractions in the maximization is interesting, but the assumption that actors plan all of their future extractions and are able to follow their plans is not applicable in anarchy. The implied assumption that the actors plan their future ﬁghts and are able to follow their plan seems even less applicable for real world conﬂict.

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

The reaction functions when Fi b Ni holds for each group are given

701

Next, we decide which group wins G, the conﬂict spoils, as follows: Let

by: F1 ≡ Z1 =

(

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 −F2 α2 β1 + F2 N2 α1 α2 β1 β2 +F2 N1 α1 α2 β21 −F22 α1 α2 β1 β2 +F22 α22 β21 α1 β1

ð10Þ

1

P1 = 0

and F2 ≡ Z2 =

1 α2 β2

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ −F1 α1 β2 + F1 N1 α1 α2 β1 β2 +F1 N2 α1 α2 β22 −F12 α1 α2 β1 β2 +F12 α21 β22 :

ð11Þ Therefore the reaction function for group i is Fi =

Zi Ni

if if

Zi bNi : Zi N Ni

ð12Þ

Since Z1 and Z2 are both convex and both are increasing with the ﬁghting effort of the other group, F2 and F1 respectively, there are ﬁve potential equilibria: (1) F1 = F2 = 0; (2) both groups are fully mobilized for ﬁghting, F1 = N1 and F2 = N2; (3) F1 b N1 and F2 b N2, an interior solution; (4) F1 = N1 and F2 b N2; and (5) F1 b N1 and F2 = N2. From Eq. (7), if F1 = F2 = 0, then E(Y1) = E(Y2) = 0. This cannot be an equilibrium since when F2 = 0, for example, a small deviation ξ toward allocating effort to ﬁghting by group 1 will increase its expected payoff from 0 to β1S(N1 − ξ) + β2S(N2) N 0. The situation F1 = N1 and F2 = N2 also cannot be an equilibrium. In this case, the partial derivatives of E(Yi) with respect to Ei are positive: ∂EðYi Þ αi Ni = ðβ SÞ N 0 αi Ni + αj Nj i ∂Ei Fi = Ni ;Fj = Nj

i; j = f1; 2g; i≠j:

Each group can increase its expected payoff by allocating more of its power to resource extraction, and so{F1 = N1, F2 = N2} is not an equilibrium. Thus the solutions to Eq. (7) are either the interior solution with F1 b N1 and F2 b N2, or else one group is fully mobilized for ﬁghting and the other is not: F1 = N1 and F2 b N2; or F1 b N1 and F2 = N2. Solving the system of Eqs. (10) and (11), the interior solution is: pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ α2 β2 ðβ1 N1 + β2 N2 Þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ F1 = p 2 β2 α1 β1 + β1 α2 β2

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ α1 β1 ðβ1 N1 + β2 N2 Þ ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : and F2 = p 2 β2 α1 β1 + β1 α2 β2

ð13Þ If group i is fully mobilized, Eq. (12) shows that Fi = Ni and Fj = −

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ﬃ αi Ni − αi Ni αi Ni + αj Nj

i; j = f1; 2g; i≠j:

αj

ð14Þ

Using Eqs. (1), (2), and (13), the total extracted resource in the interior solution is given by: G = G1 + G2 =

Sβ1 N1 + Sβ2 N2 : 2

ð15Þ

If group i is fully mobilized for ﬁghting, Ei = 0, and total extraction is given by: G = Sβj Nj −Fj ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r 0 ﬃ1 α N − α N α N + α N i i i i i i j j B C C i; j = f1; 2g; i≠j: = Sβj B @Nj + A αj ð16Þ

if

α1 F1 Nν α1 F1 + α2 F2 ; and otherwise

ð17Þ

P2 = 1−P1 where ν is a random draw from a uniform distribution over the interval [0, 1]. If P1 = 1, group 1 wins the conﬂict spoils: G1 = G and G2 = 0. Otherwise, group 2 wins: G1 = 0 and G2 = G. We assume that each group sees its power decline every period due to normal wear and tear and conﬂict damages, but converts its conﬂict spoils (if it wins) to power. This process is described for each group by the following two equations of motion: dNi = Ni εi + ϕi Gi dt

i = f1; 2g:

ð18Þ

The term Niε, where εi b 0, represents the decline in power due to the destruction and casualties caused by the ﬁghting and due to physical depreciation. The term ϕiGi, where ϕi N 0, models the conversion of the spoils to power, representing such mechanisms as using extracted resources to refurbish depreciated and damaged military, extractive, and human capital stocks, or selling extracted resources to a third party and using the proceeds to purchase military or extraction capitals, or ﬁnance ﬁghter training and recruitment activities. Next, we model the dynamics of the resource composite, S. To simplify the math, we assumed that the model includes only one resource, representing a composite of renewable and nonrenewable components (see Eq. (2)). We now assume that the natural growth of the renewable component is logistic. For the nonrenewable component, we assume that the groups do not discover new stocks. The nonrenewable dynamics are thus one of decline. The composite dynamics aggregate the two dynamics, where the resource extraction G subsumes both components. Under these assumptions, the dynamics of resource composite are given by the following expression:

dS S = rS 1− −G; dt K

ð19Þ

where r is the intrinsic growth rate of the renewable component, and K is the carrying capacity of the renewable component. If the resource composite does not include a renewable component, r = 0 in Eq. (19). In this case, the resource stock declines over time. As a result, the groups' powers and ﬁghting intensities decline in the model. Eventually, the resource stock will be zero, at which time the conﬂict must end. We do not analyze nonrenewable resources beyond this point, though we revisit the issue in the conclusion. Eqs. (18) and (19) form a three-equation system of differential equations that is a variation of the ecological predator–prey model. The groups' power stocks, Ni, represent the predators, and the resource composite stock, S, represents the prey. The solution of the model proceeds as follows. In each period we: 1. Compute the ﬁghting effort using Eq. (14) if one group is fully mobilized for ﬁghting. 2. Compute the ﬁghting effort using Eq. (13) if no group is fully mobilized for ﬁghting. 3. Award the conﬂict spoils to group 1 or 2 according to Eq. (17), where ν is drawn from the same seed. 4. Compute the conﬂict spoils using Eq. (15) or (16), depending on whether group i is fully mobilized.

