A weighted power distribution mechanism under fuzzy behavior systems

Abstract

In real situations, players might represent administrative areas of different scales; players might have different activity abilities. Thus, we propose an extension of the Banzhaf-Owen index in the framework of fuzzy transferable-utility games by considering supreme-utilities and weights simultaneously, which we name the weighted fuzzy Banzhaf-Owen index. Here we adopt three existing notions from traditional game theory and reinterpret them in the framework of fuzzy transferable-utility games. The first one is that this weighted index could be represented as an alternative formulation in terms of excess functions. The second is that, based on an reduced game and related consistency, we offer an axiomatic result to present the rationality of this weighted index. Finally, we introduce two dynamic processes to illustrate that this weighted index could be reached by players who start from an arbitrary efficient payoff vector and make successive adjustments.

Keywords

Weight the weighted fuzzy Banzhaf-Owen index excess function dynamic process 91A 91B

1 Introduction

In the framework of transferable-utility (TU) games, the power indices have been defined to measure the political power of each member of a voting system. A member in a voting system is, e.g., a party in a parliament or a country in a confederation. Each member will have a certain number of votes, and so its power will be different. Results of the power indices may be found in, e.g., Algaba [1], van den Brink and van der Laan [6], Cheng et al. [8], Dubey and Shapley [10], Hwang and Liao [14], Lehrer [17], Liao et al. [20] and so on. Banzhaf [4] defined a power index in the framework of voting games that was essentially identical to that given by Coleman [9]. This index was later extended to arbitrary games by Owen [25, 26]. Here we focus on the notion of the Banzhaf-Owen index due to Owen [25, 26]. Briefly speaking, the Banzhaf-Owen index is a power allocation rule that gathers each member’s marginal contribution from all coalitions in which he/she/it has joined. It is known that the Banzhaf-Owen index does not necessarily distribute all the utility over all players in the grand coalition. Thus, the efficient Banzhaf-Owen index and related results were proposed by Hwang and Liao [16] and Liao et al. [21] respectively.

In a traditional TU game, each member is either fully involved or not involved at all in participation with some other members. However, each member always takes different activity levels to participate in real situations. The theory of fuzzy TU games commenced with the investigation of Aubin [2, 3] where the opinions of a fuzzy TU game and the fuzzy core are introduced. In a fuzzy TU game, each member is allowed to participate with infinite many different activity levels (or decisions). Many fuzzy solution concepts have been applied wildly, e.g., Bashar et al. [5], Butnariu and Kroupa [7], Hwang [12], Li and Zhang [18], Meng and Zhang [23], Nykyforchyn and Mykytsey [24], Shazia and Akbar [29], Tijs et al. [30], and so on. By focusing on supreme-utilities among fuzzy level (decision) vectors, Liao and Chung [19] proposed the fuzzy Banzhaf-Owen index, the fuzzy efficient Banzhaf-Owen index and related results in the framework of fuzzy TU games.

Based on the notion of the efficient Banzhaf-Owen index, all players first receive their marginal contributions from all coalitions in which they have participated, and further allocate the remaining utilities equally. That is, any additional fixed utility (e.g. the cost of a common facility) should be distributed equally among the players who are concerned. In many applications, however, the efficient Banzhaf-Owen index seems unrealistic for the situation that is being modeled. Players might represent constituencies of different sizes; players might have different bargaining abilities. Also, lack of symmetry may arise when different bargaining abilities for different players are modeled. In line with the above interpretations, we would now desire that any additional fixed utility could be distributed among the players in proportion to their weights. Weights come up naturally in the framework of utilities allocation. For example, we may be dealing with a problem of utility allocation among investment projects. Then the weights could be associated to the profitability of the different projects. Weights are also included in contracts signed by the owners of a condominium and used to divide the cost of building or maintaining common facilities. Another example is data or patent pooling among firms where the size of the firms, measured for instance by their market shares, are natural weights.

The above mentioned results raise one question in the framework of fuzzy TU games:

whether the efficient Banzhaf-Owen index could be proposed by applying weights in the framework of fuzzy TU games.

The paper is devoted to investigate the question. In order to modify the discrimination among players, we apply weights to propose different results as follows.

In Section 2, we adopt supreme-utilities and weights to propose the weighted fuzzy Banzhaf-Owen index. Further, we present alternative formulation for the weighted fuzzy Banzhaf-Owen index in terms of excess functions. The excess of a coalition could be treated as the variation between the productivity and the total payoff of the coalition.

In Section 3, we adopt the efficiency-sum-reduced game to characterize the weighted fuzzy Banzhaf-Owen index. In Section 4, we propose dynamic processes to illustrate that the weighted fuzzy Banzhaf-Owen index can be approached by players who start from an arbitrary efficient payoff vector. Further, some more interpretations are also provided throughout this paper.

2 The weighted fuzzy Banzhaf-Owen index

2.1 Preliminaries

Let U be the universe of members. For i ∈ U and b_i ∈ (0, 1], B_i = [0, b_i] could be treated as the level (decision) space of member i and $B_{i}^{+} = (0, b_{i}]$ , where 0 denotes no participation. Let $B^{N} = \prod_{i \in N} B_{i}$ be the product set of the level (decision) spaces of all members of N. 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 members, $b = (b_{i})_{i \in N} \in \prod_{i \in N} B_{i}^{+}$ is the vector that presents the highest levels for each member, 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 members can gain when each member 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.

