Sufficient conditions for unique stable sets in three agent pillage games

Colin Rowat, Manfred Kerber

Abstract: Highlights • Pillage games allow externalities between coalitions and allow power to depend on resources. • They are richer than games in characteristic or partition function form. • When n = 3 , three axioms ensure stable sets’ uniqueness, and restrict them to no more than 15 elements. • We present an algorithm for deciding existence and computing stable sets under these conditions. Abstract Pillage games (Jordan, 2006a) have two features that make them richer than cooperative games in either characteristic or partition function form: they allow power externalities between coalitions; they allow resources to contribute to coalitions’ power as well as to their utility. Extending von Neumann and Morgenstern’s analysis of three agent games in characteristic function form to anonymous pillage games, we characterise the core for any number of agents; for three agents, all anonymous pillage games with an empty core represent the same dominance relation. When a stable set exists, and the game also satisfies a continuity and a responsiveness axiom, it is unique and contains no more than 15 elements, a tight bound. By contrast, stable sets in three agent games in characteristic or partition function form may not be unique, and may contain continua. Finally, we provide an algorithm for computing the stable set, and can easily decide non-existence. Thus, in addition to offering attractive modelling possibilities, pillage games seem well behaved and analytically tractable, overcoming a difficulty that has long impeded use of cooperative game theory’s flexibility.