Operational externalities and counter-terrorism

Abstract

In a structure involving two independent terror outfits operating in a country, we study possible implications for counter-terrorism (CT) strategy in the presence and absence of operational externalities. Inter alia, the analysis suggests a possible explanation for the widespread application of defensive CT measures and the sparing use of offensive CT. But confidence-building measures come to be ineffective against resource-constrained outfits, irrespective of the nature and magnitude of externalities. Offensive measures against resource-abundant outfits, appear to be successful in reducing the total number of terror strikes only when strong negative externalities prevail.

Appendices

Appendix 1

$T_{i}$’s optimization problem is to maximize its utility (1), with respect to its resource constraint (5), and non-negativity constraints $X_{i} \ge 0$ and $A_{i} \ge 0$. This is equivalent to the unconstrained maximization of the Lagrangean function:

$$ L = X_{i} + \alpha_{i} v_{i} (A_{i} ) + \lambda \{ R_{i} - X_{i} - \beta_{i} C_{i} (A_{i} ,A_{j} )\} + \gamma X_{i} + \mu A_{i} , $$

(15)

where $\lambda$, $\gamma$ and $\mu$ are non-negative Lagrangean multipliers.

Solving the FOCs, the slope for the reaction function can be obtained as:

$$ \frac{{{\text{d}}A_{i} }}{{{\text{d}}A_{j} }} = - \,\beta_{i} \frac{{\frac{{\partial^{2} C_{i} }}{{\partial A_{i} \partial A_{j} }}}}{{\beta_{i} \frac{{\partial^{2} C_{i} }}{{\partial A_{i}^{2} }} - \frac{1}{1 + \gamma }\alpha_{i} v_{i}^{\prime \prime } }}, $$

(16)

where $\gamma = 0$ when the resource constraint (5) is not binding, and $\gamma > 0$ when (5) is binding. Invoking Eq. (8), the result follows. Q.E.D.

Appendix 2

In this scenario, $T_{i}$’s budget constraint is given by Eq. (2). However, its utility is:

$$ U_{i} = X_{i} + \alpha_{i} v_{i} (A_{i,} A_{j} ),\quad \frac{{\partial v_{i} (A_{i} ,A_{j} )}}{{\partial A_{i} }} > 0,\quad \frac{{\partial^{2} v_{i} (A_{i} , A_{j} )}}{{\partial A_{i}^{2} }} \le 0,\quad \forall A_{i} ,A_{j} \ge 0, $$

(17)

$T_{i}$’s optimization problem is to maximize its utility (17) subject to its budget constraint (2). Substituting $X_{i}$ in (17) using (2), we can rewrite the utility maximization problem as:

$$ {\text{Max}}_{{A_{i} }} U_{i} = R_{i} - \beta_{i} C_{i} (A_{i} ) + \alpha_{i} v_{i} (A_{i} ,A_{j} ). $$

(18)

If an interior optimum exists, the first-order condition (FOC) is:

$$ - \,\beta_{i} C_{i}^{\prime } (A_{i} ) + \alpha_{i} \frac{{\partial v_{i} (A_{i} ,A_{j} )}}{{\partial A_{i} }} = 0. $$

(19)

From (17), the best-response (or reaction) function of each outfit $i( \ne j = 1,2)$, $A_{i} = A_{i} (A_{j} )$, can be obtained. Also, along the reaction function, $\frac{{{\text{d}}A_{i} }}{{{\text{d}}A_{j} }} = \alpha_{i} \frac{{\frac{{\partial^{2} v_{i} }}{{\partial A_{i} \partial A_{j} }}}}{{\beta_{i} C_{i}^{\prime \prime } - \alpha_{i} \frac{{\partial^{2} v_{i} }}{{\partial A_{i}^{2} }}}}$. The SOC ensures that the denominator is positive. Therefore, the reaction functions are positively (negatively) sloped if $\frac{{\partial^{2} v_{i} }}{{\partial A_{i} \partial A_{j} }} > 0$ (< $0$), i.e., if an outfits’s terror activities impose a positive (negative) externality on the utility of the other outfit’s terror activities. In this case, the numbers of attacks conducted by the outfits are strategic complements (substitutes). Q.E.D.