Definition 1. The efficient fuzzy Banzhaf-Owen index (Liao and Chung [19]), $\bar{η}$ , is the solution on Γ which associates with (N, b, v) ∈ Γ and each player i ∈ N the value

$\begin{matrix} \bar{η_{i}} (N, b, v) = η_{i} (N, b, v) + \frac{1}{| N |} \\ \cdot [v_{*} (N) - \sum_{k \in N} η_{k} (N, b, v)], \end{matrix}$ (1) where $η_{i} (N, b, v) = \sum_{\overset{S \subseteq N}{i \in S}} [v_{*} (S) - v_{*} (S \ {i})]$ is the fuzzy Banzhaf-Owen index of i. It is shown that the fuzzy Banzhaf-Owen index violates EFF, and the efficient fuzzy Banzhaf-Owen index satisfies EFF.

Let (N, b, v) ∈ Γ. A function $w : N \to ℝ^{+}$ is called a weight function if w is a non-negative function. In different situations, players in N could be assigned different weights by weight functions. These weights could be interpreted as a-priori measures of importance; they are taken to reflect considerations not captured by the characteristic function. For example, we may be dealing with a problem of cost allocation among investment projects. Then the weights could be associated to the profitability of the different projects. In a problem of allocating travel costs among various institutions visited (cf. Shapley [28]), the weights may be the number of days spent at each one. The weighted fuzzy Banzhaf-Owen index is defined as follows.

Definition 2. The weighted fuzzy Banzhaf-Owen index (WFBOI), $\bar{η^{w}}$ , is the solution which associates with all (N, b, v) ∈ Γ, weight function w and players i ∈ N the value

$\begin{matrix} \bar{η_{i}^{w}} (N, b, v) = η_{i} (N, b, v) + \frac{w (i)}{| N |_{w}} \\ \cdot [v_{*} (N) - \sum_{k \in N} η_{k} (N, b, v)], \end{matrix}$ (2) where |S|_w = ∑_i∈Sw (i) for all S ⊆ N. By the definition of $\bar{η^{w}}$ , all players first receive their marginal contributions from all coalitions, and further allocate the remaining utilities proportionally by applying weights.

A solution ψ satisfies efficiency (EFF) if ∑_i∈Nψ_i (N, b, v) = v_* (N) for all (N, b, v) ∈ Γ. Property EFF asserts that all players distribute all the utility completely.

Lemma 1. The WFBOI $\bar{η^{w}}$ satisfies EFF.

Proof. By Definition 2, it is easy to show that the WFBOI satisfies EFF.■

2.2 Motivating and practical examples

Here we provide a brief motivating example of 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 members of a management system (N, b, v). The function v could be treated as an utility function which assigns to each level vector α = (α_i) _i∈N ∈ B^N the worth that the members can obtain when each member i participates at operation strategy α_i ∈ B_i in (N, b, v). Modeled in this way, the management system (N, b, v) could be considered as a fuzzy TU game, with v being each characteristic function and B_i being the set of all operation strategies of the member i. As mentioned in Introduction, however, it may not be appropriate in many conditions if any additional fixed utility should be distributed equally among the players who are concerned. Thus, it is reasonable that weights are assigned to players and any fixed utility should be divided according to these weights.

In the following, we also provide a practical application of power distribution in a national parliament. Let N = {1, 2, ⋯ , n} be a set of all members of a national parliament of a certain country. In the national parliament of a certain country, all the members of the parliament are elected by voting or recommendation by parties. All members have the power to propose, discuss, establish, and veto all bills. They dedicate different levels of attention and participation to different bills 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 political parties. For the aforementioned reasons, strategies adopted by each member of the parliament show distinct levels of participation and certain amounts of ambiguity. The function v could be treated as a power function which assigns to each level vector α = (α_i) _i∈N ∈ B^N the power that the members can dedicate when each member i participates at operation strategy α_i ∈ B_i. Modeled in this way, the national parliament operational system (N, b, v) could be considered as a fuzzy TU game, with v being each characteristic function and B_i being the set of all operation strategies of the member i. To evaluate the influence of each member in the national parliament, using the power indicators we proposed, we first assess the supreme influence each parliament member has accumulated over previous bill meetings based on various and ambiguous behaviors, which is the the fuzzy Banzhaf-Owen index mentioned in Definition 1. As each parliament member has different academic expertise and represents different public opinions, they naturally carry different levels of importance; hence, it makes complete sense for them to derive different weights by a weight function w. The remaining shared power distribution should also be allocated in proportion to the weight derived for each member, which is the weighted fuzzy Banzhaf-Owen index mentioned in Definition 2. We provide an example with real data as follows. Let (N, b, v) ∈ Γ with N = {i, j, k}, B_i = [0, 0.91] , B_j = [0, 0.83] , B_k = [0, 1] and $w : N \to ℝ^{+}$ with w (i) =3, w (j) =5, w (k) =2. Further, let v_* (N) =9, v_* ({i}) =5, v_* ({j}) = -3, v_* ({k}) =4, v_* ({i, j}) =8, v_* ({i, k}) = -2, v_* ({j, k}) =3 and v_* (∅) =0. By Definitions 1 and 2, $\begin{matrix} η_{i} (N, v) = 16, η_{j} (N, v) = 10, η_{k} (N, v) = 4, \\ \bar{η_{i}} (N, v) = 9, \bar{η_{j}} (N, v) = 3, \bar{η_{k}} (N, v) = - 3, \\ \bar{η_{i}^{w}} (N, v) = 9.7, \bar{η_{j}^{w}} (N, v) = - 0.5, \bar{η_{k}^{w}} (N, v) = - 0.2 . \end{matrix}$

Subsequently, we would like to show that the weighted fuzzy Banzhaf-Owen index could provide “optimal allocation mechanisms” among all members, in the sense that this organization can get payoff from each combination of operation strategies of all members under fuzzy behavior.

