Sustainability for utility allocation: a game-theoretical mechanism

Abstract

By focusing on various influences arose from environmental change, sustainability has become a major conception among many fields, including utility allocation. On the other hand, game-theoretical methods have always been adopted to analyze the reasonability of utility allocation rules. In many real-world situations, however, participants and its energetic levels (decisions) should be essential factors simultaneously. By focusing on both the participants and its energetic levels (decisions), we introduce the restrained core to investigate utility allocation under fuzzy transferable-utility (TU) models. In order to analyze the reasonability for the restrained core, two axiomatic results are further provided by applying several types of reductions. Since the restrained core infringes a specific converse steadiness property, a converse steady enlargement of the restrained core is also introduced to investigate how extensive the violation of this specific converse steadiness property is. This converse steady enlargement is smallest converse steady measuration that contains the restrained core.

Keywords

Sustainability the core fuzzy TU models the restrained core reduction converse steadiness

1 Introduction

Due to the constant renovation of the trend of real-world environmental change, allocation concepts of a combination of different theoretical fields, including utility allocation, have become a major conception in the context of sustainability. In the framework of utility allocation, the use of several notions could promote the allocation efficiency no matter for improvement of allocation methods or management techniques of utility. On the other hand, the axiomatic results of game-theoretical allocations could be always adopted to analyze various interaction relationship and related models among agents and coalitions by applying mathematical results. In addition to theoretical analysis, game-theoretical results also have been applied to offer optimal outcomes or equilibrium situations for many real-word models.

A measuration is introduced through all the subsets of the collection of participants on standard transferable-utility (TU) cases. This indicates that the choices available for each participant are either to participate completely or not to participate at all. In real-word situations, however, participants always could adopt different energetic levels to participate. The framework of fuzzy (TU) models was initially developed with investigation of Aubin [1, 2]. He claimed to permit the participants to pick different energetic levels in participated situations. Therefore, it is reasonable that the domain of the measuration could be extended to permit fuzzy coalitions.

In view of various measuration or allocation rules in game theory, the core-concepts have been applied comprehensively. Based on the notions of reduced models and related properties of steadiness and its converse, several axiomatic results of the core-concepts have been proposed on standard TU models. Many axiomatic results could be found in Peleg [10], Tadenuma [12], Serrano and Volij [11], and so on. By collecting the distinction among every energetic levels for each participant, Aubin [1, 2] defined an extended core concept under fuzzy TU models. Hwang [6] and Hwang and Liao [7] proposed several extended reductions to analyze the fuzzy core concept due to Aubin [1, 2] respectively. Related investigations of the fuzzy core concepts also could be found in Butnariu [3], Chen et al. [4], Hwang and Liao [8], Tijs et al. [13] and Yu and Zhang [14].

The above existing results generalize one motivation under fuzzy TU models:

whether different core concepts could be proposed to be sustainable utility allocation mechanism by considering the participants and its energetic levels simultaneously.

This article is devoted to analyze the motivation. Different from the existing results under fuzzy TU models, the main results are introduced as follows.

In Section 2, we propose the restrained core by focusing on both the participants and its energetic levels simultaneously.

In order to analyze the reasonability for the restrained core, we firstly extend the reductions introduced by Davis and Maschler [5], Moulin [9] and Serrano and Volij [11] to fuzzy TU models in Sections 3 and 4 respectively. Further, we adopt the extended Davis and Maschler’s [5] reduction and the extended Serrano and Volij’s [11] reduction to characterize the restrained core in Sections 3 and 4 respectively.

In Section 4, we manifest that the restrained core infringes a specific converse steadiness related to the extended Moulin [9] reduction. In Section 5, we introduce to minimally enlarge the restrained core so as to regain this specific converse steadiness, that is, the smallest converse steady measuration that contains the restrained core. Some more discussions and comparisons are also provided in Section 6.

2 Preliminaries

2.1 Definitions and notations

Let UP be the universe of participants. For m ∈ UP and b_m ∈ [0, 1], we set B_m = [0, b_m] as the energetic level (strategy) set of participant m, where 0 means null level. For P ⊆ UP, let $B^{P} = \prod_{m \in P} B_{m}$ be the product set of the energetic level sets for participants in P. for every H ⊆ P, $θ^{H} \in ℝ^{P}$ is the vector with $θ_{m}^{H} = 1$ if m ∈ H, and $θ_{m}^{H} = 0$ if m ∈ P \ T. Signify 0_P the zero vector in $ℝ^{P}$ .

A fuzzy TU model is a triple (P, b, η), where P ⊆ UP, P≠ ∅ is a finite collection of participants, b = (b_m) _m∈P ∈ [0, 1] ^P is the vector that displays the highest level of energy for each participant, and $v : B^{P} \to ℝ$ is a map with η (0_P) =0 which allots to each energetic vector ω = (ω_m) _m∈P ∈ B^P the merit that the participants can receive when each participant i partakes at energetic level ω_i. A fuzzy TU model (P, b, η) will be denoted by its map η if no confusion can emerge. Signify the family of all fuzzy TU models by Λ.

Let (P, b, η) ∈ Λ, we define $C^{P} = {(m, k_{m}) ∣ m \in P, k_{m} \in B_{η}^{+}}$ , where $B_{m}^{+} = (0, b_{m}]$ . Given (P, b, η) ∈ Λ, ω ∈ B^P and H ⊆ P, we define A (ω) = {m ∈ P|ω_m ≠ 0} and signify $ω_{H} \in ℝ^{H}$ to be the restriction of ω to H. Let m ∈ P, we propose the notation ω_-m to stand for ω_P\{m} and let $ρ = (ω_{- m}, k_{m}) \in ℝ^{P}$ be defined by ρ_-m = ω_-m and ρ_m = k_m.

