Elbadawi and Sotto (2015) establish the link between natural resources and violence and role of institutions using game theoretic framework. We have adapted the framework to explain not only the role of institutions in violence but also the effect of third-party and external interference.

## 4.1. The Game

We begin by explaining our baseline game theoretical framework as presented by Elbadawi and Sotto (2015). The game under discussion deals with inter-group dynamics in which we consider two groups A and B. We factor in a power asymmetry in terms of violence capacity as well as existing resource-base, with group A being more powerful than group B. Let yA and yB represent inalienable endowment of group A and B respectively. Let N denote total population with NA and NB being sizes of group A and B respectively. For now, we assume that:

N = NA + NB (1)

Further, we introduce the existence of natural resource deposits with returns denoted by Z. If the resources are distributed equally every individual would receive:

z = Z/N (2)

Following Elbadawi and Soto (2015), we start with a two-stage sequential game led by group A. If group A chooses to accept equal distribution of resources, group B does not have any reason to retaliate. Therefore, we will have a conflict-less outcome with groups’ payoffs being:

yi + z where i= {A, B} (3)

However, if the group A opts for forceful appropriation, group B will have reason to retaliate. Therefore, in this case group B faces the option to either endure appropriation or retaliate. The final payoffs in this case depend on group B’s response. If the group B decides not to retaliate but instead to endure appropriation, the payoff for Group A will be

$${V\left({C}_{A}\right)}_{(C,P)}=\left(1-\delta \right)\left[{y}_{A}+\frac{Z}{{N}_{A}}\right]$$

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And value of payoff for Group B will be

$${V\left({C}_{B}\right)}_{(C,P)}=\left(1-\delta \right){y}_{B}$$

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Where, δ is the proportional loss to the economy and 0 < δ < 1, assuming symmetric distribution of costs among the group members. If, on the other hand, group B decides to retaliate the payoffs will be different. Here, since Group A is more powerful it will still obtain a larger share (α) of returns from natural resources but owing to the retaliation group B will also get a share (1-α) of natural resources. However, retaliation will also increase the cost of conflict from δ to Δ, i.e. \(\delta <\varDelta\). This cost involves not only the cost of enforcement but also the cost of resources destroyed because of conflict. In this scenario Group A’s payoff for conflict will be:

$${V\left({C}_{A}\right)}_{(C,C)}=\left(1-\varDelta \right)\left[{y}_{A}+\alpha \frac{Z}{{N}_{A}}\right]$$

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And Group B’s payoffs in case of conflict would be

$${V\left({C}_{B}\right)}_{(C,C)}=\left(1-\varDelta \right)\left[{y}_{B}+\left(1-\alpha \right)\frac{Z}{{N}_{B}}\right]$$

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The game is expressed in Fig. 1

Whether Group B will retaliate depends on its expectation regarding the probability of success. Denoting 1- π as the expected probability of success of Group B, with π being the perceived probability of success of group A, if the value of π is high enough Group B will choose not to retaliate. As for high values of π, group B will expect that group A will succeed in appropriating completely and by retaliating the perceived payoff would be lower than the payoff if group B decides not to retaliate owing to the higher cost of conflict in case of retaliation (Δ).

For Group A the expected payoff from not appropriating (choosing peace) remains: \(EV\left({P}_{A}\right)={y}_{A}+z\) (8)

If Group A chooses to appropriate, its expected payoffs for conflict will be:

$$EV\left({C}_{A}\right)=\pi \left[\left(1-\delta \right)\left[{y}_{A}+\frac{Z}{{N}_{A}}\right]\right]+(1-\pi )\left[\left(1-\varDelta \right)\left[{y}_{A}+\alpha \frac{Z}{{N}_{A}}\right]\right]$$

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For group A to choose to appropriate and risk instigating a conflict:\(EV\left({C}_{A}\right)>EV\left({P}_{A}\right)\)

i.e.