2.3 Alternative formulation

In the following, we present an alternative formulation for the WFBOI in terms of excess functions. If $x \in ℝ^{N}$ and S ⊆ N, write x_S for the restriction of x to S and write x (S) = ∑_i∈Sx_i. Denote that X (N, b, v) = {x ∈ I R^N|x (N) = v_* (N)} for all (N, b, v) ∈ Γ. The excess of a coalition S ⊆ N at x is the real number e (S, v, x) = v_* (S) - x (S).

Lemma 2. Let (N, b, v) ∈ Γ, x ∈ X (N, b, v) and w be weight function. Then $\begin{matrix} w (j) \sum_{S \subseteq N \ {i}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {i}, v, \frac{x}{2^{| N | - 1}})] \\ = w (i) \sum_{S \subseteq N \ {j}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {j}, v, \frac{x}{2^{| N | - 1}})] \\ \forall i, j \in N \Leftrightarrow x = \bar{η^{w}} (N, b, v) . \end{matrix}$

Proof. Let (N, b, v) ∈ Γ, x ∈ X (N, b, v) and w be a weight function. For all i, j ∈ N, $\begin{matrix} w (j) \sum_{S \subseteq N \ {i}} [e (S, v, \frac{x}{2^{| N | - 1}}) \\ - e (S \cup {i}, v, \frac{x}{2^{| N | - 1}})] \\ = w (i) \sum_{S \subseteq N \ {j}} [e (S, v, \frac{x}{2^{| N | - 1}}) \\ - e (S \cup {j}, v, \frac{x}{2^{| N | - 1}})] \end{matrix}$ $\begin{matrix} \Leftrightarrow \\ w (j) \sum_{S \subseteq N \ {i}} [v_{*} (S) - \frac{x}{2^{| N | - 1}} (S) + \frac{x}{2^{| N | - 1}} \\ (S \cup {i}) - v_{*} (S \cup {i})] \\ = w (i) \sum_{S \subseteq N \ {j}} [v_{*} (S) - \frac{x}{2^{| N | - 1}} (S) + \frac{x}{2^{| N | - 1}} \\ (S \cup {j}) - v_{*} (S \cup {j})] \\ \Leftrightarrow \\ w (j) \sum_{S \subseteq N \ {i}} [\frac{x_{i}}{2^{| N | - 1}} - v_{*} (S \cup {i}) + v_{*} (S)] \\ = w (i) \sum_{S \subseteq N \ {j}} [\frac{x_{j}}{2^{| N | - 1}} - v_{*} (S \cup {j}) + v_{*} (S)] \\ \Leftrightarrow \\ w (j) [x_{i} - \sum_{S \subseteq N \ {i}} [v_{*} (S \cup {i}) - v_{*} (S)]] \\ = w (i) [x_{j} - \sum_{S \subseteq N \ {j}} [v_{*} (S \cup {j}) - v_{*} (S)]] \\ \Leftrightarrow \\ w (j) \cdot [x_{i} - η_{i} (N, b, v)] = w (i) \cdot [x_{j} - η_{j} (N, b, v)] . \end{matrix}$ (3) By Definition 2,

$\begin{matrix} w (j) \cdot [\bar{η_{i}^{w}} (N, b, v) - η_{i} (N, b, v)] = w (i) \\ \cdot [\bar{η_{j}^{w}} (N, b, v) - η_{j} (N, b, v)] . \end{matrix}$ (4) By Equations (3) and (4), $[x_{i} - \bar{η_{i}^{w}} (N, b, v)] \sum_{j \in N} w (j) = w (i) \sum_{j \in N} [x_{j} - \bar{η_{j}^{w}} (N, b, v)] .$ Since x ∈ X (N, b, v) and $\bar{η^{w}}$ satisfies EFF, $[x_{i} - \bar{η_{i}^{w}} (N, b, v)] \cdot | N |_{w} = w (i) \cdot [v_{*} (N) - v_{*} (N)] = 0 .$ Therefore, $x_{i} = \bar{η_{i}^{w}} (N, b, v)$ for all i ∈ N.■

3 Axiomatic results

In this section, we adopt the efficiency-average-reduced game to characterize the WFBOI.

Definition 3. (Liao and Chung [19]) Let (N, b, v) ∈ Γ, S ⊆ N and ψ be a solution. The efficiency-sum-reduced game (S, b_S, v^S,ψ) with respect to ψ and S is defined by $v^{S, ψ} (α) = {\begin{matrix} 0 & if α = 0_{S}, \\ v_{*} (N) - \sum_{i \in N \ S} ψ_{i} (N, b, v) & if A (α) = S, \\ \sum_{Q \subseteq N \ S} [v_{*} (A (α) \cup Q) - \sum_{i \in Q} ψ_{i} (N, b, v)] & otherwise . \end{matrix}$

The efficiency-sum-reduction asserts that given a proposed payoff vector ψ (N, b, v), the worth of the coalition A (α) ⊆ S in (S, b_S, v^S,ψ) is computed under the assumption that A (α) can secure the cooperation of any subgroup Q of N \ S, provided each member of Q receives its component of ψ (N, b, v). After these payments are made, what remains for A (α) is the value v_* (A (α) ∪ Q) - ∑_i∈Qψ_i (N, b, v). Summing behavior on the part of A (α) involves finding the sum of the values v_* (A (α) ∪ Q) - ∑_i∈Qψ_i (N, b, v) for all Q ⊆ N \ S.