A payoff vector of (P, b, η) is a mapping $x : C^{P} \to ℝ$ , where x_{m,k_m} signifies the payoff to participant m corresponding to the energetic level k_m. For convenience, we define x_m,0 = 0 for each m ∈ P. For ω ∈ B^P, we define x (ω) = ∑_m∈Px_{m,ω_m}. Further, x is

effective (ECE) if x (b) = η (b).

restrained (RE) if for every m ∈ P and for every k_m ∈ [0, b_m), x_{m,k_m} ≤ sup {v (ω_-m, k_m) | ω ∈ B^P}.

personally reasonable (PR) if for every (m, k_m) ∈ C^P, x_{m,k_m} ≥ η (k_mθ^{m}).

coalitionally reasonable (CR) if for every ω ∈ B^P, x (ω) ≥ η (ω).

ECE states that all participants allocate whole usability entirely. RE states that each payoff should be bounded. PR asserts that the payoff for each participant is at least the benefit that the participant could receive when the participant adopts each level to operate alone. CR states that the payoff for each coalition is at least the benefit that the coalition could receive under each coalitional behavior. The collection of practical vectors of (P, b, η) is signified by $F^{*} (P, b, η) = {x \in ℝ^{C^{P}} | x (b) \leq η (b)},$ whereas F (P, b, η) means the collection of all effective vectors of (P, b, η). Moreover, x is an imputation of (P, b, η) if it is ECE, RE and PR, and the collection of imputations of (P, b, η) is signified by IT (P, b, η). A measuration on Λ is a map κ which associates with each (P, b, η) ∈ Λ a subset κ (P, b, η) ⊆ F^* (P, b, η). Here we propose an extended core by adopting both the participants and the energetic levels on Λ.

Definition 1. 1 The restrained core Δ_R (P, b, η) of (P, b, η) consists of all x ∈ IT (P, b, η) that are CR, i.e., the restrained core consists of all vectors that are ECE, RE and CR.

Remark 1. Clearly, the restrained core is proposed to determine a value for a given participant operating at a given level. By summarizing the contribution of each participant for every managing levels, Aubin [1, 2] proposed an extended core Δ_A as follows. for every (P, θ^P, η) ∈ Λ, $\begin{matrix} Δ_{A} (P, θ^{P}, η) \\ = & {x \in ℝ^{P} | \sum_{m \in P} x_{m} = η (θ^{P}), \\ \sum_{m \in P} ω_{m} x_{m} \geq η (ω) \forall ω \in [0, 1]^{P}} . \end{matrix}$

2.2 Motivating and practical examples

First we provide a brief motivating example of fuzzy TU models in the setting of “source management". Let P = {1, 2, ⋯ , p} be a set of all participants of a source management system (P, b, η). The function η could be treated as an usability function which assigns to each energetic level vector α = (α_p) _p∈P ∈ B^P the worth that the participants can obtain when each participant p participates at energetic level α_p ∈ B_p in (P, b, η). Modeled in this way, the source management system (P, b, η) could be considered as a fuzzy TU model, with η being each characteristic function and B_p being the set of all energetic levels of the participant p.

It known that strategical management efficiency plays essential roles under sustainable management processes. In the following, we also provide a practical application of management power distribution in a source management association, such electricity companies, waterworks and so on. Let P = {1, 2, ⋯ , p} be a set of all participants of steering committee of a source management association. In this steering committee, all the participants are elected by voting or recommendation by main departments (or teams). All participants have the power to propose, discuss, establish, and veto all palns (or rules). They dedicate different levels of attention and participation to different plans depending on their academic expertise and the public opinion they represent. The level of involvement is also closely associated with the alliance strategy formed for the interests of different departments. For the aforementioned reasons, strategies adopted by each participant of the steering committee show distinct levels of participation and certain amounts of ambiguity. The function η could be treated as a power function which assigns to each level vector α = (α_p) _p∈P ∈ B^P the power that the participants can dedicate when each participant p participates at operational strategy α_p ∈ B_p. Modeled in this way, the source management steering committee (P, b, η) could be considered as a fuzzy TU model, with η being each characteristic function and B_p being the set of all operational strategies of the participant p. To evaluate the influence of each participant in the source management steering committee, using the power indicators we proposed, we first assess the influence each participant has arisen over previous bill meetings based on various strategies, which is the the outcomes mentioned in Definition 1.

