Preference revelation games and strong cores ofallocation problems with indivisibilities∗
Koji TAKAMIYA†
Economics, Niigata University
August 14 2008

Abstract
This note considers the equilibrium outcomes of the preference revelation gamesin the general model of indivisible good allocation introduced by Sönmez (1999). We
adopt the concepts of coalitional equilibria and cores which are both defined in terms of
the weak deviation or blocking by a prescribed class of admissible coalitions. We provethat if the solution which induces preference revelation games is individually rationaland Pareto optimal and the class of admissible coalitions is “monotonic,” then the setof coalitional equilibrium outcomes coincides with the core. And we point out thatthe preceding analysis in the context of marriage problems (Gale and Shapley, 1962)is hardly extended to the general model.
JEL Classification— C71, C72, C78, D71, D78.
Keywords— generalized indivisible good allocation problem, preference revelation
game, strict G-proof Nash equilibrium, G-strong core.

1

Introduction

In this note we consider the equilibrium outcomes of the preference revelation games in
the general allocation model of indivisible goods introduced by Sönmez (1999). Thismodel includes, as special cases, some well-known problems such as marriage problems
and roommate problems (Gale and Shapley, 1962), housing markets (Shapley and Scarf,
1974), and hedonic coalition formation games (Banerjee, Konishi and Sönmez (2001) and
Bogomolnaia and Jackson (2002)).
In the context of marriage problems, the equilibria of preference revelation games havebeen studied extensively. Among those studies, Sönmez (1997) provides one of the mostgeneral results: The set of G-proof Nash equilibrium outcomes of the preference revelation
games induced by an individually rational and Pareto optimal solution coincides with
the G-weak core. Here G represents a class of coalitions, and these equilibrium and core∗This note is based on Chapter 4 of my dissertation submitted to Hokkaido University. I am grateful to
Tayfun Sönmez for introducing some of his works which motivated this research. I am indebted to Eiichi
Miyagawa, Ryo-ichi Nagahisa, Toyotaka Sakai, Takashi Shimizu, Ken-ichi Shimomura, Takuma Wakayama
for useful comments, and in particular to Hiroshi Ono and Tomoichi Shinotsuka for detailed comments
and discussion. I am very thankful to an anonymous referee for various comments which have significantly
improved the presentation of this paper. Part of this research has been done while I was a member of
ISER (Osaka University). I am deeply grateful for the excellent research environment of the institute.
And I thank seminar participants at Otaru University of Commerce, RIEB (Kobe University) and Kansai
University. All errors are my own responsibility.
†
Faculty of Economics, Niigata University, 8050 2-no-cho Ikarashi Nishi-ku Niigata JAPAN, 950-2181,
tel: +81-25-262-6498, E-mail: takamiya@econ.niigata-u.ac.jp.

1

concepts are defined in the fashion that the deviation or blocking power is restricted tothe coalitions belonging to G.
In this note we examine in the general setting the relationship between coalitional equilibria and cores similar to Sönmez’s result in marriage problems. We find that Sönmez’s
result is not generalized to the present model, but rather there are significant relationships
between the refined versions of his coalitional equilibrium and core concepts, namely thestrict G-proof Nash equilibrium and the G-strong core. These versions of the coalitional
equilibrium and the core are defined in terms of the weak notions of deviation or blocking
whereas Sönmez’s versions are defined by the strong notions. We prove that with theseconcepts of the equilibrium and the core, for any G, the equilibrium outcomes are contained in the core, and if G is monotonic, then the set of equilibrium outcomes and thecore coincides with each other.1 As a special case, our results imply that the set of strict
strong Nash equilibrium outcomes equals the strong core.2

