Games and Economic Behavior 108 (2018) 466–477

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Games and Economic Behavior

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Shapley’s conjecture on the cores of abstract market games ✩

Zhigang Cao a, Chengzhong Qin b, Xiaoguang Yang c,∗a Department of Economics, School of Economics and Management, Beijing Jiaotong University, Beijing 100044, Chinab Department of Economics, University of California, Santa Barbara, CA 93106, USAc MADIS, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, 100190, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 August 2016Available online 18 August 2017

JEL classification:C71C78

Keywords:Assignment gamesComplementsSubstitutesCoreConvex games

Shapley (1955) introduced the model of an abstract market game as a generalization of the assignment game model, among several other models. He conjectured that abstract market games possess non-empty cores. We analyze properties of abstract market games and provide a proof of this conjecture for cases with four or fewer players. We show by example that, in general, the structure of an abstract market game is not strong enough to guarantee the nonemptiness of the core. We establish supplemental conditions for the conjecture to hold. Our supplemental conditions are satisfied by the assignment games and abstract market games with one side consisting of a single player as with package auction games in Ausubel and Milgrom (2002).

© 2017 Elsevier Inc. All rights reserved.

1. Introduction

The assignment game model has become one of the most fruitful models of cooperative games (Shapley and Shubik, 1972; Roth and Sotomayor, 1990). An assignment game has two sides such that any two players on different sides can jointly create a certain nonnegative value but those on the same side cannot. While researchers in optimization are interested in how to match the players into disjoint pairs to maximize the total value, game theorists are more concerned with the question of how to match the players into disjoint pairs in a stable way. Shapley and Shubik proved that the core of an assignment game is non-empty and well structured.

In his investigation of the relationship between cooperative games and markets, Shapley (1955) considered a richer class of two-sided games, which he named as abstract market games as an extension of the assignment game model, among several other ones. Roughly speaking, an abstract market game is a coalitional game in which players are divided into two sides, such that any two players are “substitutes” when they belong to the same side and “complements” when they belong to different sides. Shapley conjectured that abstract market games possess non-empty cores.

In this paper, we analyze properties of abstract market games and provide a proof of Shapley’s conjecture for cases with no more than four players. We show by example that in general, the structure of an abstract market game is not strong

✩ This work was supported by the National Natural Science Foundation of China (NSFC) under grant number 11471326. Part of the work was done during the first author’s visit to UCSB and at time when he was associated with Academy of Mathematics and Systems Science, Chinese Academy of Sciences. We gratefully acknowledge helpful comments from Rabah Amir, Mamoru Kaneko, and participants at the 9th Pan-Pacific Game Theory Conference. We also thank David Levine and two anonymous referees for their comments and suggestions that helped to improve the paper.

* Corresponding author.E-mail addresses: zhigangcao@amss.ac.cn (Z. Cao), qin@econ.ucsb.edu (C. Qin), xgyang@iss.ac.cn (X. Yang).

http://dx.doi.org/10.1016/j.geb.2017.08.0060899-8256/© 2017 Elsevier Inc. All rights reserved.

Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477 467

enough to guarantee the nonemptiness of the core. We then establish supplemental conditions for the conjecture to hold. One supplemental condition involves the use of a reduced game, which guarantees the existence of optimal core elements for at least one side of abstract market games. The assignment games and abstract market games with one side having a single player are shown to satisfy this condition. Another supplemental condition is established for symmetric abstract market games to have non-empty cores.

Abstract market games with one side having a single player are similar to coalitional games associated with package auctions in Ausubel and Milgrom (2002) (see also Chapter 8 in Milgrom, 2004). As shown in Demange et al. (1986), the seller-minimum core allocation is a generalization of the second-price auction of Vickrey (1961). Furthermore, dynamic auction mechanisms resembling the deferred-acceptance algorithm of Gale and Shapley (1962) for stable matching problems can also be designed for assignment problems (Demange et al., 1986). We hope that our analysis of abstract market games will find additional useful applications to the auction theory.

The largely ignored abstract market game model and the conjecture of Shapley (1955) are important to game theory for the following reasons. First, the abstract market game model serves as a general framework for several important models including: (i) the assignment game model (Shapley and Shubik, 1972), which is a special case of the abstract market game model, in that only pairs of players are essential coalitions; (ii) the stable marriage model (Gale and Shapley, 1962), which can be considered as a non-transferable utility counterpart of the assignment game model; and (iii) the convex game model (Shapley, 1971), which is also a special case of the abstract market game model with complementary players only. Second, Shapley (1955) was the first paper to apply the notion of the core.1 Third, Shapley (1955) was the first paper introducing complementarity and substitutability within the context of the cooperative game theory. While the notions of strategic complementarity and substitutability have been extensively analyzed and widely applied in the noncooperative game theory, their earlier cooperative counterparts have not received comparable treatments.2

The rest of the paper is organized as follows. Section 2 introduces notations and preliminaries for abstract market games. Sections 3 and 4 present results for the cases with four or fewer players and the general case, respectively. Section 5concludes the paper with some further discussions. A proof of Shapley’s conjecture for the case with four or fewer players and two additional properties of abstract market games in connection with the field of combinatorial optimization are organized in Appendix A and Appendix B, respectively.

