Essential µ-compatible subgames for obtaining a von
Neumann-Morgenstern in an assignment game.
Keisuke Bando
Yakuma Furusawa
Discussion Paper No. 2016-05
February 16, 2016
Essential µ-compatible subgames for obtaining a von
Neumann-Morgenstern stable set in an assignment game
Keisuke Bando∗
Yakuma Furusawa
February 16, 2016
Abstract
We study von Neumann-Morgenstern (vNM) stable sets in an assignment
game. Núñez and Rafels (2013) have shown that for any given optimal matching
µ, the union of the extended cores of all µ-compatible subgames is a vNM stable
set. Typically, the set of all µ-compatible subgames includes many elements,
most of which are inessential for obtaining the vNM stable set. We introduce
the notion of essential µ-compatible subgames, without which one cannot obtain
the vNM stable set found by Núñez and Rafels (2013). We provide an algorithm
to find a collection of essential µ-compatible subgames for obtaining the vNM
stable set under a mild assumption for the valuation matrix. Our simulation
result reveals that the average number of essential µ-compatible subgames is
significantly lower than that of all µ-compatible subgames.
Keywords:
JEL Classification: C71, D43, D45
∗
Department of Social Engineering, Graduate School of Decision Science and Technology,
Tokyo Institute of Technology, 2-12-1 Ookayama, Meguro-ku, Tokyo 152-8552 Japan; E-mail:
[email protected]
2
1
Introduction
We study the assignment game introduced by Shapley and Shubik (1972). An assignment game describes a two-sided matching market, where players are partitioned
into buyers and sellers. Each seller owns one indivisible good, and each buyer wants
to buy at most one indivisible good. Indivisible goods are exchanged between sellers
and buyers through monetary transfers. Shapley and Shubik (1972) showed that
the core is nonempty and equivalent to the competitive equilibrium in assignment
games. Moreover, they showed that buyer-optimal and seller-optimal payoffs exist
in the core.
Shapley and Shubik (1972) pointed out, however, that the core does not sufficiently describe possible outcomes that could occur in reality. They considered
von Neumann-Morgenstern (vNM) stable set, the solution concept introduced by
von Neumann and Morgenstern (1953) in cooperative game theory, as a prospective
alternative of the core. It is well known that if a vNM stable exists, it includes the
core.1 Thus, vNM stable sets can describe a wider range of possible outcomes than
the core does.
Shubik (1984) initiated the study of vNM stable sets in assignment games. Typically, there may exist multiple vNM stable sets. He focused on the existence of a
vNM stable set that is compatible with an optimal matching that maximizes the
total surplus from exchanges between sellers and buyers.
Shubik (1984) introduced the notion of µ-compatible subgames to obtain a vNM
stable set that is compatible with an optimal matching µ (vNM stable set with µ for
short). A µ-compatible subgame is a subgame constructed from the original game
by removing a set of players Q with the following two properties that (i) for each
buyer and each seller in Q, they are not matched in µ, and (ii) even if all members
in Q are removed from the original game, the restriction of µ into the remaining
players is still optimal.
Using the core of a µ-compatible subgame, we can define the set of feasible
payoffs (called the extended core) of the original game. The extend core can be
interpreted as a collusion among buyers or sellers, which will be explained in the
next section. Shubik (1984) expected that the union of the extended cores of all
µ-compatible subgames would be a vNM stable set with µ.
Núñez and Rafels (2013) completed the proof of the claim by Shubik (1984).
Specifically, they proved that given a fixed optimal matching µ, there exists a unique
vNM stable set with µ, which is the union of the extended cores of all µ-compatible
subgames.
1
Lucas (1968) showed that the nonemptiness of the core generally does not imply the existence
of a vNM stable set.
3
The vNM stable sets in assignment games were also studied by Solymosi and
Raghavan (2001) and Bednay (2014). Solymosi and Raghavan (2001) provided a
necessary and sufficient condition under which the core is a unique vNM stable set.
Bednay (2014) characterized all vNM stable sets in an assignment game with one
seller.
Our study builds on the work of Shubik (1984) and Núñez and Rafels (2013).
Their studies reveal that the vNM stable set with µ is characterized by the extended
cores of all µ-compatible subgames. Typically, the set of all µ-compatible subgames
includes many elements, including those inessetial for obtaining the vNM stable set
with µ. This is, there may exist another collection of µ-compatible subgames C
whose extend cores are the vNM stable set. In this case, we say that C obtains
the vNM stable set with µ. Moreover, the characterization result of Núñez and
Rafels (2013) does not tell us how we find a collection of µ-compatible subgames
that obtains the vNM stable set.
To overcome these problems, we introduce the notion of essential µ-compatible
subgames for obtaining the vNM stable set, and provide an algorithm to find such
games. This paper mainly considers a symmetric assignment game in that the
number of buyers and sellers is equal because Núñez and Rafels (2013) showed that
the vNM stable set of an asymmetric game can be reduced to that of a symmetric
game.
A collection of µ-compatible subgames D, which obtains the vNM stable set
with µ, is said to be essential if D is contained in C whenever there is another
collection of µ-compatible subgames C that obtains the vNM stable set with µ. In a
symmetric assignment game, under a mild assumption for the valuation matrix, we
show the existence of a collection of essential µ-compatible subgames on the basis
of an algorithm, which we call the buyer-side (or seller-side) procedure. We then
extend this result into an asymmetric market using the reduction result of Núñez
and Rafels (2013). We will give an economic interpretation of our main result in the
next section.
The imposed assumption in obtaining our main result is that each column and
row in the valuation matrix is constituted from different positive numbers. While
this is a mild assumption, our main result does not hold without it. Technically,
our proof relies on the main result of Núñez and Rafels (2013); however, except for
this, we use only some elemental properties of the core, whose proofs are in Roth
and Sotomayor (1990).
We also analyze the number of essential µ-compatible subgames. Unfortunately,
in extreme cases, this number exponentially grows, and hence our algorithm is not a
polynomial-time algorithm. However, our simulation study reveals that the average
4
number of essential µ-compatible subgames is significantly lower than that of all
µ-compatible subgames.
Note that our study is inspired by analyses of the vNM stable set in marriage
problems (two-sided matching markets without monetary transfers) introduced by
Gale and Shapley (1962). Ehlers (2007) initiated the study of vNM stable sets in
marriage problems, and showed that a set of matchings is vNM stable only if it
is a maximal distributive lattice with certain properties. Wako (2010) proved the
existence and uniqueness of the vNM stable set by providing a polynomial-time
algorithm to produce a preference profile whose core coincides with the vNM stable
set. Bando (2014) showed that the man (or woman)-optimal matching in the vNM
stable set coincides with the matching obtained from the efficiency-adjusted deferred
acceptance mechanism (EADAM) proposed by Kesten (2010) in the context of school
choice problems. Thus, EADAM is related to the vNM stable set.
Our approach is intuitively similar to the EADAM. In school choice problems, the
definition of Pareto efficiency involves only students’ preferences because the schools
themselves do not have preferences. In this criterion, the student-optimal stable
matching produced by the deferred acceptance (DA) algorithm may not be Pareto
efficient. The EADAM introuced by Kesten (2010) finds a Pareto efficient matching
that Pareto dominates the matching produced by the student-optimal DA algorithm.
Tang and Yu (2014) provided a simplified version of the EADAM.2 Roughly, their
algorithm iteratively finds the student-optimal stable matching while removing students who match with underdemanded schools. Underdemanded schools are those
schools that no students strictly prefer over their assignments in the student-optimal
stable matching.
The buyer-side procedure proposed in this paper iteratively finds the buyeroptimal stable payoff while removing buyers who (i) buy underdemanded objects
for which the price is zero and (ii) raise the equilibrium prices of other objects.
This paper proceeds as follows. In the next section, we provide an interpretation of the vNM stable set found by Núñez and Rafels (2013) through a simple
example. In Section 3, we provide a formal description of the assignment game and
summarize the results by Núñez and Rafels (2013). We also introduce the notion of
essential µ-compatible subgames. In Section 4, we provide an algorithm to find such
subgames for a symmetric game. In Section 5, we analyze the number of essential
µ-compatible subgames. In Section 6, we extend our main result into an asymmetric
market. We also show that our main result does not hold without the assumption
for the valuation matrix. Section 7 concludes our study. All proofs in this paper are
provided in the Appendices.
2
Bando (2014) independently proposed a simplified version of the EADAM.
5
2
Interpretation of vNM stable sets in assignment games
In this section, we informally discuss an interpretation of the vNM stable set found
by Núñez and Rafels (2013). We also provide an interpretation of our main result.
In an assignment game, the vNM stable set describes a collusion among buyers
or sellers. To observe this, consider the following 2 × 2 assignment game:
1′ 2′
1 50 30
2 30 20
We refer to players 1 and 2 as buyers and 1′ and 2′ as sellers. The above matrix
indicates that, for example, the surplus generated by buyer 1 and seller 2 is 50 if
they match. The optimal matching is given by µ = {(1, 1′ ), (2, 2′ )} in which buyer 1
matches with seller 1′ and buyer 2 matches with seller 2′ . Even if buyer 2 leaves the
game, {(1, 1′ )} is still optimal in the remaining game, which is called a µ-compatible
subgame. Note that buyer 1 does not satisfy this property because µ = {(2, 1′ )} is
optimal in the subgame buyer 1 has left.
A stable payoff is given by a non-negative vector (u1 , u2 , v1′ , v2′ ) that satisfies the
following conditions:
u1 + v1′ = 50, u2 + v2′ = 20,
(1)
u1 + v2′ ≥ 30, u2 + v1′ ≥ 30,
(2)
where ui is buyer i’s payoff (i = 1, 2) and vj is seller j’s payoff (j = 1′ , 2′ ). Condition (1) means that transfers are made within the pairs in the optimal matching.
Condition (2) guarantees that both buyer 1 and seller 2′ cannot be strictly better
off no matter what monetary transfers they may trade with, and similarly for buyer
2 and seller 1′ . In other words, no pair can block the outcome.
Note that if we consider each seller’s payoff as a price, then (1) and (2) mean
that each buyer chooses their preferred good given the price vector (v1′ , v2′ ), and
hence represents a competitive equilibrium.
The set of stable payoffs is said to be the core, which is given by Figure 1a. The
payoff vector (40, 20, 10, 0) is called the buyer-optimal stable payoff. On the other
hand, from the result of Núñez and Rafels (2013), the shape of the vNM stable set
with µ is given in Figure 1b. This set can be regarded as “the core + line A + line
B.”
To obtain an intuition about the vNM set, remove buyer 2 from the original game.
We can expect that a stable payoff of the subgame that buyer 2 has left is realized,
i.e., buyer 1 trades with seller 1′ at price v1′ with 0 ≤ v1′ ≤ 30. Therefore, buyer
6
v1’
v1’
u2
u2
v2’
0
A
20
20
10
30
20
40
v2’
0
50
u1
10
B
(a) The core
30
20
40
50
u1
(b) The vNM stable set
Figure 1: The core and the vNM stable set
1’s payoff would increase and he has an incentive to ask buyer 2 not to participate
in the competition. Buyer 2 has an incentive to agree with buyer 1’s invitation in
that even after a trade between buyer 1 and seller 1′ in the core of the subgame that
buyer 2 has left is realized, buyer 2 can still realize the same trade as that in the
buyer-optimal stable payoff of the original game, i.e., buyer 2 can trade with seller
2′ at price 0. Therefore, there is a reasonable incentive to form a collusion such that
buyer 1 asks for buyer 2 not to participate in the competition and buyer 2 agrees.
The vNM stable set given in Figure 2 describes such a collusion. More specifically,
line A in Figure 1b is called an extended core, which represents the payoff vectors
such that buyer 1 gets a payoff in the core of the subgame that buyer 2 has left, and
buyer 2 gets the buyer-optimal payoff (20) of the original game. Symmetrically, line
B can be interpreted as a collusion among sellers.
Of course, buyer 2 also has an incentive to ask buyer 1 not to participate in
the competition, because buyer 2 can trade with seller 1′ . In contrast to the above
case, buyer 1 may not agree to buyer 2’s invitation because the same trade as in the
buyer-optimal stable payoff of the original game can never be realized after buyer 2
trades with seller 1′ . Such a difference is due to the fact that the subgame buyer 1
has left does not satisfy the µ-compatibility.
In an n × m assignment game, the vNM stable set is obtained by all possible
reasonable collusions such that players ask one another not to participate in the
competition. However, when the number of players increases, we do not need to
consider all possible collusions to obtain the vNM stable set. Our main result provides an algorithmic way to find collusions that are sufficient for obtaining the vNM
stable set. Moreover, these collusions turn out to be necessary for this purpose.
7
3
Preliminaries
3.1
Assignment games
Let M and M ′ be finite disjoint sets of players. We refer to M as a set of buyers
and M ′ as a set of sellers. For each buyer-seller pair (i, j), aij is a nonnegative value
that (i, j) can create if they match, and A := (aij )(i,j)∈M ×M ′ is called a valuation
matrix. The value of a single player l ∈ M ∪ M ′ is zero, which is denoted by
all := 0. We say that (M, M ′ , A) is an assignment game. The assignment game
(M, M ′ , A) is symmetric if |M | = |M ′ |, asymmetric if |M | ̸= |M ′ | and general if it is
either symmetric or asymmetric. We will mainly focus on symmetric games in our
analysis.
The following assumptions will be imposed to obtain our main result:
• valuation matrix A is positive if aij > 0 for all (i, j) ∈ M × M ′ .
• valuation matrix A is not indifferent if (i) for each i ∈ M , aij ̸= aij ′ for any
distinct j, j ′ ∈ M ′ and (ii) for each j ∈ M ′ , aij ̸= ai′ j for any distinct i, i′ ∈ M .
While these are mild assumptions, both of them are crucial to obtain our main
result.
Let us consider a general assignment game (M, M ′ , A). A matching µ is a subset
of M × M ′ such that each player appears in at most one pair in µ. For each i ∈ M
and j ∈ M ′ , we denote µ(i) = j (or equivalently µ(j) = i) when (i, j) ∈ µ, and
µ(i) = i or µ(j) = j when i or j is unmatched at µ. Let M(M, M ′ ) be the set of all
matchings. Given a matching µ, for each I ⊆ M and each J ⊆ M ′ , we denote by
µ−I∪J the restriction of µ into (M \ I) × (M ′ \ J), i.e.,
µ−I∪J := {(i, j) ∈ µ|i ∈ M \ I and j ∈ M \ J}.