2

Definitions

2.1

Model

We follow the definitions by Sönmez (1999). A generalized indivisible good allocation
problem (or a problem, simply) is a list (N, ω, Af , R). Here N is the nonempty finite
set of individuals. A coalition is a nonempty subset of N . For i ∈ N , ω(i) denotes theinitial S
endowment of individual i. Each ω(i) is a finite set. For S ⊂ N , ω(S) denotes
the set {ω(i) | i ∈ S}.3 An allocation is a correspondence x : N →→ ω(N ) satisfying(i) for any i, j ∈ N with i 6= j, x(i) ∩ x(j) = ∅, and (ii) x(N ) = ω(N ). Here the condition(i) means that no good can be owned by two or more individuals at the same time, andthe condition (ii) means no free disposal. Denote the set of all allocations by A0 . Af is
the set of feasible allocations, which is a nonempty subset of A0 satisfying ω ∈ Af .
R = (Ri )i∈N is a preference profile. Each Ri is a weak ordering on Af .4 xRi y meansthat to individual i, x is at least as good as y. As usual, I i and P i respectively denotethe symmetric (“indifferent”) and the asymmetric (“strictly prefers”) parts of Ri . For
B ⊂ Af , top Ri (B) denotes the set {x ∈ B | ∀y ∈ B : xRi y}.
For each i ∈ N , Di denotes the nonempty set
preferences of i. For a
Qof admissiblei
S
coalition S, D denotes the Cartesian product i∈S D . Let D denote DN , the domainof preferences. A solution is a single-valued function ϕ : D → Af .
One important example of preference domains is the domain P defined as follows: Foreach i ∈ N , let P i denote the set of all preference relations in which (i) individual i has
strict preferences over own assignments (i.e. xI i y ⇒ x(i) = y(i)),
Q andi (ii) there is noi
consumption externality (i.e. x(i) = y(i) ⇒ xI y). Let P denote i∈N P . In most of thestudies on marriage problems (Gale and Shapley, 1962) or housing markets (Shapley and
Scarf, 1974), the domain P is assumed.
1
Here that G is monotonic means that if a coalition S belongs to G, then any supercoalition of S also
belongs to G.
2
The strict strong Nash equilibrium is a variant of the strong Nash equilibrium and is defined in termsof the weak deviation.
3
Throughout the paper, inclusion “⊂” is weak.
4
A weak ordering is a complete and transitive binary relation.

2

2.2

G-weak and -strong cores

Let a problem (N, ω, Af , R) and a nonempty class of coalitions G be given. Let x, y ∈ Af ,and let S be a coalition. Then we say that x weakly dominates y via S under R if

x(S) = ω(S) & ∀i ∈ S : xRi y & ∃j ∈ S : xP j y .
(1)
The G-strong core is the set of the allocations which are not weakly dominated by anyother allocation via any coalition belonging to G. The G-strong core correspondence
on D is the correspondence C G : D →→ Af such that for each R ∈ D, C G (R) is theG-strong core of the problem (N, ω, Af , R). We say that x strongly dominates y via S
under R ifx(S) = ω(S) & ∀i ∈ S : xP i y .
(2)
The G-weak core is the set of the allocations which are not strongly dominated by anyother allocation via any coalition belonging to G. The G-weak core correspondence isdefined analogously and denoted by C W G . If G = 2N \ {∅}, then the G-strong core (G-weak
core, respectively) is called the strong core (weak core, respectively). Denote the strong
core correspondence by C, and the weak core correspondence by C W .
Call G monotonic if for any two coalitions T and S, if S ∈ G and S ⊂ T , then T ∈ G.
An allocation x is Pareto optimal under R if there is no allocation which weakly
dominates x via N . And x is individually rational under R if for any i ∈ N , xRi ω.
I(R) denotes the set of individually rational allocations under R. Note that any strong core
allocation is both individually rational and Pareto optimal whereas a weak core allocation
is individually rational but not necessarily Pareto optimal.
Call a solution ϕ Pareto optimal (individually rational, respectively) if for any
R ∈ D, the allocation ϕ(R) is Pareto optimal (individually rational, respectively) underR.