2. Preliminaries

A coalitional game with transferable utility (a coalitional game in short) is a pair (N, v), where N is the (finite) set of n players and v : 2N → R is the characteristic function such that v(∅) = 0. To save space, we frequently write coalitions {1, 2}, {2, 3, 4}, and so on as 12 and 234. A coalitional game (N, v) is superadditive if for all disjoint coalitions S, T ⊆ N , v(S ∪ T ) ≥ v(S) + v(T ). It is superadditive at N if v(N) ≥ ∑

1≤�≤m v(S�) whenever {S1, S2, . . . , Sm} forms a partition of N . Let RN denote the n-dimensional Euclidean space of vectors with coordinates indexed by players in N . Vectors in RN are payoff allocations. For x ∈ R

N and S ⊆ N , x(S) denotes the sum of xi over players i ∈ S . The core of (N, v) is defined as the following set of payoff allocations

C(N, v) = {x ∈RN : x(S) ≥ v(S),∀S ⊂ N, x(N) = v(N)}.

A non-empty family of coalitions B ⊆ 2N is balanced if there exist balancing coefficients δS ≥ 0, S ∈ B, such that

∑S∈B:S�i

δS = 1, i ∈ N.

The well-known Bondareva–Shapley Theorem states that the core of (N, v) is non-empty if and only if it is balanced:

v(N) ≥∑S∈B

δS v(S) (1)

for all balanced families B and balancing coefficients (δS )S∈B (Bondareva, 1963; Shapley, 1967).Given coalitions S, T ⊆ N , S \ T denotes the set-theoretic difference between coalitions S and T . We use �i = v(N) −

v(N \ i) to denote the marginal contribution of player i ∈ N to the grand coalition N . If (N, v) is balanced, then the payoff that player i gets in the core C(N, v) is at most �i , because otherwise coalition N \ i can improve upon the corresponding payoff allocations. We say that a core allocation x ∈ C(N, v) is i-optimal if xi = �i and S-optimal if x is i-optimal for all i ∈ S ⊆ N .

1 See Zhao (2017) for a review on the early history of the core.2 For the extensive literature on non-cooperative games with strategic complementarity and substitutability, the reader is referred to Topkis (1979);

Bulow et al. (1985); Vives (1990); Milgrom and Shannon (1994); Topkis (1998); Vives (1999); Amir (2005).

468 Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477

2.1. Abstract market games

Shapley (1955) defined an abstract market game by imposing certain “convexity-concavity conditions” on the characteris-tic functions to be mentioned below. He subsequently explained these conditions in terms of players’ marginal contributions to coalitions (Shapley, 1955, p. 4). These interpretations make it clear that the convexity-concavity conditions correspond to the “complementarity-substitutability relationships” between the players as specified in a later paper (Shapley, 1962). Due to their greater familiarity, we present the definition of an abstract market game by imposing the complementarity-substitutability relationships. We need the following further notations. For i, j ∈ N and S ⊆ N \ i j, we write S + i ≡ S ∪ i and S + i j ≡ S ∪ i j.

Definition 1. (Shapley, 1955) Given a coalitional game (N, v), players i, j ∈ N are complements if the presence of one of them increases the other’s marginal contributions:

v(S + i j) − v(S + i) ≥ v(S + j) − v(S) (2)

for all S ⊆ N \ i j. They are substitutes if the presence of one of them reduces the other’s marginal contributions:3

v(S + i j) − v(S + i) ≤ v(S + j) − v(S) (3)

for all S ⊆ N \ i j.

Notice that (2) and (3) are respectively equivalent to

v(S + i j) − v(S) ≥ [v(S + i) − v(S)] + [v(S + j) − v(S)]and

v(S + i j) − v(S) ≤ [v(S + i) − v(S)] + [v(S + j) − v(S)].That is, two players are complements (substitutes) if their joint marginal contribution to each coalition containing both of them is no less (no more) than the sum of the marginal contributions of each player alone, which are achieved when the other player is removed from the coalition. The convexity (concavity) of a coalitional game introduced in Shapley (1971) is equivalent to the players all being complements (substitutes).

Definition 2. (Shapley, 1955) A coalitional game (N, v) is an abstract market game if it is superadditive at N and players can be partitioned into two sides (disjoint coalitions), A and B , such that any two players are substitutes if they are on the same side and complements if they are on opposite sides.

Superadditivity at N is not explicitly stated in Shapley’s original definition of an abstract market game. However, the complementarity-substitutability relationships do not imply superadditivity at N . For a numerical example, consider a coali-tional game (N, v) where N = A ∪ B with A = 12, B = 34, and (i) v(S) = 3 for S = 14, 23; (ii) v(S) = 6 for S = 13, 24; (iii) v(S) = 6 for |S| = 3; (iv) v(N) = 10; and (v) v(S) = 0 whenever either S ∩ A = ∅ or S ∩ B = ∅. It can be checked that the complementarity-substitutability relationships are satisfied. Nonetheless, the game is not superadditive at N because 10 = v(N) < v(13) + v(24) = 12.

Due to the convexity-concavity conditions, abstract market games satisfy similar properties of convex and concave games. These properties are potentially useful for further research on abstract markets games. For that reason, we establish two such properties in Appendix B.

2.2. Assignment games

An assignment game is a coalitional game that arises from an assignment problem with two sides, A and B , such that any two players on different sides can jointly create a certain nonnegative value, but those on the same side cannot. The worth of a multi-player coalition in general is the maximum value among all the total values from various assignments of players of the coalition on one side to those within the same coalition on the opposite side (Shapley and Shubik, 1972).

The following properties of assignment games are helpful to our later discussions. The core of an assignment game C(N, v) is non-empty and polarized, in that there exist two extreme core allocations, with one being unanimously the best for players on side A and the worst for those on side B and the other unanimously the best for those on side B and the worst for players on side A, among core allocations (Shapley and Shubik, 1972). In addition, as shown in Demange (1982), the two extreme core allocations are respectively A-optimal (with players in A all receiving their marginal contributions to

3 It is worth pointing out that in cooperative game theory a popular definition for two players i and j to be substitutes requires v(S + i) = v(S + j) for all S ⊆ N \ i j. That is, to be substitutes, two players must have the same marginal contributions to all coalitions not containing them. We refer the reader to Peleg and Sudhölter (2007) for applications.

Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477 469

the grand coalition) and B-optimal (with players in B all receiving their marginal contributions to the grand coalition). Let ∨be the coordinative-wise maximum and ∧ the coordinate-wise minimum. Shapley and Shubik (1972) proved that the core of an assignment game has a lattice structure in the following sense: (xA ∨ y A, xB ∧ yB) ∈ C(N, v) and (xA ∧ y A, xB ∨ yB) ∈C(N, v) for all (xA, xB), (y A, yB) ∈ C(N, v). Payoff vector (xA ∨ y A, xB ∧ yB) is also referred to as the “join” of (xA, xB) and (y A, yB), and (xA ∧ y A, xB ∨ yB) the “meet” of (xA, xB) and (y A, yB).

3. The case with four or fewer players

With the set of players N partitioned into side A and side B , we write S A = S ∩ A and S B = S ∩ B to denote the partition of coalition S into sub-coalitions of players S A on side A and S B on side B . The following theorem shows that Shapley’s conjecture holds for the case with at most four players.

Theorem 1. Abstract market games with four or fewer players have non-empty cores.

A proof of Theorem 1 is organized in Appendix A. Unlike the assignment games, balanced abstract market games do not necessarily have polarized cores. Example 1 below provides an illustration.

Example 1. Consider a coalitional game with two sides A = 12, B = 34, and characteristic function

v(S) =

⎧⎪⎪⎪⎨⎪⎪⎪⎩

0 if |S A | = 0 or |S B | = 0;3 if (|S A |, |S B |) = (1,1);5 if (|S A |, |S B |) = (1,2), (2,1);9 if (|S A |, |S B |) = (2,2).

It can be checked that this is an abstract market game. Since (0, 1, 4, 4) is in the core of this game, the minimum core payoff for player 1 is 0. By symmetry, the minimum core payoff for each of the other players is also 0. However, there is no core allocation at which the two players in A obtain their minimum payoff simultaneously. To see this, suppose on the contrary that (0, 0, x3, x4) is in the core. Assume w.l.o.g. that x3 ≤ x4. Then, x3 ≤ 4.5, which implies that coalition 123 can improve upon the payoff allocation.

4. The general case

We begin with counterexamples in Subsection 4.1. We establish a supplemental condition with which an optimal core allocation for one side exists in Subsection 4.2. This condition is satisfied by assignment games and by games with one side occupied by a single player. In Subsection 4.3, we establish a supplemental condition for symmetric abstract market games to have non-empty cores.

4.1. Counterexamples

Abstract market games in general have more complex multi-player coalitions than assignment games. As a result, the Shapley’s conjecture is not always valid without supplemental conditions. The following is a 5-person counterexample.

Example 2. Consider a coalitional game (N, v) with N = A ∪ B , A = 123, B = 45, and characteristic function

v(S) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩

0 if |S A | = 0 or |S B | = 0;3 if (|S A |, |S B |) = (1,1);5 if (|S A |, |S B |) = (2,1), (1,2);9 if (|S A |, |S B |) = (2,2);5 if (|S A |, |S B |) = (3,1);9 if (|S A |, |S B |) = (3,2).

It is clear that this game is superadditive. Moreover, as shown in Table 1, the game is an abstract market game. However, the family {134, 235, 1245} is balanced with balancing coefficients 0.5, 0.5, 0.5, but 0.5(v(134) + v(235) + v(1245)) = 0.5 ×(5 + 5 + 9) = 9.5 > v(12345) = 9. By the Bondareva–Shapley Theorem, the core of the game is empty.

Players on the same side of the game in Example 2 are symmetric, in the sense that the worth of each coalition only depends on the numbers of players on each side. Nonetheless, one side of the game is thicker than the other side in that |A| > |B|. The following is an extended counterexample with |A| = |B|.

470 Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477

Table 1The complementarity-substitutability relationships be-tween players in Example 2. Here, �A(�, r) = v(S) −v(S ′) and �B (�, r) = v(S) − v(S ′′), for S, S ′, S ′′ ⊆ Nsuch that |S A | = |S ′′

A | = �, |S B | = |S ′B | = r, |S ′

A | = � − 1, and |S ′′

B | = r − 1.

� �A(�,0) �A(�,1) �A(�,2)

1 0 3 52 0 2 43 0 0 0

r �B (0, r) �B (1, r) �B (2, r) �B (3, r)

1 0 3 5 52 0 2 4 4

Example 3. Consider a coalitional game (N, v) where N = A ∪ B , A = 123, B = 456, v(S) = 5 if (|S A |, |S B |) = (1, 3), 9if (|S A |, |S B |) = (2, 3), 12 if (|S A |, |S B |) = (3, 3), and v(S) is the same as in Example 2 otherwise. As with the game in Example 2, this game is an abstract market game. Due to symmetry, if the core were non-empty, then the payoff vector with each player receiving a payoff of 2 would be in the core. But this payoff vector cannot be in the core, because any coalition composed of two sellers and two buyers can improve upon.

4.2. A- and B-optimal core allocations

Motivated by the existence of both A- and B-optimal core allocations for assignment games, we establish a supplemental condition on abstract market games that guarantees the existence of optimal core allocations for at least one side. To this end, given an abstract market game (N, v) with sides A and B , define for S = A, B , v S : 2S −→ R by

v S(R) = maxT ⊆N\S

⎡⎣v(R ∪ T ) −

∑j∈T

� j

⎤⎦ , R ⊆ S. (4)

It can be checked that v A(∅) = v B(∅) = 0 and hence, (A, v A) and (B, v B) are coalitional games.4 In (A, v A), players can get help in the creation of value from any coalition of players in B by agreeing to let them each receive the marginal contributions to the grand coalition. An analogous interpretation applies to (B, v B ).

