Terrorists’ Eradication versus Perpetual Terror
War
Revised Version
Bertrand Crettez
∗†Naila Hayek
‡June 2013
Abstract We study an infinite horizon sequential dynamic game where the players are
a government and an international terrorist organization. We provide conditions for the existence of equilibria in which the terrorists’ resources are totally destroyed by a government strike. Specifically, we study strong eradication equilibria in which the government strike annihilates the terrorists’ resources, preventing the terrorists from acting. We also pay attention to weak eradication equilibria in which the terrorists’ resources are destroyed, but in which the initial value of the terrorists’ strike is nevertheless positive. We also show that there is an equilibrium in which there is perpetual war between the government and the terrorists. Perpetual war can only co-exist with weak eradication equilibria. For these cases, we provide conditions under which the government would be better off in a weak eradication equilibrium.
Keywords Differential games. Counterterror measures. Nash equilibrium. AMS Classification: 49N90, 93C55, 34K35
∗We thank Susan Crettez and Emmanuela Carbonara for helpful remarks and two referees for stimulating comments on a previous version of this work.
†Université Panthéon-Assas, Paris II, Laboratoire d’économie du droit, ERMES EA 4441, [email protected].
1
Introduction
This paper studies a dynamic game where the players are a government and an international terrorist organization. It studies the existence of equilibria in which the terrorists’ resources are totally destroyed by a government strike.
The eradication of terrorists’ resources is studied in literature which does not rely on dynamic games. The main result of this literature is that eradication of terrorists’ resources is either unlikely or is achieved in the long run. In Keohane and Zeckhauser [1], eradication of terrorists’ resources may be achieved albeit rarely (this is because removing the terrorists’ resources involves bearing a fixed cost). Caulkinset al [2] show that a terror organization may be nearly
eradicated at steady-state. Kress and Szechtman [3] show that eradication of insurgency by force only is almost impossible. These authors pay attention to the role of intelligence and the positive effect of collateral damages on the recruitment of insurgents. Eradication is difficult to achieve because the “attrition to the insurgency . . . is offset by increased recruitment.” The destruction of the terrorists’ resources is an equilibrium property of some dynamic games studied in the literature. This property appears in a finite time horizon model in Behrens
et al. [4] who rely on the notion of incentive Stackelberg strategies. Eradication equilibria
also arise in Hausken and Zhuang [5] in a finite time horizon game with two state variables (namely the resources of the government and the terrorists), where the players are myopic (a two-stage game is repeated a finite number of times). They find that eradication of terrorists’ resources is an equilibrium outcome if the government’s resources are relatively high1.
The present paper studies eradication equilibria in a sequential infinite horizon dynamic game. It relies on Novaket al. [6] model in which the terrorists’ resources are the only state variable.
These resources can be either the number of terrorists, or their financial assets, and evolve across time as a function of terrorists’ actions and counter-terror actions of the government.
1In a series of recent papers, Hausken, Zhuang and their co-authors study the conditions under which
Novak et al. [6] compare the open-loop Nash and Stackelberg solution of a differential game
where the players are a terrorist organization and the government. They study the conditions under which the Stackelberg leader will act more cautiously than in the Nash equilibria, and if it pays to have a first-mover advantage.
The present paper departs from Novak et al. in that the law of motion of the terrorists’
resources is richer and the objectives are different. As for the law of motion, whereas in Novak et al. the resources of the terrorist organization explode if either the terrorists or
the government don’t act, this is no longer true here. As for the objectives, we assume that the objectives are linear, and that the government’s action affects directly the International Terror Organization’s objective (and not only its resources). The new assumptions enable us to prove the existence of Nash eradication equilibria (which cannot exist in Novak et al.).
The present paper also differs from Behrens et al. [4] since: 1) we use an infinite horizon
model, 2) we only consider Nash equilibria (as opposed to incentive Stackelberg strategies). It also differs from Hausken and Zhuang [5] (and Hausken, [12]) because in our setting, 1) the decision horizon is infinite, 2) the players are not myopic (i.e., they have perfect foresight),
3) there is a unique state variable and we do not rely on contest success functions to model the result of the interactions. Finally, in the Hausken and Zhuang paper, the government maximizes the probability that its assets are not damaged, and the terrorist maximizes the probability of damages (we use different objectives).
We also show that there may be multiple equilibria: there may be either perpetual war or instant eradication of the terrorists (the perpetual war equilibrium is similar to the Nash equilibrium studied by Novak et al., [6]). We compare the values of the objectives of
the government with perpetual war and eradication. We present conditions under which eradication equilibria are preferred to perpetual war by the government.
2
The Model
We consider a two-player infinite horizon sequential dynamic game with a single state variable. The two players are a government and an International Terror Organization (hereafter ITO). The state variable is the ITO’s resources, denotedxt. The strategic variable of the government
is denoted ut, while that of the ITO is vt. Both represent the levels of strikes devoted by each
player to hit the other one.
The dynamics of the ITO’s resources is written as follows:
xt+1 = (1 +r)xt−h(ut, vt), (1)
where h:R2 →R is specified as follows:
h(ut, vt) = cut+dvt+utvtθ (2)
The dynamics of the terrorists resources depends on the past level of these resources (through the growth rate r), and on the levels of both their strikes and those of the government
(through the function h).
All the parameters r,c, d, and θ are real numbers. We shall need the next assumption:
Assumption 1. Both the parameters d andr are positive. Furthermore, the parameters θ
and satisfy 0< <1< θ.
The requirement thatrbe positive implies that when neither the government nor the terrorists
strike, the terrorists’ resources always increase at rater. This is in contrast with Caulkins
et al. [2] where the terrorists’ resources follow a logistic growth so that the growth rate of
the resources’ stock goes to zero when its size is higher than a certain threshold value. Our specification, which follows in part that of Novaket al [6], considerably simplifies the analysis
of the equilibria. Next, we allow cto be either positive or non-positive. If cis positive, this
means that the higher the strategic effort ut of the government at date t, the lower the level
[3], the ITO’s resources increase directly with government’s efforts (or do not depend on them). This may happen when, due to collateral damages, the ITO gains support as the government strikes the ITO.2 Even in this case, a government’s strike has a negative effect
on the ITO’s resources due to the terms uvθ. In effect, the cost of an action for the ITO in
terms of lost resources —dvt+utvθt— increases with the government’s effort: launching an
attack is more and more difficult.
The condition d positive implies that the size of the ITO’s resources decreases with the size
of its effort.
The dynamics of the ITO’s resources is similar to the specification of Novak et al. [6] except that we have introduced the linear terms cu+dv. We believe that it is relevant to introduce
them because, without these terms it would be sufficient for the ITO not to act (i.e., so set
vt = 0), whatever the government’s choice, to let its resources grow without limits. This can
be considered as an awkward conclusion.
We denote by Adm(ˆv) the set of processes(x, u)satisfying
xt+1 = (1 +r)xt−cut−dvˆt−utˆvtθ
xt≥0, ut≥0, ∀t≥0
and we denote by Adm(ˆu) the set of processes (x, v) satisfying
xt+1 = (1 +r)xt−cuˆt−dvt−uˆtvtθ
xt ≥0, vt≥0, ∀t≥0.
We assume that the per-period-utility of the government at any date t is:
−xt−αut−βvt (3)
2In this case, it would be more realistic to assume that casualties caused by government’s actions result in
and the per-period-utility of the ITO at date t is:
xt−γut+δvt (4)
Assumption 2. The parametersα, β, γ, δ are all positive.
According to assumption 2, the government’s effort negatively affects both per-period-utilities. The negative effect of a government’s strike on the ITO may be due to the casualties and the political cost born by the terrorists from being hit. As for the government, the negative effect stems from the fact that counterterror measures are costly (on this more below). The ITO’s efforts affect the government’s period-utility in a negative way and its per-period-utility in a positive way. The first effect may be due to the casualties and the political cost born by the government from being hit by the terrorists. The second effect reflects the fact that the ITO derives a political benefit from striking the government (for instance it may value the deaths that are direct consequences of its strikes).
