Discussion Papers In Economics
And Business

A Dynamic Model of Conflict and Cooperation
Wolfgang Eggert

Jun-ichi Itaya

Kazuo Mino

Discussion Paper 08-27

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Dynamic Model of Conflict and Cooperation
Wolfgang Eggert

Jun-ichi Itaya

Kazuo Mino

Discussion Paper 08-27

July 2008

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANA Dynamic Model of Conflict and Cooperation∗
Wolfgang Eggert†

Jun-ichi Itaya‡

Kazuo Mino§

July 9, 2008

Abstract
We introduce a common-pool contest into a continuous-time, diﬀerential game setting
to model the dynamic behavior of agents facing a trade-oﬀ between socially productive
activities and appropriation. We are able to identify multiple Markov perfect equilibrium
strategies that are nonlinear in a state space, thus leading the economy to a state where‘partial cooperation’ occurs. We show that such cooperation can be seen as a response toconflict. We also discuss the consequences of changes in the eﬀectiveness of appropriation,
the number of contenders, and the rate of time preferences on contest equilibria.
Keywords: Conflict, Cooperation, Diﬀerential Game, Markov Perfect Equilibrium, Nonlinear Markov strategy
JEL classifications: D 74, L 11∗

Earlier versions of the paper have been presented at the meeting of the Association for Public Economic
Theory in Beijin, the International Conference on Economic Theory in Kyoto, and a CESifo Area Conference
on Applied Microeconomics. We wish to thank Makoto Yano, David , and seminar participants and Herbert
Dawid for useful discussions and are indebted to the ifo Institute for Economic Research in Munich and the
University of Hokkaido for support. The second and third author also acknowledge financial support by Grantin-Aid for Scientific Research, Socienty for the Promotions of Science in Japan (#16530117 and #17530232).
†
University of Paderborn, Warburgerstr. 100, 33098 Paderborn, Germany and ifo Institute for Economic
Research at the University of Munich. Tel: +49-5251-60-5002; E-mail: Wolfgang.Eggert@uni-paderborn.de‡
Graduate School of Economics and Business Admministration and CESifo, Hokkaido University, Sapporo,
060-0809, Japan. Tel:+81-11-706-2858; Fax:+81-11-706-4947; E-mail: itaya@econ.hokudai.ac.jp§
Graduate School of Economics, Osaka University, Osaka 657-8501, Japan. Tel/Fax:+81-6-6850-5232; Email: mino@econ.osaka-u.ac.jp

1

Introduction

There is a relatively small but growing literature in political economics initiated by Hirshleifer(1991, 1995), Skaperdas (1992) and Grossman and Kim (1995). Their models share fourcommon features. First, they postulate that conflict arises from the choice of rational andself-interested agents. Second, a well-defined and enforced property right over, at least, somegoods do not exist. Third, the agents are assumed to be myopic in a way that they maximize
only the current payoﬀ. Fourth, their model is static. This paper conducts the analysis ofconflict by extending their static models to a dynamic one.
Hirshleifer (1995) takes an initial step towards a dynamic approach by recognizing successive iterations of the one-shot game, and focuses on the convergent point of such iterations(he calls such a fixed point ‘a steady state’). Nevertheless, Maxwell and Reuveny (2005, p.31)
correctly point out that "However, this approach is not fully dynamic: it does not specify
equations of motion for any variables, time is not a variable in the model, and the condition
for dynamic stability is not derived based on standard dynamic analysis".
In response to such long-term desires, there have been several papers which attempt to construct a dynamic variation of the one-shot conflicting game analyzed by the above-mentioned
authors. Garfinkel (1990) examines a dynamic model in which agents make choices between
productive and fighting activities. She uses a repeated game setting where threats and punishments are available. Existence of cooperative (or disarmament) equilibria can be establishedusing Folk Theorem arguments. Skaperdas and Syropoulos (1996) discuss a two-period modelof conflict in which time-dependence is introduced by the assumption that second period resources of each agent are increasing in first-period’s payoﬀ. As a result, ‘the shadow of the
future’ may impede the possibilities for cooperation. In other words, competing agents engage
more in appropriation in order to capture a bigger share of today’s pie. The equilibrium solution concept we employ in this paper allows us to identify possible cooperative outcomes asa result of decentralized decision-making by agents, without having to rely on the Folk Theorem of repeated games or enforceable commitments. Nevertheless, since the one-shot gameis repeated every period due to the nature of the repeated game, it would be unsatisfactory1to describe true dynamic situations which are not ‘stationary’. More recently, Maxwell andReuveny (2005) construct a conflict model with two competing groups in which each group’s
population and a stock of common (natural)-resources both change over time. Since three nonlinear diﬀerential equations characterizing the dynamic paths of these stock variables do notallow an analytical solution, they resort to numerical simulations. These exercises reveal thatmild conflict activity depresses the use of natural resources for production, thus possibly creating a Pareto improvement compared to cooperative situations where there is no appropriate
activity, and, moreover, tends to reduce the volatility of those stocks through the transition.
Although their model generates interesting insights, they still assume that agents are myopic.
The authors in the literature have called for a full dynamic and multi-period model of the
Skaperdas-Hirshleifer-Grossman and Kim-based literature which incorporates the behavior of
non-myopic agents who taking into account the consequences of their future actions, which isalso left as an open question in Maxwell and Reuveny (2005).1 The goal of this paper is toaccomplish this task.
We develop a forward-looking agent-based infinite horizon general-equilibrium model tostudy the dynamic evolution of self-enforcing property rights. There are various ways ofextending one-shot, static models of Skaperdas, Hirshleifer, and Grossman and Kim to a
dynamic setting. Following their models, we first assume that the initial resource endowment
is fixed over time. This assumption would be defended either by interpreting the initialresource endowment as a time or labor supply, or by assuming the fixed population in order tokeep the model tractable. The relevant state variable in our dynamic model is a durable stock
which accumulates through time according to the production process using collective eﬀorts
of all parties involved. This durable stock is exhaustible or rival in the sense that one agent’s
1More recently, there is another class of dynamic conflicting models that include, e.g., Gradstein (2003)
and Gonsalez (2007). There are several important diﬀerences between the models in these papers and the one
in ours. First, in their models a flow of the output produced each period is subject to predation, while inour model a stock variable is subject to predation. Secondary and more importantly, those papers investigate
the relationship between conflict and economic growth in the standard growth model based explicitly on
the investment and saving decisions of a large number of economic agents. Hence, their models are mostly
concerned with the macroeconomic consequences, such as growth eﬀects of insecure property rights. Since
our model is a straightforward dynamic extension of Grossman, Hershleifer and Skaperdas which allows forstatic interaction among a small number of economic agents, it enables one to directly compare our resultswith those in static conflicting models and thus to highlight the strategic role of appropriation among those
few agents in the intertemporal context.