Notice that the construction of subgame (S, v S ) in (4) slightly differs from the construction of the reduced game based on player set S and payoff allocation (� j) j∈N in Davis and Maschler (1965). For the latter construction, v S (S) takes the following value:5

v S(S) = v(N) −∑

j∈N\S

� j . (5)

Lemma 1 below shows that with the condition that players on the same side are substitutes, (5) is automatically implied by (4).

Lemma 1. Let (N, v) be an abstract market game with sides A and B. Then, v S(S) satisfies (5) for S = A, B.

Proof. Due to symmetry, it suffices to prove the lemma for the case with S = A. Notice that to show v A(A) satisfies (5), we need to show that

v(A ∪ B) −∑j∈B

� j ≥ v(A ∪ S B) −∑j∈S B

� j, S B ⊆ B,

which is equivalent to

v(A ∪ B) − v(A ∪ S B) ≥∑

j∈B\S B

� j, S B ⊆ B. (6)

4 To prove v A(∅) = 0, it suffices to show that v(S B) − ∑j∈S B

� j ≤ 0 for all non-empty S B ⊆ B . This is true, because (i) v(S B ) ≤ ∑j∈S B

v( j) as players in S B are substitutes, and (ii) � j = v(N) − v(N \ j) ≥ v( j) due to superadditivity at N . We can show v B (∅) = 0 in the same way.

5 Davis and Maschler (1965) was the first to consider reduced games and reduced game property. It is known that many cooperative solutions satisfy the reduced game property. We refer the reader to Peleg (1986) and Driessen (1991) for more details.

Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477 471

The above inequality obviously holds when |B \ S B | ≤ 1. Suppose now |B \ S B | ≥ 2. Rename the players in B \ S B as 1, 2, . . . , s. Since players 1 and 2 are substitutes, we have �2 ≤ v(N \ 1) − v(N \ 12). In general, for each 2 ≤ j ≤ s, we have � j ≤v(N \ 1, . . . , j − 1) − v(N \ 1, . . . , j). Adding up these inequalities with �1 = v(N) − v(N \ 1) establishes (6). �

It turns out that the nonemptiness of the core of (A, v A) (resp. (B, v B)) and the existence of a B-optimal (resp. A-optimal) core allocation of (N, v) are equivalent.

Theorem 2. Let (N, v) be an abstract market game with sides A and B. Then, for S = A, B, (N, v) possesses an N \ S-optimal core allocation if and only if (S, v S) is balanced.

Proof. Due to symmetry, we only need to prove the theorem for the case with S = A. To this end, notice first that it suffices to prove that y ∈ R

A is in the core of (A, v A) if and only if z ≡ (y, �B) is in the core of (N, v), where �B = (� j) j∈B . Equivalently, it suffices to show that (a) for all S A ⊆ A, y(S A) ≥ v A(S A) if and only if z(S) ≥ v(S) for all S ⊆ N; (b) y(A) = v A(A) if and only if z(N) = v(N).

For S ⊆ N , z(S) = y(S A) + ∑j∈S B

� j , implying z(S) − v(S) = y(S A) − (v(S) − ∑j∈S B

� j). Thus, by (4), z(S) ≥ v(S) for all S ⊆ N if and only if y(S A) ≥ v A(S A) for all S A ⊆ A. This proves statement (a). By Lemma 1, z(N) = y(A) + ∑

j∈B � j =y(A) − (v(N) − ∑

j∈B � j) + v(N) = y(A) − v A(A) + v(N). It follows that statement (b) also holds. �Theorem 2 shows that a sufficient supplemental condition for an abstract market game to possess an optimal core alloca-

tion for one side is that at least one of the two reduced games (A, v A) and (B, v B) are balanced. Though this supplemental condition seems to be ad hoc, it is satisfied by two important subclasses of familiar abstract market games as shown in the following two propositions.

Since the core of each assignment game has a lattice structure, the corresponding (A, v A) and (B, v B) are both balanced. The following proposition shows that they are rather special balanced reduced games.

Proposition 1. Let (N, v) be an assignment game with sides A and B. Then, both (A, v A) and (B, v B) are inessential and, hence, balanced: v A(S A) = ∑

i∈S Av A(i) for S A ⊆ A and v B(S B) = ∑

j∈S Bv B( j) for S B ⊆ B.

Proof. As shown in Shapley and Shubik (1972) and Demange (1982), there exists a unique B-optimal core allocation, which we denote by (δA, �B). Furthermore, for i ∈ A, δi + � j = v(i j) if π(i) = j (i and j are paired in π ) for some j ∈ B at an optimal assignment π supporting (δA, �B), and δi = 0 if π(i) �= j for all j ∈ B .

For S A ⊆ A, since (δA, �B) is in the core,

v(S A ∪ S B) ≤∑i∈S A

δi +∑j∈S B

� j

for all S B ⊆ B . On the other hand, for S B = {π(i)|i ∈ S A},

v(S A ∪ S B) =∑i∈S A

δi +∑j∈S B

� j.

It follows that

v A(S A) =∑i∈S A

δi . (7)

By (7), v A(i) = δi for i ∈ A and v A(S A) = ∑i∈S A

v A(i). A similar proof establishes that (B, v B ) is also inessential. �Remark 1. Note that the four-player game in Example 1 is not an assignment game, but it nonetheless has both A-optimal and B-optimal core allocations. Thus, the subclass of abstract market games covered by the supplemental condition in Theorem 2 is broader than the class of assignment games. Example 4 below further demonstrates that this supplemental condition also covers abstract games that may have either an A-optimal or a B-optimal core allocation but not both.

