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Stable sets in three agent pillage games∗

Manfred Kerber † Colin Rowat‡

June 30, 2009

Abstract

Jordan [2006, “Pillage and property”, JET] characterises stable sets for three special cases of ‘pillage games’. For anonymous, three agent pillage games we show that: when the core is non-empty, it must take one of five forms; all such 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 assumption, it is unique and contains no more than 15 elements. This result uses a three step procedure: first, if a single agent can defend all of the resources against the other two, these allocations belong to the stable set; dominance is then transi- tive on the loci of allocations on which the most powerful agent can, with any ally, dominate the third, adding the maximal elements of this set to the stable set; finally, if any allocations remain undominated or not included, the game over the remain- ing allocations is equivalent to the ‘majority pillage game’, which has a unique sta- ble set [Jordan and Obadia, 2004, “Stable sets in majority pillage games”, mimeo]. Non-existence always reflects conditions on the loci of allocations along which the most powerful agent needs an ally. The analysis unifies the results in Jordan [2006] when n = 3.

Key words: pillage, cooperative game theory, core, stable sets, algorithm

JEL classification numbers: C63; C71; P14

This file: 090506comp-pill.tex

1 Introduction Jordan [2006a] introduced ‘pillage games’, a subset of abstract games in which coali- tions’ power increases monotonically in both coalitional membership, and members’ resources. The outcome of the contest of power between opposing coalitions represents the dominance relation usually taken as primitive in an abstract cooperative game.

As this contest of power generates externalities, pillage games are distinct from games in characteristic function form, the best known class of cooperative games. As coalitions’ power depends not just on their membership, but also on the resources that ∗We are grateful to Jim Jordan and Herakles Polemarchakis for useful conversations, to seminar audiences

at ESEM 2008, FEMES 2008, LAMES 2008, and to the ESRC for funding under its World Economy and Finance programme (RES-156-25-0022). Rowat thanks Birkbeck for its hospitality. †School of Computer Science, University of Birmingham; [email protected] ‡Department of Economics, University of Birmingham; [email protected]

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they hold, pillage games are also distinct from games in partition function form, the second best known class.

In studying contests of power, pillage games may be compared to games in which contest success functions convert investments in rent-seeking into a share of the spoils q.v. [Skaperdas, 1992]. Both approaches are required to make assumptions about how such contests occur: in the former case, a power function is axiomatically derived; in the latter, a contest success function is [Skaperdas, 1996], and a game form specified. Thus, while both make assumptions about the technology of power that largely remain untested, the latter also makes assumptions about game forms; insofar as real conflict is often characterised by not obeying any pre-specified rules, this may be a source of non-robustness.

The perhaps larger difference between the two approaches can be seen in their treat- ment of the costs of contests of power. While the non-cooperative approach assumes that rent-seeking is costly, the pillage literature (and variants such as Piccione and Ru- binstein [2007]) assume otherwise. In the conflict modeled by pillage games, the more powerful party is recognised as such, and not resisted by the less powerful. This, there- fore, seems consistent with environments in which there is a high degree of common knowledge about bargaining strengths; such environments may therefore include the contests of power undertaken by political factions in advanced democracies.

Jordan [2006a] presented results for both the core (the set of undominated allo- cations) and stable sets (no allocation in the set dominates any other; any allocation outside the set is dominated by one in it).1 For the former, he derived a general con- dition satisfied by any core allocation in any pillage games. For the latter, he provided results in a number of special cases: in the first, a unique stable set was characterised; in the second, partial characterisation was possible; in the third, a non-existence result was presented.

A number of natural questions were, however, left open. Is an explicit character- isation of the core possible? How are the apparently dissimilar results for stable sets related? How many other classes of stable sets are possible? What determines unique- ness or existence? Can any insight be gained into which classes of power functions represent the same dominance relations? Do any of the insights gained here aid analy- sis of games with multiple goods, in which both trade and pillage are possible [Rowat, 2009]?

These questions may be difficult to answer given the intractability of stable sets. Shapley [1959] showed that stable sets could be constructed with arbitrary closed com- ponents, by appropriate choice of the abstract game. While von Neumann and Mor- genstern [1944] assumed that stable sets existed for games in characteristic function form - the focus of their analysis - it took a quarter century for a counterexample to be found [Lucas, 1968].2 However, Jordan [2006a] proved that a set is stable if there is a consistent expectation for which it is the (farsighted) core in expectation.3 Thus, to the extent that the core and forward looking agents are compelling, the stable sets bear study.

In the context of pillage games, there has been cause for optimism. Theoretically, the monotonicity axioms that underpin dominance impose considerable structure on

1Richardson [1953] noted that stable sets appear as a Punktbasis zweiter Art – a point basis of the second type – in König [1936]. In von Neumann and Morgenstern [1944] they were simply called ‘solutions’.

2For abstract cooperative games, we are grateful to Oleg Itskhoki for noting that Condorcet cycles pro- duce counterexamples: let x, y and z be all possible imputations and x K y K z K x the only dominance relations on them; then no stable set exists.

3See Anesi [2006] and the references therein for more motivations of stable sets based on farsightedness.

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pillage games relative to general abstract games. Empirically, no examples of multiple stable sets have been found for pillage games. Theoretically, the cardinality of stable sets in pillage games is not only finite [Jordan, 2006a], but bounded by a Ramsey number [Kerber and Rowat, 2009].

This paper addresses the above questions in the case of anonymous, three agent, one-good pillage games. When the core is non-empty, it must take one of five forms; these include configurations not seen in Jordan’s examples. Turning to stable sets, all power functions yielding empty cores represent the same dominance relation, and therefore yield the same stable set.

When the core is not empty, it is known from Jordan [2006a] that it must contain the ‘tyrannical’ allocations – those giving all of society’s resources to a single agent; further, those agents are able to defend their allocations, alone, against all other agents. As core allocations necessarily belong to any stable set, these three allocations there- fore seed our candidate stable sets.

As a coalition’s power is monotonic in its resource holdings, decreasing the tyrant’s holdings may eventually yield a balance of power with the other two agents. When power is continuous in resources, allocations along this locus – which may not exist – are only dominated by allocations in turn dominated by tyrannical allocations. Thus, ensuring external stability on this locus requires inclusion of allocations from the locus itself. Further, as dominance is transitive along this locus, those allocations allowing external stability are unique. For each agent i, there are up to three of these.

Further decreasing the tyrant’s holdings may then yield allocations for which any two agents can defeat a single agent. Over such allocations, dominance is shown to be equivalent to that in the ‘majority pillage game’, in which any two agents can always defeat a third; when singleton coalitions oppose each other, the wealthier agent wins. As Jordan and Obadia [2004] prove the majority pillage game has a unique stable set of three allocations, there is a unique stable set on this domain.

When power satisfies a responsiveness axiom, in addition to anonymity and con- tinuity, this three step procedure both forces a unique stable set, when one exists, and sets an upper bound of fifteen allocations on it. It also allows us to locate the source of non-existence results along the locus of allocations along which the most powerful agent needs an ally: non-existence either reflects the absence of a maximal element along this locus, a new application of an old result [von Neumann and Morgenstern, 1944, §65.4.2], or the dominance of maximal elements by a tyrannical allocation. The non-existence result in Jordan [2006a] arose from the latter mechanism; the former is only possible when the power function is discontinuous in resource holdings, a feature not present in any of Jordan’s examples.

The procedure allows us to present an alternative algorithm to that in Roth [1976], applied to pillage games by Jordan and Obadia [2004]. That was incomplete in a number of respects: it