The Nonemptiness of the Inner Core

Abstract

We prove that if a non-transferable utility (NTU) game is cardinally balanced and if, at every individually rational and efficient payoff vector, every non-zero normal vector to the set of payoff vectors feasible for the grand coalition is strictly positive, then the inner core is nonempty. The condition on normal vectors is satisfied if the set of payoff vectors feasible for the grand coalition is non-leveled. An NTU game generated by an exchange economy where every consumer has a continuous, concave, and strongly monotone utility function satisfies our sufficient condition. Our proof relies on Qin’s theorem on the nonemptiness of the inner core.

JEL Classification: C62, C71

Mathematics Subject Classification (2010): 91A12, 91B50

Article type: Research Article

Received: February 27, 2019

Revised: July 24, 2019

1 Introduction

The inner core is a solution concept for a non-transferable utility (NTU) game. Players’ utilities are not transferable, but an inner core payoff vector is stable even if players can transfer their utilities with some fictitious rates λ. Namely, at an inner core payoff vector, any coalition cannot improve upon the λ-weighted sum of utilities. This stability is stronger than the one required for the core. Thus, the inner core is a refinement of the core.

The inner core is of significance in the relations to (1) Walrasian equilibria for some economies, and to (2) the strictly inhibitive set of payoff vectors that are stable against randomized blocking plans. An exchange economy or a production economy generates a unique NTU game, but an NTU game may be generated by multiple economies. Billera [3] proposed a method how to induce a production economy from a totally cardinally balanced NTU game so that the economy actually generates the given NTU game. Qin [14] proved that the inner core of a totally cardinally balanced NTU game coincides with the set of utility vectors at Walrasian equilibria for Billera’s induced production economy.¹ Inoue [9] constructed a coalition production economy generating a given NTU game and proved that the inner core coincides with the set of utility vectors at Walrasian equilibria for his induced coalition production economy.

When a payoff vector can be improved upon by multiple coalitions, it is not clear which coalition actually improves upon the payoff vector. Myerson [12, Section 9.8] considered the following scenario. A mediator who is not a player of a given game randomly chooses a coalition and payoffs to the members of the coalition. Every player accepts the mediator’s randomized blocking plan if, conditional on being a member of a coalition, the expected payoff to him is better than the given payoff to him. The strictly inhibitive set is the set of payoff vectors that are feasible for the grand coalition and that are stable against randomized blocking plans. Qin [13] proved that the inner core is always a subset of the strictly inhibitive set and, in some classes of NTU games, these two sets coincide.

A sufficient condition for the inner core to be nonempty was given by Qin [15, Theorem 1]. Although his sufficient condition is mathematically general, it is not easy to check whether a given NTU game satisfies the condition. Accordingly, it is useful to give classes of NTU games satisfying Qin’s sufficient condition. Qin [15, Corollary 1] proved that every cardinally balanced-with-slack NTU game satisfies his own sufficient condition. We give another class of NTU games satisfying Qin’s sufficient condition: If an NTU game is cardinally balanced and if, at every individually rational and efficient payoff vector, every non-zero normal vector to the set of payoff vectors feasible for the grand coalition is strictly positive, then the NTU game satisfies Qin’s sufficient condition for the inner core to be nonempty. The condition on normal vectors is met if the set of payoff vectors feasible for the grand coalition is non-leveled. Our class of NTU games with the nonempty inner core contains NTU games generated by exchange economies where every consumer has a continuous, concave, and strongly monotone utility function. de Clippel and Minelli [8] proved that the utility vector at a Walrasian equilibrium for such an exchange economy is always in the inner core and, therefore, the nonemptiness of the inner core follows from the well-known existence theorem of a Walrasian equilibrium. Hence, our theorem extends de Clippel and Minelli’s class of NTU games with the nonempty inner core.

In the present paper, we prove our theorem on the nonemptiness of the inner core by applying Qin’s [15] theorem. There are two other methods of proof of our theorem. The first way is due to Aubin [2, Theorem 2.1, Proposition 2.3]. Aubin adopted an abstract description of an NTU game. If we adopt a specified “representative function” of an NTU game, what Aubin called “an equilibrium for a representative function” turns out to be an inner core payoff vector. Since Aubin proved the existence of an equilibrium for a representative function, he indeed proved the nonemptiness of the inner core. At an inner core payoff vector, fictitious transfer rates λ of utilities must be strictly positive. Aubin first finds nonnegative fictitious transfer rates λ with certain properties and then provides a sufficient condition (condition (c) of Proposition 2 in Sect. 3) for λ to be strictly positive.² As discussed in Sect. 3, Aubin’s sufficient condition for λ to be strictly positive can be slightly weakened. The second way is due to Inoue [11] who took two steps as well as Aubin [2]. Inoue [11] first gives a new coincidence theorem, a synthesis of Brouwer’s fixed point theorem and a stronger separation theorem for convex sets due to Debreu and Schmeidler [7], and then, by applying the coincidence theorem, he obtains nonnegative fictitious transfer rates of utilities with certain properties. Inoue’s [11] second step of the proof is the same as Aubin’s [2].

The present paper is organized as follows. In Sect. 2, we give a precise description of an NTU game and the definition of the inner core. Also, we give Qin’s sufficient condition for the inner core to be nonempty (Theorem 1). In Sect. 3, we provide characterizations of the efficient surface with strictly positive normal vectors. In Sect. 4, we prove the nonemptiness of the inner core (Theorem 2). In Sect. 5, we prove that our class of NTU games with the nonempty inner core contains de Clippel and Minelli’s class of NTU games generated by exchange economies with continuous, concave, and strongly monotone utility functions. In Sect. 6, we give some remarks.

2 NTU Games and the Inner Core

We begin with some notation. We follow the notation of Qin [15]. Let N = {1, …, n} be a set with n ≥ 2 elements and let ${\mathbb {R}}^N$ be the n-dimensional Euclidean space of vectors x with coordinates x _i indexed by i ∈ N. The inner product of x and y in ${\mathbb {R}}^N$ is denoted by x ⋅ y, i.e., x ⋅ y =∑_{i ∈ N}x _iy _i. For $x,y\in {\mathbb {R}}^N$, we write x ≥ y if x _i ≥ y _i for every i ∈ N; x ≫ y if x _i > y _i for every i ∈ N. The symbol 0 denotes the origin in ${\mathbb {R}}^N$ as well as the real number zero. Let ${\mathbb {R}}^N_{++}=\{x\in {\mathbb {R}}^N\,|\, x\gg 0\}$. For a nonempty subset S of N, let ${\mathbb {R}}^S=\{x\in {\mathbb {R}}^N\,|\, x_i=0\ \text{for every }i\in N\setminus S\}$, let ${\mathbb {R}}^S_+=\{x\in {\mathbb {R}}^S\,|\, x_i\ge 0\ \text{for every }i\in S\}$, and let $e^S\in {\mathbb {R}}^N$ be the characteristic vector of S, i.e., $e^S_i=1$ if i ∈ S and 0 otherwise. For $x\in {\mathbb {R}}^N$, x ^S denotes the projection of x to ${\mathbb {R}}^S$. Let $\Delta =\{\lambda \in {\mathbb {R}}^N_+\,|\, \sum _{i\in N}\lambda _i=1\}$, Δ^∘ = {λ ∈ Δ | λ ≫ 0}, and, for every m ≥ n, Δ_1∕m = {λ ∈ Δ | λ _i ≥ 1∕m for every i ∈ N}.

We regard N as the set of n players. Let ${\mathcal {N}}$ be the family of all nonempty subsets of N, i.e., ${\mathcal {N}}=\{S\subseteq N\,|\, S\neq \emptyset \}$. Elements in ${\mathcal {N}}$ are called coalitions. A non-transferable utility game (NTU game, for short) with n players is a correspondence $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ such that, for every $S\in {\mathcal {N}}$, V (S) is a nonempty subset of ${\mathbb {R}}^S$ and satisfies $V(S)-{\mathbb {R}}^S_+=V(S)$. An NTU game is compactly generated if, for every $S\in {\mathcal {N}}$, there exists a nonempty compact subset C _S of ${\mathbb {R}}^S$ with $V(S)=C_S-{\mathbb {R}}^S_+$. In the present paper, we consider only compactly generated NTU games V with V (N) convex.

The core is the set of payoff vectors which is feasible for the grand coalition N and which cannot be improved upon by any coalition. By adopting a different notion of improvement by a coalition, we can define the inner core.

Definition 1

• (1) The coreC(V ) of NTU game V is the set of payoff vectors $u\in {\mathbb {R}}^N$ such that u ∈ V (N) and there exists no $S\in {\mathcal {N}}$ and u′∈ V (S) with $u^{\prime }_i>u_i$ for every i ∈ S.

• (2) The inner coreIC(V ) of NTU game V is the set of payoff vectors $u\in {\mathbb {R}}^N$ such that u ∈ V (N) and there exists $\lambda \in {\mathbb {R}}^N_{++}$ such that, for every $S\in {\mathcal {N}}$ and every u′∈ V (S), λ ^S ⋅ u ≥ λ ^S ⋅ u′ holds.