2.3

Preference revelation games

Let a list (N, ω, Af , D) and a solution ϕ : D → Af be given. Let R ∈ D. A preference
revelation game induced by ϕ is the strategic game (D, ϕ, R), where each Di is thestrategy space of individual i, and ϕ is the outcome function.
Let a game (D, ϕ, R) be given. Then a strategy profile R? ∈ D is a G-proof Nashequilibrium if for any S ∈ G and any R0S ∈ DS ,∀i ∈ S : ϕ(R?−S , R0S )Ri ϕ(R? ) ⇒ ∃j ∈ S : ϕ(R?−S , R0S )I j ϕ(R? ) .
(3)
And R? ∈ D is a strict G-proof Nash equilibrium if for any S ∈ G and any R0S ∈ DS ,∀i ∈ S : ϕ(R?−S , R0S )Ri ϕ(R? ) ⇒ ∀i ∈ S : ϕ(R?−S , R0S )I i ϕ(R? ) .
(4)
If G = 2N \ {∅}, then a G-proof Nash equilibrium (strict G-proof Nash equilibrium, respectively) is called a strong Nash equilibrium (strict strong Nash equilibrium,
respectively). If G = {{i} | i ∈ N }, then a (strict) G-proof Nash equilibrium is a Nashequilibrium.
Denote by N G (D, ϕ, R) the set of G-proof Nash equilibria, by sN G (D, ϕ, R) the set ofstrict G-proof Nash equilibria, by SN (D, ϕ, R) the set of strong Nash equilibria, and by
N ash(D, ϕ, R) the set of Nash equilibria .

3

3

Results

3.1

Assumptions

In the following, let a list (N, ω, Af , D) be given. In all of our results, we assume that
Conditions A and B in the below are satisfied.
Condition A For each i ∈ N , Di satisfies the following:(i) There is no consumption externality, that is, for any Ri ∈ Di and any x, y ∈ Af ,x(i) = y(i) ⇒ xI i y.
(ii) For any x ∈ Af , there is some Ri ∈ Di for which(a) top Ri (Af ) = {z | z(i) = x(i)}, and(b) ω(i) 6= x(i) ⇒ top Ri (Af \ top Ri (Af )) = {z | z(i) = ω(i)}.
Condition B The set Af satisfies the following: Let S be a coalition, and let x, y ∈ Af .
Then if x(S) = ω(S) = y(S) (which implies x(N \ S) = ω(N \ S) = y(N \ S)), then Afcontains the allocation v such that ∀i ∈ S : v(i) = x(i) and ∀i ∈ N \ S : v(i) = y(i) .
Note that the domain P defined in Sec. 2.1 satisfies Condition A. Condition B is satisfied in various problems such as marriage problems, roommate problems (Gale and Shapley,1962), housing markets (Shapley and Scarf, 1974), and hedonic coalition formation games(Banerjee, Konishi and Sönmez (2001) and Bogomolnaia and Jackson (2002)).

3.2

Statements of results

Theorem Let ϕ be a solution that is individually rational and Pareto optimal.
(A) For any R ∈ D, ϕ(sN G (D, ϕ, R)) ⊂ C G (R).
(B) If G is monotonic, then for any R ∈ D, ϕ(sN G (D, ϕ, R)) = C G (R).
Remarks (1) Clearly if G = 2N \{∅}, then G is monotonic. Thus it is immediate from ourtheorem that the set of strict strong Nash equilibrium outcomes coincides with the strong
core. This generalizes the same observation made in the context of marriage problems in
the preference revelation games induced by stable solutions (Shin and Suh, 1996).5(2) In (B) of our theorem, the monotonicity of G is essential. Without this assumption,
the equivalence does not hold even in the context of marriage problems, where the equivalence result of the weak versions of the coalitional equilibrium and the core holds true.
(Sönmez, 1997) (see (i) of (I) in Sec. 3.3). A counter-example is provided in Example 1
in Appendix.
(3) In our theorem, the strict G-proof Nash equilibrium cannot be replaced with the
G-proof Nash equilibrium. A counter-example is provided in Example 2(i) in Appendix.

3.3

Extendability of previous results

We note which parts of the previous results in marriage problems (other than the one by
Shin and Suh (1996) in the above Remark (1)) can be extended to the present model. Thisis concerned with the relationship between the G-proof Nash equilibrium and the G-weakcore rather than their strict or strong versions appearing in our Theorem.
(I) In the context of marriage problems,
5Note that in marriage problems, stable solutions are equivalent to both strong core solutions and weakcore solutions.

