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Dynamic Games and Applications

Erschienen in:

15.04.2016

Intertemporal Versus Spatial Externalities in Counterterror Policy Games

verfasst von: João Ricardo Faria, Emilson C. D. Silva, Daniel G. Arce

Erschienen in: Dynamic Games and Applications | Ausgabe 3/2017

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Abstract

This paper studies a dynamic game between two national governments that fight a common terrorist organization that is seeking to mount a transnational terror campaign. It is the first examination that combines the temporal externalities associated with a sustained campaign with the spatial externalities that occur when the effects of one government’s counterterror policy spill over into another country. We consider two types of noncooperative behavior; one in which national authorities are sensitive to the reactions of the terrorists on foreign soil and another in which they are insensitive. It is shown that foreign terrorist sensitivity is preferred to insensitivity. Moreover, unilaterally accounting for terrorist reactions on foreign soil can be preferred to full policy coordination between governments. This then feeds into policy recommendations as to when each nation finds it desirable to coordinate transnational counterterror policy.

Vorheriger Artikel On Nonzero-Sum Game Considered on Solutions of a Hybrid System with Frequent Random Jumps

Nächster Artikel Approachability in Stackelberg Stochastic Games with Vector Costs

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In this sense, our model is consistent with economic models of habit formation (see, e.g., [6, 10, 11, 17, 29, 36]). See also Rozen [35].

No a priori assumptions are made, however, about the existence of first- or second-mover advantages.

See also Das [16] for a model of terrorist cycles. Behrens et al. [7] examine a Stackelberg dynamic game of international terrorism.

In addition, an attack such as the Black September kidnaping of Israeli athletes at the 1972 Munich Olympic Games would qualify as a success even though few terrorists (and no athletes) survived. It certainly brought the cause of the Palestinians to prominence. By contrast, the successful extraction of hostages in a skyjacking [such as Lufthansa flight 181 during 1977 or Air France Flight 139 in 1976 (the Raid on Entebbe)] would not be deemed as a success because interventions such as these typically encourage other nations to develop similar highly trained rescue teams in order to credibly commit to a policy of no negotiation in future skyjackings.

In the games examined below, the terrorist organization makes its decisions after it observes governmental policy commitments. It is, therefore, able to calculate whether it is optimal to supply attacks in any given period.

Alternatively, one can assume that the government announces the counterterrorism effort rates, $\left\{ {\varphi _t } \right\} _{t=0,\ldots ,\infty }$. Since $g_t =\varphi _t a_t $ for $\forall t$, this alternative game yields the same subgame-perfect Nash equilibrium as the one examined in the text. It is more convenient to treat the effort levels rather than the effort rates as the variables controlled by the government.

Our analysis focuses on the steady state because it is the usual way to carry out the comparative statics analysis for stable equilibria according to Samuelson’s correspondence principle. One can also analyze the path and speed of convergence to the steady- state equilibrium (see Footnote 11).

Dynamic optimization in discrete time methods can be found in macroeconomic textbooks such as Obstfeld and Rogoff [32, Chapter 2] as well as in environmental economics textbooks, e.g., Conrad [12, Chapter 1]. Euler equations are the basic first-order necessary conditions in the calculus of variations. It is worth noting that one of the most common methods to analyze convergence paths and speed to the steady state is through log-linearization method of the dynamic system formed by the Euler equations and dynamic constraints (e.g., Barro and Sala-i-Martin, [4, pp. 110-118]); however, the log- linearization applies only to the vicinity of the steady state.

See Barros et al. [5] and Faria and Arce [20‐22] for alternative frameworks that rationalize the equation that governs the time path of terrorist activity in a nation.

The payoff function for the terrorist organization is

$$\begin{aligned}&\bar{{Y}}+\sum _{t=0}^\infty \rho ^{t}\left\{ a_{1,t} \left( {b+m\left( {a_{1,t-1} -g_{1,t-1} -\theta g_{2,t-1} } \right) } \right) \right. \nonumber \\&\left. \quad +\,a_{2,t} \left( b+m\left( {a_{2,t-1} -g_{2,t-1}-\theta g_{1,t-1} } \right) \right) -\frac{a_{1,t}^2 +a_{2,t}^2 }{2}-b\left( {1+\theta } \right) \left( {g_{1,t} +g_{2,t} } \right) \right\} , \end{aligned}$$

where $\bar{{Y}}={\bar{{y}}}/{\left( {1-\rho } \right) }$ if $a_{j,t} >0$, $j=1,2$, $\forall t$. Since the terrorist organization takes $g_{j,t} $ as given for $j=1,2$ and $\forall t$, the term $b\left( {1+\theta } \right) \left( {g_{1,t} +g_{2,t} } \right) $ is constant and can be omitted from the relevant objective function. Note that the objective function is strict concave with respect to $a_{j,t} ,j=1,2$. Hence, the first-order conditions are necessary and sufficient for a global maximum.

Recall that as the objective function is strictly convex and the constraints are given by affine functions, the first-order conditions are necessary and sufficient for a global minimum.

The equilibrium characterized by insensitivity in our context can be understood as a corner solution.

As each nation’s objective function is strictly convex and the constraints are given by affine functions, the first-order conditions are necessary and sufficient for a global minimum in each constrained minimization problem.

If, in addition, each sensitive nation were altruistic and were to care about the social costs incurred by the other nation just as much as they care about their own social costs, the problem solved by each sensitive nation would be mathematically identical to the problem solved by the international coalition.

It is important to note that the functions described by Eqs. (5c) and (5d) are derived from the terrorist organization’s best-response functions. They do not represent the nations’ best-response functions. They should be understood as “feedback functions.” The nations’ best-response functions are determined by taking the feedback functions into account.

This inequality is reached by multiplying $a^{n}$ by $(1-\theta )/(1-\theta )$ and then canceling like terms.

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Titel: Intertemporal Versus Spatial Externalities in Counterterror Policy Games
verfasst von: João Ricardo Faria
Emilson C. D. Silva
Daniel G. Arce
Publikationsdatum: 15.04.2016
Verlag: Springer US
Erschienen in: Dynamic Games and Applications / Ausgabe 3/2017
Print ISSN: 2153-0785
Elektronische ISSN: 2153-0793
DOI: https://doi.org/10.1007/s13235-016-0188-0

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