By definition, we have IC(V ) ⊆ C(V ). The vector $\lambda \in {\mathbb {R}}^N_{++}$ in the definition of the inner core represents fictitious transfer rates of utilities among players. Note that u ∈ IC(V ) if and only if u ∈ V (N) ∩ C(V _λ) for some $\lambda \in {\mathbb {R}}^N_{++}$, where $V_\lambda :{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ is the λ-transfer game defined by

$$\displaystyle \begin{aligned} V_\lambda(S)=\left\{u\in {\mathbb{R}}^S\,\left|\, \lambda\cdot u\le \lambda\cdot u'\ \text{for some }u'\in V(S)\right.\right\}\quad \text{for every }S\in {\mathcal{N}}. \end{aligned}$$

Note also that we can restrict the space of fictitious transfer rates to Δ^∘.

For $\beta \in {\mathbb {R}}^N_+$, let

$$\displaystyle \begin{aligned} \Gamma (\beta)=\left\{\gamma=(\gamma_S)_{S\in {\mathcal{N}}}\,\left|\, \gamma_S\ge 0\ \text{for every }S\in {\mathcal{N}}\text{ and}\ \sum_{S\in {\mathcal{N}}}\gamma_S\,e^S=\beta\right\}\right. \end{aligned}$$

and let $\widehat {\Gamma }(\beta )=\{\gamma \in \Gamma (\beta )\,|\, \gamma _N=0\}$. Note that Γ(e ^N) is the set of balancing vectors of weights. An NTU game V is cardinally balanced if, for every γ ∈ Γ(e ^N),

$$\displaystyle \begin{aligned} \sum_{S\in {\mathcal{N}}}\gamma_S\,V(S)\subseteq V(N). \end{aligned}$$

This notion of balancedness is stronger than the balancedness due to Scarf [16]. Scarf’s ordinal balancedness is sufficient for the nonemptiness of the core. For our theorem (Theorem 2) on the nonemptiness of the inner core, this stronger balancedness is assumed.

Before we give a condition equivalent to the cardinal balancedness, we introduce some notation. Let $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ be a compactly generated NTU game with V (N) convex. For every $S\in {\mathcal {N}}$, let C _S be a compact subset of ${\mathbb {R}}^S$ generating V (S), i.e., $V(S)=C_S-{\mathbb {R}}^S_+$, and let C _N be also convex. For every $\lambda \in {\mathbb {R}}^N_+$ and every $S\in {\mathcal {N}}$, define

$$\displaystyle \begin{aligned} v_\lambda(S)=\max \left\{\lambda\cdot u\,\left|\, u\in V(S)\right\}\right.=\max\left\{\lambda\cdot u\,\left|\, u\in C_S\right\}\right.. \end{aligned}$$

Note that, by Berge’s maximum theorem, ${\mathbb {R}}^N_+\ni \lambda \mapsto v_\lambda (S)\in {\mathbb {R}}$ is continuous. For every i ∈ N, let $b_i\in {\mathbb {R}}$ be the utility level that player i can achieve by himself, i.e,

$$\displaystyle \begin{aligned} b_i=\max\left\{u_i\in {\mathbb{R}}\,\left|\, u\in V(\{i\})\right\}\right.. \end{aligned}$$

Define a correspondence $B:{\mathbb {R}}^N_{++}\twoheadrightarrow {\mathbb {R}}^N$ by

$$\displaystyle \begin{aligned} B(\lambda)=\left\{u\in V(N)\,\left|\, \lambda\cdot u=v_\lambda (N)\right\}\right.=\left\{u\in C_N\,\left|\, \lambda\cdot u=v_\lambda (N)\right\}\right.. \end{aligned}$$

Note that B(λ) is positively homogeneous of degree zero.

The following proposition gives a condition equivalent to the cardinal balancedness.

Proposition 1

Let $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$be a compactly generated NTU game with V (N) convex. Then, V is cardinally balanced if and only if, for every λ ∈ Δ ^∘and every $\gamma \in \widehat {\Gamma }(e^N)$, $\sum _{S\in {\mathcal {N}}}\gamma _S\,v_{\lambda }(S)\le v_{\lambda }(N)$.

This equivalence is due to Shapley (see Qin [15, Proposition 1]).

For m ≥ n, define continuous functions $p^m:\Delta \to {\mathbb {R}}^N_{++}$ and $\beta ^m:\Delta \to {\mathbb {R}}^N_+\setminus \{0\}$ by

$$\displaystyle \begin{aligned} p^m_j(\lambda)=\left\{ \begin{array}{ll} \lambda_j & \text{if }\lambda_j\ge 1/m,\\ 1/m & \text{if }\lambda_j<1/m, \end{array} \right. \end{aligned}$$

and

$$\displaystyle \begin{aligned} \beta^m_j(\lambda)=\frac{\lambda_j}{p^m_j(\lambda)}=\left\{ \begin{array}{ll} 1 & \text{if }\lambda_j\ge 1/m,\\ m\lambda_j & \text{if }\lambda_j<1/m. \end{array} \right. \end{aligned}$$

We are now ready to give Qin’s theorem [15, Theorem 1] on the nonemptiness of the inner core. Inoue [11] gives another proof to the theorem.

Theorem 1 (Qin)

Let $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$be a compactly generated NTU game with V (N) convex. If there exists m ≥ n such that

• (i) for every λ ∈ Δ _1∕mand every $\gamma \in \widehat {\Gamma }(e^N)$, $\sum _{S\in {\mathcal {N}}}\gamma _S\,v_\lambda (S)\le v_\lambda (N)$, and

• (ii) for every λ ∈ Δ ^∘∖ Δ _1∕mand every $\gamma \in \widehat {\Gamma }(\beta ^m(\lambda ))$, there exists u ∈ B(p ^m(λ)) such that $\sum _{S\in {\mathcal {N}}}\gamma _S\,v_{p^m(\lambda )}(S)\le \lambda \cdot u$,

then the inner core IC(V ) of V is nonempty.

By Proposition 1, condition (i) is weaker than the cardinal balancedness. Qin [15, Corollary 1] gives a class of NTU games that satisfy both conditions (i) and (ii) for some m: If a compactly generated NTU game V with V (N) convex is cardinally balanced with slack, i.e., for every $\gamma \in \widehat {\Gamma }(e^N)$, $\sum _{S\in {\mathcal {N}}}\gamma _S\, V(S)$ is a subset of the interior of V (N), then V satisfies conditions (i) and (ii) for some m ≥ n, and, therefore, the inner core of V is nonempty.

Theorem 2 in Sect. 4 gives another class of NTU games with the nonempty inner core.

3 Characterization of Efficient Surface with Strictly Positive Normal Vectors

In the next section, we prove the nonemptiness of the inner core for an NTU game V where the normal cone to V (N) at any individually rational and efficient payoff vector is a subset of ${\mathbb {R}}^N_{++}\cup \{0\}$. In this section, we provide characterizations of the set of individually rational and efficient payoff vectors with the above mentioned property.

Let V (N) be a nonempty, closed, convex subset of ${\mathbb {R}}^N$ generated by a compact set $C_N\subseteq {\mathbb {R}}^N$, i.e., $V(N)=C_N-{\mathbb {R}}^N_+$. Let $b\in {\mathbb {R}}^N$ be such that {x ∈ V (N) | x ≥ b}≠ ∅. Define

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \mathrm{Eff}(V(N),b)\\ &\displaystyle &\displaystyle =\left\{x\in V(N)\,\left|\, x\ge b,\ \text{there exists no}\ x'\in V(N)\ \mbox{with}\ x'\ge x\ \mbox{and}\ x'\neq x\right\}\right. \end{array} \end{aligned} $$

and

$$\displaystyle \begin{aligned} \mathrm{Eff_w}(V(N),b)=\left\{x\in V(N)\,\left|\, x\ge b,\ \mbox{there exists no}\ x'\in V(N)\ \mbox{with}\ x'\gg x\right\}\right.. \end{aligned}$$

The set Eff(V (N), b) (resp. Eff_w(V (N), b)) is the set of individually rational and efficient payoff vectors (resp. individually rational and weakly efficient payoff vectors).

Remark 1

• (1) ∅ ≠ Eff(V (N), b) ⊆Eff_w(V (N), b).

• (2) Eff_w(V (N), b) is compact.

The nonemptiness of Eff(V (N), b) in statement (1) follows from the assumption that V (N) is compactly generated and {x ∈ V (N) | x ≥ b}≠ ∅. Statement (2) can be easily shown.

The following lemma gives a characterization of Eff_w(V (N), b) by normal vectors to V (N).³

Lemma 1

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \mathrm{Eff_w}(V(N),b)\\ &\displaystyle &\displaystyle \begin{array}{ll} =\{x\in V(N)\,|&\displaystyle x\ge b,\\ &\displaystyle \mathrm{there\ exists}\ \lambda\in \Delta\ \mathrm{{such\ that,\ for\ every}}\ y\in V(N),\ \lambda\cdot x\ge \lambda\cdot y\}. \end{array} \end{array} \end{aligned} $$

Proof

Let x ∈Eff_w(V (N), b). Then, x ∈ V (N) and x ≥ b. Furthermore, $V(N)\cap (\{x\}+{\mathbb {R}}^N_{++})=\emptyset$. By the separation theorem for convex sets, there exists $\lambda \in {\mathbb {R}}^N\setminus \{0\}$ such that, for every y ∈ V (N) and every $z\in \{x\}+{\mathbb {R}}^N_{++}$, λ ⋅ z ≥ λ ⋅ y. Then, we have $\lambda \in {\mathbb {R}}^N_+\setminus \{0\}$. By normalization, we may assume that λ ∈ Δ. Since the inner product is continuous, we have λ ⋅ x ≥ λ ⋅ y for every y ∈ V (N).