$$\pi \left[\left(1-\delta \right)\left[{y}_{A}+\frac{Z}{{N}_{A}}\right]\right]+\left(1-\pi \right)\left[\left(1-\varDelta \right)\left[{y}_{A}+\alpha \frac{Z}{{N}_{A}}\right]\right]>{y}_{A}+z$$

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If we solve the inequality for π we get:

$$\pi >\frac{\varDelta {y}_{A}+z\left[1-\left(\left(1-\varDelta \right)\alpha \frac{N}{{N}_{A}}\right)\right]}{{y}_{A}\left(\varDelta -\delta \right)+\frac{zN}{{N}_{A}}\left[\left(1-\delta \right)-\alpha \left(1-\varDelta \right)\right]}= \tilde{\pi }$$

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\(\tilde{\pi }\) is the reservation value or threshold such that if the perceived probability (π) exceed \(\tilde{\pi }\), Group A will initiate conflict by appropriating natural resource rents. For group B the expected value of peace or conflict depends on the Group A’s strategy. From Group B’s point of view if π is high then group A will choose to appropriate and if it is not high enough group A will choose not to appropriate. Therefore:

$$EV\left({P}_{B}\right)=\left(1-\pi \right)\left({y}_{B}+z\right)+\pi \left(1-\delta \right){y}_{B}$$

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The expected value of conflict for group B is strictly conditional on group A’s choice of conflict:

$$EV\left({C}_{B}\right)=\left(1-\pi \right)\left(1-\varDelta \right)\left({y}_{B}+\left(1-\alpha \right)\frac{zN}{{N}_{B}}\right)$$

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For group B the conflict strategy will dominate if\(EV\left({C}_{B}\right)>EV\left({P}_{B}\right)\)

Solving we get:

$$\pi <1-\frac{\left(1-\delta \right){y}_{B}}{\left(1-\varDelta -\delta \right){y}_{B}-z\left[1-\left(1-\varDelta \right)\left(1-\alpha \right)\frac{N}{{N}_{B}}\right]}=\ddot{\pi }$$

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Group B will not retaliate unless the probability of group A’s success in conflict is less than \(\ddot{\pi }\). Conflict will be the dominant equilibrium strategy if:

$$\tilde{\pi }<\pi \left(\tilde{\pi }, \ddot{\pi }|I\right)<\ddot{\pi }$$

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Which means that

$$\tilde{\pi }<\ddot{\pi }$$

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Elbadawi and Soto (2015) were of the view that π represents strength of institutions and its higher value depicts weakness of institutions or failure of institutions in preventing violence. However, it can be an indicator of military ability, economic capacity and access to resources as well. Therefore, we postulate that strength of institutions needs to be explicitly factored into the analysis instead of incorporating as part of a plethora of possible aspects and the next section does exactly that.

## 4.2. Incorporating Institutional Cost of Violence

In this section we have introduced an explicit institutional cost of aggression (λ). This cost will be accrued for engaging in violence may it be appropriation or retaliation. While this will not affect the payoff for the group that chooses peace, the payoff of opting for appropriation or retaliation will be discounted by λ, which would be added to the cost of violence by the aggressor.

If the group B decides not to retaliate but instead to endure appropriation, the payoff for Group A will be \({\stackrel{´}{V}\left({C}_{A}\right)}_{(C,P)}=\left(1-\delta -\lambda \right)\left[{y}_{A}+\frac{Z}{{N}_{A}}\right]\) (3.17)

And value of payoff for Group B will remain unchanged i.e. (3.5).

If the group B decides to retaliate the payoffs will be different. In this scenario Group A’s payoff will be:

$${\stackrel{´}{V}\left({C}_{A}\right)}_{(C,C)}=\left(1-\varDelta -\lambda \right)\left[{y}_{A}+\alpha \frac{Z}{{N}_{A}}\right]$$

3.18

And Group B’s payoffs will be \({\stackrel{´}{V}\left({C}_{B}\right)}_{(C,C)}=\left(1-\varDelta -\lambda \right)\left[{y}_{B}+\left(1-\alpha \right)\frac{Z}{{N}_{B}}\right]\) (3.19)

Figure 2 represents the modified game with institutional costs:

