Power mensuration mechanism under fuzzy behavior and multicriteria situation

Abstract

By considering supreme-utilities among fuzzy level (decision) vectors, we introduce an index and its efficient extension to investigate power mensuration mechanism in the framework of multicriteria fuzzy transferable-utility (TU) games. We also adopt reduced games and related consistency to analyze the rationality for these two indexes. Further, alternative formulation for the efficient index is also proposed by applying excess function. Based on reduced game and excess function, we introduce different dynamic processes for the efficient index.

Keywords

Multicriteria fuzzy TU games supreme-utility reduced game excess function dynamic process

1 Introduction

In the framework of transferable-utility (TU) games, the power indexes have been defined to measure the political power of each agent of a voting system. A agent in a voting system is, e.g., a party in a parliament or a country in a confederation. Each agent will have a certain number of votes, and so its power will be different. Results of the power indexes may be found in, e.g., Dubey and Shapley [8], Haller [10], Lehrer [12], van den Brink and van der Laan [5] and so on.

The theory of fuzzy TU games commenced with the investigation of Aubin [1, 2] where the opinions of a fuzzy TU game and the fuzzy core are introduced. Many fuzzy solutions have been applied wildly, e.g., Borkotokey and Mesiar [4], Butnariu and Kroupa [6], Hwang [11], Li and Zhang [13], Meng and Zhang [15], Tijs et al. [25], and so on.

Banzhaf [3] introduced a power index in the context of voting games that was essentially identical to that given by Coleman [7]. This value was later on extended to arbitrary games by Owen [18, 19]. In this work, we focus on the Banzhaf-Coleman index. Briefly speaking, the Banzhaf-Coleman index is a mechanism that collects each agent’s average marginal contribution from all coalitions in which he/she/it has participated. Several fuzzy Banzhaf values have been proposed. Related results could be found in, e.g., Gallego et al. [9], Meng et al. [16], Tan et al. [24], and so on.

In the axiomatic formulation for allocation rules, consistency is an important property. Consistency states the independence of a value with respect to fixing some agents with their assigned payoffs. It asserts that the recommendation made for any problem should always agree with the recommendation made in the subproblem that appears when the payoffs of some agents are settled on. It has been introduced in different ways depending upon how the payoffs of the agents that "leave the bargaining" are defined. This property has been investigated in various problems by applying reduced games always. In addition to axiomatizations for allocation rules, dynamic processes can be described that lead the agents to that allocation, starting from an arbitrary efficient payoff vector. The foundation of a dynamic theory was laid by Stearns [22].

In various areas, from sciences to engineering, industry and the social sciences, researchers face an increasing need to consider multiple goals optimally in their decision processes. Related problems include analyzing distribution tradeoffs, selecting optimal decision or process designs, or any other application where you need an optimal solution with tradeoffs between two or more goals in a grand coalition. In many cases these real world optimal problems could be formulated as multicriteria mathematical optimization models. The solutions of such models requires appropriate techniques to provide optima results that - unlike traditional viewpoints or methods - adopt several properties of the goals into account. Here we would like to provide the necessary mathematical foundation of multicriteria optimal solutions to analyze problems with multiple goals. The above pre-existing statements raise one motivation:

whether the power indexes could be extended under fuzzy behavior and multicriteria situation simultaneously.

The paper is devoted to answer the question. Different from the framework of fuzzy TU games, we consider the framework of multicriteria fuzzy TU games in Section 2. A power index and its efficient extension, the fuzzy Banzhaf-Coleman index and the fuzzy efficient Banzhaf-Coleman index, are further proposed by applying supreme-utilities among fuzzy level vectors on multicriteria fuzzy TU games. In order to present the rationalities for these two indexes, we propose an extended reduction to characterize these two indexes in Section 3. In order to establish the dynamic processes of the fuzzy efficient Banzhaf-Coleman index, we present alternative formulation for the fuzzy efficient Banzhaf-Coleman index in terms of excess functions. In Section 4, we adopt reduction and excess function to show that the fuzzy efficient Banzhaf-Coleman index can be reached by agents who start from an arbitrary efficient payoff vector.

2 The fuzzy Banzhaf-Coleman index and its efficient extension

Let U be the universe of agents. For i ∈ U and b_i ∈ (0, 1], B_i = [0, b_i] could be treated as the level (decision) space of agent i and $B_{i}^{+} = (0, b_{i}]$ , where 0 denotes no participation. For N ⊆ U, N≠ ∅, let $B^{N} = \prod_{i \in N} B_{i}$ be the product set of the level (decision) spaces of all agents of N. A fuzzy coalition is a vector α ∈ B^N. The i-th coordinate α_i of α is called the participation level of player i in the fuzzy coalition α. A fuzzy coalition could be described as a collection of economic agents, i.e., players, who transfer fractions of their representation to a collective decision maker, the fuzzy coalition. The term fuzzy coalition also arises when the possibility of graduating the membership of a player in a coalition is considered. For all T ⊆ N, we define θ^T ∈ B^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 $ℝ^{N}$ . For $m \in ℕ$ , let 0_m be the zero vector in $ℝ^{m}$ and $ℕ_{m} = {1, 2, \dots, m}$ .

A fuzzy TU game 1 is a triple (N, b, v), where N is a non-empty and finite set of agents, $b = (b_{i})_{i \in N} \in \prod_{i \in N} B_{i}^{+}$ is the vector that presents the highest levels for each agent, and $v : B^{N} \to ℝ$ is a characteristic mapping with v (0_N) =0 which assigns to each α = (α_i) _i∈N ∈ B^N the worth that the agents can gain when each agent i participates at level α_i. Given a fuzzy TU game (N, b, v) and α ∈ B^N, we write A (α) = {i ∈ N| α_i ≠ 0} and α_T to be the restriction of α at T for each T ⊆ N. Further, we define $v_{*} (T) = sup_{α \in B^{N}} {v (α) | A (α) = T}$ is the supreme-utility 2 among all action vector α with A (α) = T. A multicriteria fuzzy TU game is a triple (N, b, V^m), where $m \in ℕ$ , $V^{m} = (v^{t})_{t \in ℕ_{m}}$ and (N, b, v^t) is a fuzzy TU game for all $t \in ℕ_{m}$ . Denote the collection of all multicriteria fuzzy TU games by Γ.

