Stable sets in three agent pillage games∗
Manfred Kerber † Colin Rowat‡
June 30, 2009
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 
when n = 3.
Key words: pillage, cooperative game theory, core, stable sets, algorithm
JEL classification numbers: C63; C71; P14
This file: 090506comp-pill.tex
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]
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
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 ) 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
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,
These questions may be difficult to answer given the intractability of stable sets.
Shapley  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  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
In the context of pillage games, there has been cause for optimism. Theoretically,
the monotonicity axioms that underpin dominance impose considerable structure on
1Richardson  noted that stable sets appear as a Punktbasis zweiter Art – a point basis of the second
type – in König . In von Neumann and Morgenstern  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  and the references therein for more motivations of stable sets based on farsightedness.
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  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 ,
applied to pillage games by Jordan and Obadia . That was incomplete in a
number of respects: it