Two-Department Open-Loop Nash Equilibrium Scenario
This subsection presents a dynamic game-theoretical model of aid allocation. Based on evidence from the literature review, there are two essential priorities in the allocation of aid. Homeland security and geopolitical interests are first, and commercial interests are second. Furthermore, it is assumed that a fixed share of the gross domestic product (GDP), \(\psi \in \left(0, s\right)\), of a donor country is annually spent on development aid. If the GDP of the donor country is denoted by \({Y}_{d}(t)\), then the donor spends in absolute terms \(A\left(t\right)=\psi {Y}_{d}(t)\) on foreign aid. It is assumed that foreign aid reimbursement is managed by a specialized aid agency. This agency is dominated by two groups organized in two separate departments. The first department focuses on the issues of homeland security and geopolitical issues (Player 1). The second department represents the commercial interests of the donor economy (Player 2). The share of the aid budget \({u}_{1}(t)\in \left(l,h\right)\) is spent on security and geostrategic targets whereby \(0<l<h<1\). \({u}_{2}(t)\in \left[0,\theta \right]\) is the proportion of the remainder spent on projects which are entirely dedicated to long-term socioeconomic development. The remainder, i.e., \(\left[{1-u}_{2}(t)\right]\), is spent on projects that focus on the commercial interests of donors seeking markets or resources. Based on these assumptions, the basic technical relationships of the foreign aid system can be formulated as follows: \(Gross \;Foreign\; Aid \;Budget\equiv A\left(t\right),\)
\(Dono{r}^{{^{\prime}}}s\; Security\; and\; Geopolitics\; Budget\equiv A\left(t\right){u}_{1}\left(t\right),\)
\(Dono{r}^{{^{\prime}}}s \;Commercial \;Interests \;Budget = A\left(t\right)\left(1-{u}_{1}\left(t\right)\right)\left(1-{u}_{2}\left(t\right)\right);\) and
$$Net\; Development\; Aid\equiv \dot{A}=A\left(t\right)\left(1-{u}_{1}\left(t\right)\right){u}_{2}\left(t\right).$$
(1)
Both departments try to maximize their budgets. Following Hoel (1978), a linear relationship is assumed between the size of the budget of the respective team and its utility. Both Player 1 and Player 2 discount their utilities.
The utility function of the Security and Geopolitics Department, i.e., Player 1, is determined by the integral of the discounted revenue flows as follows:\({J}_{1}={\int }_{0}^{T}{e}^{-{\rho }_{1}t}\left[A(t){u}_{1}(t)\right]dt.\)
The utility function of the Commerce Department, i.e., Player 2, squares with the following equation: \({J}_{2}={\int }_{0}^{T}{e}^{-{\rho }_{2}t}\left[A\left(t\right)(1-{u}_{1}(t))(1-{u}_{2}\left(t\right))\right]dt.\)
To determine the optimal pathways of the control variables of Player 1, \({u}_{1}\left(t\right)\), and Player 2, \({u}_{2}(t)\), the Hamiltonian functions must be set. The current and future utility of Player 1 can be expressed via the Hamiltonian function, \({H}_{1}\), formulated in Eq. 7. The optimal path for \({u}_{1}\left(t\right)\) (given \({u}_{2}\left(t\right))\) will maximize \({H}_{1}\) at every instant of time (Dorfman 1969) is : \({H}_{1}={e}^{-{\rho }_{1}t}\left[A(t){u}_{1}\left(t\right)\right]+{\lambda }_{1}\left(t\right)\left[A\left(t\right)\left(1-{u}_{1}\left(t\right)\right){u}_{2}\left(t\right)\right],\)
whereby \({\lambda }_{1}\) is a co-state variable indicating the shadow value of the marginal increase of development aid for Player 1. The possible reason for the positive valuation of development by the security team is that economic development reinforces security aid by consolidating the social basis for the emerging geopolitical influence of donor countries.
According to Pontryagin’s maximum principle, the co-state variable \({\lambda }_{1}\) must satisfy the following condition:
$${\dot{\lambda }}_{1}=-\frac{\partial {H}_{1}}{\partial A}=-{e}^{-{\rho }_{1}t}{u}_{1}\left(t\right)-{\lambda }_{1}\left(t\right)\psi \left(1-{u}_{1}\left(t\right)\right){u}_{2}\left(t\right).$$
(2)
In this problem, \({\dot{\lambda }}_{1}(t)\) represents the change in the marginal contribution of purely socioeconomic development aid to the utility of Player 1 in terms of the extension of geopolitical influence at time \(t\). Since Player 1 does not look beyond the planning horizon of \(T\), the marginal value of an investment at the end of the program is zero. Hence, the transversality condition is commensurate with \({\lambda }_{1}\left(T\right)=0\).