Let (N, b, V^m) ∈ Γ. A payoff vector of (N, b, V^m) is a vector $x = (x^{t})_{t \in ℕ_{m}}$ and $x^{t} = (x_{i}^{t})_{i \in N} \in ℝ^{N}$ , where $x_{i}^{t}$ denotes the payoff to agent i in (N, b, v^t) for all $t \in ℕ_{m}$ and for all i ∈ N. A payoff vector x of (N, b, V^m) is multicriteria efficient if $\sum_{i \in N} x_{i}^{t} = v_{*}^{t} (N)$ for all $t \in ℕ_{m}$ . The collection of all multicriteria efficient vector of (N, b, V^m) is denoted by E (N, b, V^m). A solution is a map σ assigning to each (N, b, V^m) ∈ Γ an element $σ (N, b, V^{m}) = (σ^{t} (N, b, V^{m}))_{t \in ℕ_{m}},$ where $σ^{t} (N, b, V^{m}) = (σ_{i}^{t} (N, b, V^{m}))_{i \in N} \in ℝ^{N}$ and $σ_{i}^{t} (N, b, V^{m})$ is the payoff of the agent i assigned by σ in (N, b, v^t).

Next, we provide the fuzzy Banzhaf-Coleman index and the fuzzy efficient Banzhaf-Coleman index under multicriteria situation.

Definition 1. The fuzzy Banzhaf-Coleman index (FBCI), β, is defined by $β_{i}^{t} (N, b, V^{m}) = \frac{1}{2^{| N | - 1}} \sum_{\overset{S \subseteq N}{i \in S}} [v_{*}^{t} (S) - v_{*}^{t} (S \ {i})]$ for all (N, b, V^m) ∈ Γ, for all $t \in ℕ_{m}$ and for all i ∈ N. Under the solution β, all agents receive their average marginal contributions related to supreme-utilities in each S ⊆ N respectively.

A solution σ satisfies multicriteria efficiency (MEFF) if for all (N, b, V^m) ∈ Γ and for all $t \in ℕ_{m}$ , $\sum_{i \in N} σ_{i} (N, b, V^{m}) = v_{*}^{t} (N)$ . Property MEFF asserts that all agents allocate all the utility completely. It is easy to check that the FBCI violates MEFF. Therefore, we consider an efficient normalization as follows.

Definition 2. The fuzzy efficient Banzhaf-Coleman index (FEBCI), $\bar{β}$ , is defined by $\begin{matrix} \bar{β_{i}^{t}} (N, b, V^{m}) \\ = & β_{i}^{t} (N, b, V^{m}) + \frac{1}{| N |} \cdot [v_{*}^{t} (N) - \sum_{k \in N} β_{k}^{t} (N, b, V^{m})] \end{matrix}$ for all (N, b, V^m) ∈ Γ, for all $t \in ℕ_{m}$ and for all i ∈ N.

Lemma 1. The FEBCI satisfies MEFF on Γ.

Proof. For all (N, b, V^m) ∈ Γ and for all $t \in ℕ_{m}$ , $\begin{matrix} \sum_{i \in N} \bar{β_{i}^{t}} (N, b, V^{m}) \\ = & \sum_{i \in N} [β_{i}^{t} (N, b, V^{m}) + \frac{1}{| N |} \cdot [v_{*}^{t} (N) \\ - \sum_{k \in N} β_{k}^{t} (N, b, V^{m})]] \end{matrix}$ $\begin{matrix} = & \sum_{i \in N} β_{i}^{t} (N, b, V^{m}) + \frac{| N |}{| N |} \cdot [v_{*}^{t} (N) \\ - \sum_{k \in N} β_{k}^{t} (N, b, V^{m})] \\ = & \sum_{i \in N} β_{i}^{t} (N, b, V^{m}) + v_{*}^{t} (N) - \sum_{k \in N} β_{k}^{t} (N, b, V^{m}) \\ = & v_{*}^{t} (N) . \end{matrix}$

Thus, the FEBCI satisfies MEFF on Γ. ■

As we mention in Introduction, multicriteria analysis (also known as multi-objective analysis, multiattribute analysis, and so on) is an area of multiple criteria analysis that is concerned with problems involving more than one goal to be optimized simultaneously. Multicriteria analysis has been applied in many fields, including economics, politics, engineering, logistics where optimal decisions need to be adopted in the presence of trade-offsbetween two or more goals in a grand coalition. For example, minimizing cost while maximizing comfort while buying a heater, and maximizing efficiency whilst minimizing energies consumption and emission of pollutants are examples of multicriteria optimal problems involving two and three goals respectively. In various problems, there can be more than three goals. On the other hand, each agent could be allowed to participate with infinite various actions in real situations respectively. Therefore, we consider the framework of multicriteria fuzzy TU games in this paper.

Here we provide a brief application of multicriteria fuzzy TU games in the setting of “management". This kind of problem can be formulated as follows. Let N = {1, 2, ⋯ , n} be a set of all agents of a grand management system (N, b, V^m). The function v^t could be treated as an utility function which assigns to each level vector α = (α_i) _i∈N ∈ B^N the worth that the agents can obtain when each agent i participates at operation strategy α_i ∈ B_i in the sub-management system (N, b, v^t). Modeled in this way, the grand management system (N, b, V^m) could be considered as a multicriteria fuzzy TU game, with v^t being each characteristic function and B_i being the set of all operation strategies of the agent i. In the following sections, we would like to show that the FBCI and the FEBCI could provide “optimal allocation mechanisms" among all agents, in the sense that this organization can get payoff from each combination of operation strategies of all agents under fuzzy behavior and multicriteria situation.

3 Axiomatizations

In this section, we show that there exists corresponding reduced games that could be adopted to analyze the FBCI and the FEBCI.