A matching µ is optimal in (M, M ′ , A) if it is a solution of
∑
max
µ′ ∈M(M,M ′ )
aij ,
(i,j)∈µ′
∑
where (i,j)∈∅ aij := 0. We denote the optimal value by v(M ∪ M ′ ).
′
We say that (u, v) ∈ RM × RM is a payoff vector, where ui represents a payoff
of i ∈ M and vj represents a payoff of j ∈ M ′ . A payoff vector (u, v) is feasible if
∑
∑
′
′
i∈M ui +
j∈M ′ vj ≤ v(M ∪ M ), and is compatible with a matching µ if for each
i ∈ M and j ∈ M ′ , (i) (i, j) ∈ µ implies ui + vj = aij , (ii) µ(i) = i implies ui = 0,
and (iii) µ(j) = j implies vj = 0. A feasible payoff vector (u, v) is stable if it satisfies
the following two conditions:
8
• for all i ∈ M and all j ∈ M ′ , ui ≥ 0 and vj ≥ 0 (individual rationality).
• for all i ∈ M and all j ∈ M ′ , ui + vj ≥ aij (no blocking condition).
The set of all stable payoffs is said to be the core for (M, M ′ , A), which is denoted
by C(M, M ′ , A).
The following property is a well-known fact.
Fact 1 (Roth and Sotomayor (1990)). (i) If a feasible payoff vector (u, v) is stable
and compatible with a matching µ, then µ is optimal. (ii) Any stable payoff is
compatible with all optimal matchings.
Shapley and Shubik (1972) showed that the core is nonempty. They also showed
that two extreme stable payoffs exist in the core; i.e, there are stable payoffs (ū, v),
(u, v̄) ∈ C(M, M ′ , A) such that for all i ∈ M and j ∈ M ′ , ūi ≥ ui ≥ ui and
v̄j ≥ vj ≥ v j for all (u, v) ∈ C(M, M ′ , A). We refer to (ū, v) as the buyer-optimal
stable payoff and (u, v̄) as the seller-optimal stable payoff. Following Núñez and
Rafels (2008), we say that a buyer i ∈ M is active if ūi > ui , and inactive if ūi = ui .
Similarly, a seller j ∈ M ′ is active if v̄j > v j , and inactive if v̄j = v j . We will use
the activity (or inactivity) to define our procedure.
Let I ⊊ M and J ⊊ M ′ . By removing all players in I ∪ J from the original game,
we can define a subgame (M \ I, M ′ \ J, A−I∪J ) where A−I∪J is the restriction of A
into M \ I × M ′ \ J, i.e., A−I∪J := (aij )(i,j)∈M \I × M ′ \J .3 For simplicity, we write
(M \ I, M ′ \ J, A) instead of (M \ I, M ′ \ J, A−(I∪J) ). The core for (M \ I, M ′ \ J, A)
is denoted by C(M \ I, M ′ \ J, A). The buyer- and seller-optimal stable payoffs in
(M \ I, M ′ \ J, A) are denoted by (ūI∪J , v I∪J ) and (uI∪J , v̄ I∪J ) respectively.
3.2
Essential µ-compatible subgames
In this section, we introduce the vNM stable set and summarize the main results of
Núñez and Rafels (2013). We then introduce the notion of essential µ-compatible
subgames.
Consider a general assignment game (M, M ′ , A). The vNM stable sets are defined
on the basis of the dominance relation over imputations. A payoff vector (u, v) is
an imputation if it satisfies (i) ui ≥ 0 and vj ≥ 0 for all i ∈ M and all j ∈ M ′ and
∑
∑
(ii) i∈M ui + j∈M ′ vj = v(M ∪ M ′ ). We denote the set of all imputations by
I. For any two imputations (u, v) and (u′ , v ′ ), (u′ , v ′ ) dominates (u, v) if there is
3
In Núñez and Rafels (2013), the definition of the subgame includes null games, i.e., removing
I = M or J = M ′ may be possible. However, we omit the null games because they are redundant
to the analysis.
9
(i, j) ∈ M × M ′ such that u′i > ui , vj′ > vj and u′i + vj′ ≤ aij .4 It is a well-known
fact that the core is characterized by the set of imputations that are not dominated
by any other imputations.
A set of imputations V ⊆ I is a vNM stable set if it satisfies the following criteria:
• for any (u, v) and (u′ , v ′ ) ∈ V , (u′ , v ′ ) does not dominate (u, v) (Internal stability)
• for any (u, v) ∈ I \ V , there is (u′ , v ′ ) ∈ V that dominates (u, v) (External
stability).
Note that every vNM stable set includes the core.
We focus on a vNM stable set that is compatible with an optimal matching as
in Shubik (1984) and Núñez and Rafels (2013). Formally, a vNM stable set V is
compatible with an optimal matching µ if for any (u, v) ∈ V , (u, v) is compatible
with µ.
Núñez and Rafels (2013) showed that for any given optimal matching µ, there
exists a unique vNM stable set with µ. The key notions in obtaining their result
are µ-compatible subgames and extended cores, which were originally introduced by
Shubik (1984). For each I ⊊ M and each J ⊊ M ′ , I ∪ J is said to be a µ-compatible
set if (i) for each i ∈ I and j ∈ J, (i, j) ∈ µ implies that aij = 0, and (ii) µ−I∪J
is optimal in (M \ I, M ′ \ J, A). We denote by C µ the set of all µ-compatible sets.
For each I ∪ J ∈ C µ , we say that (M \ I, M ′ \ J, A) is a µ-compatible subgame. We
can identify a µ-compatible set with a µ-compatible subgame. From the definition,
∅ ∈ C µ holds; thus, the original game is a µ-compatible subgame.
It should be remarked that the original definition of µ-compatible subgames,
which is provided by Shubik (1984) and Núñez and Rafels (2013), is defined on the
basis of a characteristic function. In Appendix B, it is shown that our definition and
the original definition are equivalent.
A µ-compatible set may generally include both of buyers and sellers. However,
we can show that in a symmetric game with a positive valuation matrix, any µcompatible set contains only buyers or only sellers (See Lemma 1 in Appendix A1).
Technically, this is the primary reason for assuming positivity.
We next introduce the extended core for a µ-compatible subgame (M \ I, M ′ \
J, A). Let (u, v) be a stable payoff in (M \ I, M ′ \ J, A). As µ−I∪J is still optimal in
4
The original definition of the dominance relation is based on a characteristic function. See Roth
and Sotomayor (1990) for more details.
10
(M \ I, M ′ \ J, A), from Fact 1-(ii), we have that
(1) (i, j) ∈ µ−I∪J implies ui + vj = aij ,
(2) for any i ∈ M \ I with µ(i) = i or µ(i) ∈ J, ui = 0, and
(3) for any j ∈ M ′ \ J with µ(j) = j or µ(j) ∈ I, vj = 0.
Therefore, a payoff vector (u∗ , v ∗ ) of the original game (M, M ′ , A), which is defined
by
u∗i := ui for all i ∈ M \ I, and vj∗ := vj for all j ∈ M \ J,
u∗i := aiµ(i) for all i ∈ I, and vj∗ := aµ(j)j for all j ∈ M \ J,
is compatible with µ and hence an imputation. In other words, (u∗ , v ∗ ) is constructed
by (i) all players l in I ∪ J that leave the original game agreeing with alµ(l) or
aµ(l)l and (ii) the remaining players receiving a stable payoff in the remaining game
(M \ I, M ′ \ J, A).
From the above argument,
}
{
′ \ J, A),
′
(u
,
v
)
∈
C(M
\
I,
M
′
M \I M \J
Ĉ(I ∪ J) := (u, v) ∈ R|M | × R|M |
ui = aiµ(i) ∀i ∈ I, and vj = aµ(j)j ∀j ∈ J
′
is a set of imputations in the original market, where for any vector (u, v) ∈ RM ×RM ,
(uM \I , vM ′ \J ) := ((ui )i∈M \I , (vj )j∈M \J ). We say that Ĉ(I ∪ J) is an extended core.
The extended core can be interpreted as a collusion such that players in M \I ∪M ′ \J
ask for players in I ∪ J not to participate in the competition and players in I ∪ J
agree with this invitation.
The vNM stable set with µ is characterized by the union of the extended cores
of all µ-compatible subgames.
Theorem 1 (Núñez and Rafels (2013)). The union of the extended cores of all
µ-compatible subgames,
∪
Ĉ(I ∪ J),
I∪J∈C µ
is the unique vNM stable set with µ.
Note that the above theorem holds for general assignment games. Núñez and
Rafels (2013) showed that the vNM stable set of an asymmetric market can be
reduced to that of a symmetric market by removing unmatched players in the optimal
matching. This will be discussed in Section 6.1 in more detail.
We now introduce the notion of essential µ-compatible subgames to obtain the
vNM stable set. A collection of µ-compatible subgames C ⊆ C µ obtains V µ (the
11
∪
vNM stable set with µ) if Q∈C Ĉ(Q) = V µ . We say that C ⊆ C µ is a collection of
essential µ-compatible subgames for obtaining V µ if (i) C obtains V µ and (ii) C ⊆ C ′
for all C ′ ⊆ C µ such that C ′ obtains V µ . In other words, a collection of essential
µ-compatible subgames is necessary and sufficient for obtaining the vNM stable set
with µ. By definition, when a collection of essential µ-compatible subgames exists,
it is uniquely determined.
For example, in the 2 × 2 game provided in Section 2, the set of all µ-compatible
subgames is given by C µ = {∅, {2}, {2′ }}. In Figure 1b, line A and line B represent
Ĉ({2}) and Ĉ({2′ }) respectively. Clearly, C µ itself is the collection of essential µcompatible subgames for obtaining the vNM stable set. The following example
graphically illustrates that in a 3 × 3 game, the collection of essential µ-compatible
subgames is strictly contained in that of all µ-compatible subgames.
Example 1. Let M = {1, 2, 3} and M ′
given by
1′
1 30
2 17
3 37
= {1′ , 2′ , 3′ }. The valuation matrix A is
2′ 3′
13 10
25 14
30 25
The unique optimal matching is given by {(1, 1′ ), (2, 2′ ), (3, 3′ )}. The shape of the
core is given by Figure 2a.
(a) The core
(b) The vNM stable set
Figure 2: The core and the vNM stable set in Example 1
12
In this example, the set of all µ-compatibale subgames is given by
C µ = {∅, {3}, {2, 3}, {1, 3}, {1′ }, {2′ }, {1′ , 2′ }}.
From Theorem 1, the union of extended cores of all µ-compatible subgames coincides
with V µ (the vNM stable set with µ) which is given by Figure 2b. Note that V µ can
be regarded as “the core + hexagon ABCDEF + line HM ”. On the other hand,
the collection of essential µ-compatible subgames for obtaining V µ is given by
C = {∅, {3}, {1′ , 2′ }}.
We can confirm this fact from Figure 2b as bellow. First, hexagon ABCDEF which
is on {(u1 , u2 , u3 )|u3 = a33′ = 30} represents Ĉ({3}). On the other hand, line
AB and line AF represent Ĉ({2, 3}) and Ĉ({1, 3}) respectively. Thus, Ĉ({2, 3}) ∪
Ĉ({1, 3}) ⊆ Ĉ({3}). Second, line HM represents Ĉ({1′ , 2′ }). On the other hand,
trinangle GHI which is on {(u1 , u2 , u3 )|u2 = 0(v2′ = 25)} represents Ĉ({2}′ ) and
pentagon KJHIL which is on {(u1 , u2 , u3 )|u1 = 0(v1′ = 30)} represents Ĉ({1′ }).5
Thus, Ĉ({1′ }) ∪ Ĉ({2′ }) ⊆ Ĉ(∅). From the above argument, C = {∅, {3}, {1′ , 2′ }}
obtains V µ . Moreover, V µ is never obtained by a collection of µ-comatible subgames
that does not contain some element in C. Therefore, C is the collection of essential
µ-compatible subgames.
□
In the above example, there exists a collecton of essential µ-comaptible subgames
for obtaining the vNM stable set. In the next section, we show, by providing an
algorithm, the existence of it for any symmetric game whose valuation matrix is
positive and not indifferent.
Unfortunately, for an asymmetric game, a collection of essential µ-compatible
subgames may not exist. However, we can naturally modify the definition of essentiality for asymmetric cases with the reduction result of Núñez and Rafels (2013).
The existence of the modified version is then directly implied by the existence result
for a symmetric game. This will be discussed in Section 6.1 in more detail.
4
Main results
Here, we provide a procedure to find the essential µ-compatible subgames for a symmetric game whose valuation matrix is positive and not indifferent. This procedure
mainly consists of an algorithm that we call the buyer- (or seller-) side procedure.
This procedure iteratively removes a set of essential buyers (sellers) from the original game.
5
Precisely, G = (4, 0, 11), H = (0, 0, 11), I = (0, 0, 7), J = (0, 4, 15), K = (0, 10, 15) and L =
(0, 4, 7).
13
To define the notion of a set of essential buyers (or sellers), we need to introduce
a property related to the extreme stable payoffs. Consider a general assignment
game (M, M ′ , A) and an optimal matching µ in (M ′ , M, A). For each j ∈ M ′ with
vj > 0 and each i ∈ M , we say that i is connected from j at µ in (M, M ′ , A) if
there is a sequence of distinct players j0 = j, i1 , j1 , · · · , in , jn , in+1 = i with n ≥ 0,
{i1 , · · · , in } ⊆ M and {j1 , · · · , jn } ⊆ M ′ such that
(i) µ(ik′ ) = jk′ for all k ′ = 1, · · · , n,
(ii) v jk′ > 0 for all k ′ = 1, · · · , n,
(iii) ūik+1 + v jk = aik+1 jk for all k = 0, · · · , n,
(iv) ūi = aiµ(i) .
Note that from Fact 1-(ii), we have (i) ūik′ + v ik′ = aik′ jk′ for all k ′ = 1, · · · , n and
(ii) µ(i) ∈ M ′ implies v µ(i) = 0. Symmetrically, for each buyer i who gets a positive
seller-optimal stable payoff, we can define sellers who are connected from i at µ.
The following fact is crucial for defining our procedure:
Fact 2 (Roth and Sotomayor (1990)). (i) For each j ∈ M ′ with v j > 0, there
exist buyers who are connected from j at µ in (M, M ′ , A).
(ii) For each i ∈ M with ui > 0, there exist sellers who are connected from i at µ
in (M, M ′ , A).