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5. Solve one of the following systems of differential equations for one period.11 System 1: Neither group is fully mobilized Sβi Ni + Sβj Nj , and group j gets nothing. Group i wins Gi = 2 βi Ni + βj Nj dNi = Ni εi + ϕi S dt 2 dNj = Nj εj dt βi SNi + βj SNj dS S = rS 1− − dt K 2

dN = Nεj + ϕj βj SN dt

and

dS S = rS 1− −βj SN: dt K

23

4. Simulations

ð20Þ

i; j = f1; 2g; i≠j: System 2: Group i is fully mobilized and victorious 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ1 α N − α N α N + α N i i i i i i j j B C Group i wins Gi = Sβj @Nj + A αj and group j gets nothing. rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ1 αi Ni − αi Ni αi Ni + αj Nj B C dNi C; = Ni εi + ϕi Sβj B @Nj + A dt αj 0

dNj = Nj εj dt

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r 0 ﬃ1 αi Ni − αi Ni αi Ni + αj Nj B C dS S C = rS 1− −Sβj B @Nj + A dt K αj

i; j = f1; 2g; i≠j: ð21Þ System 3: Group i is fully mobilized and group j is victorious 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ1 α N − α N α N + α N i i i i i i j j B C Group j wins Gj = Sβj @Nj + A, αj and group i gets nothing. dNi = Ni εi dt

power to extraction, and converts the extraction to power.12 The system dynamics are given by:

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ﬃ1 αi Ni − αi Ni αi Ni + αj Nj B C dNj C = Nj εj + ϕj Sβj B @Nj + A dt αj 0

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r 0 ﬃ1 αi Ni − αi Ni αi Ni + αj Nj B C dS S C = rS 1− −Sβj B @Nj + A dt K αj i; j = f1; 2g; i≠j: ð22Þ System 4: Group i surrenders A group surrenders when its power falls below a critical level after losing many successive engagements. Unable to augment its power, the group becomes weaker due to depreciation and ﬁghting damages. Though not our focus, we assume the winner absorbs the loser. The parameters of the combined group are set equal to the winner's parameters. We assume that the combined group allocates all its 11 Although we present four systems, we actually describe eight systems of differential equations since i or j can each be 1 or 2.

Our systems of nonlinear stochastic differential equations cannot be solved analytically and so for the rest of the analysis we have to rely on numerical simulations.13 This method requires parameters, but there are many parameters to choose from, which may affect the outcome. Some studies choose synthetic parameters, seeking to illustrate some tendencies (e.g., Hirshleifer, 1995; Anderton et al., 1999). Alternatively, one may rely on real-world records for some cases, which is the approach we take here, but this approach too requires us to make a choice. In principle, we could try to parameterize the model for some country, but this is very hard to do since, despite its complexity, our model is stylized. Our approach is to rely on Vitousek's (2002) logic. Islands, he writes, offer an opportunity to study human-ecological systems in a simpler setup, and consider the results in the context of a more complex setup. Kirch (1997), Diamond (2000), Flenley and Bahn (2003), Kirch and Kahn (2007) and others take this approach for historical Polynesia. The case of Easter Island provides a good example. Many years ago it was a fertile land, with a large palm forest and nesting birds, and an agricultural society that also ﬁshed and ate birds and had little or no interaction with outsiders. When the Dutch arrived in 1722 they found a few thousand people living in extreme poverty. The place was barren, though monumental stone statues rested on large platforms throughout the island. The residents believed a powerful spirit told the statues to walk to their places. Most scholars believe the Easter Island society collapsed since it overexploited its natural resources. It is argued that the historical islanders apparently used tree trunks as rollers to move the statues from the queries to their ﬁnal location. Naturally, the process required many tree trunks. As the forest vanished, food output fell due to land erosion, ﬁshing declined since fewer canoes could be produced, and there were fewer birds to eat.14 Our model captures the predator–prey nature of this history, but the Easter Island story was probably more involved. Many scholars suggest that Polynesia had no strong central authorities and property rights. Society was organized in clans led by dominant chiefs. Conﬂict between well-organized groups over resource extraction was apparently endemic, including on Easter Island. The winner in any one encounter probably seized the loser's resource extraction, and a new conﬂict cycle ensued.15 We calibrate the model for Easter Island since it seems to provide a natural experiment of human–resource interaction involving ﬁghting over resource extraction against a background apparently in line with 12 Other scenarios are also possible. For example, the combined group may limit extraction to some level, or may crash the resource too fast. In the latter if it depletes dS S = rS 1− case, the model collapses to and S grows until it reaches K. dt K 13 Since we have three equation systems, the phase diagram and local stability methods are not tractable. Regardless, these methods study the dynamics very close to steady state(s), provided there is a steady state, but we are interested in the trajectory over time. 14 For examples of works on Easter Island, see Kirch (1984), Ponting (1991), Owsley et al. (1994), Van Tilberg (1994), Hunt and Lipo (2001), Flenley and Bahn (2003), and Diamond (2005). 15 For examples discussing conﬂict on Easter Island and in Polynesia, see Buck (1932), Goldman (1955), Kirch (1984), Ponting (1991), Keegan (1993), Earle (1997), Kolb and Dixon (2002), and Herman (2003). On the luck of central authority or property rights, see, e.g., Ponting (1991), Van Tilberg (1994), Brander and Taylor (1998), and Luterbacher (2001).

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

4.1. Illustrative Tracks In Fig. 1, the groups do not fully mobilize or surrender. Since they are identical, wins are initially driven by chance. Victories increase power. As a result, the resource falls and the conﬂict spoils and power fall. Early victories have small impacts since power and therefore the spoils are small. When power grows, resource extraction and the impact of winning grow. Beginning in period 31, group 2 wins and accumulates power. While weaker, group 1 wins beginning in period 36 and its power rises, but group 2 wins more until period 86, after which they win with relatively equal frequency. In Fig. 2, the resource trajectory (not shown) resembles the one in Fig. 2. Group 1 fully mobilizes for ﬁghting in periods 15, 35, 43–47, 116–119 and 122, as its power falls below its unconstrained best

12000 10000 Group 1 Power Group 2 Power Resource Stock

Power

8000 6000 4000 2000 0

0

20

40

60

80

100

120

Period Fig. 1. A case without full mobilization and surrender.

140

Power, Fighting Effort

9000 8000 7000 6000 Group 1 Power Group 1 Fighting Effort

5000 4000 3000 2000 1000 0

0

20

40

60

80

100

120

140

Period Fig. 2. A case with full mobilization and without surrender.

250

Power, Fighting Effort

our assumptions. The literature offers a calibration for this case, though without considering conﬂict. Our strategy is thus to rely on this set for the non-conﬂict parameters, and to rely on the Hirshleifer literature for the conﬂict parameters. In Polynesia, the resource composite was primarily renewable. Since the ﬁghting and extraction activities were people-intensive, we can think about power as measured in people. This conceptualization may not seem applicable today, but people are still by far the most important component of all armies. In a Polynesian setup, εi, the depreciation and destruction of power, becomes the net decline in people due to injury and death; and ϕi, the factor converting the conﬂict spoils to power, becomes the growth in the number of people due to recruitment and procreation. For the non-conﬂict parameters, we rely on the Brander and Taylor (1998) model of Easter Island, though conﬂict is not one of its features. They think about the island's resources as purely renewable. For our simulations, we do so as well. They chose K = 12, 000, noting that the carrying capacity is a matter of scaling; N(0) = 40, which is in the 20–100 range suggested in the literature for the number of settlers; S(0) = K, as the resource was fully in place upon arrival; r = 4% per decade, which, they argue, is about right for the island; ε = − 0.1 and ϕ = 4, indicating the population falls 10% each decade if K S = 0, and grows if S≳ ; β = 0.00001, so when S = K the group 2 extracts its subsistence using 20% of its people, in line with estimates indicating a surplus of people when S was large. For our base case, we assume identical groups with ε1 = ε2 = − 0.1; ϕ1 = ϕ2 = 4, β1 = β2 = 0.00001, N1(0) = N2(0) = 40. The surrender threshold is taken to be 1, which is intuitive given that power is equated with people. The choice of α1 and α2, the ﬁghting efﬁciency parameters, is a matter of scaling. We use α1 = α2 = 1, as in Hirshleifer (1988, 1995). Brander and Taylor (1998) simulate their model for 140 decades. Since we rely on their calibration, we use 140 decades for the sake of comparison.

