Representation of transferable utility games by coalition production economies Tomoki Inoue∗† March 12, 2011

Abstract We prove that, by the method of construction of a coalition production economy due to Sun et al. [Sun, N., Trockel, W., Yang, Z., 2008. Competitive outcomes and endogenous coalition formation in an n-person game. Journal of Mathematical Economics 44, 853-860], every transferable utility (TU) game can be generated by a coalition production economy. Namely, for every TU game, we can construct a coalition production economy that generates the given game. We briefly discuss the relationship between the core of a given TU game and the set of Walrasian payoff vectors for the induced coalition production economy. JEL classification: C71, D51 Keywords: Coalition production economy, transferable utility game, core, coalition formation, Walrasian equilibrium without double-jobbing ∗

School of Business Administration, Faculty of Urban Liberal Arts, Tokyo Metropolitan University,

1-1 Minami Osawa, Hachioji, Tokyo 192-0397, Japan; [email protected]. † I would like to thank Volker B¨ ohm, Toshiyuki Hirai, Joachim Rosenm¨ uller, Manabu Toda, and Walter Trockel for their stimulating comments. I have also benefited from comments by seminar participants at Bielefeld University, University of Tsukuba, and Waseda University.

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1

Introduction

A transferable utility (TU) game is a reduced form in the sense that it is derived from models with richer structure such as exchange economies or games in strategic form. Shapley and Shubik [18] proved that a TU game is generated by an exchange economy if and only if it is totally balanced. In an exchange economy, an allocation feasible in disjoint segmented markets is also feasible in the union of those markets. This property gives rise to the superadditivity of TU games generated by exchange economies. The total balancedness is more general than the superadditivity. In a coalition production economy, in contrast, different coalitions can access to different production sets and a feasible allocation in segmented markets need not be feasible in the union of those markets any longer. A simple example is an economy with decreasing returns to cooperation in production. As a result, a coalition production economy can generate even a nonsuperadditive TU game. Since an exchange economy is a special case of a coalition production economy, the class of TU games generated by coalition production economies must be larger than the class of TU games generated by exchange economies. Actually, by using a method of construction of a coalition production economy from a TU game due to Sun et al. [21], we prove that any TU game can be generated by a coalition production economy (Theorem 2). Our representation result of a TU game by a coalition production economy is useful when we need a richer structure in models than a TU game. One example was given by Sun et al. [21]. They explained the formation of coalitions at a core payoff vector of a TU game as the formation of firms in a market equilibrium for the coalition production economy induced from the given TU game. The core of a TU game supposes the situation where the grand coalition is formed. Sun et al. [21] focused on non-cohesive TU games where the worth of the grand coalition N is less than the sum of the worths of coalitions in some partition of N . In a non-cohesive TU game, if players do not have to form the grand coalition N , they form several disjoint coalitions constituting a partition of N . Accordingly, we can discuss the formation of coalitions at a payoff vector in the so-called

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core of the cohesive cover.1 Sun et al. [21] induced a coalition production economy from a TU game and defined a market equilibrium with indivisible labor inputs where every agent works for exactly one firm. We call such an equilibrium a Walrasian equilibrium without double-jobbing. When all members of S work for the firm corresponding to S in the induced coalition production economy, we can regard firm S as being formed. Sun et al. [21] proved that the core of the cohesive cover of a TU game coincides with the set of Walrasian payoff vectors without double-jobbing for the induced coalition production economy. Thus, Sun et al. [21] explained which coalitions are formed at a core payoff vector through which firms are formed in the associated Walrasian equilibrium without double-jobbing for the induced coalition production economy. Since we prove that the coalition production economy induced from a TU game generates the given TU game (Theorem 1), it is reasonable to regard the formation of firms in a Walrasian equilibrium without double-jobbing for the induced economy as the formation of coalitions at a core payoff vector. As a market equilibrium for the induced coalition production economy, we also consider a Walrasian equilibrium with divisible labor inputs where agents can work for several firms. Even if labor inputs are divisible, the induced coalition production economy generates a given TU game (Theorem 1). This paper is organized as follows: In Section 2, we prove that the induced coalition production economy due to Sun et al. [21] actually generates a given TU game, regardless of whether labor inputs are indivisible or divisible. In Section 3, we briefly discuss the relationship between the core of a given TU game and the set of Walrasian payoff vectors (without double-jobbing) for the induced coalition production economy. 1

