Journal of Economic Theory 131 (2006) 26 – 44
www.elsevier.com/locate/jet
Pillage and property
J.S. Jordan∗
Department of Economics, The Pennsylvania State University, 613 Kern Building,
University Park, PA 16802, USA
Received 10 April 2001; final version received 25 May 2005
Available online 10 August 2005
Abstract
This paper introduces a class of coalitional games, called pillage games, as a model of Hobbesian
anarchy. Any coalition can pillage, costlessly and with certainty, any less powerful coalition. Power
is endogenous, so a pillage game does not have a characteristic function, but pillage provides a
domination concept that defines a stable set, which represents an endogenous balance of power.
Every stable set contains only finitely many allocations, and can be represented as a farsighted core.
Additional results are obtained for particular games, including the game in which the power of each
coalition is determined by its total wealth.
© 2005 Elsevier Inc. All rights reserved.
JEL classification: C70; C71; D74
Keywords: Allocation by force; Coalitional games; Core; Stable set
Property is theft.
Proudhon 1
Property is not theft but a good deal of theft becomes property.
R.H. Tawney1
∗ Fax: +1 814 863 4775.
E-mail address: [email protected].
1 Bazelon [2, p. 55].
0022-0531/$ - see front matter © 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.jet.2005.05.008
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
27
1. Introduction
Some distinction between property and theft is a precondition of economic activity. In
the absence of property rights, the strong can despoil the weak. There is strength in numbers
but no security. Individuals can combine their forces to raid others, but are then vulnerable
to treachery among themselves. As Hobbes envisioned this kind of anarchy,
...there be no Property, no Dominion, no Mine and Thine distinct; but onely that to be
every mans, that he can get; and for so long as he can keep it (Leviathan, Part I, Ch.
13, italics in original).
Against this alternative the social benefit of property rights is obvious, even apart from
potential gains from trade and investment. But Tawney’s remark quoted above suggests an
inherent instability. Any individual or coalition strong enough to create property rights has
an incentive to make exceptions in its favor. The question is whether a stable distribution
of wealth can emerge as an endogenous balance of power.
There is a substantial literature on allocation by force. As exemplified by Skaperdas
[13], these papers pose noncooperative games in which various individuals have technologies for appropriating the property of others. Appropriation is costly and its outcome may
be uncertain. An equilibrium can involve a positive level of continuing conflict, no conflict or extortion (see also Hirshleifer [8] and other papers by Garfinkel and Skaperdas
[4], Usher [15], Hirshleifer [7], Skaperdas [14], and Konrad and Skaperdas [10]). This
approach enables the explicit analysis of the effect of conflict technologies on the welfare and other properties of the Nash equilibrium allocations. In keeping with the noncooperative game methodology, this literature limits or precludes entirely the formation of
coalitions.
The development of social institutions is naturally thought of as a group activity, so it
seems useful to allow the formation of coalitions in modeling the emergence of property
rights. The present paper introduces a class of coalitional games, called pillage games, representing Hobbesian anarchy. There is a single commodity, wealth, which must be allocated
among a finite number of individuals. Opportunities for pillage are represented by a power
function that specifies the power of each coalition as depending at least partly on the wealth
of its members. Any coalition can take the wealth of any less powerful coalition. Since
power is endogenous, a pillage game does not have a characteristic function, but pillage
itself constitutes a binary domination relation over allocations that suffices to define the
concepts of core and stable set (von Neumann–Morgenstern solution). A core allocation,
by definition, is an allocation immune to pillage. Thus, at a core allocation, wealth is defended by the power of its owners without the aid of any social construct. As one would
expect, the core is typically small and often empty. Proposition 2.6 shows that if no tyrannical allocation, in which one player owns everything, can be defended by its owner, then
the core is empty.
A stable set is defined as a set of allocations that is internally stable, in the sense that
a stable allocation cannot dominate another stable allocation, and externally stable, in the
sense that any nonstable allocation is dominated by some stable allocation. A stable set
embodies social expectations that are consistent in the sense that pillaging a stable allocation
only invites further pillage resulting in another stable allocation. In this sense a stable set
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J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
can include allocations that are supported by social expectations rather than power alone.
However, the range of allocations that can be stable is not large. Theorem 2.9 states that
any stable set must be finite.
Section 3 characterizes the core and stable set of the pillage game in which the power of
each coalition is equal to its total wealth. In this game the core consists of the tyrannical
allocations together with the allocations in which two players each own half the aggregate
wealth. The stable set, which is unique, consists of all allocations in which each player
whose wealth is positive owns a fraction of the aggregate wealth that is a power of 21 . For
example, with aggregate wealth normalized to unity, ( 41 , 0, 21 , 18 , 18 ) is a stable allocation
for five players.
Section 4 discusses some pillage games in which the power of a coalition depends on its
size as well as its total wealth. When power is a weighted sum of wealth and size, the pillage
game is a kind of plurality voting game in which wealth adds to voting strength. In particular,
suppose that total wealth is normalized to unity and that the power of each coalition is equal
1
to its total wealth plus a weight v times the number of its members. If v > n−2
, where n
is the total number of players, then tyrannical allocations cannot be defended, so the core
is empty. If v > 1, so that one player’s vote counts more than aggregate wealth, and the
number of players is odd, there is a stable set consisting of all allocations in which the
aggregate wealth is divided equally among a minimal majority, that is, a coalition of n+1
2
players. This is one of the stable sets found by von Neumann and Morgenstern [16] for the
majority game. However, since the voting strength of wealth is positive and only a plurality
is necessary for pillage, most of the stable sets for the majority game are not internally
stable for the pillage game. In the case of a Cobb–Douglas power function, in which the
power of a coalition is its total wealth times its size, the core consists of the tyrannical
allocations, because even a large coalition has no power without wealth. However, in this
case Proposition 4.6 shows that no stable set exists. The same is true of the weighted voting
game when v is positive but small.