4(i) if ϕ is individually rational and Pareto optimal, then for any R ∈ P, ϕ(N G (P, ϕ, R)) =C W G (R) (Sönmez, 1997). This result yields the two results in (ii) of (I) in the below
in the two extreme cases, G = {{i} | i ∈ N } and G = 2N \ {∅}.
(ii) If ϕ is a stable solution, then for any R ∈ P, ϕ(N ash(P, ϕ, R)) = I(R) (Alcalde,
1996) and ϕ(SN (P, ϕ, R)) = C W (R) (Shin and Suh, 1996).6(II) In our general model,(i) only one direction of inclusion in Sönmez’s result is generalized. That is, it holds
true that for any R ∈ D,
ϕ(N G (D, ϕ, R)) ⊂ C W G (R).

(5)

This can be proved by the same line as the proof of (A) of our Theorem so we omit
the proof.
(ii) The reverse inclusion of (5) does not hold in either of the two extreme cases, G ={{i} | i ∈ N } and G = 2N \ {∅}, even if ϕ is restricted to strong core solutions. (Acounter-example is Example 2(ii) in Appendix.) Thus neither of the two results in(ii) of (I) above is generalized to our model.

3.4

Implications

The significance of our results is that they shed light on the consequence of the manipulation by misrepresenting preferences. In the present model, Sönmez (1999) andTakamiya (2003) have characterized non-manipulable solutions: Given that the strong
core is nonempty for each preference profile, a solution is strategy-proof, individually rational and Pareto optimal if, and only if, the solution is a selection from the strong core
correspondence and the correspondence is essentially single-valued.7 We can draw two
implications from our results and the above characterization: (i) In most cases, the strong
core correspondence is neither nonempty-valued nor essentially single-valued so it is typical that solutions are open to manipulations. But the consequences of manipulations areagain related to strong cores if the solution is individually rational and Pareto optimal.
(ii) In the rare cases where the strong core correspondence is both nonempty-valued andessentially single-valued (as in the case of housing markets), even if a solution is manipulable, as a result of manipulations strong core allocations are likely to be realized if thesolution is individually rational and Pareto optimal.8

3.5

Proof of Theorem

Proof of (A). Suppose that for some R ∈ D, ϕ(sN G (D, ϕ, R)) 6⊂ C G (R). Then for someR0 ∈ D, R0 ∈ sN G (D, ϕ, R) and ϕ(R0 ) 6∈ C G (R). Thus there is some x ∈ Af which weakly
dominates ϕ(R0 ) via some S ∈ G under R. Choose R?S ∈ DS satisfying for each i ∈ S,
top R?i (Af ) = {z | z(i) = x(i)}, and
?i

f

?i

(6)
f

ω(i) 6= x(i) ⇒ top R (A \ top R (A )) = {z | z(i) = ω(i)}.
6(7)

Note that Shin and Suh (1996) provides two distinct results, the one in Remark (1) in Sec. 3.2 andthe other one stated here.
7
The strong core correspondence is essentially single-valued if for each preference profile any two allocations in the strong core are Pareto indifferent.
8
Recently Bochet and Sakai (2007) have made a similar observation in the division problem with singlepeaked preferences (Sprumont, 1991). In this problem, it is known that the uniform rule is strategy-proof.
They show that even if the solution is not strategy-proof, any strong Nash equilibrium outcome coincides
with the uniform rule allocation if the solution satisfies some regular properties.

5Such R?S exists in DS by Condition A. Since ϕ is individually rational, for each i ∈ S,
ϕ(R0−S , R?S )(i) equals x(i) or ω(i).
Denote ϕ(R0−S , R?S ) by y. Now we prove ∀i ∈ S : x(i) = y(i). Suppose that for somei ∈ S, x(i) 6= y(i) (which implies y(i) = ω(i) 6= x(i)). Since x weakly dominates ϕ(R0 ) via
S under R, it holds true that x(S) = ω(S). And y(S) = ω(S) since ∀i ∈ S : y(i) = x(i) or= ω(i). Then consider an allocation v which satisfies∀i ∈ S : v(i) = x(i), and(8)∀i ∈ N \ S : v(i) = y(i).