Arnold and Schwalbe [1] also considered the core of the cohesive cover to discuss endogenous coalition

formation in a dynamic model.

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2

Representation of TU games by coalition production economies

Let N = {1, . . . , n} be the set of n players or n agents and let N = {S ⊆ N | S 6= ∅} be the family of coalitions. A transferable utility game (TU game, for short) with n players is a real-valued function on N . A typical TU game is denoted by v : N → R. Every agent in a coalition production economy plays two roles: one is the role as a consumer and the other is the role as a member of production units. Formally, a coalition production economy with n agents is a collection of the commodity space RL , where L is the set of commodities, agents’ characteristics (X i , ui , ω i )i∈N , and coalitions’ production sets (Y S )S∈N satisfying the following conditions. For every agent i ∈ N , consumption set X i ⊆ RL is nonempty, closed, convex, and bounded from below; utility function ui : X i → R is continuous and concave; endowment vector ω i is in RL . For every coalition S ∈ N , its production set Y S ⊆ RL is nonempty, closed, convex, and satisfies

P P i i + Y S 6= ∅. A coalition production economy Y S ∩ RL+ = {0} and i∈S ω i∈S X ∩ is denoted by E = (RL , (X i , ui , ω i )i∈N , (Y S )S∈N ). An exchange economy is a coalition production economy with Y S = {0} for every S ∈ N . The production set Y S of coalition S represents the set of net output vectors which can be achieved through a joint action by all members of S. Thus, we distinguish between what the members of S can produce when they act as one production unit and what they can produce when they form several production subunits. For disjoint coalitions S1 and S2 , if y 1 ∈ Y S1 and y 2 ∈ Y S2 , production vector y 1 + y 2 can be achieved through two production processes conducted by members of S1 and by members of S2 . When all members of S1 ∪ S2 engage in a joint action, however, y 1 + y 2 need not be produced. Thus, (Y S )S∈N need not be superadditive. In general, Y S is determined not only by the technological knowledge of members of S but also by organizational and institutional features inherent to S.2 For example, the technology of coalition S is protected with a patent and it can be employed only when all members of S make an agreement. In this 2

Boehm [6] carefully discussed this point.

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case, even though all members of S are members of T , i.e., S ⊆ T , if some members of S refuse to use their technology for members in T \ S, coalition T cannot employ the technology of coalition S. We first induce two coalition production economies E1 (v) and E2 (v) with divisible labor inputs from a nonnegative TU game v : N → R+ . Economies E1 (v) and E2 (v) have differences only in coalitions’ production sets and the given game v affects only production sets. For h ∈ {1, 2}, define Eh (v) = (RN × R, (X i , ui , ω i )i∈N , (YhS (v))S∈N ) by, for every i ∈ N , X i = {0} × R+ ⊆ RN × R, ui : X i → R with ui (0, x) = x, ω i = (e(i), 0) ∈ RN × R where e(i) ∈ RN is the ith unit vector; for every S ∈ N ,   N S S Y1 (v) = (y, z) ∈ R × R y ∈ −R+ , 0 ≤ z ≤ v(S) min |yj | j∈S

and  Y2S (v) = λ(−e(S), z) ∈ RN × R λ ∈ R+ , 0 ≤ z ≤ v(S) , N is the where RS+ = {x = (xi )i∈N ∈ RN + | xi = 0 for every i ∈ N \ S} and e(S) ∈ R