Subsequently, we introduced reduced game and related consistency as follows. Let ψ be a solution, (N, b, V^m) ∈ Γ and S ⊆ N. The 1-reduced game $(S, b_{S}, V_{S, ψ}^{1, m})$ is defined by $V_{S, ψ}^{1, m} = (v_{S, ψ}^{1, t})_{t \in ℕ_{m}}$ and $\begin{matrix} v_{S, ψ}^{1, t} (α) = \\ {\begin{matrix} 0 & , α = 0_{S}, \\ \frac{1}{2^{| N \ S |}} \sum_{Q \subseteq N \ S} [v_{*}^{t} (A (α) \cup Q) \\ - \sum_{i \in Q} ψ_{i}^{t} (N, b, V^{m})] & , otherwise . \end{matrix} \end{matrix}$ ψ satisfies 1-consistency (1CON) if $ψ_{i}^{t} (S, b_{S}, V_{S, ψ}^{1, m}) = ψ_{i}^{t} (N, b, V^{m})$ for all (N, b, V^m) ∈ Γ, for all S ⊆ N with |S|=2, for all $t \in ℕ_{m}$ and for all i ∈ S. Further, ψ satisfies 1-standard for games (1SG) if ψ (N, b, V^m) = β (N, b, V^m) for all (N, b, V^m) ∈ Γ with |N|≤2. Next, we characterize the FBCI by applying the properties of 1CON and 1SG.

Lemma 2.

The FBCI satisfies 1CON on Γ.

On Γ, the FBCI is the only solution satisfying 1SG and 1CON.

Proof. To prove result 1, let (N, b, V^m) ∈ Γ and S ⊆ N. If |N|=1, then the proof is completed. Assume that |N|≥2 and S = {i, j} for some i, j ∈ N. For all $t \in ℕ_{m}$ and for all i ∈ S, $\begin{matrix} β_{i}^{t} (S, b_{S}, V_{S, β}^{1, m}) \\ = & \frac{1}{2^{| S | - 1}} \cdot \sum_{\overset{T \subseteq S}{i \in T}} [(v_{S, β}^{1, t})_{*} (T) - (v_{S, β}^{1, t})_{*} (T \ {i})] \\ = & \frac{1}{2^{| S | - 1} \cdot 2^{| N \ S |}} \cdot \sum_{\overset{T \subseteq S}{i \in T}} \sum_{Q \subseteq N \ S} [v_{*}^{t} (T \cup Q) \\ - v_{*}^{t} (T \ {i} \cup Q)] \\ = & \frac{1}{2^{| N | - 1}} \cdot \sum_{\overset{K \subseteq N}{i \in K}} [v (K) - v (K \ {i})] \\ = & β_{i}^{t} (N, b, V^{m}) . \end{matrix}$

Hence, β satisfies 1CON.

By result 1, β satisfies 1CON on Γ. Clearly, β satisfies 1SG on Γ. To prove uniqueness of result 2, suppose ψ satisfies 1SG and 1CON. Let (N, b, v) ∈ Γ. If |N|≤2, then ψ (N, b, v) = β (N, b, v) by 1SG. The case |N|>2: Let i ∈ N and $t \in ℕ_{m}$ , and let S ⊂ N with |S|=2 and i ∈ S. Then, $\begin{matrix} ψ_{i}^{t} (N, b, V^{m}) & = ψ_{i}^{t} (S, b_{S}, V_{S, ψ}^{1, m}) (by 1 CON) \\ = β_{i}^{t} (S, b_{S}, V_{S, ψ}^{1, m}) (by 1 SG) \\ = \frac{1}{2^{| S | - 1}} \cdot \sum_{\overset{T \subseteq S}{i \in T}} [(v_{S, ψ}^{1, t})_{*} (T) \\ - (v_{S, ψ}^{1, t})_{*} (T \ {i})] \\ = \frac{1}{2^{| S | - 1} \cdot 2^{| N \ S |}} \sum_{\overset{T \subseteq S}{i \in T}} \sum_{Q \subseteq N \ S} [v_{*}^{t} (T \cup Q) \\ - v_{*}^{t} ((T \ {i}) \cup Q)] \\ = \frac{1}{2^{| N | - 1}} \cdot \sum_{\overset{K \subseteq N}{i \in K}} v_{*}^{t} (K) - v_{*}^{t} (K \ {i}) \\ = β_{i}^{t} (N, b, V^{m}) . \end{matrix}$

Thus, ψ (N, b, V^m) = β (N, b, V^m). ■

The following examples are to show that each of the axioms adopted in Lemma 2 is logically independent of the remaining axioms.

Example 1. Define a solution ψ by for all (N, b, V^m) ∈ Γ, for all $t \in ℕ_{m}$ and for all i ∈ N, $ψ_{i}^{t} (N, b, V^{m}) = 0$ . Clearly, ψ satisfies 1CON, but it violates 1SG.

Example 2. Define a solution ψ by for all (N, b, V^m) ∈ Γ, for all $t \in ℕ_{m}$ and for all i ∈ N, $ψ_{i}^{t} (N, b, V^{m}) = {\begin{matrix} β_{i}^{t} (N, b, V^{m}) & if | N | \leq 2, \\ 0 & otherwise . \end{matrix}$

On Γ, ψ satisfies 1SG, but it violates 1CON.

Unfortunately, it is easy to check that the index $\bar{β}$ violates 1CON. Therefore, we consider the 2-reduced game as follows. Let ψ be a solution, (N, b, V^m) ∈ Γ and S ⊆ N. The 2-reduced game $(S, b_{S}, V_{S, ψ}^{2, m})$ is defined by $V_{S, ψ}^{2, m} = (v_{S, ψ}^{2, t})_{t \in ℕ_{m}}$ and $\begin{matrix} v_{S, ψ}^{2, t} (α) = \\ {\begin{matrix} 0 & , α = 0_{S}, \\ v_{*}^{t} (N) - \sum_{i \in N \ S} ψ_{i}^{t} (N, b, V^{m}) & , A (α) = S, \\ \frac{1}{2^{| N \ S |}} \sum_{Q \subseteq N \ S} [v_{*}^{t} (A (α) \cup Q) \\ - \sum_{i \in Q} ψ_{i}^{t} (N, b, V^{m})] & , otherwise . \end{matrix} \end{matrix}$ ψ satisfies 2-consistency (2CON) if $ψ_{i}^{t} (S, b_{S}, V_{S, ψ}^{2, m}) = ψ_{i}^{t} (N, b, V^{m})$ for all (N, b, V^m) ∈ Γ, for all S ⊆ N with |S|=2, for all $t \in ℕ_{m}$ and for all i ∈ S. Further, ψ satisfies 2-standard for games (2SG) if $ψ (N, b, V^{m}) = \bar{β} (N, b, V^{m})$ for all (N, b, V^m) ∈ Γ with |N|≤2. Next, we characterize the FEBCI by applying the properties of 2CON and 2SG.

