The supreme-consistent value for fuzzy transferable-utility games: Alternative formulation,dynamic process and axiomatizations 1

Abstract

In the framework of fuzzy transferable-utility games, we adopt excess functions to propose alternative formulation and related dynamic process for the supreme-consistent value. In order to investigate the rationality for allocation mechanisms, we characterize the supreme-consistent value by means of different properties.

Keywords

Fuzzy transferable-utility games the supreme-consistent value excess function dynamic process

1 Introduction

On standard transferable-utility (TU) games, a characteristic function is defined over all the set of players. This means that the options available for each player are either to participate fully in a coalition or not to participate at all. A fuzzy TU game is a natural extension of a standard TU game in which each player is allowed to participate with infinite various activity levels respectively. Several fuzzy solutions and related results have been proposed in the literature, such as Branzei et al. [3], Butnariu and Kroupa [5], Hwang [7], Hwang and Liao [8], Li and Zhang [9], Liao et al. [10], Meng and Zhang [12], Tsurumi et al. [14] and so on. Here we focus on the supreme-consistent value due to Liao et al. [10]. Liao et al. [10] characterized the supreme-consistent value by means of the properties of efficiency, fuzzy balanced contributions, fuzzy path independence, standard for games and consistency. Further, a potential approach and a dynamic result for fuzzy solutions are also provided.

The above pre-existing results raise one motivation in the framework of fuzzy TU games:

whether some more axiomatic results and dynamic processes for the supreme-consistent value could be proposed by different viewpoints.

The paper is devoted to answer the question. Different from the results introduced by Liao et al. [10], some more results of the supreme-consistent value are proposed in this paper.

In order to investigate the rationality for allocation mechanisms, we characterize the supreme-consistent value by means of the properties of efficiency, fuzzy symmetry, fuzzy covariance and consistency in Section 4.

The individual rationality is an important property in many axiomatizations for set-valued solutions. In Section 4, we also introduce the fuzzy individual rationality for single-valued solutions to characterize the supreme-consistent value.

Based on different viewpoint, we introduce alternative formulation for the supreme-consistent value in terms of excess functions. Different from pre-existing results for the supreme-consistent value, we adopt excess functions to define different correction function and related dynamic process in Section 5. This dynamic process shows that the supreme-consistent value can be reached by players who start from an arbitrary efficient payoff vector. Some more statements and comparisons are also provided in detail.

2 Preliminaries

Let U be the universe of agents. For i ∈ U and d_i ∈ (0, 1], D_i = [0, d_i] could be treated as the action (decision) space of agent i and $D_{i}^{+} = (0, d_{i}]$ , where 0 denotes no participation. Let $D^{N} = \prod_{i \in N} D_{i}$ be the product set of the action spaces of all agents of N. For all T ⊆ N, we define θ^T ∈ D^N is the vector with $θ_{i}^{T} = 1$ if i ∈ T, and $θ_{i}^{T} = 0$ if i ∈ N \ T. Denote 0_N the zero vector in $ℝ^{N}$ .

A fuzzy TU game 2 is a triple (N, d, v), where N is a non-empty and finite set of agents, d = (d_i) _i∈N ∈ [0, 1] ^N be the vector that describes the highest level for each agent, and $v : D^{N} \to ℝ$ is a utility function with v (0_N) =0 which assigns to each α ∈ D^N the worth that the agents can gain when each agent i plays at level α_i. Given a fuzzy TU game (N, d, v) and α ∈ D^N, we write (N, α, v) for the fuzzy TU subgame obtained by restricting v to {β ∈ D^N ∣ β_i ≤ α_i ∀ i ∈ N}. Let A (α) = {i ∈ N|α_i ≠ 0} and α_T be the restriction of α at T for each T ⊆ N.

Denote the class of all fuzzy TU games by Γ. A solution on Γ is a map ψ assigning to each (N, d, v) ∈ Γ an element $ψ (N, d, v) = (ψ_{i} (N, d, v))_{i \in N} \in ℝ^{N} .$

Here ψ_i (N, d, v) is the payoff of the agent i when i participate in game v.

In the framework of fuzzy TU games, Liao et al. [10] proposed the supreme-consistent value as follows.