characteristic vector of S, i.e., e(S)i = 1 if i ∈ S and 0 otherwise. Note that, for every S ∈ N , both Y1S (v) and Y2S (v) are closed convex cones with vertex (0, 0) and satisfy
× R = {(0, 0)} for h ∈ {1, 2}. Hence, E1 (v) and E2 (v) satisfy all the YhS (v) ∩ RN + + requirements for a coalition production economy. The first economy E1 (v) is essentially the same as the induced coalition production economy due to Sun et al. [21].3,4 The second economy E2 (v) is a slight modification of the first economy E1 (v), but E2 (v) explicitly describes the cooperation among members of a coalition (see the next paragraph) and E2 (v) has a relation to Billera’s [4] production 3

In the induced coalition production economy due to Sun et al. [21], every agent gets utility from

his personalized output commodity. This is not essential and, in E1 (v), every agent gets utility from the unique output commodity. Consequently, the number of commodities is reduced from 2n to n + 1. 4 Sun et al. [21] constructed a coalition production economy from the cohesive cover of a given nonnegative TU game, but their construction can be applied to every nonnegative TU game. The cohesive P cover v c of a TU game v is a TU game such that v c (S) = v(S) if S ( N and v c (N ) = max S∈π v(S) where the maximum is run over all partitions π of N . The terminology “cohesiveness” is due to Osborne and Rubinstein [15].

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economy induced from a non-transferable utility (NTU) game (see Remark 1). In both coalition production economies, every agent has the same consumption set and the same utility function. Agents are distinguished only by their endowment vectors. From the form of production sets, the first n commodities are inputs and the last commodity is an output. Since the ith input is initially owned only by agent i, input i can be interpreted as agent i’s labor. Thus, Y1S (v) or Y2S (v) represents a production set of firm S for which agents of S work. For firm S to produce the output, economy E2 (v) requires that all members of S devote their labors to firm S in the same proportion of their labors. If all members of S devote the same units λ of their labors, then they earn in total λv(S) units of the output. Therefore, Y2S (v) explicitly describes an agreement on a joint action among members of S and meets the case where the technology of firm S can be employed only when unanimous agreement among members of S is made. In economy E1 (v), in contrast, members of S can devote their labors in different proportions to firm S. In such a case, since agents’ labors are complementary, the resulting production vector is inefficient. Note that, in both production sets Y1S (v) and Y2S (v), agents’ labor inputs or labor time are divisible. Every agent can devote any proportion of his labor input to any firm. In the next section, we consider two kinds of market equilibria for our induced coalition production economy. One is a Walrasian equilibrium where every agent can work for several firms as long as his total labor time is equal to his endowed labor time. The other is a Walrasian equilibrium without double-jobbing where every agent works full-time for exactly one firm.5 When we adopt a Walrasian equilibrium without double-jobbing as a market equilibrium, the corresponding TU game should be generated by a coalition production economy with indivisible labor inputs. For h ∈ {1, 2}, let Eh∗ (v) be a coalition production economy which is obtained by introducing the indivisibility of agents’ labor inputs in economy Eh (v). The production set of coalition S for Eh∗ (v) is the same cone YhS (v) as for economy Eh (v) and we represent the indivisibility of agents’ labor inputs by the feasibility of allo5

The precise definition of a Walrasian equilibrium (without double-jobbing) is given in Section 3.

