On the position value for communication situations with fuzzy coalition

Abstract

Borm et al. (1992) characterized the position value for cooperative games with deterministic communication structure introduced by Myerson (1977). Now we follow the Myerson model to consider the uncertainty about the participation levels of players. In this setting, the position value defined in terms of its crisp version is associated with partitions by levels of players and different ways to obtain these partitions lead to different position values with particular forms. We provide several characterizations of the specific position values from the perspective of generalized component efficiency and balanced link contributions, link potential function, and effort function linked to the Choquet partition and multilinear partition respectively.

Keywords

Fuzzy cooperative game communication structure position value fuzzy core

1 Introduction

In various economic and social situations, it often occurs that a group of agents achieve a certain worth produced by forming one coalition when each agent agrees to cooperate, which can be appropriately formalized through cooperative games. In this class of games, it is usually assumed that any coalition can form. However, sometimes the cooperation of some possible coalitions is restricted by hieratical, or technical structures and so on. Myerson [11] considered this situation and first introduced the communication structure. A communication structure can be described by an undirected graph with players as vertices and feasible bilateral communication relationships as links. In order to measure the impact of communication restrictions on players, Myseson defined a graph game associated to the corresponding communication situation. The Myerson value for a communication situation is the Shapley value of its graph game. Myerson [11, 12] and Borm et al. [2] both gave different valid characterizations of the Myerson value for communication situations.

An alternative allocation rule for communication situations is the position value introduced by Meessen [7] and Borm et al. [2] with the definition of link game. While the Myerson value focuses on the role of players in establishing communication within the various possible coalitions, the position value focuses on the role of links. The power of each link is justified through the Shapley value of the link game. Borm et al. [2] provided a characterization of the position value restricted on cycle-free communication structures. Slikker [13] gave another characterization of the position value for general communication situations. Afterwards, Slikker [14] and Kamijo [6] again presented the uniqueness of the position value by means of the potential function and effortfunction.

Aubin [1] considered that there are some situations where players do not fully participate in a coalition, but to a certain extent and introduced the fuzzy cooperative games. Later on, more scholars concentrated on the solution concepts for fuzzy cooperative games, such as the Shapley value [3 , 16], core [15, 18] and so on.

Jiménez-Losada et al. [5] considered the uncertainty about the capacities of communications among players. In their model, the communication level between any two players is required not greater than the minimum of their participation levels. It is a very strong constraint for taking this model in application.

A crisp communication situation loses its applicability in dealing with allocation issues when some agents may only offer a part of resources in a production economy with the left as other constructions, or not be sure whether to join in a project due to the uncertain environment. Thus, it is necessary to study the uncertainty of participation levels of players for communication situations. Here we extend the model of Myerson [11] and present a framework of communication situation with fuzzy coalition. In our model, the communication strength of a link is measured by the Shapley value of a newly defined link game relative to partitions by levels of players. Assume each player has the veto power of the use of any link that he is an endpoint of, it seems reasonable to distribute equally the income of each link among its two endpoints. The total amount that a player obtains in this way is defined as the so-called position value for the player in the communication situation with fuzzy coalition. We provide different position values linked to two particular level partitions: Choquet partition and multilinear partition. Moreover, the characterizations of specific position values are also presented from different angles inspired by Slikker [13, 14] and Kamijo [6].

The remainder of this paper is organized as follows. In Section 2, we prepare some basic knowledge to allow readers to follow the paper: cooperative games, communication structures and fuzzy cooperative games. In Sections 3 and 4, we define the position value for communication situations with fuzzy coalition and obtain its several axiomatizations based on both of Choquet and multilinear model.

2 Preliminaries

2.1 Cooperative games and communication structures

A cooperative game is defined by a pair (N, v), where N is a finite set of players and $v : 2^{N} \to ℝ$ is the corresponding characteristic function such that v (∅) =0. We identify the game (N, v) with its characteristic function v and denote by G^N all the cooperative games with player set N. A subset S of N is called a coalition and the associated real number v (S) is called the worth of coalition S when the players in S work together. The number of players in S is denoted by |S|. The restriction of a game v to coalition S denoted by v_S is defined by v_S (T) = v (S ∩ T) for all T ⊆ N. If v (S ∪ T) + v (S ∩ T) ≥ v (S) + v (T) for all S, T ⊆ N, we say a cooperative game v is convex, and especially, superadditive when S∩ T = ∅.