Definition 2.1. The supreme-consistent value, φ, is the map which associates to each (N, d, v) ∈ Γ and each i ∈ N the value $\begin{matrix} φ_{i} (N, d, v) \\ = \sum_{\overset{S \subseteq N}{i \in S}} \frac{(| S | - 1)! (| N | - | S |)!}{| N |!} \\ \cdot [v^{*} (S) - v^{*} (S \ {i})] . \end{matrix}$

For all S ⊆ N, $v^{*} (S) = sup_{α \in D^{N}} {v (α) | A (α) = S}$ is the supreme-utility 2 among all action vector with A (α) = S.

Remark 2.1. Without loss of generality, one could assume that A (d) = N for all (N, d, v) ∈ Γ.

3 Alternative axiomatizations

In this section, we provided fuzzy extensions of the axiomatic result due to Maschler and Owen [11]. We firstly recall some property and results proposed by Liao et al. [10]. Let ψ be a solution. ψ satisfies efficiency (EFF) if ∑_i∈Nψ_i (N, d, v) = v^* (N) for all (N, d, v) ∈ Γ. Weak efficiency (WEFF) asserts that for all (N, d, v) ∈ Γ with |N|=1, ψ satisfies EFF. ψ satisfies standard for games (SFG) if ψ (N, d, v) = φ (N, d, v) for all (N, d, v) ∈ Γ with |N|=2.

Let (N, d, v) ∈ Γ, S⊆ N, S ≠ ∅, and ψ be a solution. Liao et al. [10] defined the reduced game $(S, d_{S}, v_{S, d}^{ψ})$ as follows. For all α ∈ D^S, $\begin{matrix} v_{S, d}^{ψ} (α) & = & v^{*} (A (α, d_{N \ S})) \\ - \sum_{k \in N \ S} ψ_{k} (N, (α, d_{N \ S}), v) . \end{matrix}$ ψ satisfies consistency (CON) if $ψ_{i} (S, d_{S}, v_{S, d}^{ψ}) = ψ_{i} (N, d, v)$ for all (N, d, v) ∈ Γ, for all S ⊆ N with S≠ ∅ and for all i ∈ S.

Remark 3.1. Liao et al. [10] showed that the supreme-consistent value is a unique solution satisfying SFG and CON. They also showed that if a solution ψ satisfies WEFF and CON, then it satisfies EFF.

Inspired by Maschler and Owen [11], we propose some axioms to characterize the supreme-consistent value. Let (N, d, v) ∈ Γ and i, j ∈ N. i, j are called symmetric players in (N, d, v) if v^* (S ∪ {i}) = v^* (S ∪ {j}) for all S ⊆ N \ {i, j}. A solution ψ satisfies fuzzy symmetry (FSYM) if i, j are symmetric players in (N, d, v) implies that ψ_i (N, d, v) = ψ_j (N, d, v).

Let (N, d, v) , (N, d, w) ∈ Γ and $b \in ℝ^{N}$ . (N, d, v) is b-equivalent to (N, d, w) if v^* (S) = w^* (S) + ∑_i∈Sb_i for all S ⊆ N. A solution ψ satisfies fuzzy covariance (FCOV) if for all (N, d, v) , (N, d, w) ∈ Γ, (N, d, v) is b-equivalent to (N, d, w) implies that ψ (N, d, v) = ψ (N, d, w) + b.

Property FSYM asserts that if the abilities are the same, then the payoffs should be coincident. Property FCOV can be interpreted as an extremely weak kind of additivity.

Lemma 3.1. If a solution ψ on Γ satisfies EFF, FSYM and FCOV, then ψ satisfies SFG.