The finiteness of stable sets ensures that stable allocations are locally unique, so it is
natural to ask whether they are also “locally stable”. That is, if a stable allocation is slightly
perturbed to a nonstable allocation, will the new allocation be dominated by the nearby
stable allocation? Section 5 answers this question in the negative. Theorem 5.2 states that
any nontyrannical stable allocation is “locally unstable” in the sense that arbitrarily small
perturbations result in allocations that are dominated not by the initial stable allocation
but only by other necessarily distant stable allocations. In this sense, the balance of power
represented by a stable set is locally determinate but, except for the trivial case of tyranny,
precarious.
von Neumann and Morgenstern interpret their solutions as “standards of behavior” for
players in a game, reasoning that the properties of internal and external stability assure
that “once they are generally accepted they overrule everything else and no part of them
can be overruled within the accepted standards” [16, p. 42]. For general coalitional games,
Harsanyi [6] argues that this interpretation is questionable. For pillage games, however,
the von Neumann–Morgenstern interpretation has a straightforward justification, which is
formalized in Section 6. Theorem 6.4 shows that if players hold common expectations
concerning which future acts of pillage, if any, will occur at each allocation, then a stable
set corresponds to the set of allocations that no coalition can improve upon when the future
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
29
implications of any pillage are taken into account. In this sense, a stable set of a pillage
game is a farsighted core.
2. Pillage games
The environment of a pillage game contains a fixed amount of wealth to be allocated
among a fixed population of players. Total wealth is normalized to unity.
2.1 Definitions. The set of players is the finite set I = {1, . . . , n}, where n2. Subsets will
n
be
called coalitions. The set of allocations is the set A = {w ∈ R : wi 0 for each i, and
i wi = 1}.
A pillage game is specified by a power function that determines which acts of pillage are
feasible. The general definition given below includes some natural properties. The power
of a coalition does not decrease if new members are added (p.1) or if the wealth of some
members is increased without decreasing the wealth of other members (p.2). Increasing
the wealth of every member increases the power of the coalition (p.3). This requirement
excludes power functions that are independent of wealth, such as those that depend on
coalitional size alone.
2.2 Definitions. The power function is a function : 2I × A → R satisfying
(p.1) if C ⊂ C then (C , w)(C, w) for all w;
(p.2) if wi wi for all i ∈ C then (C, w ) (C, w); and
(p.3) if C = ∅ and wi > wi for all i ∈ C then (C, w ) > (C, w).
An allocation w dominates an allocation w if
(W, w) > (L, w),
(d)
where W = {i : wi > wi } and L = {i : wi < wi }.
The role of domination (d) in a pillage game is analogous to the role of blocking in the
usual coalitional game model of an exchange economy, but with an important distinction.
Blocking, in an exchange economy, allows a coalition to veto an undesired reallocation
of the initial endowment. Domination, in a pillage game, allows a coalition to force a
desired reallocation of the initial endowment. This gives domination a more sequential
interpretation, which is represented explicitly in Section 6. The definitions of the core
in the two settings are exactly analogous. The core of an exchange economy is the set
of unblocked allocations, and the core of a pillage game (Definition 2.3) is the set of
undominated allocations. However, while the core of an exchange economy is a central
concept of general competitive analysis, the core of a pillage game is frequently empty, and
even when nonempty is typically too small to be an appealing solution concept.
Domination is interpreted to mean that coalition W has the power to move the allocation
from w to w by pillaging L. The power of the players whose wealth is unchanged is ignored
in (d). The coalition {i : wi = wi } is implicitly assumed to remain neutral. This property of
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J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
(d) will be referred to informally as the neutrality assumption. The neutrality assumption
may seem unnatural, since (p.3) implies that any act of pillage increases the power of the
pillaging coalition, W , thereby increasing the vulnerability of the neutral players. Thus it
may seem more natural to assume that any act of pillage is opposed by the entire complementary coalition I \ W , which can change the stable sets, as will be noted in Sections 3
and 4 with respect to specific games. The neutrality assumption is imposed here with the
intention of deriving the opposition of unaffected players as an implicit property of a solution concept rather than imposing it in the definition of the game. That is to say, some acts
of pillage may imperil neutral players, while others may not. It will be shown in Section 6
that if neutrality is replaced by opposition in those cases where neutral players expect to be
subsequently pillaged, then a stable set contains exactly those allocations at which pillage is
thereby precluded. In this sense, a stable set derived under the neutrality assumption gives
rise to expectations that modify neutrality so as to make the stable set a farsighted core.
Pillage games are perhaps most easily understood by comparing them with majority
games. In the usual majority game, in which each player has one vote, the power of each
coalition is equal to the number of its members. The majority power function, m (C, w) =
#C, satisfies (p.1) and (p.2) but violates (p.3). Even so, domination for the majority game
is not m (W, w) > m (L, w), as it would be under (d), but m (W, w) > n2 , which violates
the neutrality assumption. Thus the neutrality assumption, as well as the dependence of
power on wealth, distinguishes pillage games from majority games. The comparison with
majority games is discussed further in Section 4.
The power function is used only to define domination. Any monotone transformation
of the power function yields essentially the same pillage game. Indeed, the definition of a
pillage game could be generalized by replacing the concept of power function with appropriate assumptions on the domination relation in lieu of (p.1–3), but the resulting definition
would be less transparent.
Domination, as defined by (d), is asymmetric but typically not acyclic. A pillage game is
a special case of an abstract game, defined by Lucas [11] as a set with an irreflexive binary
relation. A coalitional game with a characteristic function has a derived domination relation
that is irreflexive but can fail to be asymmetric. The definitions of core and stable set stated
below follow Lucas [11] and von Neumann and Morgenstern [16, Chapter XII].
2.3 Definitions. The core is the set of undominated allocations. A set S of allocations is a
stable set if it satisfies
(IS) (internal stability) no element of S is dominated by an element of S; and
(ES) (external stability) every element of A \ S is dominated by some element of S.
For ease of exposition, the term stable allocation will be used informally to mean an
element of a stable set. This is somewhat misleading since stable sets, unlike the core or the
set of Nash equilibria of a noncooperative game, are not defined pointwise. Whether or not
a particular allocation is stable depends on the entire stable set rather than the allocation
alone.
The core is unique, but may be empty. External stability implies that a stable set cannot
be empty, but a stable set may fail to exist, even when the core is nonempty. The following
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
31
proposition records for later reference two facts that are immediate implications of the
definitions of core and stable set (e.g., Lucas [11, pp. 547–548]).
2.4 Proposition. Every stable set contains the core. If the core satisfies external stability
then the core is the unique stable set.
The core of a pillage game is the set of allocations that can be defended by force. By
the definition of domination, the undominated allocations are the allocations at which no
acts of pillage are feasible. Thus no social institution or equilibrium concept is needed to
protect the distribution of wealth in a core allocation. Proposition 2.6 derives the implication
that each player who holds wealth in a core allocation must have at least as much power
as the coalition containing every other player. This is most likely to happen at a tyrannical
allocation, which gives all of the wealth to one player. Proposition 2.6(b) shows that in order
for the core to be nonempty, it must contain a tyrannical allocation for at least one player.