The solution concept in a cooperative game defines how to allocate the profit v (N) of grand coalition N among the players. Therefore, a solution is a vector $x \in ℝ^{n}$ and satisfies that ∑_i∈Nx_i = v (N). As the important solution concepts for cooperative games, the Shapley value of a game v is defined for all player i ∈ N as $\begin{matrix} {Sh}_{i} (v) & = & \sum_{S \subseteq N; i \in S} \frac{(| N | - | S |)! (| S | - 1)!}{| N |!} \\ \cdot [v (S) - v (S ∖ {i})] \end{matrix}$ and a core of a game v is the set $\begin{matrix} C (v) & = & {x \in ℝ^{n} | \sum_{i \in N} x_{i} = v (N), \\ \sum_{i \in S} x_{i} \geq v (S) for any S \subseteq N} . \end{matrix}$

In the classical cooperative game theory models, we often tactically assume that any coalition can form. Myerson [11] in 1977 considered the intermediate possibilities between no cooperation and universal cooperation, and described this kind of communication relationships as a communication structure.

Denote $L = {{i, j} | i \in N, j \in N, i \neq j}$ . Myerson defined the communication structure over N as an undirected graph (N, L) where $L \subseteq L$ is the collection of feasible communication links among players in N. The set of communication structures over N is denoted by CS^N. Particularly, $(N, L)$ is a complete graph representing a total cooperation of the player set N. Let g = (S, A) be a graph. A subgraph of g is another graph g′ = (S′, A′) with S′ ⊆ S and A′ ⊆ A, denoted by g′ ⊆ g. If T is a coalition, the restriction to T of g is denoted by g_T = (T, A (T) ) with A (T) = {{i, j} | {i, j} ∈ A, i, j ∈ T, i ≠ j}. A path of graph g is a sequence of m vertices (i₁, i₂, …, i_m) satisfying that {i_k, i_k+1} ∈ A is different for each k = 1, 2, …, m - 1. A cycle in graph g is a path (i₁, i₂, …, i_m) where m ≥ 4 and i_m = i₁. Two players i, j ∈ S are connected if there exists a path containing both of them. The graph g is connected if any pair of players in g are connected. We say H is a component in graph g if g_H is the maximally connected subgraph of g and denote the set of components in g = (S, A) by S/A. The set of all adjacent links of i ∈ N with the set E ⊆ L as the formed links is written as E_i = {{i, j} | {i, j} ∈ E, i, j ∈ N, i ≠ j}. We use the notation N (A) = {i| {i, j} ∈ A forsome j} as the set of players who have at least one link in g and | · | as the cardinality of a set.

Throughout this paper, we suppose that any game v is zero-normalized, that is, v ({i}) =0 for any i ∈ N. We denote the collection of all zero-normalized games by $G_{0}^{N}$ .

It is necessary to discuss what profit should be allocated in order to obtain a solution of the game v for any communication structure among the players. An allocation rule of a game v for communication structures is a function $ψ^{v} : {CS}^{N} \to ℝ^{n}$ . Two famous allocation rules for communication situations are the Myerson value and position value, which are provided from different viewpoints.

Myerson [11] in 1977 introduced the Myerson value which is an extension of the Shapley value. Let $G^{N} = {g | g \subseteq (N, L)}$ , he defined a measure of the profit by the game v as $r (S, A) = \sum_{H \in S / A} v (H)$ for any $(S, A) \in G^{N}$ , reflecting the reward obtained by the coalition S posessing the communication links A. Given a communication structure (N, L) ∈ CS^N, he defined the graph game (N, v^L), v^L for short, by $v^{L} (S) = r (S, L (S))$ for any S ⊆ N. The Myerson value for every (N, L) ∈ CS^N is defined as $μ_{i}^{v} (N, L) = {Sh}_{i} (v^{L}), {for any i \in N .$

Different from the Myerson value, Meessen [7] and Borm et al. [2] proposed the position value which more focuses on the role of links. Given a communication structure (N, L) ∈ CS^N, the link game (L, r^v), denoted by r^v for simplicity, is given by $r^{v} (E) = r (N, E)$ for any E ⊆ L. Let V^N be the set of all link games over N. We denote the restriction of r^v to coalition T ⊆ N by $r_{T}^{v} (E) = r^{v} (E \cap L (T))$ for any E ⊆ L.

The position value for any (N, L) ∈ CS^N is defined as $π_{i}^{v} (N, L) = \frac{1}{2} \sum_{l \in L_{i}} {Sh}_{l} (r^{v}), {for any i \in N .$

As it is known to all, the set of all functions ${r_{A} | A \subseteq L}$ constructs a basis of the linear space V^N where r_A is defined for any $A \subseteq L$ as $r_{A} (E) = {\begin{matrix} 1, & if E \supseteq A, \\ 0, & otherwise . \end{matrix}$

Thus, any link game r^v can be written as a unique linear combination of $r_{A} (A \subseteq L)$ , i.e.,

r^{v} = \sum_{A \subseteq L} α_{A} \cdot r_{A},

where α_A = ∑_A′⊆A (-1) ^|A|-|A′|r^v (A) is known as the Harsanyi dividend of the link set A.

