Games and Economic Behavior 44 (2003) 379–389
www.elsevier.com/locate/geb

The weak core of simple games with ordinal
preferences: implementation in Nash equilibrium
Tomoichi Shinotsuka a,∗ and Koji Takamiya ba Department of Economics, Otaru University of Commerce, 3-5-21, Midori, Otaru 047-8501, Japanb Institute of Social and Economic Research, Osaka University, 6-1, Mihogaoka, Ibaraki,

Osaka 567-0047, Japan
Received 11 January 2002

Abstract
In a simple game, coalitions belonging to a given class are “absolutely powerful” while others
have no power. We attempt to make this distinction operational. Toward this end, we propose two
axioms on social choice correspondences, Strong Non-Discrimination and Exclusion. Strong NonDiscrimination describes circumstances under which certain coalitions, the losing coalitions, have noinfluence over social choice. Exclusion requires that there are situations in which certain coalitions,
the winning coalitions, can exercise their power. We show that the weak core correspondence isthe minimal correspondence satisfying Maskin Monotonicity and Strong Non-Discrimination. We
also show that the weak core is the unique correspondence satisfying Nash implementability, Strong
Non-Discrimination, and Exclusion.
 2003 Elsevier Inc. All rights reserved.
JEL classification: C71; C72; D71; D78

0. Introduction
This paper studies Nash-implementation of social choice correspondences (SCC for
short) on the class of simple games with ordinal preferences. In a simple game, coalitions
belonging to a given class are “absolutely powerful” while others have no decision
power. In this paper, we attempt to make this distinction operational by proposing two
axioms on social choice correspondences, Strong Non-Discrimination and Exclusion.
* Corresponding author.

E-mail address: shino@res.otaru-uc.ac.jp (T. Shinotsuka).
0899-8256/03/$ – see front matter  2003 Elsevier Inc. All rights reserved.
doi:10.1016/S0899-8256(03)00055-1

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T. Shinotsuka, K. Takamiya / Games and Economic Behavior 44 (2003) 379–389

Strong Non-Discrimination describes circumstances under which certain coalitions, the“losing” coalitions, have no influence over social choice: for any such coalition and any
two alternatives, if all the individuals outside the coalition are indifferent between thesealternatives, then the SCC should not discriminate between them, in the sense that one
alternative is chosen, i.e., contained in the image of the SCC, if and only if the otheralternative is chosen. Strong Non-Discrimination is a strengthening of a condition used indifferent contexts (Thomson, 1987; Gevers, 1986; and Nagahisa, 1991, 1994). Exclusion
requires that there are situations in which certain coalitions, the “winning” coalitions,can exercise their power: for any such coalition, if all the members of the coalition haveidentical, nontrivial preferences (in the sense that at least two alternatives are not judged
indifferent), and, furthermore, all the members of the complementary coalition have trivial
preferences (for them all the alternatives are indifferent), then the coalition has the powerof excluding at least one alternative.
Some remarks are in order on Strong Non-Discrimination and Exclusion. The former
cannot be interpreted as saying that losing coalitions have no power. In fact, it tells us
nothing if some member of the complementary coalition finds one alternative preferable
to another. Similarly, Exclusion demands that a SCC should grant a winning coalition a“right” to exclude some alternative only in the very special case when all of its members
have common preferences and all the members of the complementary coalition regard allalternatives indifferent. Therefore, Exclusion does not seem to grant unlimited power to thewinning coalitions. However, we will see that Strong Non-Discrimination and Exclusion
become considerably stronger when the important axiom of Maskin Monotonicity is
imposed as well.
Recall that a SCC satisfies Maskin Monotonicity if it preserves the desirability ofan allocation under transformations of preferences that raise the relative ranking of
the allocation. Maskin Monotonicity and Strong Non-Discrimination imply that losing
coalitions have no veto power (Lemma 3.10). Furthermore, these three axioms together
imply that winning coalitions are “all-powerful” and losing coalitions are “completely
powerless” (Corollary 3.6).
Recall the distinction between the strong core and the weak core of simple games. An
alternative is in the strong core of a simple game if there exist no winning coalition andanother alternative that is at least as good for all members of the winning coalition andstrictly preferred by some member of the winning coalition. An alternative is in the weakcore if there exist no winning coalition and another alternative that is strictly preferred by
all members of the winning coalition. The strong core satisfies Strong Non-Discriminationand Exclusion but violates Maskin Monotonicity (see Remark 3.7). Hence, the distinction
between the weak core and the strong core is critical.
We show that the weak core correspondence is the minimal correspondence satisfying
Maskin Monotonicity and Strong Non-Discrimination. This is the central result in thispaper but, strictly speaking, it is not a full characterization of the weak core. Toward thisend, we work with a domain on which the weak core is nonempty. Then, we show thatthe weak core is the unique correspondence satisfying Maskin Monotonicity, Strong NonDiscrimination, and Exclusion. It is well known that Maskin Monotonicity is necessary