Here we provide an application with real data as follows. Let (P, b, η) ∈ Λ with P = {m, n} and b = (b_m, b_n) = (0.92, 0.87). For every ω ∈ B^P, $η (ω) = {\begin{matrix} 4 & , if ω = (0.92, 0.87), \\ 0 & , if ω_{m} = 0 or ω_{n} = 0, \\ ω_{m} + ω_{n} + 1 & , otherwise . \end{matrix}$ If a payoff vector x is ECE and CR, then x is ECE and PR. So, ${\begin{matrix} x_{m, 0.92} + x_{n, 0.87} = 4, \\ x_{m, 0.92} \geq η (0.92, 0) = 0, \\ x_{n, 0.87} \geq η (0, 0.87) = 0 . \end{matrix}$ (1) If x is RE, then for all k_m ∈ (0, 0.92) and for all k_n ∈ (0, 0.87), ${\begin{matrix} x_{m, k_{m}} & \leq sup {v (k_{m}, k_{n}) | k_{n} \in B_{n}} \\ = sup {k_{m} + k_{n} + 1 | k_{n} \in B_{n}} \\ = k_{m} + 0.87 + 1, \\ x_{n, k_{n}} & \leq sup {v (k_{m}, k_{n}) | k_{m} \in B_{m}} \\ = sup {k_{m} + k_{n} + 1 | k_{m} \in B_{m}} \\ = k_{n} + 0.92 + 1 . \end{matrix}$ (2) If x is CR, then x is PR. So, for all k_m ∈ (0, 0.92) and for all k_n ∈ (0, 0.87), ${\begin{matrix} x_{m, k_{m}} \geq η (k_{m}, 0) = 0, \\ x_{n, k_{n}} \geq η (0, k_{n}) = 0, \\ x_{m, k_{m}} + x_{n, k_{n}} \geq η (k_{n}, k_{n}) = k_{m} + k_{n} + 1 . \end{matrix}$ (3) By definition of η and equation (3), for all k_m ∈ (0, 0.92) and for all k_n ∈ (0, 0.87), ${\begin{matrix} x_{m, k_{m}} + x_{n, k_{n}} \geq η (k_{n}, k_{n}) = k_{m} + 0.87 + 1, \\ x_{m, k_{m}} + x_{n, k_{n}} \geq η (k_{n}, k_{n}) = 0.92 + k_{n} + 1 . \end{matrix}$ (4) By definition of payoff vector and equations (1), (2), (3) and (4), ${\begin{matrix} x_{m, 0.92} + x_{n, 0.87} = 4, \\ x_{m, 0.92} \geq 0, \\ x_{n, 0.87} \geq 0, \\ x_{m, k_{m}} = k_{m} + 1.87 & for all k_{m} \in (0, 0.92), \\ x_{n, k_{n}} = k_{n} + 1.92 & for all k_{n} \in (0, 0.87), \\ x_{m, 0} = 0 = x_{n, 0} . \end{matrix}$ (5) if x ∈ Δ_R (P, b, η) = {x| x isECE, REandCR}.

In the following sections, we would like to demonstrate that the restrained core could present “optimal distribution mechanism" among all participants, in the sense that this organization can receive remuneration from each combination of energetic levels of all participants under fuzzy transferable-utility behavior.

3 Axioms and axiomatic result

In order to analyze the restrained core, some more properties are needed. Let κ be a measuration on Λ.

Completeness (CPS): for every (P, b, η) ∈ Λ, κ (P, b, η) ⊆ F (P, b, η).

Personal reasonability (PRY): for every (P, b, η) ∈ Λ and for every x ∈ κ (P, b, η), x is PR.

Restrictionism (REM): for every (P, b, η) ∈ Λ and for every x ∈ κ (P, b, η), x is RE.

Single-agent reasonability (SARY): for every (P, b, η) ∈ Λ with |P|=1, κ (P, b, η) = IT (P, b, η).

Subsequently, we introduce an extension of the reduction of Davis and Maschler [5] under fuzzy TU models.

Definition 2. Let (P, b, η) ∈ Λ, S ∈ 2^P \ {∅} and $x \in ℝ^{P}$ . The reduced model related to S and x is the model 2 $(S, b_{S}, η_{S}^{x})$ where $η_{S}^{x} (ω) = {\begin{matrix} 0 & if ω = 0_{S}, \\ η (b) - x (b_{P \ S}) & if ω = b_{S}, \\ sup_{ρ \in B^{P \ S}} {η (ω, ρ) - x (ρ)} & otherwise . \end{matrix}$

Let κ be a measuration on Λ. Steadiness states that if x is ordered by κ for a fuzzy TU model (P, b, η), then the projection x_S should be ordered by κ for the reduction related to S and x. Thus, the projection x_S should be steady with the expectancies of the participators of S as reflected by its reduction. Converse steadiness states that if the projection of an imputation x to each coalition S is steady with the expectancies of the participators of S as reflected by its reduction then x itself should be advocated for entire model.

Here we denote x|_S = (x_{m,k_m}) _{(m,k_m)∈C^S} for every (P, b, η) ∈ Λ, for every $x \in ℝ^{P}$ and for every S ∈ 2^P \ {∅}. Let κ be a measuration on Λ.

Steadiness (SDS): for every (P, b, η) ∈ Λ, for every S ∈ 2^P \ {∅} and for every x ∈ κ (P, b, η), $x |_{S} \in κ (S, b_{S}, η_{S}^{x})$ .

Converse steadiness (CSDS): for every (P, b, η) ∈ Λ with |P|≥2 and for every x ∈ IT (P, b, η), if $x |_{S} \in κ (S, b_{S}, η_{S}^{x})$ for every S ⊂ P with 0 < |S| < |P|, then x ∈ κ (P, b, η).

Next, we axiomatize the restrained core by adopting PRY, REM, SARY, SDS and CSDS.

Lemma 1. Let (P, b, η) ∈ Λ, S ∈ 2^P \ {∅} and $x \in ℝ^{P}$ .

If x is ECE in (P, b, η), then x|_S is ECE in $(S, b_{S}, η_{S}^{x})$ .

If x is CR in (P, b, η), then x|_S is CR in $(S, b_{S}, η_{S}^{x})$ .

If x is RE in (P, b, η), then x|_S is RE in $(S, b_{S}, η_{S}^{x})$ .