2.5 Definition. For each i, let ei denote the allocation that gives everything to player i, that
is, eii = 1. The ei ’s are called tyrannical allocations.
2.6 Proposition. The core is the set
{w : {i : wi > 0} = {i : ({i}, w)(I \ {i}, w)}} .
In particular,
(a) for each i, the core contains ei if and only if ({i}, ei ) (I \ {i}, ei ); and
(b) if ({i}, ei ) < (I \ {i}, ei ) for all i, then the core is empty.
Proof. Let w be any allocation, and suppose that for some player i, wi > 0 but ({i}, w) <
(I \ {i}, w). Then w is dominated by the allocation w defined by wi = 0 and wj = wj +
wi
n−1 for all j = i, so w is not in the core. Alternatively, suppose that ({i}, w) (I \{i}, w)
for every player i such that wi > 0. Let w be any other allocation, and let W = {i : wi > wi }
and L = {i : wi < wi }. Then L ⊂ {i : wi > 0}. Let i ∈ L. Then property (p.1) implies
(L, w)({i}, w)(I \ {i}, w)(W, w),
so w is undominated. This proves the first assertion.
Assertion (a) follows directly from the first assertion. To prove (b), note that property (p.2)
implies that for each w and each i, ({i}, w)({i}, ei ) and (I \ {i}, w) (I \ {i}, ei ).
Therefore, if the set {i : ({i}, ei ) (I \ {i}, ei )} is empty, then for every w the set
{i : ({i}, w)(I \ {i}, w)} is empty and, by the first assertion, the core is empty. If there are only two players, domination is transitive, so it is not surprising that a general
characterization of the core and stable set can be given for this case.
2.7 Proposition. Suppose that n = 2. Then the core is nonempty and equal to the unique
stable set. If there is some wo ∈ A satisfying 1 > w1o > 0 and ({1}, w o ) = ({2}, w o ),
then the core is the set {e1 , e2 , wo }. Otherwise the core is a subset of {e1 , e2 }.
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J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
Proof. First, suppose that for each allocation w with 1 > w1 > 0, ({1}, w) > ({2}, w).
Then by (p.2), e1 dominates every such w and ({1}, e1 ) > ({2}, e1 ). If ({1}, e2 ) >
({2}, e2 ) then e1 dominates every w = e1 , so the singleton {e1 } is the core and the unique
stable set. If ({1}, e2 ) ({2}, e2 ) then e1 and e2 are undominated, and e1 dominates every
other allocation, so the set {e1 , e2 } is the core and the unique stable set. The case in which
({2}, w) > ({1}, w) for each allocation w with 1 > w1 > 0 is symmetric. Now suppose
there is some allocation wo satisfying 1 > w1o > 0 and ({1}, w o ) = ({2}, w o ). Then w o
is undominated. Let w , w ∈ A with 1 > w1 > w1o > w 1 > 0. Then (p.3) implies that
({1}, e1 ) > ({1}, w ) > ({1}, wo ) > ({1}, w ) > ({1}, e2 )
(1)
({2}, e2 ) > ({2}, w ) > ({2}, wo ) > ({2}, w ) > ({2}, e1 ).
(2)
and
Since ({1}, wo ) = ({2}, w o ), (1) and (2) imply that e1 and e2 are undominated, that e1
dominates w and that e2 dominates w . Hence the core is the set {e1 , e2 , wo }, and every
other allocation is dominated by either e1 or e2 , so Proposition 2.4 implies that the core is
the unique stable set. Stable sets in pillage games having more than two players typically differ from the core,
and can be difficult to find or characterize. One example of a pillage game for which the
stable set coincides with the core is the game in which the power of each coalition is equal to
the wealth of its wealthiest member. This power function, which is introduced primarily for
illustration, depicts a situation in which each coalition is represented by a single member,
who competes with the representatives of other coalitions on the basis of personal wealth.
2.8 Example. Define the power function max by
max (C, w) = max{wi : i ∈ C}.
For each nonempty coalition C, let eC denote the allocation
1/#C if i ∈ C;
eiC =
0
otherwise,
where #C denotes the number of players C. Let E denote the set of allocations eC for all
nonempty C. Then E is the core and the unique stable set for the power function max .
Proof. It follows directly from the definition of max that the allocations in E are undom/ E, eiT > wi for all i ∈ T .
inated. Let w ∈
/ E. Let T = {i : wi wj for all j }. Since w ∈
T
Moreover, (T , w) > (C, w) for every C ⊂ I \ T , so e dominates w. Hence E is the
core and E satisfies external stability, so E is the unique stable set by Proposition 2.4. Analogous power functions can be defined using other order statistics, such as the mean
or minimum wealth level of a coalition. These two power functions violate property (p.1),
since adding a player with lesser wealth can reduce a coalition’s power. However, they
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
33
are easily seen to have the same core and unique stable set as max , for the same reason.
Here again, unless wealth is shared equally by all players who possess a positive amount,
the coalition consisting of the player or players with the greatest wealth can pillage the
complementary coalition.
The final result of this section shows that for pillage games, internal stability alone is
a restrictive concept in the sense that any internally stable set must exclude almost every
distribution of wealth.
2.9 Theorem. An internally stable set can contain at most finitely many allocations.
Proof. Suppose by way of contradiction that S is an infinite set satisfying (IS). Let {w k } be
an infinite sequence in S. Choosing a subsequence if necessary, we can assume that for each
i, wik is either strictly increasing, strictly decreasing or constant over k. Let W = {i : wik
is strictly increasing} and L = {i : wik is strictly decreasing}. For each k, (IS) implies that
wk+1 does not dominate w k , so
(W, wk )(L, wk ),
(3)
and wk does not dominate w k+1 , so
(W, wk+1 ) (L, wk+1 ).
(4)
For each k > 1, (3) and (4) both apply since k = (k − 1) + 1, so
(W, w k ) = (L, wk ).
(5)
But this is impossible because the definitions of W and L together with (p.3) imply
that the left-hand side of (5) is strictly increasing in k and the right-hand side is strictly
decreasing. 2.10 Corollary. The core contains at most finitely many allocations.