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for Nash implementability. It turns out that the weak core is the unique correspondence
satisfying Nash implementability, Strong Non-Discrimination, and Exclusion.1
Maskin monotonicity delivers a striking result to social choice functions (singletonvalued social choice correspondences). Indeed, Saijo (1987) showed that a social choicefunction satisfying a “dual dominance” condition is Maskin monotonic if and only if it isconstant. As Saijo remarked, a social choice correspondence satisfies dual dominance if
the domain of the correspondence contains the profile of trivial preferences. In this paper,
we require that the domain of social choice correspondences satisfy a certain “richness”
condition that implies that the profile of trivial preferences belongs to the domain. The
weak core correspondence is Maskin monotonic but the weak core of the profile of trivial
preferences is equal to the set of all alternatives. Hence the weak core correspondence
is not singleton-valued. Therefore, Saijo’s theorem does not apply to the weak core
correspondence.
Implementability of the core has been investigated in different contexts.2 Kara and
Sönmez (1996) studied two-sided matching problems. They showed that the weak core
correspondence is Nash implementable by means of Danilov’s (1992) and Yamato’s (1992)
results. We employ the same technique to show that the weak core is Nash implementablein our framework, too. Unlike Kara and Sönmez, we can also apply Maskin’s result on
Nash implementability under the additional assumption that every singleton coalition islosing (Theorem 3.11). Sönmez (1996) studied generalized matching problems including
housing markets and marriage problems. He generalized the results of Kara and Sönmez
to the class of generalized matching problems.
Another strand of the literature deals with mechanisms implementing the corein economic environments. Wilson (1978) studied standard exchange economies. He
proposed a two-stage, competitive bidding game in which one subgame perfect equilibriumoutcome is always in the weak core of the exchange economy. The unfortunate partof Wilson’s mechanism is that it only partially implements the weak core. Kalai
et al. (1979) studied public good economies. They defined a strategic form mechanismthat fully implements the weak core of the exchange economy in strong equilibrium.
The mechanism, however, implements the individually rational correspondence in Nashequilibrium. Serrano and Vohra (1997) worked with private ownership economies
including public goods. They constructed an extensive form mechanism that implements
the weak core in subgame prefect equilibrium.
The plan of the paper is as follows: The next section provides preliminary definitions.
Section 2 introduces the axioms for SCC’s. Section 3 states the main results. And Section 4
gives concluding remarks.

1 An anonymous referee suggested calling coalitions in the given class “privileged.” Following this suggestion,
we summarize our results as follows. Implementability implies that a “privileged” coalition must be “winning”(all-powerful) after all.
2 For excellent surveys on implementation problems, see Maskin (1985), Moore (1992), and Jackson (2001).