703

200 Group 1 Power Group 1 Fighting Effort

150 100 50 0

0

20

40

60

80

100

120

140

Period Fig. 3. A case with full mobilization and surrender.

response to group 2's ﬁghting. Still, group 1 wins some battles; its probability of victory is lower than for group 2, but it is not zero. Group 1 is able to win a victory without too much time fully mobilized. After two periods of full mobilization group 1 wins the resource and group 1's power increases to where full mobilization is unnecessary. Fig. 3 presents a case with surrender. Group 1 is fully mobilized in period 4, wins a few battles, as evident by its growing power, and loses, which forces it to fully mobilize in period 20. At this point group 1 enters a vicious cycle. Although fully mobilized, each period the probability of winning becomes more and more remote. If group 1 could win the spoils extracted by group 2 it could make substantial gains and cease full mobilization, but it never does. It surrenders after period 68 when its power falls below the critical level.16 Since these ﬁgures are based on a calibration for Easter Island, it is interesting to examine them in this context. The studies we cited suggest the settlers arrived in 400 to 700 A.D. The maximum population ranged from 7000 to 20,000, and the peak occurred in the years 1100 to 1500. When the Dutch discovered the island in 1722, they estimated there were 3000 people. In 1774, Captain Cook estimated there were 2000 people. The carbon dating records suggest a noticeable decline in forest cover circa 900 A.D. In Fig. 1, the population peaks at about 14,000 around period 53, or 530 years after arrival. Toward the end of the track, the number of people is 2000. In the Brander and Taylor no-conﬂict interpretation, the population peaks at about 10,000 around period 80, or 800 years after arrival, and is 3800 toward the end of the track. Both of these tracks are consistent with anthropological and archeological records, 16 Though not of a central interest here and therefore not shown, upon group 1's surrender, group 2 allocates all of its power to resource extraction. Its power grows until it overtaxes the resource stock and subsequently declines as the resource stock declines. As the extraction pressure on the resource stock eases, it grows back.

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R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

but the Brander and Taylor model does not include conﬂict over resource extraction, which was apparently endemic on the island. Our model may thus offer an equally plausible interpretation of the key social forces that played a role in the demise of the Easter Island society. 4.2. Comparative Dynamics Figs. 1–3 illustrate the dynamics for one calibration. This subsection examines the dynamics when we vary the calibration. We could change several parameters at the same time, but it would be difﬁcult to identify causes and effects since our system is interconnected. We therefore change one parameter at a time, holding the others constant. Next, we need numerical changes and, as before, there are many possibilities. Since we calibrated the model for Easter Island, we use it as a base case and increase, one at a time, the carrying capacity (K) by 66.666%, the resource growth (r) by 50%, group 2's ﬁghting capabilities (α2) by 50%, group 2's extraction efﬁciency (β2) by 100%, group 2's power accumulation capabilities (ϕ2) by 50%, and group 2's initial power (N2(0)) by 100%. These changes seem plausible, but they are ultimately arbitrary, aimed at illustrating the direction of the dynamic tendencies, not their size.17 The results can be presented in plots, but since the model includes a chance element, we would need many plots, which would make it hard to follow. We present average results computed based on a large sample of tracks using the same seed in Eq. (17). We decide to calculate our average results based on 2000 tracks because we ﬁnd that the averages barely change anymore if we increase the number of tracks. For a larger K or r, the groups' results differ only due to chance, as they are homogenous. A larger K raises the incidence of full mobilization (FM) (Table A1), makes earlier its ﬁrst instance (Table A2) and raises its duration (Table A3), suggesting a more forceful and volatile conﬂict. A larger r has a similar effect, except the duration of FM declines. This happens because a larger r increases the conﬂict spoils, enabling a quicker exit from FM. When α2 increases, group 2 allocates less power to ﬁghting in equilibrium, and more to extraction, since its marginal return to ﬁghting declines (see Eq. (13)). Group 1 allocates more power to ﬁghting, facing the larger extraction of group 2 and seeking to compensate for its weakness. As a result, group 2 is less likely to meet its available power constraint and its FM incidence declines, its ﬁrst FM occurs later, and its FM duration falls. Group 1 exhibits the opposite effects, as it ﬁghts harder. When β2 rises, group 2 allocates less power to ﬁghting in equilibrium, and more to extraction, since its marginal return to ﬁghting declines (see Eq. (13)); group 1 exhibits the opposite effect. As a result, group 2's FM incidence falls and its ﬁrst FM occurs later. Group 1's FM incidence rises and its ﬁrst FM occurs sooner. When group 2 is fully mobilized, the spoils are smaller than when group 1 is fully mobilized since group 1 extracts less due to its smaller extraction allocation. Since it is now harder for group 2 to exit the FM state when it wins and converts the spoils to power, its FM duration increases; group 1's FM duration declines. With a larger ϕ2, group 2's power grows faster over time when it wins. Since it now allocates relatively less effort to ﬁghting, its FM incidence and duration decline and its ﬁrst FM occurs later. Group 1 needs to ﬁght harder, facing a stronger opponent, and therefore its FM incidence and duration increase and its ﬁrst FM occurs earlier. When N2(0) is larger, group 2 enters FM less frequently and starts to do so later, having more initial power; group 1 does the opposite. Both groups exhibit a larger FM duration since the spoils are larger 17 From here on, the phrases “compared with the base case” and “average” are dropped but are understood.

Table A1 Incidence of full mobilization in runs where neither surrenders. Experiment

Group 1 [%]

Group 2 [%]

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

74.013 79.275 76.757 95.537 90.209 97.625 79.353

73.193 78.050 76.603 38.470 66.149 44.834 65.639

Table A2 Average ﬁrst time period of entering full mobilization. Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