2

use of the stock does diminish its availability to other agents, and is open to appropriation byrivals due to the lack of well-defined or enforceable property rights. Hence each of the agents is
tempted by the immediate benefit attainable from capturing the stock. Natural resources andland in primitive historical societies are examples of such durable stocks or disputed wealth.
We model the incentives of agents to exert eﬀort in an attempt to defend their claims on
the stock and challenge the claims of others. All agents who succumb to the temptation reduce
their help in production of the common-pool stock to increase their eﬀorts to convert claimson the common stock into eﬀective property rights. More specifically, agents derive utility(or a payoﬀ) from owning the stock of durable good and, at every instant in time, choose
how to allocate an endowment between appropriation of the common-pool stock (creating
property rights) and participating in the production process to accumulate the common-poolstock in the economy. The production and appropriation decisions made independently andnoncooperatively by each of the contenders jointly determine the evolution of the commonly
accessible stock.
We present a tractable version of a diﬀerential game formulation of this model of conflict
between several agents who attempt to appropriate a common-pool durable stock over an infinite horizon. The solution concept employed is Markov perfect (MP) equilibrium, restricting
strategies to be functions of the current payoﬀ-relevant state variable. Not all the strategies
that describe a solution of the intertemporal optimizing problem of an agent are MP equilibria. The key to determining which describe equilibrium outcomes is subgame perfection
over the global domain of a state variable. In spite of this natural but stringent requirement,
there are multiple non-linear MP equilibrium strategies in our model. This multiplicity ofstrategies has the following characteristics and implications. First, most of those solutions
commonly reveal that initially poor countries will exhibit an increase in appropriation as theaggregate stock of durable good gets larger until a steady state is reached. Second, on theother hand, in economies with an aﬄuent endowment of natural resources the ‘marginal gain’of appropriation is high and agents substitute appropriation for production for a while until
the state variable reaches a threshold level. From that threshold onwards, agents choose toengage in production activity to some extent until a steady state is reached where the output
3

of production is only just suﬃcient to replace the stock of durable goods. This result relates
to the observation that rent-seeking activities in rich countries may result in deindustrialization as suggested by the literature on the resource curse (e.g., Sachs and Warner, 1999; Auty,
2001).2 Third, our results also reveal that even if two economies are ‘similar’ in terms of theinitial levels of common-pool endowment, production technology or preferences of agents, theeconomies converge to diﬀerent steady states as well as follow diﬀerent transitional paths. Put
it diﬀerently, the model predicts that ‘similar’ countries converge to a low-income steady statewith more unstable property rights, and some converge to a high income equilibrium withmore stable property rights.3 Which equilibria is realized in the long run depends on whether
the coordination regarding the expectations formed players successfully is achieved or not.
Neither of the above-mentioned one-shot models has addressed this indeterminacy feature.
Fourth, in the long run property rights may be ‘partially’ enforced in the sense that appropriation and productive activities coexist, so that neither a totally peaceful (disarmed)
equilibrium nor a full-fighting equilibrium emerges as a long run outcome. The degree of ‘partially’ cooperation would vary depending on which steady state is reached among a continuumof steady states.
The organization of the paper is as follows. The next section describes the basic model.
Section 3 conducts comparative static analysis with respect to several principle structural parameters. Section 4 derives an eﬃcient solution (i.e., cooperative solution) as a reference path.
Section 5 concludes the paper. Some mathematical proofs will be given in the appendices.

2

The Model

Consider an infinite horizon economy populated by n ≥ 2 agents who strategically interact.
Each of the agents derives utility from the consumption (or services) of a common-pool as2

There is evidence that resource abundance in the definition used by Sachs and Warner (1999) is associatedwith civil war (e.g., Collier and Hoeﬄer, 2004; Hodler, 2006).
3
Auty (2001) argues that experiences in diﬀerent countries are complex and diverse. Some countries like
Malaysia, Australia, Norway, Botswana and Canada appear to have used their resources judiciously, whereas
countries like Nigeria, Mexico and Venezuela seem to have squandered their oil windfalls. According to
Acemoglu et. al. (2001) the limiting force of conflict is institutional quality as a key driver for economic growthand prosperity.

4

set (such as land territories and natural resources) or (tangible and intangible) the stock of
durables. We want our model to capture the role of productive and aggressive activities withthe understanding that aggressive investment causes an inward shift of the aggregate production possibility frontier. Accordingly, we use a setup where appropriation and production are
two substitutable investment choices. Specifically, let an individual decide at each point intime how much resources to devote for appropriation ai ≥ 0 and production li ≥ 0. The
individual resource (e.g. time) constraint is:(1)

ai + li = ei ,where ei is the endowment of a fill-in activity that is not subject to appropriation.4 We will
set ei = 1 for ease of exposition. The time arguments have been suppressed in this and all
subsequent equations.
The common-pool stock is subject to appropriation. The stock is generated by accumulation of output. Output is produced with a linear production technology:

Y (l1 , . . . , ln ) =

n
X

lj ,

(2)

j=1

which captures the idea that higher productive eﬀorts by agents cause an outward shift ofthe production possibility frontier for the economy as a whole. The output of production canbe stored to augment the common-pool stock. However, storage entails costs such that thestock Z evolves according toŻ = Y (l1 , . . . , ln ) − δZ,(3)where δ ∈ (0, 1) is the rate at which output will depreciate if stored for future consumption, Żdenotes the change of Z over time and Z (0) ≥ 0 is the initial stock.
A main ingredient of the model is the conflict technology which, for any given valuesof a1 , . . . , an , determines each agent’s probability of winning sole possession in obtaining the4

The standard assumption that each agent has some essential property rights is implicit in this formulation.
Individual’s labor supply is such an example. Maxwell and Reuveny (2005) further assume that labor supply
is growing over time as a result of the growth of population. However, in order to keep the model simple, we
assume that the population of agents remains constant over time.