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cations within a coalition. This formulation of cone production sets follows McKenzie’s [14] introduction of entrepreneurial factor as a marketed commodity; the entrepreneurial factor is indivisible in the sense that its supply is always one unit, but the production set is formulated as a cone. In our economies, every agent i can work for any firm S with S 3 i and, therefore, he chooses a firm for which he will work. Therefore, the indivisibility of agents’ labor inputs is modeled as follows: whenever agent i works for firm S, he has to devote all his labor time to firm S. We adopt the following feasibility of allocations within a coalition for Eh (v) and for Eh∗ (v). For h ∈ {1, 2} and every S ∈ N , FEh (v) (S) denotes the set of feasible S-allocations for Eh (v) with divisible labor inputs, i.e., FEh (v) (S) = {(0, xi )i∈S | (0, xi ) ∈ X i for every i ∈ S and

P

i∈S

((0, xi ) − ω i ) ∈ YhS (v)},

and FEh∗ (v) (S) denotes the set of feasible S-allocations for Eh∗ (v) with indivisible labor inputs, i.e., FEh∗ (v) (S) = {(0, xi )i∈S | (0, xi ) ∈ X i for every i ∈ S and
P i i S S N \S × R }. i∈S ((0, x ) − ω ) ∈ Yh (v) ∩ {−1, 0} × {0} Note that both sets FEh (v) (S) and FEh∗ (v) (S) are nonempty and compact. The next lemma clarifies the relation of production sets Y1S (v) and Y2S (v) with divisible labor inputs or with indivisible labor inputs. Lemma 1. Let v : N → R+ and let S ∈ N . (i) Y2S (v) ⊆ Y1S (v). (ii) If (y, z) ∈ Y1S (v), then (y, z) ≤ (− minj∈S |yj |e(S), z) ∈ Y2S (v). If (y, z) ∈ Y1S (v) ∩ ({−1, 0}S × {0}N \S × R), then (y, z) ≤ (− minj∈S |yj |e(S), z) ∈ Y2S (v) ∩ ({−1, 0}S × {0}N \S × R). This lemma can be shown straightforwardly. For h ∈ {1, 2}, coalition production economies Eh (v) with divisible labor inputs and Eh∗ (v) with indivisible labor inputs generate TU games vEh (v) : N → R and vEh∗ (v) : N → R, 7

respectively, by defining, for every S ∈ N , ( ) X vEh (v) (S) = max ui (0, xi ) (0, xi )i∈S ∈ FEh (v) (S) i∈S

and

( X

vEh∗ (v) (S) = max

i∈S

) ui (0, xi ) (0, xi )i∈S ∈ FEh∗ (v) (S) .

The next theorem states that the TU games generated by coalition production economies E1 (v) and E2 (v) with divisible labor inputs or by E1∗ (v) and E2∗ (v) with indivisible labor inputs are all equal to the original nonnegative game v. Theorem 1. Let v : N → R+ . Then, v = vE1 (v) = vE2 (v) = vE1∗ (v) = vE2∗ (v) . Proof. Since Y2S (v) ∩ ({−1, 0}S × {0}N \S × R) ⊆ Y1S (v) ∩ ({−1, 0}S × {0}N \S × R) ⊆ Y1S (v) and Y2S (v) ∩ ({−1, 0}S × {0}N \S × R) ⊆ Y2S (v) ⊆ Y1S (v) for every S ∈ N by Lemma 1 (i), we have FE2∗ (v) (S) ⊆ FE1∗ (v) (S) ⊆ FE1 (v) (S) and FE2∗ (v) (S) ⊆ FE2 (v) (S) ⊆ FE1 (v) (S). Therefore, vE2∗ (v) (S) ≤ vE1∗ (v) (S) ≤ vE1 (v) (S) and vE2∗ (v) (S) ≤ vE2 (v) (S) ≤ vE1 (v) (S) for every S ∈ N . Hence, it is enough to prove the inequalities vE1 (v) (S) ≤ v(S) ≤ vE2∗ (v) (S) for every S ∈ N . Let S ∈ N . We first prove that vE1 (v) (S) ≤ v(S). By the definition of vE1 (v) (S), there exists (0, xi )i∈S ∈ FE1 (v) (S) such that vE1 (v) (S) =

X

ui (0, xi ) =

X

i∈S

xi .

i∈S

Since (0, xi )i∈S ∈ FE1 (v) (S), there exists (y, z) ∈ Y1S (v) such that ! (y, z) =

