Pillage games with multiple stable sets

We prove that pillage games (Jordan in J Econ Theory 131.1:26–44, 2006, “Pillage and property”, JET) can have multiple stable sets, constructing pillage games with up to \(2^{\tfrac{n-1}{3}}\) stable sets, when the number of agents, \(n\), exceeds four. We do so by violating the anonymity axiom common to the existing literature to establish a power dichotomy: for all but a small exceptional set of endowments, powerful agents can overcome all the others; within the exceptional set, the lesser agents can defend their resources. Once the allocations giving powerful agents all resources are included in a candidate stable set, deriving the rest proceeds by considering dominance relations over the finite exceptional sets—reminiscent of stable sets’ derivation in classical cooperative game theory. We also construct a multi-good pillage game with only three agents that also has two stable sets.