A solution ψ satisfies bilateral efficiency-sum-consistency (BESCON) if ψ_i (S, b_S, v^S,ψ) = ψ_i (N, b, v) for all (N, b, v) ∈ Γ with |N|≥2, for all S ⊆ N with |S|=2 and for all i ∈ S.

Lemma 3. The WFBOI $\bar{η^{w}}$ satisfies BESCON.

Proof. Let (N, b, v) ∈ Γ, S ⊆ N with |S|=2 and w be a weight function. Let $x = \bar{η^{w}} (N, b, v)$ . Suppose S = {i, j} then $\begin{matrix} \sum_{T \subseteq S \ {i}} [e (T, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2^{| S | - 1}}) - e (T \cup {i}, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2^{| S | - 1}})] \\ = & [e ({j}, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2}) - e (S, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2})] + [e (\emptyset, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2}) - e ({i}, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2})] \\ = & (v_{*}^{S, \bar{η^{w}}} ({j}) - \frac{x_{j}}{2}) - (v_{*}^{S, \bar{η^{w}}} (S) - \frac{x_{S}}{2} (S)) + 0 - (v_{*}^{S, \bar{η^{w}}} ({i}) - \frac{x_{i}}{2}) \\ = & (v_{*}^{S, \bar{η^{w}}} ({j}) - \frac{x_{j}}{2}) - 0 + 0 - (v_{*}^{S, \bar{η^{w}}} ({i}) - \frac{x_{i}}{2}) \\ = & ([\sum_{Q \subseteq N \ S} [v_{*} ({j} \cup Q) - \sum_{k \in Q} \frac{x_{k}}{2}]] - \frac{x_{j}}{2}) - ([\sum_{Q \subseteq N \ S} [v_{*} ({i} \cup Q) - \sum_{k \in Q} \frac{x_{k}}{2}]] - \frac{x_{i}}{2}) \\ = & \sum_{Q \subseteq N \ S} ([v_{*} ({j} \cup Q) - \frac{x_{j}}{2^{| N | - 1}}] - [v_{*} ({i} \cup Q) - \frac{x_{i}}{2^{| N | - 1}}]) \\ = & \sum_{Q \subseteq N \ S} ([v_{*} ({j} \cup Q) - \sum_{k \in Q} \frac{x_{k}}{2^{| N | - 1}} - \frac{x_{j}}{2^{| N | - 1}}] - [v_{*} ({i} \cup Q) - \sum_{k \in Q} \frac{x_{k}}{2^{| N | - 1}} - \frac{x_{i}}{2^{| N | - 1}}]) \\ = & \sum_{Q \subseteq N \ S} ([v_{*} ({j} \cup Q) - \sum_{k \in {j} \cup Q} \frac{x_{k}}{2^{| N | - 1}}] - [v_{*} ({i} \cup Q) - \sum_{k \in {j} \cup Q} \frac{x_{k}}{2^{| N | - 1}}]) \\ = & \sum_{Q \subseteq N \ S} [(e ({j} \cup Q, v, \frac{x}{2^{| N | - 1}}) - (e ({i} \cup Q, v, \frac{x}{2^{| N | - 1}})] \\ = & \sum_{Q \subseteq N \ {i}} [(e (Q, v, \frac{x}{2^{| N | - 1}}) - (e (Q \cup {i}, v, \frac{x}{2^{| N | - 1}})] \end{matrix}$ (5) Similar to Equation (5), $\begin{matrix} \sum_{T \subseteq S \ {j}} [e (T, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2^{| S | - 1}}) - e (T \cup {j}, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2^{| S | - 1}})] \\ = & \sum_{Q \subseteq N \ {j}} [(e (Q, v, \frac{x}{2^{| N | - 1}}) - (e (Q \cup {j}, v, \frac{x}{2^{| N | - 1}})] \end{matrix}$ By EFF of $\bar{η^{w}}$ and Definition 3, $x_{S} \in X (S, b_{S}, v^{S, \bar{η^{w}}})$ . By Lemma 2, $\begin{matrix} w (j) \cdot \sum_{T \subseteq S \ {i}} [e (T, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2^{| S | - 1}}) - e (T \cup {i}, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2^{| S | - 1}})] \\ = & w (j) \cdot \sum_{Q \subseteq N \ {i}} [(e (Q, v, \frac{x}{2^{| N | - 1}}) - (e (Q \cup {i}, v, \frac{x}{2^{| N | - 1}})] \\ (by Equation (5)) \\ = & w (i) \cdot \sum_{Q \subseteq N \ {j}} [(e (Q, v, \frac{x}{2^{| N | - 1}}) - (e (Q \cup {j}, v, \frac{x}{2^{| N | - 1}})] \\ (by Lemma 2) \\ = & w (i) \cdot \sum_{T \subseteq S \ {j}} [e (T, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2^{| S | - 1}}) - e (T \cup {j}, v^{S, \bar{η^{w}}}, \frac{x_{S}}{2^{| S | - 1}})] . \\ (similar to Equation (5)) \end{matrix}$ By Lemma 2 and $x_{S} \in X (S, b_{S}, v^{S, \bar{η^{w}}})$ , we have that $x_{S} = \bar{η^{w}} (S, b_{S}, v^{S, \bar{η^{w}}})$ . Hence, $\bar{η^{w}}$ satisfies BESCON.■

Inspired by Hart and Mas-Colell [11], we provide an axiomatic result of the WFBOI as follows. A solution ψ satisfies weighted Banzhaf-Owen standard for games (WBSFG) if $ψ (N, b, v) = \bar{η^{w}} (N, b, v)$ for all (N, b, v) ∈ Γ with |N|≤2. Property WBSFG is a generalization of the two-person standardness axiom of Hart and Mas-Colell [11].