X

i

(0, x ) − ω

i




=

−e(S),

i∈S

X

x

i

.

i∈S

Hence, we have vE1 (v) (S) =

X

xi = z ≤ v(S) min |yj | = v(S). j∈S

i∈S

We next prove that v(S) ≤ vE2∗ (v) (S). For every i ∈ S, define (0, xi ) ∈ X i by xi = v(S)/|S|. Since ! X i∈S

(0, xi ) − ω


i

=

−e(S),

X

xi


= (−e(S), v(S)) ∈ Y2S (v)∩ {−1, 0}S × {0}N \S × R ,

i∈S

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we have (0, xi )i∈S ∈ FE2∗ (v) (S). Therefore, vE2∗ (v) (S) ≥

X

ui (0, xi ) =

i∈S

X

xi = v(S).

i∈S

We can prove that every TU game can be generated by a coalition production economy with divisible labor inputs and by a coalition production economy with indivisible labor inputs, regardless of whether it is nonnegative or not. Theorem 2. Every TU game v : N → R can be generated by a coalition production economy with divisible labor inputs and by a coalition production economy with indivisible labor inputs. Proof. Let v : N → R. Since N is finite, for a large positive vector c ∈ RN , we have P (v + c)(S) = v(S) + i∈S ci ≥ 0 for every S ∈ N . Thus, by Theorem 1, v + c can be generated by coalition production economies E1 (v + c) and E2 (v + c) with divisible labor inputs and also by coalition production economies E1∗ (v + c) and E2∗ (v + c) with indivisible labor inputs. In these economies, agent i’s utility function ui : X i → R is given by ui (0, x) = x. Define u¯i : X i → R by u¯i (0, x) = x − ci . For h ∈ {1, 2}, let E¯h (v + c) and E¯h∗ (v + c) be coalition production economies where agent i’s utility function ui of Eh (v + c) and Eh∗ (v + c), respectively, is replaced with u¯i . It can be easily shown that coalition production economies E¯1 (v + c), E¯2 (v + c), E¯1∗ (v + c), and E¯2∗ (v + c) generate v. Remark 1. Billera [4] proved that the class of totally cardinally balanced NTU games coincides with the class of NTU games generated by production economies where every agent has the common convex cone production set. When we apply Billera’s construction of a production economy to a TU game v : N → R+ , the common production set among S P agents is the convex hull of S∈N Y2S (v), which is equal to S∈N Y2S (v). Therefore, by decomposing Billera’s production set, we obtain coalitions’ production sets in our second economy E2 (v).

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Remark 2. Billera’s [4] induced production economy can be converted to an exchange economy by Rader’s [16] method (see also Billera and Bixby [5]). In contrast, our induced coalition production economies E1 (v) and E2 (v) cannot be converted to exchange economies even if a given TU game is totally balanced. Remark 3. A similar construction method to our second economy E2 (v) can be applied to NTU games. Inoue [13] proved that any compactly generated NTU game can be generated by a coalition production economy. Remark 4. Bejan and G´omez [3] induce a production economy from a TU game by a method similar to E1 (v). They explicitly describe the ownership of each firm and assume that if firm S is owned by some members of firm T , i.e., T ) S, then firm T can employ the technology of firm S. Consequently, the production set of coalition T turns out to P be the form S⊆T Y1S (v) and their induced private ownership economy generates only a totally balanced TU game.

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Equivalence between the core and Walrasian payoff vectors

We briefly discuss the relationship between the core of a TU game and the set of Walrasian payoff vectors for our induced coalition production economies. It is well known that in an economy with infinitely many agents, the set of core allocations coincides with the set of Walrasian allocations (see Debreu and Scarf [8] and Aumann [2] for an exchange economy and see Hildenbrand [10] for a coalition production economy). Sun et al. [21] started with a TU game v rather than an economy, and considered the relationship between the core of the given game v and the set of Walrasian payoff vectors without double-jobbing for induced coalition production economy E1∗ (v).6 6

Sun et al. [21] follows Shapley and Shubik [19] who proved that, given a totally balanced TU game,

its core coincides with the set of Walrasian payoff vectors for the exchange economy induced by Shapley and Shubik [18].