We next prove the inverse inclusion. Let x ∈ V (N) be such that x ≥ b and there exists λ ∈ Δ such that, for every y ∈ V (N), λ ⋅ x ≥ λ ⋅ y. Suppose, to the contrary, that x∉Eff_w(V (N), b). Then, there exists x′∈ V (N) with x′≫ x. Thus, we have λ ⋅ x′ > λ ⋅ x, a contradiction. Hence, we have x ∈Eff_w(V (N), b). □

The following proposition gives a characterization of the individually rational and efficient surface such that every non-zero normal vector to V (N) at every point of the surface is strictly positive.

Proposition 2

The following three conditions are equivalent.

• (a) Let x ∈Eff(V (N), b) and $\lambda \in {\mathbb {R}}^N\setminus \{0\}$be such that λ ⋅ x =max_{y ∈ V (N)}λ ⋅ y. Then, λ ≫ 0. Namely, for every x ∈Eff(V (N), b), the normal cone to V (N) at x is a subset of ${\mathbb {R}}^N_{++}\cup \{0\}$.⁴

• (b) There exists $b'\in {\mathbb {R}}^N$such that b′≪ b and Eff(V (N), b′) = Eff_w(V (N), b′).

• (c) For every x ∈ V (N) with x ≥ b and every $S\in {\mathcal {N}}$with $S\subsetneq N$, there exist ε, δ > 0 such that x − ε e ^S + δ e ^N∖S ∈ V (N).

Condition (b) holds if V (N) is non-leveled, i.e., x = y whenever x and y are on the boundary of V (N) and x ≥ y. Condition (c) is due to Aubin [2, Proposition 2.3]. Although Eff(V (N), b) need not be closed (see Arrow et al. [1] for such an example), condition (a) implies that Eff(V (N), b) is closed as shown by the next lemma.

Lemma 2

Condition (a) of Proposition 2 implies that

$$\displaystyle \begin{aligned} \mathrm{Eff}(V(N),b)=\mathrm{Eff_w}(V(N),b). \end{aligned}$$

Therefore, under condition (a) , Eff(V (N), b) is compact.

Proof of Lemma 2

Suppose, to the contrary, that $\mathrm {Eff}(V(N),b)\subsetneq \mathrm {Eff_w}(V(N),b)$. Then, there exists x ∈Eff_w(V (N), b) with x∉Eff(V (N), b). By Lemma 1, there exists λ ∈ Δ such that, for every y ∈ V (N), λ ⋅ x ≥ λ ⋅ y.

Claim 1

λ _i = 0 for some i ∈ N.

Proof of Claim 1

Suppose, to the contrary, that λ ≫ 0. Since x ∈ V (N), x ≥ b, and x∉Eff(V (N), b), there exists x′∈ V (N) with x′≥ x and x′≠ x. Therefore, we have λ ⋅ x′ > λ ⋅ x, a contradiction. Thus, λ _i = 0 for some i ∈ N. □

Define S = {i ∈ N | λ _i = 0}. By Claim 1 above, we have S ≠ ∅. Define

$$\displaystyle \begin{aligned} A=\left\{y\in V(N)\,\left|\, y^{N\setminus S}=x^{N\setminus S},\ y^S\ge x^S\right\}\right.. \end{aligned}$$

Since A is nonempty and compact, A has a maximal element with respect to ≥, i.e., there exists $\bar {y}\in A$ such that there exists no y′∈ A with $y'\ge \bar {y}$ and $y'\neq \bar {y}$. Note that, for every x′∈ V (N) with x′≥ x, we have x′∈ A, since λ ⋅ x =max_{y ∈ V (N)}λ ⋅ y.

Claim 2

$\bar {y}\in \mathrm {Eff}(V(N),b)$.

Proof of Claim 2

Suppose, to the contrary, that $\bar {y}\not \in \mathrm {Eff}(V(N),b)$. Then, there exists y′∈ V (N) with $y'\ge \bar {y}$ and $y'\neq \bar {y}$. Since $y'\ge \bar {y}\ge x$, we have y′∈ A. This contradicts that $\bar {y}$ is a maximal element in A. Thus, $\bar {y}\in \mathrm {Eff}(V(N),b)$. □

Since $\bar {y}^{N\setminus S}=x^{N\setminus S}$, we have $\lambda \cdot \bar {y}=\lambda \cdot x=\max _{y\in V(N)}\lambda \cdot y$. Thus, by condition (a), we have λ ≫ 0, which contradicts that λ _i = 0 for every i ∈ S. Therefore, Eff(V (N), b) = Eff_w(V (N), b). The compactness of Eff(V (N), b) follows from Remark 1. This completes the proof of Lemma 2. □

Remark 2

By Lemma 2, we can replace Eff(V (N), b) by Eff_w(V (N), b) in condition (a) of Proposition 2.

We are now ready to prove Proposition 2.

Proof of Proposition 2

(a) ⇒ (b): Suppose, to the contrary, that for every $r\in \mathbb {N}$,

$$\displaystyle \begin{aligned} \mathrm{Eff}(V(N),b-1/r\,e^N)\subsetneq \mathrm{Eff_w}(V(N),b-1/r\,e^N). \end{aligned}$$

Then, for every $r\in \mathbb {N}$, there exists x ^r ∈Eff_w(V (N), b − 1∕re ^N) such that x ^r∉Eff(V (N), b − 1∕re ^N). By Lemma 1, for every $r\in \mathbb {N}$, there exists λ ^r ∈ Δ with λ ^r ⋅ x ^r =max_{y ∈ V (N)}λ ^r ⋅ y. For every $r\in \mathbb {N}$, define $I_r=\{i\in N\,|\, \lambda ^r_i=0\}$. Since x ^r∉Eff(V (N), b − 1∕re ^N), we have I _r ≠ ∅ for every $r\in \mathbb {N}$.⁵ Since N has finitely many nonempty subsets, there exists $S\in {\mathcal {N}}$ such that I _r = S for infinitely many r. By passing to a subsequence if necessary, we may assume that I _r = S for every r. Since both sequences (x ^r)_r and (λ ^r)_r are bounded, by passing to further subsequences if necessary, we have

$$\displaystyle \begin{aligned} x^r\to x\quad \mbox{and}\quad \lambda^r\to \lambda \in \Delta. \end{aligned}$$

Then, x ∈ V (N) and x ≥ b. Thus, we have also that x ∈Eff_w(V (N), b). By Lemma 2, x ∈Eff(V (N), b). Since $\lambda ^r_i=0$ for every i ∈ S and every r, we have λ _i = 0 for every i ∈ S. Since $\tilde {\lambda }\mapsto \max _{y\in C_N}\tilde {\lambda }\cdot y$ is continuous, from

$$\displaystyle \begin{aligned} \lambda^r\cdot x^r=\max_{y\in V(N)}\lambda^r\cdot y=\max_{y\in C_N}\lambda^r\cdot y, \end{aligned}$$

it follows that

$$\displaystyle \begin{aligned} \lambda\cdot x=\max_{y\in C_N}\lambda\cdot y=\max_{y\in V(N)}\lambda\cdot y. \end{aligned}$$

Since λ _i = 0 for every i ∈ S, we have a contradiction.(b) ⇒ (c): Let x ∈ V (N) with x ≥ b and let $\emptyset \neq S\subsetneq N$. Since b′≪ b, there exists ε > 0 such that x − εe ^S ≥ b′. Since x − εe ^S∉Eff(V (N), b′) = Eff_w(V (N), b′) and $V(N)-{\mathbb {R}}^N_+=V(N)$, there exists δ > 0 such that x − εe ^S + δe ^N ∈ V (N). Since x − εe ^S + δe ^N∖S ≤ x − εe ^S + δe ^N, we have x − εe ^S + δe ^N∖S ∈ V (N).(c) ⇒ (a): Let x ∈Eff(V (N), b) and $\lambda \in {\mathbb {R}}^N\setminus \{0\}$ be such that λ ⋅ x =max_{y ∈ V (N)}λ ⋅ y. Since $V(N)=C_N-{\mathbb {R}}^N_+$, we have $\lambda \in {\mathbb {R}}^N_+$. Suppose, to the contrary, that λ _i = 0 for some i ∈ N. By condition (c), there exist ε, δ > 0 such that x − εe ^{i} + δe ^N∖{i}∈ V (N). Since

$$\displaystyle \begin{aligned} \lambda\cdot \left(x-\varepsilon\,e^{\{i\}}+\delta\,e^{N\setminus \{i\}}\right)=\lambda \cdot x+\delta\sum_{j\in N\setminus \{i\}}\lambda_j>\lambda\cdot x, \end{aligned}$$

we have a contradiction. Thus, λ ≫ 0. □

Remark 3

If a compactly generated NTU game V satisfies one of the conditions of Proposition 2, its inner core IC(V ) is closed.