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1. Notations and definitions
We fix the set X of (social) alternatives (or outcomes), and the finite set N of individuals. Denote by  the set of complete and transitive binary relations on X. Let i ∈ N , andR i ∈ . We denote by R i individual i’s preference relation. For x, y ∈ X, xR i y meansthat to individual i, x is at least as good as y. We denote the asymmetric part of R i by P i ,and the symmetric part by I i . We say that R i ∈  is trivial if individual i is indifferent
among all alternatives, that is, for all x, y ∈ X, xI i y. Denote by R0i individual i’s trivial
preference relation. We call R = (R i )i∈N ∈ N a preference profile. The trivial preference
profile R0 consists of the trivial preference relation of each individual, i.e., R0 = (R0i )i∈N .
1.1. Simple games
Let W be a nonempty subset of 2N − {φ}. We call a subset of N a coalition. A coalitionin W is a winning coalition and a coalition not in W is a losing coalition. Let R be apreference profile. For simplicity, we fix W from now on. Hence, we identify a simple gamewith a preference profile. Let x, y ∈ X and let S ⊂ N . We say that x strongly dominates
y via S if S is winning and for all i ∈ S, xP i y. We say that x weakly dominates y via S
if S is winning, x is at least as good as y for every individual in S, and x is preferred bysome individual in S. The weak core of a simple game R, denoted by C (R), is the set of
all alternatives that are not strongly dominated by any other alternative. The strong coreof a simple game R, denoted by SC (R), is the set of all alternatives that are not weakly
dominated by any other alternative.
1.2. Implementation
For each i ∈ N , let D i be a nonempty subset of , interpreted as the set of admissible
preferences of individual i. For a coalition S, D S denotes the Cartesian product i∈S D i .
A social choice correspondence (henceforth, SCC) is a nonempty-valued correspondence
ϕ : D N → X. We call D N the domain of ϕ. For R i ∈  and x ∈ X, let L(R i , x) ={y ∈ X | xR i y}. For preference profiles R and R , call R a monotonic transformationof R at x if for all i ∈ N : L(R i , x) ⊂ L(R i , x). Denote by MT(R, x) the set of all
monotonic transformations of R at x. Let ϕ be a SCC. In order to freely perform monotonic
transformations, the domain D N should satisfy a certain “richness” condition. That is,
we say that D N is closed under monotonic transformations (or simply, closed) if for all
R ∈ D N and x ∈ X, MT(R, x) ⊂ D N . Under this assumption, the domain D N contains thetrivial profile R0 because ϕ is nonempty-valued, and R
0 is a monotonic transformation of
any R at any x.3 A game form is a list (S, g) with S = i∈N S i , where each S i is a strategy
space for individual i, and g : S → X is an outcome function. A game is a list (S, g, R),
where (S, g) is a game form, and R ∈ N . A strategy profile s ∈ S is a Nash equilibrium of
the game (S, g, R) if there is no i ∈ N such that for some t i ∈ S i : g(t i , s −i )P i g(s). Denoteby NE((S, g, R)) the set of Nash equilibria of the game (S, g, R). Let ϕ be a SCC. Then
3 The richness condition has another implication. See Appendix A.

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the game form (S, g) Nash-implements ϕ if for all R ∈ D N : ϕ(R) = g(NE((S, g, R))).
Say that ϕ is Nash-implementable if there exists a game form which Nash-implements ϕ.
Necessary and sufficient conditions for Nash-implementation have been investigated indetail. A fundamental property is as follows. Let ϕ be a SCC.
Maskin Monotonicity (MMON): For all R and R ∈ D N , if x ∈ ϕ(R) and R ∈ MT(R, x),then x ∈ ϕ(R ).
Maskin (1999) proved that if a SCC is Nash-implementable, then it satisfies MMON.
The converse of this claim does not hold true. Danilov (1992) and Yamato (1992)
developed a sharper condition.4 Let i ∈ N and let Y ⊂ X. Let us denote Ess(i; ϕ; Y ) ={y ∈ Y | there exists R ∈ D N such that y ∈ ϕ(R) and L(R i , y) ⊂ Y }.
Essential Monotonicity (EMON): For all R and R ∈ D N , if x ∈ ϕ(R) and for all i ∈ N ,Ess(i; ϕ; L(R i , x)) ⊂ L(R i , x), then x ∈ ϕ(R ).
Yamato (1992) proved that(i) if |N|  3, any SCC satisfying EMON is Nash-implementable, and(ii) under a certain mild condition imposed on admissible preferences, any Nashimplementable SCC satisfies EMON.