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The precise definition of a Walrasian equilibrium without double-jobbing for Eh (v) is as follows: Definition 1. A tuple ((ˆ q , pˆ), (0, xˆi )i∈N , (ˆ y S , zˆS )S∈N ) of a price vector (ˆ q , pˆ) ∈ RN × R, agents’ consumption vectors (0, xˆi )i∈N and coalitions’ production vectors (ˆ y S , zˆS )S∈N is a Walrasian equilibrium without double-jobbing for Eh∗ (v), h = 1 or 2, if (1) (ˆ q , pˆ) ∈ RN + × R+ ; (2) for every S ∈ N , (ˆ y S , zˆS ) ∈ YhS (v) and (ˆ q , pˆ) · (ˆ y S , zˆS ) = max(y,z)∈YhS (v) (ˆ q , pˆ) · (y, z) = 0; (3) for every i ∈ N , (0, xˆi ) maximizes ui in the budget set {(0, x) ∈ X i | (ˆ q , pˆ) · (0, x) ≤ (ˆ q , pˆ) · ω i }; (4)

ˆi ) = i∈N (0, x

P

i i∈N ω +

P

P

y S , zˆS ); S∈N (ˆ

and

(5) for every i ∈ N , there exists a unique S ∈ N such that i ∈ S, yˆiS = −1, and yˆiT = 0 for every T 6= S. The vector (ui (0, xˆi ))i∈N = (ˆ xi )i∈N is called a Walrasian payoff vector without doublejobbing of Eh∗ (v). The set of all Walrasian payoff vectors without double-jobbing of Eh∗ (v) is denoted by Wh∗ (v). Condition (5) represents the no-double-jobbing condition; every agent must work fulltime for exactly one firm. Since the feasibility of allocations for Eh∗ (v) adopts the same indivisibility of agents’ labor inputs as a Walrasian equilibrium without double-jobbing, we can regard vEh∗ (v) as a TU game generated by Eh∗ (v). Although agents’ labor inputs are indivisible, production set YhS (v) of coalition S is modeled as a cone, and, therefore, the maximum profit for firm S is zero. As we mentioned earlier, this formulation of cone production sets follows McKenzie [14]. Moreover, by this formulation, we can avoid a problem of how to divide a positive profit among laborers if the maximum profit is positive.

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In induced economy E1∗ (v), since several firms can conduct production processes simultaneously and since agents do not have to act as one production unit of the grand coalition, the resource-feasibility condition (4) requires that the aggregate net trade vector P P ˆi ) − ω i ) be in S∈N Y1S (v).7 If all members of S work full-time for firm S, we i∈N ((0, x can interpret that coalition S is formed. Thus, we can discuss the coalition formation at a payoff vector by seeing which firms are active in the associated Walrasian equilibrium without double-jobbing.8 Sun et al. [21] mainly considered non-cohesive TU games,9 because in a cohesive TU game, forming the grand coalition is most efficient and there is nothing to argue about the coalition formation. They proved that, given a TU game v, the core of the cohesive cover v c of v, ( C(v c ) =

x ∈ RN

) X X xi ≥ v(S) for every S ∈ N \ {N } , xi = v c (N ) and i∈N

i∈S

coincides with the set W1∗ (v) of Walrasian payoff vectors without double-jobbing for E1∗ (v). If the cohesive cover v c is not balanced, its core is empty but the equivalence still holds.10 The family of active firms, or, equivalently, the family of formed coalitions, in a Walrasian 7