Here, we consider a symmetric assignment game (M, M ′ , A). Let µ be an optimal
matching in (M, M ′ , A), C µ be the set of all µ-compatible subgames and V µ be the
vNM stable set with µ. We will focus only on the buyer-side procedure because the
seller-side procedure can be symmetrically defined by exchanging the roles of sellers
and buyers.
We first introduce the notion of a set of essential buyers for each µ-compatible
subgame. Let Q ∈ C µ be a µ-compatible set with Q ⊆ M . Consider the µ-compatible
subgame (M \ Q, M ′ , A). Suppose that there exists a seller j ∈ M ′ with v Q
j > 0.
From Fact 2-(i), there exist buyers in M \ Q who are connected from j at µ−Q in
(M \ Q, M ′ , A) because µ−Q is optimal in (M \ Q, M ′ , A). We denote by Pj (Q) the
set of all buyers who are connected from j at µ−Q in (M \ Q, M ′ , A). We say that
S ⊆ M \ Q is a set of essential buyers in (M \ Q, M ′ , A) if for some j ′ ∈ M ′ with
′
vQ
j ′ > 0, S = Pj (Q) (in this case, we say that S is a set of essential buyers from
′
j ). Note that there may exist multiple sets of essential buyers when many sellers
get positives in the buyer-optimal stable payoff. Under positivity, we can show the
following result.
For each j ∈ M ′ with v Q
j > 0, Q ∪ Pj (Q) is also a µ-compatible set.
14
(3)
The proof is given in Appendix A1 (Lemma 3). This result enables us to inductively
construct a collection of µ-compatible sets from ∅ ∈ C µ .
We now define the buyer-side procedure. This procedure works as follows.
• in first step, find a set of essential buyers, say S1 , in (M, M ′ , A) and remove
S1 from (M, M ′ , A).
• in the next step, find a set of essential buyers, say S2 , in (M \ S1 , M ′ ) and
remove S2 from (M \ S1 , M ′ ), and so on.
We repeat this procedure until there is no set of essential buyers, i.e., every seller
gets 0 in the buyer-optimal stable payoff. This procedure terminates in finite steps
because M is a finite set and yields a collection of sets of buyers {∅, S1 , S1 ∪ S2 , · · · }.
By (3), every element in {∅, S1 , S1 ∪ S2 , · · · } is a µ-compatible set. However, it
may be insufficient to obtain the vNM stable set with µ. To do so, we need to
consider all possible µ-compatible sets that are produced by the above procedure.
This is the primary reason for the number of essential µ-compatible subgames to
grow exponentially large.
A formal description of the buyer-side procedure is given as follows:
0 := {∅} and proceed to Step 1.
• Step 0: Define DM
• Step 1: If v j = 0 for all j ∈ M ′ , this algorithm terminates. Otherwise, define
1
DM
:= {S| S = Pj (∅) for some j ∈ M ′ with v j > 0}
and proceed to the next step.
k−1
• Step k (k ≥ 2): If, for all j ∈ M ′ and for all Q ∈ DM
, vQ
j = 0, then this
algorithm terminates. Otherwise, define
k−1
k
DM
:= {S| S = Q ∪ Pj (Q) for some Q ∈ DM
and for some j ∈ M ′ with v Q
j >0 }
and proceed to the next step.
Let k ∗ > 0 be the termination of the buyer-side procedure. Define
µ
DM
:=
∪
k
DM
.
k∈{0,1,··· ,k∗ −1}
µ
By (3), for each Q ∈ DM
, Q is a µ-compatible set. By exchanging the roles of
µ
buyers and sellers, we can also define the seller-side procedure. We denote by DM
′
the collection of sets of sellers generated by the seller-side procedure. Define Dµ :=
µ
µ
DM
∪ DM
′ . We then obtain the following result.
15
Proposition 1. Suppose that A is positive. Then, Dµ obtains V µ .
In Appendix A2, we will show that Dµ obtains V µ using Theorem 1. That is,
it will be shown that for any µ-compatible set QC ∈ C µ , there is QD ∈ Dµ such
that Ĉ(Qc ) ⊆ Ĉ(QD ). To get this result, we do not require the no indifference
condition. Without positivity, however, this proposition does not hold because we
must consider µ-compatible sets that include both buyers and sellers to obtain V µ
(See Example 5 in Section 6.2).
While Dµ obtains V µ , Dµ may include elements that are inessential for obtaining
µ
V µ . Therefore, we slightly modify DM
to
µ
µ
D̄M
:= {Q ∈ DM
| all buyers in M \ Q are active in (M \ Q, M ′ , A)}.
This means that we consider only Q ∈ DM such that no buyer gets a constant
µ
µ
µ
payoff in the core. Clearly, D̄M
⊆ DM
holds. Moreover, under positivity, ∅ ∈ D̄M
also holds. In other words, all buyers are active in the original game (See Lemma 7 in
Appendix A2). Typically, except for degenerate cases, each buyer’s payoff in the core
is not constant. Therefore, in almost all cases, we can expect that this modification
µ
does not affect DM
. However, theoretically, we must conduct this modification to
obtain essential µ-compatible subgames.
µ
µ
′
Symmetrically, we can define D̄M
′ from DM ′ by excluding the subgames Q ∈
µ
µ
µ
′
′
µ
DM
′ such that some sellers are inactive in (M, M \Q , A). Defining D̄ := D̄M ∪ D̄M ′ ,
we can obtain the following result.
Proposition 2. Suppose that A is positive and not indifferent. Then, D̄µ obtains
V µ.
We require the no indifference condition to prove Proposition 2. In fact, without
the no indifference condition, D̄µ may not obtain V µ . Moreover, a collection of
essential µ-compatible subgames may not exist without this condition. An example
will be given in Section 6.2 (Example 6).
From definition, D̄µ may include many elements. Nevertheless, we can obtain
the following result.
Theorem 2. Suppose that A is positive and not indifferent. Then, D̄µ is the collection of essential µ-compatible subgames for obtaining V µ .
Here, we outline the proof of Theorem 2. From Proposition 2, it remains to be
shown that for every Q ∈ D̄µ ,
∃(u∗ , v ∗ ) ∈ Ĉ(Q) such that (u∗ , v ∗ ) ∈
/
∪
Q′ ∈C µ \{Q}
16
Ĉ(Q′ ).
µ
To obtain this result, the following property of DM
, which we call the strict effectiveness property, is crucial:
µ
S
for any Q ∈ DM
and any S ∈ C µ with S ⊊ Q, ∃i ∈ M \ Q such that ūQ
i > ūi .
That is, when all buyers in Q \ S come back to the subgame (M \ Q, M ′ , A), there
will be at least one buyer in M \ Q whose optimal stable payoff strictly decreases.
µ
µ
Q
Then, consider any Q ∈ D̄M
. From the definition of D̄M
, we have ūQ
i > ui for
all i ∈ M \ Q. Therefore, for sufficiently small ϵ > 0, the payoff vector (u∗ , v ∗ ) which
is defined by
Q
∗
u∗i = ūQ
i − ϵ for all i ∈ M \ Q and v j = v j + ϵ for all j ∈ µ(M \ Q) ,
u∗i = aiµ(i) for all i ∈ Q and vj∗ = 0 for all j ∈ µ(Q),
is in the extended core Ĉ(Q). By the strict effectiveness property and the definition
∪
of the extended core, it can be shown that (u∗ , v ∗ ) ∈
/ Q′ ∈C µ \{Q} Ĉ(Q′ ).
In the 3×3 game provided in Example 1, (i) the buyer-side procedure terminates
µ
at Step 2 with DM
= {∅, {3}}, (ii) the seller-side procedure terminates at Step 2
µ
′
with DM ′ = {∅, {1 , 2′ }} and (iii) Dµ = D̄µ . The following example shows that
D̄µ ⊊ Dµ even if a valuation matrix is positive and not indifferent.
Example 2. Let M = {1, 2, 3, 4} and M ′ = {1′ , 2′ , 3′ , 4′ }. The valuation matrix A
is given by
1′ 2′ 3′ 4′
1 10 4 5 1
2 15 10 4 2
3 2
1 6 9
4 1
2 3 10
The unique optimal matching is given by µ := {(1, 1′ ), (2, 2′ ), (3, 3′ ), (4, 4′ )}. We
demonstrate the buyer-side procedure as below. To identify buyers who are connected from a seller, a graph representation is useful: for each i ∈ M and j ∈ M ′ ,
i → j if and only if µ(i) = j, and j → i if and only if ūi + v j = aij and µ(i) ̸= j.
Then, i is connected from j with v j > 0 if and only if there exists a path of distinct
players j → i1 → j1 → · · · → in → jn → i such that (i) v jk′ > 0 for all k ′ = 1, · · · , n
and (ii) ūiµ(i) = aiµ(i) .
Step 1: Figure 3 is the graph representation of the original game, where (·)
denotes each player’s buyer-optimal stable payoff.
Sellers 1′ and 4′ get positive values in the buyer-optimal stable payoff. The set
of essential buyers from 1′ is given by P1′ (∅) = {2}, and that from 4′ is given by
1 = {{2}, {3}}. Note that 4′ → 3 → 3′ → 1 → 1′ → 2
P4′ (∅) = {3}. Therefore, DM
and ū2 = a22′ hold. However, 2 ∈
/ P4′ (∅) because seller 3′ gets 0.
17
(5)
1’
(0)
2’
(0)
3’
(3)
4’
1
(5)
2
(10)
3
(6)
4
(7)
Figure 3: The graph representation of (M, M ′ , A)
Step 2: We need to consider two subgames (M \{2}, M ′ , A) and (M \{3}, M ′ , A)
whose graph representations are given in Figure 4.
(0)
1’
1
(10)
(0)
2’
(0)
3’
(3)
4’
(5)
1’
(0)
2’
3
(6)
4
(7)
1
(5)
2
(10)
(a) (M \ {2}, M ′ , A)
(0)
3’
(0)
4’
4
(10)
(b) (M \ {3}, M ′ , A)
Figure 4: The graph representations
In (M \ {2}, M ′ , A), P4′ ({2}) = {3} is the set of essential buyers. In (M \
2 = {{2, 3}}.
{3}, M ′ , A), P1′ ({3}) = {2} is the set of essential buyers. Therefore, DM
In (M \{3}, M ′ , A), buyer 2 is inactive. To see this, note that µ′ = {(1, 3′ ), (2, 1′ ), (4, 4′ )}
is also optimal in (M \ {3}, M ′ , A). In µ′ , seller 2 is unmatched and hence v̄2′ =
v 2′ = 0 from Fact 1-(ii). Again, from Fact 1-(ii), ū2 + v 2′ = u2 + v̄2′ = 10, which
implies that ū2 = u2 = 10. On the other hand, in (M \ {2}, M ′ , A), all buyers are
active. For example, the following payoff vector is stable:
u1 = 4, u3 = 1, u4 = 2, v1′ = 6, v2′ = 0, v3′ = 5, v4′ = 8.
Step 3: We need to consider the subgame (M \ {2, 3}, M ′ , A). It is easy to see
that all sellers get 0 in the buyer-optimal stable payoff. Therefore, the buyer-side
procedure terminates at this step.
The output of the buyer side-procedure is given by:
µ
0
1
2
DM
= DM
∪ DM
∪ DM
= {∅, {2}, {3}, {2, 3}}.
As there exists an inactive buyer in (M \ {3}, M ′ , A), we have
µ
D̄M
= {∅, {2}, {2, 3}}.
18
µ
µ
′
The seller-side procedure terminates at step 2 and DM
′ = D̄M ′ = {∅, {1 }}. In
this example, D̄µ = {∅, {2}, {2, 3} {1′ }} is the collection of essential µ-compatible
subgames for obtaining the vNM stable set with µ.
□
5
Number of essential µ-compatible subgames
In this section, we analyze the number of essential µ-compatible subgames (|D̄µ |).
We first show that it grows exponentially large in extreme cases. In constrast, our
simulation study reveals that the number of essential µ-compatible subgames (|D̄µ |)
is significantly lower than that of all µ-compatible subgames (|C µ |).
The following example illustrates that |D̄µ | grows exponentially large. Specifically, for sets of buyers and sellers M and M ′ , resepectively, with |M | = |M ′ | = 2n,
µ
there exists a valuation matrix A such that |D̄M
| = 2n .
Example 3. Let M = {1, 2, · · · , 2n} and M ′ = {1′ , 2′ , · · · , 2n′ } (n ≥ 1). The
valuation matrix A = (aij )(i,j)∈M ×M ′ is given by

50
30
ϵ31
ϵ31
..
.
30
20
ϵ34
ϵ34
..
.











 ϵ
 2n−11 ϵ2n−12
ϵ2n1
ϵ2n2
ϵ13 ϵ14
ϵ23 ϵ24
50 30
30 20
..
..
.
.
... ...
... ...
...
...
...
...
..
.
ϵ12n−1
ϵ22n−1
ϵ32n−1
ϵ42n−1
..
.
...
...
50
30
ϵ12n
ϵ22n
ϵ32n
ϵ42n
..
.












30 

20
We assume that for each i and j, 0 < ϵij < 20. We can also assume that A
satisfies the no indifference condition. The unique optimal matching is given by
µ
µ
µ = {(1, 1′ ), (2, 2′ ), . . . , (2n, 2n′ )}. We show that DM
= D̄M
= 2E in this game,
where E := {2, 4, · · · , 2n} is the set of even numbers in M .
In this game, for all i = 2, 4, · · · , 2n, we have
i′ − 1 i′
i − 1 50
30
i
30
20
The essential µ-compatible subgames of the original game will be characterized by
those of the above 2 × 2 games. As we saw in Section 2, the buyer-optimal stable
payoff in this 2 × 2 game is given by:
ūi−1 = 40, ūi = 20, v i−1′ = 10, v i′ = 0.
19
Note that {i} is the set of essential buyers in this 2 × 2 game. Moreover, the buyeroptimal stable payoff in the µ-compatible subgame ({i − 1}, {i − 1′ , i′ }, A) is given
by:
ūi−1 = 50, v i−1′ = 0, v i′ = 0
We next analyze the original game. The buyer-optimal stable payoff is characterized by the above 2 × 2 game, and given by (u, v) such that, for all i ∈ E,
ūi−1 = 40, ūi = 20, v i−1′ = 10, v i′ = 0.
Therefore, the collection of sets of essential buyers is given by {{2}, {4}, · · · , {2n}}.