Example 4. Consider a four-player coalitional game (N, v) where N = A ∪ B , A = 12, B = 34, and

v(S) =

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

7 if S = N;5 if |S| = 3;4 if S = 13,23;3 if S = 14,24;0 otherwise.

472 Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477

Note that this is not an assignment game because v(123) > max{v(13), v(23)}. But it can be checked that (N, v) is an abstract market game. Furthermore, it can be verified that v B (3) = 2, v B(4) = 1, v B(34) = 3, and hence the reduced game (B, v B) is balanced. It follows that (N, v) possesses an A-optimal core allocation. However, 3 = v A(12) < v A(1) + v A(2) =2 + 2, implying that (N, v) does not possess a B-optimal core allocation.

Another subclass consists of abstract market games with one of the two sides composed of a single player. As shown in the following proposition, the cores of these abstract market games have a unique structure that is similar to part (3) of Theorem 7 in Ausubel and Milgrom (2002).

Proposition 2. Let (N, v) be an abstract market game with two sides A and B such that |A| = 1. Then, (A, v A) is balanced and the core of (N, v) is given by

C(N, v) ={

x ∈RN :

∑i∈N

xi = v(N), v( j) ≤ x j ≤ � j,∀ j ∈ B

}, (8)

which is non-empty and polarized.

Proof. Since (A, v A) is a single player game, it is obviously balanced. The proof of (8) is similar to the proof of Lemma 1and essentially the same as the proof of Theorem 7 in Ausubel and Milgrom (2002). We provide it for completeness.

Suppose w.l.o.g. that A = 1 and B = 23 . . .n. Denote the right-hand side of (8) as C . That C(N, v) ⊆ C is obvious. It remains to show that the converse also holds. To this end, let x ∈ C be given. Since players in B are substitutes, x(R) ≥∑

j∈R v( j) ≥ v(R) for all R ⊆ B .

Suppose now S = 1∪ R with R ⊆ B . W.l.o.g., let R = {2, 3, . . . , r}. Since players in B are substitutes, � j = v(N) − v(N \ j) ≤v(N \ r + 1 . . . j − 1) − v(N \ r + 1 . . . j) for all j ≥ r + 1. Notice that N \ r + 1 . . . j − 1 = N for j = r + 1. We have

x(S) = v(N) − x(B \ R)

≥ v(N) −∑

j∈B\R

[v(N) − v(N \ j)]

≥ v(N) −n∑

j=r+1

[v(N \ r + 1 . . . j − 1) − v(N \ r + 1 . . . j)]

= v(N \ r + 1 . . .n)

= v(S).

This establishes (8). The nonemptiness and polarization of the core follow automatically from (8). �It can be seen from the proof of Proposition 2 that complementarity is not needed when |A| = 1.

Remark 2. Ausubel and Milgrom (2002) assumed v(S) = 0 for S consisting of buyers only, which is natural for their context. Together with this assumption, the main condition they worked with, “buyer-submodularity”, is equivalent to the condition that “buyers are substitutes”. From Theorem 2 it follows that this assumption is inessential.

The core in (8) may not have a lattice structure as with the cores of the assignment games. The following example provides an illustration.

Example 5. Suppose there are three players in total: A = 1 and B = 23. Consider characteristic function v(1) = v(2) = v(3) =v(23) = 0, v(12) = v(13) = 2 and v(123) = 4. It can be checked that this is an abstract market game. The payoff vectors (2, 2, 0) and (2, 0, 2) are in the core. However, neither (2 ∨ 2, (2, 0) ∧ (0, 2)) = (2, 0, 0) nor (2 ∧ 2, (2, 0) ∨ (0, 2)) = (2, 2, 2)

is in the core.

4.3. Dual-symmetry

We consider a notion of symmetry for abstract market games as defined below.

Definition 3. A coalitional game (N, v) is dual-symmetric if players can be partitioned into two disjoint groups A and Bsuch that for some real-valued function f : {0, 1, . . . , |A|} × {0, 1, . . . , |B|} −→R,

v(S) = f (|S A |, |S B |)for S ⊆ N . It is completely dual-symmetric if in addition |A| = |B| and f (|S A |, |S B |) = f (|S B |, |S A |) for S ⊆ N .

Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477 473

There is a critical difference between dual-symmetric and the usual symmetric games. While the worth of each coalition is determined solely by the total number of players of the coalition in the usual symmetric games, we need to distinguish players on different sides in (non-degenerate) dual-symmetric games. Notice that Examples 1 and 3 are completely dual-symmetric, Examples 2 and 5 are dual-symmetric but not completely dual-symmetric, and Example 4 is not dual-symmetric. Example 3 provides an illustration of a completely dual-symmetric abstract market game having an empty core.

A bivariate function f (·, ·) is coordinate-wise concave if both f (p, ·) and f (·, q) are concave for all p, q in the correspond-ing domains. The following lemma shows that for dual-symmetric games, the complementarity-substitutability relationships between the players are equivalent to supermodularity and coordinate-wise concavity of the corresponding bivariate func-tion f .

Lemma 2. Let (N, v) be dual-symmetric with respect to partition {A, B} of N. Then, the complementarity-substitutability rela-tionships between the players hold in (N, v) if and only if f (|S A |, |S B |) = v(S) is supermodular and coordinate-wise concave on {0, 1, . . . , |A|} × {0, 1, . . . , |B|}.