52.804 45.834 51.523 23.326 24.607 17.306 42.335

54.491 47.733 52.918 95.613 62.245 84.833 67.477

across the board, increasing the dominance of the victor. Group 2 exhibits a larger rise in FM duration than group 1. Since group 1 has less power, the spoils rise less when group 2 is in FM than when group 1 is in FM, making it easier for group 1 to exit from FM compared to group 2. We turn to incidence of surrender in Table A4. When K or r rises in Table A4, the size of the spoils and the impact of defeats are larger and so the incidence of surrender rises slightly. When α2, ϕ2, or N2(0) rises, group 2's incidence of surrender falls and that of group 1 rises. When β2 is larger, group 2 allocates less power to ﬁghting and more to extraction; group 1 does the opposite. Group 2 now has a smaller chance of winning larger spoils, and group 1 has a larger chance. These factors work against each other. In our case, the incidence of surrender falls for both groups—the larger total spoils can keep both groups alive longer provided both groups get some access to it. If, however, β2 is much larger than β1 group 2 might well surrender before it can ever gain from its better extraction. Table A5 examines the surrender time. For runs without surrender, the time is set to 140 (the track's end). This generates numbers close to 140 since surrender is rare (Table A4). Increasing K or r leads to an earlier surrender time, as the conﬂict spoils are larger and the ﬁghting intensiﬁes. When α2, ϕ2 or N2(0) is larger, group 2 surrenders later; group 1 exhibits the opposite effect. When β2 is larger, the surrender time declines for both groups. Reﬂecting the above competing effects, in our case group 2 surrenders later than group 1. Table A6 examines the peak conﬂict or ﬁghting allocation. With a larger K or r, the peak conﬂict rises for both groups due to the larger conﬂict spoils. When α2 rises, group 2's ﬁghting effort and therefore peak conﬂict fall. Group 1 ﬁghts harder and its peak rises but only slightly. The greater probability of losing conﬂicts means group 1 generally has less to allocate to ﬁghting. When β2 is larger, the larger

Table A3 Average duration of full mobilization. Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

3.246 4.247 3.048 4.349 2.717 4.381 3.277

3.009 4.148 2.935 2.286 4.545 2.799 3.143

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712 Table A4 Incidence of surrender.

705

Table A7 Average duration of intense conﬂict.

Experiment

Group 1 [%]

Group 2 [%]

Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

1.400 3.300 1.450 5.650 1.800 15.250 2.100

1.050 2.850 1.100 0.250 2.050 0.550 0.550

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

0.007 12.814 0.024 0.020 7.288 6.501 0.008

0.009 12.821 0.029 0.000 0.978 8.028 0.005

Table A5 Average time period of surrender.

5. Extending the Base Model

Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

138.797 138.112 138.756 135.210 131.218 131.179 138.049

139.101 138.169 139.056 139.779 136.544 139.491 139.550

Table A6 Average maximum conﬂict. Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

3431.409 6041.927 3841.422 3558.484 5184.579 4767.900 3416.859

3441.061 6051.899 3847.464 3035.029 4007.566 5343.069 3436.858

In the base model, groups access a common pool resource and win based on a uniform distribution. This section changes these assumptions. 5.1. Power-based Resource Access Eq. (2) assumes open access to the resources, but groups ﬁghting over the extracted resource may try to block access to the resource. We model this notion by assuming access depends on power: Sit = Nit/ (N1t + N2t)St. Consequently, the extraction is given by: Git = βi

Fit

18 Whereas group 1's share of ﬁghting allocation is larger than group 2's share when group 1's ﬁghting efﬁciency is smaller than that of group 2, which is an equilibrium effect, it has a smaller peak ﬁghting than group 2, which is an equation of motion effect. 19 The ﬁgure 5000 is about half of the simulations' peak power. As long as this threshold is reasonably large, the results hold qualitatively.

ð24Þ

i = f1; 2g:

The power-based access in Eq. (24) changes the optimization problem in Eq. (7) to: max

spoils drive a larger peak, though less so for group 2 as it allocates less to conﬂict. When ϕ2 is larger, group 2's peak rises, inducing a similar though weaker effect for group 1.18 With a larger N2(0), group 2 can afford a smaller peak conﬂict, driving a similar though smaller effect for group 1. Table A7 examines the duration of intense conﬂict, deﬁned as ﬁghting effort larger than 5000.19 For runs without intense conﬂict, the duration is set to 0. This creates small numbers since intense conﬂict is rare. With a larger K or r, the duration rises for both groups due to the larger conﬂict spoils. When α2 rises, group 2's ﬁghting allocation falls, reducing its duration of intense conﬂict. Group 1's duration rises as its ﬁghting effort rises. When β2 grows, both durations rise, though more so for group 1 whose ﬁghting effort rises. When ϕ2 is large, the durations rise, though more so for group 2 whose power is larger. Finally, with a larger N2(0), group 2 can afford to ﬁght intensely for a shorter time; group 1 exhibits the opposite effect. In sum, a larger carrying capacity or resource growth intensiﬁes the conﬂict and makes the groups more prone to FM and surrender. A group with more ﬁghting capabilities ﬁghts less intensely and is less prone to FM and surrender than its rival. A group better at extraction exhibits mixed effects: it is less prone to FM, ﬁghts more intensely, though less than its rival, and may be more or less prone to surrender. A group better at accumulating power ﬁghts more intensely and is less prone to FM and surrender. A group initially more powerful ﬁghts less intensely and is less prone to FM and surrender.

Nit SE ; N1t + N2t t it

αi Fi N1 N2 β1 SðN1 −F1 Þ + β2 SðN2 −F2 Þ α1 F1 + α2 F2 N1 + N2 N1 + N2

s:t: Fi ≥ 0; Ni −Fi ≥ 0

i = f1; 2g

Proceeding as in the base model, the reaction functions when Fi b Ni, i = {1, 2} are:

F1 =

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 0 β21 N12 α22 F22 +β21 N13 α1 α2 F2 +β1 N1 α1 β2 N22 α2 F2 −β1 N1 α1 β2 N2 F22 α2 1 @ A −α2 F2 β1 + α1 β1 N1

ð25Þ

F2 =

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ1 0 β22 N22 α21 F12 +β2 N2 α2 β1 N12 α1 F1 −β2 N2 α2 β1 N1 F12 α1 +β22 N23 α2 α1 F1 1 @ A: −α1 F1 β2 + α2 β2 N2

ð26Þ Solving the system of equations given by Eqs. (25) and (26), the interior equilibrium ﬁghting efforts are: pﬃﬃﬃﬃﬃﬃ 2 2 α2 β1 N1 + β2 N2 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ and F1 = pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ α1 β2 N1 + α2 β1 N2 2 β1 N1 pﬃﬃﬃﬃﬃﬃ α1 β1 N12 + β2 N22 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ : F2 = pﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃpﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ α1 β2 N1 + α2 β1 N2 2 β2 N2

ð27Þ

If group i is fully mobilized and group j is not, the equilibrium ﬁghting efforts are:

Fi = Ni

and Fj = −

i; j = f1; 2g; i≠j:

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ αi Ni − αi Ni αi Ni + αj Nj αj

ð28Þ

706

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

Table B1 Incidence of full mobilization in runs where neither side surrenders.

Table B3 Average duration of full mobilization.