Proof. Let (P, b, η) ∈ Λ, S ∈ 2^P \ {∅} and $x \in ℝ^{P}$ . To verify (1), let x ∈ F (P, b, η). By definition of $η_{S}^{x}$ , $η_{S}^{x} (b_{S}) = η (b) - x (b_{P \ S}) = x (b_{S}) .$ That is, $x |_{S} \in F (S, b_{S}, η_{S}^{x})$ .

To verify (2), let x be CR in (P, b, η). for every ω ∈ B^S, $\begin{matrix} η_{S}^{x} (ω) - x (ω) & = sup_{ρ \in B^{P \ S}} {η (ω, ρ) - x (ρ)} - x (ω) \\ = sup_{ρ \in B^{P \ S}} {η (ω, ρ) - x (ρ) - x (ω)} \\ = sup_{ρ \in B^{P \ S}} {η (ω, ρ) - x (ω, ρ)} . \end{matrix}$ Since x is CR, η (ω, ρ) - x (ω, ρ) ≤0. So, $x (ω) \geq η_{S}^{x} (ω)$ for every ω ∈ B^P\S. Hence, x|_S is CR in $(S, b_{S}, η_{S}^{x})$ . It is easy to finish the proof of (3) by definition of RE.■

Lemma 2. On Λ, the restrained core Δ_R suits SDS.

Proof. It is easy to manifest that the restrained core Δ_R suits SDS by Lemma 1.■

Lemma 3. On Λ, the restrained core Δ_R suits CSDS.

Proof. Let (P, b, η) ∈ Λ with |P|≥2 and let x ∈ IT (P, b, η). Assume that $(S, b_{S}, η_{S}^{x}) \in Λ$ and $x |_{S} \in Δ_{R} (S, b_{S}, η_{S}^{x})$ for every S ⊂ P with 0 < |S| < |P|. We will manifest that x ∈ Δ_R (P, b, η). Since x ∈ IT (P, b, η), it remains to manifest that x (ω) ≥ η (ω) for every ω ∈ B^P. Two conditions could be distinguished:

Condition (1): |P|=2:

Assume that P = {m, n}. Let ω = (ω_m, ω_n) ∈ B^P.

Consider ω = (ω_m, ω_n) ∈ B^P with ω_k = 0 or ω_k = b_k where k = m, n. Since x ∈ IT (P, b, η), x (ω) ≥ η (ω).

Consider ω = (ω_m, ω_n) ∈ B^P with ω_m ≠ b_m. Then $\begin{matrix} x_{m, ω_{m}} & \geq η_{{m}}^{x} (ω_{m}) \\ (by x |_{{m}} \in Δ_{R} ({m}, b_{m}, η_{{m}}^{x})) \\ \geq η (ω_{m}, ω_{n}) - x_{n, ω_{n}} . \end{matrix}$ Hence, x (ω) ≥ η (ω).

Consider ω = (ω_m, ω_n) ∈ B^P with ω_n ≠ b_n. The proof is similar to the above proofs by applying the reduction $({n}, b_{n}, η_{{n}}^{x})$ .

Condition (2): |P|>2:

Let ω ∈ B^P \ {0_P}. If ω = b, then we complete the proof by CPS and PRY of IT (P, b, η). Assume that ω ≠ b.

Assume that there exists m ∈ P such that 0 < ω_m < b_m. Consider the reduction $({m}, b_{m}, η_{{m}}^{x})$ . Then $\begin{matrix} x_{m, ω_{m}} & \geq η_{{m}}^{x} (ω_{m}) \\ (by x |_{{m}} \in Δ_{R} ({m}, b_{m}, η_{{m}}^{x})) \\ \geq η (ω_{m}, ω_{P \ {m}}) - x (ω_{P \ {m}}) \\ = η (ω) - x (ω_{P \ {m}}) . \end{matrix}$ Hence, x (ω) ≥ η (ω).

Assume that ω_t = 0 or ω_t = b_t for every t ∈ P. Let m, n ∈ P with ω_m = b_m and ω_n = 0. Consider the reduction $({m, n}, (b_{m}, b_{n}), η_{{m, n}}^{x})$ . Then

\begin{matrix} x_{m, ω_{m}} & \geq η_{{m, n}}^{x} (ω_{m}, 0) \\ (by x |_{{m, n}} \in Δ_{R} ({m, n}, (b_{m}, b_{n}), η_{{m, n}}^{x})) \\ \geq η (ω_{m}, 0, ω_{P \ {m}}) - x (ω_{P \ {m}}) \\ = η (ω) - x (ω_{P \ {m}}) . \\ (Since ω_{n} = 0) \end{matrix}

Hence, x (ω) ≥ η (ω).■

Lemma 4. If a measuration κ suits SARY and SDS, then it also suits CPS.

Proof. Let (P, b, η) ∈ Λ, x ∈ κ (P, b, η) and m ∈ P. Consider the reduction $({m}, b_{m}, η_{{m}}^{x})$ . Clearly, $η_{{m}}^{x} (b_{m}) = η (b) - x (b_{P \ {m}})$ . By SDS of κ, $x |_{{m}} \in κ ({m}, b_{m}, η_{{m}}^{x})$ . By SARY of κ, $x_{m, b_{m}} \geq η_{{m}}^{x} (b_{m}) = η (b) - x (b_{P \ {m}})$ . Thus, x (b) ≥ η (b). Since κ be a measuration on Λ, κ (P, b, η) ⊆ E^* (P, b, η). Thus, x (b) ≤ η (b). Since x (b) ≥ η (b) and x (b) ≤ η (b), x (b) = η (b).■

Theorem 1. A measuration κ suits PRY, REM, SARY, SDS and CSDS if and only if κ (P, b, η) = Δ_R (P, b, η) for every (P, b, η) ∈ Λ.