We have here assumed that the ITO’s resources enter with a negative sign in the government’s per-period-utility and with a positive sign in the ITO’s one. At date 0 the ITO’s resources
are given, but this is no more true from date1on. So the government may seek to decrease or
to limit the growth of the future value of the ITO’s resources because without resources, the ITO cannot strike. Therefore, we can consider that reducing the level of ITO’s resources is an intermediate objective to avoid future strikes. For opposite reasons, the ITO may consider that having more resources improves its relative strength and future capacity to strike. We have not introduced the government’s resources in the analysis. Certainly, fighting terrorism is costly. But the issue of the dynamics of the government’s resources, and specifically the dynamics of the public debt does not seem to be relevant for the analysis, at least on a first approach. This is in contrast with the ITO, whose operations seem limited by the size of its resources. Our modelling choice of not taking into account the governement’s resources is in line with some papers in the literature, e.g., [11] (as noted in the introduction,
(page 328), “[this] is realistic because the government has the capacity needed to compile the resources needed for optimizing behavior; for example, transfer funds from other government branches". In the present paper, the government is assumed to have access to an unlimited amount of resources. Using these resources, however, is not without cost, this is why the level of its strikes affects negatively its objective. This negative effect could reflect an opportunity cost, because resources used to strike the ITO could be spent to achieve different aims (i.e.,
public health, education, and so forth).
The per-period-utilities used in this paper are slightly at variance with the specification used in Novaket al. [6]. In [6], the function h positively enters in the government’s per-period-utility
(whereas we assume thatxtenters negatively), while the government’s strikeutdoes not affect
the ITO’s one. Our per-period-utilities are in the spirit of [4]. The government’s objective is the same as that of [4] (except that in the present paper the decision horizon is infinite). The only difference between the ITO’s objectives used in this paper and in [4] is that the government’s strike enters negatively in the ITO’s objective in this paper, whereas it enters positively in [4]3.
The per-period utility functions of the government are discounted with a discount factor ρ1, while that of the ITO are discounted with a discount factor ρ2.
Assumption 3 . The discount factors ρi satisfy ρi >0, and (1 +r)ρi <1,i= 1,2.
This assumption is necessary for the existence of a perpetual war equilibrium4. It will be
discussed after the statement of Theorem 3.1.
Assumption 3 is not always sufficient to ensure that the sum of the discounted values of the per-period utilities of the ITO converges in R (especially in the case wherec is negative, see
below). We therefore give a definition which addresses this issue.
Definition 2.1. An overtaking Nash equilibrium is a pair of strategies (ˆu,ˆv) such that
3The reason why the government’s strike enters positively in the ITO’s objective in [4] is that “the ITO might have political objectives aiming at inducing “excessive” counter-attacks by the West (page 230)”.
(ˆx,ˆu)∈Adm(ˆv), (ˆx,ˆv)∈Adm(ˆu) and which satisfies
lim sup T→+∞
( T
X
t=0
ρt1[−xt−αut−βvˆt]− T
X
t=0
ρ1t[−xˆt−αuˆt−βˆvt])≤0
∀(x, u)∈Adm(ˆv)
(5)
and
lim sup T→+∞(
T
X
t=0
ρt2[xt−γuˆt+δvt]− T
X
t=0
ρt2[ˆxt−γuˆt+δˆvt])≤0
∀(x, v)∈Adm(ˆu).
(6)
By taking lim inf
T→+∞ instead of lim supT→+∞
in this definition we obtain the definition of a weak
overtaking Nash equilibrium.
The overtaking maximality notion can be found in Carlson et al. [13] and is due to Von
Weizacker.
We note that an overtaking Nash equilibrium is a weak overtaking Nash equilibrium. We also observe that when the series of the discounted per-period utility functions of the government and the ITO converge in R for all admissible paths, then the two definitions reduce to
the definition of the usual Nash equilibrium. This is the case when a government’s strike does not have collateral effects (c > 0). Indeed for all t ≥ 0 we have xt ≤ (1 + r)tx0,
ut ≤(1 +r)t+1x0/c, and vt ≤ (1 +r)t+1x0/d. SoP+t=0∞ρt1| −xt−αut−βvt| ≤ P+t=0∞(ρ1(1 +
r))tx
0[1 +α(1 +r)/c+β(1 +r)/d] which converges sinceρ1(1 +r)<1. The same holds true for the ITO’s objective. Therefore in this case, the overtaking Nash equilibrium and the weak overtaking Nash equilibrium collapse to the usual notion of a Nash equilibrium which solves the following problem:
maximizeu J1(u,ˆv) = +∞
X
t=0
ρt1[−xt−αut−βˆvt]
(x, u)∈Adm(ˆv)
(7)
and
maximizev J2(ˆu, v) = +∞
X
t=0
ρt2[xt−γuˆt+δvt]
(x, v)∈Adm(ˆu)
We observe that the game is state separable (i.e., the state equation and the objective
functional are linear in the state variables and have no (multiplicative) interaction between control variables and state variables). As noted in Novaket al., “state separable games exhibit
a special structure which allows an analytical characterization of Nash-solutions” (see also Dockneret al. [14],[15]5).
Before studying the interactions between the government and the ITO, an explanation is in order.
We have assumed that both the government and ITO consider the resource constraints of the former in the same way (recall the definitions of Adm(ˆu) and Adm(ˆv)), page 5. In
particular, both consider that (1 +r)xt−h(ut, vt)must always be non-negative. While this
is a relevant assumption for the ITO (which is resource-constrained), this is disputable for the government. For instance, let us assume that the ITO has chosen a level of strike at date t such that xt+1 is positive if the government does not strike (i.e., vt < (1+dr)xt). The
government could then perceive the motion of the terrorists’ resources as being given by
xt+1 = max{0,(1 +r)xt−h(ut, vt)}. That is, the level of its strike may not necessarily be
bounded. In effect, the government’s effort may be much higher than what is necessary to eradicate the terrorists’ resources. Nevertheless allowing this in our game would not change the result because levels of efforts higher than what is actually necessary to eradicate the terrorists’ resources would never be chosen by the government as they are costly. So, the government would not waste resources. Therefore, the constraints(1 +r)xt−h(ut, vt)≥0
which we have considered in the government problem is not restrictive.
3
Perpetual Terror War Equilibrium
In this section, we study the case where a war between the government and the ITO lasts forever. This kind of equilibrium has already been shown to exist by Novak et al. [6]. The
next Theorem states a set of conditions under which a perpetual war is possible in our setting. It also ensures that there is a unique equilibrium with perpetual war and constant levels of
5State-separable games have important properties. Notably, their open-loop Nash equilibria are Markov
strikes when government strike have no collateral effects.
Theorem 3.1. There is a Nash equilibrium (respectively, overtaking Nash equilibrium or
weak overtaking Nash equilibrium) in which war is perpetual and the values of the strikes are
constant if and only if αÄ1−(1 +r)ρ1
ä
−cρ1 >0, δ
Ä
1−(1 +r)ρ2
ä
−dρ2 >0, and x0 ≥ h(ˆu,rvˆ).
The optimal strategies of this Nash equilibrium are given by:
ˆ
ut= ˆu=
δÄ1−(1 +r)ρ2
ä
−dρ2
ρ2θ
Ñ
ρ1
αÄ1−(1 +r)ρ1
ä
−cρ1
éθ−1
θ
θ +θ−1
, t ∈N (9)
ˆ
vt = ˆv =
Ñ
αÄ1−(1 +r)ρ1
ä
−cρ1
ρ1
é1
θ
ˆ
u1−θ, t∈N (10)
and the ITO’s stock of resources is given by
ˆ
xt= (1 +r)t
ñ x0−
h(ˆu,vˆ)
r ô
+ h(ˆu,ˆv)
r , t∈N (11)
When there are no collateral effects (c > 0), this Nash equilibrium is the unique one with
perpetual war.
Thus under the provided conditions, both the government and the ITO strike forever and the sizes of the strikes are constant across time. Moreover, terrorists’ resources grow without bounds except in the special case x0 = h(ˆu,rvˆ) where they remain constant. We notice in equation (11) that a perpetual terror war only occurs if the initial value of the ITO’s resources is sufficiently high,i.e., if:
x0 ≥
h(ˆu,vˆ)
r (12)
Otherwise the efforts made by the government and those of the terrorists deplete the latter’s resources, making a perpetual war unsustainable. Eradication is therefore inevitable (see the next section).
The inequalities αÄ1−(1 +r)ρ1
ä
−cρ1 >0 andδ
Ä
1−(1 +r)ρ2
ä
government considers as given the action of the terrorists. If at datetthe government changes
the value of its strike uˆ by an amount4u (in such a way thatuˆ+4u is non negative), then
it must bear a direct decrease in its objective, which is equal toα4u (the bigger the strike,
the higher the loss born by the government). An increase of uˆ by 4u also leads to a change
in the value of xt+1, which is approximately equal to 4x≡ −(c+(ˆu)−1(ˆv)θ)4u. Indeed,
a government’s strike directly decreases the resources of the terrorists by an amount equal to c4u. But there is also an indirect effect because an increase in ut raises the cost of the
terrorists’ actions (by an amount equal to (ˆu)−1(ˆv)θ4u). The combined effects at date t+ 1
which work through the change in xt+1, affect the government’s objective by an amount equal
toρ1(c+(ˆu)−1(ˆv)θ)4u. The change in the ITO’s resources at date t+ 1 affects the whole intertemporal objective of the government (this contribution is measured by the implicit price
q1 which is introduced in the appendix). This first order intertemporal effect is equal to:
ρ14x
Ä
1 +ρ1(1 +r) +ρ21(1 +r)2+. . .