5

stock Z in a given period. To model this probability for agent i, a natural assumption is thatthe probability is increasing in aggressive investment of agent i, the fraction of time player idevotes to aggression, and decreasing in the sum of aggressive investment of all agents. A
plausible form of the conflict technology is the Tullock contest success function (Tullock,1980; Hirshleifer, 1991 and 1995; Gonzalez, 2007). In its standard formulation this function
reads:pi (a1 , . . . , an ) =

⎧
.³
´
⎪
P
⎪
⎨ari
ari + nj6=i arj
for ai > 0
⎪
⎪
⎩1/n

(4)

for ai = 0 ∀i

where the parameter r captures the eﬀectiveness of aggression. From the contest successfunction (4) we obtain the relative success of contender i in the contest. Alternatively, thecontest success function (4) may be interpreted as a sharing rule, or ownership of assets that
depends on the respective eﬀorts of aggression. It is natural to assume in the analysis thateach agent has an equal access to the prize when agents do not engage in aggressive behavior;
hence the assumption that pi (0, . . . , 0) = 1/n will be in force throughout the analysis.
The instantaneous expected payoﬀ to each agent is given by pi (a1 , . . . , an ) Z.5 Each of theagents chooses the streams of ai and li to maximize the discounted value of total expected
payoﬀs subject to the feasibility conditions introduced in (1)-(4):

max
ai

Ż =

Z ∞

pi (a1 , . . . , an ) Ze−ρt dt subject to

0

n
X
j=1

(1 − aj ) − δZ, Z (0) = Z0 ≥ 0,

0 ≤ ai (t) ≤ 1 for(5)∀t ∈ [0, ∞) ,where ρ > 0 is the rate of time preference.
5

Alternatively, one may view the prize as flow services, such as output or utility from the stock variable Z
rather than Z itself. To clarify this we need to introduce a concave function, say u (Z) instead of Z. However,this complication does not aﬀect our results at all.

6

2.1

Solution Concept

We solve the diﬀerential game using the notion of a (stationary) MP Nash equilibrium, because
we think that this equilibrium concept captures the essential strategic interactions over time.
MP strategies are decision rules such that each agent’s decision is the best response to those of
the other players, conditional on the current payoﬀ-relevant state variable Z (see, e.g., Chapter
4 in Docker et al., 2000). Markovian strategies rule out path dependence in the sense thatthey depend only on the current values of the state variables rather than strategy choices inhistory. As a result, it does not matter how one gets to a particular point, only that one getsthere.
MP equilibrium strategies must satisfy the Hamiltonian-Jacobi-Bellman equation given by:ρVi (Z) = maxai ∈[0,1]"

pi (a1 , . . . , an ) Z + Vi0 (Z)

( n
X
j=1

(1 − aj ) − δZ

)#

,

(6)

where Vi denotes the maximum value agent i attributes to the game that starts at Z. Notice
that

⎧
⎪
⎪
⎨n = 2 ∧ r > 0,n (r − 1) − 2r
∂ 2 piZ = r (n − 1)
Z < 0 for2⎪∂ai
n3 a2i⎪⎩n > 2 ∧ 0 < r < n/(n − 2),(7)

implying that the r.h.s. of (6) is concave in ai ∈ [0, 1]. We assume that r < 1/(n − 1) inwhat follows,6 guaranteeing not only that the second-order condition (7) holds but also thatthe linear strategy of each agent, which plays an important role in the later analysis, is anonnegative value. The first-order necessary condition for agent’s choice of appropriation is
given by⎧⎪⎪⎪= 0 =⇒ ai ∈ [0, 1] ,
⎪
⎪
⎪
⎨
∂pi
Z − Vi0 (Z) > 0 =⇒ ai = 1,
⎪
∂ai
⎪
⎪
⎪
⎪
⎪
⎩< 0 =⇒ ai = 0,(8)

the l.h.s. of which is evaluated for all ai ∈ [0, 1]. According to (8), each agent, when choosing ai ,6

Tullock (1980) assumes the same condition in his two-agent, rent-seeking game. Hirshleifer (1991, 1995)
and Gonzalez (2007) also assume that r < 1 in their two-agent games. Condition r < n/(n − 1) reduces tor < 1 when n = 2.

7

trades the marginal increase in expected payoﬀ from an increase in appropriation against themarginal loss in the discounted value of the future stream of payoﬀs which results from a
reduction of productive eﬀort. If the payoﬀ gain from an increase in ai is larger than the
payoﬀ loss implied by the decrease in li for all levels of ai , then agent i will rationally devote
all resources to appropriation. In contrast, the agent chooses ai = 0 in cases where thediscounted marginal gain from productive investment exceeds the instantaneous marginal gainfrom aggressive behavior for all levels of ai .

2.2

Equilibrium

We can then make use of (8) to characterize subgame perfect equilibria of the diﬀerential game.
Since we have started our analysis assuming identical agents, a natural focus is on symmetricequilibria. The symmetry assumption allows us to drop the subscript i in the subsequentdiscussion, and we will suppress this index unless strictly necessary for expositional clarity.
Let us first analyze interior solutions of ai . Diﬀerentiation of the interior first-order condition in (8) gives

V 00 (Z) =nX∂ 2 pi 0∂pi
r (n − 1) 0
r (n − 1).
aj (Z)Z +
=−
a
(Z)Z
+
2 a2
2a
∂a
∂a
∂a
n
n
i
j
i
j=1

(9)

At an interior solution of a (Z) we may apply the envelope theorem to characterize a0 (Z).
Using the symmetry assumption, we obtain
1 r (n − 1)
+ 2[(1 − a (Z)) n − (ρ + 2δ) Z]nn a (Z)0.
a (Z) =r (n − 1)
Z (n − δZ)n2 a (Z)2(10)

We will employ phase-plane methods to characterize the qualitative solution of the nonlinear diﬀerential equation (10) and the associated MP strategies. For this purpose we have to
identify the steady state locus where Ż = 0, called C1 in the following. Let us denote by C2
the loci where a0 (Z) goes to plus infinity, and by C3 the loci where a0 (Z) equals zero in the8

(Z, a) space:

C1 := {(Z, a) : Ż = (1 − a (Z)) n − δZ = 0},
C2 := {(Z, a) : a0 (Z) → ±∞},

(11)

C3 := {(Z, a) : a0 (Z) = 0}.
The steady-state line C1 is a downward-sloping, straight line in the (Z, a) space. It intersects
the vertical axis at point (0, 1) and the horizontal axis at point (n/δ, 0). Turn to C2 . Setting
the denominator in (10) equal to zero, we obtain a vertical line at point (n/δ, 0). The locus C3
is obtained by setting the numerator in (10) equal to zero. Solving for a gives the following
locus:a=−r (n − 1) ρ + 2δr (n − 1)+Z.
1 − r (n − 1) 1 − r (n − 1) n

(12)