2. New axioms for social choice correspondences
In this section, we propose new axioms that describe how SCC’s depend on the class of
winning coalitions. Before we get to those, we start with a discussion of a standard axiom.
The first axiom says that if all individuals are indifferent between two alternatives, thenthe SCC should not treat the alternatives differently.
Non-Discrimination (ND): For all R ∈ D N , x, y ∈ X, if for all i ∈ N , xI i y, then x ∈ ϕ(R)if and only if y ∈ ϕ(R).
The axiom appears in axiomatic characterizations of the Walrasian correspondence (seeThomson (1987), Gevers (1986), and Nagahisa (1991, 1994)). We will see that ND isalso useful in the context of axiomatic studies of the weak core of simple games (seeLemma 3.4). We also consider a strengthening of ND to obtain further results.
The next axiom says that if all the individuals outside a losing coalition find twoalternatives indifferent, then the SCC should not treat the alternatives differently.
Strong Non-Discrimination (SND): For all R ∈ D N , x, y ∈ X, and S ⊂ N , if for all/ W, then x ∈ ϕ(R) if and only if y ∈ ϕ(R).
i ∈ N\S, xI i y, and S ∈4 For other refined conditions of Maskin monotonicity, see, e.g., Maskin (1985), Williams (1986), Repullo(1987), Saijo (1988), McKelvey (1989), Moore and Repullo (1990), Sjöström (1991), and Ziad (1998).

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In other words, the SCC treats two alternatives symmetrically, independently of thepreferences of the members of the losing coalition, as long as all the members of thecomplementary coalition are indifferent between the alternatives. Hence, losing coalitions
have no power then. However, SND is far from saying that losing coalitions are absolutely
powerless. This is because SND is not applicable if two people outside a losing coalition
have different preferences over the two alternatives.
The connection between ND and SND is straightforward. Since W does not containthe empty set, SND implies ND. The converse does not hold (for example, the Pareto
correspondence violates SND but does satisfy ND).
The next axiom says that a winning coalition can exclude at least one alternative if all
the members of the coalition have identical nontrivial preferences and all the members of
the complementary coalition are indifferent among all the alternatives.
Exclusion (EX): For all S ∈ W and R S ∈ D S , if for all i, j ∈ S, R i = R j , and for all i ∈ S,
R i = R0i , then ϕ(R S , R0−S ) = X.
The axiom cannot be interpreted as saying that winning coalitions are all-powerful forthe following reasons. First, excluding at least one alternative is not equivalent to picking
the best alternative under the foregoing circumstances. Second, if some member of thecoalition N\S has nontrivial preferences, EX does not apply.

3. Main results
In this section, we assume that every profile in the domain D N of SCC’s underconsideration has a nonempty weak core and that D N is closed under monotonic
transformations. We start with the following important observation.
Lemma 3.1. The weak core correspondence satisfies MMON.
/ C(R ).
Proof. Let R, R be two profiles and x ∈ X be such that R ∈ MT(R, x) and x ∈Thus under R , x is dominated by some other y ∈ X via some S ∈ W . That is, for all i ∈ S,
yP i x. Since R ∈ MT(R, x), this implies for all i ∈ S, yP i x. Thus x is dominated by yunder R. But this says that x ∈/ C(R). Therefore, C satisfies MMON. ✷Theorem 3.2. If a SCC ϕ satisfies MMON and SND, then ϕ is a supercorrespondence of
the weak core C , that is, for any R ∈ D N , ϕ(R) ⊃ C(R).
Proof. Let x ∈ X and R ∈ D N be such that x ∈ C(R). Suppose that x ∈/ ϕ(R). Lety ∈ ϕ(R), T − = {i ∈ N | xR i y}, and T + = {i ∈ N | yP i x}. Let R be the profile obtainedby raising y to the same indifference class as x and keeping the other indifference classes
intact for each member of T − : For all i ∈ T − , xI i y and for all v, w ∈ X\{y}: vR i w if andonly if vR i w and vI i w if and only if vI i w, and for all i ∈ T + , R i = R i .
Note that R ∈ MT(R, x). Since D N is closed, R ∈ D N . By MMON, y ∈ ϕ(R ).
Further, since L(R i , x) = L(R i , x) for each i ∈ N , we have R ∈ MT(R , x). Again, by