Proof

Let $b\in {\mathbb {R}}^N$ be such that $b_i=\max \{u_i\in {\mathbb {R}}\,|\, u\in V(\{i\})\}$ for every i ∈ N. Let (x ^r)_r be a sequence in IC(V ) such that x ^r → x. Since x ^r ∈Eff(V (N), b) for every r and since, by Lemma 2, Eff(V (N), b) is closed, we have x ∈Eff(V (N), b). For every r, let λ ^r ∈ Δ^∘ be such that, for every $S\in {\mathcal {N}}$, $\lambda ^{r,S}\cdot x^r\ge v_{\lambda ^r}(S)$. Since (λ ^r)_r is a bounded sequence, by passing to a subsequence if necessary, we have λ ^r → λ ∈ Δ. Since $\Delta \ni \tilde {\lambda }\mapsto v_{\tilde {\lambda }}(S)\in {\mathbb {R}}$ is continuous, we have λ ^S ⋅ x ≥ v _λ(S) for every $S\in {\mathcal {N}}$. Thus, in particular, λ ⋅ x =max_{y ∈ V (N)}λ ⋅ y. By condition (a) of Proposition 2, we have λ ∈ Δ^∘. Therefore, we have x ∈ IC(V ). Hence, IC(V ) is closed. □

The following example illustrates that condition (a), (b), or (c) of Proposition 2 can be weakened for the nonemptiness of the inner core.

Example 1

Let N = {1, 2} and let $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ be such that $b_i=\max \{u_i\in {\mathbb {R}}\,|\, u\in V(\{i\})\}=1$ for i ∈ N and

$$\displaystyle \begin{aligned} V(N)=\left\{u\in {\mathbb{R}}^N\,\left|\, u_1+u_2\le 3,\ u_1\le 2,\ u_2\le 3\right.\right\}. \end{aligned}$$

Then, V is compactly generated and V (N) is convex. Note that x := (2, 1) ∈Eff(V (N), b) (see Fig. 1). The vector λ := (1, 0) is normal to V (N) at x. Thus, condition (a) of Proposition 2 is violated. In this example, however, V (N) can be extended to V ′(N) such that the pair (V ′(N), b) satisfies condition (a) of Proposition 2 and the inner core of the extended NTU game V ′ is not larger than the inner core of V . Therefore, we can weaken the conditions of Proposition 2 sufficient for the nonemptiness of the inner core of V .

Fig. 1

V of Example 1

For example, define

$$\displaystyle \begin{aligned} V'(N)=\left\{u\in {\mathbb{R}}^N\,\left|\, u_1+u_2\le 3,\ u_1\le 3,\ u_2\le 3\right.\right\}. \end{aligned}$$

The shaded region in Fig. 1 is extended. The pair (V ′(N), b) satisfies condition (a) of Proposition 2. Any payoff vector y in the extended region is not in the inner core of the new game V ′, because it violates the individual rationality. Thus, the inner core IC(V ′) of V ′ is not larger than the inner core IC(V ) of V . By the extension of V (N), since the normal cone to the set of payoff vectors feasible for N at x becomes smaller,⁶IC(V ′) can be smaller than IC(V ).⁷ In this example, however, both inner cores are the same.

Condition (d) of the following proposition represents that, for some extension V ′(N) of V (N), every non-zero normal vector to V ′(N) at every payoff vector in Eff(V ′(N), b) is strictly positive.

Proposition 3

The following two conditions are equivalent.

• (d) There exists a nonempty, closed, convex subset V ′(N) of ${\mathbb {R}}^N$such that V (N) ⊆ V ′(N), V ′(N) is generated by a compact set $C^{\prime }_N$, {x ∈ V (N) | x ≥ b} = {x ∈ V ′(N) | x ≥ b}, and condition (a) of Proposition 2 holds for (V ′(N), b), i.e.,

  $$\displaystyle \begin{aligned} \left[x\in \mathrm{Eff}(V'(N),b),\ \lambda\in {\mathbb{R}}^N\setminus \{0\},\ \mathit{\mbox{and}}\ \lambda\cdot x=\max_{y\in V'(N)}\lambda\cdot y\right]\ \mathit{\mbox{implies}}\ \lambda\gg 0. \end{aligned}$$

• (e) There exists a compact subset K of Δ ^∘such that for every x ∈Eff_w(V (N), b), there exists λ ∈ K with λ ⋅ x =max_{y ∈ V (N)}λ ⋅ y.

Clearly, condition (a) of Proposition 2 implies condition (d). Thus, by Propositions 2 and 3, we have

$$\displaystyle \begin{aligned} \mathrm{(a)}\ \Leftrightarrow\ \mathrm{(b)}\ \Leftrightarrow\ \mathrm{(c)}\ \Rightarrow\ \mathrm{(d)}\ \Leftrightarrow\ \mathrm{(e)}. \end{aligned}$$

Proof of Proposition 3

(d) ⇒ (e): Define a correspondence $\xi :\mathrm {Eff}(V'(N),b)\twoheadrightarrow \Delta$ by

$$\displaystyle \begin{aligned} \xi(x)=\left\{\lambda\in \Delta\,\left|\, \lambda\cdot x=\max_{y\in V'(N)}\lambda\cdot y\right\}\right.. \end{aligned}$$

Then, by Lemma 1, ξ is nonempty-valued. By Lemma 2, condition (d) implies that Eff(V ′(N), b) = Eff_w(V ′(N), b) and this common set is compact. It is clear that ξ is compact-valued and upper hemi-continuous. Thus, ξ(Eff(V ′(N), b)) is compact.

Let K = ξ(Eff(V ′(N), b)). By condition (d), we have K ⊆ Δ^∘. From {x ∈ V (N) | x ≥ b} = {x ∈ V ′(N) | x ≥ b}, it follows that

$$\displaystyle \begin{aligned} \mathrm{Eff_w}(V(N),b)=\mathrm{Eff_w}(V'(N),b)=\mathrm{Eff}(V'(N),b). \end{aligned}$$

Therefore, for every x ∈Eff_w(V (N), b), there exists λ ∈ K with

$$\displaystyle \begin{aligned} \lambda\cdot x=\max_{y\in V'(N)}\lambda\cdot y\ge \max_{y\in V(N)}\lambda\cdot y. \end{aligned}$$

Since x ∈ V (N), we have λ ⋅ x =max_{y ∈ V (N)}λ ⋅ y.(e) ⇒ (d): By condition (e), for every x ∈Eff_w(V (N), b), we choose and fix a λ ^x ∈ K with λ ^x ⋅ x =max_{y ∈ V (N)}λ ^x ⋅ y. Let $b'\in {\mathbb {R}}^N$ be such that b′≪ b and b′≤ y for every y ∈ C _N, where C _N is a compact set generating V (N). Define

$$\displaystyle \begin{aligned} C^{\prime}_N=\left\{y\in {\mathbb{R}}^N\,\left|\, y\ge b'\ \text{and, for every }x\in \mathrm{Eff_w}(V(N),b),\ \lambda^x\cdot y\le \lambda^x\cdot x\right.\right\}. \end{aligned}$$

Then, $C_N\subseteq C_N^{\prime }$ and thus $C_N^{\prime }$ is nonempty. We have also that $C_N^{\prime }$ is compact and convex. Define $V'(N)=C^{\prime }_N-{\mathbb {R}}^N_+$. Then, V ′(N) is nonempty, closed, convex, and satisfies V (N) ⊆ V ′(N). Therefore, {y ∈ V (N) | y ≥ b}⊆{y ∈ V ′(N) | y ≥ b}.

Claim 3

{y ∈ V ′(N) | y ≥ b}⊆{y ∈ V (N) | y ≥ b}.

Proof of Claim 3

Let $\bar {y}\in V'(N)$ be such that $\bar {y}\ge b$. Suppose, to the contrary, that $\bar {y}\not \in V(N)$. Let $\bar {z}\in {\mathbb {R}}^N$ be the closest point in {y ∈ V (N) | y ≥ b} from $\bar {y}$, i.e., $\bar {z}\in V(N)$, $\bar {z}\ge b$, and

$$\displaystyle \begin{aligned} \|\bar{y}-\bar{z}\|=\min\left\{\|\bar{y}-z\|\,\left|\, z\in V(N),\ z\ge b\right\}\right., \end{aligned}$$

where ∥⋅∥ stands for the Euclidean norm. Since {y ∈ V (N) | y ≥ b} is nonempty, compact, and convex, $\bar {z}$ is uniquely determined.

We prove that $\bar {y}-\bar {z}\in {\mathbb {R}}^N_+\setminus \{0\}$. Since $\bar {y}\not \in V(N)$ and $\bar {z}\in V(N)$, we have $\bar {y}-\bar {z}\neq 0$. Suppose, to the contrary, that $\bar {y}_i<\bar {z}_i$ for some i ∈ N. Define $\widehat {z}\in {\mathbb {R}}^N$ by

$$\displaystyle \begin{aligned} \widehat{z}_j= \left\{ \begin{array}{ll} \bar{z}_j & \mbox{if}\ j\neq i,\\ \bar{y}_i & \mbox{if}\ j=i. \end{array} \right. \end{aligned}$$

Since $\bar {z}\ge b$ and $\bar {y}\ge b$, we have $\widehat {z}\ge b$. We have also that $\widehat {z}\in \{\bar {z}\}-{\mathbb {R}}^N_+\subseteq V(N)-{\mathbb {R}}^N_+=V(N)$. Since $\|\bar {y}-\widehat {z}\|<\|\bar {y}-\bar {z}\|$, we have a contradiction. Hence, $\bar {y}-\bar {z}\in {\mathbb {R}}^N_+\setminus \{0\}$.