Experiment

Group 1 [%]

Group 2 [%]

Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

94.619 96.713 94.721 99.895 97.835 99.943 100.000

93.909 96.023 94.061 61.675 89.062 59.943 91.582

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

2.472 2.939 2.471 3.368 2.691 3.962 2.467

2.298 2.980 2.304 1.818 2.729 2.072 2.340

When neither group is fully mobilized, the conﬂict spoils are given by: 2 2 1 S β1 N1 + β2 N2 : G= 2 N1 + N2

ð29Þ

When group i is fully mobilized and group j is not, the conﬂict spoils are given by: rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ1 αi Ni − αi Ni αi Ni + αj Nj B C Ni BN + C G = Sβj A N1 + N2 @ j αj 0

ð30Þ

i; j = f1; 2g; i≠j: Comparing Eqs. (15) to (29), or Eqs. (16) to (30), the power-based access spoils are smaller than the open access spoils. Victory is determined by Eq. (17). Substituting the modiﬁed spoils expressions in the equations of motion (18) and (19), we get the four dynamic systems shown in Appendix A.1. For the comparative dynamics, we use the base model's calibration and parameter changes and examine average comparative dynamics compared with the base case, for 2000 tracks of 140 periods. The results generally mimic those for the base model, though there are some differences, which we discuss below. In Tables B1–B3, FM is more frequent, begins earlier, and lasts for a shorter period, compared with the base model. Power-based access reduces the conﬂict spoils and makes it harder to maintain dominance. When N2(0) is larger in Tables B1–B2, the power gap forces FM for group at the start of each track. In Table B3, group 1's FM duration falls, unlike in Table A3, since the initial power advantage dissipates more quickly. In Table B2, when β2 is larger, group 2's FM occurs earlier, unlike in Table A2. Group 2 still allocates less power to ﬁghting, delaying FM, but the conﬂict spoils are smaller, expediting FM. In Tables B4–B5 surrender is less frequent and begins later than in Tables A4–A5. By reducing the spoils, power-based access makes it harder for one group to obtain complete dominance. When α2, ϕ2, and N2(0) are larger, group 1 surrenders more often than group 2. In these cases, group 2 is more able to restrict the access to the resource for group 1, a feature not included in the base model.

Table B2 Average ﬁrst time period of entering full mobilization, weighted.

In Tables B6–B7, the level and duration of peak conﬂict are smaller than in Tables A4–A5, as the spoils are smaller. When N2(0) is larger, the peak conﬂict rises for both groups, unlike in the base model. Since group 1 fully mobilizes at the start of each run, group 2 also fully mobilizes more frequently. As a result, the incidence of peak conﬂict rises. In sum, compared with an open access model, a power-based access model exhibits a smaller conﬂict spoil, which increases the incidence of FM and reduces the time of its onset and duration. Surrender occurs later and less frequently, and the duration and level of the peak conﬂict decline. 5.2. Conﬂict Decisiveness In the base model, the stochastic element comes from a uniform distribution. This section changes this distribution to make the dominance of a stronger group more decisiveness. As before, Pi = αiFi/(αiFi + αjFj) measures the relative conﬂict power of group i, and group i is dominant if αiFi N αjFj. Recalling Eq. (17), we can write for the base model: p

i χðυÞdυ; Prð group i winsÞ = ∫−∞

ð31Þ

where Pr denotes probability and χ is the characteristic function of the interval [0, 1]. We seek an alternative to χ such that when group i is dominant (pi N 0.5) it is relatively more likely to win. A normal distribution with a mean of 0.5 as in the uniform distribution might seem a good

Table B4 Incidence of surrender. Experiment

Group 1 [%]

Group 2 [%]

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

0.700 2.900 0.700 4.450 3.950 11.550 1.450

0.800 2.800 0.800 0.050 1.000 0.200 0.550

Table B5 Average time period of surrender.

Experiment

Group 1

Group 2

Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

43.278 30.391 43.206 20.753 20.039 15.206 0.000

45.960 32.576 45.684 87.579 40.493 78.398 52.947

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

139.643 138.687 139.643 137.398 134.793 133.648 138.829

139.546 138.860 139.546 139.949 138.751 139.826 139.723

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712 Table B6 Average maximum conﬂict.

707

Table C1 Incidence of full mobilization in runs where neither surrenders.

Experiment

Group 1

Group 2

Experiment

Group 1 [%]

Group 2 [%]

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

708.385 5256.594 797.361 1076.918 3373.978 3335.080 860.857

703.475 5243.148 789.486 891.667 2576.911 3302.807 867.946

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

88.844 90.890 90.652 99.541 96.384 99.593 94.985

87.986 89.573 89.742 52.167 76.349 55.226 82.928

The optimization problem facing each group is now given by:

candidate, but it is problematic. To see that, we substitute its characteristic function in Eq. (31): Prð group i winsÞ =

pi ∫−∞

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e 2πσ 2

−ðυ−μ Þ2 2σ 2

max

!

ð32Þ

dυ:

Consider a case where Fi = 0, in which pi = 0. For the uniform distribution assumption, we obtain: 0

Prð group i winsÞj pi = 0 = ∫−∞ χðυÞdυ = 0;

ð33Þ

or group i never wins if it does not ﬁght, which is both intuitive and accords with real world conﬂict. For the normal distribution, however, when Fi = pi = 0, we get: Prð group i winsÞj pi = 0 =

0 ∫−∞

1 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ e 2πσ 2

−ðυ−μ Þ2 2σ 2

!

dυ N 0;

ð34Þ

or group i faces a positive probability of victory even if does not ﬁght. This possibility does not ﬁt real-world conﬂict. One can modify the normal distribution by setting it to zero outside the interval [0, 1] and rescaling the new function to make the area underneath it equal to 1, but this approach makes the math nearly intractable. Seeking a simpler function that makes the conﬂict more decisive by placing greater weight on the likelihood of victory for a dominant group, we use a parabolic characteristic function. g ðυÞ =

6υ−6υ2 0

for 0≤υ≤1 : otherwise

ð35Þ

The area under this function equals 1, making it a valid probability measure. It is also symmetric around a mean of 0.5, and it places a higher probability of victory for group i when pi N 0.5 than does the uniform distribution, which is what we need. Appendix A.2 shows that using g(υ) is equivalent to using a uniform distribution on the interval [0, 1] while replacing the base model's contest success function (5) by the function Pit = Prð group i winsÞ =

α2i Fi2 ðα1 F1 + 3α2 F2 Þ ðα1 F1 + α2 F2 Þ3

i = f1; 2g:

ð36Þ

Fi

α2i Fi2 ðα1 F1 + 3α2 F2 Þ S½β1 ðN1 −F1 Þ + β2 ðN2 −F2 Þ ðα1 F1 + α2 F2 Þ3

s:t:Fi ≥ 0; Ni −Fi ≥ 0

ð37Þ

i = f1; 2g:

The interior and FM equilibrium solutions and their respective dynamic systems are shown in Appendix A.3. For the simulations, we use the base model calibration and parametric changes and compute averages for 2000 tracks of 140 periods each, compared with the base case. The results generally resemble those for the base model. We discuss the differences below. In Tables C1–C3, FM occurs more frequently and begins earlier than for the base model. The effects of improving group 2's ﬁghting resemble Tables A1–A3, but they are more pronounced since the conﬂict form is more decisive. Compared with the base model, group 1's FM duration rises relatively more when α2, ϕ2, and N2(0) rise because group 2's likelihood of victory now rises more, pushing group 1 more to its limit. Two effects in Table C3 differ from Table A3. Group 1's FM duration rises when β2 rises since the gain from group 2's larger extraction in Table A3 is outweighed by the tendency to ﬁght harder for a more decisive conﬂict form. When N2(0) rises, group 2's FM duration falls because group 2's dominance has a relatively more decisive effect. Surrender is less frequent in Table C4 than in Table A4, and occurs later in Table C5 than in Table A5 because the likelihood of victory rises when a group is more dominant. The effects on group 1 are now more pronounced since it is less likely to win overall. The effects for group 2 are less pronounced than in the base model since group 1 now faces a larger probability of winning when it is dominant due to the chance element in the model.