ä
= ρ14x
1−ρ1(1 +r) (13)
Thus, the government does not change the level of its strike if:
−α4u+ρ1
Ä
c+(ˆu)−1(ˆv)θä4u 1−ρ1(1 +r)
= 0 (14)
which implies that:
−αÄ1−ρ1(1 +r)
ä
+ρ1
Ä
c+(ˆu)−1(ˆv)θä= 0. (15)
A necessary condition for this equation to be satisfied is therefore:
αÄ1−(1 +r)ρ1
ä
−ρ1c >0 (16)
holds. Otherwise, the positive first-order effect of a decrease in the ITO’s resources would be infinite, and the government would be eager to eradicate theses resources. Perpetual war, which requires limited strikes, would then be impossible.
The same kind of explanation can be used for inequality δÄ1−(1 +r)ρ2
ä
−dρ2 >0. The inequality only holds when (1 +r)ρ2 <1, that is when Assumption 3 is satisfied.
When there are collateral effects (when c is negative), condition (16) is always satisfied when
Assumption 3 holds. Its interpretation, however should be adapted accordingly.
When there is a perpetual war, the ITO’s resources grow without limits except in case
x0 =h(ˆu,ˆv). This is due to the specification of the resources’ law of motions, which is such that its non-trivial steady-state is unstable.
Perpetual wars are not the only outcome of the infinite interactions between the government and the ITO. As will be seen in the next section, equilibria with depletion of the ITO’s resources in finite time may also exist.
4
Eradication Equilibria
We first state two definitions of an eradication or annihilation equilibrium.
Definition 4.1. A strong eradication equilibrium is a Nash equilibrium in which the ITO’s
resources are destroyed at date 1 (x1 = 0), the government’s action takes its greatest possible
value at date 0, the ITO does not act at date 0 and the war is over.
In this definition, eradication of the ITO’s resources arises as a result of a government’s strike
only. This is why we call this strong eradication.
Notice that in case there are no collateral effects (i.e., c is positive), if the ITO’s resources
are destroyed at date 1 (x1 = 0) they will remain nil forever (xt = 0, t≥ 1) and both the
government’s and the ITO’s actions will stop (ut= 0, vt= 0, t≥1).
The depletion of the ITO’s resources can also be due to the fact that the ITO decides to use all of its resources at date 0. But this situation can hardly be qualified as being an
Beyond these two extreme cases, annihilation can happen because of the joint actions of the
government and the ITO. This motivates the definition of a weak eradication equilibrium.
Definition 4.2. A weak eradication equilibrium is a Nash equilibrium in which both the
government and the ITO strike at date 0, the ITO’s resources are depleted at date1 (x1 = 0)
and remain nil, and the war is over.
The next Proposition highlights the fact thatd >0is an essential assumption for the existence
of strong or weak equilibria.
Proposition 4.1. If d= 0, then eradication equilibria do not exist.
Proof. In an eradication equilibrium, by assumption the war is over from date 1 on (there
are no more strikes). When d = 0, let us show that the best response of the ITO when
ut= 0 for all t≥1cannot be vt= 0 for allt ≥1. In this case, the dynamics of xt reduces to
xt+1 = (1 +r)xt, for t≥1. This implies that the choices of vt,t ≥1, are not constrained. It
is easy to check that the strategy v such that v0 ≥0 and vt = 0 for all t≥ 1 is not a best
response. Any strategy v˜ such that v˜0 =v0 and v˜t =w for all t≥1 (where w is a positive
constant) satisfies
J2(u,˜v) =x0−γu0+δv0+ +∞
X
t=1
ρt2[xt+δw]> x0−γu0+δv0+ +∞
X
t=1
ρt2xt =J2(u, v).
So there cannot be eradication equilibria.
Next we provide a characterization for the existence of strong or weak eradication equilibria.
Theorem 4.1.
(i) A strong eradication equilibrium exists if and only if c >0andαÄ1−(1+r)ρ1
ä
−cρ1 ≤0.
In such an equilibrium, the values of the strikes at date0 are given by:
u0 =(1 +r)
x0
c (17)
(ii) A weak eradication equilibrium exists if and only if (c≥0and δÄ1−(1+r)ρ2
ä
−dρ2 >0)
or (c <0, δÄ1−(1 +r)ρ2
ä
−dρ2 >0 and x0 is small enough).
In such an equilibrium, when c ≥ 0, the values of the strikes at date 0 are such that
u0 6= 0, v0 6= 0, x1 = 0 and:
u0 = min
αÄ1−ρ1(1 +r)
ä
−ρ1c
ρ1v0θ
1
−1
, M(v0) ,
v0 = min
δ−ρ2d
Ä
1 + δ
d(1 +r)
ä
ρ2θu0
Ä
1 + δ
d(1 +r)
ä
1
θ−1
, N(u0) ,
if αÄ1−ρ1(1 +r)
ä
−ρ1c > 0 and
u0 =M(v0)
v0 = min
δ−ρ2d
Ä
1 + δd(1 +r)ä
ρ2θu0
Ä
1 + δ
d(1 +r)
ä
1
θ−1
, N(u0) ,
if αÄ1−ρ1(1 +r)
ä
−ρ1c≤0,
where v0 (resp. u0) being given, M(v0) (resp. N(v0)) is the unique non negative value of u0
(resp. v0) such that: cu0 +dv0+u0v0θ = (1 +r)x0 and where v0 is higher than a certain
threshold v0. And when c < 0, the values of the strikes at date 0 are such that x1 = 0 and:
u0 =
αÄ1−ρ1(1 +r)
ä
−ρ1c
ρ1v0θ
1
−1
,
v0 = min
δ−ρ2d
Ä
1 + δ
d(1 +r)
ä
ρ2θu0
Ä
1 + δ
d(1 +r)
ä
1
θ−1
, N(u0) . (19)
We notice that both strong and weak eradication equilibria may co-exist. Let us now comment on result (i). To understand the conditions on αÄ1−(1 +r)ρ1
ä
−cρ1, let us assume that the
resources. Let us then assume that the government increases its effort u0 at datet ≥0 by a feasible positive amount 4u. The objective of the government then decreases directly by α4u. Of course, x1 now decreases (exactly) by an amount equal to c4u (by assumption we are in the case where c is positive). In view of the law of motion of xt, x2 increases by
(1 +r)c4uand so forth. The sequence of the decreases in the future values of x leads to an
increase in the value of the government objective by:
ρ1c4u+ρ21c(1 +r)4u+ρ31c(1 +r)34u+· · ·=
ρ1c4u
1−ρ1(1 +r) (20) It is in the government’s interest to increase ut at date 0(in fact at any date where xt is not
nil) iff:
−α4u+ ρ1c4u 1−ρ1(1 +r)
>0 (21)
or:
αÄ1−ρ1(1 +r)
ä
−ρ1c <0 (22)
which is the condition used in the statement of Theorem 4.1 (if the above expression is nil, the government is indifferent to increasing or decreasing the level of its strike). This condition gives a necessary and sufficient condition for the government to always be willing to strike if the ITO does or cannot strike.
The above condition is likely to be satisfied when the government discount factor ρ1 is high, because then the government cares of the future terrorists’ strikes. It is also likely to be satisfied when the impact of a government’s strike on the terrorists’ resources (c) is positive
and high (it is never satisfied when there are collateral effects, or when the government’s strike is inefficient), and when the cost of a strike (α) is low. The condition is also satisfied
when the terrorists’ resources grow at a fast rate, that is when r is high.
Let us now turn to result(ii). To interpret the inequality assume that the government and
increases at date 0by an amount δ4v. But the increase in the ITO’s strike decreases the
value of its resources at date 1 by an amount approximately equal to: 4x1 ≡(d+θu0v0θ−1)4v. This decrease in the ITO’s resources at date 1 leads itself to a decrease in these resources at dates 2 and so forth. The impact of these decreases on the ITO’s objective is approximately equal to ρ24x1+ρ22(1 +r)4x1+. . .. Overall, the impact of an increase in the ITO’s strike at date0 leads to a change in the ITO’s objective which is approximately equal to:
δ4v− ρ2(d+θu
0vθ0−1)
1−ρ2(1 +r) 4
v (23)
Therefore, a necessary condition for the ITO to deplete its resources at date 1is: δÄ1−ρ2(1 +
r))−ρ2d >0.