In order to establish consistency of the FEBCI, it will be useful to present alternative formulation for the FEBCI in terms of excess. Let (N, b, V^m) ∈ Γ, S ⊆ N and x be a payoff vector in (N, b, V^m). Define that $x^{t} (S) = \sum_{i \in S} x_{i}^{t}$ for all $t \in ℕ_{m}$ . The excess of a coalition S ⊆ N at x is the real number $\begin{matrix} e (S, V^{m}, x) = (e (S, v^{t}, x^{t}))_{t \in ℕ_{m}}, & and \\ e (S, v^{t}, x^{t}) = v_{*}^{t} (S) - x^{t} (S) . \end{matrix}$ (1)

The value e (S, v^t, x^t) can be treated as the complaint of coalition S when all agents receive their payoffs from x^t in (N, b, v^t).

Lemma 3. Let (N, b, V^m) ∈ Γ and x ∈ E (N, b, V^m). Then for all i, j ∈ N and $t \in ℕ_{m}$ $\begin{matrix} \sum_{S \subseteq N \ {i, j}} e (S \cup {i}, v^{t}, x^{t}) \\ = \sum_{S \subseteq N \ {i, j}} e (S \cup {j}, v^{t}, x^{t}) \\ \Leftrightarrow & x = \bar{β} (N, b, V^{m}) . \end{matrix}$

Proof. Let (N, b, V^m) ∈ Γ and x ∈ E (N, b, V^m). For all $t \in ℕ_{m}$ and for all i, j ∈ N, $\begin{matrix} \sum_{S \subseteq N \ {i, j}} e (S \cup {i}, v^{t}, x^{t}) \\ = \sum_{S \subseteq N \ {i, j}} e (S \cup {j}, v^{t}, x^{t}) \\ \Leftrightarrow \end{matrix}$ $\begin{matrix} \sum_{S \subseteq N \ {i, j}} [v_{*}^{t} (S \cup {i}) - x^{t} (S \cup {i})] \\ = \sum_{S \subseteq N \ {i, j}} [v_{*}^{t} (S \cup {j}) - x^{t} (S \cup {j})] \\ \Leftrightarrow \\ [\sum_{S \subseteq N \ {i, j}} v_{*}^{t} (S \cup {i})] - 2^{| N | - 2} \cdot x_{i}^{t} \\ = [\sum_{S \subseteq N \ {i, j}} v_{*}^{t} (S \cup {j})] - 2^{| N | - 2} \cdot x_{j}^{t} \\ \Leftrightarrow \\ x_{i}^{t} - x_{j}^{t} \\ = \frac{1}{2^{| N | - 2}} \sum_{S \subseteq N \ {i, j}} [v_{*}^{t} (S \cup {i}) - v_{*}^{t} (S \cup {j})] . \end{matrix}$ (2)

By definition of $\bar{β}$ , $\begin{matrix} \bar{β_{i}^{t}} (N, b, V^{m}) - \bar{β_{j}^{t}} (N, b, V^{m}) \\ = & \frac{1}{2^{| N | - 2}} \sum_{S \subseteq N \ {i, j}} [v_{*}^{t} (S \cup {i}) - v_{*}^{t} (S \cup {j})] \end{matrix}$ (3)

By Equations (2) and (3), for all i, j ∈ N, $x_{i}^{t} - x_{j}^{t} = \bar{β_{i}^{t}} (N, b, V^{m}) - \bar{β_{j}^{t}} (N, b, V^{m}) .$

Hence, $\sum_{j \in N} [x_{i}^{t} - x_{j}^{t}] = \sum_{j \in N} [\bar{β_{i}^{t}} (N, b, V^{m}) - \bar{β_{j}^{t}} (N, b, V^{m})] .$

That is, $| N | \cdot x_{i}^{t} - \sum_{j \in N} x_{j}^{t} = | N | \cdot \bar{β_{i}^{t}} (N, b,$ $V^{m}) - \sum_{j \in N} \bar{β_{j}^{t}} (N, b, V^{m})$ . Since x ∈ E (N, b, V^m) and $\bar{β}$ satisfies MEFF, $| N | \cdot x_{i}^{t} - v_{*}^{t} (N) = | N | \cdot \bar{β_{i}^{t}} (N, b, V^{m}) - v_{*}^{t} (N)$ . Therefore, $x_{i}^{t} = \bar{β_{i}^{t}} (N, b, V^{m})$ for all $t \in ℕ_{m}$ and for all i ∈ N, i.e., $x = \bar{β} (N, b, V^{m})$ . ■

Remark 1. It is easy to check that $\begin{matrix} \sum_{S \subseteq N \ {i, j}} e (S \ {i}, V^{m}, β (N, b, V^{m})) \\ = & \sum_{S \subseteq N \ {i, j}} e (S \ {j}, V^{m}, β (N, b, V^{m})) \end{matrix}$ for all (N, b, V^m) ∈ Γ and for all i, j ∈ N.

Theorem 1.

The FEBCI satisfies 2CON on Γ.

If ψ satisfies 2SG and 2CON, then it also satisfies MEFF.

On Γ, the FEBCI is the only solution satisfying 2SG and 2CON.