Hildenbrand [10] and Sondermann [20] adopted a different resource-feasibility condition. They require

that the aggregate net trade vector be in the production set Y1N (v) of the grand coalition. Since they assume that coalitions’ production sets are additive or superadditive, a larger coalition can conduct a more efficient production process. In contrast, in induced economy E1∗ (v), since (Y1S (v))S∈N need not be superadditive, agents can achieve a more efficient production vector by forming several subcoalitions of N than by forming the grand coalition. Ichiishi [11] and Greenberg [9] adopted the same resource-feasibility condition as ours. 8 Ichiishi [11] and Greenberg [9] explain the formation of firms in a Walrasian equilibrium for a labormanaged economy. A labor-managed economy is an economy where members of coalition S are not owners of firm S but they work for firm S. Hence, our induced economies Eh (v) and Eh∗ (v) (h = 1, 2) are labor-managed economies. Ichiishi [11] and Greenberg [9] assumed that agents’ labor inputs are not marketed and their wages are rewards that they earn by participating in production units. In contrast, in Eh (v) and Eh∗ (v), agents’ labor inputs are marketed and their wages are determined to clear the labor markets. 9 For the definition of the cohesiveness of a TU game, see footnote 4. 10 Thus, the balancedness assumption in Theorem 3.2 of Sun et al. [21] is dispensable.

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equilibrium without double-jobbing for E1∗ (v) need not be a partition of N . If v({i}) = 0, it is indifferent for agent i to work for an inactive firm S such that some agent j ∈ S does not work for firm S and to work for firm {i}, because both firms produce zero output.11 If v(S) > 0 for every S ∈ N , the family of active firms in a Walrasian equilibrium without double-jobbing for E1∗ (v) is a partition of N . In any case, the aggregate net trade P vector i∈N ((0, xˆi ) − ω i ) of a Walrasian equilibrium without double-jobbing for E1∗ (v) is P in S∈π Y1S (v) for some partition π of N . Since the aggregate net trade vector may not S P be in Y1N (v) but must be in π S∈π Y1S (v), the set W1∗ (v) of Walrasian payoff vectors without double-jobbing for E1∗ (v) coincides the core C(v c ) of the cohesive cover of v, rather than the core C(v) of v. In a Walrasian equilibrium without double-jobbing for E2∗ (v), the family of active firms is always a partition of N . Also, we have the same equivalence between the core C(v c ) of the cohesive cover v c of v and the set W2∗ (v) of Walrasian payoff vectors without double-jobbing for E2∗ (v). Thus, in particular, two induced economies with indivisible labor inputs have the same set of Walrasian payoff vectors without double-jobbing, i.e., W1∗ (v) = W2∗ (v). Inoue [12] considered the case where agents can work for several firms fractionally. A Walrasian equilibrium for E1 (v) or E2 (v) with divisible labor inputs is defined by conditions (1)-(4) of Definition 1. Clearly, the set of Walrasian payoff vectors for Eh (v) is larger than the set of Walrasian payoff vectors without double-jobbing for Eh∗ (v). Since agents’ labor inputs are divisible both in a Walrasian equilibrium for Eh (v) and in a feasible allocation within a coalition for Eh (v), we can regard vEh (v) as a TU game generated by Eh (v). The two induced economies E1 (v) and E2 (v) with divisible labor inputs give the same set of Walrasian payoff vectors and it coincides with the core of the balanced cover of v. Hence, by Bondareva-Shapley theorem (see Bondareva [7] and Shapley [17]), both induced coalition production economies E1 (v) and E2 (v) have always a Walrasian equilibrium. Given a payoff vector in the core of the balanced cover v b of a TU game v, if agent i works for firm S for the same ratio as the balancing vector of weights achieving 11

For details, see Inoue [12, Example 1].

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v b (N ), then the resulting state turns out to be a Walrasian equilibrium.

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[21] Sun, N., Trockel, W., Yang, Z., 2008. Competitive outcomes and endogenous coalition formation in an n-person game. Journal of Mathematical Economics 44, 853-860.

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