µ
Now, we show that DM
= 2E . Let Q ⊆ E with E \ Q ̸= ∅. From the above
argument, it is sufficient to show that, in the subgame (M \ Q, M ′ , A), the collection
of sets of essential buyers is given by {{2}, {4}, · · · , {2n}} \ {{i}|i ∈ Q}. The buyeroptimal stable payoff in (M \ Q, M ′ , A) is as follows:
Q
Q
Q
for all i ∈ E \ Q, ūQ
i−1 = 40, ūi = 20, v i−1′ = 10, v i′ = 0, and
Q
Q
for all i ∈ Q, ūQ
i−1 = 50, v i−1′ = 0, v i′ = 0.
Therefore, in the subgame (M \ Q, M ′ , A), the collection of sets of essential buyers
coincides with {{2}, {4}, · · · , {2n}} \ {{i}|i ∈ Q}.
µ
We finally show that DM
= D̄µ . Let Q ⊆ E with E \ Q ̸= ∅ and ϵ∗ := maxij ϵij .
Define a payoff vector in (M \ Q, M ′ , A) as follows:
for all i ∈ E \ Q, ui−1 = 10 + ϵ∗ , ui = ϵ∗ , vi−1′ = 40 − ϵ∗ , vi′ = 20 − ϵ∗ , and
for all i ∈ Q, ui−1 = 30, vi−1′ = 20, vi′ = 0.
It is straightforward to check that (u, v) is stable in (M \ Q, M ′ , A) and ūQ
i > ui for
µ
µ
E
all i ∈ M \ Q. This implies that DM = D̄M = 2 . Therefore, we have |D̄µ | = 2n in
this example.
□
We next compare |D̄µ | with |C µ | through simulation analysis. For simplicity,
we compare |Dµ | with |C µ | instead of |D̄µ |. Specifically, we conduct the following
simulation.
• fix a market size n := |M | = |M ′ |.
• randomly generate an n × n valuation matrix A until A satisfies the no indifference condition. Each element of A is independently drawn from a uniform
distribution over {1, 2, · · · , 1000}.
• For (M, M ′ , A), find Dµ and C µ , and then calculate |Dµ | and |C µ |.
20
For each of n = 5, 10 and 15, we examine 1000 instances. For each market size,
the average numbers of |Dµ | and |C µ | are summarized in Table 1 and the distribution
of (|C µ |, |Dµ |) is given in Figure 5. As a whole, |Dµ | is significantly lower than |C µ |.
For example, when n = 15, the average number of |Dµ | is 39.046, while that of |C µ |
is 3153.7. Moreover, the maximal number of |Dµ | in all instances is 171 while that
of |Dµ | is 15903. We can expect such differences to become large as the market size
increases.
Table 1: Average numbers of |Dµ | and |C µ |.
Market size
Dµ
Average number of
Average number of C µ
n=5
n = 10
n = 15
5.203
30.312
15.675
318.184
39.046
3153.7
12
40
35
10
30
8
25
20
6
15
4
10
2
5
0
0
10
20
30
40
50
60
70
0
200
(a) n = 5
400
600
800
1000
1200
(b) n = 10
180
160
140
120
100
80
60
40
20
0
0
2000
4000
6000
8000
10000
12000
14000
16000
(c) n = 15
Figure 5: Distributions of (|C µ |, |Dµ |) where the horizontal line represents |C µ | and
the vertex line represents |Dµ | for each market size.
21
1400
1600
6
6.1
Discussions
Extension to asymmetric assignment games
Here, we introduce the reduction result shown by Núñez and Rafels (2013); namely,
the vNM stable set of an asymmetric game can be characterized by that of a symmetric game by removing unmatched players from the former under an optimal
matching. We then extend our main result into an asymmetric game.
Let (M, M ′ , A) be an asymmetric assignment game and µ be an optimal matching in (M, M ′ , A). Without loss of generality, we assume that |M | > |M ′ |. For
simplicity, we also assume that A is positive and hence no seller is unmatched in µ.
We denote by C µ the set of all µ-compatible subgames, and by V µ the vNM stable
set with µ in (M, M ′ , A).
Let U ⊆ M be the set of buyers who are unmatched at µ. Then, (M \ U, M ′ , A)
is a symmetric assignment game. Moreover, µ is still optimal in (M \ U, M ′ , A). We
µ
µ
denote by C−U
the set of all µ-compatible subgames and by V−U
the vNM stable set
′
with µ in the reduced game (M \ U, M , A). We define
µ
CUµ := {Q ∪ U |Q ∈ C−U
}.
Clearly, CUµ ⊆ C µ holds. Then, the vNM stable set with µ for the original game
(M, M ′ , A) is characterized by the vNM stable set with µ for the reduced symmetric
game (M \ U, M ′ , A).
Theorem 3 (Núñez and Rafels (2013)).
∪
Ĉ(Q)
µ
Q∈CU
or equivalently
′
µ
{(u, v) ∈ RM × RM |(uM \U , v) ∈ V−U
, and ui = 0 ∀i ∈ U }
is the unique vNM stable set with µ in (M, M ′ , A).
As noted in Section 3.2, a collection of essential µ-compatible subgames may not
exist in an asymmetric game (the example will be given later). However, we can
naturally modify the definition of essentiality using Theorem 3 as follows: C ⊆ CUµ is
a collection of U -essential µ-compatible subgames for obtaining V µ if (i) C obtains
V µ and (ii) C ⊆ C ′ for all C ′ ⊆ CUµ that obtains V µ .
Then, from Theorem 3, C ⊆ CUµ is a collection of U -essential µ-compatible subgames for obtaining V µ if and only if {Q \ U |Q ∈ C} is a collection of essential
µ
µ-compatible subgames for obtaining V−U
. Moreover, the collection of U -essential
22
µ-comaptible subgames is a minimal collection of µ-comaptible subgames for obtainµ
ing V µ . That is, if C ⊆ C−U
is a collection of U -essential µ-compatible subgames for
µ
obtaining V , then there is no C ′ ⊊ C that obtains V µ .
Finally, the following example illustrates that a collection of essential µ-compatible
subgames does not exist in an asymmetric market.
Example 4. Let M = {1, 2, 3} and M ′ =
by:
1′
1 50
2 30
3 1
{1, 2}. The valuation matrix A is given
2′
30
20
2
Clearly, µ = {(1, 1′ ), (2, 2′ )} is the unique optimal matching. Therefore, the reduced
symmetric game is given by:
1′ 2′
1 50 30
2 30 20
As observed in Section 2, a collection of essential µ-compatible subgames in the
reduced game is given by {∅, {2}, {2′ }}. Therefore, {{3}, {2, 3}, {2′ , 3}} is the collection of U -essential µ-compatible subgames for obtaining the vNM stable set with
µ. This implies that it is a minimal collection of µ-compatible subgames.
We now show that a collection of essential µ-compatible subgames does not
exist. It is sufficient to show that there exists another minimal collection of µcompatible subgames. Note that {2′ } is a µ-compatible set in the original game.
Moreover, Ĉ({2′ }) coincides with Ĉ({2′ , 3}). This implies that {{3}, {2, 3}, {2′ }} is
also a minimal collection of µ-compatible subgames of the original game. Therefore,
a collection of essential µ-compatible subgames does not exist in this example.
□
6.2
Without the positivity and no indifference conditions
In this section, we demonstrate that Theorem 2 do not hold when the positivity and
no indifference conditions are not assumed.
To obtain the vNM stable set under the positivity condition, it is sufficient to
consider µ-compatible sets that include only buyers or only sellers. However, without
positivity, we must consider µ-compatible sets that include both sellers and buyers
to obtain the vNM stable set. In this case, Dµ may not obtain the vNM stable set.
The following example illustrates this fact.
23
Example 5. Let M = {1, 2, 3, 4} and
is given by:
1′
1 30
2 50
3 0
4 0
M ′ = {1′ , 2′ , 3′ , 4′ }. The valuation matrix A
2′ 3′ 4′
0
0
0
30 22 0
0 30 0
0 50 30
The unique optimal matching is given by µ := {(1, 1′ ), (2, 2′ ), (3, 3′ ), (4, 4′ )}. Let V µ
be the vNM stable set with µ. In this game, {4, 1′ } is a µ-compatible set. We will
show that without Ĉ({4, 1′ }), V µ is never obtained.
To see this, consider the following imputation (u, v):
u1 = 0, u2 = 11, u3 = 19, u4 = 30, v1′ = 30, v2′ = 19, v3′ = 11, v4′ = 0.
It is straightforward to show that (u, v) ∈ Ĉ({4, 1′ }). We will show that for any
µ-compatible set Q, if (u, v) ∈ Ĉ(Q), then we must have Q = {4, 1′ }. This property
directly implies that we cannot obtain V µ without Ĉ({4, 1′ }). Let Q be any µcompatible set with (u, v) ∈ Ĉ(Q). By u2 , v2′ < a22′ and u3 , u3′ < a33′ , we have that
l∈
/ Q for all l = 2, 3, 2′ , 3′ .
Next, we show that 4 ∈ Q and 1′ ∈ Q. Otherwise, we have 4 ∈
/ Q or 1′ ∈
/ Q.
′
Suppose that 4 ∈
/ Q. From u4 + v3′ = 30 + 11 < a43′ = 50 and 3 ∈
/ Q, we have
′
(uM \Q , vM \Q ) ∈
/ C(M \ Q, M \ Q, A). This contradicts the fact that (u, v) ∈ Ĉ(Q).
Suppose that 1′ ∈
/ Q. From u2 + v1′ = 11 + 30 < a21′ = 50 and 2 ∈
/ Q, we have
′
(uM \Q , vM \Q ) ∈
/ C(M \ Q, M \ Q, A). This contradicts the fact that (u, v) ∈ Ĉ(Q).
Therefore, 4 ∈ Q and 1′ ∈ Q.
From the above argument, we have {4, 1′ } ⊆ Q ⊆ {1, 4, 1′ , 4′ }. Then, the definition of µ-compatibility implies that Q = {4, 1′ }.
□
The no indifference condition guarantees that the modified versions of the buyerand seller-side procedures (D¯µ ) also obtains the vNM stable set (Proposition 2).
Without the no indifference condition, this result does not hold, and, moreover, a
collection of essential µ-compatible subgames may not exist. The following example
illustrates these facts.
Example 6. Let M = {1, 2, 3} and M ′
given by:
1′
1 35
2 10
3 16
= {1′ , 2′ , 3′ }. The valuation matrix A is
2′ 3′
10 20
10 10
1 10
24
The unique optimal matching is given by µ := {(1, 1′ ), (2, 2′ ), (3, 3′ )}. In this game,
µ
the buyer-side procedure terminates at step 2 and yields DM
= {∅, {3}}; the sellerµ
′
side procedure terminates at step 2 and yields DM ′ = {∅, {3 }}. Moreover, we have
µ
′
µ = {∅, {3}, {3′ }} obtains the vNM stable
D̄M
′ = {∅, {3 }}. From Proposition 1, D
set with µ which is denoted by V µ . We will show that (i) D̄µ does not obtain V µ ,
and (ii) a collection of essential µ-compatible subgames does not exist.
(i): In the subgame (M \ {3}, M ′ , A), µ′ = {(1, 1′ ), (2, 3′ )} is also optimal and
hence v̄2′ = v 2′ = 0 from Fact 1-(ii). Again, from Fact 1-(ii), we have ū2 = u2 = 10.
This implies that buyer 2 is inactive in (M \ {3}, M ′ , A). Therefore, we have that
µ
D̄M
= {∅} and hence D̄µ = {∅, {3′ }}. However, D̄µ does not obtain the vNM stable
set with µ. For example, the buyer-optimal stable payoff in Ĉ({3}), i.e.,
u1 = 35, u2 = 10, u3 = 10, v1′ = 0, v2′ = 0, v3′ = 0,
is not in Ĉ(∅) ∪ Ĉ({3′ }). Therefore, Ĉ(∅) ∪ Ĉ({3′ ) ̸= V µ .
µ
(ii): We first show that DM
= {∅, {3}, {3′ }} is a collection of minimal µcompatible subgames for obtaining V µ . Thereby, it is sufficient to show that Ĉ(∅) ∪
Ĉ({3}) ̸= V µ , Ĉ(∅) ∪ Ĉ({3′ }) ̸= V µ and Ĉ({3}) ∪ Ĉ({3′ ) ̸= V µ . We have already
shown that Ĉ(∅) ∪ Ĉ({3′ }) ̸= V µ . To see that Ĉ(∅) ∪ Ĉ({3}) ̸= V µ , consider the
seller-optimal stable payoff in Ĉ({3′ }), i.e.,
u1 = 0, u2 = 0, u3 = 0, v1′ = 35, v2′ = 10, v3′ = 10.
Then, this payoff vector is not contained in both Ĉ(∅) and Ĉ({3}). Therefore,
Ĉ(∅) ∪ Ĉ({3′ }) ̸= V µ . To see that Ĉ({3}) ∪ Ĉ({3′ }) ̸= V µ , consider the following
imputation:
u1 = 28, u2 = 9, u3 = 9, v1′ = 7, v2′ = 1, v3′ = 1.
Then, it is straightforward to show that (u, v) ∈ Ĉ(∅). However, (u, v) ∈
/ Ĉ({3})
′
′
′
because u3 = 10 for all (u , v ) ∈ Ĉ({3}). Similarly, we can get (u, v) ∈
/ Ĉ({3′ }).
Therefore, Ĉ({3}) ∪ Ĉ({3′ }) ̸= V µ .
Finally, we show that there is another collection of minimal µ-compatible subgames for obtaining V µ . Consider (M \ {2, 3}, M ′ , A), which is µ-compatible. Moreµ
over, Ĉ({2, 3}) coincides with Ĉ({3}). Therefore, DM
= {∅, {2, 3}, {3′ }} is also a
collection of minimal µ-compatible subgames to obtain V µ . This implies that a
collection of essential µ-compatible subgames does not exist in this example.
□
7
Concluding Remarks
We have presented an algorithm for finding the collection of essential µ-compatible
subgames under the positivity and no indifference conditions.
25
We briefly discuss future research on the vNM stable set for assignment games.
The assignment game introduced by Shapley and Shubik (1972) implicitly assumes
that all players’ utility can be measured by money. In other words, all agents have
quasi linear utility functions. However, Demange and Gale (1985) presented a model
of the assignment game wherein players’ utility functions may not be quasi liner, and
investigated properties of the core. It is interesting to analyze whether vNM stable
sets exist in the model of Demange and Gale (1985). We hope that the approach
taken in this paper will be useful for analyzing vNM stable sets in the assignment
games with general utility functions.