Proof. Assume that a solution ψ satisfies EFF, FSYM and FCOV. Given (N, d, v) ∈ Γ with N = {i, j} for some i ≠ j. We define a game (N, d, w) to be that w^* (S) = v^* (S) - ∑_i∈Sv^* ({i}) forall S ⊆ N. So, $w^{*} ({i, j}) = v^{*} ({i, j}) - v^{*} ({i}) - v^{*} ({j}),$ and $\begin{matrix} w^{*} ({i}) & = & v^{*} ({i}) - v^{*} ({i}) = 0 \\ = & v^{*} ({j}) - v^{*} ({j}) = w^{*} ({j}) . \end{matrix}$

By FSYM of ψ, ψ_i (N, d, w) = ψ_j (N, d, w). By EFF of ψ, w^* ({i, j}) = ψ_i (N, d, w) + ψ_j (N, d, w) =2 · ψ_i (N, d, w). Therefore, $\begin{matrix} ψ_{i} (N, d, w) & = \frac{w^{*} ({i, j})}{2} \\ = \frac{1}{2} \cdot [v^{*} ({i, j}) - v^{*} ({i}) - v^{*} ({j})] . \end{matrix}$

By FCOV of ψ, $\begin{matrix} ψ_{i} (N, d, v) \\ = v^{*} ({i}) + \frac{1}{2} \cdot [v^{*} ({i, j}) - v^{*} ({i}) - v^{*} ({j})] . \end{matrix}$

By definition of φ, ψ_i (N, d, v) = φ_i (N, d, v). Hence, ψ satisfies SFG. ■

Theorem 3.1.

On Γ, the supreme-consistent value is the only solution satisfying EFF, FSYM, FCOV and CON.

On Γ, the supreme-consistent value is the only solution satisfying WEFF, FSYM, FCOV and CON.

Proof. The proofs follow from Remark 2 and Lemma 1. The following examples are to show that each of the axioms used in Theorem 1 is logically independent of the remaining axioms. ■

Example 3.1. Define a solution ψ to be $ψ_{i} (N, d, v) = \frac{v^{*} (N)}{| N |}$ for all (N, d, v) ∈ Γ and for all i ∈ N. Clearly, ψ satisfies EFF (WEFF), FSYM and CON, but it violates FCOV.

Example 3.2. Define a solution ψ to be $ψ_{i} (N, d, v) = \sum_{\overset{S \subseteq N}{i \in S}} \frac{w_{i} \cdot C_{S}^{v}}{\sum_{k \in S} w_{k}}$ for all (N, d, v) ∈ Γ and for all i ∈ N, where w = {w_t|t ∈ U} be a collection of positive real numbers and $C_{S}^{v} = \sum_{T \subseteq S} (- 1)^{| S | - | T |} v^{*} (T)$ . Clearly, ψ satisfies EFF (WEFF), FCOV, CON, but it violates FSYM.

Example 3.3. Define a solution ψ to be $ψ_{i} (N, d, v) = v^{*} ({i}) + \frac{1}{| N |} \cdot [v^{*} (N) - \sum_{k \in N} v^{*} ({k})]$ for all (N, d, v) ∈ Γ and for all i ∈ N. Clearly, ψ satisfies EFF (WEFF), FSYM and FCOV, but it violates CON.

Example 3.4. Define a solution ψ to be $ψ_{i} (N, d, v) = φ_{i} (N, d, v) + ε$ for all (N, d, v) ∈ Γ and for all i ∈ N, where $ε \in ℝ \ {0}$ . Clearly, ψ satisfies FSYM, FCOV and CON, but it violates EFF (WEFF).

4 Fuzzy individual rationality

In this section, we propose different axiomatizations of the supreme-consistent value. Let ψ be a solution. ψ satisfies fuzzy non-overload (FNO) if ∑_i∈Nψ_i (N, d, v) ≤ v^* (N) for all (N, d, v) ∈ Γ. ψ satisfies fuzzy individual rationality (FIR) if ψ_i (N, d, v) ≥ v^* ({i}) for all (N, d, v) ∈ Γ and for all i ∈ N.

Individual rationality asserts that the payoff of each player assigned by cooperation may be better than the utility of each player when he works alone. Individual rationality has been adopted in many axiomatizations for set-valued solutions extensively.

A game (N, d, v) ∈ Γ is a-inessential if there exists $a \in ℝ^{N}$ such that v^* (S) = ∑_i∈Sa_i for all S ⊆ N. A solution ψ satisfies fuzzy inessential games (FIEG) if a game (N, d, v) ∈ Γ is a-inessential implies that ψ (N, d, v) = a. Weak fuzzy inessential games (WFIEG) asserts that ψ satisfies FIEG for all (N, d, v) ∈ Γ with |N|=1.