Lemma 4. If a solution ψ satisfies WBSFG and BESCON, then it satisfies EFF.

Proof. Suppose ψ satisfies WBSFG and BESCON. Let (N, b, v) ∈ Γ. If |N|≤2, then ψ satisfies EFF by BESCON of ψ. Suppose |N|>2, i, j ∈ N and S = {i, j}. Since ψ satisfies EFF in two-person games, $ψ_{i} (S, b_{S}, v^{S, ψ}) + ψ_{j} (S, b_{S}, v^{S, ψ}) = v_{*}^{S, ψ} (S) = v_{*} (N) - \sum_{k \neq i, j} ψ_{k} (N, b, v) .$ (6) By BESCON of ψ, $ψ_{t} (S, b_{S}, v^{S, ψ}) = ψ_{t} (N, b, v) for all t \in S .$ (7) By (6) and (7), v_* (N) = ∑_k∈Nψ_k (N, b, v), i.e., ψ satisfies EFF.■

Theorem 1. A solution ψ satisfies WBSFG and BESCON if and only if $ψ = \bar{η^{w}}$ .

Proof. By Lemma 3, $\bar{η^{w}}$ satisfies BESCON. Clearly, $\bar{η^{w}}$ satisfies WBSFG.

To prove uniqueness, suppose ψ satisfies WBSFG and BESCON. By Lemma 4, ψ satisfies EFF. Let (N, b, v) ∈ Γ. If |N|≤2, it is trivial that $ψ (N, b, v) = \bar{η^{w}} (N, b, v)$ by SFG. Assume that |N|>2. Let i ∈ N and S = {i, j} for some j ∈ N \ {i}. Then $\begin{matrix} ψ_{i} (N, b, v) - \bar{η_{i}^{w}} (N, b, v) \\ = & ψ_{i} (S, b_{S}, v^{S, ψ}) - \bar{η_{i}^{w}} (S, v^{S, \bar{η^{w}}}) (by BESCON of ψ, \bar{η^{w}}) \\ = & \bar{η_{i}^{w}} (S, b_{S}, v^{S, ψ}) - \bar{η_{i}^{w}} (S, v^{S, \bar{η^{w}}}) (by WBSFG of ψ, \bar{η^{w}}) \\ = & η_{i}^{w} (S, b_{S}, v^{S, ψ}) + \frac{w (i)}{| S |_{w}} \cdot [v_{*}^{S, ψ} (S) - [η_{i}^{w} (S, b_{S}, v^{S, ψ}) + η_{j}^{w} (S, b_{S}, v^{S, ψ})]] \\ - η_{i}^{w} (S, b_{S}, v^{S, \bar{η^{w}}}) - \frac{w (i)}{| S |_{w}} \cdot [v_{*}^{S, \bar{η^{w}}} (S) - [η_{i}^{w} (S, b_{S}, v^{S, \bar{η^{w}}}) + η_{j}^{w} (S, b_{S}, v^{S, \bar{η^{w}}})]] \\ = & [v_{*}^{S, ψ} (S) + v_{*}^{S, ψ} ({i}) - v_{*}^{S, ψ} ({j})] + \frac{w (i)}{| S |_{w}} \cdot [- v_{*}^{S, ψ} (S)] \\ - [v_{*}^{S, \bar{η^{w}}} (S) + v_{*}^{S, \bar{η^{w}}} ({i}) - v_{*}^{S, \bar{η^{w}}} ({j})] - \frac{w (i)}{| S |_{w}} \cdot [- v_{*}^{S, \bar{η^{w}}} (S)] . \end{matrix}$ (8) By definitions of v^S,ψ and $v^{S, \bar{η^{w}}}$ , $\begin{matrix} v_{*}^{S, ψ} ({i}) - v_{*}^{S, ψ} ({j}) & = \sum_{Q \subseteq N \ S} [v_{*} ({i} \cup Q) - v_{*} ({j} \cup Q)] \\ = v_{*}^{S, \bar{η^{w}}} ({i}) - v_{*}^{S, \bar{η^{w}}} ({j}) . \end{matrix}$ (9) By Equations (8) and (9), $\begin{matrix} ψ_{i} (N, b, v) - \bar{η_{i}^{w}} (N, b, v) = [1 - \frac{w (i)}{| S |_{w}}] \cdot [v_{*}^{S, ψ} (S) - v_{*}^{S, \bar{η^{w}}} (S)] \\ = & \frac{w (j)}{| S |_{w}} \cdot [ψ_{i} (N, b, v) + ψ_{j} (N, b, v) - \bar{η_{i}^{w}} (N, b, v) - \bar{η_{j}^{w}} (N, b, v)] . \end{matrix}$ That is, $w (i) \cdot [ψ_{i} (N, b, v) - \bar{η_{i}^{w}} (N, b, v)] = w (j) \cdot [ψ_{j} (N, b, v) - \bar{η_{j}^{w}} (N, b, v)] .$ By EFF of ψ and $\bar{η^{w}}$ , $\begin{matrix} 0 & = v_{*} (N) - v_{*} (N) = \sum_{j \in N} [ψ_{j} (N, b, v) - \bar{η_{j}^{w}} (N, b, v)] \\ = w (i) \cdot [ψ_{i} (N, b, v) - \bar{η_{i}^{w}} (N, b, v)] \sum_{j \in N} \frac{1}{w (j)} . \end{matrix}$ Hence, $ψ_{i} (N, b, v) = \bar{η_{i}^{w}} (N, b, v)$ for all i ∈ N.■

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

Example 1. Define a solution ψ by for all (N, b, v) ∈ Γ and for all i ∈ N, $ψ_{i} (N, b, v) = \frac{v_{*} (N)}{| N |} .$ Clearly, ψ satisfies BESCON, but it violates WBSFG.