We next prove that $\bar {z}\in \mathrm {Eff_w}(V(N),b)$. Suppose, to the contrary, that $\bar {z}\not \in \mathrm {Eff_w}(V(N),b)$. Then, there exists z′∈ V (N) with $z'\gg \bar {z}$. Since $\bar {y}-\bar {z}\in {\mathbb {R}}^N_+\setminus \{0\}$, $\bar {y}_k>\bar {z}_k$ for some k ∈ N. Define $\tilde {z}\in {\mathbb {R}}^N$ by

$$\displaystyle \begin{aligned} \tilde{z}_j= \left\{ \begin{array}{ll} \bar{z}_j & \text{if }j\neq k,\\ \min \{\bar{y}_k, z^{\prime}_k\} & \text{if }j=k. \end{array} \right. \end{aligned}$$

Since $z'\gg \bar {z}\ge b$ and $\bar {y}\ge b$, we have $\tilde {z}\ge b$. We have also that $\tilde {z}\in \{z'\}-{\mathbb {R}}^N_+\subseteq V(N)-{\mathbb {R}}^N_+=V(N)$. Since $\|\bar {y}-\tilde {z}\|<\|\bar {y}-\bar {z}\|$, we have a contradiction. Hence, $\bar {z}\in \mathrm {Eff_w}(V(N),b)$.

From $\bar {z}\in \mathrm {Eff_w}(V(N),b)$, it follows that $\lambda ^{\bar {z}}\in K\subseteq \Delta ^\circ$ and $\lambda ^{\bar {z}}\cdot \bar {z}=\max _{y\in V(N)}\lambda ^{\bar {z}}\cdot y$. Since $\lambda ^{\bar {z}}\gg 0$ and $\bar {y}-\bar {z}\in {\mathbb {R}}^N_+\setminus \{0\}$, we have $\lambda ^{\bar {z}}\cdot \bar {y}>\lambda ^{\bar {z}}\cdot \bar {z}$. Thus, by the definition of V ′(N), we have $\bar {y}\not \in V'(N)$, which is a contradiction. Therefore, we have proven that {y ∈ V ′(N) | y ≥ b}⊆{y ∈ V (N) | y ≥ b}. □

It remains to prove that condition (a) of Proposition 2 holds for (V ′(N), b). Let x ∈Eff(V ′(N), b) and $\lambda \in {\mathbb {R}}^N\setminus \{0\}$ be such that $\lambda \cdot x=\max _{y\in V'(N)}\lambda \cdot y$. Since $V'(N)-{\mathbb {R}}^N_+=V'(N)$, we have $\lambda \in {\mathbb {R}}^N_+$. Suppose, to the contrary, that λ _i = 0 for some i ∈ N. Since b′≪ b, we have x − αe ^{i}≥ b′ for sufficiently small α > 0. Since K ⊆ Δ^∘ is compact, there exists δ > 0 such that for every λ′∈ K and every j ∈ N, $\lambda ^{\prime }_j>\delta$ holds. Thus, for every z ∈Eff_w(V (N), b), we have $\lambda ^z\cdot \left (\alpha \,e^{\{i\}}\right )=\alpha \lambda ^z_i>\alpha \delta$. For every z ∈Eff_w(V (N), b), since x ∈ V ′(N), we have λ ^z ⋅ z ≥ λ ^z ⋅ x. Let l ∈ N be such that λ _l > 0. Then, for every z ∈Eff_w(V (N), b),

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lambda^z\cdot \left(x-\alpha \,e^{\{i\}}+\alpha \delta \,e^{\{l\}}\right) &\displaystyle \le&\displaystyle \lambda^z\cdot z+\lambda^z\cdot \left(-\alpha \,e^{\{i\}}+\alpha \delta \,e^{\{l\}}\right)\\ &\displaystyle <&\displaystyle \lambda^z\cdot z-\alpha \delta+\alpha \delta=\lambda^z\cdot z. \end{array} \end{aligned} $$

Since x − αe ^{i} + αδe ^{l}≥ b′, we have

$$\displaystyle \begin{aligned} x-\alpha \,e^{\{i\}}+\alpha \delta \,e^{\{l\}}\in C^{\prime}_N\subseteq V'(N). \end{aligned}$$

We have also that

$$\displaystyle \begin{aligned} \lambda\cdot \left(x-\alpha \,e^{\{i\}}+\alpha \delta \,e^{\{l\}}\right)=\lambda \cdot x+\alpha \delta \lambda_l>\lambda\cdot x, \end{aligned}$$

which contradicts that $\lambda \cdot x=\max _{y\in V'(N)}\lambda \cdot y$. Therefore, λ ≫ 0. This completes the proof of Proposition 3. □

The following example inspired by Qin [13, Example 2] illustrates that (1) IC(V ′) can be strictly smaller than IC(V ) and (2) the inner core IC(V ) of V need not be closed when V satisfies condition (d) or (e) of Proposition 3. Recall that if V satisfies one of the conditions of Proposition 2, IC(V ) is closed (see Remark 3).

Example 2

Let N = {1, 2, 3}. An NTU game $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} V(\{1,2\}) &\displaystyle =&\displaystyle \{u\in {\mathbb{R}}^{\{1,2\}}_+\,|\, u_1^2+u_2^2\le 4/25\}-{\mathbb{R}}^{\{1,2\}}_+, \\ V(N) &\displaystyle =&\displaystyle \{u\in {\mathbb{R}}^N\,|\, u\cdot e^N=1\ \mbox{and}\ u\ge 0\}-{\mathbb{R}}^N_+,\ \mbox{and}\\ V(S) &\displaystyle =&\displaystyle \{(0,0,0)\}-{\mathbb{R}}^S_+\quad \text{for any other coalition }S. \end{array} \end{aligned} $$

Then, V is compactly generated, V (N) is convex, and $b=0\in {\mathbb {R}}^N$.

Note that, for every t ∈ (2∕5, 1], we have x(t) := (0, t, 1 − t) ∈ IC(V ) as shown below. Let t ∈ (2∕5, 1]. Then, x(t) ∈ V (N). For $\lambda :=(\lambda _1, (1-\lambda _1)/2,(1-\lambda _1)/2)\in {\mathbb {R}}^N$ with λ ₁ ∈ (0, 1∕3), let

$$\displaystyle \begin{aligned} z=\left(\frac{4\lambda_1}{5\sqrt{1-2\lambda_1+5\lambda_1^2}},\ \frac{2(1-\lambda_1)}{5\sqrt{1-2\lambda_1+5\lambda_1^2}},\ 0\right)\in V(\{1,2\}). \end{aligned}$$

Then, v _λ({1, 2}) = λ ^{{1, 2}}⋅ z. Thus,

$$\displaystyle \begin{aligned} \lambda^{\{1,2\}}\cdot x(t)-v_\lambda(\{1,2\})=\lambda^{\{1,2\}}\cdot x(t)-\lambda^{\{1,2\}}\cdot z=\frac{1-\lambda_1}{2}\cdot t-\frac{\sqrt{1-2\lambda_1+5\lambda_1^2}}{5}. \end{aligned}$$

For sufficiently small λ ₁ > 0, e.g., λ ₁ = (t − 2∕5)∕2, we have

$$\displaystyle \begin{aligned} \frac{1-\lambda_1}{2}\cdot t-\frac{\sqrt{1-2\lambda_1+5\lambda_1^2}}{5}>0. \end{aligned}$$

Thus, v _λ({1, 2}) ≤ λ ^{{1, 2}}⋅ x(t) for sufficiently small λ ₁ > 0. Since 0 < λ ₁ < 1∕3, we have v _λ(N) = (1 − λ ₁)∕2 = λ ⋅ x(t). Since v _λ(S) = 0 for any other coalition S, it is clear that v _λ(S) = 0 ≤ λ ^S ⋅ x(t). Thus, for every t ∈ (2∕5, 1], x(t) ∈ IC(V ) holds.

Let x ^∗ := (0, 1∕2, 1∕2) ∈ IC(V ). Note that x ^∗∈Eff(V (N), 0) and the vector (0, 1∕2, 1∕2) ∈ Δ is normal to V (N) at x ^∗. Thus, the pair (V (N), 0) does not satisfy condition (a) of Proposition 2. Define $V':{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ by

$$\displaystyle \begin{aligned} V'(N)=\left\{u\in {\mathbb{R}}^N\,\left|\, u\cdot e^N=1\quad \mbox{and}\quad u\ge -1/5\,e^N\right\}\right.-{\mathbb{R}}^N_+ \end{aligned}$$

and V ′(S) = V (S) for every $S\in {\mathcal {N}}\setminus \{N\}$. Then, the pair (V ′(N), 0) satisfies condition (d) of Proposition 3.

We prove that x ^∗∉IC(V ′). Since μ := (1∕3, 1∕3, 1∕3) ∈ Δ is a unique normal vector to V ′(N) at x ^∗ and since

$$\displaystyle \begin{aligned} \max_{u\in V'(\{1,2\})}\mu^{\{1,2\}}\cdot u=\frac{1}{3}\cdot \frac{\sqrt{2}}{5}+\frac{1}{3}\cdot \frac{\sqrt{2}}{5}=\frac{2\sqrt{2}}{15}>\frac{1}{6}=\mu^{\{1,2\}}\cdot x^*, \end{aligned}$$

we have x ^∗∉IC(V ′). Hence, $IC(V')\subsetneq IC(V)$.