Lemma 4.1. Let ψ be a solution. ψ satisfies WEFF if and only if ψ satisfies WFIEG.

Proof. Let ψ be a solution and (N, d, v) ∈ Γ with N = {i}. $\begin{matrix} ψ satisfies WEFF & \Leftrightarrow ψ_{i} (N, d, v) = v^{*} ({i}) \\ \Leftrightarrow ψ satisfies WFIEG . \end{matrix}$

The proof is completed. ■

Remark 4.1. By Remark 2, Lemma 2 and definition of EFF, the supreme-consistent value satisfies WFIEG and FNO.

Let (N, d, v) ∈ Γ. (N, d, v) is said to be a supreme-convex fuzzy TU game if v^* (S ∪ T) ≥ v^* (S) + v^* (T) - v^* (S ∩ T) for all S, T ⊆ N. Denote the class of all supreme-convex fuzzy TU games by Γ_C.

Lemma 4.2. The supreme-consistent value satisfies FIR on Γ_C.

Proof. Let (N, d, v) ∈ Γ_C. If N = {i}, then φ_i (N, d, v) = v^* ({i}) by EFF of φ. Hence, φ_i (N, d, v) ≥ v^* ({i}). Assumed that |N|≥2. Let i ∈ N. Since (N, d, v) ∈ Γ_C,

$\begin{matrix} v^{*} (S) - v^{*} (S \ {i}) \\ \geq v^{*} (S \ {i}) + v^{*} ({i}) - v^{*} (S \ {i}) \\ = v^{*} ({i}) . \end{matrix}$ (1) for all S ⊆ N with i ∈ S. By definition of φ, $\begin{matrix} φ_{i} (N, d, v) \\ = \sum_{\overset{S \subseteq N}{i \in S}} \frac{(| S | - 1)! (| N | - | S |)!}{| N |!} \cdot [v^{*} (S) - v^{*} (S \ {i})] \\ \geq \sum_{\overset{S \subseteq N}{i \in S}} \frac{(| S | - 1)! (| N | - | S |)!}{| N |!} \cdot v^{*} ({i}) (by Equation (1)) \\ = (\sum_{\overset{S \subseteq N, i \in S}{| S | = 1}} + \dots + \sum_{\overset{S \subseteq N, i \in S}{| S | = | N |}}) \frac{(| S | - 1)! (| N | - | S |)!}{| N |!} \cdot v^{*} ({i}) \\ = \sum_{t = 0}^{| N | - 1} \frac{(| N | - 1)!}{(| N | - t - 1)! t!} \cdot \frac{(| N | - t - 1)! t!}{| N |!} \cdot v^{*} ({i}) \\ = \sum_{t = 0}^{| N | - 1} \frac{1}{| N |} \cdot v^{*} ({i}) \\ = v^{*} ({i}) . \end{matrix}$

Hence, the supreme-consistent value satisfies FIR on Γ_C. ■

Lemma 4.3. If a solution ψ on Γ satisfies FNO, FIR and CON, then ψ satisfies EFF.

Proof. Let ψ be a solution satisfying FNO, FIR and CON. Let (N, d, v) ∈ Γ with N = {i}. By FNO of ψ, $ψ_{i} (N, d, v) \leq v^{*} ({i}) .$ (2)

By FIR of ψ, $ψ_{i} (N, d, v) \geq v^{*} ({i}) .$ (3)

By Equations (2) and (3), ψ satisfies WEFF. The remaining proofs are completed by Remark 2. ■

Theorem 4.1. Let ψ be a solution.

On Γ, ψ satisfies WFIEG, FSYM, FCOV and CON if and only if ψ = φ.

On Γ_C, ψ satisfies FIR, FNO, FSYM, FCOV and CON if and only if ψ = φ.