Appendix A: Proofs
A1: Basic Lemmas
In this section, we introduce basic lemmas to obtain our main results (Proposition
1, 2, and Theorem 2).
Let (M, M ′ , A) be a symmetric assignment game, µ be an optimal matching in
(M, M ′ , A) and C µ be the set of all µ-compatible subgames. Throughout this section,
we assume that A is positive and hence no player is unmatched at µ.
We first introduce a well-known comparative static result of the core: When
some buyers leave the game, the remaining buyers are never worse off and all sellers
are never better off in the extreme stable payoffs.
Fact 3 (Demange and Gale (1985)). Let Q, Q′ ⊊ M with Q ⊆ Q′ . Then, we have
that
′
′
Q
Q
Q
′
ūQ
i ≥ ūi and ui ≥ ui for all i ∈ M \ Q ,
′
′
Q
′
v̄jQ ≥ v̄jQ and v Q
j ≥ v j for all j ∈ M .
The following lemma states that any µ-compatible set includes only buyers or
only sellers.
Lemma 1. For every Q ∈ C µ , either Q ⊆ M or Q ⊆ M ′ .
Proof. Let Q ∈ C µ . Suppose that Q contains both sellers and buyers, i.e, Q ∩ M ̸= ∅
and Q ∩ M ′ ̸= ∅. Take any i ∈ Q ∩ M and j ∈ Q ∩ M ′ . By positivity, aiµ(i) > 0
and aµ(j)j > 0. From the definition of Q, this implies that µ(i) ∈
/ Q and µ(j) ∈
/ Q.
′
′
Therefore, µ := µ−Q ∪ {(µ(i), µ(j))} is a feasible matching in (M \ Q, M \ Q, A).
By positivity, we have that
∑
∑
∑
aij =
aij + aµ(j)µ(i) >
aij .
(i,j)∈µ′
(i,j)∈µ−Q
(i,j)∈µ−Q
This contradicts the definiton of µ-compatible set.
26
From here, we introduce properties related to a µ-compatible set of buyers. All
of the following lemmas symmetrically hold if we consider µ-compatible set of sellers.
For any Q ∈ C µ with Q ⊆ M , we say that i ∈ M \Q is a 0-buyer in (M \Q, M ′ , A)
if ūQ
i = aiµ(i) , i.e, buyer i gets µ(i)’s object with price 0 at the buyer-optimal
stable payoff. The following lemma states that any set of 0-buyers, say S, in a
µ-compatible subgame (M \ Q, M ′ , A), the restriction of the buyer-optimal stable
payoff in (M \ Q, M ′ , A) into M \ (Q ∪ S) ∪ M ′ is stable in (M \ (Q ∪ S), M ′ , A),
and Q ∪ S is also µ-compatible.
Lemma 2. Let Q ∈ C µ with Q ⊆ M . Take any S ⊆ M \Q with Q∪S ⊊ M . Suppose
′
that ūQ
i = aiµ(i) for all i ∈ S. Define a payoff vector (u, v) in (M \ (Q ∪ S), M , A)
by
Q
′
ui := ūQ
i for all i ∈ M \ (Q ∪ S), and vj := v j for all j ∈ M .
Then, (u, v) is compatible with µ−Q∪S and stable in (M \ (Q ∪ S), M ′ , A). Moreover,
Q ∪ S ∈ Cµ.
Proof. We first show that (u, v) is compatible with µ−Q∪S in (M \ (Q ∪ S), M ′ , A).
Take any (i, j) ∈ µ−Q∪S . This implies that (i, j) ∈ µ−Q . By the µ-compatibality of
Q
Q, we have ui +vj = ūQ
i +v j = aij . Consider any unmatched player j in µ−Q∪S such
that j ∈ M \ (Q ∪ S) ∪ M ′ . Then, j ∈ µ(S) or j ∈ µ(Q). We will show that vj = 0.
Suppose that j ∈ µ(S). Define i := µ(j) ∈ S. Then, we have (i, j) ∈ µ−Q . By the
Q
Q
µ-compatibality of Q, we have ūQ
i + v j = aij . From the assumption of ūi = aij ,
we can obtain vj = v Q
j = 0. Suppose that j ∈ µ(Q). By the µ-compatibality
of Q, we have that vj = v Q
j = 0. Therefore, (u, v) is compatible with µ−(Q∪S) .
Q
Q
The stability of (ū , v ) in (M \ Q, M ′ , A) directly implies the stability of (u, v)
in (M \ (Q ∪ S), M ′ , A). From Fact 1-(i), this implies that µ−Q∪S is optimal in
(M \ (Q ∪ S), M ′ , A) and hence Q ∪ S is a µ-compatible set.
Note that for any µ-compatible subgame (M \ Q, M ′ , A) and any j ∈ M ′ with
> 0, all buyers in Pj (Q) are 0-buyers in (M \ Q, M ′ , A). We also have that
µ(j) ∈
/ Q ∪ Pj (Q) and hence Q ∪ Pj (Q) ⊊ M . Therefore, Lemma 2 implies the
following lemma which guarantees that every element in Dµ is µ-compatible.
vQ
j
µ
Lemma 3. For any Q ∈ C µ and any j ∈ M ′ with v Q
j > 0, Q ∪ Pj (Q) ∈ C .
We next introduce two lemmas related to a set of essential buyers. The following lemma states that when a set of essential buyers from j is removed from a
µ-compatible subgame, the buyer-optimal stable payoff of j strictly decreases.
Lemma 4. Let Q ∈ C µ with Q ⊆ M . Consider any seller j ∗ ∈ M ′ with vQ
j ∗ > 0 and
Q
Q∗
∗
Q := Q ∪ Pj ∗ (Q). Then, we have that v j ∗ > v j ∗ .
27
∗
∗
Proof. By the optimality of (ūQ , v Q ), it is sufficient to show that there exists a
stable payoff (u, v) in (M \ Q∗ , M ′ , A) such that v Q
j ∗ > vj ∗ .
′
∗
Let M0 := µ(Q ) be the set of unmatched sellers at µ−Q∗ and M1′ := M ′ \ M0′ be
the set of sellers who match with some buyer in M \ Q∗ at µ−Q∗ . Note that v Q
j =0
′
for all j ∈ M0 by the µ-compatibility of Q and the definition of Pj ∗ (Q). Define
′
M11
:= {j ∈ M ′ |v Q
j > 0 and Pj (Q) ⊆ Pj ∗ (Q)}.
′ . Because v Q > 0 for all j ∈ M ′ , M ′ ⊆ M ′ holds. Let M ′ :=
Clearly, j ∗ ∈ M11
11
11
1
12
j
′ . Then, M ′ is partitioned into M ′ , M ′ and M ′ . We denote µ(M ′ ) by
M1′ \ M11
0
11
12
11
′ ) by M . Then, M \ Q∗ is partitioned into M
M11 and µ(M12
12
11 and M12 .
′ ,
We next show that for all i ∈ M12 and all j ∈ M11
Q
ūQ
i + v j > aij .
(4)
′ such that ūQ + v Q ≤ a .
Suppose not. Then, there exist i ∈ M12 and j ∈ M11
ij
i
j
Q
Q
Q
′
Q
Because (ū , v ) is stable in (M \ Q, M , A), from i ∈ M \ Q, we have ūi + v j ≥ aij
Q
Q
Q
and hence ūQ
i + v j = aij . By the µ-compatibility of Q, we have ūi + v µ(i) = aiµ(i) .
Q
By the individual rationality, we have that ūQ
i = aiµ(i) or ūi < aiµ(i) . Suppose that
Q
Q
Q
ūQ
i = aiµ(i) . From ūi + v j = aij and v j > 0, i is connected from j at µ−Q in
′ implies that i ∈ P ∗ (Q) and hence i ∈ Q∗ . However,
(M \ Q, M ′ , A). Then, j ∈ M11
j
this contradicts the fact that i ∈ M12 . Therefore, we have ūQ
i < aiµ(i) which implies
Q
′
′
/ M11 , there exists i ∈
/ Pj ∗ (Q) that is connected from
v µ(i) > 0. Because µ(i) ∈
Q
Q
′ / P ∗ (Q) is also
µ(i) at µ−Q in (M \ Q, M ′ , A). By ūQ
j
i + v j = aij and vj > 0, i ∈
′
connected from j at µ−Q in (M \ Q, M , A). However, this contradicts the fact that
′ . Therefore, we can obtain (4).
j ∈ M11
We now construct a stable payoff (u, v) in (M \ Q∗ , M ′ ) such that v Q
j ∗ > vj ∗ .
Q
′
By (4) and the fact that vj > 0 for all j ∈ M11 , there exists ϵ > 0 such that (i)
Q
Q
′
′
ūQ
i + v j − ϵ > aij for all i ∈ M12 and all j ∈ M11 and (ii) v j − ϵ > 0 for all j ∈ M11 .
Define a payoff vector (u, v) in (M \ Q∗ , M ′ , A) by
Q
′
ui = ūQ
i + ϵ if i ∈ M11 and vj = v j − ϵ if j ∈ M11 ,
Q
′
ui = ūQ
i if i ∈ M12 and vj = v j if j ∈ M12 , and
′
vj = v Q
j (= 0) for all j ∈ M0 .
′ ,
It is straightforward to check that (u, v) is stable in (M \ Q∗ , M ′ , A). By j ∗ ∈ M11
Q
we have that v j ∗ > vj ∗ . This completes the proof.
Finally, the following lemma shows that a converse of Lemma 4 holds: Given a
µ-compatible subgame, if seller j’s buyer-optimal stable payoff strictly decreases by
removing a set of 0-buyers S, then S must contain the set of essential buyers from
j.
28
Lemma 5. Let Q ∈ C µ with Q ⊆ M . Take any S ⊆ M \Q with Q∪S ⊊ M . Suppose
Q
Q∪S
′
, then Pj (Q) ⊆ S.
that ūQ
i = aiµ(i) for all i ∈ S. For any j ∈ M , if v j > v j
Q∪S
. This implies that v Q
Proof. Let j ∈ M ′ . Suppose that v Q
j > vj
j > 0 and hence
Pj (Q) is well-defined. We will show that Pj (Q) ⊆ S. Pick any i ∈ Pj (Q). Then, i (in
M \ Q) is connected from j at µ−Q in (M \ Q, M ′ , A). That is, there is a sequence of
distinct players j0 = j, i1 , j1 , · · · , in , jn , in+1 = i with n ≥ 0 and {i1 , · · · , in } ⊆ M \Q
and {j1 , · · · , jn } ⊆ M ′ such that
(i) µ(ik′ ) = jk′ for all k ′ = 1, · · · , n,
′
(ii) v Q
jk > 0 for all k = 1, · · · , n,
Q
(iii) ūQ
ik+1 + v jk = aik+1 jk for all k = 0, · · · , n,
(iv) ūQ
i = aiµ(i) .
Q
Q
Q
′
By ūQ
ik′ + v jk′ = aik′ jk′ and vjk′ > 0, we have that ūik′ < aik′ jk′ for all k = 1, · · · , n.
From the assumption of S, we have that ik′ ∈
/ S for all k ′ = 1, · · · , n and hence
{i1 , · · · , in } ⊆ M \ (Q ∪ S).
Q∪S
Q∪S
. By v Q
We next show that for all j ∈ {j0 , j1 , · · · , jn }, v Q
j > vj
j0 > v j0 , it is
Q∪S
Q∪S
implies v Q
sufficient to show that for any k ∈ {0, 1, · · · , n−1}, v Q
jk > v jk
jk+1 > v jk+1 .
Q∪S
Let k ∈ {0, 1, · · · , n − 1}. Suppose that v Q
jk > v jk . By ik+1 ∈ {i1 , · · · , in }, the
Q
stability of (ūQ∪S , v Q∪S ) implies that ūiQ∪S
+v Q∪S
≥ aik+1 jk . By ūQ
jk
ik +1 +v jk = aik+1 jk
k+1
Q∪S
Q∪S
Q
and v Q
jk > v jk , we can obtain ūik+1 > ūik+1 . Note that Q ∪ S is µ-compatible from
Q∪S
Q
Q
Lemma 2. Therefore, we have that ūQ∪S
ik+1 + v jk+1 = aik+1 jk+1 . By ūik+1 + v jk+1 =
Q
Q
Q∪S
aik+1 jk+1 and ūQ∪S
ik+1 > ūik+1 , we can obtain v̄jk+1 > v jk+1 .
Finally, we show i(= in+1 ) ∈ S. Suppose that i ∈
/ S. By i ∈ Pj (Q), we have
Q∪S
Q∪S
), ūQ∪S
+ v Q∪S
i ∈
/ Q ∪ S. Therefore, by the stability of (ū
,v
≥ aijn . By
i
jn
Q
Q
Q∪S
Q∪S
Q
Q
ūi + v jn = aijn and v jn > v jn , we have that ūi
> ūi = aiµ(i) . However,
Q∪S
this implies that v µ(i) < 0 by the µ-compatibility of Q ∪ S. This contradicts the
individual rationality of (ūQ∪S , v Q∪S ). Therefore, we have i ∈ S which implies
Pj (Q) ⊆ S.
A2: Proof of Proposition 1
We first introduce a useful lemma to prove Proposition 1.
Lemma 6. Let Q ⊆ C µ with Q ⊆ M . Take any S ⊆ M \ Q with Q ∪ S ⊊ M .
Q
Q∪S
Suppose that ūQ
for all j ∈ M ′ , then
i = aiµ(i) for all i ∈ S. If v j = v j
Ĉ(Q ∪ S) ⊆ Ĉ(Q).
29
Q∪S
Proof. We will show that Ĉ(Q ∪ S) ⊈ Ĉ(Q) implies v Q
for some j ∗ ∈ M ′ .
j ∗ ̸= v j ∗
By Ĉ(Q ∪ S) ⊈ Ĉ(Q), there exists (u, v) ∈ Ĉ(Q ∪ S) such that (u, v) ∈
/ Ĉ(Q). By
(u, v) ∈ Ĉ(Q ∪ S), we have that (i) (uM \(Q∪S) , v) is stable in (M \ (Q ∪ S), M ′ , A)
and (ii) ui = aiµ(i) for all i ∈ Q ∪ S. Since ui = aiµ(i) for all i ∈ S, it is easy to
check that (uM \Q , v) is compatible with µ−Q in (M \ Q, M ′ , A). This implies that
(uM \Q , v) is feasible and individually rational in (M \ Q, M ′ , A).
We first show that there exist i∗ ∈ S and j ∗ ∈ M ′ such that ui∗ + vj ∗ < ai∗ j ∗ .