Appendix 3

Differentiating the FOCs for $T_{i}$ [given by Eq. (7)] and $T_{j}$ [obtained by interchanging subscripts $i$ and $j$ in Eq. (7)] with respect to $\alpha_{i}$ ($i \ne j = 1,2$), and solving the resulting pair of Equations, we obtain

$$ \frac{{{\text{d}}A_{i}^{*} }}{{{\text{d}}\alpha_{i} }} = - \frac{{v_{i}^{\prime } \left( {\alpha_{j} v_{j}^{\prime \prime } - \beta_{j} \frac{{\partial^{2} C_{j} }}{{\partial A_{j}^{2} }}} \right)}}{H} > 0, $$

(20)

invoking the SOC, Eqs. (1) and (9). Under positive externalities,

$$ \frac{{{\text{d}}A_{j}^{*} }}{{{\text{d}}\alpha_{i} }} = - \,\beta_{j} v_{i}^{\prime } \frac{{\frac{{\partial^{2} C_{j} }}{{\partial A_{j} \partial A_{i} }}}}{H} > 0. $$

(21)

Obviously, the total number of attacks also increases if $\alpha_{i}$ increases, since

$$ \frac{{{\text{d}}(A_{i}^{*} + A_{j}^{*} )}}{{{\text{d}}\alpha_{i} }} = - \frac{{v_{i}^{\prime } \left( {\alpha_{j} v_{j}^{\prime \prime } - \beta_{j} \frac{{\partial^{2} C_{j} }}{{\partial A_{j}^{2} }}} \right) + \frac{{\partial^{2} C_{j} }}{{\partial A_{j} \partial A_{i} }}}}{H} > 0. $$

(22)

In the context of $\beta_{i}$ ($i \ne j = 1,2$), Eq. (7) can similarly be utilized to obtain

$$ \frac{{{\text{d}}A_{i}^{*} }}{{{\text{d}}\beta_{i} }} = - \frac{{\left( {\alpha_{j} v_{j}^{\prime \prime } - \beta_{j} \frac{{\partial^{2} C_{j} }}{{\partial A_{j}^{2} }}} \right)\frac{{\partial C_{i} }}{{\partial A_{i} }}}}{H} < 0. $$

(23)

And under positive externalities,

$$ \frac{{{\text{d}}A_{j}^{*} }}{{{\text{d}}\beta_{i} }} = \beta_{j} \frac{{\frac{{\partial^{2} C_{j} }}{{\partial A_{j} \partial A_{i} }}\frac{{\partial C_{i} }}{{\partial A_{i} }}}}{H} < 0. $$

(24)

Obviously, the total number of attacks also decreases if $\beta_{i}$ increases, since

$$ \frac{{{\text{d}}(A_{i}^{*} + A_{j}^{*} )}}{{{\text{d}}\beta_{i} }} = - \frac{{\left( {\alpha_{j} v_{j}^{\prime \prime } - \beta_{j} \frac{{\partial^{2} C_{j} }}{{\partial A_{j}^{2} }}} \right)\frac{{\partial C_{i} }}{{\partial A_{i} }} + \frac{{\partial^{2} C_{j} }}{{\partial A_{j} \partial A_{i} }}\frac{{\partial C_{i} }}{{\partial A_{i} }}}}{H} > 0. $$

(25)

Lastly, for ($i \ne j = 1,2$), $\frac{{{\text{d}}A_{i}^{*} }}{{{\text{d}}R_{i} }} = \frac{{{\text{d}}A_{j}^{*} }}{{{\text{d}}R_{i} }} = \frac{{{\text{d}}(A_{i}^{*} + A_{j}^{*} )}}{{{\text{d}}R_{i} }} = 0$. Q.E.D.