The finiteness of all stable sets is unusual in coalitional games. As noted by Lucas [11,
p. 578], most games in characteristic function form have no finite stable sets.
3. Wealth is power
This section characterizes the stable set when the power of each coalition is simply its
total wealth.
3.1 Definitions. Define the power function w by w (C, w) =
tion w dominates an allocation w if
wi >
wi ,
i∈W
i∈L
where W = {i : wi > wi } and L = {i : wi < wi }.
i∈C
wi . Then an alloca(dw )
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J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
A number 0 x 1 is dyadic if x = 0 or x = 2−k for some nonnegative integer k. An
allocation w is dyadic if each wi is dyadic. Let D denote the set of dyadic allocations. For
each positive integer k, let Dk = {w : w is dyadic and for each i, if wi > 0, then wi 2−k }.
Then Dk ⊂ Dk+1 for each k, and D = ∪k Dk .
It is easily seen that the core consists of the tyrannical allocations together with
allocations that give half the total wealth to each of two players. In the above notation
this is the set D1 .
3.2 Proposition. For the power function w , the core is the set D1 .
The unique stable set for this pillage game is the set of all dyadic allocations, D. The
internal stability of D follows from the fact that, in moving from one dyadic allocation
to another, the wealth of each of the winners is at least doubled, so the losers must have
at least as much initial wealth as the winners. This is established in Lemma 3.4. The
proof of uniqueness uses an induction argument that is reminiscent of Roth’s algorithm for
constructing a “supercore” of an abstract game [12,1].
3.3 Theorem. For the power function w , D is the unique stable set.
3.4 Lemma. The set D is internally stable.
Proof. Let w , w ∈ D and let W ={i : wi > w
i } and L = {i : wi < wi }. Then for each
i ∈ W , wi 2wi . This implies that i∈L wi i∈W wi , so w does not dominate w. Lemmas 3.6, 3.7 and 3.8 show that D is externally stable by showing how to construct
a dyadic allocation dominating any given nondyadic allocation. This is illustrated by the
1 1 1
, 12 , 24 ). The first
following example. Let w denote the nondyadic allocation ( 21 , 16 , 18 , 12
1
1
nondyadic wi is 6 , so increase this to the dyadic number 4 . Continue increasing each
successive player’s allocation to the next highest dyadic number until the total wealth is
exhausted. In this case, this is accomplished by increasing w3 to 41 , resulting in the dyadic
allocation w = ( 21 , 41 , 41 , 0, 0, 0). Since w2 +w3 > w4 +w5 +w6 , w dominates w. Lemma
3.8 shows that this construction is general.
3.5 Definitions. For each 0 < x < 1, define d(x) = min{y : y > x and y is dyadic}. An
allocation w is in standard form if w1 · · · wn .
3.6 Lemma. Let w be an allocation in standard form, let m = max{i : wi > 0}, and
let 1 < j m such that wi is dyadic for all i < j . Then m
i=j wi wj −1 . In particular, if
j = m, then wm = wm−1 .
Proof. By the definition of j, wj −1 = 2−k for some positive integer k, and for each i < j −1,
j −1
wi = 2−ki for some ki k. Therefore i=1 wi = p2−k for some integer 0 < p < 2k , so
m
k
−k
−k
i=j wi = (2 − p)2 2 = wj −1 . In particular, if j = m, wm wm−1 . Since w is in
standard form, this proves the final assertion. J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
35
3.7 Lemma. Let w be a nondyadic allocation
Let j = min{i : wi is not
in standard form.
dyadic}. Then there is some j such that i=j d(wi ) = ni=j wi .
Proof. The existence of the desired will be proved by induction on m − j , where m =
max{i : wi > 0}. By the final assertion of Lemma 3.6, the case j = m is impossible. Suppose
that j = m − 1. Then wm−2 d(wm−1 ) > wm−1 wm > 0. Then Lemma 3.6 implies that
wm−1 + wm d(wm−1 ), so wm d(wm−1 ) − wm−1 . Applying the final assertion of Lemma
3.6 to the allocation (w1 , . . . , wm−2 , d(wm−1 ), wm − (d(wm−1 ) − wm−1 ), 0, . . . , 0) shows
that wm − (d(wm−1 ) − wm−1 ) = 0, so the lemma is satisfied by = m − 1 = j .
Now suppose that the desired exists when j = m − p, and suppose that j = m −
p − 1. If wm d(wj ) − wj , the induction hypothesis implies that the desired exists for
the allocation (w1 , . . . , wj −1 , d(wj ), wj +1 , . . . , wm−1 , wm − (d(wj ) − wj ), 0, . . . , 0),
for which j is larger since d(wj ) is dyadic. The same has the desired property for w.
If wm < d(wj ) − wj , the induction hypothesis implies that the desired exists for the
allocation (w1 , . . . , wj −1 , wj + wm , wj +1 , . . . , wm−1 , 0, . . . , 0), for which m is smaller.
Since d(wj ) > wj + wm , d(wj ) = d(wj + wm ), so the same has the desired property
for w, which completes the proof. 3.8 Lemma. Let w be a nondyadic allocation in standard form. Let j = min{i : wi
is not dyadic}, and let j be given by Lemma 3.7. Then the dyadic allocation w =
(w1 , . . . , wj −1 , d(wj ), . . . , d(w ), 0, . . . , 0) dominates w.
Proof. Let W = {i : wi > wi }, and let L = {i : wi < wi }. Then W = {i : j i } and
L = {i : i > and wi > 0}. By the definition of d(·), wi d(wi ) − wi for all i, with strict
inequality for i = j , since wj is not dyadic. Therefore
i∈W
wi >
i=j
(d(wi ) − wi ) =
i>
wi =
wi ,
i∈L
where the middle equality follows from the fact that
w dominates w. i=j
d(wi ) =
n
i=j
wi . Therefore
Lemma 3.8 shows that the set D of dyadic allocations is externally stable, so it remains
to show that the stable set is unique. The main step in proving uniqueness is Lemma 3.9,
which shows that if an allocation w o ∈ Dk+1 is dominated by any allocation w, then w
is in turn dominated by an allocation in Dk . This allows uniqueness to be established by
induction on k using an argument similar to Roth’s algorithm for constructing a “supercore”
of an abstract game [12,1].
3.9 Lemma. For any k 1, let w o ∈ Dk+1 and let w be any allocation that dominates w o .
Then w is nondyadic. Permuting player indices if necessary, we can assume that w is in
standard form. Let w be the allocation given by Lemma 3.8. Then w ∈ Dk .