The choice of the value of \({u}_{1}\left(t\right)\) from the possible range of \(\left[l,h\right]\) depends upon the value of the product of the co-state variable, \({\lambda }_{1}(t)\), and the control variable of the commerce team, \({u}_{2}(t)\). The optimality conditions of Pontryagin type for Nash equilibria are given by:\({u}_{1}=\left\{\begin{array}{c}l\\ h\end{array}\right\} if {\lambda }_{1}{u}_{2}{e}^{{\rho }_{1}t}\left\{\begin{array}{c}>1\\ <1\end{array}\right\}.\)
The Hamiltonian function of Player 2, i.e., the Commerce Department, has the following form:
$${H}_{2}={e}^{-{\rho }_{1}t}\left[A(t)(1-{u}_{1}\left(t\right))(1-{u}_{2}(t))\right]+{\lambda }_{2}\left[A\left(t\right)(1-{u}_{1}\left(t\right)){u}_{2}\left(t\right)\right].$$
(3)
The co-state variable, \({\lambda }_{2}(t)\), is the marginal contribution of development commitment to the Commerce Department’s utility. The path of \({\lambda }_{2}\) is determined by the differential:
$$\begin{aligned}{\dot{\lambda }}_{2}&=-\frac{\partial {H}_{2}}{\partial A}=-{e}^{-{\rho }_{2}t}\left[(1-{u}_{1}\left(t\right))(1-{u}_{2}(t))\right]-{\lambda }_{2}\left[\left(1-{u}_{1}\left(t\right)\right){u}_{2}\left(t\right)\right]\\&=-\left(1-{u}_{1}\left(t\right)\right)\left({e}^{-{\rho }_{2}t}\left(1-{u}_{2}\left(t\right)\right)+{\lambda }_{2}{u}_{2}\left(t\right)\right)\\&=-\left(1-{u}_{1}\left(t\right)\right)\left({e}^{-{\rho }_{2}t}-{u}_{2}\left(t\right)\left({\lambda }_{2}-1\right)\right).\end{aligned}$$
(4)
\({H}_{2}\) is maximized by the values of \({u}_{2}(t)\) that satisfy the following conditions:\({u}_{2}=\left\{\begin{array}{c}0\\ 1\end{array}\right\} if {\lambda }_{2}{e}^{{\rho }_{2}t}\left\{\begin{array}{c}<1\\ >1\end{array}\right\} {u}_{2}=\left\{\begin{array}{c}0\\ 1\end{array}\right\} if {\lambda }_{2}{e}^{{\rho }_{2}t}\left\{\begin{array}{c}<1\\ <1\end{array}\right\}.\)
Without discounting, the optimality condition would be\({u}_{2}=\left\{\begin{array}{c}0\\ 1\end{array}\right\} if {\lambda }_{2}\left\{\begin{array}{c}<1\\ >1\end{array}\right\}.\)
Hence, four possible scenarios are in line with the optimality conditions:
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I.
\({u}_{1}=l\); \({u}_{2}=0\); if \({\lambda }_{1}{u}_{2}{e}^{{\rho }_{1}t}>1\) and \({\lambda }_{2}{e}^{{\rho }_{2}t}<1\) (unfeasible scenario)
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II.
\({u}_{1}=h\); \({u}_{2}=0\); if \({\lambda }_{1}{u}_{2}{e}^{{\rho }_{1}t}<1\) and \({\lambda }_{2}{e}^{{\rho }_{2}t}<1\) (nondevelopment security mode)
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III.
\({u}_{1}=l\); \({u}_{2}=1\); if \({\lambda }_{1}{u}_{2}{e}^{{\rho }_{1}t}>1\) and \({\lambda }_{2}{e}^{{\rho }_{2}t}>1\) (development mode)
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IV.
\({u}_{1}=h\); \({u}_{2}=1\); if \({\lambda }_{1}{u}_{2}{e}^{{\rho }_{1}t}<1\) and \({\lambda }_{2}{e}^{{\rho }_{2}t}>1\) (security and development mode)
It is obvious from inspection that scenario I is not feasible because \({u}_{2}=0\) and \({\lambda }_{1}{u}_{2}{e}^{{\rho }_{1}t}>1\) are mutually incompatible. This leaves only three scenarios to be examined.
An analysis of this kind of problem starts with scrutiny of the end phase rather than the initial phase (Lancaster 1973). There is a boundary condition \({\lambda }_{2}{e}^{{\rho }_{2}t}=0\), and \({\lambda }_{2}\) must be a continuous function of time. Thus, \({\lambda }_{2}{e}^{{\rho }_{2}t}<1\) must hold at the end and for some period before the end. Let \(\overline{t }\) denote the beginning of the period for which \({\lambda }_{2}{e}^{{\rho }_{2}t}<1\) (if \({\lambda }_{2}{e}^{{\rho }_{2}t}<1\) everywhere, then \(\overline{t }=0).\) In this final phase, combination II holds. Now, the following equations are produced by inserting the relevant values of the control variables in the differential Eqs. (1), (2), and (4):
\(\dot{A}=0, A\left(t\right)=A\left(\overline{t }\right) \forall t\in \left[\overline{t },T\right],\)
\({\dot{\lambda }}_{1}=-{e}^{-{\rho }_{1}t}h, { \lambda }_{1}\left(t\right)={\lambda }_{1}\left(\overline{t }\right)-{e}^{-{\rho }_{1}t}h\left(t-\overline{t }\right) \forall t\in \left[\overline{t },T\right],\) and
\({\dot{\lambda }}_{2}=-{e}^{-{\rho }_{2}t}\left(1-h\right), {\lambda }_{2}\left(t\right)={\lambda }_{2}\left(\overline{t }\right)-{e}^{-{\rho }_{2}t}\left(1-h\right)\left(t-\overline{t }\right) \forall t\in \left[\overline{t },T\right].\)
The last two equations show that both co-state variables decrease linearly in the final phase. This and the transversality condition, \({\lambda }_{2}\left(T\right)=0\), enable the calculation of \(\overline{t }\). This is the time point at which \({\lambda }_{2}\left(\overline{t }\right)=1\). To this end, t in Eq. (3) is replaced by T, and to make use of the transversality condition, \({\lambda }_{2}\left(T\right)\) is replaced by zero.