Proof. To verify result 1, let (N, b, V^m) ∈ Γ and S ⊆ N. If |N|=1, then the proof is completed. Assume that|N|≥2, $x = \bar{β} (N, b, V^{m})$ and S = {i, j} for some i, j ∈ N. For all $t \in ℕ_{m}$ and for all l ∈ S, $\begin{matrix} e ({l}, v_{S, x}^{2, t}, x_{S}^{t}) \\ = & v_{S, x}^{2, t} ({l}) - x_{l}^{t} \\ = & (\frac{1}{2^{| N \ S |}} \cdot \sum_{Q \subseteq N \ S} [v_{*}^{t} ({l} \cup Q) - \sum_{k \in Q} x_{k}^{t}]) - x_{l}^{t} \\ = & \frac{1}{2^{| N \ S |}} \cdot \sum_{Q \subseteq N \ S} (v_{*}^{t} ({l} \cup Q) - \sum_{k \in Q} x_{k}^{t} - x_{l}^{t}) \\ = & \frac{1}{2^{| N \ S |}} \cdot \sum_{Q \subseteq N \ S} [v_{*}^{t} ({l} \cup Q) - \sum_{k \in {l} \cup Q} x_{k}^{t}] \\ = & \frac{1}{2^{| N \ S |}} \cdot \sum_{Q \subseteq N \ {i, j}} e (Q \cup {l}, v^{t}, x^{t}) . \end{matrix}$ (4)

Since $\bar{β}$ satisfies MEFF, $x_{S}^{t} \in E (S, b_{S}, v_{S, x}^{2, t})$ by definition of 2-reduced game. In addition, by equation (4) and Lemma 3, $\begin{matrix} e ({i}, v_{S, x}^{2, t}, x_{S}^{t}) \\ = & \frac{1}{2^{| N \ S |}} \cdot \sum_{Q \subseteq N \ {i, j}} e (Q \cup {i}, v^{t}, x^{t}) & (by equation (4)) \\ = & \frac{1}{2^{| N \ S |}} \cdot \sum_{Q \subseteq N \ {i, j}} e (Q \cup {j}, v^{t}, x^{t}) & (by Lemma 3) \\ = & e ({j}, v_{S, x}^{2, t}, x_{S}^{t}) . & (by equation (4)) \end{matrix}$

So, $x_{S} = \bar{β} (S, b_{S}, V_{S, \bar{β}}^{2, m})$ . That is, $\bar{β}$ satisfies 2CON.

To prove result 2, suppose ψ satisfies 2SG and 2CON. Let (N, b, V^m) ∈ Γ and $t \in ℕ_{m}$ . If |N|≤2, it is trivial that ψ satisfies MEFF by 2SG. The case |N|>2: Assume, on the contrary, that there exists (N, b, V^m) ∈ Γ such that $\sum_{i \in N} ψ_{i}^{t} (N, b, V^{m}) \neq v_{*}^{t} (N)$ . This means that there exist i ∈ N and j ∈ N such that $[v_{*}^{t} (N) - \sum_{k \in N \ {i, j}} ψ_{k}^{t} (N, b, V^{m})] \neq [ψ_{i}^{t} (N, b, V^{m}) + ψ_{j}^{t}$ (N, b, V^m)]. By 2CON and ψ satisfies MEFF for two-person games, this contradicts with $\begin{matrix} ψ_{i}^{t} (N, b, V^{m}) + ψ_{j}^{t} (N, b, V^{m}) \\ = & ψ_{i}^{t} ({i, j}, b_{{i, j}}, v_{{i, j}, ψ}^{2, t}) + ψ_{j}^{t} ({i, j}, b_{{i, j}}, v_{{i, j}, ψ}^{2, t}) \\ = & v_{*}^{t} (N) - \sum_{k \in N \ {i, j}} ψ_{k}^{t} (N, b, V^{m}) . \end{matrix}$

Hence ψ satisfies MEFF.

To prove result 3, $\bar{β}$ satisfies 2CON by result 1. Clearly, $\bar{β}$ satisfies 2SG. To prove uniqueness, suppose ψ satisfies 2SG and 2CON, hence by result 2, ψ also satisfies MEFF. Let (N, b, V^m) ∈ Γ. If |N|≤2, it is trivial that $ψ (N, b, V^{m}) = \bar{β} (N, b, V^{m})$ by 2SG. The case |N|>2: Let i ∈ N, $t \in ℕ_{m}$ and S = {i, j} for some j ∈ N \ {i}. Then $\begin{matrix} ψ_{i}^{t} (N, b, V^{m}) - \bar{β_{i}^{t}} (N, b, V^{m}) \\ = & ψ_{i}^{t} (S, b_{S}, V_{S, ψ}^{2, m}) - \bar{β_{i}^{t}} (S, b_{S}, V_{S, \bar{β}}^{2, m}) \\ (by 2 CON of ψ, \bar{β}) \\ = & \bar{β_{i}^{t}} (S, b_{S}, V_{S, ψ}^{2, m}) - \bar{β_{i}^{t}} (S, b_{S}, V_{S, \bar{β}}^{2, m}) \\ (by 2 SG of ψ, \bar{β}) \\ = & \frac{1}{2} \cdot [(v_{S, ψ}^{2, t})_{*} (S) + (v_{S, ψ}^{2, t})_{*} ({i}) - (v_{S, ψ}^{2, t})_{*} ({j})] \\ - \frac{1}{2} \cdot [(v_{S, \bar{β}}^{2, t})_{*} (S) + (v_{S, \bar{β}}^{2, t})_{*} ({i}) - (v_{S, \bar{β}}^{2, t})_{*} ({j})] . \end{matrix}$ (5)

By definitions of $v_{S, ψ}^{2, t}$ and $v_{S, \bar{β}}^{2, t}$ , $\begin{matrix} (v_{S, ψ}^{2, t})_{*} ({i}) - (v_{S, ψ}^{2, t})_{*} ({j}) \\ = & \frac{1}{2^{| N \ S |}} \sum_{Q \subseteq N \ S} [v_{*}^{t} ({i} \cup Q) - v_{*}^{t} ({j} \cup Q)] \\ = & (v_{S, \bar{β}}^{2, t})_{*} ({i}) - (v_{S, \bar{β}}^{2, t})_{*} ({j}) . \end{matrix}$ (6)