For Group A the expected payoffs from not appropriating remain the same (3.8). If Group A chooses to appropriate, its expected payoffs will be:

$$\stackrel{´}{EV}\left({C}_{A}\right)=\pi \left[\left(1-\delta -\lambda \right)\left[{y}_{A}+\frac{Z}{{N}_{A}}\right]\right]+(1-\pi )\left[\left(1-\varDelta -\lambda \right)\left[{y}_{A}+\alpha \frac{Z}{{N}_{A}}\right]\right]$$

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For group A to choose Conflict:\(\stackrel{´}{EV}\left({C}_{A}\right)>\stackrel{´}{EV}\left({P}_{A}\right)\)

If we solve for π we get: \(\pi >\frac{\left(\varDelta +\lambda \right){y}_{A}+z\left[1-\left(\left(1-\varDelta -\lambda \right)\alpha \frac{N}{{N}_{A}}\right)\right]}{{y}_{A}\left(\varDelta -\delta \right)+\frac{zN}{{N}_{A}}\left[\left(1-\delta -\lambda \right)-\alpha \left(1-\varDelta -\lambda \right)\right]}= \widehat{\pi }\) (20)

\(\widehat{\pi }\) is the reservation value or threshold such that if the perceived probability (π) exceeds \(\widehat{\pi }\), Group A will initiate conflict.

$$\frac{\partial \widehat{\pi }}{\partial \lambda }>0$$

Improvement in institutional constraints will increase the threshold minimum probability of success that would induce Group A to opt for appropriation.

For group B the expected value of peace or conflict depends on the Group A’s strategy. From Group B’s point of view if π is high then group A will choose to appropriate and if it is not high enough group A will choose not to appropriate. Therefore:

$$\stackrel{´}{EV}\left({P}_{B}\right)=\left(1-\pi \right)\left({y}_{B}+z\right)+\pi \left(1-\delta \right){y}_{B}$$

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The expected value of conflict is strictly conditional on group A’s choice of conflict:

$$\stackrel{´}{EV}\left({C}_{B}\right)=\left(1-\pi \right)\left(1-\varDelta -\lambda \right)\left({y}_{B}+\left(1-\alpha \right)\frac{zN}{{N}_{B}}\right)$$

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For group B the conflict strategy will dominate if\(\stackrel{´}{EV}\left({C}_{B}\right)>\stackrel{´}{EV}\left({P}_{B}\right)\)

Solving we get: \(\pi <1-\frac{\left(1-\delta \right){y}_{B}}{\left(1-\varDelta -\delta -\lambda \right){y}_{B}-z\left[1-\left(1-\varDelta -\lambda \right)\left(1-\alpha \right)\frac{N}{{N}_{B}}\right]}=\stackrel{\prime }{\pi }\) (23)

Group B will not retaliate unless the probability of group A’s success in conflict is less than \(\stackrel{\prime }{\pi }\).

$$\frac{\partial \stackrel{\prime }{\pi }}{\partial \lambda }<0$$

Improvement in institutional constraints will reduce the threshold probability of success of Group A that would induce Group B to opt for retaliation. Conflict will be the dominant equilibrium strategy if: \(\widehat{\pi }<\pi \left(\widehat{\pi }, \stackrel{\prime }{\pi }\right)<\stackrel{\prime }{\pi }\) (24)

Increase in institutional cost would make the conflict unlikely by narrowing the range of π within which conflict would become the dominant equilibrium strategy. As presence of strong institutions of accountability would increase the cost of rebelling for group B and at the same time would make a favorable outcome from exploitation less likely for group A.

## 4.3. Adding Outsider Influence

In this section we incorporate external influence in the model. As explained before external influence incorporates outsider’s greed mechanism, in which foreign vested interests (at times implicitly) arm and support rebels’ groups for continued instability. This (partially) compensates the rebels for costs potential incurred during conflict. For this purpose, we incorporate the term ε in the previous setup. ε is a positive parameter that adds to the resources available to rebel group in case of violent conflict, hereby, compensating for costs of conflict (Δ + λ). This will change the Group B’s payoffs for conflict to:

$${\stackrel{´}{V}\left({C}_{B}\right)}_{(C,C)}=\left(1-\varDelta -\lambda \right)\left[{y}_{B}\left(1+\epsilon \right)+\left(1-\alpha \right)\frac{Z}{{N}_{B}}\right]$$

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Where, \(\epsilon >0\).