Proof. The proofs of this theorem could be completed by Theorem 1, Remarks 2, 3 and Lemmas 1, 2, 3, 4. ■

5 Excess formulation and related dynamic process

In general, dynamic processes can be described that lead the players to a stable and reasonable solution, starting from an arbitrary efficient payoff vector. Based on the axiomatic justifications in Remark 2 and Theorems 1, 2, it is shown that the supreme-consistent value is in the sense a stable solution that satisfies several reasonable properties. In this section, we present a dynamic process for the supreme-consistent value. In order to find the dynamic process that lead the players to the supreme-consistent value, we present alternative formulation for the supreme-consistent value in terms of excess as follows.

The set of preimputations of game (N, d, v) is denoted by X (N, d, v) = {x ∈ I R^N|∑_i∈Nx_i = v^* (N)}. If $x \in ℝ^{N}$ and S ⊆ N, we writex (S) = ∑_i∈Sx_i. The excess of a coalition S ⊆ N at x is the real number $e (S, v, x) = v^{*} (S) - x (S) .$ (4)

The excess function e (S, v, x) can be treated as the complaint of coalition S when all agents receive their payoffs from x in (N, d, v).

Lemma 5.1. Let (N, d, v) ∈ Γ and x ∈ X (N, d, v). Then $\sum_{S \subseteq N \ {i, j}} e (S \cup {i}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}}) = \sum_{S \subseteq N \ {i, j}} e (S \cup {j}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}})$ for all i, j ∈ N if and only if x = φ (N, d, v), where $a_{S} = \frac{| S |! (| N | - | S | - 2)!}{(| N | - 1)!}$ for all S ⊆ N.

Proof. Let (N, d, v) ∈ Γ and x ∈ X (N, d, v). For all i, j ∈ N,

$\begin{matrix} \sum_{S \subseteq N \ {i, j}} e (S \cup {i}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}}) \\ = \sum_{S \subseteq N \ {i, j}} e (S \cup {j}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}}) \\ \Leftrightarrow \\ \sum_{S \subseteq N \ {i, j}} [v^{*} (S \cup {i}) - \frac{x (S \cup {i})}{2^{| N | - 2} \cdot a_{S}}] \\ = \sum_{S \subseteq N \ {i, j}} [v^{*} (S \cup {j}) - \frac{x (S \cup {j})}{2^{| N | - 2} \cdot a_{S}}] \\ \Leftrightarrow \\ \sum_{S \subseteq N \ {i, j}} [v^{*} (S \cup {i}) - \frac{x_{i}}{2^{| N | - 2} \cdot a_{S}}] \\ = \sum_{S \subseteq N \ {i, j}} [v^{*} (S \cup {j}) - \frac{x_{j}}{2^{| N | - 2} \cdot a_{S}}] \\ \Leftrightarrow \\ x_{i} - x_{j} \\ = \sum_{S \subseteq N \ {i, j}} a_{S} [v^{*} (S \cup {i}) - v^{*} (S \cup {j})] . \end{matrix}$ (5)

By Definition 1,

$\begin{matrix} φ_{i} (N, d, v) - φ_{j} (N, d, v) \\ = \sum_{S \subseteq N \ {i, j}} a_{S} [v^{*} (S \cup {i}) - v^{*} (S \cup {j})] . \end{matrix}$ (6)

By Equations (5) and (6), $x_{i} - x_{j} = φ_{i} (N, d, v) - φ_{j} (N, d, v) .$

Thus, $\begin{matrix} | N | \cdot x_{i} - \sum_{j \in N} x_{j} = | N | \cdot φ_{i} (N, d, v) \\ - \sum_{j \in N} φ_{j} (N, d, v) . \end{matrix}$

Since x ∈ X (N, d, v) and φ satisfies EFF, $| N | \cdot [x_{i} - φ_{i} (N, d, v)] = v^{*} (N) - v^{*} (N) = 0 .$

Therefore, x_i = φ_i (N, d, v) for all i ∈ N. ■

By applying the excess formulation of the supreme-consistent value, we define a correction function as follows.