By equation (6), equation (5) becomes $\begin{matrix} ψ_{i}^{t} (N, b, V^{m}) - \bar{β_{i}^{t}} (N, b, V^{m}) \\ = & \frac{1}{2} \cdot ((v_{S, ψ}^{2, t})_{*} (S) - (v_{S, \bar{β^{t}}}^{2, t})_{*} (S)) \\ = & \frac{1}{2} \cdot (ψ_{i}^{t} (N, b, V^{m}) + ψ_{j}^{t} (N, b, V^{m}) - \bar{β_{i}^{t}} (N, b, V^{m}) \\ + \bar{β_{j}^{t}} (N, b, V^{m})) . \end{matrix}$

That is, for all i, j ∈ N, $\begin{matrix} ψ_{i}^{t} (N, b, V^{m}) - ψ_{j}^{t} (N, b, V^{m}) \\ = & \bar{β_{i}^{t}} (N, b, V^{m}) - \bar{β_{j}^{t}} (N, b, V^{m}) . \end{matrix}$

By MEFF of ψ and $\bar{β}$ , $\begin{matrix} | N | \cdot ψ_{i}^{t} (N, b, V^{m}) - v_{*}^{t} (N) \\ = & | N | \cdot \bar{β_{i}^{t}} (N, b, V^{m}) - v_{*}^{t} (N) . \end{matrix}$

Thus, $ψ_{i}^{t} (N, b, V^{m}) = \bar{β_{i}^{t}} (N, b, V^{m})$ for all i ∈ N, i.e., $ψ (N, b, V^{m}) = \bar{β} (N, b, V^{m})$ . ■

The following examples are to show that each of the axioms adopted in Theorem 1 is logically independent of the remaining axioms.

Example 3. Define a solution ψ by for all (N, b, V^m) ∈ Γ, for all $t \in ℕ_{m}$ and for all i ∈ N, $ψ_{i}^{t} (N, b, V^{m}) = 0$ . Clearly, ψ satisfies 2CON, but it violates 2SG.

Example 4. Define a solution ψ by for all (N, b, V^m) ∈ Γ, for all $t \in ℕ_{m}$ and for all i ∈ N, $ψ_{i}^{t} (N, b, V^{m}) = {\begin{matrix} \bar{β_{i}^{t}} (N, b, V^{m}) & , | N | \leq 2, \\ 0 & , otherwise . \end{matrix}$ On Γ, ψ satisfies 2SG, but it violates 2CON.

4 Dynamic processes

In this section, we adopt excess function and reduction to propose dynamic processes for the FEBCI.

In order to establish the dynamic processes of the FEBCI, we firstly define correction function by means of excess functions. The correction function is based on the idea that, each agent shortens the complaint relating to his own and others’ non-participation, and adopts these regulations to correct the original payoff.

Definition 3. Let (N, b, V^m) ∈ Γ and i ∈ N. The correction function is defined to be $f = (f^{t})_{t \in ℕ_{m}}$ , where $f^{t} = (f_{i}^{t})_{i \in N}$ and $f_{i}^{t} : E (N, b, V^{m}) \to ℝ$ is define by $\begin{matrix} f_{i}^{t} (x) \\ = & x_{i}^{t} + w \sum_{j \in N \ {i}} \frac{1}{2^{| N | - 2}} \sum_{Q \subseteq N \ {i, j}} [e (Q \ {j}, v^{t}, x^{t}) \\ - e (Q \ {i}, v^{t}, x^{t})], \end{matrix}$ where $w \in ℝ$ is a fixed postive number, which reflects the assumption that agent i does not ask for full correction (when w = 1) but only (usually) a fraction of it. Define [x] ⁰ = x, [x] ¹ = f ([x] ⁰) , ⋯ , [x] ^q = f ([x] ^q-1) for all $q \in ℕ$ .

Lemma 4. f (x) ∈ E (N, b, V^m) for all (N, b, V^m) ∈ Γ and for all x ∈ E (N, b, V^m).

Proof. Let (N, b, V^m) ∈ Γ, $t \in ℕ_{m}$ , i, j ∈ N and x ∈ E (N, b, V^m). Similar to equation (2), $\begin{matrix} \sum_{j \in N \ {i}} \frac{1}{2^{| N | - 2}} \sum_{Q \subseteq N \ {i, j}} [e (Q \ {j}, v^{t}, x^{t}) \\ - e (Q \ {i}, v^{t}, x^{t})] \\ = & \sum_{j \in N \ {i}} [(\frac{1}{2^{| N | - 2}} \sum_{Q \subseteq N \ {i, j}} [v^{t} (Q \ {j}) \\ - v^{t} (Q \ {i})]) - x_{i}^{t} + x_{j}^{t}] . \end{matrix}$ (7)

By definition of $\bar{β}$ , $\begin{matrix} \bar{β_{i}^{t}} (N, b, V^{m}) - \bar{β_{j}^{t}} (N, b, V^{m}) \\ = & \frac{1}{2^{| N | - 2}} \sum_{Q \subseteq N \ {i, j}} [v^{t} (Q \ {j}) - v^{t} (Q \ {i})] . \end{matrix}$ (8)

By Equations (7) and (8), $\begin{matrix} \sum_{j \in N \ {i}} \frac{1}{2^{| N | - 2}} \sum_{Q \subseteq N \ {i, j}} [e (Q \ {j}, v^{t}, x^{t}) \\ - e (Q \ {i}, v^{t}, x^{t})] \\ = \sum_{j \in N \ {i}} (\bar{β_{i}^{t}} (N, b, V^{m}) - \bar{β_{j}^{t}} (N, b, V^{m}) - x_{i}^{t} + x_{j}^{t}) \\ = ((| N | - 1) (\bar{β_{i}^{t}} (N, b, V^{m}) \\ - x_{i}^{t}) - \sum_{j \in N \ {i}} \bar{β_{j}^{t}} (N, b, V^{m}) + \sum_{j \in N \ {i}} x_{j}^{t}) \\ = (| N | (\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}) - v_{*}^{t} (N) + v_{*}^{t} (N)) \\ (by MEFF of \bar{β}, x \in E (N, b, V^{m})) \\ = | N | \cdot (\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}) . \end{matrix}$ (9)