The analysis of Group A remains that same. The analysis for Group B changes in this case, in which expected return in case for peace would be:

$$\stackrel{´}{EV}\left({P}_{B}\right)=\left(1-\pi \right)\left({y}_{B}+z\right)+\pi \left(1-\delta \right){y}_{B}$$

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The expected value of conflict is strictly conditional on group A’s choice of conflict:

$$\stackrel{´}{EV}\left({C}_{B}\right)=\left(1-\pi \right)\left(1-\varDelta -\lambda \right)\left({y}_{B}\left(1+\epsilon \right)+\left(1-\alpha \right)\frac{zN}{{N}_{B}}\right)$$

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For group B the conflict strategy will dominate if\(\stackrel{´}{EV}\left({C}_{B}\right)>\stackrel{´}{EV}\left({P}_{B}\right)\)

Solving we get: \(\pi <1-\frac{\left(1-\delta \right){y}_{B}}{\left(1-\varDelta -\lambda \right){y}_{B}\left(1+\epsilon \right)-z\left[1-\left(1-\varDelta -\lambda \right)\left(1-\alpha \right)\frac{N}{{N}_{B}}\right]}=\stackrel{\prime }{\pi }\) (28)

Group B will not retaliate unless the probability of group A’s success in conflict is less than \(\stackrel{\prime }{\pi }\).

\(\frac{\partial \stackrel{\prime }{\pi }}{\partial \lambda }<0\) and \(\frac{\partial \stackrel{\prime }{\pi }}{\partial \epsilon }>0\)

An increase in external influence will raise the lower threshold for sustained conflict scenario, thereby making conflict more likely. Further, we have also established that

$$\frac{{\partial }^{2}\stackrel{\prime }{\pi }}{\partial \epsilon \partial \lambda }>0$$

Hence proving that in the presence of external interference the violence inhibiting impact of institutional accountability will be reduced. If we revisit the condition for conflict equilibrium

$$\widehat{\pi }<\pi \left(\widehat{\pi }, \stackrel{\prime }{\pi }\right)<\stackrel{\prime }{\pi }$$

We can see the third-party interference increases the threshold value \(\stackrel{\prime }{\pi }\) expanding the range of π within which conflict would become the dominant equilibrium strategy.

The game explains not only that the existence of natural resources creates motivation for exploitation by the dominant coalition but also that institutional accountability can be instrumental in all parties opting for peaceful resolution of distributional concerns giving credence to both weak states and grievance hypotheses. Strong and indiscriminate institutional accountability would penalize the incumbent group (Group A) for reneging and exploiting group B. At the same time group B would also have to face accountability for rebelling and violence. This would discount the expected benefits accrued from a violent conflict and would reduce the motivation for violence.

Further, we demonstrated that external interference that may exploit the existing grievances by reducing cost for rebelling would undermine institutional effectiveness in preventing violence. This shows an interplay between outsider’s greed and rebels’ greed mechanisms in perpetuation of political violence. The two dynamics are mutually reinforcing and can effectively neutralize the impact of institutional accountability in preventing violence. This posits a serious issue for countries with existing cross-border conflicts, where the internal cleavages can be manipulated and exacerbated by the external antagonists. Therefore, the work presents a case for actively seeking regional peacekeeping along with strengthening of domestic institutions for accountability for violence and exploitation. Hence domestic institutional reforms should work in stride with a peacekeeping foreign policy. Further, the interests of multinational organizations have to be factored into any attempts at peacekeeping in the country. The most effective design would be having an institutional framework that would penalize not only acts of aggression but also exploitation by the dominant coalition, hereby eliminating the core cause for conflict.