In fact, an alternative expression for the Myerson value is $μ_{i}^{v} (N, L) = \sum_{A \subseteq L; A_{i} \neq \emptyset} α_{A} \cdot \frac{1}{| N (A) |}, {for any i \in N,$ and for position value is $π_{i}^{v} (N, L) = \sum_{A \subseteq L; A_{i} \neq \emptyset} α_{A} \cdot \frac{| A_{i} |}{2 | A |}, {for any i \in N .$

2.2 Fuzzy cooperative games

There are some situations where players do not fully participate in a coalition, but take action according to the level of participation. Aubin [1] studied the problem at first by the proposal of fuzzy coalition. We will use the word “crisp” to differentiate the fuzzy things and non-fuzzy things.

Let N be a finite set of players. A fuzzy set in N is a function U : N → [0, 1]. We denote $F^{N}$ as the family of fuzzy sets in N. A fuzzy coalition is defined as a fuzzy set $U \in F^{N}$ , where U_i represents the participation level of the player i ∈ N in this coalition. We define the carrier of fuzzy coalition U by U^cr = {i ∈ N|U_i ≠ 0} and denote Q (U) = {U_i|U_i > 0, i ∈ N}. Take $O, R \in F^{N}$ , we use the notation O ≤ R if O_i = R_i or 0 for all i ∈ N and denote O + R = (O₁ + R₁, O₂ + R₂, …, O_n + R_n) if O_i + R_i ≤ 1 for any i ∈ N. Each fuzzy coalition e^S with $e_{i}^{S} = 1$ if i ∈ S and otherwise $e_{i}^{S} = 0$ corresponds to the situation where the players within S fully cooperate. We write eⁱ instead of e^{i}.

A fuzzy cooperative game is a function $v^{f} : F^{N} \to ℝ$ such that v^f (e^∅) =0. The set of fuzzy cooperative games is denoted by FG^N. The associated crisp gamew ∈ G^N of v^f is defined as w (S) = v^f (e^S) for each S ⊆ N.

In the following context, we fix $U \in F^{N}$ with |Q (U) | = m. We write the non-zero elements in Q (U) in a increasing order h₁ < h₂ < ⋯ < h_m and h₀ = 0. In 1980, Butnariu [3] firstly introduced a limited subclass of fuzzy cooperative games as follows.

Definition 2.1. A fuzzy cooperative game v^f is said to be with proportional form if and only if $v^{f} (U) = \sum_{k = 1}^{m} h_{k} \cdot w ([U]^{h_{k}})$ for any $U \in F^{N}$ , here [U] ^{h
_k} is the set of players in fuzzy coalition U with the same participation level h_k, i.e., [U] ^{h
_k} = {i|i ∈ N, U_i = h_k} for each k ∈ {1, 2, …, m}.

However, most of fuzzy cooperative games with proportional form are neither monotone non-decreasing nor continuous with regards to the rate of participation of players. In 2001 Tsurumi et al. [16] proposed a new class of fuzzy cooperativegames.

Definition 2.2. A fuzzy cooperative game v^f is said to be with Choquet integral formif and only if $v^{f} (U) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot w ([U]_{h_{k}})$ for any $U \in F^{N},$ where [U] _{h
_k} is the set of players in fuzzy coalition U with the participation level not smaller than h_k, i.e., [U] _{h
_k} = {i|i ∈ N, U_i ≥ h_k} for each k = 1, 2, …, m.

Denote the set of fuzzy cooperative games with Choquet integral form by ${FG}_{c}^{N}$ . Tsurumi et al. [16] also gave the expression of the Shapley value with fuzzy coalition U over ${FG}_{c}^{N}$ , i.e., $φ^{ch} (v^{f}) (U) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot Sh (w) ([U]_{h_{k}}),$ where Sh (w) ([U] _{h
_k}) is the Shapley value of the restriction to coalition [U] _{h
_k} of the associated crisp game w.

Later, Meng and Zhang [10] introduced the fuzzy cooperative games with multilinear extensionform.

Definition 2.3. A fuzzy cooperative game v^f is said to be with multilinear extension formif and only if $v^{f} (U) = \sum_{T \subseteq U^{cr}} \prod_{i \in T} \prod_{i \in U^{cr} ∖ T} (1 - U_{i}) \cdot w (T)$ for any $U \in F^{N}$ .

