Conflict Externalization and the Quest for Peace: Theory and Case Evidence from Colombia

Hector Galindo-Silva

doi:10.1515/peps-2020-0010

Article

Conflict Externalization and the Quest for Peace: Theory and Case Evidence from Colombia

Hector Galindo-Silva

Published/Copyright: October 12, 2020

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From the journal Peace Economics, Peace Science and Public Policy Volume 27 Issue 1

Abstract

I study the relationship between the likelihood of a violent domestic conflict and the risk that such a conflict “externalizes” (i.e. spreads to another country by creating an international dispute). I consider a situation in which a domestic conflict between a government and a rebel group has the potential to externalize. I show that the risk of externalization increases the likelihood of a peaceful outcome, but only if the government is sufficiently powerful relative to the rebels, the risk of externalization is sufficiently high, and the foreign actor who can intervene in the domestic conflict is sufficiently uninterested in material costs and benefits. I show how this model helps to understand the recent and successful peace process between the Colombian government and the country’s most powerful rebel group, the Revolutionary Armed Forces of Colombia (FARC).

Keywords: conflict; externalization; peace talks

JEL Classification: H72; D72

Corresponding author: Hector Galindo-Silva, Departamento de Economía, Pontificia Universidad Javeriana, Bogota, Colombia, E-mail: galindoh@javeriana.edu.co.

Acknowledgments

I am grateful to Raphael Godefroy, David Karp, Alessandro Riboni, the editor and two anonymous referees for helpful comments and suggestions. All errors or omissions are mine.

Appendix

Proof of Proposition 1

I solve the game by backward induction. Having examined F’s decision about whether to intervene in the main text, I must now examine the other groups’ optimal decisions about whether to attack each other. I start by showing that, under Assumption 1, peace (i.e. [p, p]) always constitutes a Nash equilibrium of the game. Then, I show that under Assumptions 2 and 3, there is a threshold value for ϕ that determines whether war (i.e. [a, a]) is also a Nash equilibrium of the game. I conclude by showing that neither (a, p) nor (p, a) can constitute a Nash equilibrium.

First, define π(p,p)−π(a,p)=Z(G)−(1−ϕ) (1−W(G)) Z(G+L)+C, where π(p,p) and π(a,p) are from Table 1, and where I have used the fact that Φ=ϕ+(1−ϕ) W(G). Now note that if π(p,p)>π(a,p), then when the rebels seek peace, the government’s best response is also to seek peace. In the last expression, note that π(p,p)>π(a,p) if and only if

(1)C>(1−ϕ)(1−W(G))Z(G+L)−Z(G).

Now note that since Z (0) = 0, Z (x) = 1 for all x≥G‾, and Z′>0 and Z′′≤0 for all x∈(0,G‾), then Assumption 1 implies that C > Z (x + L) − Z (x) for x ≥ 0. In particular, Assumption 1 implies that C > Z (G + L) − Z (G) for G∈(L,G‾). Thus, since (1−ϕ) (1−W(G))≤1 for all G∈(L,G‾), we have that Eq. (1) holds for all G∈(L,G‾).

Now define ρ (p, p) − ρ (p, a) = −Z (G) + Z(G − L) + C, where ρ (p, p) and ρ (p, a) are from Table 1, and where I have used the fact that Φ=ϕ+(1−ϕ) W(G). Note that if ρ (p, p) > ρ (p, a), the rebels’ best response is to seek peace, given that the government also seeks peace. In the last expression, note that ρ (p, p) > ρ (p, a) if and only if

(2)C>Z(G)−Z(G−L).

Since C > Z(x + L) − Z(x) for all x ≥ 0 (which follows from Assumption 1, as previously shown), then choosing x = G − L for G∈(L,G‾), we have that Eq. (2) holds for all G∈(L,G‾).

We therefore have that peace (i.e. [p, p]) constitutes a Nash equilibrium of the game. This proves Proposition 1.(i).

Now I establish the conditions under which war (i.e. [a, a]) is also a Nash equilibrium of the game. To do this, first define D≡π(p,a)−π(a,a), and note that if D > 0, then the government’s best response to a rebel attack is to choose peace, and if D < 0, the government’s best response is to attack. From Table 1, note that D is equal to

(3)D=Z(G−L)−(1−ϕ)(1−W(G))Z(G)

where I have used the fact that Φ=ϕ+(1−ϕ) W(G). Differentiating Eq. (3) with respect to G, we get

(4)dDdG=Z′(G−L)−(1−ϕ)(1−W(G))Z′(G)+(1−ϕ)W′(G)Z(G)

which, rearranging the terms, is equivalent to

(5)dDdG=[Z′(G−L)−(1−ϕ)Z′(G)]+(1−ϕ)W′(G)Z(G)[1+W(G)Z′(G)W′(G)Z(G)].