Example 2. Define a solution ψ by for all (N, b, v) ∈ Γ and for all i ∈ N, $ψ_{i} (N, b, v) = {\begin{matrix} \bar{η_{i}^{w}} (N, b, v) & , if | N | \leq 2, \\ 0 & , otherwise . \end{matrix}$ Clearly, ψ satisfies WBSFG, but it violates BESCON.

4 Dynamic results

In this section, we introduce two dynamic processes of the WFBOI by applying excess functions and reductions.

In the following, we adopt excess functions to propose a correction function and related dynamic process for the WFBOI.

Definition 4. Let (N, b, v) ∈ Γ, i ∈ N and w be a weight function. The e-correction function $f_{i}^{\bar{η^{w}}} : X (N, b, v) \to ℝ$ is defined by $\begin{matrix} f_{i}^{\bar{η^{w}}} (x) = x_{i} + t \sum_{j \in N \ {i}} (w (i) \sum_{S \subseteq N \ {j}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {j}, v, \frac{x}{2^{| N | - 1}})] \\ - w (j) \sum_{S \subseteq N \ {i}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {i}, v, \frac{x}{2^{| N | - 1}})]), \end{matrix}$ where t ∈ (0, ∞), which reflects the assumption that player i does not ask for full correction (when t = 1) but only (usually) a fraction of it.

When a player withdraws from the coalitions he/she/it joined, some complaints may be occurred from other players. The e-correction function is based on the idea that, each agent shortens the weighted excess relating to his own and others’ non-participation in all coalitions, and adopts these regulations to correct the original payoff.

The following lemma shows that the e-correction function is well-defined, i.e., the efficiency is preserved under the e-correction function.

Lemma 5. Let (N, b, v) ∈ Γ, w be a weight function and $f^{\bar{η^{w}}} = (f_{i}^{\bar{η^{w}}})_{i \in N}$ . If x ∈ X (N, b, v), then $f^{\bar{η^{w}}} (x) \in X (N, b, v)$ .

Proof. Let (N, b, v) ∈ Γ, i, j ∈ N, x ∈ X (N, b, v) and w be a weight function. Similar to the Equation (3), $\begin{matrix} w (i) \sum_{S \subseteq N \ {j}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {j}, v, \frac{x}{2^{| N | - 1}})] \\ - w (j) \sum_{S \subseteq N \ {i}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {i}, v, \frac{x}{2^{| N | - 1}})] \\ = & w (i) [x_{j} - \bar{η_{j}^{w}} (N, b, v)] - w (j) [x_{i} - \bar{η_{i}^{w}} (N, b, v)] . \end{matrix}$ (10) By Equation (10), $\begin{matrix} \sum_{j \in N \ {i}} (w (i) \sum_{S \subseteq N \ {j}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {j}, v, \frac{x}{2^{| N | - 1}})] \\ - w (j) \sum_{S \subseteq N \ {i}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {i}, v, \frac{x}{2^{| N | - 1}})]) \\ = & w (i) \sum_{j \in N \ {i}} [x_{j} - \bar{η_{j}^{w}} (N, b, v)] - [x_{i} - \bar{η_{i}^{w}} (N, b, v)] \sum_{j \in N \ {i}} w (j) \\ = & w (i) \cdot [v_{*} (N) - v_{*} (N)] - [x_{i} - \bar{η_{i}^{w}} (N, b, v)] \cdot | N |_{w} \\ (by EFF of \bar{η^{w}}, x \in X (N, b, v)) \\ = & | N |_{w} \cdot (\bar{η_{i}^{w}} (N, b, v) - x_{i}) . \end{matrix}$ (11) Moreover, $\begin{matrix} \sum_{i \in N} | N |_{w} \cdot (\bar{η_{i}^{w}} (N, b, v) - x_{i}) = | N |_{w} \cdot (v_{*} (N) - v_{*} (N)) (by EFF of \bar{η^{w}}, x \in X (N, b, v)) = 0 . \end{matrix}$ (12) So we have that $\begin{matrix} \sum_{i \in N} f_{i}^{\bar{η^{w}}} (x) = \sum_{i \in N} [x_{i} + t \sum_{j \in N \ {i}} (w (i) \sum_{S \subseteq N \ {j}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {j}, v, \frac{x}{2^{| N | - 1}})] \\ - w (j) \sum_{S \subseteq N \ {i}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {i}, v, \frac{x}{2^{| N | - 1}})])] \\ = & v_{*} (N) + t \cdot 0 (by equations (11), (12) and x \in X (N, b, v)) = v_{*} (N) . \end{matrix}$ Hence, $f^{\bar{η^{w}}} (x) \in X (N, b, v)$ if x ∈ X (N, b, v).■

Based on Lemma 5, we can define $x^{0} = x, x^{1} = f^{\bar{η^{w}}} (x^{0}), \dots, x^{q} = f^{\bar{η^{w}}} (x^{q - 1})$ for all (N, b, v) ∈ Γ, for all x ∈ X (N, b, v) and for all $q \in ℕ$ . Next, we adopt the correction function to propose a dynamic process.