Proof. By Lemma 3.4, w is nondyadic. Let W + = {i : wi > wio > 0}. Then for each
i ∈ W + , wi > wio 2−(k+1) . Let be given by Lemma 3.8. If i o for some i o ∈ W + ,
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J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
then since w is in standard form, the definition of w implies that wi w = d(w ) 2−k
for all 1i . Since wi = 0 for all i > , it follows that w ∈ Dk .
Now suppose that > i for all i ∈ W + . If w 2−(k+1) , then for all i , wi w =
d(w )2−k , so w ∈ Dk . Therefore, it remains only to eliminate the possibility that w <
2−(k+1) , which, since w is in standard form, reduces to the case in which wi < 2−(k+1)
for all i . For each i, define (wi − wio )+ = max{0, wi − wio } and (wio − wi )+ =
max{0, wio − wi }. For each i, since w o is dyadic, the definition of w implies that if wi > wio
then (wi −wio )+ = wi −wio wio . Also, if i ∈ W + then i < , so the definition of w implies
that wi wi > wio . Therefore i (wi − wio )+ W + (wi − wio )+ W + wio = (W, w o ),
where W = {i : wi > wio } = W + ∪ {i : wi > wio = 0}. Let L = {i : wi < wio }. For any i,
suppose that wio > wi . If i < then the definition of wi implies that wi wi , so i ∈ L. If
i then wi < 2−(k+1) . Since wio > wi , wio > 0. Therefore wio > wi , since w o ∈ Dk+1 ,
so again i ∈ L. Therefore i (wio − wi )+ = L (wio − wi )+ L wio = (L, wo ). Since w
dominates wo , (W, w o ) > (L, wo ), so
i (wi − wio )+ (W, wo ) > (L, wo ) i (wio − wi )+ .
However, since both w and wo sum to unity, i (wi − wio )+ = i (wio − wi )+ . This
contradiction eliminates the possibility that w < 2−(k+1) . Proof of Theorem 3.3. By Lemma 3.4, D is internally stable, and the external stability
of D follows from Lemma 3.8. To prove uniqueness, let S be a stable set. Since D1 is the
core, D1 ⊂ S. For any k 1, suppose that Dk ⊂ S. Let w o ∈ Dk+1 . By Lemma 3.9, any
allocation w that dominates wo is in turn dominated by some w ∈ Dk . Since S is internally
stable, w ∈
/ S. Therefore wo is not dominated by any allocation in S. Since S is externally
o
stable, w ∈ S, so Dk+1 ⊂ S. Therefore, by induction on k, D ⊂ S. Let w ∈ S. If w ∈
/ D,
then by Lemma 3.8, w is dominated by an allocation in D. Since S is internally stable and
D ⊂ S, it follows that w ∈
/ S. This contradiction proves S ⊂ D. The fact that all stable sets are finite places an upper bound on the required k. More specifically, Proposition 3.10 shows that for any w ∈ D, min{wi : wi > 0}2−(n−1) , so D =
Dn−1 , where n is the number of players. The dyadic allocation ( 21 , 41 , 18 , . . . , ( 21 )n−1 , ( 21 )n−1 )
shows that the bound n − 1 is tight.
3.10 Proposition. D = Dn−1 . Moreover, D ⊂ Dk for any k < n − 1.
Proof. Let w ∈ D in standard form, and let m = max{i : wi > 0}. By Lemma 3.6,
i j wi wj −1 for each j m. Thus 2wm wm−1 + wm , 4wm wm−2 + wm−1 + wm ,
and continuing in this fashion, 2m−1 wm i wi = 1, so wm 2−(m−1) . Since m n,
wm 2−(n−1) , so w ∈ Dn−1 . Hence Dk = Dn−1 for all k > n − 1, so D = Dn−1 .
The dyadic allocation wo = ( 21 , 41 , 18 , . . . , ( 21 )n−1 , ( 21 )n−1 ) ∈ Dn−1 , but w o ∈
/ Dk for any
k < n − 1, so D ⊂ Dk for any k < n − 1. The stability of the set D depends on the neutrality assumption. In particular, suppose
that the neutrality assumption is replaced by the assumption that any pillage is opposed by
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
37
all players who do not benefit from it. Domination is then defined by
i∈W
wi >
1
.
2
(dw )
This results in the same core as above, but the stable set is significantly different. The
domination relation (dw ) is a proper subset of (dw ), so D remains internally stable but is
no longer externally stable. Jordan [9] shows that the new stable set, which is again unique,
consists of two parts. One part is the subset of dyadic allocations that divide the total wealth
equally among any set of 2k players for any k. The other part consists of all allocations in
which wi = 21 for some player i. Under (dw ), the presence of a player with half the wealth
prevents pillage among the others. Moreover, none of the other players will join the wealthy
player in pillage, since doing so would leave the wealthy player with enough power to take
everything.
4. Strength in numbers
This section gives some partial results when the power of a coalition depends on its size
as well as its total wealth. The simplest such power function is a weighted sum of the two.
4.1 Definitions. Given any real number v 0, define the power function v by
v (C, w) = v#C +
wi ,
i∈C
where #C denotes the number of players in C.
The parameter v represents the power of each player’s vote, measured in multiples of
aggregate wealth. If v = 0 then v reduces to w . If v is small but positive, tyrannical
allocations can be defended but the allocation ( 21 , 21 , 0, . . . , 0) is no longer in the core,
since it is dominated by ( 43 , 0, 41 , 0, . . . , 0), in particular. If v is large enough, tyrannical
allocations cannot be defended and the core is empty.
4.2 Proposition. Suppose that n > 2 and the power function is v .
1
(a) If 0 < v n−2
, then the core is the set {ei : 1 i n}.
1
(b) If n−2 < v then the core is empty.
1
then for each i, v ({i}, ei ) v (C, ei ) for every C ⊂ I \ {i}, so ei is
Proof. If v n−2
undominated. Let w be any allocation with wi = 1 for all i. To complete the proof of
(a), it suffices to show that w is dominated. Assume without loss of generality that w is
in standard form, so that 1 > w1 · · · wn . Then w2 > 0. If v > 0, v ({1, 3}, w) >
v ({2}, w), so the allocation (w1 + w22 , 0, w3 + w22 , w4 , . . . , wn ) dominates w, which
1
< v then for each i, v (I \ {i}, ei ) > v ({i}, ei ), so (b) follows from
proves (a). If n−2
Proposition 2.6 (b). 38
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
If v > 1, a single vote is more powerful than the aggregate wealth. In this case the pillage
game is sufficiently similar to a majority game that the allocations which divide wealth
equally among a minimal majority constitute a stable set.