In 2009, Yu and Zhang [18] defined the imputation setI^U (v^f) of the fuzzy cooperative game v^f with fuzzy coalition U as a set of n-dimensional payoff vectors x satisfying that

x_i = 0 when i ∉ U^cr,

∑_i∈Nx_i = v^f (U),

x_i ≥ v^f (U_i).

and the fuzzy core as

\begin{matrix} {\tilde{C}}^{U} (v^{f}) & = & {x \in ℝ^{n} | \sum_{i \in N} x_{i} = v^{f} (U), \\ \sum_{i \in (U_{T})^{cr}} x_{i} \geq v^{f} (U_{T}) for any T \subseteq N} \end{matrix}

where

U_{T} \in F^{N}

is defined by

(U_{T})_{i} = {\begin{matrix} U_{i}, & if i \in T, \\ 0, & otherwise . \end{matrix}

3 Position value for communication situations with fuzzy coalition

Here we pay attention to the uncertainty of participation levels of players and perform our study on a framework of communication structure with fuzzy coalition.

A communication structure with fuzzy coalition is described as a triple γ = (U, N, L) where (N, L) ∈ CS^N and U is a fuzzy coalition with U_i > 0 being interpreted as the participation level of player i ∈ N. The set of communication structures with fuzzy coalition over N is denoted by CSF^N. Let ${GF}^{N} = {(W, S, A) | W \leq U, W^{cr} = S, and (S, A) \subseteq (N, L)}$ , then $(W, S, A) \in {GF}^{N}$ is said to be a graph with fuzzy coalition and (S, A) is called its associated crisp graph. The basic concepts of a graph with fuzzy coalition and its associated crisp graph about “connected”, “component”and so on from graph theory are consistent.

To provide an allocation rule in this setting, following Myerson we need to measure the potential profit of players using their communication abilities and participation levels by a game v.

Definition 3.1. A function $\tilde{r} : {GF}^{N} \to ℝ$ is said to be a measure of profit obtained for all the players in a communication structure with fuzzy coalition (U, N, L) by v if it satisfies

$\tilde{r} (W, S, A) = r (S, A)$ when W = e^S,

$\tilde{r} (W, S, A) \geq \tilde{r} (W, S, A ∖ {l})$ for any {l} ∈ A,

$\tilde{r} (W, S, A) = \sum_{H \in S / A} \tilde{r} (W, H, A (H))$ .

Let g = (S, A) be a crisp graph, we assert that k · g = (ke^S, S, A), k ∈ (0, 1]. Given two graphs with fuzzy coalition (W¹, S¹, A¹) and (W², S², A²), if $W_{i}^{1} + W_{i}^{2} \leq 1$ for all i ∈ N we define the operation $\begin{matrix} (W^{1}, S^{1}, A^{1}) + (W^{2}, S^{2}, A^{2}) \\ = (W^{1} + W^{2}, S^{1} \cup S^{2}, A^{1} \cup A^{2}) . \end{matrix}$

Now we begin to consider several games relevant to profit measure served for the main results. For any (U, N, L) ∈ CSF^N, firstly we take a fuzzy game $\tilde{v^{L}} (W) = \tilde{r} (W, W^{cr}, L), for all W \leq U .$

Again we define an auxiliary crisp game, called the link game with fuzzy coalition, i.e., for any E ⊆ L, $r^{U, v} (E) = \tilde{r} (U, N, E),$ abbreviated to r^U. If we take U = e^N, obviously r^U = r^v. In addition, $r^{\bar{v}}$ is called the included link game where $\bar{v} (S) = \tilde{v^{L}} (U_{S}) .$

An allocation rule for communication structures with fuzzy coalition by v is a function $Ψ^{v} : {CSF}^{N} \to ℝ^{n}$ such that $\sum_{i \in N} Ψ_{i}^{v} (U, N, L) = r^{U} (L)$ for all (U, N, L) ∈ CSF^N.

Definition 3.2. Given a profit measure $\tilde{r}$ , for any (U, N, L) ∈ CSF^N by v, the $\tilde{r}$ -position value is defined as

$π_{i}^{\tilde{r}, v} (U, N, L) = \frac{1}{2} \sum_{l \in L_{i}} {Sh}_{l} (r^{U}), {for any i \in N .$

Apparently, the $\tilde{r}$ -position value is an allocation rule, defined in the spirit of crisp position value. Definition 3.2 implies that when a communication structure with fuzzy coalition is degenerated into a crisp communication structure, the $\tilde{r}$ -position value corresponds to the crisp position value, i.e., $π^{\tilde{r}, v} (U, N, L) = π^{v} (N, L)$ if U = e^N.

Remark 1. Ghintran [4] took into account the asymmetries among players and proposed the weighted position values by dividing the Shapley value of a link in the link game (Borm et al. [2]) unequally between its two incident players. However, in our model, the newly defined link game has covered the information of players’ participation levels and the $\tilde{r}$ -position value is obtained by dividing the Shapley value of each link equally between its two endpoints.