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MMON, x ∈/ ϕ(R ). Thus ϕ treats x and y differently under R . Suppose that T + is losing.
/ ϕ(R ), which is a contradiction. Hence, T +Then, by SND either x, y ∈ ϕ(R ) or x, y ∈is winning. Thus under R , y dominates x via T + . This says that x ∈/ C(R ). Note thatR ∈ MT(R, x) since L(R i , x) = L(R i , x) for all i ∈ N . Lemma 3.1 tells us that C satisfies
MMON. Thus x ∈/ C(R). This is a contradiction. ✷Remark 3.3. In Theorem 3.2 above, the set inclusion may be strict. Consider the SCC thatchooses the whole set of alternatives X for every profile. This SCC satisfies both MMON
and SND.
Lemma 3.4. If a SCC ϕ satisfies MMON, ND, and EX, then ϕ satisfies the following
property:() For all S ⊂ N , R S ∈ D S , and x ∈ X, if S ∈ W and for all i ∈ S, y, z ∈ X\{x}, xP i y
and yI i z, then ϕ(R S , R0−S ) = {x}.
Proof. Let S ∈ W and x ∈ X. Let R be a profile such that for all i ∈ S, y, z ∈ X\{x}, xP i y
and yI i z, and for all i ∈ N\S, R i = R0i .
Let R ∗ ∈ D N and v ∈ ϕ(R ∗ ). Since R0 ∈ MT(R ∗ , v) and D N is closed, R0 ∈ D N . Thenby MMON, v ∈ ϕ(R0 ). Since x is indifferent to v for any i ∈ N at R0 , by ND x ∈ ϕ(R0 ).
Note that R ∈ MT(R0 , x). Then since D N is closed, R ∈ D N . Then, by MMON, x ∈ ϕ(R).
On the other hand, by EX, ϕ(R) = X. That is, there is some alternative that does not belongto ϕ(R). By construction, under R, any two alternatives other than x are indifferent forany member of N . Thus ND implies that any alternative other than x is excluded from
ϕ(R). ✷
Theorem 3.5. If a SCC ϕ satisfies MMON, ND, and EX, then ϕ is a subcorrespondence of
the weak core, that is, for any R ∈ D N , ϕ(R) ⊂ C (R).
/ C (R). Then there exists
Proof. Let x ∈ X and R ∈ D N . Assume that x ∈ ϕ(R), but x ∈S ∈ W and y ∈ X such that for all i ∈ S, yP i x. Let R be a profile such that for all i ∈ S,v, w ∈ X\{y}, yP i v and vI i w, and for all i ∈ N\S, R i = R0i .
Note that R ∈ MT(R, x). Then by MMON, we have x ∈ ϕ(R ). By Lemma 3.4,
ϕ(R ) = {y}. This implies x = y, a contradiction. ✷The following result is immediate from the foregoing theorems.
Corollary 3.6. W is the unique SCC that satisfies MMON, SND, and EX.
Remark 3.7. The above Corollary 3.6 characterizes the weak core of simple games.
This result is tight in the sense that the three properties MMON, SND, and EX are
logically independent. The example in Remark 3.3 establishes the independence of EX.
For the independence of SND, fix x ∈ X and let ϕ(R) = {x} for all R. Then, ϕ obviously
satisfies MMON and EX but violates SND. To show the independence of MMON, notethat SC satisfies SND and EX but violates MMON. To illustrate a violation of MMON,