Proof. By Lemmas 2, 3, the restrained core suits SDS and CSDS. Clearly, the restrained core suits PRY, REM and SARY.

To manifest the uniqueness, assume that a measuration κ suits PRY, REM, SARY, SDS and CSDS. Let(P, b, η) ∈ Λ. The proof will progress by induction on |P|. If |P|=1, then κ (P, b, η) = Δ_R (P, b, η) by SARY of κ. Assume that κ (P, b, η) = Δ_R (P, b, η) if |P| < k, k ≥ 2.

The condition |P| = k:

First we manifest that κ (P, b, η) ⊆ Δ_R (P, b, η). Let x ∈ κ (P, b, η). Since κ suits SARY and SDS, κ suits CPS by Lemma 4. This implies that x ∈ F (P, b, η). So, x ∈ IT (P, b, η) by PRY and REM of κ. By SDS of κ, $x |_{S} \in κ (S, b_{S}, η_{S}^{x})$ for every S ⊂ P with 0 < |S| < |P|. Based on the induction hypothesis, $x |_{S} \in κ (S, b_{S}, η_{S}^{x}) = Δ_{R} (S, b_{S}, η_{S}^{x})$ for every S ⊂ P with 0 < |S| < |P|. By CSDS of the restrained core, x ∈ Δ_R (P, b, η). The remaining processes could be manifested alike by exchanging the positions of κ and Δ_R. Hence, κ (P, b, η) = Δ_R (P, b, η).■

Next, we apply some examples to present that each of the properties adopted in Theorem 1 is logically independent of the others.

Example 1. Let κ¹ (P, b, η) =∅ for every (P, b, η) ∈ Λ. Then κ¹ suits PRY, REM, SDS and CSDS, but it infringes SARY.

Example 2. Let κ² (P, b, η) = IT (P, b, η) for every (P, b, η) ∈ Λ. Then κ² suits PRY, SARY, REM and CSDS, but it infringes SDS.

Example 3. Define the measuration κ³ to be that $\begin{matrix} (P, b, η) = \\ {\begin{matrix} IT (P, b, η) & if | P | = 1, \\ {x \in F (P, b, η) | x is RE} & otherwise . \end{matrix} \end{matrix}$ Then κ³ suits SARY, REM, SDS and CSDS, but it infringes PRY.

Example 4. Define the measuration κ⁴ to be that $\begin{matrix} κ^{4} (P, b, η) = \\ {\begin{matrix} IT (P, b, η) & if | P | = 1, \\ {x \in F (P, b, η) | x is CR} & otherwise . \end{matrix} \end{matrix}$ Then κ⁴ suits SARY, PRY, SDS and CSDS, but it infringes REM.

Example 5. Define the measuration κ⁵ to be that $κ^{5} (P, b, η) = {\begin{matrix} IT (P, b, η) & if | P | = 1, \\ \emptyset & otherwise . \end{matrix}$ Then κ⁵ suits SARY, PRY, REM and SDS, but it infringes CSDS.

4 Alternative reductions

Inspired by the outcome of Serrano and Volij [11], we propose alternative reduction to axiomatize the restrained core.

Definition 3. Let (P, b, η) ∈ Λ, S ∈ 2^P \ {∅} and $x \in ℝ^{P}$ . The revised reduced model related to S and x is the model (S, b_S, η_S,x) where $η_{S, x} (ω) = {\begin{matrix} 0 & if ω = 0_{S}, \\ sup_{ρ \in B^{P \ S}} {η (ω, ρ) - x (ρ)} & otherwise . \end{matrix}$

The notion of the revised reduction was initially defined by Serrano and Volij [11]. The only dissimilitude among this and the reduction defined in Definition 2 is the fact that the coalition S is also permitted to conceive promising interaction with arbitrary coalition inside P \ S. In order to arrive at the supreme worth, all coalitions formed by the participants of P \ S should to be considered.

Revised steadiness (RSDS): for every (P, b, η) ∈ Λ, for every S ∈ 2^P \ {∅} and for every x ∈ κ (P, b, η), x|_S ∈ κ (S, b_S, η_S,x).

Converse revised steadiness (CRSDS): for every (P, b, η) ∈ Λ with |P|≥2 and for every x ∈ F (P, b, η), if x|_S ∈ κ (S, b_S, η_S,x) for every S ⊂ N with 0 < |S| < |P|, then x ∈ κ (P, b, η).

The statements for RSDS abd CRSDS are similar to those given in Section 3. An axiomatic outcome of the restrained core are as follows.

Lemma 5.

On Λ, the restrained core Δ_R suits RSDS.

If a measuration κ suits SARY and SDS, then it also suits CPS.

Proof. The proofs are similar to Lemmas 2, 4.■

Lemma 6. On Λ, the restrained core Δ_R suits CRSDS.