In this paper, let $W \in F^{N}$ be fixed with |Q (W) | = d, Q (W) = {l₁, l₂, …, l_d} and 0 = l₀ < l₁ < l₂ < ⋯ < l_d. In the following, we will provide a specific method to measure the profit for communication situations with fuzzy coalition by introducing the concept of level partition.

Definition 3.3. A level partition of $(W, S, A) \in {GF}^{N}$ is a finite sequence $(s_{k}, g_{k})_{k = 1}^{d}$ where s_k ∈ (0, 1], g_k = (T_k, E_k) with T_k ⊆ S, and E_k ⊆ A, satisfying that $\sum_{k = 1}^{d} s_{k} \cdot g_{k} = (W, S, A^{'})$ , A′ ⊆ A .

Select a level partition $lp = (s_{k}, g_{k})_{k = 1}^{d}$ , for all $(W, S, A) \in {GF}^{N}$ we use the following measure defined by v, ${\tilde{r}}^{lp} (W, S, A) = \sum_{k = 1}^{d} s_{k} \cdot r (T_{k}, E_{k}) .$

However, ${\tilde{r}}^{lp}$ is not always a profit measure, i.e., satisfying the three conditions in Definition 3.1. In the next subsections, we are devoted to the position values over CSF^N based on both of Choquet model and multilinear model.

3.1 On Choquet model

We consider such a level partition, called the Choquet partition, i.e., $cp (W, S, A) = (l_{k} - l_{k - 1}, ([W]_{l_{k}}, A))_{k = 1}^{d},$ for any $(W, S, A) \in {GF}^{N}$ .

Choquet partition attracts those players who have a common part of participation levels to be together, which is indeed a level partition since $\sum_{k = 1}^{d} [l_{k} - l_{k - 1}] \cdot ([W]_{l_{k}}, A) = (W, S, A) .$

As a result, we derive that if $\tilde{r} = {\tilde{r}}^{cp}$ , ${\tilde{r}}^{cp} (W, S, A) = \sum_{k = 1}^{d} [l_{k} - l_{k - 1}] \cdot r ([W]_{l_{k}}, A)$ and then $\begin{matrix} r^{U} (E) & = & {\tilde{r}}^{cp} (U, N, E) \\ = & \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot r ([U]_{h_{k}}, E) \\ = & \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot r_{[U]_{h_{k}}}^{v} (E), \end{matrix}$ where $r_{[U]_{h_{k}}}^{v}$ is the restriction to coalition [U] _{h
_k} of the crisp link game r^v. Apparently, ${\tilde{r}}^{cp}$ is a profit measure according to the Definition 3.1. A specific allocation rule with Choquet integral form, called the ${\tilde{r}}^{cp}$ -position value, is presented immediately for any i ∈ N, $π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot π_{i}^{v} ([U]_{h_{k}}, L) .$ (1)

It is noted that the ${\tilde{r}}^{cp}$ -position value is in fact a linear combination of crisp position value. Now, we provide two similar properties as crisp position value for an allocation rule over CSF^N.

Choquet-component efficiency: for any component H ∈ N/L and (U, N, L) ∈ CSF^N, $\sum_{i \in H} Ψ_{i}^{v} (U, N, L) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}} \cap H) .$

f-balanced link contributions: for all (U, N, L) ∈ CSF^N and i, j ∈ N, $\begin{matrix} \sum_{l \in L_{j}} [Ψ_{i}^{v} (U, N, L) - Ψ_{i}^{v} (U, N, L ∖ {l})] \\ = \sum_{l \in L_{i}} [Ψ_{j}^{v} (U, N, L) - Ψ_{j} (U, N, L ∖ {l})] . \end{matrix}$

In the following, we will axiomatize the ${\tilde{r}}^{cp}$ -position value with Choquet-component efficiency and f-balanced link contributions in terms of Slikker [13].

Theorem 3.4. There exists one unique allocation rule ${\tilde{r}}^{cp}$ -position value satisfying Choquet-component efficiency and f-balanced link contributions based on Choquet model.

Proof. (1) First, we prove that $π^{{\tilde{r}}^{cp}, v}$ satisfies Choquet-component efficiency and f-balanced link contributions.