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let X = {x, y}, N = {1, 2}, W = {N}, xP1 y, yP2 x, xI1 y, yP2 x. Then, x ∈ ϕ(R), R ∈MT(R, x), but x ∈/ ϕ(R ).
Theorem 3.8. Assume that |N|  3. Then C is Nash-implementable.
Proof. In view of Yamato (1992), it suffices to show that C satisfies EMON. Suppose
otherwise. Then,(∗) there exist R, R ∈ D N such that x ∈ C(R) and for all i ∈ N , Ess(i; C; L(R i , x)) ⊂L(R i , x) but x ∈/ C(R ).
/ C (R )) in (∗) above implies
Since C satisfies MMON (Lemma 3.1), (x ∈ C (R) and x ∈/ MT(R, x). Hence, there exists i ∈ N such that L(R i , x)\L(R i , x) = φ. Letthat R ∈z ∈ L(R i , x)\L(R i , x). By (∗), z ∈/ L(R i , x) implies z ∈/ Ess(i; C; L(R i , x)). Therefore,
for some i ∈ N ,/ C(R ).
(∗∗) for all R ∈ D N , if L(R i , z) ⊂ L(R i , x), then z ∈Now let R ∗ be a profile such that for all j ∈ N , xI ∗j z and for all v and w ∈ X\{z},vR ∗j w if and only if vR j w.
Thus for each j ∈ N , we have L(R j , x) ∪ {z} = L(R ∗j , x). This implies L(R j , x) ⊂L(R ∗j , x). Thus, R ∗ ∈ MT(R, x). Since x ∈ C(R) and C satisfies MMON, we havex ∈ C(R ∗ ). Since L(R i , x) contains z by assumption, L(R i , x) = L(R ∗i , x). Since xI ∗i z
by construction, L(R ∗i , z) = L(R ∗i , x). Therefore L(R ∗i , z) = L(R i , x). This equality and(∗∗) together imply z ∈/ C(R ∗ ). On the other hand, x ∈ C (R ∗ ) and for all j ∈ N , xI ∗j z.
∗
Hence, z ∈ C(R ). This is a contradiction. ✷Now we conclude the following.
Theorem 3.9. Assume that |N|  3. Then C is the unique SCC satisfying Nash
implementability, SND and EX.
We discuss the logical relations between Maskin’s ‘no veto power’ condition and SND
and EX. Let ϕ denote a SCC. Let S be a coalition. Say that S has veto power if there are/ ϕ(R) and for all j ∈ N\S, L(R j , x) = X. Maskinsome R ∈ D N and x ∈ X such that x ∈(1999) introduced the following definition.
No Veto Power (NVP): For all i ∈ N , R ∈ D N , and x ∈ X, if for all j ∈ N\{i}, L(R j , x) =X, then x ∈ ϕ(R).
It directly follows from the definition that if a SCC satisfies EX, then every winningcoalition has veto power.
Lemma 3.10. If a SCC ϕ satisfies MMON and SND, then no losing coalition has vetopower.

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Proof. Let ϕ satisfy MMON and SND. Let S be a losing coalition. Suppose that S has
veto power. Then there are R ∈ D N and x ∈ X such that for all j ∈ N\S, L(R j , x) = X
and x ∈/ ϕ(R). Since ϕ satisfies SND, S ∈/ W implies ϕ(R S , R0−S ) = X. Since L(R j , x) =j
X = L(R0 , x), R ∈ MT((R S , R0−S ), x). Thus MMON implies x ∈ ϕ(R). This contradictsx∈/ ϕ(R). ✷Maskin (1999) proved that if a SCC satisfies both MMON and NVP, and |N|  3, thenit is Nash implementable. Combining Lemma 3.10 with Maskin’s result, we obtain thefollowing.
Theorem 3.11. Assume |N|  3. Let ϕ be a SCC. Assume that there does not exist i ∈ Nsuch that {i} ∈ W . Then if ϕ satisfies MMON and SND, then ϕ is Nash-implementable.
Remark 3.12. Theorems 3.2 and 3.11 immediately imply the following: Let there be atleast three individuals. Assume there does not exist a winning coalition consisting of only
one individual. Then if a SCC satisfies MMON and SND, then it is Nash implementable
and it is a supercorrespondence of the weak core.