Moreover, $\begin{matrix} \sum_{i \in N} \sum_{j \in N \ {i}} \frac{1}{2^{| N | - 2}} \sum_{Q \subseteq N \ {i, j}} [e (Q \ {j}, v^{t}, x^{t}) \\ - e (Q \ {i}, v^{t}, x^{t})] \\ = \sum_{i \in N} | N | \cdot (\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}) \\ = | N | \cdot (\sum_{i \in N} \bar{β_{i}^{t}} (N, b, V^{m}) - \sum_{i \in N} x_{i}^{t}) \\ = | N | \cdot (v_{*}^{t} (N) - v_{*}^{t} (N)) \\ (by MEFF of \bar{β}, x \in E (N, b, V^{m})) \\ = 0 . \end{matrix}$ (10)

So we have that $\begin{matrix} \sum_{i \in N} f_{i}^{t} (x) \\ = & \sum_{i \in N} [x_{i}^{t} + w \sum_{j \in N \ {i}} \frac{1}{2^{| N | - 2}} \\ \cdot \sum_{Q \subseteq N \ {i, j}} [e (Q \ {j}, v^{t}, x^{t}) - e (Q \ {i}, v^{t}, x^{t})]] \\ = & \sum_{i \in N} x_{i}^{t} + w \sum_{i \in N} \sum_{j \in N \ {i}} \frac{1}{2^{| N | - 2}} \\ \cdot \sum_{Q \subseteq N \ {i, j}} [e (Q \ {j}, v^{t}, x^{t}) - e (Q \ {i}, v^{t}, x^{t})] \\ = & v_{*}^{t} (N) + 0 \\ (by equation (10) and x \in E (N, b, V^{m})) \\ = & v_{*}^{t} (N) . \end{matrix}$ Hence, f (x) ∈ E (N, b, V^m) if x ∈ E (N, b, V^m). ■

Theorem 2. Let (N, b, V^m) ∈ Γ. If $0 < t < \frac{2}{| N |}$ , then ${[x]^{q}}_{q = 1}^{\infty}$ converges geometrically to $\bar{β} (N, b, V^{m})$ for each x ∈ E (N, b, V^m).

Proof. Let (N, b, V^m) ∈ Γ, $t \in ℕ_{m}$ , i ∈ N and x ∈ E (N, b, V^m). By equation (9) and definition of f, $\begin{matrix} f_{i}^{t} (x) - x_{i}^{t} \\ = & w \sum_{j \in N \ {i}} \frac{1}{2^{| N | - 2}} \sum_{Q \subseteq N \ {i, j}} [e (Q \ {j}, v^{t}, x^{t}) \\ - e (Q \ {i}, v^{t}, x^{t})] \\ = & w \cdot | N | \cdot (\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}) . \end{matrix}$

Hence, $\begin{matrix} \bar{β_{i}^{t}} (N, b, V^{m}) - f_{i}^{t} (x) \\ = & \bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t} + x_{i}^{t} - f_{i}^{t} (x) \\ = & \bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t} - w \cdot | N | \cdot (\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}) \\ = & (1 - w \cdot | N |) [\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}] . \end{matrix}$

So, for all $q \in ℕ$ , $\bar{β} (N, b, V^{m}) - [x]^{q} = (1 - w \cdot | N |)^{q} [\bar{β} (N, b, V^{m}) - x] .$

If $0 < w < \frac{2}{| N |}$ , then -1 < (1 - w · |N|) < 1 and ${[x]^{q}}_{q = 1}^{\infty}$ converges geometrically to $\bar{β} (N, b, V^{m})$ . ■

Inspired by Maschler and Owen [14], we will find a dynamic process under reductions.

Definition 4. Let ψ be a solution, (N, b, V^m) ∈ Γ, S ⊆ N and x ∈ E (N, b, V^m). The (x, ψ)-reduced game $(S, b_{S}, V_{ψ, S, x}^{r, m})$ is given by $V_{ψ, S, x}^{r, m} = (v_{ψ, S, x}^{r, t})_{t \in ℕ_{m}}$ and for all T ⊆ S, $v_{ψ, S, x}^{r, t} (α) = {\begin{matrix} v_{*}^{t} (N) - \sum_{i \in N \ S} x_{i}^{t} & A (α) = S, \\ v_{S, ψ}^{2, t} (α) & otherwise . \end{matrix}$

Inspired by Maschler and Owen [14], we also define different correction function as follow. The R-correction function to be $g = (g^{t})_{t \in ℕ_{m}}$ , where $g^{t} = (g_{i}^{t})_{i \in N}$ and $g_{i}^{t} : E (N, b, V^{m}) \to ℝ$ is define by $g_{i}^{t} (x) = x_{i}^{t} + w \sum_{k \in N \ {i}} (\bar{β_{i}^{t}} ({i, k}, v_{\bar{β}, {i, k}, x}^{t}) - x_{i}^{t}) .$ Define [η] ⁰ = x, [η] ¹ = g ([η] ⁰) , ⋯ , [η] ^q = g ([η] ^q-1) for all $q \in ℕ$ .

Lemma 5. g (x) ∈ E (N, b, V^m) for all (N, b, V^m) ∈ Γ and for all x ∈ E (N, b, V^m).