Using 1−r (n − 1) > 0, (12) shows that the straight line C3 has a positive slope and a negativeintercept on the vertical axis, as shown in Figs. 1 and 2. Moreover, the point of intersection
between the straight lines C2 and C3 , labelled E, is situated in the nonnegative region ofthe (Z, a) plane:(ZE , aE ) =µ¶n r (ρ + δ) (n − 1),,
δ [1 − r (n − 1)] δ(13)which is called ‘a singular point’. Note, however, that since point E may be located below or
above the resource constraint (1), the value of aE may or may not be less than 1. Dependingon this value we can draw two diagrams such as in Figs.1 and 2. Moreover, it follows from (3)that any strategy a (Z) above line C1 implies that Z declines in time, while any strategy a (Z)below line C1 entails an increase of Z over time.
Collecting the arguments, we can illustrate an uncountable number of the hyperbolic curves
corresponding to the solutions satisfying the HJB equation (6) in Figs. 1 and 2. These figures
display representatives of those integral curves that are divided into five types of the familiesof strategies. Arrows on the families of integral curves aj , j = 1, . . . , 4, and aL illustrate theevolution of Z over time. In particular, by direct integration of (10) and manipulating we can

9Figure 1: Phase diagram when aE < 1.

10

Figure 2: Phase diagram when aE > 1.

11

obtain a general solution to (10):ρ+δ(n − 1) rZ (n − δZ) δ (ρ + δ)a (Z) =

ρ+δ

,

(14)

n [1 − r (n − 1)] (n − δZ) δ + (n − 1) r (ρ + δ) c1
where c1 represents an arbitrary constant of integration and may take a positive, zero or
negative value. When c1 = 0, (14) simplifies toaL (Z) =r (n − 1) (ρ + δ)Z.
[1 − r (n − 1)] n(15)

It is easy to confirm by the well-established guessing method for a value function that strategy aL stands for the linear strategy (see Appendix B). Moreover, the left branch of thelinear strategy aL to the left of the steady state line C1 starts from the origin, and thenreaches point S on the steady state line C1 , while its right branch starts from any initialvalue Z0 > ZS [if we do not here take into account the resource constraint (1)], then reaching
point S also. Moreover, it can be verified by substitution that the linear strategy aL also goes
through the singular point E.
The a4 - (a3 -) family of strategies represents the solution curve of (14) coupled with c1 > 0
when n < δZ (n > δZ), while the a1 - (a2 -) family of strategies represents the solution curveof (14) coupled with c1 < 0 when n < δZ (n > δZ). Moreover, the left-branch of the a4 family of strategies also starts from the origin, while its right-branch starts from any initialvalue Z0 < ZE , both of which reach the same point on line C1 . The left branch of the a1 -familyof strategies starts from the origin and reaches a point on the steady state line C1 , while its
right branch starts from point (n/δ, 0) and reaches the same point on line C1 ; therefore, thosestrategies never hit the horizontal axis except for the origin and point (n/δ, 0). On the other
hand, when the a2 - and a3 -families of strategies start from any initial value Z0 > ZE , the a2 family of strategies approaches point (n/δ, 0), while the a3 -family of strategies goes to plus
infinity, as illustrated in Figs.1 and 2.
Nevertheless, not all integral curves in Figs.1 and 2 are qualified as MP equilibrium strategies. There are three additional requirements which have to be met. The first prerequisite isthat strategies should not violate the resource constraint (1). This implies that a (Z) should12be bounded to the nonnegative region below a horizontal line with intercept 1 in Figs.1 and 2.
The second requirement is that strategies should cover the entire (or global) domain [0, ∞)in a continuous way.7 At first glance this requirement seems to eliminate all strategies aj ,j = 1, . . . , 4, and aL . Nevertheless, strategies can potentially be continuously extended either
by the cornered strategy a = 1 along the resource constraint (1), and/or by the non-aggressive
strategy a = 0 on the horizontal axis (see Rowat, 2007). Both potential extensions are triggeredby the corner solutions where the equality in (8) does not apply. Despite this, the strategiesof the a3 -family that does not reach the resource constraint (1) are immediately eliminatedbecause they can neither cover the global domain [0, ∞) by themselves nor be extended byany strategy in a continuous way.
Furthermore, the non-aggressive strategy a(Z) = 0 on the horizontal axis is eliminated,since this strategy does not satisfy (8) for Z ∈ (0, ∞), as shown in Lemma 1 in Appendix C.
As a result, extending the a1 - and a2 -families of strategies by the patching strategy a = 0 to
the global domain is not possible.
Turn to the linear strategy b
aL , where the hat indicates those strategies extended by thepatching cornered strategy a = 1. Since strategy b
aL can continuously pass through point E

in Fig. 1, the coordinates of which are given by (13), strategy b
aL is continuos over the entire
domain [0, ∞). This property is also obtained in the case illustrated in Fig. 2 where strategy baL

does not go through point E. Here, the patching strategy a = 1 instead of the interioraL is continuos over the entire domain of Z
strategy aL will cross locus C2 and thus strategy b

in Fig.2 as well. Taken together, the extended linear strategy b
aL survives as a candidate for asubgame perfect strategy which will be discussed below.

The third and final requirement is subgame perfection. We have to show that there do not
aL is stable in the sense that from anexist profitable deviations from strategy b
aL . Strategy b
arbitrary initial value of Z strategy b
aL can reach the steady state point S in the long run. As aresult, the convergence towards the finite steady state point S, together with (B4), ensures that
7

Tsutsui and Mino (1990), Itaya and Shimomura (2001) and Rubio and Casino (2002) restrict the state
space in order to get continuos, stable and subgame perfect equilibrium strategies. In particular, Tsutsui
and Mino treat the domain of a state variable as endogenous to get diﬀerent stable Markov perfect strategies
associated with diﬀerent steady states. Unfortunately, this approach prevents comparison of payoﬀs between
strategies.

13the value function associated with strategy b
aL is bounded. Hence we can apply the suﬃciency

conditions in Theorem 3 of Rowat’s (2007) to demonstrate its subgame perfectness.