The above condition is likely to be satisfied when the ITO’s discount factor is low (so that what the ITO cares of, is the instant impact of a strike). It is also likely to be satisfied when the immediate gain of a strike is high (that is when δ is high). Also, the cost in terms of
future decreases in x must be low (that isd must not be too high).
To prove that the necessary condition is also sufficient one must take into account the strategic interactions between the government and the ITO. The study of weak eradication equilibria shows that there are two different cases.
– Case αÄ1−(1 +r)ρ1
ä
−cρ1 >0
In that case, we know that there is no strong eradication equilibrium. There is however a continuum of weak eradication equilibria which are characterized by the fact that the ITO’s strike is higher than a threshold v0 and lower than (1 +r)x0/d (the level above which the resources are eliminated).
To understand this, first notice that when the ITO does not strike, then nor does the government under our assumptions (see the discussion above of point (i) of Theorem 4.1).
case, assuming that the ITO never strikes from date 1on, the objective of the government
reduces to: −x0−αu0−βv0−
ρ1[(1+r)x0−cu0−dv0−u0v0θ]
1−ρ1(1+r) . So, when the ITO strikev0 is positive,
the marginal gain associated to the government strike u0 may be arbitrarily high when u0 is close to 0. This is because a marginal increase in u0 decreases sharply x1 the value of the terrorists’ resources at date 1, whereas its cost at0 is finite (this cost being equal toα).
However, the value of the government’s strike at date 0 may be insufficient to eradicate the
remaining part of the terrorists’ resources. Such an eradication appears therefore only if the level of ITO’s strike is higher than a thresholdv0. This case is shown in figure 1 where all
the weak eradication equilibria locate on the grey part of the graph of the couples (u0, v0) which eliminate the ITO’s resources6.
Interestingly, a weak eradication equilibrium can be obtained even if a government’s strike has no effect (because cbeing nil), and even when it has a positive impact (c being negative)
on the terrorists’ resources (due to collateral effects). In that case, when the ITO strikes, a small increase in the government’s strike from0raises directly the ITO’s resources by a finite
margin equal to −c. However, this marginal increase in the government’s strike may also
indirectly decrease the ITO’s resources by a large amount (large enough to compensate the
first effect). Thus, while a government’s strike directly increases the ITO’s resources because of collateral effects, it can also increase sharply the cost of action of the ITO, so that the net effect on the ITO’s resources is negative. Such negative net effect of a government’s strike may thus sustain a weak eradication equilibrium.
– Case αÄ1−(1 +r)ρ1
ä
−cρ1 ≤0
In this case, the government always finds valuable to strike whether the ITO strikes or not. Therefore, whatever the level of the ITO’s strike is, since the government always finds valuable to strike, the terrorists’ resources are always eradicated. As a result, the set of values of v0 that can be sustained in a weak eradication equilibrium is therefore equal to]0,(1 +r)x0/d[.
To conclude our discussion of Theorem 4.1, weak and strong eradication may co-exist. When this happens, as in the preceding case, a strong eradication equilibrium is a limit case of
6The values of the parameters used in the figure are as follows: α= 2,β = 1,c= 1,d= 1,δ= 1,r= 0.01,
v
u0
v0
0 = (1 +r)x0−h(u0, v0)
v0 = [α(1−ρ1(1+ρ1r))−ρ1cu10−]1/θ
v0 = [
δ−ρ2d
Ä
1+δd(1+r)
ä
ρ2θu0
Ä
1+δ d(1+r)
ä]1/(θ−1)
Figure 1 – Existence of weak eradication equilibria, inexistence of a strong eradication equilibrium
weak eradication equilibria.
5
Eradication versus Perpetual War
In this section we compare the values of the government’s objective both in the cases of perpetual war and weak eradication. According to Theorems 3.1 and 4.1, the conditions under which there exist both a perpetual war and a strong eradication equilibrium are incompatible7. Nevertheless, both perpetual war and weak eradication equilibria exist when
c≥0, αÄ1−(1 +r)ρ1
ä
−cρ1 >0andδ
Ä
1−(1 +r)ρ2
ä
−dρ2 >0(and the value of the ITO’s resources is high enough). However both perpetual war and weak eradication equilibria are incompatible when c <0 as shown in Lemma C.1 in the appendix8
Let(ˆu,ˆv) denote the perpetual war equilibrium and (u, v) a weak eradication equilibrium.
The optimal values of the government and the ITO objectives in a perpetual war are given by:
J1(ˆu,ˆv) = −
x0
1−ρ1(1 +r)−
αuˆ+βvˆ 1−ρ1
+ ρ1h(ˆu,ˆv) (1−ρ1)
Ä
1−ρ1(1 +r)
ä (24)
J2(ˆu,vˆ) =
x0
1−ρ2(1 +r) −
γuˆ−δvˆ 1−ρ2 −
ρ2h(ˆu,vˆ)
(1−ρ2)
Ä
1−ρ2(1 +r)
ä (25)
and in the case of a weak eradication equilibrium, the values of the objectives are:
J1(u, v) = −x0−αu0−βv0 (26)
J2(u, v) = x0−γu0+δv0 (27)
7For a strong eradication equilibrium to exist it must be in the government’s interest to destroy any
positive amount of the ITO’s resources (this is the conditionα 1−(1 +r)ρ1
−cρ1<0in Theorem 4.1, when
c >0). But for a perpetual war equilibrium to exist, the ITO’s resources must never be eliminated (this is the conditionα 1−(1 +r)ρ1
−cρ1>0 in Theorem 3.1). Both conditions are therefore incompatible. 8For a weak eradication equilibrium to exist, the level of the ITO’s resources must not be too high. Indeed,
where the pair (u0, v0)is such that x1 = 0 and is given as in Theorem 4.1 (ii).
From the government’s viewpoint, the comparison between perpetual war and eradication hinges on the following elements. On the one hand, eradication is costly at the time of the attack but it eliminates future terrorists’ attacks. On the other hand, in the case of a perpetual war, the government has to bear an indefinite stream of terrorists’ attacks.
In the next result we show that the part of the loss incurred by the government due to the ITO’s resources xt, is lower in an eradication equilibrium than in a perpetual war. Indeed,
by inspecting J1(ˆu,vˆ) and J1(u, v), we have:
Lemma 5.1.
x0 <
Ä x0−
ρ1h(ˆu,vˆ)
1−ρ1
ä 1
1−ρ1(1 +r) (28)
Proof. Since h(ˆu,rˆv) ≤x0 and (1−ρ1r)(1+r) >1 we have
x0 >
h(ˆu,ˆv)
(1−ρ1)(1 +r) (29)
so the inequality follows.
The next lemma compares the values of the strikes in a perpetual war and in a weak eradication equilibrium.
Lemma 5.2. Let us assume that c ≥ 0. Under the hypotheses of Theorem 3.1 we have:
u0 ≤uˆ, and v0 ≥vˆ.
Proof. See Appendix.
Proposition gives sufficient conditions for the government to be better off in a weak eradication equilibrium.
Proposition 5.1. Let c≥0, d >0, αÄ1−(1 +r)ρ1
ä
−cρ1 >0, δ
Ä
1−(1 +r)ρ2
ä
−dρ2 >0,
ˆ
x0 is sufficiently high (xˆ0 ≥ h(ˆu,rˆv)), anduˆandˆv are given by (6) and (7). Then the government
is better off in a weak eradication equilibrium ifv0 ≤ 1−ˆvρ1. If not, this is the case when β ≥A
where
A=
x0ρ1(1+r)
1−ρ1(1+r) −
ρ1h(ˆu,ˆv)
(1−ρ1)
Ä
1−ρ1(1+r)
ä−α(u0− uˆ 1−ρ1)
(v0− 1−ˆvρ1)
Proof. We have:
J1(ˆu,vˆ)−J1(u, v) = −
x0
1−ρ1(1 +r)−
αuˆ+βvˆ 1−ρ1
+ ρ1h(ˆu,ˆv) (1−ρ1)
Ä
1−ρ1(1 +r)
ä +x0+αu0+βv0
= −x0ρ1(1 +r) 1−ρ1(1 +r)
+ ρ1h(ˆu,vˆ) (1−ρ1)
Ä
1−ρ1(1 +r)
ä+α(u0−
ˆ
u
1−ρ1
) +β(v0−
ˆ
v
1−ρ1
)
Using Lemma 5.1 we have −x0ρ1(1 +r)
1−ρ1(1 +r)
+ ρ1h(ˆu,vˆ) (1−ρ1)
Ä
1−ρ1(1 +r)
ä <0and using Lemma 5.2
we have u0−
ˆ
u
1−ρ1 ≤
0.