Proof. Suppose f is supermodular and coordinate-wise concave. Let i, j be two players of coalitional game (N, v) and S ≡ (S A, S B) ⊆ N \ i j. If i and j are on the same side of the game, say A, then v(S + i j) − v(S + i) = f (|S A | + 2, |S B |) −f (|S A | + 1, |S B |) ≤ f (|S A | + 1, |S B |) − f (|S A |, |S B |) = v(S + j) − v(S), where the inequality is due to the coordinate-wise concavity of f . If i and j are on different sides of the game, say i ∈ A and j ∈ B , then v(S + i j) − v(S + i) = f (|S A | +1, |S B | +1) − f (|S A | +1, |S B |) ≥ f (|S A |, |S B | +1) − f (|S A |, |S B |) = v(S + j) − v(S), where the inequality is due to the supermodularity of f . Therefore, the “if” part is correct. The “only if” part can be similarly proved. �

Note that if an abstract market game (N, v) is dual-symmetric and f (p, q) = 0 when pq = 0, then f (1, 1)(p ∧ q) ≤f (p, q) ≤ f (1, 1)pq for all 0 ≤ p ≤ |A| and 0 ≤ q ≤ |B|.6 This means that the worth of a coalition in a dual-symmetric abstract market game (under the mild condition that coalitions consisting of players on one side have zero worth) grows at a medium rate as its size increases.

Theorem 3. Let (N, v) be dual-symmetric with two sides A and B and let f be the bivariate function implied by the dual-symmetry of the game.

(a) A sufficient condition for the core of (N, v) to be non-empty is that f (p, q)/(p + q) reaches its maximum at p = |A| and q = |B|.(b) If in addition (N, v) is a completely dual-symmetric abstract market game, then a necessary and sufficient condition for the core

of (N, v) to be non-empty is that f (p, p)/p reaches its maximum at p = |A|.

Proof. Suppose f (p, q)/(p +q) reaches its maximum at p +q = |A| +|B|. Then, for all integers p and q such that 0 ≤ p ≤ |A|and 0 ≤ q ≤ |B|, the condition that f (p, q)/(p + q) reaches its maximum at p + q = |A| + |B| implies that

f (p,q) ≤ p + q

|A| + |B| f (|A|, |B|). (9)

Consider payoff allocation x ∈ RN where xi = f (|A|, |B|)/(|A| + |B|) for i ∈ N . Then, for any coalition S ⊆ N , equation (9)

implies that

x(S) = |S A | + |S B ||A| + |B| f (|A|, |B|) ≥ f (|S A |, |S B |).

This establishes part (a).Suppose now (N, v) is a completely dual-symmetric abstract market game. In this case, |A| = |B|. By Lemma 2, f is

supermodular. Consequently,

f (p,q) = 1

2( f (p,q) + f (q, p)) ≤ 1

2( f (p, p) + f (q,q)).

It follows that f (p, q)/(p + q) reaches its maximum at p = q = |A| if and only if f (p, p)/p reaches its maximum at p = |A|. Thus, the sufficiency of the condition follows from part (a). By the complete dual-symmetry, if the core is non-empty, then the symmetric payoff allocation x with xi = f (|A|, |A|)/(2|A|) for all i ∈ N is in the core. Consequently, f (p, p)/p must reach its maximum at p = |A|. This establishes the necessity. The proof of part (b) is thus completed. �

6 The simple proof is as follows. Since f (p, q) = 0 when p ∧ q = 0, it can be observed that coordinate-wise concavity implies coordinate-wise subad-

ditivity. Hence, f (p, q) ≤p︷ ︸︸ ︷

f (1,q) + · · · + f (1,q) = pf (1, q) ≤ p(

q︷ ︸︸ ︷f (1,1) + · · · + f (1,1)) = f (1, 1)pq. On the other hand, due to superadditivity, f (p, q) ≥

p∧q︷ ︸︸ ︷f (1,1) + · · · + f (1,1)+ f (p − p ∧ q, q − p ∧ q) = f (1, 1)(p ∧ q).

474 Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477

Note that the coordinate-wise concavity of f is not used in the proof of Theorem 3(b), and the condition in Theo-rem 3(b) is satisfied when g(p) ≡ f (p, p) is convex in {1, 2, . . . , |A|}. We end this section by remarking that the class of dual-symmetric abstract market games includes some familiar examples. In particular, two well-known functional forms for dual-symmetric abstract market games in the following examples imply the nonemptiness of the core.

Example 6. Let f (p, q) = p ∧ q be a Leontief production function. Then, the resulting dual-symmetric game, which is an abstract market game known as the gloves game, possesses a nonempty core (Shapley, 1955).

Example 7. Let f (p, q) = pαqβ be a Cobb–Douglas production function with 1/2 ≤ α, β ≤ 1, and (N, v) be a corresponding symmetric two-sided game with |A| = |B|. Then ∂2 f /∂ p∂q = αβpα−1qβ−1 ≥ 0, ∂2 f /∂ p2 = −α(1 − α)pα−2qβ ≤ 0, and ∂2 f /∂q2 = −β(1 − β)pαqβ−2 ≤ 0. Thus, f is supermodular and coordinate-wise concave. It follows from Lemma 2 that the complementarity-substitutability relationships hold in (N, v). Furthermore, f is superadditive, implying that (N, v) is a dual-symmetric abstract market game.7 Since it is checkable that f (p, q)/(p + q) reaches its maximum at p = q = |A|, (N, v) is balanced.