\(\begin{aligned}0 = 1-{e}^{-{\rho }_{2}t}\left(1-h\right)\left(T-\overline{t }\right) \to {e}^{-{\rho }_{2}t}\left(1-h\right)\left(T-\overline{t }\right) = 1 \to \left(T-\overline{t }\right)=\frac{1}{{e}^{-{\rho }_{2}t}\left(1-h\right)} \to \overline{t }=T-\frac{1}{{e}^{-{\rho }_{2}t}\left(1-h\right)}.\end{aligned}\)
Given \(T>\frac{1}{{e}^{-{\rho }_{1}t}(1-h)}\), the system enters its final phase at time \(\overline{t }\) given by Eq. 4. Before \(\overline{t }\), \({\lambda }_{2}\left(t\right)>1\). Thus, this phase will be defined by combination III or IV. Since in both these combinations \({u}_{2}=1\), in this phase \({\lambda }_{1}{u}_{2}={\lambda }_{1}\), and thus the value of \({u}_{1}\) depends on whether \({\lambda }_{1}\) is greater or less than unity. Going back to the final phase, \({\lambda }_{1}\left(T\right)=0\) and \({\lambda }_{1}\) declines at the time rate \({e}^{-{\rho }_{1}t}h\) during this phase. At \(\overline{t }\), therefore, the value of \({\lambda }_{1}\) is given by \({\lambda }_{1}\left(\overline{t }\right)={e}^{-{\rho }_{1}t}h\left(T-\overline{t }\right)=\frac{h}{1-h}\). Now, \(h\) is the maximum ratio of security-related foreign aid to the total aid budget, and is assumed to be greater than 0.5. Hence, \(\frac{h}{1-h}>1\) and \({\lambda }_{1}\left(\overline{t }\right)>1\). This implies that the penultimate phase will be characterized by combination III (\({u}_{1}=l, {u}_{2}=1\)). Insertion of the control variable values into the differential equations for \({\lambda }_{1}(t)\) and \({\lambda }_{2}(t)\) shows that both co-state variables are greater than zero. Thus, there is no phase before that represented by combination III and the solution consists of two phases, commencing with III, then switching to II at time \(\overline{t }\). In other words, the security department spends minimally and capitalists accumulate up to the time \(\overline{t }\). Then, both groups switch to maximum levels of expenditures to assure homeland security’s and donor’s economic interests overseas.
Interpretation of the Open Loop Nash Solution
The model implies that during the first phase of development, cooperation within the terrorism-or war-ravaged regions, the donors devote a minimum share of the aid budget to assure a basic level of security. The rest of the aid budget is dedicated to purely developmental projects. This phase ends after assuring the necessary level of security, which prevents social unrest and enables economic recovery. The overwhelming share of aid consists of humanitarian aid and aid for the development of basic infrastructure. The beginning of the second phase corresponds with positive growth rates and political stabilization. At this stage, the aid budget is divided between geopolitical and commercial expenditures. Geopolitical expenditures assure the steady or growing political influence within the hypothetical developing country. Commercial expenditures assure resource and market seeking. Hence, during the second phase, foreign aid, in its essence, does not fulfill the development aspiration of ODA, but rather investment in the improvement of foreign geopolitical and economic interests of donor countries.
During the first phase, it is not possible to enhance any lasting geopolitical or commercial interest. In terms of political stability, it is a kind of fragility phase. Hence, both departments direct their available resources toward recovery of basic statehood and socioeconomic stabilization. Hence, during the first phase, there is plenty of aid for economic recovery and promotion of basic institutions to assure political stability. These institutions are the prerequisite for proliferation of the donor’s geopolitical dominance in the aftermath of state fragility. During the second phase, foreign aid is dedicated solely to the enhancement of commercial interests and geopolitical project costs for the enhancement of geopolitical influence within developing areas.
Hence, if it is hypothesized that terrorism incidents are a measure of political fragility, then following the proposed differential game model, a positive relationship between terrorism incidents and foreign aid action during political fragility and initial post-conflict recovery phases is conceivable.