So ifv0 ≤ 1−ˆvρ1 then J1(ˆu,ˆv)≤J1(u, v).
If not, the last expression is positive when β ≥A and negative otherwise where
A=
x0ρ1(1+r)
1−ρ1(1+r) −
ρ1h(ˆu,ˆv)
(1−ρ1)
Ä
1−ρ1(1+r)
ä−α(u0− uˆ 1−ρ1)
(v0− 1−ˆvρ1)
Interestingly, the parameter β, which represents the cost of ITO’s strike on the per-period
government utility does not enter into the values of uˆand ˆv, nor into the optimal values ofu0 and v0 in a weak eradication equilibrium (this stems from the fact that the terrorists’ strike
vt enters the government’s objective in a linear way). Therefore, from the above inequality,
preferred to a perpetual war. This arises provided that the cost of the terrorists’ strike is high enough for the government.
6
Concluding Remarks
In this paper, we have studied the dynamic interactions between a government and an international terror organization. A key feature of these interactions is the dynamics of the terrorists’ resources which is described by a difference equation. The dynamics of these resources depends negatively on the strength of the strike of the terrorists and the government (assuming that there are no collateral effects).
In this setting we have shown that there is an equilibrium in which there is a perpetual war between the government and the terrorists. This result has already been found in the literature (in a slightly different setup). However, a major difference between this paper and existing studies is that in our setting there are also eradication equilibria. Depending on the values taken by the parameters, there may be a strong eradication equilibrium in which the government’s strike annihilates the terrorists’ resources, preventing the terrorists from acting. There may also be weak eradication equilibria in which the terrorists’ resources are destroyed, but in which the initial value of the terrorists’ strike is nevertheless positive.
We have also shown that both perpertual war and strong eradication equilibria can never co-exist. Likewise, when the government’s strikes generate collateral effects, both perpetual war and weak eradication equilibria can never co-exist.
For these cases in which both equilibria with perpetual war and weak eradication equilibria co-exist, we have provided conditions under which the government would be better off in a weak eradication equilibrium.
weak eradication equilibrium). Since in the alternative policy, i.e. engaging in a perpetual
war, the ITO will also strike, the government must face the fact that avoiding casualties is not possible.
We have finally also shown that there are multiple weak eradication equilibria, and we are therefore in a situation of indeterminacy since we do not know how the two players may coordinate their actions.
A
Proof of Theorem 3.1
This proof studies separately the casesc >0andc≤0and relies on several lemmas. Forc >0,
Lemma A.1 shows that if a sequence of vectors ((ut)t,(vt)t) with positive coordinates is a
Nash equilibrium then (together with adjoint variables) it is a solution of a dynamical system. Lemma A.2 shows that the conditions(C): [αÄ1−(1+r)ρ1
ä
−cρ1 >0,δ
Ä
1−(1+r)ρ2
ä
−dρ2 >0 and x0 ≥ h(ˆu,rˆv)] are necessary and sufficient for the existence of a solution of that dynamical system. Hence conditions(C) are necessarily satisfied in a Nash equilibrium with perpetual
war. Lemma A.3 together with Lemma A2 prove that these conditions are also sufficient. Forc≤0 a similar procedure is followed.
A.1
The Case
c >
0
Lemma A.1. If (ˆu,ˆv) is a Nash equilibrium such that for all t ∈N, xˆt>0, uˆt>0, vˆt>0,
then there exist (q1
t)t≥1, (q2t)t≥1 such that:
qt1 =−1 + (1 +r)ρ1qt1+1 (30)
−α=ρ1qt1+1
Ä
c+uˆt−1vˆθtä (31)
qt2 = 1 + (1 +r)ρ2qt2+1 (32)
δ =ρ2qt2+1
Ä
d+θuˆtvˆθt−1ä (33)
Proof. Set
Φ1(x
t, ut, vt)≡ −xt−αut−βvt (34)
Φ2(x
t, ut, vt)≡xt−γut+δvt (35)
ft(xt, ut, vt)≡(1 +r)xt−cut−dvt−utvtθ (36)
Set also
Ht1(xt, ut, vt, p1t+1)≡ρt1Φ1(xt, ut, vt) +p1t+1ft(xt, ut, vt) (37)
Since (ˆu,ˆv) is a Nash equilibrium,(ˆut)t is a solution of the following optimal control problem:
maximizeu J1(u,ˆv) = +∞
X
t=0
ρt1Φ1(xt, ut,vˆt)
subject to xt+1 = (1 +r)xt−cut−dvˆt−utvˆθt
xt ≥0, ut ≥0, ∀t≥0.
and (ˆvt)t is a solution of:
maximizev J2(ˆu, v) = +∞
X
t=0
ρt2Φ2(xt,uˆt, vt)
subject to xt+1 = (1 +r)xt−cuˆt−dvt−uˆtvtθ
xt≥0vt≥0, ∀t≥0.
Notice that, given vˆ, for all t ∈ N, Φ1 and ft are of class C1 with respect to (xt, ut) on
R∗+×R∗+ and Dutft(ˆxt,uˆt,vˆt) =−c−uˆ
−1
t vˆt 6= 0.
Notice also that, given uˆ, for all t∈N, Φ2 and ft are of class C1 with respect to (xt, vt) on
R∗+×R∗+ and Dvtft(ˆxt,uˆt,ˆvt) =−d−θuˆ
tˆvtθ−1 6= 0.
Our hypotheses allow us to use a necessary condition theorem for infinite-horizon optimal control discrete time problems (see Blot-Chebbi [16], Theorem 4). Applying this theorem to each problem provides the existence of adjoint variables (p1t)t≥1, and (p2t)t≥1 such that :
p1t =DxtH 1
t(ˆxt,uˆt,vˆt, p1t+1), t≥1 (39)
DutH 1
t(ˆxt,uˆt,vˆt, p1t+1) = 0, t≥0 (40)
p2t =DxtH 2
t(ˆxt,uˆt,vˆt, p2t+1), t≥1 (41)
DvtH 2
or:
p1t =−ρt1+ (1 +r)p1t+1 (43)
−αρt1 =p1t+1Äc+uˆt−1vˆθtä (44)
p2t =ρt2+ (1 +r)p2t+1 (45) δρ2t =p2t+1Äd+θuˆtvˆθt−1ä (46)
Now we make the following changes of variables:
qt1 := p
1
t
ρt
1
(47)
qt2 := p
2
t
ρt
2
. (48)
which implies (30)-(33).
Consider the following system (S):
qt1 =−1 + (1 +r)ρ1qt1+1
−α =ρ1qt1+1
Ä
c+ut−1vtθ
ä
qt2 = 1 + (1 +r)ρ2qt2+1
δ =ρ2qt2+1
Ä
d+θutvtθ−1
ä
xt+1 = (1 +r)xt−cut−dvt−utvtθ
(S)
Lemma A.2. System (S) has a unique solution (satisfyingxt >0, ut>0, vt>0) if and only
if αÄ1−(1 +r)ρ1
ä
−cρ1 >0, δ
Ä
1−(1 +r)ρ2
ä
given by:
qt1 = 1 (1 +r)ρ1−1
, ∀t ≥1 (49)
qt2 = −1 (1 +r)ρ2−1
, ∀t ≥1 (50)
ˆ
ut= ˆu, ∀t ≥0 (51)
ˆ
vt= ˆv, ∀t≥0 (52)
where
ˆ
u=
δÄ1−(1 +r)ρ2
ä
−dρ2
ρ2θ
Å ρ
1
αÄ1−(1 +r)ρ1
ä
−cρ1
ãθ−1
θ
θ +θ−1
(53)
ˆ
v =
Ñ
αÄ1−(1 +r)ρ1
ä
−cρ1
ρ1
é1
θ
ˆ
u1−θ (54)
and
ˆ
xt= (1 +r)t
ñ x0−
h(ˆu,vˆ)
r ô
+ h(ˆu,ˆv)
r , t∈N (55)
Proof. Since q1
t =−1 + (1 +r)ρ1q1t+1 we have:
q1
t+1 =
1 (1 +r)ρ1
q1
t + 1
(1 +r)ρ1 (56)
so:
q1t+1 =
Ç
1 (1 +r)ρ1
åtÇ
q11− 1
(1 +r)ρ1−1
å
+ 1
(1 +r)ρ1−1 (57) Similarly:
q2t+1 =
Ç
1 (1 +r)ρ2
åtÇ
q12− 1
1−(1 +r)ρ2
å
+ 1
1−(1 +r)ρ2 (58) The sequences (q1
we have |q1
t+1| = ρ1(c+uαt−1vtθ) <
α
ρ1c and using the fourth equation of system (S) we have |q2
t+1|= ρ2(d+θuδtvtθ−1) <
δ ρ2d.