Choquet-component efficiency: for any component H ∈ N/L and (U, N, L) ∈ CSF^N, by the definition of $π^{{\tilde{r}}^{cp}, v}$ and efficiency of Shapley value we have $\begin{matrix} \sum_{i \in H} π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) \\ = \sum_{i \in H} \frac{1}{2} \sum_{l \in L_{i}} {Sh}_{l} (r^{U}) \\ = \sum_{l \in L (H)} {Sh}_{l} (r^{U}) \\ \overset{*}{=} r^{U} (L (H)) \\ = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}} \cap H), \end{matrix}$ where the “*” part is true with the fact that for any l ∈ L (H), A ⊆ L ∖ {l}, $\begin{matrix} r^{U} (A \cup {l}) - r^{U} (A) \\ = r^{U} ((A \cap L (H)) \cup {l}) - r^{U} (A \cap L (H)) . \end{matrix}$

f-balanced link contributions: by simple calculations for any (U, N, L)∈ CSF^N and i, j ∈ N, we have $\begin{matrix} \sum_{l \in L_{j}} [π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) - π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L ∖ {l})] \\ = \sum_{l \in L_{j}} [\sum_{A \subseteq L} r_{A} \cdot \frac{| A_{i} |}{2 | A |} - \sum_{A \subseteq L ∖ l} r_{A} \cdot \frac{| A_{i} |}{2 | A |}] \\ = \sum_{l \in L_{j}} \sum_{A \subseteq L; l \in A} r_{A} \cdot \frac{| A_{i} |}{2 | A |} \\ = \sum_{A \subseteq L} \sum_{l \in A_{j}} r_{A} \cdot \frac{| A_{i} |}{2 | A |} \\ = \sum_{A \subseteq L} r_{A} \cdot \frac{| A_{i} | | A_{j} |}{2 | A |} \\ = \sum_{l \in L_{i}} [π^{{\tilde{r}}^{cp}, v} (U, N, L) - π^{{\tilde{r}}^{cp}, v} (U, N, L ∖ {l})] . \end{matrix}$

(2) Let Ψ^v be an allocation rule over CSF^N which satisfies Choquet-component efficiency and f-balanced link contributions. We verify that $Ψ^{v} (U, N, L) = π^{{\tilde{r}}^{cp}, v} (U, N, L)$ for any (U, N, L) ∈ CSF^N by induction on |L|.

First, if |L|=0, we obtain that $Ψ^{v} (U, N, L) = π^{{\tilde{r}}^{cp}, v} (U, N, L)$ by Choquet-component efficiency.

Second, let $k \in ℤ_{+}$ , we assume that when |L| ≤ k - 1, $Ψ^{v} (U, N, L) = π^{{\tilde{r}}^{cp}, v} (U, N, L)$ . Now, we only need to prove that when |L| = k, Ψ^v (U, N, L) $= π^{{\tilde{r}}^{cp}, v} (U, N, L)$ also holds. The process of proof is as follows.

Given a component H ∈ N/L. If |H|=1, it follows directly from Choquet-component efficiency that $Ψ^{v} (U, N, L) = π^{{\tilde{r}}^{cp}, v} (U, N, L)$ . If |H|≥2, denote H = {1, 2, …, h}. Using Choquet-component efficiency and f-balanced link contributions, we have the system of equalities, ${\begin{matrix} | L_{2} | Ψ_{1}^{v} (U, N, L) - | L_{1} | Ψ_{2}^{v} (U, N, L) \\ = \sum_{l \in L_{2}} Ψ_{1}^{v} (U, N, L ∖ {l}) - \sum_{l \in L_{1}} Ψ_{2}^{v} (U, N, L ∖ {l}), \\ ⋮ \\ | L_{h} | Ψ_{1}^{v} (U, N, L) - | L_{1} | Ψ_{h}^{v} (U, N, L) \\ = \sum_{l \in L_{h}} Ψ_{1}^{v} (U, N, L ∖ {l}) - \sum_{l \in L_{1}} Ψ_{h}^{v} (U, N, L ∖ {l}), \\ \sum_{i \in H} Ψ_{i}^{v} (U, N, L) \\ = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}} \cap H) . \end{matrix}$

It is straightforward to get that the coefficient determinant of this system is non-negative. Consequently, this set of equations with h variables, $Ψ_{1}^{v} (U, N, L)$ , …, $Ψ_{h}^{v} (U, N, L)$ , has a unique solution. Meanwhile, because $π^{{\tilde{r}}^{cp}, v}$ also satisfies Choquet-component efficiency and f-balanced link contribution, $π^{{\tilde{r}}^{cp}, v}$ is exactly the unique solution.

Hence, we conclude that ${\tilde{r}}^{cp}$ -position value is the unique allocation rule satisfying Choquet-component efficiency and f-balanced link contributions. □

Example 1. Consider a river with a number of tributaries, along which four villages are distributed. Now theses villages decide to develop the river for irrigation and tourism in order to obtain a certain economic benefits. However, every of villages could not offer all the human resources and material resources to take part in the cooperation. Let the villages be represented by the nodes of a graph and rivers between them by links. U = (0.6, 0.5, 0.5, 0.6) is a fuzzy coalition with the coordinate meaning the participation fraction of the corresponding village. This situation can be modeled by a communication structure with fuzzy coalition γ = (U, {1, 2, 3, 4} , {{1, 2} , {2, 3} , {1, 4} }) in Fig. 1. Given a game v ∈ G₀^N with the value v (S) for any S ⊆ {1, 2, 3, 4} being the worth gained by common cooperation of villages in S: $\begin{matrix} v (12) = 60, v (14) = 100, v (23) = 50, \\ v (124) = 100, v (123) = 120, v (1234) = 200 . \end{matrix}$

Fig.1

(N, L) in Example 1.