We finally prove that IC(V ) is not closed. Let y := (0, 2∕5, 3∕5). For every λ ∈ Δ^∘, we have v _λ({1, 2}) > λ ^{{1, 2}}⋅ y. Thus, y∉IC(V ). Since x(t) ∈ IC(V ) for every t ∈ (2∕5, 1] and x(t) → y as t → 2∕5, IC(V ) is not closed. Therefore, even if an NTU game satisfies condition (d) or (e) of Proposition 3, its inner core need not be closed.

4 Nonemptiness of the Inner Core

We give the main result.

Theorem 2

Let $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$be a compactly generated NTU game with V (N) convex. If V is cardinally balanced and if V satisfies condition (d) or (e) of Proposition 3, then the inner core IC(V ) of V is nonempty.

We prove this theorem by applying Theorem 1.

Proof of Theorem 2

Note first that the inner core satisfies the following covariance property.

Lemma 3

Let $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ be an NTU game and let $a\in {\mathbb {R}}^N$ . Define an NTU game $V+a:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ by

$$\displaystyle \begin{aligned} (V+a)(S)=V(S)+\{a^S\}\quad \mathit{\text{for every }}S\in {\mathcal{N}}. \end{aligned}$$

Then, IC(V + a) = IC(V ) + {a}.

This can be easily shown, so we omit the proof.

Since, by Proposition 3, conditions (d) and (e) are equivalent, we may assume that V satisfies condition (e). Let $b'\in {\mathbb {R}}^N$ be such that b′≪ b and, for every $S\in {\mathcal {N}}$ and every y ∈ C _S, b′≤ y, where C _S is a compact subset of ${\mathbb {R}}^S$ with $V(S)=C_S-{\mathbb {R}}^S_+$. We define V ′ as in the proof of (e) ⇒ (d). By condition (e), for every x ∈Eff_w(V (N), b), we choose and fix a λ ^x ∈ K with λ ^x ⋅ x =max_{y ∈ V (N)}λ ^x ⋅ y. Define

$$\displaystyle \begin{aligned} C^{\prime}_N=\{y\in {\mathbb{R}}^N\,|\, y\ge b'\ \text{and, for every }x\in \mathrm{Eff_w}(V(N),b), \lambda^x\cdot y\le \lambda^x\cdot x\}, \end{aligned}$$

$V'(N)=C^{\prime }_N-{\mathbb {R}}^N_+$, and V ′(S) = V (S) for every $S\in {\mathcal {N}}$ with $S\subsetneq N$. By the proof of (e) ⇒ (d) of Proposition 3, V ′(N) satisfies all the properties of condition (d). Define an NTU game $\widehat {V}$ by $\widehat {V}=V'-b'$. Then, $\widehat {V}(N)$ is convex, $\widehat {V}$ is cardinally balanced,⁸ and, for every $S\in {\mathcal {N}}$, $\widehat {V}(S)$ is generated by a compact subset of ${\mathbb {R}}^S_+$. Since $IC(\widehat {V}+b')=IC(V')\subseteq IC(V)$, by Lemma 3, it suffices to prove that $IC(\widehat {V})\neq \emptyset$. We will prove that $\widehat {V}$ satisfies all the conditions of Theorem 1. Since $\widehat {V}$ is cardinally balanced, by Proposition 1, condition (i) of Theorem 1 is met for $\widehat {V}$ and for every m ≥ n. It remains to prove that condition (ii) is met for $\widehat {V}$ and for some m ≥ n, i.e., there exists m ≥ n such that, for every λ ∈ Δ^∘∖ Δ_1∕m and every $\gamma \in \widehat {\Gamma }(\beta ^m(\lambda ))$, there exists y ∈ B(p ^m(λ)) with $\sum _{S\in {\mathcal {N}}}\gamma _S\,v_{p^m(\lambda )}(S)\le \lambda \cdot y$, where B(p ^m(λ)) and $v_{p^m(\lambda )}$ are defined for NTU game $\widehat {V}$.

Since, by the definition of b′, for every $S\in {\mathcal {N}}\setminus \{N\}$ and every y ∈ C _S, y ≥ b′ holds and since, for every $y\in C^{\prime }_N$, y ≥ b′ holds, for every m ≥ n, every λ ∈ Δ^∘∖ Δ_1∕m, and every $S\in {\mathcal {N}}$, we have $v_{p^m(\lambda )}(S)\ge 0$. Since 0 ≤ β ^m(λ) ≤ e ^N for every λ ∈ Δ, $\widehat {V}$ is cardinally balanced, and $v_{p^m(\lambda )}(S)\ge 0$ for every $S\in {\mathcal {N}}$, we have, for every m ≥ n, every λ ∈ Δ^∘∖ Δ_1∕m, every $\gamma \in \widehat {\Gamma }(\beta ^m(\lambda ))$, and every y ∈ B(p ^m(λ)),

$$\displaystyle \begin{aligned} \sum_{S\in {\mathcal{N}}}\gamma_S\,v_{p^m(\lambda)}(S)\le v_{p^m(\lambda)}(N)=p^m(\lambda)\cdot y. \end{aligned}$$

Thus, it suffices to prove that there exists m ≥ n such that, for every λ ∈ Δ^∘∖ Δ_1∕m and every y ∈ B(p ^m(λ)), p ^m(λ) ⋅ y ≤ λ ⋅ y holds.

Since K ⊆ Δ^∘ is compact, there exists k ≥ n with K ⊆ Δ_1∕k. Let $m\in \mathbb {N}$ be such that m > (n − 1)(k − n + 1) + 1.

Claim 4

m > k.

Proof of Claim 4

Since k ≥ n ≥ 2, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} m-k &\displaystyle >&\displaystyle (n-1)(k-n+1)+1-k=k(n-2)-(n-1)^2+1\\ &\displaystyle \ge &\displaystyle n(n-2)-(n-1)^2+1 =0. \end{array} \end{aligned} $$

Hence, m > k. □

Claim 5

Let λ ∈ Δ^∘∖ Δ_1∕m and $x\in \mathrm {Eff_w}(\widehat {V}(N),0)$ be such that

$$\displaystyle \begin{aligned} \lambda\cdot x=\max_{y\in \widehat{V}(N)}\lambda\cdot y. \end{aligned}$$

Then, for every i ∈ N with λ _i < 1∕m, x _i = 0 holds.

Proof of Claim 5

Suppose, to the contrary, that x _i > 0 for some i ∈ N with λ _i < 1∕m. Since λ ∈ Δ and λ _i < 1∕m, there exists j ∈ N ∖{i} such that

$$\displaystyle \begin{aligned} \lambda_j>\frac{1-\frac{1}{m}}{n-1}=\frac{m-1}{m(n-1)}. \end{aligned}$$

Define

$$\displaystyle \begin{aligned} E=\left\{y\in {\mathbb{R}}^N_+\,\left|\, y^{N\setminus \{i,j\}}=x^{N\setminus \{i,j\}},\ \mu\cdot y\le \mu\cdot x\quad \text{for every }\mu\in \Delta_{1/k}\right\}\right.. \end{aligned}$$

We first prove that $E\subseteq C^{\prime }_N-\{b'\}$. Let y ∈ E. Since $x\in \widehat {V}(N)=C^{\prime }_N-\{b'\}-{\mathbb {R}}^N_+$, we have $x+b'\in C^{\prime }_N-{\mathbb {R}}^N_+$. Hence, by the definition of $C^{\prime }_N$, for every z ∈Eff_w(V (N), b), λ ^z ⋅ (x + b′) ≤ λ ^z ⋅ z holds. Since λ ^z ∈ K ⊆ Δ_1∕k,

$$\displaystyle \begin{aligned} \lambda^z\cdot (y+b')\le \lambda^z\cdot (x+b')\le \lambda^z\cdot z. \end{aligned}$$

Since y ≥ 0, we have y + b′≥ b′. Thus, $y+b'\in C^{\prime }_N$. Hence, $y\in C^{\prime }_N-\{b'\}$ and $E\subseteq C^{\prime }_N-\{b'\}$.

For every l ∈ N, define μ ^{k, l} ∈ Δ_1∕k by $\mu ^{k,l}_t=1/k$ for t ∈ N ∖{l} and $\mu ^{k,l}_l=1-(n-1)/k$. Then, {μ ^{k, 1}, …, μ ^{k, n}} is the set of all extreme points of Δ_1∕k. Therefore,

$$\displaystyle \begin{aligned} \begin{array}{rcl} E &\displaystyle =&\displaystyle \left\{y\in {\mathbb{R}}^N_+\,\left|\, y^{N\setminus \{i,j\}}=x^{N\setminus \{i,j\}},\ \mu^{k,l}\cdot y\le \mu^{k,l}\cdot x\quad \text{for every }l\in N\right\}\right.\\ &\displaystyle =&\displaystyle \left\{y\in {\mathbb{R}}^N_+\,\left|\, y^{N\setminus \{i,j\}}=x^{N\setminus \{i,j\}},\ \mu^{k,i}\cdot y\le \mu^{k,i}\cdot x,\ \mu^{k,j}\cdot y\le \mu^{k,j}\cdot x\right\}\right.\\ &\displaystyle =&\displaystyle \{y\in {\mathbb{R}}^N_+\,|\, y^{N\setminus \{i,j\}}=x^{N\setminus \{i,j\}},\ (k-n+1)y_i+y_j\le (k-n+1)x_i+x_j,\\ &\displaystyle &\displaystyle \qquad \qquad \ y_i+(k-n+1)y_j\le x_i+(k-n+1)x_j\}. \end{array} \end{aligned} $$