Applying the above suﬃciency conditions and using the cornered strategy a = 1, we canconstruct more subgame perfect equilibrium strategies. As shown in Lemma 2 of Appendix
C, the range of the state variable Z where the cornered strategy a = 1 remains as an optimal
one (i.e., (∂pi /∂ai ) Z > Vi0 (Z)) depends on the chosen value of c3 in (C6). In particular, thecornered strategy a = 1 remains subgame perfect over the domain (0, ∞) if the constant c3
is larger than the infimum of the set of c3 satisfying inequality (C7) in Appendix C (see Fig.
3). When c3 is smaller than that infimum, more of the a4 -families of strategies patched bystrategy a = 1 will be subgame perfect as c3 increases to that infimum, as shown in Fig.3.
In Fig.1 (i.e., when aE < 1) the strategy a4 patched by the cornered strategy a = 1 isqualified as a subgame perfect equilibrium strategy over the domain (0, ∞) if c3 is greaterthan that infimum. If this is a case, the linear MP strategy ends up with the highest level ofsteady state durable goods stock. On the other hand, in Fig.2 the patched strategy â4 is still
qualified if c3 is greater than that critical value. In addition, the strategies left oﬀ the strategyâ1 which is tangent to the constraint a = 1 and then connected with the strategy a = 1 atthat tangent point in Fig.2 are also subgame perfect. As a result, there are non-linear MP
strategies which entail higher levels of durable goods stock in the steady state such as Ẑ as
compared to that supported by the linear strategy âL .
We summarize with the following theorem:

Proposition 1 Consider the dynamic game defined by (5) and assume that the constant ofintegration c3 is chosen so as to satisfy inequality (C7) in Appendix C. Symmetric Nashequilibria in continuos and asymptotically stable Markov perfect strategies must satisfy thefollowing:(i)If aE 5 1, thus r (ρ + δ) (n − 1) 5 [1 − r (n − 1)] δ, then there exist uncountably many
non-linear Markov perfect strategies and the unique linear Markov perfect equilibrium strategy,
coupled with strategy a(Z) = 1, that are globally defined over the entire domain of a state space,leading to steady state equilibria ranging over (0, ZS ]; and14

(ii)If aE > 1, thus r (ρ + δ) (n − 1) > [1 − r (n − 1)] δ, then there exist uncountably many nonlinear Markov perfect strategies, coupled with strategy a(Z) = 1, that are globally defined over³ithe entire domain of a state space, leading to steady state equilibria ranging over 0, Ẑ withZS < Ẑ < ZE , whereZE =nr (n − 1) (δ + ρ)and ZS =.
δ
(n − 1) rρ + ρ(16)

Proposition 1 implies that there exist multiple MP strategies in the space of nonlinear
strategies including the linear strategy, even in the case where the domain is globally defined.
It can be best understood that multiplicity of MP strategies and of the associated steady state
equilibria arise from the incomplete transversality condition, as pointed out by Tsutsui and
Mino (1990). Since there are unaccountably many asymptotically stable nonlinear Markov
strategies that converge to diﬀerent steady states, it is impossible to uniquely pin down theconstant of integration c1 in (14) in addition to the constant c3 in (C6) of Appendix C,
so that we are unable to identify the unique path supported by a particular nonlinear MP
strategy which converges the unique steady state. The emergence of a continuum of steadystates supported by multiple non-linear MP equilibrium strategies significantly distinguishes
our results not only from the results of Hirshleifer (1991, 1995) and Skaperdas (1992) using
a static model but also from those of Maxwell and Reuveny’s (2005) using a dynamic modelwith myopic agents in which the unique one-shot Nash equilibrium prevails or is repeatedevery period.
Another important aspect of Proposition 1 is that given any initial stock of Z, the economy
approaches the range of steady state equilibria where the common-pool stock takes a positive value and individual aggressiveness takes an intermediate value between zero and one.
In this sense, (implicit) ‘partial cooperation’ can be seen as a best response to the risk ofappropriation. In aﬄuent economies where the level of the stock variable is suﬃciently large,
investment in aggression reaches the maximum possible level (i.e., a = 1) in finite time. It thenis decreasing until the steady state S is reached. Put diﬀerently, in aﬄuent societies where
there is a large amount of the common pool stock, a full fighting strategy will be rationallyand inevitably chosen during the transition to the steady state. On the other hand, if theinitial stock level is relatively low at the start of the game investment in aggressive behavior
15monotonically increases toward the steady state S over time. That is, as the common-poolstock Z gets larger over time, the contenders will become greedier, because the marginal gainof appropriation will be higher. In the long run the economy will reach a situation where ‘partial cooperation’ prevails in the sense that every agent chooses to contribute to the productionof the common-pool stock Z to some extent. Although such ‘partial cooperation’ is also foundin the static models of Hirshleifer and Skaperdas, diﬀering degrees of ‘partial cooperation’depending on the coordination of expectations formed by agents in the present model neveremerge in their models.

3

Comparative Static Analysis

In this section we discuss the eﬀects of a change in the model parameters on the transition pathof the linear strategy aL as well as on the associated long-run equilibrium point S, since othernon-linear MP strategies display almost the same comparative statics properties. Consider firstthe eﬀects of a change in the productivity (or eﬀectiveness) of conflict technology. The recent
developments in computer networks and their applications mentioned in the introduction
are examples of technological change that potentially puts at risk the intellectual property
rights of the software and music industry. In the model, a change in the productivity of theconflict technology is captured by a change in r. The shift of point S can be calculated by
diﬀerentiating (15) and ZS in (16) with respect to the parameter r, respectively:daS= δ (δ + ρ) (n − 1) ∆−2 > 0,drdZS
= −n (n − 1) (δ + ρ) ∆−2 < 0,drwhere aS ≡ aL (ZS ) and ∆ ≡ (n − 1) rρ + δ > 0. Although an increase in r does not aﬀect
the line C1 , this increase strengthens the intensity of appropriation associated with every levelof the common-pool stock Z during the transition path, thus making the linear strategy (15)steeper. Since the productivity of appropriation becomes more eﬀective with higher r, allcompeting agents engage in more aggressive behavior in the hope of capturing more resources.

16This finding is quite intuitive, and is also consistent with the static conflict models of Hirshleifer(1991, 1995).
An increase in the number of agents augments the aggregate endowment in proportion to n,which causes an outward shift of the aggregate resource constraint C1 (i.e., scale eﬀect), sinceeach entrant provides one additional unit of the endowment. The larger aggregate endowment
will increase the payoﬀ each agent can expect to obtain from a given investment in aggression,
thereby intensifying each agent’s aggressive behavior and thus making the linear strategy aL
steeper. Since the former eﬀect causes a counter-clockwise turn of line C1 around point (0, 1),
these two eﬀects together intensify individual appropriation, but the long run eﬀect on thecommon-pool stock Z is ambiguous:8
daS
dn
dZS
dn

= r (δ + ρ) δ∆−2 > 0,=£¤
{1 − r (n − 1) − nr} (−rρ + δ) − ρn2 r2 ∆−2 R 0.