Proof. Let (P, b, η) ∈ Λ with |P|≥2 and let x ∈ F (P, b, η). Assume that (S, b_S, η_S,x) ∈ Λ and x|_S ∈ Δ_R (S, b_S, η_S,x) for every S ⊂ P with 0 < |S| < |P|. We will manifest that x ∈ Δ_R (P, b, η), i.e., x (ω) ≥ η (ω) for every ω ∈ B^P. Let m ∈ P and ω ∈ B^P. Consider the reduction ({m} , b_m, η_{m},x). Then $\begin{matrix} x_{m, ω_{m}} & \geq η_{{m}, x} (ω_{m}) \\ (by x |_{{m}} \in Δ_{R} ({m}, b_{m}, η_{{m}, x})) \\ = sup_{ρ \in B^{P \ {m}}} {η (ω_{m}, ρ) - x (ρ)} \\ \geq η (ω_{m}, ω_{P \ {m}}) - x (ω_{P \ {m}}) . \end{matrix}$ Hence, x (ω) ≥ η (ω).■

Theorem 2. A measuration κ suits REM, SARY, RSDS and CRSDS if and only if κ (P, b, η) = Δ_R (P, b, η) for every (P, b, η) ∈ Λ.

Proof. By Lemmas 5, 6, the restrained core suits RSDS and CRSDS. And it is known that the restrained core suits REM and SARY. The remaining proof is similar to Theorem 1.■

Next, we apply some examples to present that each of the properties adopted in Theorem 2 is logically independent of the others.

Example 6. Let κ⁶ (P, b, η) =∅ for every (P, b, η) ∈ Λ. Then κ⁶ suits REM, RSDS and RCSDS, but it infringes SARY.

Example 7. Define the measuration κ⁷ to be that $\begin{matrix} κ^{7} (P, b, η) = \\ {\begin{matrix} IT (P, b, η) & if | P | = 1, \\ {x \in F (P, b, η) | x is CR} & otherwise . \end{matrix} \end{matrix}$ Then κ⁷ suits SARY, RSDS and CRSDS, but it infringes REM.

Example 8. Let κ⁸ (P, b, η) = IT (P, b, η) for every (P, b, η) ∈ Λ. Then κ⁸ suits SARY, REM and CRSDS, but it infringes RSDS.

Example 9. Define the measuration κ⁹ to be that $κ^{9} (P, b, η) = {\begin{matrix} IT (P, b, η) & if | P | = 1, \\ \emptyset & otherwise . \end{matrix}$ Then κ⁹ suits SARY, REM and RSDS, but it infringes CRSDS.

Here we extend the reductions of Davis and Maschler [5] and Serrano and Volij [11] to fuzzy TU models. This raises the guess whether the other types of reduction that have been adopted in the axiomatic results of the standard TU core could be applied under fuzzy TU models. The reduction due to Moulin [9] is one worth considering. A natural extended Moulin reduction is as follows. Given (P, b, η) ∈ Λ, S ∈ 2^P \ {∅} and $x \in ℝ^{P}$ , we define the M-reduced model related to S and x as the model $(S, b_{S}, η_{S, x}^{M})$ given by $η_{S, x}^{M} (ω) = {\begin{matrix} 0 & if ω = 0_{S}, \\ η (ω, b_{P \ S}) - x (b_{S}) & otherwise . \end{matrix}$

“M-reduction" instead of “reduction", we propose the M-steadiness (MSDS) and the converse M-steadiness (CMSDS). Based on Theorems 1, 2, the restrained core could be axiomatized by means of previous reductions. However, based on the M-reduction, this conclusion is incorrect. The reason is that the restrained core infringes converse M-steadiness. The following example illustrates this fact.

Example 10. Let (P, b, η) be a model, where P = {m, n}, b = (1, 1) and η is defined as follows. For every ω ∈ B^P, $η (ω) = {\begin{matrix} 4 & , if ω = (1, 1), \\ 0 & , if ω_{m} = 0 or ω_{n} = 0, \\ ω_{m} + ω_{n} + 1 & , otherwise . \end{matrix}$ Let $x \in ℝ^{P}$ with $x_{m, k} = \frac{3}{2}$ and $x_{n, k} = \frac{5}{2}$ for every k ∈ (0, 1]. Clearly, x ∈ IT (P, b, η), $η_{{m}, x}^{M} (1) = \frac{3}{2}$ , $η_{{n}, x}^{M} (1) = \frac{5}{2}$ and $η_{{m}, x}^{M} (t) = t - \frac{1}{2}$ , $η_{{n}, x}^{η} (t) = t + \frac{1}{2}$ for every t ∈ (0, 1). Clearly, $x |_{{m}} \in Δ_{R} ({m}, η_{{m}, x}^{M})$ and $x |_{{n}} \in Δ_{R} ({n}, η_{{n}, x}^{M})$ . Taking $ω = (\frac{2}{3}, \frac{1}{3})$ , $\begin{matrix} x (ω) = \frac{2}{3} \cdot \frac{3}{2} + \frac{1}{3} \cdot \frac{5}{2} = \frac{11}{6} < \frac{2}{3} + \frac{1}{3} + 1 \\ = η (\frac{2}{3}, \frac{1}{3}) = η (ω) . \end{matrix}$ ∥Thus, x ∉ Δ_R (P, b, η). The restrained core infringes CMSDS.

Remark 2. Similar to Lemma 2, the restrained core suits MSDS.

5 The minimal converse M-steady enlargement

Here we introduce the minimal core and manifest that the minimal core coincides with the minimal converse M-steady enlargement of the restrained core. Further, a parallel analogue of Theorems 1, 2 is provided.