5. Discussions

Sotomayor (1992) proposed the model of a multi-partner assignment game as an extension of the assignment game model. In a multi-partner assignment game, each player on one side may form multiple partnerships with players on the other side, subject to a capacity constraint on the total number of partners the player can have. Recall that, in assignment games, a payoff allocation is in the core if and only if it is stable in that there exists no blocking pair (Shapley and Shubik, 1972). In contrast, the core may not coincide with the set of stable allocations in the multi-partner assignment game, although the set of stable allocations still has a lattice structure (Sotomayor, 1999). Using a method similar to Shapley (1962), it can be shown that the multi-partner assignment game is also an abstract market game. Other extensions of the assignment game model include Kelso and Crawford (1982), Fujishige and Tamura (2007), and Massó and Neme (2014). It would be interesting to investigate possible connections between these extensions and abstract market games.

As shown in this paper, abstract market games do not necessarily have non-empty cores, which stand in sharp contrast to the extensions of the assignment game model mentioned above. The reason is that abstract market games have more complex essential coalitions. Future research includes finding necessary and sufficient supplemental conditions for general abstract market games to possess non-empty cores, and analyzing structural properties of the core. This paper is a first attempt in this line of research.

Appendix A. Proof of Theorem 1

Proof. When min{|A|, |B|} = 0, the core is non-empty, because the superadditivity at N implies that the game is inessen-tial when players are all substitutes. The case with min{|A|, |B|} = 1 is covered by Proposition 2. Therefore, it remains to consider the case |A| = |B| = 2.

A family of coalitions of players is called minimal balanced if it is balanced and has no proper balanced subfamilies. As is well-known, a game is balanced if and only if condition (1) holds for all minimal balanced families (Bondareva, 1963; Shapley, 1967).

For four-player games, there are nine types of minimal balanced families: (a) {12, 34}, (b) {123, 4}, (c) {12, 3, 4}, (d) {1, 2, 3, 4}, (e) {123, 124, 34}, (f) {12, 13, 23, 4}, (g) {123, 14, 24, 3}, (h) {123, 14, 24, 34}, and (i) {123, 124, 134, 234}(see Shapley, 1967). By superadditivity at N , the balancedness condition (1) is satisfied in types (a)–(d), because balanced families form partitions of N . It remains to prove the theorem for types (e)–(i). Balancing coefficients are easily seen and for that reason, they will not be explicitly stated.

For type (e), if players 3 and 4 are on the same side of A and B , then v(34) ≤ v(3) + v(4), and hence v(123) + v(124) +v(34) ≤ (v(123) + v(4)) + (v(124) + v(3)) ≤ 2v(1234), where the second inequality follows from the superadditivity at N . Suppose player 3 and 4 belong to different sides. Then, v(123) + v(124) − v(12) ≤ v(1234), because players 3 and 4 are complements. Consequently, v(123) + v(124) + v(34) = (v(123) + v(124) − v(12)) + (v(12) + v(34)) ≤ 2v(1234).

For type (f), suppose w.l.o.g. that A = 12 and B = 34. Then, v(12) + v(13) + v(23) + 2v(4) ≤ v(123) + v(1) + v(23) +2v(4) = (v(123) + v(4)) + (v(1) + v(23) + v(4)) ≤ 2v(1234), where the first inequality follows from the complementarity between players 2 and 3, and the second inequality follows from the superadditivity at N .

For type (g), if players 1 and 2 are on the same side of A and B , then players 3 and 4 are on the same side and v(123) + v(3) ≤ v(13) + v(23), because players 1 and 2 are substitutes. Therefore, v(123) + v(14) + v(24) + v(3) ≤ (v(14) +v(23)) + (v(24) + v(13)) ≤ 2v(1234). If players 1 and 2 are on different sides, then players 3 and 4 are also on different

7 We check in this footnote that f (p1 + p2, q1 + q2) ≥ f (p1, q1) + f (p2, q2) for all p1, q1, p2, q2 ≥ 0. When min{p1, p2, q1, q2} ≥ 1, let p′ = p1/p2, q′ =q1/q2, this is equivalent to (1 + p′)α(1 + q′)β ≥ 1 + p′ αq′ β . Suppose w.l.o.g. that p′ ≥ q′ . Define gq′ (p′) = (1 + p′)α(1 + q′)β − p′ αq′ β − 1. Observe that gq′ (0) = (1 + q′)β − 1 ≥ 0. It suffices to show that dg/dp′ ≥ 0. It is elementary that dg/dp′ = α(1 + p′)α−1q′ β (

(1 + 1/q′)β − (1 + 1/p′)1−α), which is

nonnegative, because β ≥ 1/2 ≥ 1 − α and 1/q′ ≥ 1/p′ . When min{p1, p2, q1, q2} = 0, f (p1 + p2, q1 + q2) ≥ f (p1, q1) + f (p2, q2) is obviously true.

Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477 475

sides and v(14) + v(24) ≤ v(124) + v(4) and v(123) + v(124) ≤ v(1234) + v(12) because of the complementarity between players 1 and 2 and between 3 and 4. Consequently, v(123) + v(14) + v(24) + v(3) ≤ v(123) + v(124) + v(4) + v(3) ≤v(1234) + v(12) + v(4) + v(3) ≤ 2v(1234).

For type (h), suppose w.l.o.g. that A = 12 and B = 34. Then, v(14) + v(34) ≤ v(134) + v(4) and v(123) + v(134) ≤v(1234) + v(13) because of the complementarity between players 1 and 3 and between 2 and 4. Consequently, 2v(123) +v(14) + v(24) + v(34) = (v(14) + v(34)) + 2v(123) + v(24) ≤ v(1234) + (v(123) + v(4)) + (v(13) + v(24)) ≤ 3v(1234).