4.3 Proposition. Suppose the power function is v and n is odd. Let M denote the set
2
2
consisting of the allocation ( n+1
, . . . , n+1
, 0, . . . , 0) under all permutations of the player
set I. If v > 1 then M is a stable set.
Proof. To prove that M satisfies internal stability, let w and w be distinct allocations in
of M, #W = #L
M, let
W = {i : wi >wi } and let L = {i : wi < wi }. By the definition
and i∈W wi = 0 < i∈L wi , so v (W, w) < v (L, w). Therefore w does not dominate
2
w. To prove that M satisfies external stability, let w ∈
/ M. Then #{i : wi < n+1
} n+1
2 .
2
2
Let w ∈ M satisfy {i : wi = n+1 } ⊂ {i : wi < n+1 }, so that #{i : wi > wi } = n+1
2 >
n−1
2 #{i : wi < wi }. If v > 1 then v ({i : wi > wi }, w)−v ({i : wi < wi }, w) v −1 >
0, so w dominates w. Even when v is large, the pillage game differs from the majority game in two ways. First,
wealth still makes a small but positive contribution to a coalition’s power. Second, because of
the neutrality assumption, domination does not require the winners to constitute a majority
of all players. The winners only have to outvote the losers. These two distinctions eliminate
most of the stable sets found by von Neumann and Morgenstern [16] for the majority game.
For example, suppose n = 3 and let 0 c < 21 . Then the set {(c, (1 − c), (1 − )(1 − c)) :
0 1} is a stable set for the majority game. However, for the pillage game with v > 1,
the allocation (c, (1 − c), 0) dominates the allocation (c, 23 (1 − c), 13 (1 − c)), so this set is
not internally stable.
The Cobb–Douglas power function defined below depends on coalitional size in a way
that preserves the defensibility of tyrannical allocations, since the complete absence of
wealth makes any coalition powerless. However, despite the nonemptiness of the core,
Proposition 4.6 shows that no stable set exists for this power function. Although the power
function is continuous, domination suffers from a discontinuity that is discussed following
the result.
4.4 Definition. Define the power function c by
c (C, w) = #C
wi .
i∈C
4.5 Proposition. Suppose that n > 2 and the power function is c . Then the core is the set
{ei : 1i n}.
Proof. For each i, c ({i}, ei ) = 1 > 0 = c (C, ei ) for every C ⊂ I \ {i}, so ei is undominated. Let w be any allocation with wi = 1 for all i. To show that w is dominated, assume
without loss of generality that w is in standard form, so that 1 > w1 w2 · · · wn . Then
w2 > 0 and c ({1, 3}, w)2w1 > w2 = c ({2}, w), so the allocation (w1 + w22 , 0, w3 +
w2
2 , w4 , . . . , wn ) dominates w. J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
39
4.6 Proposition. Suppose that n > 2 and the power function is c . Then no stable set
exists.
Proof. Suppose by way of contradiction that S is a stable set. For each
the allocation
,
w
by
w1
=
n−1
n ,
w2
=
n,
and
wi
=
(1−)
(n−2)n
n−2
n−1
< 1, define
for each i > 2. Then for each
c ({1}, w ) = c ({2, . . . , n}, w )
(6)
c ({2}, w ) > c ({3, . . . , n}, w ).
(7)
and
Then for each , either w ∈ S or there is some w ∈ S that dominates w . Since wi1 = 0
for all i > 2 and c ({1}, w 1 ) > c ({2}, w 1 ), e1 dominates w 1 . The previous proposition
implies that e1 ∈ S so w 1 ∈
/ S. Let < 1 and suppose that some w ∈ S dominates w .
Let W = {i : wi > wi } and L = {i : wi < wi }. Then c (W, w ) > c (L, w ) so (6)
implies that 1 ∈
/ L. If w1 > w1 then (6) implies that e1 dominates w , which contradicts
/ W , (7) implies that 2 ∈
/ L.
the assumption that w ∈ S. Therefore w1 = w1 . Since 1 ∈
1
Hence w2 w2 . Since w = w ,
1
> w2 w2 = .
n
n
(8)
Thus, for each n−2
n−1 < < 1, either w ∈ S or there is some w ∈ S satisfying (8). This
implies that S must be infinite, which contradicts Theorem 2.9.
The proof is easily
modified to cover the more general class of Cobb–Douglas functions
(C, w) = (#C) ( i∈C wi )1− for 0 < < 1. A similar argument proves the same
result for the power function v for sufficiently small positive values of v, specifically
1
.
0 < v < 3(n−3)+1
The proof makes use of a discontinuity in domination. Specifically, the set {(w , w) :
w dominates w} is not an open subset of A × A. Small perturbations in w and w can
change the size of the winning or losing coalitions. If n = 3, for example, (1, 0, 0) dominates
( 23 , 13 , 0), but does not dominate ( 23 , 13 − , ) for > 0. Of course, the same discontinuity
occurs when the power function is v and v > 1, in which case a stable set exists (Proposition
4.3), so the discontinuity is not always fatal.
5. Local stability
The finiteness of stable sets enhances the determinacy of the solution concept. For a
given stable set, the stable allocations are at least locally unique. On the other hand, in
richer models the distribution of wealth may be continually perturbed by economic activity
or acts of nature. Such perturbations must lie outside the stable set. If a small perturbation
of a stable allocation is dominated by the original allocation, stability can be reestablished
40
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
without social upheaval. This suggests a concept of local stability that requires internal and
external stability only locally.
5.1 Definition. A set S ⊂ A is locally stable if for each w ∈ S,
(LIS) (local internal stability) there is some > 0 such that the set {w ∈ S :
||w − w|| < } satisfies internal stability (IS); and
(LES) (local external stability) for every > 0 there is some > 0 such that for each
w ∈ A satisfying ||w − w|| < either w ∈ S or w is dominated by some w ∈ S
such that ||w − w|| < .