Table C2 Average ﬁrst time period of entering full mobilization, weighted. Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

36.653 31.165 35.423 15.422 15.848 12.145 18.697

38.384 34.284 37.404 82.924 52.206 73.866 50.177

Table C3 Average duration of full mobilization.

Table B7 Average duration of intense conﬂict. Experiment

Group 1

Group 2

Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

0.003 4.335 0.012 0.009 0.717 0.949 0.000

0.001 4.298 0.007 0.000 0.285 1.437 0.004

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

3.227 3.956 3.123 4.836 3.284 5.239 3.277

3.226 3.912 3.126 2.334 3.846 2.700 3.187

708

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

Table C4 Incidence of surrender.

Table C7 Average duration of intense conﬂict.

Experiment

Group 1 [%]

Group 2 [%]

Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

0.550 2.700 0.600 1.800 3.200 13.650 1.000

0.400 2.350 0.450 0.150 0.800 0.250 0.300

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

0.067 17.784 0.198 0.139 7.940 5.896 0.063

0.076 17.753 0.195 0.024 2.099 8.919 0.082

Table C5 Average time period of surrender. Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

139.502 138.960 139.458 138.414 131.782 133.528 139.058

139.680 139.079 139.634 139.870 137.940 139.792 139.777

Compared with the base model, the peak conﬂict in Table C6 is lower, and its duration in Table C7 is larger. The more decisive conﬂict motivates allocating more power to ﬁghting and less to extraction, lowering the peak conﬂict. The duration of intense conﬂict rises for both groups since they now have a greater incentive to ﬁght intensely. Table C6 resembles Table A6, except that a larger N2(0) raises group 2's peak conﬂict. With a more decisive conﬂict, a stronger group 2 wins more battles early on and can ﬁght more intensely. Table C7 resembles Table A7, expect that a larger N2(0) reduces group 1's intense conﬂict duration. The more decisive conﬂict weakens group 1's power to the point of becoming less able to ﬁght intensely. In sum, compared with the base model, a more decisive form of conﬂict increases the incidence of FM and reduces its onset time. The duration of FM rises and more so for the weaker group. Surrender is less frequent and occurs later, the peak conﬂict declines, and the duration of intense conﬂict rises. 6. Applying the Model The story of Easter Island is famous, but apparently not unique. Other examples of societal collapse thought to have been precipitated by resource conﬂict include the Sumerian, Maya, Zulu, and Anasazi societies (e.g., Ponting, 1991; Redman; 1999; Keegan, 1993; Kirch, 1997; Diamond, 2005). Our models may apply to these cases, but this section seeks to examine how they might apply today. At ﬁrst glance, our assumptions do not seem to hold today. Unlike in our model, even the poorest and most unstable LDCs have a government, and their economies include non-resource sectors, trade with other countries, and receive foreign investments and aid. However, governments in LDCs are often weak and politically unstable. LDCs also depend more on resources, do not trade much, and receive little investment and aid. All of these factors are in line

Table C6 Average maximum conﬂict. Experiment

Group 1

Group 2

Base case Larger carrying capacity Larger resource growth Larger group 2 ﬁghting efﬁciency Larger group 2 production efﬁciency Larger group 2 power accumulation Larger group 2 initial power

3212.266 6770.642 3723.408 3516.184 5221.502 4801.032 3198.368

3214.038 6782.283 3726.916 2947.356 4392.171 5573.867 3224.641

with our model. The situation in DCs is less in line with our model, but like LDCs they too operate in international anarchy, may face domestic enforcement problems, and depend on water, agriculture, oil, and minerals. While resource conﬂicts are more prevalent in LDCs, DCs also ﬁght over resources (Section 2). Thus the gist of our assumptions seems to hold to a varying extent in the real world. As with any ecological-economic model, ours is, of course, a simpliﬁcation. Our reliance on a speciﬁc modeling approach and calibration also implies that our results may not necessarily hold equally well in all cases. To the extent that our assumptions hold, however, our results may still tell us something about the real world in the cautious sense suggested by scholars such as Kirch (1997), Brander and Taylor (1998), Diamond (2000, 2005), Vitousek (2002), Flenley and Bahn (2003), and Kirch and Kahn (2007) for applying their ﬁndings on historical Polynesia to a broader context. For example, while we have not explicitly modeled the mechanics of external intervention, our comparative dynamics can be thought of as representing several types of intervention. The results obtained for a larger S(0), K, r, or β may inform policy makers of the expected effect on the conﬂict of increasing resource-related external aid, technology, trade, and investment. The results for a larger α, ϕ, or N(0) may provide insights on the expected effect of providing more arms, military technology, and other military assistance. The spirit of the power-based access model version may represent what happens when the system becomes less anarchic in the sense that the actors are more restricted. The more decisive form of the conﬂict model version may represent what happens when there is more anarchy in the sense that the actors are more able, or less restricted, to exploit their dominance. The examination of the models' implications for policy requires a policy goal. It is reasonable to assume that a group would seek to achieve an overall victory, however the goal for society is not clear since both groups belong to society. Taking a position, we assume that society strives to reduce resource conﬂict and promote cooperation. A group seeking to achieve an overall victory should improve its ﬁghting capabilities and conversion of resource extraction to power. When it becomes dominant, or relatively stronger, the group should seek external aid and investments in the resource sector, which would make the rival relatively more prone to FM and surrender. A move to improve resource extraction, in contrast, may backﬁre. It would make a group less prone to FM, but may make it more prone to surrender. In implementing these policies, the group could sell extracted resources to a third party and use the proceeds for recruitment, training, and buying arms. Since both groups seek to win, these policies will likely be self-defeating, leading to longer and more intense conﬂicts. A society seeking to reduce resource conﬂict could join the stronger group, making its rival more prone to surrender. This approach, however, may be costly in terms of human life and economic damage, as the duration of intense ﬁghting would rise. To be sure, we do not argue that the stronger side should be assisted regardless of the ideas it stands for, but a third party threat to strengthen the stronger side may go a long way toward inducing the weaker side to come to the negotiation table. Alternatively, the society could restrict interventions that strengthen the resource sector as long as the ﬁghting continues,