A higher depreciation rate causes a reduction in the level of the common-pool stock Z
available to contenders, thereby discouraging appropriation. This negative prize eﬀect causes
a clockwise turn of line C1 around point (0, 1) (i.e., the aggregate resource constraint C1 moves
inward toward the origin). At the same time, a higher δ implies that the cost of reproducing
the common-pool stock increases relative to the cost of aggressive behavior, which in turnstrengthens an incentive for aggressive behavior, thus making the linear strategy aL steeper.
Although these two eﬀects on appropriation operate in opposite directions, the following result
indicates that the former eﬀect will outweigh the latter eﬀect in the long run:
daS
dδ
dZS
dδ

= −ρr (n − 1) [1 − r (n − 1)] ∆−2 < 0,
= −n [1 − r (n − 1)] ∆−2 < 0.

A decrease of the subjective rate of time preference makes the linear strategy aL steeper,
8This eﬀect has been also found in Result 4B of Hirshleifer (1995).

17

but it has no eﬀect on line C1 . Hence we obtain the following long run eﬀects:
daS
dρ
dZS
dρ

= r (n − 1) [1 − r (n − 1)] δ∆−2 > 0,
= −n (n − 1) [1 − r (n − 1)] r∆−2 < 0.

The economic explanation is that as contenders become more patient (i.e., smaller ρ), they
put more weight on future stocks of durable goods rather than the current one, and thus tendto spend more resources on production activity rather than on aggressive investment. This
result has apparently not been addressed by Hirshleifer (1991, 1995) and Skaperdas (1992),
who use the static conflict models. It stands in contrast to Skaperdas and Syropoulos’s (1996)
result in which the higher is the valuation of the future (i.e., smaller ρ), the stronger is theintensity of fighting. The reason for this diﬀerence is that in their two-period’s model agent’s
first-period expenditure on appropriation increases agent’s second-period payoﬀ. Rather, ourresult is similar to Garfinkel’s (1990) Folk Theorem type result in repeated games where ahigher discount factor (i.e., a smaller ρ) makes it easier to sustain cooperative outcomes. Aninterpretation of our result is that long-sighted agents become less aggressive because they are
more concerned about the future.
We may then summarize the discussion in the following proposition:
Proposition 2(i) An increase in the eﬀectiveness of aggression leads to a higher level of aggression and to alower level of the common-pool stock;(ii) an increase in the number of agents leads to a higher level of aggression, but the eﬀect on
the common-pool stock is ambiguous;(iii) an increase in the depreciation rate leads to lower levels of aggression and of the commonpool stock; and(iv)a decrease in the subjective rate of time preference (i.e., agents become more patient) leads
to a lower level of aggression and to a higher level of the common-pool stock.

18

4

The Cooperative Solution

We will characterize the explicit cooperative solution as a benchmark steady state in thefollowing. Assume an outside enforcer or centralized agency has the power to enforce every
contender to execute its command. The cooperative strategy is one for which a centralizednat the outset of theagency chooses the infinite-horizon planning profile of strategy a ∈ R+
R∞
P
game so as to maximize 0 Ze−ρt dt subject to Ż = n − nj=1 aj − δZ where aj ∈ [0, 1]

for all j. Clearly, this optimization yields a totally peaceful solution, that is, aj (t) = 0

for t ∈ [0, ∞) and all j. The result is understood by noting that expenditure on appropriation
is socially wasteful in the sense that it causes a deadweight loss because of the non-productive
use of resources. This deadweight loss should be zero in the hypothetical case where a central
agency can directly control the allocation between productive and appropriation. As a result,the superior authority should establish point (n/δ, 0) in the long run. Agents would benefit
from an enforced peaceful resolution because socially wasteful aggressive activity is completely
eliminated.
Combined with the comparative static results in the previous section, we obtain the following results:
Proposition 3 Assume that a centralized agency chooses an allocation between aggressive andproductive investment so as to maximize aggregate payoﬀ. The resulting allocation dictatesthat agents devote all resources to the socially productive activity to obtain the Pareto eﬃcient
point (n/δ, 0) in the long run. In an anarchic situation where every agent follows the Markov
perfect equilibrium behavior described in Sections 3 and 4, a decrease in either the productivityof aggressiveness, the depreciation rate, or the subjective rate of time preference moves theresulting long run equilibrium closer to the first-best one.
The nuclear nonproliferation treaty which deters the development of nuclear weapons(i.e., aggressive technology) would be socially desirable in a way that makes the long run
outcome resulting from a non-cooperative equilibrium behavior closer to the peaceful andeﬃcient one. Another example is patent law, which aims at enforcing property rights on investment return and thus limits socially wasteful activities. Patent law potentially prevents19

a rapid fall in the expected return from new innovation, which would be a consequence ofimitation by rivals. The increase in return on investment caused by secure property rights isapproximately captured by the eﬀect of a lower depreciation rate in our model.
The problem with using the cooperative solution as a benchmark is that the socially attractive steady state is not self-enforcing because it does not usually constitute a subgame perfect(Nash) equilibrium. Nevertheless, agents have to be confronted with a coordination problem
in order to select the most eﬃcient strategy in non-cooperative environments where there
are multiple equilibrium strategies. Accordingly, governments or central agencies can play animportant role in solving the coordination problem stated above. Multiplicity entails that awindow of opportunity is available for an anarchic society in the sense that the equilibrium is
not predetermined only by the stock of durable goods Z0 . A coincidental start abstracts fromall means of communication and trust between agents. But agents have an interest in avoiding the loss caused by aggression and may transcend their lack of confidence to achieve thePareto-dominant equilibrium. Benevolent governmental institutions, even when they cannot
enforce full property rights, might be important devices for solving the coordination problem,
such that the society in the best circumstance is able to coordinate on the preferred among
all feasible equilibria.