In Eq. (5), note that Z′(G−L)−(1−ϕ) Z′(G)≥0 since Z″(x)≤0 for all x∈(L,G‾). Thus, given any ϕ∈[0,1], a sufficient condition for dDdG>0 is

(6)1+W(G)Z′(G)W′(G)Z(G)<0

given that W′<0 for all x∈(0,A) where A>G‾. I will now show that Assumption 2 implies that Eq. (6) holds for all G∈(L,G‾). To see this, note that

(7)1+W(G)Z(G)Z′(G)W′(G)≤1+W(G)Z(G)k

where k≡sup{Z′(G)W′(G)|G∈(L,G‾)}. In addition, note that since W′(x)<0 and Z′(x)>0 for all x∈(0,G‾), we have that k ≤ 0, and that

(8)1+W(G)Z(G)k≤1+W(G)k≤1+W(G‾)k.

Combining Eqs. (7) and (8), we have that

(9)1+W(G)Z(G) Z′(G)W′(G)≤1+W(G‾) k .

So, if 1 + W(G‾)k < 0 (as stated in Assumption 2), then Eq. (6) holds. And if Eq. (6) holds, we have that Eq. (5) is greater than zero given any ϕ∈[0,1] and for all G∈(L,G‾).

Next, note that dDdG>0 and L<G‾ implies that

(10)D∈(−(1−ϕ) (1−W(L))Z(L),Z(G‾−L)−(1−ϕ)(1−W(G‾))Z(G‾))

for any G∈(L,G‾). Since −(1−ϕ) (1−W(L))Z(L)<0 when ϕ < 1, then, for this case, the government’s best response to a rebel attack is always to counterattack if Z(G‾−L)−(1−ϕ) (1−W(G‾)) Z(G‾)<0, or, equivalently, if

(11)Z(G‾−L)−(1−ϕ) (1−W(G‾))<0

where I have used the fact that Z(G‾)=1. To identify the conditions under which Eq. (11) holds, define

(12)ϕ‾≡1−Z(G‾−L)(1−W(G‾))

and note that under Assumption 3, ϕ‾>0. From Eq. (12), note that when ϕ<ϕ‾, Eq. (11) holds. This means that when ϕ<ϕ‾, the government’s best response to a rebel attack is to also attack, regardless of the value of G. And since the rebel’s best response to a government attack is clearly to attack, then whenever ϕ≤ϕ‾ and ϕ<1, (a, a) there is a Nash equilibrium for all G∈(L,G‾). This proves Proposition 1.(ii).

From Eq. (12), note that ϕ‾<1, so there are values of ϕ such that 1>ϕ>ϕ‾. For these values, note that Eq. (11) does not hold. Combining this observation with the fact that dDdG>0 for all G∈(L,G‾) (and given any ϕ∈[0,1]), we obtain that whenever 1>ϕ>ϕ‾, there exists Gˆ(ϕ)∈(L,G‾), defined implicitly by

(13)Z(Gˆ−L)=(1−ϕ) (1−W(Gˆ)) Z(Gˆ)

such that (a, a) constitutes a Nash equilibrium if and only if G≤Gˆ(ϕ). To see that Gˆ′(ϕ)<0, differentiate implicitly Eq. (13) with respect to ϕ to get Gˆ′(dDdG|G=Gˆ)+(1−W(Gˆ)) Z(Gˆ)=0; then note that the fact that for any ϕ∈[0,1], dDdG>0 for G∈(L,G‾), and that (1−W(Gˆ)) Z(Gˆ)>0, imply that Gˆ′<0. This proves Proposition 1.(iii).

Finally, recall from the analysis above that the best response of both the rebels and government is to seek peace when their opponent also seeks peace. This implies that neither (a, p) nor (p, a) can constitute a Nash equilibrium (i.e. Proposition 1.[v]). This, combined with the fact that when ϕ = 1, D = Z (G − L) > 0 for all G∈(L,G‾) proves Proposition 1.(iv).

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Received: 2020-03-05

Accepted: 2020-09-29

Published Online: 2020-10-12

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https://doi.org/10.1515/peps-2020-0010

Keywords for this article

conflict; externalization; peace talks