4. Concluding remarks
Though we require SCC’s to be nonempty-valued, this requirement played little rolein the foregoing arguments. Though the nonemptiness requirement has some conceptual
appeal, there is a room for applying Occam’s Razor. In fact, in axiomatic studies, someauthors (see footnote 5, for example) do not insist on this requirement but explore how far
they can go formally without it. In this section, we take this view.
In this section, we assume that the domain D N contains at least one simple game R
whose weak core is nonempty.
Let ϕ be a correspondence from D N to X, possibly empty valued for some R ∈ D N .
We introduce the following property:
Restricted Nonemptiness (RNEM):5 For all R ∈ D N : C(R) = φ implies ϕ(R) = φ.
Theorem 4.1. The weak core is the unique correspondence satisfying MMON, SND, EX,
and RNEM.
Proof. By the foregoing arguments, the weak core C satisfies these four properties. We
shall prove uniqueness. First, it is easy to verify that Theorem 3.2 still holds even if thenonempty-valuedness of ϕ is weakened to the assumption that ϕ satisfies RNEM. Thisestablishes ϕ ⊃ C . Second, Theorem 3.5 is also true without nonempty-valuedness or
RNEM. And in that case, nonempty-valuedness of the weak core C becomes superfluous.
5 Similar properties can be found, for example, in Norde et al. (1996), and Peleg et al. (1996). These papers
study axiomatizations of solutions to strategic form games.

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Recall that SND implies ND. Then all conditions for Theorem 3.5 are met. This provesϕ ⊂ C. ✷Remark 4.2. The properties appearing in Theorem 4.1 are logically independent. By
adapting the examples in Remark 3.7 in a straightforward manner, we can show that each of
MMON, SND, and EX is independent of the other three properties. The following example
shows RNEM is independent of the other three properties: Let ϕ be the solution such that
ϕ(R) = φ for any R ∈ D N . Then ϕ satisfies MMON, SND, and EX but violates RNEM.

Acknowledgments
The first author is very grateful to William Thomson for helpful suggestions andcomments, and to the faculty and staff members, especially Lionel W. McKenzie, of the
Department of Economics at the University of Rochester for their hospitality. The second
author thanks Hiroshi Ono for encouragement. The authors thank an associate editor ofthis journal and an anonymous referee for comments that greatly improved this paper. Allerrors are our own responsibility.
Appendix A
An anonymous referee pointed out an interesting implication of the richness condition that we imposed on thedomain D N . That is, if the domain D N is closed under monotonic transformation, then it includes all preference
profiles comprising preferences with top-ranked alternatives. Clearly, if the set of alternatives is finite, D N isunrestricted. The following proof is due to the referee.
Let R = (R 1 , . . . , R n ) be any arbitrary profile comprising of preferences with top-ranked alternatives,and let, for each i, x i denote the top-ranked alternative for individual i. We know that the trivial profile R0−{1}

belongs to D N . Let (R 1 , R0

) be a profile such that only player 1 has a (possibly) nontrivial preference R 1 .

−{1}
−{1}
Since (R 1 , R0 ) is a monotonic transformation of R0 at x 1 , (R 1 , R0 ) belongs to D N . Similarly,−{1,2}−{1}−{1,2}(R 1 , R 2 , R0
) is a monotonic transformation of (R 1 , R0 ) at x 2 , so (R 1 , R 2 , R0
) belongs to D N .
−{1,2,...,n−1}) belongs to D N . Since R = (R 1 , . . . , R n ) is a
By induction hypothesis, (R 1 , R 2 , . . . , R n−1 , R0−{1,2,...,n−1}
monotonic transformation of (R 1 , R 2 , . . . , R n−1 , R0
) at xn , it belongs to D N .

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