Calculate the link game r^U based on Choquet model and we derive that r^U ({1, 2}) =30, r^U ({2, 3}) =25, r^U ({1, 4}) =60, r^U ( {{1, 2} , {2, 3} } ) = 30, r^U ( {{1, 2} , {1, 4} } ) = 30, r^U ( {{1, 4} , {2, 3} } ) = 85, r^U ( {{1, 2} , {2, 3} , {1, 4} } ) = 110 . Therefore, the payoff vector for these four villages is $Π^{{\tilde{r}}^{cp}, v} (U, N, L) = (42 \frac{1}{12}, 27 \frac{1}{12}, 12 \frac{11}{12}, 27 \frac{11}{12}) .$

It is easy to verify that this result is consistent with that obtained by Equation (1).

In 2005, Slikker [14] provided another characterization of crisp position value using potential function. Here we extend the characterization to the setting of communication situations with fuzzy coalition.

Let P^v be a real valued function defined on CSF^N by v, we define the marginal contribution of a player to a communication situation with fuzzy coalition as the sum of marginal contribution of each of his incident links, i.e., for any i ∈ N, $\begin{matrix} D_{i} P^{v} (U, N, L) \\ = \sum_{l \in L_{i}} [P^{v} (U, N, L) - P^{v} (U, N, L ∖ {l})] . \end{matrix}$

A function $P^{v} : {CSF}^{N} \to ℝ$ is said to be a link potential function with fuzzy coalition if P^v (U, N, ∅) =0 and the sum of marginal contribution of each player with respect to D is equal to the worth produced by all the feasible links, i.e., $\sum_{i \in N} D_{i} P^{v} (U, N, L) = r^{U} (L)$ for any (U, N, L) ∈ CSF^N such that L≠ ∅.

Theorem 3.5. There exists one unique link potential function with fuzzy coalition P^v for v over CSF^N based on Choquet model. In addition, for any (U, N, L) ∈CSF^N and i ∈ N, it holds that $D_{i} P^{v} (U, N, L) = π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L)$ .

Proof. First, we prove that a link potential function with fuzzy coalition exists. Let (U, N, L) ∈ CSF^N, consider that $P^{v} (U, N, L) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{A \subseteq L ([U]_{h_{k}})} r_{A} \cdot \frac{1}{2 | A |} .$

Obviously, we have P^v (U, N, ∅) =0. Furthermore, $\begin{matrix} \sum_{i \in N} D_{i} P^{v} (U, N, L) \\ = \sum_{i \in N} \sum_{l \in L_{i}} \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot [\sum_{A \subseteq L ([U]_{h_{k}})} r_{A} \cdot \frac{1}{2 | A |} \\ - \sum_{A \subseteq L ([U]_{h_{k}}) ∖ {l}} r_{A} \cdot \frac{1}{2 | A |}] \\ = \sum_{i \in N} \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{A \subseteq L ([U]_{h_{k}})} r_{A} \cdot \frac{| A_{i} |}{2 | A |} \\ = \sum_{i \in N} \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot π_{i}^{v} ([U]_{h_{k}}, L) \\ = \sum_{k = 1}^{m} r^{v} ([U]_{h_{k}}) \\ = r^{U} (L) . \end{matrix}$

Next, it remains to show the uniqueness. According to the definition of link potential function with fuzzy coalition, we easily obtain the recurrence relation $\begin{matrix} P^{v} (U, N, L) \\ = \frac{1}{2 | L |} [r^{U} (L) + 2 \sum_{l \in L} P^{v} (U, N, L ∖ {l})] \end{matrix}$ with the initial value P^v (U, N, ∅) =0. Hence, P^v (U, N, L) can be defined uniquely in a recursive way. □

In the following, we will provide some nice results about $\tilde{r}$ -position value over CSF^N.

Proposition 3.6. Let (U, N, L) ∈ CSF^N, $\tilde{r} = {\tilde{r}}^{cp}$ , the following statements hold

If (N, L) is cycle-free, then $r^{U} (E) = r^{\bar{v}} (E)$ for any E ⊆ L.

$\sum_{i \in H} π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) = \sum_{i \in H} π_{i}^{\bar{v}} (N, L)$ for any H ∈ N/L.