Define $z^*\in {\mathbb {R}}^N$ by

$$\displaystyle \begin{aligned} \begin{array}{l} z^{*N\setminus \{i,j\}}=x^{N\setminus \{i,j\}},\\ z^*_i=0,\\ z^*_j=x_j+\frac{1}{k-n+1}x_i \end{array} \end{aligned}$$

(See Fig. 2). Then, $z^*\in E\subseteq C^{\prime }_N-\{b'\}\subseteq \widehat {V}(N)$. Since $\lambda _j>\frac {m-1}{m(n-1)}$ and m > (n − 1)(k − n + 1) + 1, we have

$$\displaystyle \begin{aligned} \frac{\lambda_j}{k-n+1}>\frac{m-1}{m(n-1)(k-n+1)}>\frac{m-1}{m(m-1)}=\frac{1}{m}>\lambda_i. \end{aligned}$$

Therefore, since x _i > 0, we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lambda\cdot z^* &\displaystyle =&\displaystyle \lambda\cdot x^{N\setminus \{i,j\}}+\lambda_jx_j+\frac{\lambda_j}{k-n+1}x_i\\ &\displaystyle >&\displaystyle \lambda\cdot x^{N\setminus \{i,j\}}+\lambda_jx_j+\lambda_ix_i\\ &\displaystyle =&\displaystyle \lambda\cdot x = \max_{y\in \widehat{V}(N)}\lambda\cdot y\ge \lambda\cdot z^*, \end{array} \end{aligned} $$

a contradiction. Therefore, x _i = 0 for every i ∈ N with λ _i < 1∕m. □

Fig. 2

Definition of z ^{∗{i, j}}

Define σ ^m :  Δ → Δ_{1∕(m+n−1)} by

$$\displaystyle \begin{aligned} \sigma^m_j(\lambda)=\frac{p^m_j(\lambda)}{p^m(\lambda)\cdot e^N}. \end{aligned}$$

Since

$$\displaystyle \begin{aligned} \begin{array}{rcl} B(p^m(\lambda)) &\displaystyle =&\displaystyle \left\{y\in \widehat{V}(N)\,\left|\, p^m(\lambda)\cdot y=\max_{z\in \widehat{V}(N)}p^m(\lambda)\cdot z\right\}\right.\\ &\displaystyle =&\displaystyle \left\{y\in \widehat{V}(N)\,\left|\, \sigma^m(\lambda)\cdot y=\max_{z\in \widehat{V}(N)}\sigma^m(\lambda)\cdot z\right\}\right.\\ &\displaystyle =&\displaystyle \left\{y\in C^{\prime}_N-\{b'\}\,\left|\,\sigma^m(\lambda)\cdot y=\max_{z\in \widehat{V}(N)}\sigma^m(\lambda)\cdot z\right\}\right., \end{array} \end{aligned} $$

by Lemma 1 and the definition of b′, we have $B(p^m(\lambda ))\subseteq \mathrm {Eff_w}(\widehat {V}(N),0)$ for every λ ∈ Δ. Let λ ∈ Δ^∘∖ Δ_1∕m and let i ∈ N with λ _i < 1∕m. Then, $p^m_i(\lambda )=1/m$ and p ^m(λ) ⋅ e ^N > 1. Thus, $\sigma ^m_i(\lambda )<1/m$. By Claim 5, for every λ ∈ Δ^∘∖ Δ_1∕m, every y ∈ B(p ^m(λ)), and every i ∈ N with λ _i < 1∕m, y _i = 0 holds. Therefore, for every λ ∈ Δ^∘∖ Δ_1∕m and every y ∈ B(p ^m(λ)),

$$\displaystyle \begin{aligned} \begin{array}{rcl} p^m(\lambda)\cdot y &\displaystyle =&\displaystyle \sum_{j:\lambda_j\ge 1/m}p^m_j(\lambda)y_j+\sum_{j:\lambda_j<1/m}p^m_j(\lambda)y_j\\ &\displaystyle =&\displaystyle \sum_{j:\lambda_j\ge 1/m}p^m_j(\lambda)y_j = \sum_{j:\lambda_j\ge 1/m}\lambda_jy_j = \lambda\cdot y. \end{array} \end{aligned} $$

Hence, condition (ii) of Theorem 1 holds for $\widehat {V}$ and m. Thus, $IC(\widehat {V})\neq \emptyset$. As we mentioned before, this implies that IC(V ) ≠ ∅. □

Qin [15, Example 1] exemplifies that, if a cardinally balanced NTU game does not satisfy condition (d) or (e) of Proposition 3, then its inner core can be empty. We give another example of a totally cardinally balanced NTU game V with every V (S) polyhedral such that the inner core IC(V ) of V is empty.⁹ Billera and Bixby [4] proved that any totally cardinally balanced NTU game with every V (S) polyhedral can be generated by an exchange economy where agents’ consumption sets have the form [0, 1]^l and their utility functions are concave and continuous. Therefore, by the following example, the nonemptiness of the inner core is irrelevant to such representation of NTU games. It should be worth mentioning that in the first step of the proof of Billera and Bixby’s representation, the inner core plays an essential role. Actually, $\bar {x}$ in the proof of Lemma 3.2 of Billera and Bixby [4] is an inner core payoff vector.

Example 3

Let N = {1, 2, 3}. An NTU game $V:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ is given by

$$\displaystyle \begin{aligned} \begin{array}{rcl} V(\{i\}) &\displaystyle =&\displaystyle \{(0,0,0)\}-{\mathbb{R}}^{\{i\}}_+\quad \text{for every }i\in N,\\ V(\{1,2\})&\displaystyle =&\displaystyle \{(1,1/2,0)\}-{\mathbb{R}}^{\{1,2\}}_+,\\ V(\{1,3\})&\displaystyle =&\displaystyle \{(1,0,1)\}-{\mathbb{R}}^{\{1,3\}}_+,\\ V(\{2,3\})&\displaystyle =&\displaystyle \{(0,0,0)\}-{\mathbb{R}}^{\{2,3\}}_+,\ \mbox{and}\\ V(N)&\displaystyle =&\displaystyle \{x\in {\mathbb{R}}^N\,|\, x_i\le 1\ \text{for every }i\in N\text{ and}\ x_2+x_3\le 1\}. \end{array} \end{aligned} $$

Then, $b=0\in {\mathbb {R}}^N$. Figure 3 depicts $V(N)\cap {\mathbb {R}}^N_+$. Note that every V (S) is a polyhedron. For a balancing vector γ with γ _{1,2} = γ _{1,3} = γ _{2,3} = 1∕2 and γ _S = 0 otherwise, we have

$$\displaystyle \begin{aligned} \frac{1}{2}\,\left(1,\frac{1}{2},0\right)+\frac{1}{2}\,(1,0,1)+\frac{1}{2}\,(0,0,0)=\left(1,\frac{1}{4},\frac{1}{2}\right)\in V(N). \end{aligned}$$

For any other balancing vector γ′ of weights, we can show that $\sum _{S\in {\mathcal {N}}}\gamma ^{\prime }_S\,V(S)\subseteq V(N)$. Hence, V is cardinally balanced. Moreover, any subgame of V is cardinally balanced and, therefore, V is totally cardinally balanced. Since (1, 0, 0) is normal to V (N) at (1, 1∕2, 1∕2) ∈Eff(V (N), 0) and since (1, 1∕2, 1∕2) ≫ b, there exists no V ′(N) satisfying condition (d) of Proposition 3.

Fig. 3

V (N) of Example 3

We prove that IC(V ) = ∅. Let $\lambda \in {\mathbb {R}}^N_{++}$. Since λ ^{{1, 3}}⋅ (1, 0, 1) > λ ^{{1, 3}}⋅ x for every x ∈ V (N) with x ≥ 0 and x ≠ (1, 0, 1), payoff vector (1, 0, 1) is a unique candidate of an element of the inner core. Since (1, 1∕2, 0) ∈ V ({1, 2}) and

$$\displaystyle \begin{aligned} \lambda^{\{1,2\}}\cdot \left(1,\frac{1}{2},0\right)=\lambda_1+\frac{1}{2}\,\lambda_2>\lambda_1=\lambda^{\{1,2\}}\cdot (1,0,1), \end{aligned}$$

payoff vector (1, 0, 1) is not in the inner core. Therefore, IC(V ) = ∅.

5 NTU Games Generated by Exchange Economies

de Clippel and Minelli [8, Proposition 1] proved that, in an exchange economy where every agent has a continuous, concave, and strongly monotone utility function, the utility vector at any Walrasian allocation is always in the inner core. Since there exists a Walrasian equilibrium for such an exchange economy, the inner core is nonempty. In this section, we prove that an NTU game generated by such an exchange economy satisfies condition (a) of Proposition 2. Thus, by Theorem 2, its inner core is nonempty. Therefore, our class of NTU games with the nonempty inner core contains de Clippel and Minelli’s [8] class.