Finally, for type (i), suppose w.l.o.g. that A = 12 and B = 34. We have v(123) + v(134) ≤ v(1234) + v(13) and v(124) +v(234) ≤ v(1234) + v(24) due to the complementarity between players 2 and 4 and between 1 and 3. Therefore, v(123) +v(124) + v(134) + v(234) ≤ 2v(1234) + (v(13) + v(24)) ≤ 3v(1234). �Appendix B. Two basic properties

As conditions for set functions, the convexity-concavity conditions of abstract market games were also considered in the field of combinatorial optimization. In this appendix, we establish two properties for abstract market games in connection with this field. To this end, we follow Schrijver (1979) to refer set functions satisfying the convexity-concavity conditions dual-submodular functions. When either A or B is empty, dual-submodular games reduce to concave games. The dual-submodular games also turn out to be closely related to convex games, in the sense that the dual with respect to one side (to be defined below) is convex whenever the game is dual-submodular. Given partition {A, B} of N and S ⊆ N , we write (S A, S B) ≡ S and v(S A, S B) ≡ v(S).

Definition 4. Given a coalitional game (N, v) and a partition {A, B} of N , the B-dual of (N, v) is denoted by (N, v B) and given by

v B(S) = v(B) − v(S A, B \ S B),∀S ⊆ N. (10)

Observe that (N, v B) is indeed a coalitional game because v B (∅) = 0. It can also be seen that v B (B) = v(B) and the B-dual of (N, v B) is (N, v). Hence, this is indeed a dual relation. In addition, if A = ∅ and B = N , then v B is exactly the standard dual game (N, v∗), which is defined as v∗(S) = v(N) − v(N \ S) (see, e.g., Bilbao, 2000). As a result, our notion of B-duality is an extension of the standard duality. It is well-known that (N, v) is concave if and only if (N, v∗) is convex. Below is a generalization of this result.

Proposition 3. A coalitional game (N, v) is dual-submodular w.r.t. partition {A, B} of N if and only if (N, v B) is convex.

Proof. Suppose game (N, v) is dual-submodular w.r.t. {A, B}. For all k1, k2 ∈ N and S ≡ (S A, S B) ⊆ N \ {k1, k2}, we prove this proposition by establishing

v B(S + k1k2) + v B(S) ≥ v B(S + k1) + v B(S + k2),

which would imply the convexity of v B , in three mutually exclusive and jointly exhaustive cases of possible memberships of k1 and k2.

Case 1. k1 ∈ A and k2 ∈ A.

By definition of v B , we need to show that

v(S A + k1k2, B \ S B) + v(S A, B \ S B) ≤ v(S A + k1, B \ S B) + v(S A + k2, B \ S B).

Writing T = (S A, B \ S B), the above inequality can be rewritten as

v(T + k1k2) + v(T ) ≤ v(T + k1) + v(T + k2),

which holds because k1 and k2 are substitutes in v .

Case 2. k1 ∈ B and k2 ∈ B . This case is symmetric to Case 1.

Case 3. k1 ∈ A and k2 ∈ B .

By definition of v B , we need to show that

v(S A + k1, B \ (S B + k2)) + v(S A, B \ S B) ≤ v(S A + k1, B \ S B) + v(S A, B \ (S B + k2)).

Writing T = (S A, B \ (S B + k2)), the above inequality can be rewritten as

v(T + k1) + v(T + k2) ≤ v(T + k1k2) + v(T ),

which holds because k1 and k2 are complements in v .

476 Z. Cao et al. / Games and Economic Behavior 108 (2018) 466–477

In summary, we have proved the sufficiency. Note that in each of the three cases, the two corresponding inequalities are equivalent. This establishes the necessity. �

For our next property, we need the following notations. Given partition {A, B} of N , define as in Schrijver (1979),

S � T = (S A ∪ T A, S B ∩ T B)

and

S � T = (S A ∩ T A, S B ∪ T B),

for S, T ⊆ N . The “union” of S and T , S � T , consists of the players either in S or in T on side A and those in both Sand T on side B . In contrast, the “intersection” of S and T , S � T , consists of the players in both S and T on side A and those either in S or in T on side B . Except for the difference in the domains, these two operations are similar to the join and meet considered in Shapley and Shubik (1972). With such extended union and intersection of coalitions, our next proposition establishes an equivalence between the dual-submodularity and a concavity-like property for two-sided games.

Proposition 4. Let {A, B} be a partition of N. Then, a coalitional game (N, v) is dual-submodular w.r.t. {A, B} if and only if

v(S) + v(T ) ≥ v(S � T ) + v(S � T ), S, T ⊆ N. (11)

Proof. Suppose that v is dual-submodular w.r.t. {A, B}. Then

v(S A, S B) + v(T A, T B)

= 2v B(B) − (v B(S A, B \ S B) + v B(T A, B \ T B))

≥ 2v B(B) − (v B(S A ∪ T A, (B \ S B) ∪ (B \ T B)) + v B(S A ∩ T A, (B \ S B) ∩ (B \ T B)))

= 2v B(B) − (v B(S A ∪ T A, B \ (S B ∩ T B)) + v B(S A ∩ T A, B \ (S B ∪ T B)))

= (v B(B) − (v B(S A ∪ T A, B \ (S B ∩ T B)) + (v B(B) − v B(S A ∩ T A, B \ (S B ∪ T B)))

= v(S A ∪ T A, S B ∩ T B) + v(S A ∩ T A, S B ∪ T B)

= v(S � T ) + v(S � T ),

where the inequality is due to the convexity of v B as shown in Proposition 3, and the duality between v and v B is applied in the first and third equalities. The sufficiency follows directly from the definition of dual-submodularity and (11). �References

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