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

including trade, aid, and investment. Curtailing the supply of arms to both groups can also go a long way toward reducing the conﬂict. The more decisive conﬂict form extension further illustrates the importance of these restrictions, reducing the risk that the combatants would adopt conﬂict as a way of life and become very good at it. Representing the ﬂip side of this risk, a third party interested in reducing the conﬂict could limit access to the resource to both groups, as suggested by the power-based access extension. Finally, though not of a central interest here, one may conclude that the groups would have more resource wealth should they stop ﬁghting, cooperate in extraction, and share the proceeds. This intuition holds in our model if the ﬁghting is very destructive. Perhaps paradoxically, conﬂict that diverts power from extraction to low-intensity ﬁghting may beneﬁt a resource dependent economy more than cooperation that allocates all the power to extraction. Cooperation may deplete the resource faster than it can grow, ending in a Malthusian trap. “A small enough” conﬂict may be beneﬁcial, though allocation of power to resource and non-resource peaceoriented economic activities would be even more beneﬁcial. 7. Conclusion Our dynamic models of conﬂict over resource extraction introduce new features to the Hirshleifer and the ecological economics literatures, including victory driven by chance, though not pure chance; power asymmetries between groups; heterogeneous ﬁghting and resource extraction capabilities; power-based resource access; two forms of conﬂict decisiveness; the possibilities of FM for ﬁghting; and the possibility of the conﬂict ending endogenously via surrender. Summarizing our key ﬁndings, increasing the resource carrying capacity and growth rate intensiﬁes the ﬁghting. A group that is better at ﬁghting, accumulates power faster, or is initially more powerful ﬁghts less intensely than its rival and is less prone to FM and surrender. A group that is better at resource extraction ﬁghts more intensely, though less than its rival, and may be more prone to surrender. Powerbased resource access reduces the intensity of ﬁghting, while a more decisive form of conﬂict reduces the peak conﬂict, but prolongs periods of intense conﬂict and reduces the incidence of surrender. Taking a broader view, our models generate dynamic outcomes that are often observed in real-world conﬂict over resource extraction, including resource decline when the conﬂict intensity peaks; victories by a weaker party; the winner taking all of the conﬂict spoils in a certain engagement; repeated engagements over time; changes in the relative dominance of the ﬁghting groups; victories following defeats and vice verse; periods of FM to ﬁghting; and the surrender of one group, which ends the conﬂict. Our ﬁndings for the comparative dynamics suggested several policy implications, assuming the goal is to reduce the conﬂict. External aid intervention strengthening resource dependent economies may exacerbate ongoing resource conﬂicts and should therefore be conditioned on ending the ﬁghting. External parties can reduce the conﬂict by refusing to import resources from or export military-related goods to conﬂict regions. Third parties may be able to bring a weaker reticent group to the negotiation table by threatening to strengthen its rival. Finally, a third party can reduce the ﬁghting intensity by limiting access to the resource, though this may prolong the conﬂict over time. These policies may seem harsh. Curtailing development aid, trade, and investments, limiting access to resources, and threatening to intervene in the conﬂict are costly for all the actors involved and under certain scenarios may end up hurting non-combatants and even draw third parties into the ﬁght. Our message is not to abandon all aid, stop all trade and investment, limit resource access to everyone, and resort to brinkmanship, but rather to highlight the limitations and consequences that might result from not taking such actions. To be sure our model is simple when compared to real-world conﬂicts and as such some interventions may be less successful in

709

certain circumstances than our model suggests. Our policy recommendations may also depend to some extent on our model assumptions, like all models, though we know that their general dynamics are robust across three different variations of the model. These dynamics also depend on our choice of parameters and make more sense in the context of closely matched actors who come into conﬂict repeatedly. For example, if one group is overwhelmingly better at ﬁghting efﬁciency, the conﬂict will not last long and the dynamics are uninteresting. Alternatively, if the natural resource stock replenishes slowly enough, or does not replenish at all, the power of the two groups would ultimately decline in the model to zero due to depreciation and normal wear and tear, regardless of any conﬂict. It is in this sense that we see this article as a step along a research path rather than its ﬁnal statement. One may extend our models in a number of ways, though the math may not be easy. For example, one may model the groups as comprised of subgroups with different goals. A second extension may model the mechanics of external intervention. One may also assume that one group attacks ﬁrst, and the target decides whether to respond. Other models may introduce growing resolve and willingness to ﬁght for the group as the conﬂict intensiﬁes and the group's power declines. Going beyond the Hirshleifer approach, one may integrate Welsch's (2008) two sector static model of labor allocation to production and resource looting with our dynamic models of ﬁghting. One may further augment this model to include more than one resource stock. Perhaps mathematically more complicated, one may introduce conservation norms or property rights as means to bring peace, using Suzuki and Iwasa (2009) and BenDor et al. (2009) as a starting point. In principle, it is not clear whether the actors would voluntarily agree to change their behavior since the beneﬁts from conservation or property rights tend to accrue in the future, while groups engaged in conﬂict care urgently about the present. Regardless of the direction of future research, increasing our understanding of conﬂict over extracted resources is important. Looking ahead, climate change will likely reduce the availability of renewable resources such as fresh water, arable land, timber, ﬁsh, and commodities, and continued extraction will reduce the availability of nonrenewable resources such as oil, and some minerals and metals that currently do not have good substitutes. As supply will likely fall short of demand, more and more groups may resort to conﬂict as a way to obtain resources. Viewed in this light, our ﬁndings provide a grim glimpse of what could the future look like if societies decide to ﬁght over resources instead of devising policies to manage their decline. Appendix A A.1. Power-based Resource Access In this case, the differential equation systems for the model are given by: System 1: Neither group is fully mobilized 1 Sβi Ni + Sβj Nj Group i is victorious, receiving payoff of Gi = , 2 N1 + N2 and group j receives nothing.

dNi S Sβi Ni + Sβj Nj = Ni εi + ϕi 2 dt N1 + N 2 dNj = Nj εj dt dS S S Sβi Ni + Sβj Nj = rS 1− − dt K 2 N1 + N 2 i; j = f1; 2g; i≠j:

ðA1Þ

710

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

System 2: Group i is fully mobilized and victorious Ni i receives payof f of Gi = Sβj 0 Group qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ N1 + N2 ﬃ1 αi Ni − αi Ni αi Ni + αj Nj B C @Nj + A and Group j receives nothing. αj

0 dNi Ni B BN + = Ni εi + ϕi Sβj dt N1 +N2 @ j

α1 F1 −0 = Q−0 G α1 F1 + α2 F2

ðA6Þ

Q

= ∫ dυ:

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ1 αi Ni − αi Ni αi Ni +αj Nj C C; A αj

0

That is K

Q

0

0

∫ gðυÞdυ = ∫ dυ;

dNj = Nj εj dt

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 0 ﬃ1 αi Ni − αi Ni αi Ni +αj Nj C dS S Ni B BN + C = rS 1− −Sβj A N1 +N2 @ j dt K αj

i; j = f1; 2g; i≠j:

ðA2Þ System 3: Group i is fully mobilized and group j is victorious 0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ1 αi Ni − αi Ni αi Ni + αj Nj Ni B C Group j receives Gj =Sβj @Nj + A, αj N1 + N2

and group i receives nothing. dNi = Ni εi dt

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ1 αi Ni − αi Ni αi Ni + αj Nj B C dNj Ni BN + C = Nj εj + ϕj Sβj A dt N1 + N2 @ j αj rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 αi Ni − αi Ni αi Ni + αj Nj B C dS S Ni BN + C: = rS 1− −Sβj A dt K N1 + N2 @ j αj i; j = f1; 2g; i≠j:

ðA3Þ System 4: Group i surrenders We assume group i is absorbed by its rival, and the conﬂict ends. Group j is assumed to allocate all of its effort, N, to resource extraction and convert its entire product each period to effort. dN dS S = Nεj + ϕj βj SN and = rS 1− −βj SN: dt dt K

α1 F1 α1 F1 2 ∫α0 1 F1 + α2 F2 g ðυÞdυ = ∫α0 1 F1 + α2 F2 6ν−6ν dυ 2 3 α1 F1 α1 F1 −2 =3 α1 F1 + α2 F2 α1 F1 + α2 F2

2 2

= Thus, we use Q =

α1 F1 ðα1 F1 + 3α2 F2 Þ : ðα1 F1 + α2 F2 Þ3

α21 F12 ðα1 F1 + 3α2 F2 Þ ðα1 F1 + α2 F2 Þ3

in Eq. (37).