An exchange economy with n consumers is a list of the commodity space ${\mathbb {R}}^L$, where L is a finite set of commodities, and consumers’ characteristics (u ⁱ, ω ⁱ)_{i ∈ N} such that, for every consumer i, utility function $u^i:{\mathbb {R}}^L_+\to {\mathbb {R}}$ is continuous, concave, and strongly monotone, and endowment vector ω ⁱ is in ${\mathbb {R}}^L_+\setminus \{0\}$. An exchange economy is denoted by ${\mathcal {E}}=({\mathbb {R}}^L, (u^i,\omega ^i)_{i\in N})$. For every coalition $S\in {\mathcal {N}}$, let $F_{\mathcal {E}}(S)$ be the set of feasible S-allocations, i.e.,

$$\displaystyle \begin{aligned} F_{\mathcal{E}}(S)=\left\{(z^i)_{i\in S}\,\left|\, z^i\in {\mathbb{R}}^L_+\quad \text{for every }i\in S\text{ and}\quad \sum_{i\in S}\left(z^i-\omega^i\right)=0\right\}\right.. \end{aligned}$$

A feasible N-allocation (z ⁱ)_{i ∈ N} is a Walrasian allocation for ${\mathcal {E}}$ if there exists a price vector $p\in {\mathbb {R}}^L$ such that, for every i ∈ N, p ⋅ z ⁱ ≤ p ⋅ ω ⁱ and u ⁱ(z ⁱ) ≥ u ⁱ(y) for every $y\in {\mathbb {R}}^L_+$ with p ⋅ y ≤ p ⋅ ω ⁱ. Let $W({\mathcal {E}})$ be the set of all Walrasian allocations for ${\mathcal {E}}$.

A feasible N-allocation (z ⁱ)_{i ∈ N} is an inner core allocation for ${\mathcal {E}}$ if there exists $\lambda \in {\mathbb {R}}^N_{++}$ such that, for every $S\in {\mathcal {N}}$ and every $(y^i)_{i\in S}\in F_{{\mathcal {E}}}(S)$, ∑_{i ∈ S}λ _iu ⁱ(z ⁱ) ≥∑_{i ∈ S}λ _iu ⁱ(y ⁱ) holds. Let $IC({\mathcal {E}})$ be the set of all inner core allocations for ${\mathcal {E}}$.

An exchange economy ${\mathcal {E}}=({\mathbb {R}}^L, (u^i,\omega ^i)_{i\in N})$ generates an NTU game $V_{\mathcal {E}}:{\mathcal {N}}\twoheadrightarrow {\mathbb {R}}^N$ by defining

$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle V_{\mathcal{E}}(S)\\ &\displaystyle &\displaystyle =\left\{v\in {\mathbb{R}}^S\,\left|\, \text{there exists }(z^i)_{i\in S}\in F_{\mathcal{E}}(S)\text{ such that, for every }i\in S, v_i\le u^i(z^i)\right\}\right.. \end{array} \end{aligned} $$

Since $F_{\mathcal {E}}(S)$ is nonempty, compact, and convex for every $S\in {\mathcal {N}}$, NTU game $V_{\mathcal {E}}$ is compactly convexly generated. Note that, for every i ∈ N,

$$\displaystyle \begin{aligned} b_i=\max \{v_i\in {\mathbb{R}}\,|\, v\in V_{{\mathcal{E}}}(\{i\})\}=u^i(\omega^i), \end{aligned}$$

and $IC(V_{{\mathcal {E}}})=\{(u^i(z^i))_{i\in N}\,|\, (z^i)_{i\in N}\in IC({\mathcal {E}})\}$.

Qin [13, Remark 2] states the following without proof. For completeness, we give its proof.

Proposition 4

Let ${\mathcal {E}}=({\mathbb {R}}^L, (u^i,\omega ^i)_{i\in N})$be an exchange economy where, for every i ∈ N, $u^i:{\mathbb {R}}^L_+\to {\mathbb {R}}$is continuous, concave, and strongly monotone, and $\omega ^i\in {\mathbb {R}}^L_+\setminus \{0\}$. Then, NTU game $V_{\mathcal {E}}$generated by ${\mathcal {E}}$satisfies condition (a) of Proposition 2, i.e., if $x\in \mathrm {Eff}(V_{\mathcal {E}}(N),b)$and $\lambda \in {\mathbb {R}}^N\setminus \{0\}$satisfy $\lambda \cdot x=\max _{y\in V_{\mathcal {E}}(N)}\lambda \cdot y$, then λ ≫ 0.

Proof

Let $x\in \mathrm {Eff}(V_{\mathcal {E}}(N),b)$ and $\lambda \in {\mathbb {R}}^N\setminus \{0\}$ be such that $\lambda \cdot x=\max _{y\in V_{\mathcal {E}}(N)}\lambda \cdot y$. Since $V_{\mathcal {E}}(N)-{\mathbb {R}}^N_+=V_{\mathcal {E}}(N)$, we have $\lambda \in {\mathbb {R}}^N_+$. Suppose, to the contrary, that there exists j ∈ N with λ _j = 0. Since $\lambda \in {\mathbb {R}}^N_+\setminus \{0\}$, there exists k ∈ N with λ _k > 0. From $x\in V_{\mathcal {E}}(N)$, it follows that there exists a feasible N-allocation (z ⁱ)_{i ∈ N} such that u ⁱ(z ⁱ) ≥ x _i for every i ∈ N. Since u ^j(z ^j) ≥ x _j ≥ b _j = u ^j(ω ^j), $\omega ^j\in {\mathbb {R}}^L_+\setminus \{0\}$, and u ^j is strongly monotone, we have $z^j\in {\mathbb {R}}^L_+\setminus \{0\}$. Define an allocation (y ⁱ)_{i ∈ N} by

$$\displaystyle \begin{aligned} y^i= \left\{ \begin{array}{ll} z^i & \text{if }i\in N\setminus\{j,k\},\\ z^k+z^j & \text{if }i=k,\\ 0 & \text{if }i=j. \end{array} \right. \end{aligned}$$

Then, (y ⁱ)_{i ∈ N} is a feasible N-allocation. Therefore, $u:=(u^i(y^i))_{i\in N}\in V_{\mathcal {E}}(N)$. Since λ _j = 0, λ _k > 0, and u ^k(z ^k + z ^j) > u ^k(z ^k), we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} \lambda\cdot u &\displaystyle =&\displaystyle \sum_{i\in N}\lambda_iu^i(y^i)=\sum_{i\in N\setminus \{j,k\}}\lambda_iu^i(z^i)+\lambda_ku^k(z^k+z^j)+\lambda_ju^j(0)\\ &\displaystyle >&\displaystyle \sum_{i\in N\setminus\{j,k\}}\lambda_iu^i(z^i)+\lambda_ku^k(z^k)+\lambda_ju^j(z^j)\ge \lambda\cdot x. \end{array} \end{aligned} $$

This contradicts that $\lambda \cdot x=\max _{y\in V_{\mathcal {E}}(N)}\lambda \cdot y$. Therefore, λ ≫ 0 and hence $V_{\mathcal {E}}$ satisfies condition (a). □

Since an exchange economy with the properties in Proposition 4 generates a cardinally balanced NTU game,¹⁰ by Theorem 2, we have the following.

Corollary 1

Let ${\mathcal {E}}=({\mathbb {R}}^L, (u^i,\omega ^i)_{i\in N})$be an exchange economy where, for every i ∈ N, $u^i:{\mathbb {R}}^L_+\to {\mathbb {R}}$is continuous, concave, and strongly monotone, and $\omega ^i\in {\mathbb {R}}^L_+\setminus \{0\}$. Then, the inner core $IC(V_{\mathcal {E}})$of NTU game $V_{\mathcal {E}}$generated by ${\mathcal {E}}$is nonempty.

We can show this corollary without relying on Theorem 2. de Clippel and Minelli [8, Proposition 1] proved the inclusion $W({\mathcal {E}})\subseteq IC({\mathcal {E}})$ for an exchange economy ${\mathcal {E}}$ satisfying the properties in Corollary 1. Since there exists a Walrasian equilibrium for ${\mathcal {E}}$, the inner core of $V_{\mathcal {E}}$ is nonempty.

6 Concluding Remarks

The inner core has a relation to the strictly inhibitive set, the set of payoff vectors stable against randomized blocking plans (see Myerson [12, Section 9.8] and Qin [13]). For a compactly generated NTU game, its inner core is a subset of the strictly inhibitive set (Qin [13, Theorem 2]). Thus, our Theorem 2 gives a sufficient condition for the nonemptiness of the strictly inhibitive set. Furthermore, if V (S) is convex for every $S\in {\mathcal {N}}$ and if V satisfies one of the conditions of Proposition 2, then the inner core coincides with the strictly inhibitive set (Qin [13, Theorem 4]).

By Proposition 4, an exchange economy with continuous, concave, and strongly monotone utility functions generates an NTU game satisfying the cardinal balancedness and condition (a) of Proposition 2. Since different economies can generate the same NTU game, exchange economies without the properties in Proposition 4 or production economies may generate NTU games satisfying condition (d) or (e) of Proposition 3. Billera [3] proved that every compactly generated, totally cardinally balanced NTU game V with every V (S) convex can be generated by a production economy where every consumer has a upper semi-continuous and concave utility function on a compact convex consumption set and has his own compact convex production set. Billera’s induced production economy can be converted to an exchange economy (see Billera and Bixby [5]). Inoue [9] proved that every compactly generated NTU game can be generated by a coalition production economy. In both induced economies due to Billera [3] and due to Inoue [9], the inner core coincides with the set of utility vectors at Walrasian allocations (see Qin [14] and Inoue [9], respectively). Thus, if an NTU game satisfies all the assumptions of Theorem 2, then there exists a Walrasian equilibrium for both Billera’s and Inoue’s induced economies.