Since (1 +r)ρi <1, i = 1,2 the unique bounded solutions to the above equations are the
constant sequences:
qt1 = −1 1−ρ1(1 +r)
, ∀t ≥1 (59)
qt2 = 1 1−(1 +r)ρ2
, ∀t ≥1. (60)
Now ifαÄ1−(1 +r)ρ1
ä
−cρ1 >0,δ
Ä
1−(1 +r)ρ2
ä
−dρ2 >0andx0 ≥ h(ˆu,rˆv), then the second and the fourth equations in (S) provide uˆt and vˆt and therefore the state equation provides ˆ
xt. If α
Ä
1−(1 +r)ρ1
ä
−cρ1 ≤0 orδ
Ä
1−(1 +r)ρ2
ä
−dρ2 ≤ 0or x0 < h(ˆu,rvˆ), then there is no solution.
Lemma A.3. If αÄ1−(1 +r)ρ1
ä
−cρ1 >0 and δ
Ä
1−(1 +r)ρ2
ä
−dρ2 >0and if x0 ≥ h(ˆu,rˆv)
then a Nash equilibrium satisfying xˆt >0, uˆt> 0, vˆt >0, for all t ∈N, is given by (ˆu, ˆv)
where:
ˆ
ut= ˆu=
δÄ1−(1 +r)ρ2
ä
−dρ2
ρ2θ
Ñ
ρ1
αÄ1−(1 +r)ρ1
ä
−cρ1
éθ−1
θ
θ +θ−1
(61)
ˆ
vt= ˆv =
Ñ
αÄ1−(1 +r)ρ1
ä
−cρ1
ρ1
é1
θ
ˆ
u1−θ (62)
and xˆ is given by
ˆ
xt= (1 +r)t
ñ x0−
h(ˆu,vˆ)
r ô
+ h(ˆu,ˆv)
r , t∈N (63)
Proof. Let us consider the unique constant sequences (q1
solve System (S)together with (ˆxt)t≥0 and which are given by Lemma A.2. Set:
p1t =ρt1qt1, (64) p2t =ρt2qt2 (65)
Then (p1
t)t≥1, (p2t)t≥1, (ˆut)t≥0, (ˆvt)t≥0, (ˆxt)t≥0 satisfy (39)-(42) and (1). We notice that q1
t <0 so p1t <0 and so Ht1 is concave with respect to xt and ut. We also
notice that q2
t >0 sop2t >0so Ht2 is concave with respect to xt and vt.
In what follows, we omit the superscript 1 in H1 and p1. Givenvˆ, let (x, u) be an admissible process in the government’s problem. Using (39)-(42), for all t we have:
ρt1Φ1(ˆxt,uˆt,vˆt)−ρt1Φ1(xt, ut,vˆt) =Ht(ˆxt,uˆt,vˆt, pt+1)−pt+1ft(ˆxt,uˆt,ˆvt) (66)
−Ht(xt, ut,ˆvt, pt+1) +pt+1ft(xt, ut,ˆvt) (67) =Ht(ˆxt,uˆt,vˆt, pt+1)−Ht(xt, ut,vˆt, pt+1) (68)
−Dxt+1Ht+1(ˆxt+1,uˆt+1,ˆvt+1, pt+2)(ˆxt+1−xt+1) (69) −DutHt(ˆxt,uˆt,vˆt, pt+1)(ˆut−ut) (70)
therefore we obtain:
T
X
t=0
Ä
ρt1Φ1( ˆxt,uˆt,vˆt)−ρt1Φ1(xt, ut,vˆt)
ä
= T
X
t=0
Å
Ht(ˆxt,uˆt,vˆt, pt+1)−Ht(xt, ut,vˆt, pt+1)
−Dxt+1Ht+1(ˆxt+1,uˆt+1,vˆt+1, pt+2)(ˆxt+1−xt+1)
−DutHt(ˆxt,uˆt,vˆt, pt+1)(ˆut−ut)
ã
= T
X
t=0
Å
Ht(ˆxt,uˆt,ˆvt, pt+1)−Ht(xt, ut,vˆt, pt+1)
−DxtHt(ˆxt,uˆt,ˆvt, pt+1)(ˆxt−xt)
−DutHt(ˆxt,uˆt,vˆt, pt+1)(ˆut−ut)
ã
+pT+1(ˆxT+1−xT+1),
respect to xt and ut, we have:
Ht(ˆxt,uˆt,vˆt, pt+1)−Ht(xt, ut,vˆt, pt+1)
−DxtHt(ˆxt,uˆt,vˆt, pt+1)(ˆxt−xt)−DutHt(ˆxt,uˆt,vˆt, pt+1)(ˆut−ut)≥0 (71)
So:
T
X
t=0
Å
ρt1Φ1( ˆxt,uˆt,vˆt)−ρt1Φ1(xt, ut,ˆvt)
ã
≥pT+1(ˆxT+1−xT+1) (72)
=ρT1+1q1T+1(ˆxT+1−xT+1) (73)
= 1
(1 +r)ρ1−1
ρT1+1(ˆxT+1−xT+1) (74)
≥ 1
(1 +r)ρ1 −1
ρT1+1xˆT+1 (75)
It is easy to show that lim T→∞ρ
T+1
1 xˆT+1 = 0 using (63). Hence: we have
∞
X
t=0
Ä
ρt1Φ1( ˆxt,uˆt,vˆt)−ρt1Φ1(xt, ut,ˆvt)
ä
≥0. (76)
That is J1(ˆu,vˆ)≥J1(u,ˆv).
Now givenuˆ, let (x, v) be an admissible process in the ITO’s problem. Proceeding similarly
we obtain:
T
X
t=0
Å
ρt2Φ2( ˆxt,uˆt,vˆt)−ρt2Φ2(xt,uˆt, vt)
ã
≥p2T+1(ˆxT+1−xT+1) (77)
=ρT2+1q2T+1(ˆxT+1−xT+1) (78)
= 1
1−(1 +r)ρ2
ρT2+1(ˆxT+1−xT+1) (79)
When c≥0it is easy to show that lim T→∞ρ
T+1
2 (ˆxT+1−xT+1) = 0.
and thus lim t→∞ρ
t+1
2 xt+1 = 0. Hence:
∞
X
t=0
Ä
ρt2Φ2( ˆxt,uˆt,vˆt)−ρt2Φ2(xt,uˆt, vt)
ä
≥0. (80)
That is J2(ˆu,vˆ)≥J2(ˆu, v).
A.2
The Case
c
≤
0
Let(ˆu,ˆv)be an overtaking Nash equilibrium (respectively a weak overtaking Nash equilibrium).
We notice that, given ˆv, for all t ∈ N, Φ1 and ft are of class C1 with respect to (xt, ut) on
R∗+×R∗+ and Dxtft(ˆxt,uˆt,ˆvt) = 1 +r > 0. Therefore, we can use a necessary condition theorem for infinite-horizon optimal control discrete time problems (see Blot-Chebbi [16], Theorem 3 ).
We also notice that, given uˆ, for all t∈N, Φ2 andft are of classC1 with respect to (xt, vt)
on R∗+×R∗+ and Dvtft(ˆxt,uˆt,vˆt) = −d−θuˆ
tˆvtθ−1 6= 0. Therefore, we can use a necessary
condition theorem for infinite-horizon optimal control discrete time problems (see Blot-Chebbi [16], Theorem 4).
Applying these theorems to our problems provides the existence of λ and (p1
t)t≥1, not both nil and(p2
t)t≥1 satisfying the next equations :
λ≥0 (81)
p1t =−λρt1+ (1 +r)p1t+1 (82)
−λαρt1 =p1t+1Äc+uˆt−1vˆtθä (83)
p2t =ρt2+ (1 +r)p2t+1 (84) δρ2t =p2t+1Äd+θuˆtvˆθt−1ä (85)
qt1 := p
1
t
ρt
1
(86)
q2
t :=
p2
t
ρt
2
. (87)
We consider now the following system (S’):
q1
t =−λ+ (1 +r)ρ1q1t+1
−λα =ρ1qt1+1
Ä
c+ut−1vtθ
ä
q2
t = 1 + (1 +r)ρ2qt2+1
δ =ρ2qt2+1
Ä
d+θutvtθ−1
ä
xt+1 = (1 +r)xt−cut−dvt−utvtθ
(S’)
We notice that the sequence (q2
t)t≥1 is bounded since d >0. If λ = 0 then q1
t = (1 +r)ρ1q1t+1, so that q1t = ((1+1r)ρ1)
t−1q11, t ≥1. If q1
1 = 0, then qt1 = 0,
∀t≥1. But this is a contradiction so λ6= 0, and we may set λ = 1. Ifq1
1 6= 0, then (q1t)t≥1 is an unbounded sequence and the sequences (ˆut)t and (ˆvt)t which are constant do not depend
onq1
t.