(9)

Condition B ensures that Af contains this allocation v. Then v Pareto-dominates y. But
this contradicts the Pareto optimality of ϕ. Therefore, it holds true that ∀i ∈ S : x(i) =y(i). Recall
ϕ(R0 ) via S under R. Then it follows ∀i ∈ S : that x weaklyj dominatesi00yR ϕ(R ) & ∃j ∈ S : yP ϕ(R ) . This implies R0 6∈ sN G (D, ϕ, R), a contradiction. 2
Proof of (B). It suffices to prove that ∀R ∈ D, C G (R) ⊂ ϕ(sN G (D, ϕ, R)) under themonotonicity of G. Let R ∈ D and x ∈ C G (R). Choose R0 ∈ D such that for any i ∈ N ,top R0i (Af ) = {z | z(i) = x(i)}, and
0i

0i

f

(10)
f

ω(i) 6= x(i) ⇒ top R (A \ topR (A )) = {z | z(i) = ω(i)}.

(11)

Such R0 exists in D by Condition A. Then since ϕ is Pareto optimal, ϕ(R0 ) = x.
Now we prove R0 ∈ sN G (D, ϕ, R) (thus x ∈ ϕ(sN G (D, ϕ, R))). Suppose the contrary.
Then there is some coalition S ∈ G and R?S ∈ DS such that
∀i ∈ S : ϕ(R0−S , R?S )Ri ϕ(R0 ) & ∃j ∈ S : ϕ(R0−S , R?S )P j ϕ(R0 ) .
(12)
Denote ϕ(R0−S , R?S ) by y. Then if y(S) = ω(S), then (12) implies that y weakly dominates
x via S under R, which says x 6∈ C G (R), a contradiction. Thus it must be y(S) 6=
ω(S). Now let T denote an ⊂-minimal element of the class of coalitions {G ⊂ N | S ⊂G & y(G) = ω(G)}. Thus for any i ∈ T \ S, y(i) 6= ω(i). (Otherwise, T is not ⊂-minimal.)
Then since ϕ is individually rational, the construction of R0 implies ∀i ∈ T \S : x(i) = y(i),which implies xI i y. This and (12) together say (∀i ∈ T : yRi x) and (∃j ∈ S ⊂ T : yP j x),
and by construction y(T ) = ω(T ). This says that y weakly dominates x via T under R.
Since G is monotonic, T ∈ G. Thus it follows x 6∈ C G (R), a contradiction. This concludes
R0 ∈ sN G (D, ϕ, R). 2

Appendix
Here we provide the counter-examples in connection with Remarks (2), (3) in Sec. 3.2 and(II) in Sec. 3.3.
Example 1 This example shows that even in the marriage problem, without the monotonicity of G, (B) of Theorem does not hold true. (Remark (2) in Sec. 3.2.)
Consider the marriage problem (Gale and Shapley, 1962) with two men and two women:
N = M ∪ W where M = {1, 2} and W = {3, 4}; ∀i ∈ N : ω(i) = {i}; andn
Af = x ∈ A0 | ∀i ∈ N : #x(i) = 1 & ∀m ∈ M : x(m) ⊂ W ∪ {m} &o∀w ∈ W : x(w) ⊂ M ∪ {w} & ∀i, j ∈ N : x(i) = {j} ⇔ x(j) = {i} .
(13)
The domain D equals P.9 Let ϕ be a stable solution. Let G be {{i} | i ∈ N } ∪ {{1, 2, 3}}.
Clearly, G is not monotonic.
9

In this case, a preference relation on the feasible allocations (matchings) is identified with the corresponding relation on the set of men (women, respectively) plus herself (himself, respectively).

6Consider R ∈ P defined in the sequel, for which C G (R) 6⊂ ϕ(sN G (P, ϕ, R)): Let R ∈ P
be such that3P 1 1P 1 4,

(14)

2

2

(15)

3

3

1P 3P 2,

(16)

4

4

(17)

4P 2P 3,
2P 4P 1.
Consider the matching x ∈ Af such thatx(1) = 1,(18)x(2) = 4,(19)x(3) = 3.