Definition 5.1. Let (N, d, v) ∈ Γ and i ∈ N. The correction function $f_{i} : X (N, d, v) \to ℝ$ is defined by $\begin{matrix} f_{i} (x) & = & x_{i} + t \sum_{j \in N \ {i}} \sum_{S \subseteq N \ {i, j}} \\ [e (S \cup {j}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}}) \\ - e (S \cup {i}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}})], \end{matrix}$ where $a_{S} = \frac{| S |! (| N | - | S | - 2)!}{(| N | - 1)!}$ for all S ⊆ N and t is a fixed positive number, which reflects the assumption that player i does not ask for full correction (when t = 1) but only (usually) a fraction of it. The number t represents how much the excess is corrected. When a player withdraws from the coalitions he/she/it joined, some complaints may be occurred from other players. The correction function is devoted to regulating these complaints to be more coincident among all players. Define f = (f_i) _i∈N and x⁰ = x, x¹ = f (x⁰) , ⋯ , x^q = f (x^q-1) for all $q \in ℕ \ {0}$ .

The following lemma shows that the correction function introduced in Definition 2 is also well-defined.

Lemma 5.2. For all (N, d, v) ∈ Γ and for all x ∈ X (N, d, v), f (x) ∈ X (N, d, v).

Proof. Let (N, d, v) ∈ Γ, i, j ∈ N and x ∈ X (N, d, v). Similar to Equations (5) and (6),

$\begin{matrix} \sum_{S \subseteq N \ {i, j}} [e (S \cup {j}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}}) \\ - e (S \cup {i}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}})] \\ = x_{j} - x_{i} + \sum_{S \subseteq N \ {i, j}} a_{S} [v^{*} (S \cup {i}) - v^{*} (S \cup {j})] \\ = x_{j} - x_{i} + φ_{i} (N, d, v) - φ_{j} (N, d, v) . \end{matrix}$ (7)

By Equation (7),

$\begin{matrix} \sum_{j \in N \ {i}} \sum_{S \subseteq N \ {i, j}} [e (S \cup {j}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}}) \\ - e (S \cup {i}, v, \frac{x}{2^{| N | - 2} \cdot a_{S}})] \\ = \sum_{j \in N \ {i}} [x_{j} - x_{i} + φ_{i} (N, d, v) - φ_{j} (N, d, v)] \\ = \sum_{j \in N \ {i}} [x_{j} - φ_{j} (N, d, v)] \\ - \sum_{j \in N \ {i}} [φ_{i} (N, d, v) - x_{i}] \\ = [v^{*} (N) - v^{*} (N)] - | N | \cdot [x_{i} - φ_{i} (N, d, v)] \\ (by EFF of φ, x \in X (N, d, v)) \\ = | N | \cdot [φ_{i} (N, d, v) - x_{i}] . \end{matrix}$ (8) By Definition 2 and Equation (8),

$\begin{matrix} \sum_{i \in N} f_{i} (x) \\ = \sum_{i \in N} [x_{i} + t \cdot | N | \cdot [φ_{i} (N, d, v) - x_{i}]] \\ (by Equation (8)) \\ = v^{*} (N) + t \cdot | N | \sum_{i \in N} [φ_{i} (N, d, v) - x_{i}] \\ (by x \in X (N, d, v)) \\ = v^{*} (N) + t \cdot | N | [v^{*} (N) - v^{*} (N)] \\ (by EFF of φ and x \in X (N, d, v)) \\ = v^{*} (N) . \end{matrix}$ (9)

Hence, f (x) ∈ X (N, d, v) if x ∈ X (N, d, v). ■

In the following, we provided some discussions for the dynamic process. Let (N, d, v) ∈ Γ and x be an efficient payoff vector. By a process of induction we assume that the players have already agreed on the solution φ. Based on the statement in Definition 2, the correction function is devoted to regulating the complaints occurred from players. These modifications, done simultaneously by all 2-person coalitions, will lead to a new payoff vector x^* and the process will repeat. The hope is that it will converge and, moreover, converge to φ (N, d, v). Based on the correction function introduced in Definition 2, a dynamic process for the supreme-consistent value is also proposed.

Theorem 5.1. Let (N, d, v) ∈ Γ. If $0 < t < \frac{2}{| N |}$ , then ${x^{q}}_{q = 1}^{\infty}$ converges to φ (N, d, v) for all x ∈ X (N, d, v).