5Conclusions

The first message of this paper is that completely aggressive behavior is not necessarily arational strategy for an agent in anarchic situations. Rather, every agent will voluntarily anduniquely choose ‘partial cooperation’, in which each agent devotes his individual resource bothto productive and appropriation at the same time, even though agents act fully rational andare guided by their self-interest. The primary driving force is the durability of the commonpool stock in conjunction with the forward looking behavior of agents. These intrinsicallydynamic ingredients induce each contender to behave ‘partially cooperatively’, even without
punishments and threats, unlike Garfinkel (1990). In other words, either if the stock depreciates completely each period or if contenders have myopic foresight, they are less motivated to20

follow a cooperative behavior in producing a commonly-accessible good.
The second message is that the use of nonlinear MP strategies would provide multiplicity ofequilibrium strategies and uncountably many long run equilibria including the better outcome
supported by the linear MP strategy, which is consistent with Dockner and van Long (1993)
and Rowat (2007) in a diﬀerential game of international pollution control between two countries.9 In other words, contenders may be able to reach a self-enforcing agreement or implicit
tacit collusion which makes the long-run outcome closer to the first-best one as compared to
the linear MP strategies. However, the problem of selecting among infinitely many nonlinear
strategies clearly requires some coordination, or more explicitly, some preplay communication.
In non-cooperative environments which do not allow such preplay communication or negotiation it may be one of the primary functions of centralized institutions to play a critical role
to resolve such a coordination problem, as stated in Section 4.
In addition to the above-mentioned role of the government, the results of the present paper,
which would be the third message, suggest that the government (or central agency) should also
play the following roles in order to achieve a socially desirable outcome, and thus move thelong run outcome closer to the more eﬃcient ones. To do this, governments should attempt
to deter the development of the conflict technology, reduce the depreciation rate of commonpool assets or induce people to have longer sight. Such structural or institutional reforms,
including laws or institutional schemes, could reduce the likelihood of aggression, and thus
lead to peaceful and more eﬃcient outcomes in the long run.
The model presented in this paper should be developed further in several directions. In
particular, introducing asymmetry among agents would enable us to compare the results of thepresent model with those static models which do incorporate asymmetric agents. The ‘paradoxof power’ (Hirshleifer, 1991) may be generated in such an asymmetric dynamic conflicting
model.
9
Dockner and Long (1993) find that the first-best pollution stock can be approximated through an appropriate choice of non-linear MP strategies, when the rate of time preference should be suﬃciently small. Our
result slightly diﬀers from theirs in that the linear MP strategy itself becomes closer to the first-best outcome
as the rate of time preference is smaller.

21

Appendix A: Derivation on the HJB equation
In this appendix we show how to derive (10) in the text. Assuming an interior solution, we
solve (8) for each agent to get the optimal strategy ai = ai (Z). By substituting this optimal
strategy into (6), the HJB equation (6) associated with agent i is transformed intoρVi (Z) = pi (a1 (Z) , ..., an (Z))Z + Vi0 (Z)

" n
X
j=1

#

(1 − aj (Z)) − δZ .

(A.1)

By diﬀerentiating (A.1) with respect to Z and applying the envelope theorem to the resulting
expression, we obtain

ρVi0 (Z) =

n
X
∂pi

a0j (Z) Z + pi (.) + Vi00 (Z)
∂a
j
j=1
" n
#
X
0
0
+Vi (Z) −aj (Z) − δ .

" n
X
j=1

(1 − aj (Z)) − δZ

#
(A.2)

j=1

Substituting (8) and (9) into Vi0 (Z) and Vi00 (Z) in (A.2), respectively, and exploiting symmetry
yields

¸
∂pi
∂pi
Z−
Z a0 (Z) + p(.)+
0 = (n − 1)
∂ak
∂ai
¸
∙ 2
∂ 2 pi 0∂ pi 0a (Z) + (n − 1)
a (Z) Z [n (1 − a(Z)) − δZ]∂a2i i∂ak ∂ai k∂pi∂pi[n (1 − a(Z)) − δZ] − (δ + ρ)Z, k 6= i.
+
∂ai
∂ai
∙

(A.3)

Since the assumption of symmetry further allows us to make use of the following simple
expressions:1 ∂pi
r (n − 1) ∂pir,,== − 2,
2
n ∂ai
na
∂ak
an
2
n (r − 1) − 2r∂ pi
r2 (−n + 2)
= r (n − 1)
,
=
,
n3 a2
∂ak ∂ain3 a2

pi =∂ 2 pi
∂a2i

22

(A.4)

we substitute those expressions into (A.3) yielding

0=

1
n−1
[−r − r (n − 1)] Za0 (Z) + +2annr (n − 1)[r (−n + 2) + n (r − 1) − 2r] Z [n (1 − a(Z)) − δZ] a0 (Z)n3 a2
r (n − 1)r (n − 1)[n (1 − a(Z)) − δZ] − (δ + ρ)
Z.
+
2
na
n2 a

(A.5)

Further rearranging (A.5) gives rise to (10) in the text.

Appendix B: Linear Strategy
We will show below that (15) represents a linear strategy. Under symmetry, rewrite the HJB
equation (6) as follows:ρV (Z) = max [p (a1 , a2 , ..., an ) Z + V 0 (Z) {n (1 − a) − δZ}] .
ai ∈[0,1](B.1)

Suppose that the value function is linear, that is, V (Z) = A+BZ, where A and B are unknown
constants. Substitute this hypothetical value function into the above HJB equation to get∙¸
1ρ [A + BZ] = max Z + B {n (1 − a) − δZ} .
n

(B.2)

Substituting further the (interior) first-order condition (8) , that is, a = r (n − 1) Z/Bn2 into
a in (B.2), we obtain¶¾½ µ
r (n − 1)
1
Z − δZ .
ρA + ρBZ = Z + B n 1 −nBn2Comparing the coeﬃcient of Z and the constant in both sides of (B3) yields

ρA − Bn = 0 and ρB −1 r (n − 1)+
+ Bδ = 0.
nnSolving the above simultaneous system of equations in terms of A and B to yield

23

(B.3)

A=

n
1 − r (n − 1)
B and B =.
ρ(ρ + δ) n

Further substitution of these expressions into the first-order condition, a = r (n − 1) Z/Bn2 ,yields (15). The corresponding value function is given by
1 − r (n − 1)
V (Z) =
(ρ + δ) n

µ

¶
n
+Z .
ρ

(B.4)

Appendix C: Subgame Perfect Solutions
To prove the existence of MP Nash equilibria we apply a suﬃciency theorem stated in Theorem
3 of Rowat’s (2007). To do this, we first have to show that the strategy is feasible, which isdefined in Definition 1 of Rowat’s (2007). Although it may be appropriate to distinguishbetween the value function associated candidates strategies and that associated with the welldefined strategies like those of Rowat’s, it is omitted for the sake of notational simplicity
but with understanding that the solutions have to pass the further tests to qualify for an
equilibrium strategy satisfying the properties stated in the text.
Lemma 1 The strategy a(Z) = 0 for Z ∈ (0, ∞) is not an equilibrium strategy.
Proof. Suppose that (∂pi /∂ai ) Z − Vi0 (Z) < 0. Then it follows from (8) that ai = 0. In
this case, the HJB equation (6) becomesρVi (Z) =1
Z + Vi0 (Z) (n − δZ) .
n