Because ui = aiµ(i) for all i ∈ Q, (u, v) ∈
/ Ĉ(Q) implies that (uM \Q , v) is not stable
′
in (M \ Q, M , A). This implies that there exist i∗ ∈ M \ Q and j ∗ ∈ M ′ such
that ui∗ + vj ∗ < ai∗ j ∗ because (uM \Q , v) is feasible and individually rational in
(M \ Q, M ′ , A). The stability of (uM \(Q∪S) , v) in (M \ (Q ∪ S), M ′ , A) implies that
i∗ ∈ S.
Q∪S
Q∪S , v Q∪S ) in (M \
Finally, we show that v Q
j ∗ > v j ∗ . By the optimality of (ū
Q∪S
(Q ∪ S), M ′ , A), we have that v j ∗ ≤ vj ∗ which implies that ui∗ + v Q∪S
< ai ∗ j ∗ .
j∗
Q
Q
′
On the other hand, by the stability of (ū , v ) in (M \ Q, M , A), we have that
Q
Q
Q
Q∪S
ūQ
i∗ + v j ∗ ≥ ai∗ j ∗ . Therefore, we have that ūi∗ + v j ∗ > ui∗ + v j ∗ . From the
Q
assumption of S, we have ūQ
i∗ = ai∗ µ(i∗ ) and hence ūi∗ = ui∗ . Therefore, we can
Q∪S
obtain v Q
j ∗ > v j ∗ . This completes the proof.
We now prove Proposition 1. From Theorem 1, it is sufficient to show that for
each QC ∈ C µ , there exists QD ∈ Dµ such that Ĉ(QC ) ⊆ Ĉ(QD ). Pick any QC ∈ C µ .
By Lemma 1, QC ⊆ M or QC ⊆ M ′ . Without loss of generality, we assume that
QC ⊆ M .
µ
µ
Note that there exists an element in DM
that is contained in QC by ∅ ∈ DM
.
µ
Let QD be a maximal element in DM that is contained in QC , i.e, QD is an element
µ
µ
of DM
such that (i) QD ⊆ QC and (ii) there is no Q ∈ DM
with QD ⊊ Q ⊆ QC . We
will show that Ĉ(QC ) ⊆ Ĉ(QD ). Let S := QC \ QD . When S = ∅, then QC = QD
and hence the proof has be done. So, we assume S ̸= ∅.
D
To use Lemma 6, we first show that ūQ
= aiµ(i) for all i ∈ S. Suppose not.
i
QD
D
Then, there exists i0 ∈ S with ūi0 < ai0 µ(i0 ) . Let µ(i0 ) := j0 . By ūQ
i0 < ai0 µ(i0 ) , we
D
′
have v Q
j0 > 0. By the maximality of QD , Pj0 (QD ) ⊈ S. So, there exists i ∈ M \ QD
such that i′ ∈
/ S and i′ is connected from j0 at µ−QD in (M \ QD , M ′ , A). That is,
there is a sequence of distinct players j0 , i1 , j1 , · · · , in , jn , in+1 = i′ with n ≥ 0,
{i1 , · · · , in } ⊆ M \ QD and {j1 , · · · , jn } ⊆ M ′ such that
(i) µ(ik′ ) = jk′ for all k ′ = 1, · · · , n,
D
′
(ii) v Q
jk > 0 for all k = 1, · · · , n,
QD
D
(iii) ūQ
ik+1 + v jk = aik+1 jk for all k = 0, · · · , n,
30
D
(iv) ūQ
in+1 = ain+1 µ(in+1 ) .
Let jn+1 := µ(in+1 ). By i0 ∈ S, {0 ≤ k ≤ n + 1|ik ∈ S} ̸= ∅. Define l := max{0 ≤
k ≤ n + 1|ik ∈ S}. By in+1 ∈
/ S, n + 1 ∈
/ {0 ≤ l ≤ n + 1|il ∈ S} and hence
0 ≤ l < n + 1. From definition, for all k with n + 1 ≥ k > l, ik ∈
/ S and hence
ik ∈ M \ QC . Moreover, by il ∈ S ⊆ QC , jl is unmatched in µ−QC . Therefore,
µ′ := (µ−QC \{(il+1 , jl+1 ), (il+2 , jl+2 ) · · · , (in+1 , jn+1 )})
∪
{(il+1 , jl ), (il+2 , jl+1 ), · · · , (in+1 , jn )}
is a matching in (M \ QC , M ′ , A). We also have that
∑
aij −
(i,j)∈µ′
∑
aij = ail+1 jl + ail+2 jl+1 + · · · + ain+1 jn − (ail+1 jl+1 + ail+2 jl+2 · · · + ain+1 jn+1 ).
(i,j)∈µ−QC
Note that
ail+1 jl + ail+2 jl+1 · · · + ain+1 jn
QD
QD
QD
QD
QD
D
= (v Q
jl + ūil+1 ) + (v jl+1 + ūil+2 ) + · · · + (v jn + ūin+1 )
(from (iii))
QD
QD
QD
QD
QD
QD
QD
D
= vQ
jl + (ūil+1 + v jl+1 ) + (ūil+2 + v il+2 ) + · · · + (ūin + v jn ) + ūin+1
QD
D
= vQ
jl + ail+1 jl+1 + ail+2 jl+2 + · · · + ain jn + ūin+1
D
= vQ
jl + ail+1 jl+1 + ail+2 jl+2 + · · · + ain jn + ain+1 jn+1 .
(from (i))
(from (iv))
However, this implies that
∑
(i,j)∈µ′
aij −
∑
D
aij = v Q
jl > 0. (from (ii))
(i,j)∈µ−QC
D
This contradicts the µ-compatibility of QC . Therefore, we can obtain that ūQ
=
i
aiµ(i) for all i ∈ S.
D
C
We finally show that v Q
= vQ
for all j ∈ M ′ , which implies Ĉ(QC ) ⊆ Ĉ(QD )
j
j
D
C
from Lemma 6. Suppose not. Then, for some j ∈ M ′ , v Q
̸= v Q
j
j . From Fact
D
C
3, we have that v Q
> vQ
j
j . By Lemma 5, this implies that Pj (QD ) ⊆ S and
hence QD ∪ Pj (QD ) ⊆ QC . However, this contradicts the maximality of QD . This
completes the proof.
A3: Proof of Proposition 2
We first introduce element properties related to inactive players for a general assignment game (M, M ′ , A). Let µ be an optimal matching in (M, M ′ , A′ ). It is
straightforward to check that for each player l ∈ M ∪ M , if l is inactive, then µ(l) is
also inactive. Moreover, for each i ∈ M and j ∈ M ′ , if j is inactive and ūi +v j = aij ,
31
then i is also inactive.6 By repeating these properties, we can obtain that for any
inactive seller j ∈ M ′ with v j > 0 and any i ∈ M , if i is connected from j at µ
in (M, M ′ , A), then i is also inactive. Similarly, for any inactive buyer i ∈ M with
ui > 0 and any j ∈ M ′ , if j is connected from i at µ in (M, M ′ , A), then j is also
inactive.
Let us return to the analysis of a symmetric assignment game (M, M ′ , A) such
that A is positive. We first show that all players are active in the original game.
Lemma 7. All players are active in (M, M ′ , A).
Proof. Suppose not. Without loss of generality, there exists an inactive buyer i
because no player is unmatched at µ. We first show that there exists an inactive
buyer i′ such that ūi′ = ai′ µ(i′ ) . If ūi = aiµ(i) , the proof has been done. Otherwise,
we have that ūi < aiµ(i) which implies v µ(i) > 0. So, there exists i′ ∈ M who
is connected from µ(i). From definition, we have ūi′ = ai′ µ(i′ ) . Because µ(i) is
inactive, i′ is also inactive. We next show that there exists an inactive seller j ′ such
that v̄j ′ = aµ(j ′ )j ′ . By ūi′ = ui′ = ai′ µ(i′ ) > 0 (from positivity), there exists j ′ ∈ M ′
who is connected from i′ at µ. From definition, we have v̄j ′ = aµ(j)′ j ′ . Because i′ is
inactive, j ′ is also inactive. Then, µ(i′ ) ∈ M ′ and µ(j ′ ) ∈ M are inactive players with
v̄µ(i′ ) = v µ(i′ ) = 0 and ūµ(j ′ ) = uµ(j ′ ) = 0. Therefore, ūµ(i′ ) + v µ(j ′ ) = 0 < aµ(i′ )µ(j ′ )
by positivity. This contradicts the stability of (ū, v).
We next show that when we remove inactive 0-buyers from a µ-compatible subgame, (i) the extended core expands and (ii) the seller-optimal stable payoffs are
unchanged.
Lemma 8. Let Q ∈ C µ with Q ⊆ M . Take any S ⊆ M \ Q with Q ∪ S ⊊ M .
′
Suppose that for all i ∈ S, i is inactive with ūQ
i = aiµ(i) in (M \ Q, M , A). Then,
(i) Ĉ(Q) ⊆ Ĉ(Q ∪ S), and
Q∪S
(ii) uQ
for all i ∈ M \ (Q ∪ S).
i = ui
Proof. (i): Pick any (u, v) ∈ Ĉ(Q). By the assumption of S, we have that ui = aiµ(i)
for all i ∈ Q ∪ S. This implies that (uM \(Q∪S) , v) is compatible with µ−Q∪S , and
hence feasible in (M \ (Q ∪ S), M ′ , A). The stability of (uM \Q , v) in (M \ Q, M ′ , A)
directly implies the stability of (uM \(Q∪S) , v) in (M \ (Q ∪ S), M ′ , A). Therefore,
(u, v) ∈ Ĉ(Q ∪ S).
6
To see this, suppose that j is inactive and ūi + v j = aij , but ūi > ui . This implies that
ui + v j < aij . By v̄j = v j , we have that ui + v j = ui + v̄j < aij . This contradicts the stability of
(u, v̄). Therefore, i is inactive.
32
Q∪S
(ii): Suppose that uQ
for some i ∈ M \ (Q ∪ S). Then, by Fact 3, we
i ̸= ui
Q∪S
Q
> ui . Therefore, there exists j ∈ M ′ who is connected from i at
have that ui
µ−Q∪S in (M \ (Q ∪ S), M ′ , A). That is, there is a sequence of distinct players i0 =
i, j1 , i1 , · · · , in , jn+1 = j with n ≥ 0, {i1 , · · · , in } ⊆ M \(Q∪S) and {j1 , · · · , jn } ⊆ M ′
such that
(i) µ(ik′ ) = ik′ for all k ′ = 1, · · · , n,
+ v̄jQ∪S
= aik jk+1 for all k = 0, · · · , n, and
(ii) uQ∪S
ik
k+1
(iii) v̄jn+1 = aµ(jn+1 )jn+1 .
Note that by {i0 , · · · , in } ⊆ M \Q, the stability condition for (uQ , v̄ Q ) can be applied
to all buyers in {i0 , · · · .in }.
We first show that v̄jQ > v̄jQ∪S for all j ∈ {j1 , · · · , jn+1 } and uQ∪S
> uQ
i
i for
Q∪S
Q
all i ∈ {i0 , i1 , · · · , in }. By ui0
> ui0 , it is sufficient to show that (i) for any
Q∪S
Q
Q∪S
′
k ∈ {0, 1, · · · , n}, uik > uik implies v Q
ik+1 > v ik+1 and (ii) for any k ∈ {1, · · · , n},
Q∪S
vQ
implies uQ∪S
> uQ
i ′ > vi ′
i ′
i ′ . We first show (i). Take any k ∈ {0, 1, · · · , n}.
k
k
k
k
Q
Q
Q Q
Suppose that uQ∪S
> uQ
ik
ik . By the stability of (u , v̄ ), we have that uik + v̄jk+1 ≥
Q
Q∪S
aik jk+1 . By uQ∪S
+ v̄jQ∪S
= aik jk+1 and uQ∪S
> uQ
ik
ik
ik , we can obtain v̄jk+1 > v̄jk+1 .
k+1
We next show (ii). Take any k ′ ∈ {1, · · · , n}. Suppose that v̄jQ ′ > v̄jQ∪S
. Note that
′
k
k
Q
(ik′ , jk′ ) ∈ µ−Q . So, by the µ-compatibility of Q, we have uQ
i ′ + v̄j ′ = aik′ jk′ . From
k
k
Q∪S
uQ∪S
= aik′ jk′ and v̄jQ ′ > v̄jQ∪S
, we can obtain uQ∪S
> uQ
ik′ + v̄jk′
ik′
ik′ .
k
k′
We next show µ−Q∪S (jn+1 ) = jn+1 . Suppose that µ−Q∪S (jn+1 ) ∈ M . We
denote µ−Q∪S (jn+1 ) by in+1 . By (in+1 , jn+1 ) ∈ µ−Q∪S , we have that (in+1 , jn+1 ) ∈
Q
Q
µ−Q . By the µ-compatibility of Q, we have uQ
in+1 + v̄jn+1 = ain+1 jn+1 . By v̄jn+1 >
v̄jQ∪S
= ain+1 jn+1 , we have that uQ
in+1 < 0. However, this contradicts the individual
n+1
Q
Q
rationality of (ū , v ). Therefore, µ−Q∪S (jn+1 ) = jn+1 .
By µ−Q∪S (jn+1 ) = jn+1 , we have jn+1 ∈ µ(Q) or jn+1 ∈ µ(S). Suppose that
jn+1 ∈ µ(Q). Then, we have v̄jQn+1 = 0 by the µ-compatibility of Q. By v̄jQn+1 > v̄jQ∪S
,
n+1
Q∪S
we have 0 > v̄jn+1 . This contradicts the individual rationality of (uQ∪S , v̄ Q∪S ).
Therefore, jn+1 ∈ µ(S). From the assumption of S, jn+1 is inactive in (M \Q, M ′ , A)
with v̄jQn+1 = 0. By v̄jQn+1 > v̄jQ∪S
, we have 0 > v̄jQ∪S
. This contradicts the individual
n+1
n+1
Q∪S
Q∪S
Q∪S
, v̄
). Therefore, we have that uQ
for all i ∈ M \ (Q ∪
rationality of (u
i = ui
S).
Next lemma shows that given a µ-compatible subgame, when we remove a set of
essential buyers from an inactive seller, (i) the extended core expands and (ii) the
set of inactive sellers who gets a positive buyer-optimal payoff strictly shrinks.