For every (P, b, η) ∈ Λ, the collection of coalitions with full energetic level of personal participant, $K_{b}^{P}$ , is defined by $K_{b}^{P} = {ω \in B^{P} | ω = (b_{m}, 0_{P \ {m}}) for every m \in P} .$ The collection of coalitions with non-full energetic level of all participants, $H_{b}^{P}$ , is defined by $H_{b}^{P} = {ω \in B^{P} | ω_{m} \neq b_{m} for every m \in P} .$ Let $B_{*}^{P} = B^{P} \ (K_{b}^{P} \cup H_{b}^{P})$ , the minimal core Δ_M (P, b, η) is defined by $\begin{matrix} Δ_{M} (P, b, η) \\ = & {\begin{matrix} Δ_{R} (P, b, η) & , | P | = 1, \\ {x \in F (P, b, η) | x (ω) \geq η (ω) \forall ω \in B_{*}^{P}} & , o . w . \end{matrix} \end{matrix}$ Clearly, Δ_R (P, b, η) ⊆ Δ_M (P, b, η) for all (P, b, η) ∈ Λ.

Let Ω be the class of all measurations and Ω^M be the class of all measurations suiting CMSDS respectively. Given κ ∈ Ω, we define $\begin{matrix} Ω_{κ}^{M} \\ = & {τ \in Ω^{M} | κ (P, b, η) \subseteq τ (P, b, η) \forall (P, b, η) \in Λ} . \end{matrix}$

Furthermore, the minimal converse M-steady enlargement of κ, ${Mce}_{κ}^{M}$ , is defined by ${Mce}_{κ}^{M} = ⋂_{τ \in Ω_{κ}^{M}} τ .$

The minimal converse M-steady enlargement is well-defined and uniquely determined since the practical set suits CMSDS, and CMSDS is preserved under intersection.

Lemma 7. The minimal core Δ_M suits MSDS and CMSDS.

Proof. First, we manifest that the minimal core suits MSDS. Let (P, b, η) ∈ Λ, S ⊆ P with 0 < |S| < |P| and x ∈ Δ_M (P, b, η). By (1) and (2) of Lemma 1, $x |_{S} \in F (S, b_{S}, η_{S, x}^{M})$ and x is RE. It remains to manifest that $x (ω) \geq η_{S, x}^{M} (ω)$ for every $ω \in B_{*}^{S}$ . Let $ω \in B_{*}^{S}$ , $η_{S, x}^{M} (ω) - x (ω) = η (ω, b_{P \ S}) - x (b_{S}) - x (ω) .$ (6) Since $ω \in B_{*}^{S}$ and 0 < |S| < |P|, $(ω, b_{P \ S}) \in B_{*}^{P}$ . Since x ∈ Δ_M (P, b, η) and $(ω, b_{P \ S}) \in B_{*}^{P}$ , by equation (6), $η_{S, x}^{M} (ω) - x (ω) = η (ω, b_{P \ S}) - x (ω, b_{P \ S}) \leq 0 .$ Hence, $x |_{S} (ω) \geq η_{S, x}^{M} (ω)$ for every $ω \in B_{*}^{S}$ .

Next, we manifest that the minimal core suits CMSDS. Let (P, b, η) ∈ Λ with |P|≥2 and x ∈ IT (P, b, η). Assume that $(S, b_{S}, η_{S, x}^{M}) \in Λ$ and $x |_{S} \in Δ_{M} (S, b_{S}, η_{S, x}^{M})$ for every S ⊂ P such that 0 < |S| < |P|. Let $ω \in B_{*}^{P}$ and ω_T = b_T for some T ⊆ P with 1 ≤ |T| < |P|. Since 1 ≤ |T| < |P| and $x |_{P \ T} \in Δ_{M} (P \ T, η_{P \ T, x}^{M})$ , $x (ω_{P \ T}) \geq η_{P \ T, x}^{M} (ω_{P \ T}) = η (ω_{P \ T}, b_{T}) - x (b_{T}) .$ Thus, x (ω) ≥ η (ω). Assume that ω = b. Since x ∈ IT (P, b, η), x (b) = η (b). Hence, x (ω) ≥ η (ω) for every $ω \in B_{*}^{P}$ .■

Theorem 3. The minimal core Δ_M is the minimal converse M-steady enlargement of the restrained core.

Proof. Let (P, b, η) ∈ Λ. Since Δ_R (P, b, η) ⊆ Δ_M (P, b, η) and Δ_M suits CMSDS, ${Mce}_{Δ_{R}}^{M} (P, b, η) \subseteq Δ_{M} (P, b, η)$ by definition of ${Mce}_{Δ_{R}}^{M}$ .

Further, it remains to manifest that $Δ_{M} (P, b, η) \subseteq {Mce}_{Δ_{R}}^{M} (P, b, η)$ for every (P, b, η) ∈ Λ. Let (P, b, η) ∈ Λ. The proof proceeds by induction on |P|. Assume that |P|=1. Since $Δ_{R} (P, b, η) \subseteq {Mce}_{Δ_{R}}^{M} (P, b, η) \subseteq Δ_{M} (P, b, η)$ and Δ_M (P, b, η) = Δ_R (P, b, η) = IT (P, b, η), ${Mce}_{Δ_{R}}^{M} (P, b, η) = Δ_{M} (P, b, η)$ . Assume that $Δ_{M} (P, b, η) \subseteq {Mce}_{Δ_{R}}^{M} (P, b, η)$ if |P| ≤ l - 1, where l ≥ 2.