Local internal stability is clearly weaker than internal stability. Relative to external stability, local external stability is weaker in one respect but stronger in another. It is weaker
in that large perturbations of an allocation w ∈ S need not be dominated by an allocation
in S, and stronger in that a small perturbation of w must be dominated by an allocation in S
that is near w. The following result states that local stability is only possible at tyrannical
allocations that are strictly undominated, in the sense that ({i}, ei ) > (I \ {i}, ei ). Note
that, as a formal matter, the empty set is locally stable, which simplifies the statement of
the result.
5.2 Theorem. If S is locally stable then S ⊂ {ei : ({i}, ei ) > (I \ {i}, ei )}. If is
continuous then any subset of {ei : ({i}, ei ) > (I \ {i}, ei )} is locally stable.
Proof. Let S be a locally stable set and let w ∈ S. First, suppose by way of contradiction
that w = ei for all i. Renumbering the players, if necessary, we can assume that 1 >
w1 w2 > 0. Let q > w12 and let w q = (w1 + q1 , w2 − q1 , w3 , . . . , wn ) and wq =
(w1 − q1 , w2 + q1 , w3 , . . . , wn ). If w dominates w q then (p.2) implies that w q dominates w.
By the same reasoning, if w dominates w q then w q dominates w. Hence w fails to dominate
either w q or w q or both. If w does not dominate w q , let w q = w q ; otherwise let w q = wq .
Local external stability (LES) implies that for all q sufficiently large, there is some w q ∈ S
such that wq dominates w q and w q → w. By the previous argument, w q = w for all q,
so the set {wq }q is infinite. Local internal stability (LIS) implies that for some Q > 0, the
set {w q }q Q satisfies internal stability (IS). However, Theorem 2.9 then implies that the
set {wq }q Q is finite. This contradiction proves that w = ei for some i, say i = 1. Now
suppose by way of contradiction that ({1}, e1 ) (I \ {1}, e1 ). Then (p.2) implies that for
1
1
, . . . , (n−1)q
), so the
any q > 1, e1 does not dominate the allocation w q = (1 − q1 , (n−1)q
same reasoning as above contradicts this possibility, which proves the first assertion.
To prove the second assertion, suppose that is continuous and ({i}, ei ) > (I \ {i},
i
e ). The set {w : ({i}, w) > (I \ {i}, w)} is open, so there is some > 0 such that
ei dominates w for every w = ei satisfying ||w − ei || < . Hence the singleton {ei }
is a locally stable set, as is any set of tyrannical allocations ej satisfying ({j }, ej ) >
(I \ {j }, ej ). Theorem 5.2 is easily illustrated by two of the games studied above. Consider the wealthis-power game, w (C, w) = i∈ wi , of Section 3. In the case of four players, for example, the
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
41
only stable allocation that dominates the perturbed allocation ( 21 +, 21 −, 0, 0) is the tyrannical allocation (1, 0, 0, 0); and the only stable allocations that dominate ( 41 +, 41 , 41 , 41 −)
are ( 21 , 21 , 0, 0) and ( 21 , 0, 21 , 0). Suppose that allocations in this game are repeatedly subject
to a small exogenous perturbation followed by pillage to a stable allocation. It is easy to
see that if the small perturbations are sufficiently random, then eventually the allocation
will remain near one of the tyrannical allocations. For the three-player game, v (C, w) =
v#C+i∈C wi for v > 1, of Section 4, the only stable allocation that dominates the perturbed
allocation ( 21 + , 21 − , 0) is (0, 21 , 21 ). In this game, tyrannical allocations are dominated,
so repeated perturbations would cause the allocation to fluctuate among the three stable
allocations.
6. The core in expectation
von Neumann and Morgenstern [16] interpret a stable set as a self-enforcing “standard
of behavior.” A stable allocation w may be dominated by an allocation w , but external
stability ensures that w is in turn dominated by a stable allocation w , and internal stability
implies that w does not dominate w. Thus players are dissuaded from moving to w by the
prospect of a subsequent move to w . However, Harsanyi [6] observes that the players who
are made better off by the move to w might be still further improved by the subsequent
move to w , so the expectations implicit in the definition of a stable set can actually be
destabilizing. This can be illustrated in the wealth-is-power game studied in Section 3.
With three players, the unique stable set consists of the allocations (1, 0, 0), ( 21 , 21 , 0),
( 21 , 41 , 41 ) and their permutations. From the stable allocation w = ( 21 , 41 , 41 ), player 1 can
pillage player 3 to achieve the nonstable allocation w = ( 43 , 41 , 0), which then allows
player 1 to pillage player 2 to achieve the stable tyrannical allocation w = (1, 0, 0).
Moreover, the move from w to w does not expose player 1 to any interim risk, since
player 1 has sufficient power at w to be immune to pillage by the other two players. Thus
player 1’s expectation of the second pillage only encourages the first. In this case, Harsanyi
argues that the stable allocation w indirectly dominates w, making w an implausible
outcome.
Harsanyi [6] addresses this problem by developing a bargaining process for games in
characteristic function form. Greenberg [5] and Chwe [3] introduce forward-looking solution concepts for general coalitional games. Chwe’s concept of largest consistent set, in
particular, excludes the stable allocation w because it is indirectly dominated by w .
The neutrality assumption, however, makes indirect domination deceptive. At the stable
allocation w, player 1 can pillage player 3 only because of the assumed neutrality of player
2. If player 2 also anticipates the second pillage from w to w , then player 2 will join player
3 in opposing the first pillage, effectively preventing it. The potential effect of expectations
on the neutrality assumption is embodied in the concept of domination in expectation,
defined below. The farsighted core is defined as the set of allocations that are undominated
in expectation. Theorem 6.4 shows that for general pillage games, if expectations satisfy a
consistency condition, then the concepts of stable set and farsighted core are equivalent. In
this sense, the expectations implicit in the concept of stability prevent pillage at any stable
42
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
allocation. Under the neutrality assumption, the stable allocation w is dominated by w and
indirectly dominated by the stable allocation w , but w is undominated in expectation due
to the farsighted opposition of otherwise neutral players.
6.1 Definitions. An expectation is a function f : A → A with the property that for each
w ∈ A, f (f (w)) = f (w). Given an expectation f, let Sf denote the set of stationary
allocations, that is, Sf = {w : f (w) = w}. Note that Sf = f (A).
An expectation f represents the common belief that if an allocation w were to occur, then
w will persist if f (w) = w; otherwise w will be replaced by the allocation f (w). The
condition that f (f (w)) = f (w) means that this belief is consistent in the sense that a move
from w to f (w) will not be followed by a further move. If w is a stationary allocation,
then any player i contemplating an act of pillage that moves the allocation to some w will
anticipate the subsequent move to f (w ), and evaluate the proposed pillage by comparing
wi to f (w )i rather than wi .
