Abstract
We consider a three-party game of conflict with potential alliance formation, introducing the concept of a stag hunt alliance and using the concept as a novel solution to the alliance formation puzzle in contests. In a stag hunt alliance, allied inputs interact as (multiplicative) complements in the contest success function, reflecting the idea that allied efforts are coordinated and targeted against non-allies. Allowing for asymmetry among conflicting parties, we find conditions for stable alliance formation and show that, in some circumstances, the formation of an alliance can improve the expected payoffs of both the allied and unallied parties relative to unallied conflict. At the same time, the expected payoffs of allied parties also can be greater than their payoffs under a three-party exogenous settlement division without conflict. Hence, the ability to form stag hunt alliances may not simply change conflict structures, but may sometimes generate conflict when settlements are possible. Our results depend on how allies divide the contest prize in the event of victory, and we consider both the case of exogenous division according to a pre-specified rule and endogenous division in a second-stage intra-alliance contest. We also show that the threat of the latter can be used to establish an intuitive exogenous rule by simple bargaining. Finally, we specify conditions that determine which parties choose to ally with one another.
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Notes
The terms “contest” and “conflict” are used interchangeably herein.
That feature follows the general notion of strategic substitutes and complements originating from Bulow et al. (1985), since we treat allies as if they engage in a non-cooperative game environment even when in an alliance. In equilibrium, the additive alliance CSF then results in an outcome in which an increase in one ally’s efforts provides a direct incentive for the other to reduce theirs; the multiplicative form on the other hand has supermodular properties, so an increase in one ally’s effort increases the incentive for the other ally to provide effort.
Boudreau et al. (2017) show that noise in the CSF can also reduce substitutability in order to solve for \(s_{i}^{*}\) or \(s_{j}^{*}\) uniquely.
We thank an anonymous reviewer for their help in streamlining this proof.
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Boudreau, J.W., Rentschler, L. & Sanders, S. Stag hunt contests and alliance formation. Public Choice 179, 267–285 (2019). https://doi.org/10.1007/s11127-018-0567-8
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DOI: https://doi.org/10.1007/s11127-018-0567-8