Theorem 2. Let (N, b, v) ∈ Γ and w be a weight function. If $0 < t < \frac{2}{| N |_{w}}$ , then ${x^{q}}_{q = 1}^{\infty}$ converges geometrically to $\bar{η^{w}} (N, b, v)$ for all x ∈ X (N, b, v).

Proof. Let (N, b, v) ∈ Γ, i ∈ N, x ∈ X (N, b, v) and w be a weight function. By Equation (11) and definition of $f^{\bar{η^{w}}}$ , $\begin{matrix} f_{i}^{\bar{η^{w}}} (x) - x_{i} = t \sum_{j \in N \ {i}} (w (i) \sum_{S \subseteq N \ {j}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {j}, v, \frac{x}{2^{| N | - 1}})] \\ - w (j) \sum_{S \subseteq N \ {i}} [e (S, v, \frac{x}{2^{| N | - 1}}) - e (S \cup {i}, v, \frac{x}{2^{| N | - 1}})]) = t \cdot | N |_{w} (\bar{η_{i}^{w}} (N, b, v) - x_{i}) . \end{matrix}$ Hence, $\begin{matrix} \bar{η_{i}^{w}} (N, b, v) - f_{i}^{\bar{η^{w}}} (x) & = \bar{η_{i}^{w}} (N, b, v) - x_{i} + x_{i} - f_{i}^{\bar{η^{w}}} (x) = \bar{η_{i}^{w}} (N, b, v) - x_{i} - t \cdot | N |_{w} \cdot (\bar{η_{i}^{w}} (N, b, v) - x_{i}) \\ = (1 - t \cdot | N |_{w}) [\bar{η_{i}^{w}} (N, b, v) - x_{i}] . \end{matrix}$ For all $q \in ℕ$ , $\bar{η^{w}} (N, b, v) - x^{q} = (1 - t \cdot | N |_{w})^{q} [\bar{η^{w}} (N, b, v) - x] .$ If $0 < t < \frac{2}{| N |_{w}}$ , then -1 < (1 - t · |N|_w ) < 1 and ${x^{q}}_{q = 1}^{\infty}$ converges geometrically to $\bar{η^{w}} (N, b, v)$ .■

By applying a specific reduction, Maschler and Owen [22] defined a correction function to introduce a dynamic process for the Shapley value [27]. In the following, we propose a dynamic process by applying the notion due to Maschler and Owen [22].

Definition 5. Let ψ be a solution, (N, b, v) ∈ Γ, S ⊆ N and x ∈ X (N, b, v). The (x, ψ)-reduced game $(S, b_{S}, v_{ψ, S, x}^{r})$ is defined by $v^{ψ, S, x} (α) = {\begin{matrix} v_{*} (N) - x (N \ S) & , A (α) = S, \\ v^{S, ψ} (α) & , otherwise . \end{matrix}$ for all α ⊆ B^S. Inspired by Maschler and Owen [22], we define a correction function as follow. Let (N, b, v) ∈ Γ and w be a weight function. The R-correction function to be g = (g_i) _i∈N and $g_{i} : X (N, b, v) \to ℝ$ is define by $g_{i} (x) = x_{i} + t \sum_{k \in N \ {i}} (\bar{η_{i}^{w}} ({i, k}, b_{{i, k}} v^{\bar{η^{w}}, {i, k}, x}) - x_{i}) .$ Define x⁰ = x, x¹ = g (x⁰) , ⋯ , x^q = g (x^q-1) for all $q \in ℕ$ .

Lemma 6. g (x) ∈ X (N, b, v) for all (N, b, v) ∈ Γ and for all x ∈ X (N, b, v).

Proof. Let (N, b, v) ∈ Γ, w be a weight function, i, k ∈ N and x ∈ X (N, b, v). Let S = {i, k}, by EFF of $\bar{η^{w}}$ and Definition 5, $\bar{η_{i}^{w}} (S, b_{S}, v^{\bar{η^{w}}, S, x}) + \bar{η_{k}^{w}} (S, b_{S}, v^{\bar{η^{w}}, S, x}) = x_{i} + x_{k} .$ By Definition 5 and BESCON and WBSFG of $\bar{β}$ , $\begin{matrix} \bar{η_{i}^{w}} (S, b_{S}, v^{\bar{η^{w}}, S, x}) - \bar{η_{k}^{w}} (S, b_{S}, v^{\bar{η^{w}}, S, x}) & = \bar{η_{i}^{w}} (S, b_{S}, v^{S, \bar{η^{w}}}) - \bar{η_{k}^{w}} (S, b_{S}, v^{S, \bar{η^{w}}}) \\ = \bar{η_{i}^{w}} (N, b, v) - \bar{η_{k}^{w}} (N, b, v) . \end{matrix}$ Therefore, $\begin{matrix} 2 \cdot [\bar{η_{i}^{w}} (S, b_{S}, v^{\bar{η^{w}}, S, x}) - x_{i}] = \bar{η_{i}^{w}} (N, b, v) \\ - \bar{η_{k}^{w}} (N, b, v) - x_{i} + x_{k} . \end{matrix}$ (13) By definition of g and Equation (13), $\begin{matrix} g_{i} (x) = & x_{i} + \frac{t}{2} \cdot [\sum_{k \in N \ {i}} \bar{η_{i}^{w}} (N, b, v) - \sum_{k \in N \ {i}} x_{i} \\ - \sum_{k \in N \ {i}} \bar{η_{k}^{w}} (N, b, v) + \sum_{k \in N \ {i}} x_{k}] \\ = & x_{i} + \frac{w}{2} \cdot [(| N | - 1) \bar{η_{i}^{w}} (N, b, v) - (| N | - 1) x_{i} \\ - (v_{*} (N) - \bar{η_{i}^{w}} (N, b, v)) + (v_{*} (N) - x_{i})] \\ = & x_{i} + \frac{| N | \cdot t}{2} \cdot [\bar{η_{i}^{w}} (N, b, v) - x_{i}] . \end{matrix}$ (14) So we have that $\begin{matrix} \sum_{i \in N} g_{i} (x) & = \sum_{i \in N} [x_{i} + \frac{| N | \cdot t}{2} \cdot [\bar{η_{i}^{w}} (N, b, v) - x_{i}]] \\ = \sum_{i \in N} x_{i} + \frac{| N | \cdot t}{2} \cdot [\sum_{i \in N} \bar{η_{i}^{w}} (N, b, v) - \sum_{i \in N} x_{i}] \\ = v_{*} (N) + \frac{| N | \cdot t}{2} \cdot [v_{*} (N) - v_{*} (N)] \\ = v_{*} (N) . \end{matrix}$ Thus, g (x) ∈ X (N, b, v) for all x ∈ X (N, b, v).■