Lemma 9. Let Q ∈ C µ with Q ⊆ M . Take any j ∗ ∈ M ′ who is inactive in
∗
(M \ Q, M ′ , A). Suppose that v Q
j ∗ > 0 and Q := Q ∪ Pj ∗ (Q). Then, we have that
33
(i) Ĉ(Q) ⊆ Ĉ(Q∗ ) and (ii)
∗
∗
′
{j ∈ M ′ | j is inactive with v Q
j > 0 in (M \ Q , M , A)}
′
⊊ {j ∈ M ′ | j is inactive with v Q
j > 0 in (M \ Q, M , A)}.
Proof. (i): Note that for all i ∈ Pj ∗ (Q), i is inactive with ūi = aiµ(i) because j ∗ is
inactive. From Lemma 8-(i), we have that Ĉ(Q) ⊆ Ĉ(Q∗ ).
∗
∗
′
(ii): Let I ∗ := {j ∈ M ′ | j is inactive with v Q
j > 0 in (M \ Q , M , A)} and I :=
Q
{j ∈ M ′ | j is inactive with v j > 0 in (M \ Q, M ′ , A)}. We will show that I ∗ ⊊ I.
∗
We first show that I ∗ ⊆ I. Pick any j ∈ I ∗ . Then, j is inactive with v Q
j > 0
∗
in (M \ Q∗ , M ′ , A). By v Q
> 0, µ(j) ∈ M \ Q∗ holds and µ(j) is inactive in
j
∗
Q
Q∗
Q∗
(M \ Q∗ , M ′ , A). From Lemma 7-(ii), uQ
µ(j) = uµ(j) . By ūµ(j) = uµ(j) , we have that
∗
∗
Q
Q
Q
Q
Q
ūQ
µ(j) = uµ(j) . From Fact 3, ūµ(j) ≥ ūµ(j) which implies that uµ(j) ≥ ūµ(j) . On the
Q
Q
Q
other hand, uQ
µ(j) ≤ ūµ(j) from definition. Therefore, we have that ūµ(j) = uµ(j)
which implies that µ(j) is inactive in (M \ Q, M ′ , A). Therefore, j is inactive in
Q∗
Q
(M \ Q, M ′ , A). Moreover, from Fact 3, v Q
j ≥ v j and hence v j > 0. This implies
that j ∈ I.
To complete the proof, we need to show that I ′ ⊊ I. By the assumption, j ∗ ∈ I.
Q∗
Q∗
Q
From Lemma 4, we have that v Q
j ∗ > v j∗ . From Lemma 3, v̄j ∗ ≥ v j ∗ and hence
∗
∗
∗ / I ∗ . Therefore, we can obtain I ∗ ⊊ I.
v̄jQ∗ > v Q
j ∗ . This implies that j ∈
Next lemma states that under the no indifference condition, given a µ-compatible
subgame, if there exists an inactive buyer, then there exists an inactive seller who
gets a positive buyer-optimal stable payoff.
Lemma 10. Suppose that A is not indifferent. Consider any Q ∈ C µ with Q ⊆ M .
If there exist an inactive buyer in (M \ Q, M ′ , A), then there exists an inactive seller
j ∈ M ′ such that v Q
j > 0.
Proof. We denote by i ∈ M \ Q an inactive buyer in (M \ Q, M ′ , A). Suppose that
Q
ūQ
i < aiµ(i) . Then, µ(i) is an inactive seller with v µ(i) > 0 and the proof has been
Q
Q
done. So, we assume that ūQ
i = aiµ(i) . Then, we have ūi = ui = aiµ(i) > 0 by
Q
positivity. From Fact 2, there exists j ∈ M ′ such that j ̸= µ(i) and uQ
i + v̄j = aij ,
Q
which implies that j is also inactive in (M \ Q, M ′ , A). Therefore, uQ
i + v j = aij .
Q
By uQ
i = aiµ(i) , we have v j = aij − aiµ(i) . By the individual rationality, we have
Q
vQ
j = aij − aiµ(i) ≥ 0. By the no indifference condition, this implies that v j > 0.
This completes the proof.
We now prove Proposition 2, i.e, D̄µ also obtains V µ under the positivity and no
indifference conditions. From Propostition 1, it is sufficient to show that for every
µ
µ
µ
Q ∈ DM
, there exists Q̄ ∈ D̄M
such that Ĉ(Q) ⊆ Ĉ(Q̄). Let Q ∈ DM
. By iterative
34
µ
applications of Lemma 9, there exists Q̄ ∈ DM
such that (i) Ĉ(Q) ⊆ Ĉ(Q̄) and (ii)
Q̄
′
there exists no j ∈ M who is inactive with v j > 0 in (M \ Q̄, M ′ , A). From Lemma
10, this implies that there are no inactive buyers in (M \ Q̄, M ′ , A). Therefore,
µ
. This completes the proof.
Q̄ ∈ D̄M
A3: Proof of Theorem 1
µ
µ
To show the strict effectiveness property of DM
(or DM
′ ), we need to introduce the
following lemma which states that the intersection of µ-compatible sets of buyers is
also µ-compatible.
Lemma 11. For any Q1 and Q2 ∈ C µ with Q1 , Q2 ⊆ M , Q1 ∩ Q2 ∈ C µ .
Proof. We show that µ−Q1 ∩Q2 is optimal in (M \ (Q1 ∩ Q2 ), M ′ , A). From Fact 1-(i),
it is sufficient to show that there exists a stable payoff in (M \ (Q1 ∩ Q2 ), M ′ , A)
that is compatible with µ−Q1 ∩Q2 . Let (u1 , v 1 ) and (u2 , v 2 ) be stable payoffs in
(M \ Q1 , M ′ , A) and (M \ Q2 , M ′ , A) respectively. Note that vj1 = 0 for all j ∈
µ(Q1 ) and vj2 = 0 for all j ∈ µ(Q2 ) by the µ-comaptibility of Q1 and Q2 . Define
∗
vj∗ := max{vj1 , vj2 } for all j ∈ M ′ and u∗i := aiµ(i) − vµ(i)
for all i ∈ M \ (Q1 ∩ Q2 ).
∗
∗
′
Then, (u , v ) is a payoff vector in (M \ (Q1 ∩ Q2 ), M , A).
We first show that (u∗ , v ∗ ) is compatible with µ−Q1 ∩Q2 in (M \(Q1 ∩Q2 ), M ′ , A).
Pick any (i, j) ∈ µ−Q1 ∩Q2 . Then, i ∈ M \ (Q1 ∩ Q2 ) and µ(i) = j. From definition,
u∗i + vj∗ = aij − vj∗ + vj∗ = aij . Pick any j ∈ µ(Q1 ∩ Q2 ). Then, vj1 = vj2 = 0 and hence
vj∗ = 0. Therefore, (u∗ , v ∗ ) is compatible with µ−Q1 ∩Q2 in (M \ (Q1 ∩ Q2 ), M ′ , A).
We now show the stability of (u∗ , v ∗ ) in (M \ (Q1 ∩ Q2 ), M ′ , A). We first show
that u∗i + vj∗ ≥ aij for all i ∈ M \ (Q1 ∩ Q2 ) and all j ∈ M ′ . Let i ∈ M \ (Q1 ∩ Q2 )
and j ∈ M ′ . There are 3 cases to consider.
1
Case 1 (i ∈ Q1 and i ∈ M \ Q2 ): In this case, µ(i) ∈ µ(Q1 ) and hence vµ(i)
= 0.
∗
2
This implies that vµ(i) = vµ(i) . By i ∈ M \ Q2 , (i, µ(i)) ∈ µ−Q2 holds. By the µ2
compatibility of Q2 , u2i = aiµ(i) − vµ(i)
and hence u∗i = u2i . From the definition of v ∗ ,
we have that u∗i + vj∗ = u2i + vj∗ ≥ u2i + vj2 . The stability of (u2 , v 2 ) in (M \ Q2 , M ′ , A)
implies u2i + vj2 ≥ aij . Therefore, u∗i + vj∗ ≥ aij .
Case 2 (i ∈ Q2 and i ∈ M \ Q1 ): We can obtain u∗i + vj∗ ≥ aij by the same
argument as in Case 1.
Case 3 (i ∈
/ Q1 ∪ Q2 ): In this case, i ∈ M \ Q1 and i ∈ M \ Q2 hold. Suppose that
∗
1
1
vµ(i) = vµ(i) . By the µ-compatibility of Q1 , u1i = aiµ(i) − vµ(i)
and hence u∗i = u1i .
From the defintion of v ∗ , we have that u∗i + vj∗ = u1i + vj∗ ≥ u1i + vj1 . The stability of
(u1 , v 1 ) in (M \ Q1 , M ′ , A) implies that u1i + vj1 ≥ aij . Therefore, u∗i + vj∗ ≥ aij . In
∗
2 , we can obtain u∗ + v ∗ ≥ a by the same argument.
the case with vµ(i)
= vµ(i)
ij
i
j
35
The above analysis reveals that u∗i = u1i or u2i for all i ∈ M \ (Q1 ∩ Q2 ). This
implies that u∗i ≥ 0 for all i ∈ M \ (Q1 ∩ Q2 ). Clearly, vj∗ ≥ 0 for all j ∈ M ′ .
Therefore, (u∗ , v ∗ ) is stable in (M \ (Q1 ∩ Q2 ), M ′ , A). This completes the proof.
µ
We now prove the strict effectiveness property of DM
.
µ
Lemma 12. For every Q ∈ DM
and every S ∈ C µ with S ⊊ Q, there exists i ∈ M \Q
Q
such that ūi > ūSi .
µ
Proof. From the definition of DM
, it is sufficient to show that for every step k ≥ 0,
k
µ
every Q ∈ DM and every Q ∈ C with S ⊊ Q, there exists i ∈ M \ Q such that
S
ūQ
i > ūi . We will show this statement by induction.
0 = {∅}. Suppose that for
Clearly, in step 0, this statement holds because DM
l . That is, we assume that
step l(≥ 0), the statement of Lemma 12 is true for DM
l and every S ∈ C µ with S ⊊ Q, there exists i ∈ M \ Q such that
for every Q ∈ DM
l+1
l+1
S
µ
ūQ
i > ūi . We will show that this property holds for DM . Let Q ∈ DM and S ∈ C
l+1
l and for some j ∗ ∈ M ′
with S ⊊ Q. From the definition of DM
, for some Q∗ ∈ DM
∗
with vjQ∗ > 0, Q = Q∗ ∪ Pj ∗ (Q∗ ). There are two cases to consider: Pj (Q∗ ) ⊈ S or
Pj (Q∗ ) ⊆ S.
S
Case 1, Pj ∗ (Q∗ ) ⊈ S: Let µ(j ∗ ) := i∗ . It is sufficient to show that ūQ
i∗ > ūi∗
because i∗ ∈ M \ Q. Let S ∗ := Pj (Q∗ ) ∩ S. Then, by Pj ∗ (Q∗ ) ⊈ S, we have that
∗
S ∗ ⊊ Pj ∗ (Q∗ ). From the definiton of Pj (Q∗ ), ūQ
= aiµ(i) for all i ∈ Pj (Q∗ ) and
i
Q∗
hence ūi = aiµ(i) for all i ∈ S ∗ . From Lemma 3, Q∗ ∪ S ∗ is a µ-compatible set.
Consider the µ-compatible subgame (M \ (Q∗ ∪ S ∗ ), M ′ , A). Note that i∗ ∈
∗ ∪S ∗
∗
M \ (Q∗ ∪ S ∗ ) holds. We first show that ūQ
= ūQ
i∗
i∗ . Suppose not. Then,
∗ ∪S ∗
∗
∗
Q∗ ∪S ∗
∗
ūQ
̸= ūQ
> ūQ
i∗
i∗ . From Fact 3, ūi∗
i . By the µ-comaptibility of Q and
∗
∗
∗
∗
∗
∗
Q
Q ∪S
∪S
Q∗ ∪ S ∗ , we have that ūQ
+ vQ
= ai∗ j ∗ Therefore, we
i∗ + v j ∗ = ai∗ j ∗ and ūi∗
j∗
∗
∗
∗
Q
Q ∪S
∗
∗
have that v j ∗ > v̄j ∗
. From Lemma 5, Pj ∗ (Q ) ⊆ S . However, this contradicts
∗ ∪S ∗
∗
∗
∗
S ⊊ Pj ∗ (Q ). Therefore, we have ūQ
= ūQ
i∗
i∗ .
Q∗ ∪S ∗
S
∗
∗
We now show that ūQ
≥ ūSi∗ from
i∗ > ūi∗ . By S ⊆ Q ∪ S , we have that ūi∗
∗
∗
∗
∗
∗
Q
Q
Q
S
Fact 3. By ūiQ∗ ∪S = ūQ
i∗ , we have that ūi∗ ≥ ūi∗ . By Lemma 4, vj ∗ > vj ∗ . By the
∗
Q
Q
S
µ-compatibility of Q∗ and Q, this implies that ūQ
i∗ > ūi∗ . Therefore, ūi∗ > ūi∗ .
Case 2, Pj ∗ (Q∗ ) ⊆ S: Let S ∗ = Q∗ ∩ S. By Lemma 11, S ∗ ∈ C µ . By Pj ∗ (Q∗ ) ⊆ S
and S ⊊ Q∗ ∪ Pj ∗ (Q∗ ), we have S ∗ ⊊ Q∗ . By the µ-comaptibility of S ∗ , S and Q∗ ,
∗
∗
we have that v Sj = 0 for all j ∈ µ(S ∗ ), v Sj = 0 for all j ∈ µ(S) and v Q
j = 0 for all
j ∈ µ(Q∗ ). By µ(S ∗ ) ⊆ µ(S) and µ(S ∗ ) ⊆ µ(Q∗ ), we have that for all j ∈ µ(S ∗ ),
36
∗
∗
∗
′
v Sj = v Sj = v Q
j = 0. Define a payoff vector (u, v) in (M \ S , M , A) by:
ui := ūSi if i ∈ Q∗ \ S ∗ and vj := v Sj if j ∈ µ(Q∗ \ S ∗ ),
∗
∗
∗
ui := ūQ
if i ∈ M \ Q∗ and vj := v Q
i
j if j ∈ µ(M \ Q ),
∗
∗
∗
vj := 0(= v Sj = v Sj = v Q
j ) if j ∈ µ(S ).
We first show that (u, v) is compatible with µ−S ∗ . Pick any (i, j) ∈ µ−S ∗ . Then,
i ∈ Q∗ \ S ∗ or i ∈ M \ Q∗ . If i ∈ Q∗ \ S ∗ , then i ∈
/ S. This implies that (i, j) ∈ µ−S .
