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.
Highlights
We present Algorithm 2 to outline the structure of the coming arguments about deciding and computing stable sets in three agent pillage games satisfying the three additional axioms
We note that the algorithm implies that any two pillage games satisfying its conditions and sharing a Bi share a stable set, if they have one: the common Bi ensures that they both pass or fail the conditions in lines 1, 4 and 7 in the same way; the elements of the ensuing stable sets, if they exist, depend entirely on the geometry of the shared Bi
By contrast with the cardinality bound derived for pillage games here, for n = 3 characteristic function form2 (CF) games, the best known class of cooperative games, ‘‘stable sets are typically not unique’’, but existence is guaranteed (Lucas, 1992, pp. 562–563)
Summary
Opponents of a newly elected government pay taxes, knowing its supporters are more powerful than they are; international alliances are formed and allies supported to maintain both their allegiance and their effectiveness; firms transfer. Jordan (2006a) introduced pillage games, a class of cooperative games whose dominance relations are represented by power functions, increasing in both coalitional membership and members’ resource holdings Such games allow power to depend on both inalienable attributes and holdings of transferable resources without the imposition of restrictive game forms. Continuity and responsiveness axioms, this procedure both yields a unique stable set, when one exists, and sets a tight upper bound of 15 allocations on it This bound is much tighter than the finite bound of Jordan (2006a), the Ramsey bound of Kerber and Rowat (2011) or the doublyexponential one of Saxton (2011). This class of pillage games allows richer modelling possibilities than either CF or PF games for three agents, and allows tighter predictions—but subject to the caveat that the stable set may not exist
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