Proof. Let (N, d, v) ∈ Γ, i ∈ N and x ∈ X (N, d, v). By Definition 2 and Equation (8), f_i (x) - x_i = t · |N| · [φ_i (N, d, v) - x_i]. Hence, $\begin{matrix} φ_{i} (N, d, v) - f_{i} (x) \\ = φ_{i} (N, d, v) - x_{i} + x_{i} - f_{i} (x) \\ = φ_{i} (N, d, v) - x_{i} - t \cdot | N | \cdot [φ_{i} (N, d, v) - x_{i}] \\ = (1 - t \cdot | N |) [φ_{i} (N, d, v) - x_{i}] . \end{matrix}$

So, for all $q \in ℕ$ , $φ_{i} (N, d, v) - x_{i}^{q} = (1 - t \cdot | N |)^{q} [φ_{i} (N, d, v) - x_{i}] .$

If $0 < t < \frac{2}{| N |}$ , then -1 < (1 - t · |N|) <1 and ${x^{q}}_{q = 1}^{\infty}$ converges geometrically to φ (N, d, v). ■

Remark 5.1. Inspired by Maschler and Owen [11], Liao et al. [10] defined a correction function as follows. Define that $f_{i}^{M} (x) = x_{i} + t \sum_{j \in N \ {i}} [φ_{i} ({i, j}, d_{{i, j}}, v_{{i, j}, x}^{φ}) - x_{i}],$ for all (N, d, v) ∈ Γ, i ∈ N and x ∈ X (N, d, v), where $({i, j}, d_{{i, j}}, v_{{i, j}, x}^{φ})$ is the x-reduction which is defined by $v_{S, d}^{x} (α) = {\begin{matrix} v^{*} (N) - \sum_{i \in N \ S} x_{i}, & α = d_{S}, \\ v_{S, d}^{φ} (α), & otherwise . \end{matrix}$ for all α ∈ D^S. Define $f^{M} = (f_{i}^{M})_{i \in N}$ and y⁰ = x, y¹ = f^M (y⁰) , ⋯ , y^q = f^M (y^q-1) for all x ∈ X (N, d, v) and for all $q \in ℕ \ {0}$ .

Liao et al. [10] showed that if $0 < t < \frac{4}{| N |}$ , then ${y^{q}}_{q = 1}^{\infty}$ converges geometrically to φ (N, d, v) for all (N, d, v) ∈ Γ and for all x ∈ X (N, d, v). The only difference is that our correction function is based on “excess functions” and Liao et al.’s [10] correction function is based on “reduced games”.

6 Conclusions

In this paper, we build on the results proposed by Maschler and Owen [11] and Liao et al. [10] on fuzzy TU games. One should compare our results with these results of standard TU games.

We show that the axiomatic technique due to Maschler and Owen [11] could be applied in the framework of fuzzy TU games.

We propose the properties of the fuzzy individual rationality and the (weak) fuzzy inessential games to axiomatize the supreme-consistent value. These axiomatic results do not appear in axiomatizations of single-valued solutions for standard TU games.

Several differences among Liao et al.’s [10] work and ours are as follows.

In the framework of fuzzy TU games, we provide the properties of fuzzy symmetry, fuzzy covariance, fuzzy individual rationality and (weak) fuzzy inessential games to axiomatize the supreme-consistent value. These properties and related axiomatic results do not appear in Liao et al. [10].

Different the results proposed by Liao et al. [10], we propose alternative formulation and related dynamic result for the supreme-consistent value by applying excess functions.

These mentioned above raise two question on fuzzy TU games.

Whether some more solutions could be introduced in the framework of fuzzy TU games.

Whether alternative formulations and related results of other solutions could be described in the framework of fuzzy TU games.

To our knowledge, these issues are still open questions.

Footnotes

A fuzzy TU game, which is defined by Aubin [1, ], is a pair (N, v^a), where v^a is a mapping such that $v^{a} : [0, 1]^{N} \to ℝ$ and v^a (0_N) =0. In fact, (N, v^a) = (N, θ^N, v).

Here we consider bounded fuzzy TU games, defined as those games (N, d, v) such that, there exists $K_{v} \in ℝ$ such that v (α) ≤ K_v for all α ∈ D^N. We adopt it to ensure that v^* (S) is well-defined.

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