(C.1)

By integration and imposing symmetry, we have

V (Z) =

ρ(n − δZ)2 [n + (ρ + 2δ) Z]+ c2 (n − δZ)− δ ,n (ρ + 3δ) (ρ + 2δ)(C.2)

where c2 represents a constant of integration. When c2 6= 0, lim V (Z) = ±∞. This implies
Z→n/δ

that the strategy a(Z) = 0 for Z ∈ [0, ∞) is not an equilibrium strategy whenever c2 6= 0,
24

because the value function (C.2) associated with this strategy is unbounded at Z = n/δ andthus the strategy a(Z) = 0 for Z ∈ [0, ∞) ceases to be continuous at this point.
Next, consider the case where c2 = 0. In this case it turns out that the derivative of theresulting value function with respect to Z is bounded above:∙¸
2δnn − δZ(n − 3δZ) −< 0 for ∀Z > 0.
V (Z) =n (ρ + 3δ)ρ + 2δ
0(C.3)

On other hand, since∂pir (n − 1)
Z = limZ → ∞ for ∀Z > 0,
ai →0 ∂ai
ai →0
n2 a
lim

(C.4)

inequality (∂pi /∂ai ) Z < Vi0 (Z) never holds except for Z = 0. Hence, the cornered strategy
a(Z) = 0 is not an equilibrium strategy for Z ∈ (0, ∞).
Lemma 2 There exists a constant of integration which makes the strategy a(Z) = 1 an equilibrium strategy over Z ∈ (0, ∞).
Proof. When all players play strategy ai = 1, the HJB equation (6) becomesρV (Z) =1
Z + V 0 (Z) (−δZ) .
n

(C.5)

By integration and imposing symmetry, we haveρ

Z + Z − δ n (ρ + δ) c3,
V (Z) =n (ρ + δ)(C.6)

where c3 represents a constant of integration. When setting a = 1 in ∂pi /∂ai yields (∂pi /∂ai )Z ≡r (n − 1) Z/n2 , the first-order condition (∂pi /∂ai ) Z ≥ Vi0 (Z) only allows (C.6) to hold for thevalues of Z satisfying¸∙
ρ+δr (n − 1) ρ+2δ1
δ≤ c3 .
Z δ −Z δρn ρ + δn(C.7)

Since the first exponent inside the brackets on the left-hand side of (C.7) is smaller than thesecond exponent term, it dominates for smaller values of Z, whereas for larger values of Z
25

Figure 3: Transitions between the corner and interior solutions.
the second term overpowers it. Indeed, diﬀerentiation of the left-hand side of (C.7), which isdenoted by Φ (Z), with respect to Z yields∙¸
dΦ (Z)r (n − 1) (ρ + 2ρ)
1 ρ
=
Zδ 1−
Z ,
dZ
ρn
n
which implies that for smaller values of Z the left-hand side of (C.7) takes a positive slope,
while for larger values of Z that takes a negative slope. Noting that Φ (0) = 0, these results
together imply that the function Φ (Z) displays a U-shaped curve, as illustrated in Fig.3. This
diagram shows that for larger values of c3 , as happens with c03 in Fig.3, (C.7) is satisfied at
any value of Z and thus a (Z) = 1 is a global solution to the HJB equation (C.5), whereas forsmaller values of c3 as happens with c003 in Fig.3, it is satisfied for smaller Z, is then violated,
and finally is satisfied for larger values of Z again.

References
Acemoglu, D., S. Johnson and J.A. Robinson, 2001, The colonial origins of comparative
development: An empirical investigation, American Economic Review 91, 1369- 1401.
Auty, R.M., 2001, Resource abundance and economic development, Oxford: Oxford University
Press.
Collier, P. and A. Hoeﬄer, 2004, Greed and Grievance in Civil War, Oxford Economic Papers,
26

forthcoming.
Dockner, E. and N.V., Long, 1993, International pollution control: cooperative versus noncooperative strategies, Journal of Environmental Economics and Management 25, 13-29.
Dockner, E., Jorgensen, S., Long, N.V., and G. Sorger, 2000, Diﬀerential Games in Economics and Management Science, Cambridge University Press.
Garfinkel, M.R., 1990, Arming as a strategic investment in a cooperative equilibrium, American Economic Review 80, 50-68.
Gonzalez, F.M., 2007, Eﬀective property rights, conflict and growth, Journal of Economic
Theory 137, 127-139.
Gradstein, M., 2003, Governance and Economic Growth, World Bank Policy Research Working Paper No. 3098.
Grossman, H.I. and Kim, M., 1995, Swards or plowshares? A theory of the security of claims
to property. Journal of Political Economy 103, 275-288.
Hirshleifer, J., 1991, The paradox of power, Economics and Politics 3, 177-200.
Hirshleifer, J., 1995, Anarchy and its breakdown, Journal of Political Economy 103, 26-52.
Hodler, R., 2006, The curse of natural resources in fractionalized countries, European Economic Review, In Press.
Itaya, J. and K. Shimomura, 2001, A dynamic conjectural variations model in the private
provision of public goods: A diﬀerential game approach, Journal of Public Economics
103, 153-172.
Maxwell, J.W. and R. Reuveny, 2005, Continuing conflict, Journal of Economic Behavior &Organization 58, 30-52.
Rowat, C., 2007, Nonlinear strategies in a linear quadratic diﬀerential game with bounded
controls, Journal of Economic Dynamics & Control 31, 3179-3202.
27

Rubio, S.J. and Casino, B., 2002, A note on cooperative versus non-cooperative strategies ininternational pollution control, Resource and Energy Economics 24, 251-261.
Sachs, J.D. and A.M. Warner, 1999, The big push, natural resource booms and growth,
Journal of Development Economics 59, 43-76.
Skaperdas, S., 1992, Cooperation, conflict, and power in the absence of property rights,
American Economic Review 82, 711-739.
Skaperdas, S. and C. Syropoulos, 1996, Can the shadow of the future harm cooperation?
Journal of Economic Behavior and Organization 29, 353-372.
Tsutsui, S. and K. Mino, 1990, Nonlinear strategies in dynamic duopolist competition withsticky prices, Journal of Economic Theory 52, 136-161.
Tullock, G., 1980, Eﬃcient rent seeking, in J.M. Buchanan, R. D. Tollison, G. Tullock, eds.,
Toward a Theory of the Rent-Seeking Society, Taxes: A&M University Press, 97-112.

28