Theorem 3. Let (N, b, v) ∈ Γ and w be a weight function. If $0 < α < \frac{4}{| N |}$ , then ${x^{q}}_{q = 1}^{\infty}$ converges to $\bar{η^{w}} (N, b, v)$ for each x ∈ X (N, b, v).

Proof. Let (N, b, v) ∈ Γ, w be a weight function and x ∈ X (N, b, v). By Equation (14), $g_{i} (x) = x_{i} + \frac{| N | \cdot t}{2} \cdot [\bar{η_{i}^{w}} (N, b, v) - x_{i}]$ for all i ∈ N. Therefore, $(1 - \frac{| N | \cdot t}{2}) \cdot [\bar{η_{i}^{w}} (N, b, v) - x_{i}] = [\bar{η_{i}^{w}} (N, b, v) - g_{i} (x)]$ So, for all $q \in ℕ$ , $\bar{η^{w}} (N, b, v) - x^{q} = (1 - \frac{| N | \cdot t}{2})^{q} [\bar{η^{w}} (N, b, v) - x] .$ If $0 < t < \frac{4}{| N |}$ , then $- 1 < (1 - \frac{| N | \cdot t}{2}) < 1$ and ${x^{q}}_{q = 1}^{\infty}$ converges to $\bar{η^{w}} (N, b, v)$ for all (N, b, v) ∈ Γ, for all weight function w and for all i ∈ N.■

5 Conclusions

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

By considering supreme-utilities and weights simultaneously, we propose the weighted fuzzy Banzhaf-Owen index under fuzzy behavior.

Further, we adopt excess functions to present alternative viewpoint for the weighted fuzzy Banzhaf-Owen index.

In order to present the rationality of the weighted fuzzy Banzhaf-Owen index, we adopt the efficiency-sum-reduction and related consistency property to characterize the weighted fuzzy Banzhaf-Owen index.

By applying excess functions and reductions respectively, we propose two correction functions and related dynamic processes to illustrate that the weighted fuzzy Banzhaf-Owen index can be approached by players who start from an arbitrary efficient payoff vector.

One should compare our results with related pre-existing results.

The weighted fuzzy Banzhaf-Owen index and related results are introduced initially in the framework of fuzzy TU games. Furthermore,

Different from the Banzhaf-Owen index and other solutions on traditional TU games, we propose the weighted fuzzy Banzhaf-Owen index to analyze power distribution under fuzzy behavior.

Different from the fuzzy Banzhaf-Owen index and other solutions on fuzzy TU games, we adopt the weighted fuzzy Banzhaf-Owen index to analyze power distribution based on various weights among players.

Inspired by Maschler and Owen [22], we propose dynamic processes for the weighted fuzzy Banzhaf-Owen index. The major difference is that our e-correction function (Definition 4) is based on “excess functions”, and Maschler and Owen’s [22] correction function is based on “reductions”.

The advantages of the method presented in this paper are as follows.

Solution concepts on traditional games have only discussed participation or non-participation of all members. In this paper, we assume that all members have various and ambiguous levels of participation.

In a multitude of fuzzy game literature on solution concepts, although it might be also assumed that all members have various and ambiguous levels of participation, most literature evaluated the power or value of a given member presented with a given level of participation, such as Hwang and Liao [13, 15]. In this paper, we focus on the overall value or power each member exerts with various and ambiguous levels of participation.

Under the fuzzy efficient Banzhaf-Owen index due to Liao and Chung [19], any additional fixed utility should be distributed equally among the players who are concerned. By considering many real-world situations, we propose the weighted fuzzy Banzhaf-Owen index to allocate additional fixed utility among the players in proportion to their weights.

The disadvantages of the method presented in this paper are as follows. As stated in the advantages above, each member has various and ambiguous participation levels. Although we can evaluate the overall value or power each member exerts, it is impossible to evaluate the power or value of a given member with a given level of participation. Therefore, in future researches, we will propose different solution concepts based on the simultaneous consideration of the overall value and the specific level of participation.

The results proposed in this paper raise two questions.

Whether there exist weighted modifications and related results for some more solutions.

Whether there exist alternative formulations and related results for some more solutions.

These issues are left to the readers.

Footnotes

A fuzzy TU game, which is defined by Aubin [2, ], 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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