We however restrict our analysis to bounded sequences (q1
t)t≥1. Thus we set λ = 1 and
the system (S’) looks like the system (S) in the case c > 0. We only consider bounded
sequences(q1
t)t≥1 which gives results similar to those of Lemma A.2 (replace a unique solution by a solution in the statement of this lemma) and which is also justified by applying direct sufficient conditions with a transversality condition.
Lemma A.3 goes the same as in the case c >0until: T
X
t=0
Å
ρt1Φ1( ˆxt,uˆt,ˆvt)−ρt1Φ1(xt, ut,vˆt)
ã
≥ (1 +r1)ρ
1−1
ρT1+1xˆT+1
It is easy to show that lim T→∞ρ
T+1
1 xˆT+1 = 0using (63). Hence by takinglim inf
T→+∞
PT
t=0
Å ρt
1Φ1( ˆxt,uˆt,vˆt)−
ρt
1Φ1(xt, ut,vˆt)
ã
We also have:
T
X
t=0
Å
ρt2Φ2( ˆxt,uˆt,vˆt)−ρt2Φ2(xt,uˆt, vt)
ã
≥ 1
1−(1 +r)ρ2
ρT2+1(ˆxT+1−xT+1).
We notice thatxt+1 ≤(1 +r)xt−cuˆt= (1 +r)xt−cuˆsoxt+1 ≤(1 +r)t+1x0−cuˆPti=0(1+r)t−i andρt2+1xt+1 ≤ρ2t+1(1 +r)t+1x0−cuρˆ 2t+1(1 +r)tPti=0(1 +r)−i. Thustlim→∞ρt2+1xt+1 = 0. Hence by taking lim inf
T→+∞
PT
t=0
Å ρt
2Φ2( ˆxt,uˆt,vˆt)−ρt2Φ2(xt,uˆt, vt)
ã
we obtain that ˆv satisfies (6). So
(ˆu,vˆ)is an overtaking Nash equilibrium. We observe that by takinglim sup
T→+∞ instead oflim infT→+∞
this implies that (ˆu,ˆv)is a weak overtaking Nash equilibrium.
B
Proof of Theorem 4.1
This proof consists in checking that there is a Nash equilibrium in which xt is nil from date 1
on, andut =vt= 0, for allt ≥1.
B.1
No collateral damages (
c
>
0
)
B.1.1 Best responses of the government
First we study the best-responses functions (or correspondences) of the government assuming that: vt = 0, for all t≥1 and 0≤v0 ≤ (1+dr)x0 (so that there are feasible values of u0).
We recall that the government’s problem is as follows:
max
(ut)t J1(u, v)
xt+1 = (1 +r)xt−cut, t≥1
x1 = (1 +r)x0−cu0−dv0−u0v0θ,
We may rewrite the government’s objective as follows:
J1(u, v) = −x0−αu0−βv0+ +∞
X
t=1
ρt1[−xt−αut]
=−x0−αu0−βv0+ +∞
X
t=1
ρt1 ï
−xt+
α c Å
xt+1−(1 +r)xt
ãò
=−x0−αu0−βv0− +∞
X
t=1
ρt1xt
Ä
1 + α
c(1 +r) ä
+
+∞
X
t=1
ρt1α cxt+1
=−x0−αu0−βv0− +∞
X
t=1
ρt1xt
Ä
1 + α
c(1 +r) ä
+
+∞
X
t=2
ρt1−1α
cxt+x1 α
c −x1 α
c
=−x0−αu0−βv0−x1
α c +
+∞
X
t=1
ρt−1 1 xt
Åα
c −ρ1 Ä
1 + α
c(1 +r) äã
(88)
=−x0−αu0−βv0−((1 +r)x0−cu0−dv0−u0v0θ)
α c + +∞ X t=1
ρt1−1xt
Åα
c −ρ1 Ä
1 + α
c(1 +r) äã
=−x0(1 + (1 +r)
α
c)−v0(β−d α
c) +u0
v 0θ α c + +∞ X t=1
ρt1−1xt
ÅαÄ1−(1 +r)ρ
1
ä
−cρ1
c
ã
(89)
The government’s problem is equivalent to:
max u0≥0
(1+r)x0−cu0−dv0−u0v0θ≥0
−x0(1 + (1 +r)
α
c)−v0(β−d α
c) +u0
v
0θ
α c
+ max
(xt)t>1
0≤xt≤(1+r)t−1x1
x1=(1+r)x0−cu0−dv0−u0v0θ
+∞
X
t=1
ρt1−1xt
ÅαÄ1−(1 +r)ρ
1
ä
−cρ1
c
ã
The existence of a solution to the government’s problem is obtained by noticing that the problem is a maximization problem of a continuous function on a compact set. The proof uses the product topology defined on the space of infinite sequences of real numbers.
• If αÄ1−(1 +r)ρ1
ä
−cρ1 <0, then it is optimal to setxt = 0, for all t≥2. This requires
So the problem becomes
max
u0 g(u0)≡maxu0
−x0(1 + (1 +r)
α
c)−v0(β−d α
c) +u0
v
0θ
α c +x1
ÅαÄ1−(1 +r)ρ
1
ä
−cρ1
c
ã
= max u0
−cu0
ÅαÄ1−(1 +r)ρ
1
ä
c −ρ1 ã
+ρ1u0v0θ(1 +
α(1 +r)
c )
−x0−βv0−ρ1(1 +r)
Ä
1 + α
c(1 +r) ä
x0+ρ1dv0
Ä
1 + α
c(1 +r) ä
(90)
under the constraintsu0 ≥0 and x1 ≥0 that iscu0+dv0+u0v0θ ≤(1 +r)x0. We can see thatgis an increasing function ofu0under the hypothesisα
Ä
1−(1+r)ρ1
ä
−cρ1 <0. Thus the maximum is attained for the greatest value of u0, denoted M(v0) such that
cu0+dv0+u0v0θ = (1 +r)x0, i.e., x1 = 0. Since we have assumed that dv0 ≤(1 +r)x0,
cu0+dv0+u0v0θ = (1 +r)x0 always has a solution and it is unique. We have then shown that u0 =M(v0), ut= 0, t≥1, xt= 0,t ≥1.
• IfαÄ1−(1 +r)ρ1
ä
−cρ1 = 0, then the problem of the government reduces to
maxu0
¶
−x0(1 + (1 +r)αc)−v0(β−dαc) +u0v0θ αc
©
under the constraints u0 ≥0and x1 ≥0 (for t≥1,xt+1 = (1 +r)xt−cut≥0 and ut≥0).
If v0 6= 0, then u0 = M(v0), i.e., x1 = 0 and this implies ut = 0, t ≥ 1 and xt = 0, t ≥ 1.
Hence we get the same result as for the case αÄ1−(1 +r)ρ1
ä
−cρ1 <0.
If v0 = 0 then u0 can take all values between 0 and (1 +r)x0/c. So x1 ≥ 0 (for t ≥ 1,
xt+1 = (1 +r)xt−cut ≥0and ut ≥0).
• If αÄ1−(1 +r)ρ1
ä
implies ut= 0, t≥1. So we have from (88):
−x0−αu0 −βv0−x1
α
c + max(xt)t>1
(+∞ X
t=1
ρt−1 1 xt
Åα
c −ρ1 Ä
1 + α
c(1 +r) äã)
(91)
=−x0−αu0−βv0 −x1
α c +
+∞
X
t=1
ρt1−1(1 +r)t−1x1
Åα
c −ρ1 Ä
1 + α
c(1 +r) äã
(92)
=−x0−αu0−βv0 −x1
α c +
1 1−ρ1(1 +r)
x1
Åα
c −ρ1 Ä
1 + α
c(1 +r) äã
(93)
=−x0−αu0−βv0 −
ρ1x1
1−ρ1(1 +r)
=−x0−αu0−βv0 −ρ1
(1 +r)x0−cu0 −dv0−u0v0θ
1−ρ1(1 +r) ≡
s(u0) (94)
Therefore, the problem is now written: maxu0s(u0) under the constraints u0 ≥0and x1 ≥0.
