Patterns of Rebellion: A Model with Three Heterogeneous Challengers

Keisuke Nakao

doi:10.1515/peps-2017-0021

Article

Patterns of Rebellion: A Model with Three Heterogeneous Challengers

Keisuke Nakao

Published/Copyright: November 23, 2017

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From the journal Peace Economics, Peace Science and Public Policy Volume 24 Issue 1

Abstract

This article proposes a dynamic model of rebellion, where three players individually decide to challenge their common adversary. It is formally demonstrated that the pattern of rebellion is determined endogenously, depending on the challengers’ resolve and strength. In other words, a challenger with more resolve and/or strength tends to fight earlier than others do.

Keywords: bandwagoning; strategic coordination; rebellion

JEL Classification: D74; F51

Acknowledgement

I thank Petros Sekeris and an anonymous referee for detailed comments. All errors are my own.

Appendix

Proof of Lemma 1

(i, vi) With Inequality (1), γ acquiesces once, given hold_{α∣T_β}. Otherwise, γ fights immediately, given hold_{α∣T_β}.^[13] (ii, iii) By Assumption 2-(i), α and β’s “win” guarantees that the government is weak. Then, γ’s sequential rationality after α and β’s “win” (“loss”) is given by Assumption 3-(i) [Assumption 3-(ii)]. (iv) As α and β hold out longer, Pr(g^W∣hold_α,β∣T) converges to one by Assumption 1 and Assumption 2-(iii), and γ will fight in some period T + 1 by Assumption 3-(i). (v) T_γ is independent of T_β because γ’s payoff is independent of T_β by Assumption 2-(iv). ▪

Proof of Lemma 2

(i, ii) Player β’s decision after α’s “win” (“loss”) is given by Assumption 3-(i) [Assumption 3-(ii)]. (iii) To determine T_β, β compares its expected payoffs from fighting in all future periods ( T + τ for any τ ≥ 2). (Unlike γ, β’s incentive to fight does not monotonically increase because of γ’s lagged fighting at T_γ + 1.) (iv) Inequality (2) checks β’s incentive at T_γ. If it holds, β fights at T_γ instead of at T_γ + 1, or T_β < T_γ. ▪

Proof of Proposition 1

We check the incentive compatibility for each player. (i) For γ, given T_β, γ chooses T_γ such that T_β < T_γ if Inequality (1) holds. For β, given T_γ, β chooses T_β such that T_β < T_γ if Inequality (2) holds. (ii) No prewar coalition is feasible, because if α fights alone in the first period (i.e. Vα|1Sn(Tβ,Tγ)≥0), β and γ would like to acquiesce at least once [Assumption 4-(i)]. If α “wins,” β and γ immediately fight [Assumption 2-(i) and Assumption 3-(i)]. If α “loses,” they never fight [Assumption 3-(ii)]. ▪

Proof of Proposition 2

As in the proof of Proposition 1, we check each player’s incentive. (i) During α’s “holdout,” β and γ fight simultaneously if T_β = T_γ, which requires that neither Inequality (1) or (2) holds. (ii) If Vα|1Ca(Tβ,γ)≥0, α initiates the catalytic rebellion alone. As α fights alone in the first period, β or γ acquiesce at least once [Assumption 4-(i)]; and thus no prewar coalition is possible. ▪

Proof of Proposition 3

Players α and β, but not γ, are willing to jointly fight because (i) a rebellion is impossible without a coalition, and (ii) for α and β, the partially coordinated rebellion is no worse than permanent “acquiesce” (i.e. Vi|1Pa(Tγ)≥0 for i ∈ {α, β}). On the other hand, γ is willing to acquiesce at least once [Assumption 4-(ii)]. If α and β “win,” γ fights immediately [Assumption 2-(i) and Assumption 3-(i)]. If they “lose,” γ never fights [Assumption 3-(ii)]. ▪

Proof of Proposition 4

The three players are willing to challenge concurrently because (i, ii) no other form of rebellion is incentive compatible, but (iii) for all three of them, the fully coalitional rebellion is no worse than permanent “acquiesce” (i.e. ViFu≥0 for i ∈ {α, β, γ}). ▪

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Published Online: 2017-11-23

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https://doi.org/10.1515/peps-2017-0021

Keywords for this article

bandwagoning; strategic coordination; rebellion