A.3. Conﬂict Decisiveness When we draw ν from Eq. (35), each group solves the optimization problem (37). The solution for this problem is found by the procedure we used for our base model, though the math is much more cumbersome. If neither group is fully mobilized the interior solution for group ﬁghting efforts is given by the following ﬁghting allocations: 3N α2 α k + N1 α32 + α31 k4 N2 + 3α21 k3 N2 α2 F1 = 6 1 2 21 31kα2 α1 + 9α32 + 9k3 α31 + 31k2 α21 α2 2

F2 = 6

3

3 4

2 3

3N1 α2 α1 k + N1 α2 + α1 k N2 + 3α1 k N2 α2 : k 31kα22 α1 + 9α32 + 9k3 α31 + 31k2 α21 α2

ðA9Þ

In these expressions, k is a constant such that 3 4

2

3

2

3

β1 α1 k + 3β1 α1 α2 k −3β2 α2 α1 k−β2 α2 = 0:

This appendix shows that drawing ν from a new distribution is equivalent to drawing ν from a uniform distribution and altering the contest success function, as argued in Section 5.2. Let g be a function over the unit interval [0, 1], and let G be a K function such that GðK Þ = ∫ g ðυÞdυ, where K is a number in the unit 0

interval. This deﬁnition implies that G(0) = 0. α1 F1 , then: Next, let K = α1 F1 + α2 F2

Since k is the root of a fourth degree equation it can be written explicitly in terms of α1, α2, β1, and β2. However, this explicit solution is lengthy and here suppressed. If group i is fully mobilized, the equilibrium ﬁghting effort allocations are given by: Fi = Ni

α1 F1 α1 F1 + α2 F2

g ðυÞdυ α1 F1 −Gð0Þ α1 F1 + α2 F2 α1 F1 −0: =G α1 F1 + α2 F2

ðA8Þ

ðA4Þ

A.2. The Distribution of ν and the Contest Success Function

∫ g ðυÞdυ = ∫0 0 =G

ðA7Þ

where K and g are deﬁned as above. In the context of Section 5.1, K is our original contest success function, and g is the new distribution from which we draw ν. The probability of winning generated by this contest success function is equivalent to the probability of winning a contest deﬁned by the contest success function Q and drawing ν from a uniform distribution. We now derive the Q that mimics the outcome of a contest success function K when ν is drawn from the parabolic distribution (Eq. (35)). To compute the contest success function for our parabolic distribution, observe that

0

0

K

α1 F1 Now deﬁne a new function Q = G , and observe α1 F1 + α2 F2 that

1 Fj = 3 ðA5Þ

1 2 2 αi Ni ðΓÞ 3 αi

−

11 3

α2i Ni2 4 α i Ni

2 2 1 − 3 αj N αj αi Ni ðΓÞ 3

ðA10Þ

where Γ is a given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ Γ = 98αi Ni + 81Nj αj + 9 135α2i Ni2 + 196αi Ni Nj αj + 81Nj2 α2j :

R. Reuveny et al. / Ecological Economics 70 (2011) 698–712

The four systems of differential equations are given in this case by:

711

System 4: Group i surrenders

System 1: Neither group is fully mobilized Group i is victorious, receiving all the spoils Gi = S[βi(Ni − Fi) + βj (Nj − Fj)], where

dN dS S = Nεj + ϕj βj SN and = rS 1− −βj SN: dt dt K

3N α2 α k + N1 α32 + α31 k4 N2 + 3α21 k3 N2 α2 F1 = 6 1 2 21 31kα2 α1 + 9α32 + 9k3 α31 + 31k2 α21 α2

References

2

3

3 4

2 3

3N α α k + N1 α2 + α1 k N2 + 3α1 k N2 α2 ; F2 = 6 1 2 12 k 31kα2 α1 + 9α32 + 9k3 α31 + 31k2 α21 α2 and group j receives nothing. dNi = Ni εi + ϕi S βi ðNi −Fi Þ + βj Nj −Fj dt dNj = Nj εj dt dS S = rS 1− − βi ðNi −Fi Þ + βj Nj −Fj dt K

ðA11Þ

i; j = f1; 2g; i≠j:

System 2: Group i is fully mobilized and victorious Group i receives all the spoils Gi = Sβj 11 3

α2i Ni2

1 αj α2i Ni2 ðΓÞ 3

−

4 αi Ni 3 αj

ÞÞ

ð ð

1 1 α2i Ni2 ðΓÞ 3 Nj − − 3 αi

, and group j receives nothing. 11

0

0 1 2 2 3 B B1 αi Ni ðΓÞ dNi 11 B N − − = Ni εi + ϕi Sβj B j @ @3 dt αi 3

C 4 αi Ni C CC; − 1 AA α 3 j αj α2i Ni2 ðΓÞ 3

α2i Ni2

dNj = Nj εj dt

0 0 1 2 2 3 B B1 αi Ni ðΓÞ dS S 11 Nj −B − = rS 1− −Sβj B @ @ αi dt K 3 3

11 C 4 αi Ni C CC

2 2 1 − 3 αj AA 3 αj αi Ni ðΓÞ α2i Ni2

i; j = f1; 2g; i≠j:

ðA12Þ System 3: Group i is fully mobilized and group j is victorious

ð ð

Group j receives Gj = Sβj Nj − 4 αi Ni 3 αj

ÞÞ

1 1 α2i Ni2 ðΓÞ 3 11 − 3 3 αi

α2i Ni2

αj α2i Ni2 ðΓÞ

1

−

3

, and group i receives nothing.

dNi = Ni εi dt

ðA14Þ

11

0

0 1 1 2 2 3 B B3 αi Ni ðΓÞ dNj 11 B − = Nj εj + ϕj Sβj B @Nj −@ 3 dt αi

C 4 αi Ni C CC

1 − 3 αj AA αj α2i Ni2 ðΓÞ 3

0 0 1 1 2 2 3 B B3 αi Ni ðΓÞ dS S 11 B N − − = rS 1− −Sβj B j @ @ dt K 3 αi

2 2 αi Ni

11 C 4 αi Ni C CC

2 2 1 − 3 αj AA αj αi Ni ðΓÞ 3 α2i Ni2

i; j = f1; 2g; i≠j:

ðA13Þ

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