(20)

It is easy to show x ∈ C G (R). But x 6∈ ϕ(sN G (P, ϕ, R)), which is proved as follows: Let
R0 ∈ P be any profile such that x = ϕ(R0 ). We prove R0 6∈ sN G (P, ϕ, R). Consider the
profile (R{1,2,3} , R04 ). Under this profile, there is the unique stable matching y such thaty(1) = 3,(21)y(2) = 4.

(22)(This is because: (i) It is clear that 1 and 3 have to be matched with each other under(R{1,2,3} , R04 ). (ii) If 4 were not matched with 2 under (R{1,2,3} , R04 ), then 4 would haveto be matched with herself. But this implies that 4 would have been matched withherself under R0 . ) Then y = ϕ(R{1,2,3} , R04 ). Note that yP 1 x, yP 3 x and yI 2 x. Thus
R0 6∈ sN G (P, ϕ, R), the desired conclusion. We conclude C G (R) 6⊂ ϕ(sN G (P, ϕ, R)).
Example 2 This example shows that (i) the G-proof Nash equilibrium is not applicable to
Theorem (Remark (3) in Sec. 3.2) and (ii) the reverse inclusion of the formulae (5) does
not hold in either of the two extreme cases of G ((ii) of (II) in Sec. 3.3).
Consider the housing market (Shapley and Scarf, 1974) with three traders: N ={1, 2, 3}; ∀i ∈ N : #ω(i) = 1; and Af = {z ∈ A0 | ∀i ∈ N : #z(i) = 1}. The domain D equals P.10 In this case, the strong core correspondence is singleton-valued (Rothand Postlewaite, 1977). Denote ω(i) simply by i. Let ϕ be the strong core solution.11(i) Consider R ∈ P defined in the sequel, for which ϕ(SN (P, ϕ, R)) 6⊂ C(R): Let R ∈ P
be such that2P 1 1P 1 3,

(23)

2

2

3P 1P 2,

(24)

3

3

(25)

1P 3P 2.

Note that ϕ(R) = (2, 3, 1). Let x = (2, 1, 3)(6= ϕ(R)). And choose another profile R0 ∈ Psuch that
3P 01 2P 01 1,
02

(27)

03

03

(28)

1P 2P 3,
3P 1P 2.
10
11

(26)

02

In this case, a preference relation on Af is identified with the corresponding relation defined on ω(N ).
In this example, abusing notation, for an allocation x, x denotes the triple (x(1), x(2), x(3)).

7Then ϕ(R0 ) = x. And it is easy to show R0 ∈ SN (P, ϕ, R). Thus one has x ∈ϕ(SN (P, ϕ, R)) and x 6∈ C(R), i.e. ϕ(SN (P, ϕ, R)) 6⊂ C(R). (Also, this implies ϕ(N ash(P, ϕ, R)) 6⊂
C(R).)(ii) Consider R ∈ P defined in the sequel, for which C W (R) 6⊂ ϕ(N ash(P, ϕ, R)): Let
R ∈ P such that2P 1 1P 1 3,

(29)

2

2

1P 3P 2,

(30)

3

3

(31)

1P 3P 2.

First, note that x = (2, 3, 1) ∈ C W (R). Let R0 ∈ P be any profile such that x = ϕ(R0 ).
Then by the well-known “top trading cycle” algorithm for computing strong core allocations (Shapley and Scarf, 1974), it can be shown that any such R0 satisfies ∀i ∈N : x(i) ranks top in R0i . Now choose any R?2 ∈ P 2 which ranks the good 1 as thebest. Then ϕ(R0−2 , R?2 ) = (2, 1, 3). Now clearly, ϕ(R0−2 , R?2 )(2)P 2 ϕ(R0 )(2). This saysthat any such R0 cannot be a Nash equilibrium of the game (P, ϕ, R). Thus one has
x ∈ C W (R) and x 6∈ ϕ(N ash(P, ϕ, R)), i.e. C W (R) 6⊂ ϕ(N ash(P, ϕ, R)). This implies
I(R) 6⊂ ϕ(N ash(P, ϕ, R)) and C W (R) 6⊂ ϕ(SN (P, ϕ, R)).

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