Proof. Let (N, b, V^m) ∈ Γ, $t \in ℕ_{m}$ , i, k ∈ N and x ∈ E (N, b, V^m). Let S = {i, k}, by MEFF of $\bar{β}$ and Definition 4, $\bar{β_{i}^{t}} (S, b_{S}, V_{\bar{β}, S, x}^{r, m}) + \bar{β_{k}^{t}} (S, b_{S}, V_{\bar{β}, S, x}^{r, m}) = x_{i}^{t} + x_{k}^{t} .$

By 2CON and 2SG of $\bar{β}$ , $\begin{matrix} \bar{β_{i}^{t}} (S, b_{S}, V_{\bar{β}, S, x}^{r, m}) - \bar{β_{k}^{t}} (S, b_{S}, V_{\bar{β}, S, x}^{r, m}) \\ = & (v_{\bar{β}, S, x}^{r, t})_{*} ({i}) - (v_{\bar{β}, S, x}^{r, t})_{*} ({k}) \\ = & (v_{S, \bar{β}}^{2, t})_{*} ({i}) - (v_{S, \bar{β}}^{2, t})_{*} ({k}) \\ = & \bar{β_{i}^{t}} (S, b_{S}, V_{S, \bar{β}}^{2, m}) - \bar{β_{k}^{t}} (S, b_{S}, V_{S, \bar{β}}^{2, m}) \\ = & \bar{β_{i}^{t}} (N, b, V^{m}) - \bar{β_{k}^{t}} (N, b, V^{m}) . \end{matrix}$

Therefore, $\begin{matrix} 2 \cdot [\bar{β_{i}^{t}} (S, b_{S}, V_{\bar{β}, S, x}^{r, m}) - x_{i}^{t}] \\ = & \bar{β_{i}^{t}} (N, b, V^{m}) - \bar{β_{k}^{t}} (N, b, V^{m}) - x_{i}^{t} + x_{k}^{t} . \end{matrix}$ (11)

By definition of g and equation (11), $\begin{matrix} g_{i}^{t} (x) \\ = & x_{i}^{t} + \frac{w}{2} \cdot [\sum_{k \in N \ {i}} \bar{β_{i}^{t}} (N, b, V^{m}) - \sum_{k \in N \ {i}} x_{i}^{t} \\ - \sum_{k \in N \ {i}} \bar{β_{k}^{t}} (N, b, V^{m}) + \sum_{k \in N \ {i}} x_{k}^{t}] \\ = & x_{i}^{t} + \frac{w}{2} \cdot [\sum_{k \in N \ {i}} \bar{β_{i}^{t}} (N, b, V^{m}) - (| N | - 1) x_{i}^{t} \\ - \sum_{k \in N \ {i}} \bar{β_{k}^{t}} (N, b, v) + (v_{*}^{t} (N) - x_{i}^{t})] \\ = & x_{i}^{t} + \frac{w}{2} \cdot [(| N | - 1) \bar{β_{i}^{t}} (N, b, V^{m}) - (| N | - 1) x_{i}^{t} \\ - (v_{*}^{t} (N) - \bar{β_{i}^{t}} (N, b, V^{m})) + (v_{*}^{t} (N) - x_{i}^{t})] \\ = & x_{i}^{t} + \frac{| N | \cdot w}{2} \cdot [\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}] . \end{matrix}$ (12)

So we have that $\begin{matrix} \sum_{i \in N} g_{i}^{t} (x) \\ = & \sum_{i \in N} [x_{i}^{t} + \frac{| N | \cdot w}{2} \cdot [\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}]] \\ = & \sum_{i \in N} x_{i}^{t} + \frac{| N | \cdot w}{2} \cdot [\sum_{i \in N} \bar{β_{i}^{t}} (N, b, V^{m}) - \sum_{i \in N} x_{i}^{t}] \\ = & v_{*}^{t} (N) + \frac{| N | \cdot w}{2} \cdot [v_{*}^{t} (N) - v_{*}^{t} (N)] \\ = & v_{*}^{t} (N) . \end{matrix}$ Thus, g (x) ∈ E (N, b, V^m) for all x ∈ E (N, b, V^m). ■

Theorem 3. Let (N, b, V^m) ∈ Γ. If $0 < α < \frac{4}{| N |}$ ,then ${[η]^{q}}_{q = 1}^{\infty}$ converges to $\bar{β} (N, b, V^{m})$ for each x ∈ E (N, b, V^m).

Proof. Let (N, b, V^m) ∈ Γ, $t \in ℕ_{m}$ and x ∈ E (N, b, V^m). By equation (12), $g_{i}^{t} (x) = x_{i}^{t} + \frac{| N | \cdot w}{2} \cdot [\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}]$ for all i ∈ N. Therefore, $\begin{matrix} (1 - \frac{| N | \cdot w}{2}) \cdot [\bar{β_{i}^{t}} (N, b, V^{m}) - x_{i}^{t}] \\ = & [\bar{β_{i}^{t}} (N, b, V^{m}) - g_{i}^{t} (x)] . \end{matrix}$

So, for all $q \in ℕ$ , $\bar{β} (N, b, V^{m}) - [η]^{q} = (1 - \frac{| N | \cdot w}{2})^{q} [\bar{β} (N, b, V^{m}) - x] .$

If $0 < w < \frac{4}{| N |}$ , then $- 1 < (1 - \frac{| N | \cdot w}{2}) < 1$ and ${[η]^{q}}_{q = 1}^{\infty}$ converges to $\bar{β} (N, b, v)$ for all (N, b, V^m) ∈ Γ, for all $t \in ℕ_{m}$ and for all i ∈ N. ■

5 Concluding remarks

In this paper, we investigate the fuzzy Banzhaf-Coleman index and the fuzzy efficient Banzhaf-Coleman index. Based on reduced game, two axiomatizations for these two indexes are proposed. By applying reduction and excess function, we also introduce alternative formulations and related dynamic processes for the fuzzy efficient Banzhaf-Coleman index. One should compare our results with related pre-existing results:

The fuzzy Banzhaf-Coleman index and the fuzzy efficient Banzhaf-Coleman index are introduced initially in the framework of multicriteria fuzzy TU games.

The idea of our correction functions in Definitions 3, 4 and related dynamic processes are based on that of Maschler and Owen’s [14] dynamic results for the Shapley value [?]. The major difference is that our correction functions in Definition 4 are based on “excess function", and Maschler and Owen’s [14] correction function is based on “reduced games".

These mentioned above raise one question:

Whether there exist other normalizations and related results for some more solutions in the framework of multicriteria 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 N is a coalition and v^a is a mapping such that $v^{a} : [0, 1]^{N} ⟶ ℝ$ and v^a (0_N) =0. In fact, (N, v^a) =(N, θ^N, v).

From now on we consider bounded fuzzy TU games, defined as those games (N, b, v) such that, there exists $K_{v} \in ℝ$ such that v (α) ≤ K_v for all α ∈ B^N. We adopt it to ensure that v_* (T) is well-defined.

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