By the µ-compatibility of S, we have that ui +vj = ūSi +v Sj = aij . If i ∈ M \Q∗ , then
∗
Q∗
(i, j) ∈ µ−Q∗ . By the µ-compatibility of Q∗ , we have that ui + vj = ūQ
i + v j = aij .
The set of unmatched players in µ−S ∗ in (M \ S ∗ , M ′ , A) is equivalent to µ(S ∗ ).
Therefore, from definition, all unmatched players’ payoffs are 0 in (u, v).
Next, we show that there exist i1 ∈ Q∗ \ S ∗ and j2 ∈ µ(M \ Q∗ ) such that
∗
ūSi1 + v Q
j2 < ai1 j2 .
l , S ∗ ∈ C µ and S ∗ ⊊ Q∗ . By induction hypothesis, there exists
Note that Q∗ ∈ DM
∗
S∗
S∗ S∗
i ∈ M \ Q∗ such that ūQ
i > ūi . By the optimality of (ū , v ), this implies that
(u, v) is not stable in (M \S ∗ , M ′ , A). Therefore, there exist i1 ∈ M \S ∗ and j2 ∈ M ′
such that ui1 + vj2 < ai1 j2 , because (u, v) is compatible with µ−S ∗ and individually
rational in (M \S ∗ , M ′ , A). Moreover, we can obtain i1 ∈ Q∗ \S ∗ and j2 ∈ µ(M \Q∗ ).
To see this, consider any i ∈ M \ Q∗ and any j ∈ M ′ . If j ∈ µ(S ∗ ) ∪ µ(M \ Q∗ ),
∗
Q∗
Q∗ Q∗
∗
′
then ui + vj = ūQ
i + v j . The stability of (ū , v ) in (M \ Q , M , A) implies that
∗
∗
∗
Q
Q
S
∗
∗
∗
ui + vj = ūQ
i + v j ≥ aij . If j ∈ µ(Q \ S ), then ui + vj = ūi + v j . By j ∈ µ(Q ),
∗
∗
∗
∗
Q
Q
Q
∗
′
vQ
≥ aij and
j = 0. By the stability of (ū , v ) in (M \ Q , M , A), we have ūi
S
∗
∗
hence ui + vj = ūi + v j ≥ aij . Therefore, there exist no i ∈ M \ Q and j ∈ M ′
such that ui + vj < aij . Consider any i ∈ Q∗ \ S ∗ and any j ∈ µ(S ∗ ) ∪ µ(Q∗ \ S ∗ ).
Then, ui + vj = ūSi + v Sj . The stability of (ūS , v S ) in (M \ S, M ′ , A) implies that
ui + vj = ūSi + v Sj ≥ aij . Therefore, we must have i1 ∈ Q∗ \ S ∗ and j2 ∈ µ(M \ Q∗ ).
S
We finally show that there exists i ∈ M \ Q such that ūQ
i > ūi . Suppose not.
Q
S
S
Then, ūQ
i ≤ ūi for all i ∈ M \ Q. From Fact 3, ūi = ūi for all i ∈ M \ Q. By the
S
µ-compatibility of Q and S, we have that v Q
j = v j for all j ∈ µ(M \ Q). Consider
∗
∗
i1 ∈ Q∗ \S ∗ and j2 ∈ µ(M \Q∗ ) such that ūSi1 +v Q
j2 < ai1 j2 . From j2 ∈ µ(M \Q ), we
have that j2 ∈ µ(Pj ∗ (Q∗ )) or j2 ∈ µ(M \ Q). Suppose that j2 ∈ µ(Pj ∗ (Q∗ )). By the
∗
S
assumption of Pj ∗ (Q∗ ) ⊆ S, j2 ∈ µ(S) and hence v Sj2 = 0 ≤ v Q
j2 . Therefore, ūi1 +
v Sj2 < ai1 j2 . This contradicts the stability of (ūS , v S ) in (M \ S, M ′ , A). Therefore,
Q∗
Q
S
j2 ∈ µ(M \ Q). From Fact 3, we have that v Q
j2 ≤ v j2 . Therefore, ūi1 + v j2 < ai1 j2 .
S
S
S
By j2 ∈ µ(M \ Q), we have v Q
j2 = v j2 and hence ūi1 + v j2 < ai1 j2 . This contradicts
the stability of (ūS , v S ) in (M \ S, M ′ , A). Therefore, there exists i ∈ M \ Q such
S
that ūQ
i > ūi .
37
We now prove Theorem 1. To complete the proof, we need to show that for each
µ
µ
Q ∈ D̄M
∪ D̄M
′,
∃(u∗ , v ∗ ) ∈ Ĉ(Q) such that (u∗ , v ∗ ) ∈
/ Ĉ(Q′ ) for all Q′ ∈ C µ with Q′ ̸= Q.
(5)
µ
We will show that (5) holds for all Q ∈ D̄M
as below.
µ
From Lemma 7, ∅ ∈ D̄M . We first show that ∅ satisfies (5). By the definition of
µ
D̄M , ūi > ui for all i ∈ M . So, there exists ϵ > 0 such that ūi − ϵ > ui for all i ∈ M .
Define a payoff vector (u∗ , v ∗ ) for (M, M ′ , A) by
u∗i = ūi − ϵ for all i ∈ M , and vj∗ = v̄j + ϵ for all j ∈ M ′ .
It is straightforward to check that (u∗ , v ∗ ) is also stable in (M, M ′ , A). We also have
that aiµ(i) > u∗i > 0 for all i ∈ M and aµ(j)j > vj∗ > 0 for all j ∈ M ′ . Pick any
Q′ ∈ C µ with Q′ ̸= ∅. Then, there exists i ∈ Q ∩ M such that ui = aiµ(i) for all
(u, v) ∈ Ĉ(Q) or j ∈ Q ∩ M ′ such that vj = aµ(j)j for all (u, v) ∈ Ĉ(Q). In both
cases, we have (u∗ , v ∗ ) ∈
/ Ĉ(Q).
µ
We next show that for all Q ∈ D̄M
with Q ̸= ∅, Q satisfies (5). Consider any
µ
µ
Q ∈ DM with Q ̸= ∅. Pick any S ∈ C with S ⊊ Q. From Lemma 12, there is
Q
S
S
i ∈ M \ Q such that ūQ
i > ūi . For each i ∈ M \ Q with ūi > ūi , we can define
S
ϵS,i > 0 such that ūQ
i − ϵS,i > ūi . We define
S
E := {ϵS,i | S ⊆ C µ with S ⊊ Q and i ∈ M \ Q with ūQ
i > ūi }.
Note that E is nonempty by Lemma 12 and ∅ ∈ C µ . Clearly, E is a finite set. Define
ϵ1 := min E. Then, ϵ1 > 0 holds.
We construct (u∗ , v ∗ ) ∈ Ĉ(Q) that satisfies (5) as below. From the definition of
µ
Q
D̄M
, we have that ūQ
i > ui for all i ∈ M \ Q. The stability of (ū, v) implies that for
Q
all i ∈ M \ Q and all j ∈ µ(Q), ūQ
i + v j > aij . So, there exists ϵ > 0 such that
Q
ūQ
i − ϵ > ui for all i ∈ M \ Q,
(6)
ūQ
i
(7)
−ϵ+
vQ
j
> aij for all i ∈ M \ Q and all j ∈ µ(Q).
Then, it is straightforward to show that for any ϵ > 0 that satisfies (6) and (7), the
following payoff vector in (M \ Q, M ′ , A):
Q
ui = ūQ
i − ϵ for all i ∈ M \ Q, and vj = v j + ϵ for all j ∈ µ(M \ Q),
vj = 0(= v̄jQ = v Q
j ) for all j ∈ µ(Q),
38
is stable in (M \ Q, M ′ , A). Take any ϵ2 > 0 that satisfies (5) and (6), and define
ϵ∗ := min{ϵ1 , ϵ2 }. Clearly, ϵ∗ is positive and satisfies (5) and (6). Define a payoff
vector (u∗ , v ∗ ) in the original market as follows:
Q
∗
∗
∗
u∗i = ūQ
i − ϵ for all i ∈ M \ Q, and vj = v j + ϵ for all j ∈ µ(M \ Q),
u∗i = aiµ(i) for all i ∈ Q, and vj∗ = 0 for all j ∈ µ(Q).
Then, (u∗ , v ∗ ) ∈ Ĉ(Q).
We prove that (u∗ , v ∗ ) ∈
/ Ĉ(Q′ ) for all Q′ ∈ C µ with Q′ ̸= Q. Pick any Q′ ∈ C µ
with Q′ ̸= Q. From Lemma 3, Q′ ⊆ M or Q′ ⊆ M ′ . Suppose that Q′ ⊆ M ′ . From
Q′ 7
Fact 3, we have that for all i ∈ M \ Q, ūQ
i ≥ ūi ≥ ūi . From Lemma 12, there
Q′
exists i ∈ M \ Q such that ūQ
i > ūi and hence ūi − ϵ∅,i > ūi . From the definition
Q′
∗
Q′ Q′
of ϵ∗ , we have that that ū∗i = ūQ
i − ϵ > ūi . By the optimality of (ū , v ), we can
obtain (u∗ , v ∗ ) ∈
/ Ĉ(Q′ ).
Finally, we consider the case with Q′ ⊆ M . There are two cases to consider;
Q′ \ Q ̸= ∅ and Q′ ⊊ Q. Suppose that Q′ \ Q ̸= ∅. Pick any i ∈ Q′ \ Q. Then,
for all (u, v) ∈ Ĉ(Q′ ), ui = aiµ(i) . By i ∈ M \ Q, u∗i < aiµ(i) . This implies that
(u∗ , v ∗ ) ∈
/ Ĉ(Q′ ). Suppose that Q′ ⊊ Q. By Lemma 12, there exists i ∈ M \ Q such
Q′
Q
Q′
∗
that ūQ
i > ūi and hence ūi − ϵQ′ ,i > ūi . From the definition of ϵ , we have that
′
Q
∗
∗ ∗ / Ĉ(Q′ ) by the optimality of (ūQ′ , v Q′ ).
u∗i = ūQ
i − ϵ > ūi , and hence (u , v ) ∈
µ
Therefore, (5) holds for all Q ∈ D̄M
. By the same argument, we can show that (5)
µ
holds for all Q ∈ D̄M ′ . This completes the proof.
Appendix B: Original definition of µ-compatible subgame
In this section, we show that our definition of µ-compatible sets is equivalent to
the original definition provided byShubik (1984) and Núñez and Rafels (2013). Let
(M, M ′ , A) be a general assignment game. A characteristic function is a function
′
v : 2M ∪M → R such that for all I ⊆ M and J ⊆ M ′ ,

∑
max
if I ̸= ∅ and J ̸= ∅,
µ∈M(I,J)
(i,j)∈µ aij
v(I ∪ J) =
0
otherwise.
For any I ⊊ M and J ⊊ M ′ , I ∪ J is a µ-compatible set* if it satisifes
v(M ∪ M ′ ) = v((M \ I) ∪ (M \ J)) +
∑
i∈I:µ(i)̸=i
′
aiµ(i) +
∑
aµ(i)j .
j∈J:µ(j)̸=j
Q
Q
′
′
′
Precisely, to obtain ūi ≥ ūQ
i , we use the fact that for any Q, Q ⊊ M with Q ⊆ Q , ūi ≥ ūi
for all i ∈ M .
7
39
′
We say that (M \ I, M ′ \ J, A) is µ-compatible subgame if I ∪ J is µ-compatible set*.
Let C∗µ be the set of all µ-compatible sets* and C µ be the set of all µ-compatible sets
defined in Section 3.2. Then, we have the following proposition.
Proposition 3. C µ = C∗µ .
Proof. Let I ⊊ M and J ⊊ M ′ . From the definition of v((M \ I) ∪ (M ′ \ J)), we
have that
∑
aiµ(i) ≤ v((M \ I) ∪ (M ′ \ J)), and
(8)
i∈M \I:µ(i)∈M \J
“=” holds if and only if µ−I∪J is optimal in (M \ I, M ′ \ J, A).
From the definition of a matching, we have that
∑
∑
aµ(j)j =
aµ(j)j +
j∈J:µ(j)∈M \I
j∈J:µ(j)̸=j
=
∑
j∈J:µ(j)̸=j
∑
aµ(j)j
j∈J:µ(j)∈I
∑
aiµ(i) +
i∈M \I:µ(i)∈J
This implies that
∑
aµ(j)j ≥
∑
(9)
aµ(j)j .
j∈J:µ(j)∈I
aiµ(i) , and
(10)
i∈M \I:µ(i)∈J
“=” holds if and only if for all (i, j) ∈ µ, {i, j} ⊆ I ∪ J implies that aij = 0. (11)
We now prove that C µ = C∗µ . We first show that C µ ⊆ C∗µ . Pick any I ∪ J ∈ C∗µ .
We need to show that (i) for all (i, j) ∈ µ, {i, j} ⊆ I ∪ J implies that aij = 0, and
(ii) µ−I∪J is optimal in (M \ I, M \ J). Note that
∑
v(M ∪ M ′ ) =
aiµ(i)
i∈M :µ(i)̸=i
=
∑
i∈M \I:µ(i)∈J
∑
aiµ(i) +
i∈M \I:µ(i)∈M \J
aiµ(i) +
∑
aiµ(i) .
i∈I:µ(i)̸=i
Suppose that (i) or (ii) does not holds. By (8), (9), (10), and (11), we have that
∑
∑
v(M ∪ M ′ ) < v((M \ I) ∪ (M \ J)) +
aiµ(i) +
aµ(i)j .
i∈I:µ(i)̸=i
j∈J:µ(j)̸=j
This contradicts that I ∪ J is a µ-compatible set* and hence I ∪ J ∈ C µ .
Finally, we show that C µ ⊆ C∗µ . Pick any I ∪ J ∈ C µ . Again, by (8), (9), (10),
and (11), we have that
∑
∑
v(M ∪ M ′ ) = v((M \ I) ∪ (M \ J)) +
aµ(j)j +
aiµ(i) ,
j∈J:µ(j)̸=j
which implies I ∪ J ∈ C∗µ .
40
i∈I:µ(i)̸=i
Acknowledgements
The authors thank Tomomi Matsui for giving Example 5 of this paper and helpful
comments. We also thank Ryo Kawasaki, Yasushi Kawase, Shigeo Muto and Jun
Wako for their helpful comments and suggestions. Keisuke Bando acknowledges
the Japan Society for the Promotion of Science for financial support through the
Research Activities Start-up (No. 26885028)
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