6.2 Definitions. Let f be an expectation and let be a power function. An allocation w dominates an allocation w in expectation if (Wf , w) > (Lf , w), where Wf = {i :
f (w )i > wi } and Lf = {i : f (w )i < wi }. Define the farsighted core as the set of
allocations that are undominated in expectation, that is, Kf = {w : no w ∈ A dominates
w in expectation}.
Given any expectation f, the stationary set Sf is nonempty since Sf = f (A). The
farsighted core Kf is easily seen to be the set of all allocations that are not dominated,
in the conventional sense, by any stationary allocation. For example, if f is the identity
function, then Sf = A and Kf is the conventional core. In this example, the stationary
set is implausibly large due to the arbitrary nature of the expectation f. In particular, any
allocation that is dominated in expectation invites pillage and should not be expected to be
stationary. Conversely, any allocation that is undominated in expectation discourages pillage
and should be stationary. This additional consistency property is formalized as follows.
6.3 Definition. An expectation f is consistent if for each w ∈ A, either
(i) f (w) dominates w in expectation; or
(ii) w is undominated in expectation and f (w) = w.
Consistency is a rational expectation property. If an expectation is shared by all players,
then at any allocation w, an act of pillage is rational if it results in an allocation that dominates
w in expectation. An expectation is consistent if only rational acts of pillage are expected,
and an allocation is expected to persist only if no rational pillage is possible. Theorem 6.4
states that a set of allocations is stable if and only if it is the farsighted core for a consistent
expectation.
6.4 Theorem. If f is a consistent expectation, then Sf = Kf . A set of allocations, S, is
stable if and only if there exists a consistent expectation f such that S = Sf = Kf .
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
43
Proof. The first assertion is an immediate consequence of Definition 6.3. To prove the
second assertion, suppose that f is a consistent expectation satisfying S = Kf = Sf . Let
w ∈ S. Since w ∈ Kf , w is undominated in expectation, so w is undominated by any
w ∈ Sf , and hence w is undominated by any w ∈ S. Therefore S is internally stable. Let
w∈
/ S. Then w ∈
/ Kf , so w is dominated in expectation by some w . Thus w is dominated
by f (w ) ∈ Sf = S, so S is externally stable.
Let S be a stable set. Define an expectation f as follows. For each w ∈ S, let f (w) = w;
and for each w ∈
/ S, let f (w) = w for some w ∈ S that dominates w. The requisite
w exists because S is externally stable. Then Sf = S. Therefore, for each w ∈
/ S, f (w)
dominates w in expectation. Since S is internally stable and f (A) = S, each w ∈ S is
undominated in expectation. Therefore f is a consistent expectation with S = Sf , which
completes the proof. Given a stable set S, the expectation f constructed in the proof of Theorem 6.4 embodies the von Neumann–Morgenstern interpretation of a stable set. For w ∈ S, f (w) = w
since w is expected to persist. For w ∈
/ S, f (w) can be any allocation in S that dominates w. The selection of f (w) in this case determines which act of pillage is expected to
move the allocation into the stable set. For any such f, the stable set coincides with the farsighted core. Thus a stable set derived under the neutrality assumption gives rise to expectations that modify neutrality so as to make the stable allocations undominated in expectation.
For some acts of pillage at nonstable allocations, neutrality is not modified, in which case
domination and domination in expectation coincide. For example, in the three-player wealthis-power game, the nonstable allocation w = ( 21 , 13 , 16 ) allows player 2 to pillage player
3 to attain the stable allocation w = ( 21 , 21 , 0). Since w is expected to persist, player 1
remains neutral. Thus the concept of domination in expectation constitutes an endogenous
modification of the neutrality assumption.
The assumption that all players have the same expectation is obviously essential. It is less
obvious but also essential that the expectation is nonstochastic. Consider the wealth-is-power
game with four players. At the stable allocation w = ( 41 , 41 , 41 , 41 ), players 1, 2 and 3 can
collude to pillage player 4 and attain the nonstable allocation w = ( 13 , 13 , 13 , 0). The expectation f (w ) = ( 21 , 21 , 0, 0), for example, prevents this pillage by causing player 3 to oppose
it. Suppose instead that the players expect that w will be succeeded by an equiprobable random draw among the dominating stable allocations ( 21 , 21 , 0, 0), ( 21 , 0, 21 , 0) or (0, 21 , 21 , 0).
Then the expected-wealth vector resulting from the move to w is w itself. This implies
that if players are risk-neutral, the stable allocation w is dominated in expectation by w .
In either case, players 1, 2 and 3 know that if they collude to pillage player 4, then two of
them will subsequently collude to betray the other. If the identity of the eventual victim of
betrayal is known, as is the case under the expectation f, then the pillaging coalition {1,
2, 3} will not form. However, if the victim will be chosen by lot, the players will collude
and take their chances.
Theorem 6.4 does not extend to general coalitional games. Chwe [3] gives
some simple examples in which the concept of indirect domination is both compelling
and inconsistent with stability. The concept of domination in expectation relies
directly on the power function, which is the defining characteristic of a pillage
game.
44
J.S. Jordan / Journal of Economic Theory 131 (2006) 26 – 44
Theorem 6.4 supports the interpretation of a stable set as an endogenous balance of power.
Stable sets can be significantly larger than the core, which gives some encouragement to
the view that property rights can arise endogenously, without an external enforcement
mechanism. Unfortunately, as has been shown above, stable sets are necessarily finite, and
stable allocations are typically sensitive to small perturbations. These shortcomings cast
Theorem 6.4 as a somewhat negative result, since it implies that any set of allocations that is
stationary under consistent expectations must be a stable set. However, the dynamics implicit
in the definition of an expectation are quite restrictive. The expected allocation can depend
only on the current allocation. Thus expectations cannot depend on the past, and cannot
depend on whether particular players have previously engaged in pillage, which clearly
limits the possible uses of power to enforce property rights by punishing violators. The
extension of the model to a richer dynamic setting might well expand the set of enforceable
allocations. Of course, such extensions must still face the problem that any act of pillage
increases the power of the pillagers.
Acknowledgments
I am grateful to Michael Jung as well as an associate editor and two anonymous referees
for helpful comments that have improved this draft.
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