Since s is continuous and the set of feasible choices of u0 is compact s realizes its maximum. If v0 = 0, then s(.) is a decreasing linear function of u0 (because we are in the case
αÄ1−(1 +r)ρ1
ä
−cρ1 >0). Therefore, u0 = 0. If v0 = (1 +r)x0/d, then u0 = 0. So x1 = 0.
If v0 6= 0 andv0 6= (1 +r)x0/d, it is easy to check that u0 = 0 does not realize the maximum of s under the constraints so the value of u0 that solves the first-order condition
0 = s0(u0) =−α+
ρ1c
1−ρ1(1 +r)
+ ρ1u0 −1v
0θ
1−ρ1(1 +r) (95) is:
u0 =
ñ
α(1−ρ1(1 +r))−ρ1c
ρ1v0θ
ô1/(−1)
(96)
Sincesis strictly concave onR+ it attains its maximum at this value ifu0 satisfiesu0 ≤M(v0). If not then the maximum is attained at M(v0). So we obtain that ut = 0, t ≥ 1 and
u0 = min{ hα(1−ρ
1(1+r))−ρ1c
ρ1v0θ
i−11
, M(v0)} and, x1 = (1 +r)x0 −cu0 −dv0 −u0v0θ, xt+1 =
B.1.2 Best response of the ITO
Let us study the best response of the ITO when ut= 0 for all t≥1. We assume throughout
that 0≤u0 ≤ (1+cr)x0 (so that there are feasible values of v0). Let us consider the following problem:
max
(vt)tJ2(u, v)
xt+1 = (1 +r)xt−dvt, t≥1
x1 = (1 +r)x0−cu0−dv0−u0v0θ,
xt≥0, vt≥0, ∀t≥0.
We have:
J2(u, v) =x0−γu0+δv0+ +∞
X
t=1
ρt2[xt+δvt] (97)
=x0−γu0+δv0+ +∞
X
t=1
ρt2
ñ xt−
δ d Ä
xt+1−(1 +r)xt
äô
=x0−γu0+δv0+ +∞
X
t=1
ρt2xt
Ä
1 + δ
d(1 +r) ä
−
+∞
X
t=1
ρt2δ dxt+1
=x0−γu0+δv0+ +∞
X
t=1
ρt2xt
Ä
1 + δ
d(1 +r) ä
−
+∞
X
t=2
ρt2−1δ
dxt−x1 δ d +x1
δ d
=x0−γu0+δv0+x1
δ d−
+∞
X
t=1
ρt2−1xt
Åδ
d −ρ2 Ä
1 + δ
d(1 +r) äã
(98)
=x0−γu0+δv0+ (1 +r)x0
δ d −
Ä
cu0+dv0+u0v0θ
äδ d − +∞ X t=1
ρt2−1xt
Åδ
d −ρ2 Ä
1 + δ
d(1 +r) äã
=x0(1 + (1 +r)
δ
d)−u0(γ+c δ d)−u0
v 0θ δ d − +∞ X t=1
ρt2−1xt
ÅδÄ1−(1 +r)ρ
2
ä
−dρ2
d
ã
(99)
The ITO’s problem is equivalent to:
max v0≥0
(1+r)x0−cu0−dv0−u0v0θ≥0
x0(1 + (1 +r)
δ
d)−u0(γ+c δ d)−u0
v
0θ
δ d
+ max
(xt)t>1 0≤xt≤(1+r)t−1x1, t≥1
x1=(1+r)x0−cu0−dv0−u0v0θ −
+∞
X
t=1
ρt2−1xt
ÅδÄ1−(1 +r)ρ
2
ä
−dρ2
d
ã
• IfδÄ1−(1 +r)ρ2
ä
−dρ2 >0, then it is optimal to set xt= 0, for allt≥2, which implies
that v1 = (1 +r)x1/d and vt= 0, t≥2. So the problem becomes:
max
v0 k(v0) := maxv0
x0(1 + (1 +r)
δ
d)−u0(γ+c δ d)−u0
v
0θ
δ d −x1
ÅδÄ1−(1 +r)ρ
2
ä
−dρ2
d ã = max v0 ® v0 ñ
δ−ρ2d
Ä
1 + δ
d(1 +r) äô
−ρ2u0v0θ
Ä
1 + δ
d(1 +r) ä
+x0−γu0+ρ2(1 +r)
Ä
1 + δ
d(1 +r) ä
x0−ρ2cu0
Ä
1 + δ
d(1 +r) ä´
(100)
under the constraints v0 ≥0and x1 ≥0. So the set of feasible choices is the set of v0’s such that0≤v0 ≤N(u0)whereN(u0)is the non negative solution ofcu0+dv0+u0v0θ = (1+r)x0. Since k(.) is continuous and the set of feasible choices is compact, k realizes its maximum.
If u0 = 0, then v0 = (1 +r)x0/d. So x1 = 0.
If u0 6= 0, then the solution of the first-order condition:
0 =k0(v0) =δ−ρ2d
Ä
1 + δ
d(1 +r) ä
−ρ2θu0v0θ−1
Ä
1 + δ
d(1 +r) ä
is:
v0 =
δ−ρ2d
Ä
1 + δd(1 +r)ä
ρ2θu0
Ä
1 + dδ(1 +r)ä
1/(θ−1)
(101)
This expression is well defined since we are in the case: δÄ1−(1 +r)ρ2
ä
−dρ2 >0.
Otherwise, v0 = N(u0). So we have obtained that v0 = min{ "
δ−ρ2d
Ä
1+δd(1+r)
ä
ρ2θu0
Ä
1+δd(1+r)
ä
#θ−11
, N(u0)},
v1 = (1 +r)x1/d,vt= 0, t≥2and x1 = (1 +r)x0−cu0−dv0−u0v0θ, xt= 0, t≥2.
In particular if u0 = (1 +r)x0/c then v0 =N((1 +r)x0/c) = 0.
• IfδÄ1−(1 +r)ρ2
ä
−dρ2 = 0, the problem reduces to
maxv0
¶
x0(1 + (1 +r)δd)−u0(γ+cδd)−u0v0θ δd
©
under the constraintsv0 ≥0 and x1 ≥0. If u0 6= 0 , the maximum is reached at v0 = 0. We then have xt+1 = (1 +r)xt−dvt, t≥ 1,
x1 = (1 +r)x0−cu0. The objective does not depend on the future values of (vt)t. If u0 = 0 then v0 can take all values between 0 and (1 +r)x0/d.
• If δÄ1−(1 +r)ρ2
ä
−dρ2 <0, then it is optimal to choose xt= (1 +r)t−1x1, t≥1 and this
implies vt= 0, t≥1. So we have from equation (98):
x0−γu0+δv0+x1
δ
d −(maxxt)t>1
(+∞ X
t=1
ρt2−1xt
Åδ
d −ρ2 Ä
1 + δ
d(1 +r) äã)
=x0−γu0+δv0+x1
δ d −
x1
1−ρ2(1 +r)
Åδ
d −ρ2 Ä
1 + δ
d(1 +r) äã
=x0−γu0+δv0+
ρ2x1
1−ρ2(1 +r)
=x0−γu0+δv0+
ρ2
1−ρ2(1 +r)
Ä
(1 +r)x0−cu0−dv0−u0v0θ
ä
=v0
ñ
δ−d ρ2
1−ρ2(1 +r)
ô
− ρ2
1−ρ2(1 +r)
u0v0θ
+x0−γu0+
ρ2
1−ρ2(1 +r)
Ä
(1 +r)x0−cu0
ä
(102)
≡l(v0) (103)
The problem reduces to maxv0l(v0) under the constraints v0 ≥0 and x1 ≥0. Since under
the assumption δÄ1−(1 +r)ρ2
ä
B.1.3 Nash Equilibria
Up to now, we have studied the best responses of the government and the ITO when each of them considers that the other ceases to play from date t onward, and when they take the
other’s decision at date zero as given. We will now give the Nash equilibria obtained. These equilibria are sequences (ut)t∈N, (vt)t∈N such that ∀t ≥1, ut = 0, vt= 0, and (u0, v0) can take different values and thus provide different kinds of equilibria.
This leads us to distinguish several cases.
Case αÄ1−(1 +r)ρ1
ä
−cρ1 >0, δ
Ä
1−(1 +r)ρ2
ä
−dρ2 >0
-In this case, there are weak eradication equilibria, where the actions at t= 0 are given by all (u0, v0) satisfying: u0 6= 0, v0 6= 0 and x1 = 0
u0 = min
αÄ1−ρ1(1 +r)
ä
−ρ1c
ρ1v0θ
1
−1
, M(v0) ,
v0 = min
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