Third-party intervention in secessions

Abstract

This paper presents and solves a model of ethnically motivated warfare which treats annexations as the explicit outcomes of secessions. An ethnic group within one country may want to engage in a secession in order to join its ethnic peers in the annexing country. The annexing country decides whether or not to support the separatist ethnic group in the conflict. I use this model to discuss how equilibrium behavior depends on economies of scale in the public good provision, ethnic heterogeneity, and sanctions. Among others, I find that sanctions can have the seemingly paradoxical effect of increasing conflict intensity

Appendix

Lemma 3

Player L’s best response is $b_{L}(b_{M},b_{R})=\sqrt{(\lambda _{M}+b_{R})b_{M}\Pi _{R}}-(\lambda _{M}+b_{R})b_{M}$ if $\Pi _{R}>(\lambda _{M}+b_{R})b_{M}$ holds, and $b_{L}(b_{M},b_{R})=0$ otherwise. Player M’s best response is $b_{M}(b_{L},b_{R})=\sqrt{\frac{b_{L}}{(\lambda _{M}+b_{R})}S_{M}\Pi _{M}}-\frac{b_{L}}{(\lambda _{M}+b_{R})}$ if $S_{M}\Pi _{M}>\frac{b_{L}}{(\lambda _{M}+b_{R})}$ holds, and $b_{M}(b_{L},b_{R})=0$ otherwise. Player R’s best response is $b_{R}(b_{L},b_{M})=\sqrt{\frac{S_{R}}{1+\Psi }\frac{b_{L}}{b_{M}}\Pi _{R}}-\lambda _{M}-\frac{b_{L}}{b_{M}}$ if $\frac{S_{R}}{1+\Psi }\frac{b_{L}}{b_{M}}\Pi _{R}>(\lambda _{M}+\frac{b_{L}}{b_{M}})^{2}$ holds, and $b_{R}(b_{L},b_{M})=0$ otherwise.

Figure 4 plots the best response functions for all three players; and shows that the best response functions’ shifts to changes can be ambiguous with respect to each other.

Fig. 4

Players’ best response functions for $\lambda _{M}=1/2, S_{M}=1, t=1/2, S_{R}=3, \alpha =1/2, \Psi =1/10$. Left: Player M’s and L’s best response functions holding player R’s choice variable fixed with $b_{R}=0.1$ and $b_{R}=0.2$. Right: Player M’s and R’s best response functions holding player L’s choice variable fixed with $b_{L}=0.1$ and $b_{L}=0.2$

Player L raises $b_{L}(b_{M},b_{R})$ with diminishing marginal rate if his potential loss increases. Furthermore, player L raises $b_{L}(b_{M},b_{R})$ with increasing $b_{M},b_{R}$, and $\lambda _{M}$ if his gain from preventing a secession is sufficiently high, i.e., if $4 b_{M}<\Pi _{R}/(b_{R}+\lambda _{M})$ is satisfied. Otherwise, it is too costly for player L to fight a secession and he reduces his conflict bid.

Player M raises $b_{M}(b_{L},b_{R})$ with diminishing marginal rate if his potential gain increases. He also raises $b_{M}(b_{L},b_{R})$ with increasing $b_{L}$ if his gain from a secession is sufficiently high, i.e., if $S_{M}\Pi _{M}>4b_{L}/(\lambda _{M}+b_{R})$ is satisfied. Otherwise, it is too costly for player M to fight a secession and he reduces his conflict bid. Furthermore, player M raises $b_{M}(b_{L},b_{R})$ with increasing $b_{R}$ and $\lambda _{M}$ if his gain from winning the conflict is sufficiently low, i.e., if $S_{M}\Pi _{M}<4b_{L}/(\lambda _{M}+b_{R})$ is satisfied. Player M’s conflict bid is large if his gain from conflict is large. In this case, player R’s support allows player M to reduce his bid without imposing a loss on himself.

Player R raises $b_{R}(b_{L},b_{M})$ with diminishing marginal rate if his potential gain increases. Sanctions always reduce player R’s willingness to provide support. He also raises $b_{R}(b_{L},b_{M})$ with increasing $b_{L}$ if his gain from a secession is sufficiently high, i.e., if $S_{R}\Pi _{R}/(1+\Psi )>4b_{L}/b_{M}$ is satisfied. Otherwise, it is too costly for player R to provide support and he reduces his conflict bid. Furthermore, player R raises $b_{R}(b_{L},b_{M})$ with increasing $b_{M}$ if his gain from a successful secession is sufficiently low, i.e., if $S_{R}\Pi _{R}/(1+\Psi )<4b_{L}/b_{M}$ is satisfied.