The condition |P| = l: Let x ∈ Δ_M (P, b, η). By MSDS of Δ_M, $x_{S} \in Δ_{M} (S, b_{S}, η_{S, x}^{M})$ for every S ⊂ P with 0 < |S| < |P|. Based on induction hypotheses, $x_{S} \in Δ_{M} (S, b_{S}, η_{S, x}^{M}) \subseteq {Mce}_{Δ_{R}}^{M} (S, b_{S}, η_{S, x}^{M})$ . By CMSDS of ${Mce}_{Δ_{R}}^{M}$ , $x \in {Mce}_{Δ_{R}}^{M} (P, b, η)$ . Hence, $Δ_{M} (P, b, η) \subseteq {Mce}_{Δ_{R}}^{M} (P, b, η)$ .■

Theorem 4. A measuration κ suits REM, PRY, SARY, MSDS and CMSDS if and only if κ (P, b, η) = Δ_M (P, b, η) for every (P, b, η) ∈ Λ.

Proof. Clearly, Δ_M suit REM, SARY and PRY. By Lemma 7, Δ_M suits MSDS and CMSDS. The remaining proofs are similar to Theorem 1.■

Examples 1-5 also could be adopted to manifest that each of the properties adopted in Theorem 4 is logically independent of the others.

Remark 3. In addition to minimally enlarge the restrained core so as to recover converse M-steadiness, one would like to maximally diminish the restrained core so as to recover converse M-steadiness. Given κ ∈ Ω, we define $\begin{matrix} Φ_{κ}^{M} = \\ {τ \in Ω^{M} | κ (P, b, η) \supseteq τ (P, b, η) \forall (P, b, η) \in Λ} . \end{matrix}$

The maximal sub-measuration of κ, ${Mcs}_{κ}^{M}$ , is defined by ${Mcs}_{κ}^{M} = ⋃_{τ \in Φ_{κ}^{M}} τ .$

Since converse M-steadiness can not be preserved under union, however, the maximal sub-measuration is not certain to suit converse M-steadiness.

6 Conclusions

Different from pre-existing measuration concepts, we propose new measuration in the framework of fuzzy TU models. Several main results of this article are as follows.

The restrained core are proposed by focusing on both the participants and its energetic levels simultaneously.

The reductions due to Davis and Maschler [5], Moulin [9] and Serrano and Volij [11] are extended to characterize the restrained core on fuzzy TU models.

Since the restrained core infringes a specific converse steadiness related to the extended Moulin [9] reduction, we introduce to minimally enlarge the restrained core so as to regain this specific converse steadiness.

One should compare our results with related results due to Aubin [1, 2], Hwang [6] and Hwang and Liao [7]. There two major differences between existing results and ours.

We propose the restrained core and related results by simultaneously considering the participants and the energetic levels.

In order to investigate how extensive the violation of converse M-steadiness is, the converse M-steadiness enlargement of the restrained core and related results are proposed.

Steadiness and converse steadiness are essential to the proofs of many axiomatic processes of the core concepts. However, the restrained core proposed in this paper infringes converse M-steadiness. Therefore, we would like to axiomatize the restrained core by forsaking the properties of steadiness or its converse. We leave it to the researchers to investigate this in future studies.

Footnotes

Acknowledgment

The authors are grateful to the editor, the associate editor and the anonymous referees for very helpful suggestions and comments.

Without loss of generality, one could assume that A (b) = P.

In order to ensure that $η_{S}^{x} (ω)$ is well-defined, we consider bounded fuzzy TU models, defined as those models (P, b, η) such that, there exists $L_{η} \in ℝ$ such that η (ω) ≤ L_η for every ω ∈ B^P. Furthermore, $(S, b_{S}, η_{S}^{x}) \in Λ$ always.

References

Aubin

J.P.

, Coeur et valeur des jeux flous á paiements latéraux, Comptes Rendus de l’Académie des Sciences 279 (1974), 891–894.

Aubin

J.P.

, Cooperative fuzzy games, Mathematics of Operations Research 6 (1981), 1–13.

Butnariu

, Fuzzy games: a description of the concept, Fuzzy Sets and Systems 1 (1978), 181–192.

Chen

H.C.

, Wei

H.C.

, Li

A.T.

, Wang

W.N.

, Liao

Y.H.

, The supreme-core on multicriteria fuzzy games, Journal of Intelligent and Fuzzy Systems 38 (2020), 1753–1760.

Davis

, Maschler

, The kernel of a cooperative game, Naval Research Logistics Quarterly 12 (1965), 223–259.

Hwang

Y.A.

, Fuzzy games: a characterization of the core, Fuzzy Sets and Systems 158 (2007), 2480–2493.

Hwang

Y.A.

, Liao

Y.H.

, Max-consistency, complementconsistency, and the core of fuzzy games, Fuzzy Sets and Systems 159 (2007), 152–163.

Hwang

Y.A.

, Liao

Y.H.

, The core configuration for fuzzy games, Journal of Intelligent and Fuzzy Systems 27 (2014), 3007–3014.

Moulin

, The separability axiom and equal-sharing methods, Journal of Economic Theory 36 (1985), 120–148.

10.

Peleg

, On the reduced game property and its converse, International Journal of Game Theory 15 (1986), 187–200.

11.

Serrano

, Volij

, Axiomatizations of neoclassical concepts for economies, Journal of Mathematical Economics 30 (1998), 87–108.

12.

Tadenuma

, Reduced games, consistency, and the core, International Journal of Game Theory 20 (1992), 325–334.

13.

Tijs

, Branzei

, Ishihara

, Muto

, On cores and stable sets for fuzzy games, Fuzzy Sets and Systems 146 (2004), 285–296.

14.

, Zhang

, The fuzzy core in games with fuzzy coalitions, Journal of Computational and Applied Mathematics 20 (2009), 173–186.