If the game v is superadditive, then $C (r^{\bar{v}}) \subseteq C (r^{U}) .$

If game v is convex, and (N, L) has no cycles, then r^U is also convex and $π^{{\tilde{r}}^{cp}, v} (U, N, L) \in {\tilde{C}}^{U} (\tilde{v^{L}})$ .

Proof. (1) For any E ⊆ L, when $\tilde{r} = {\tilde{r}}^{cp}$ , by straight calculations we observe that $\begin{matrix} r^{U} (E) \\ = {\tilde{r}}^{cp} (U, N, E) \\ = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot r_{[U]_{h_{k}}}^{v} (E) \\ = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{H \in [U]_{h_{k}} / E} v (H) \\ = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{C \in N / E} \sum_{H \in [U]_{h_{k}} \cap C / E} v (H) \end{matrix}$ and $\begin{matrix} r^{\bar{v}} (E) \\ = \sum_{C \in N / E} {\tilde{r}}^{cp} (U, C, L (C)) \\ = \sum_{C \in N / E} \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}} \cap C) \\ = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{C \in N / E} \sum_{H \in U_{h_{k}} \cap C / L} v (H) . \end{matrix}$

From this, we conclude that if for each i, j ∈ [U] _{h
_k} ∩ C, C ∈ N/E, i and j are connected in the graph ([U] _{h
_k} ∩ C, E) iff i and j are connected in the graph ([U] _{h
_k} ∩ C, L), then $r^{U} = r^{\bar{v}}$ holds. Now we place our emphasis on the proof of “iff” part.

When i and j are connected in the graph ([U] _{h
_k} ∩ C, E), it is a direct consequence that i and j are also connected in the graph ([U] _{h
_k} ∩ C, L).

If we assume that i and j are connected in the graph ([U] _{h
_k} ∩ C, L), but not connected in the graph ([U] _{h
_k} ∩ C, E), then there exists a path from i to j in ([U] _{h
_k} ∩ C, L) passing the edges outside E. Further, since i, j ∈ C, C ∈ N/E, we learn that there also exists a path from i to j in ([U] _{h
_k} ∩ C, E) which deduces a contradiction with only having one path from i to j by the condition that (N, L) is cycle-free.

Hence, for any E ⊆ L, $r^{U} (E) = r^{\bar{v}} (E)$ .

(2) For each H ∈ N/L, we obtain that $\begin{matrix} \sum_{i \in H} π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) \\ = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}} \cap H) . \end{matrix}$ from the Choquet-component efficiency. Moreover, by the component efficiency of crisp position value, we have $\begin{matrix} \sum_{i \in H} π_{i}^{\bar{v}} (N, L) = \bar{v} (H) = \tilde{v^{L}} (U_{H}) \\ = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}} \cap H) . \end{matrix}$

Hence, $\sum_{i \in H} π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) = \sum_{i \in H} π_{i}^{\bar{v}} (N, L)$ for any H ∈ N/L.

(3) According to the part (1), we get that $r^{U} (E) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{C \in N / E} \sum_{H \in [U]_{h_{k}} \cap C / E} v (H)$ and $r^{\bar{v}} (E) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{C \in N / E} \sum_{H \in U_{h_{k}} \cap C / L} v (H) .$ With the fact that for each i, j ∈ [U] _{h
_k} ∩ C, the statement “i and j are connected in ([U] _{h
_k}, E)” is the unnecessary and sufficient condition of “i and j are connected in ([U] _{h
_k}, L)”, the superadditivity of game v implies that $C (r^{\bar{v}}) \subseteq C (r^{U})$ .

(4) Let game v be convex and (N, L) be cycle-free.

Using the Corollary 1 in [17], we clearly obtain that $r_{[U]_{h_{k}}}^{v}$ (k ∈ {1, 2, …, m}) is convex, namely, for each E, F ⊆ L, $r_{[U]_{h_{k}}}^{v} (E \cup F) + r_{[U]_{h_{k}}}^{v} (E \cap F) \geq r_{[U]_{h_{k}}}^{v} (E) + r_{[U]_{h_{k}}}^{v} (F)$ . Further, it follows from $r^{U} (E) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot r_{[U]_{h_{k}}}^{v} (E)$ that r^U is convex.

On the one hand, along with $\sum_{i \in N} π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{i \in N} π_{i}^{v} ([U]_{h_{k}}, L) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}})$ and $\tilde{v^{L}} (U) = \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}})$ , we have $\sum_{i \in N} π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) = \tilde{v^{L}} (U) .$ On the other hand, because (N, L) is cycle-free and v is convex, it follows that $π^{v} ([U]_{h_{k}}, L) \in C ((v^{L})_{[U]_{h_{k}}})$ from Theorem 4 in [17]. Namely, $\sum_{i \in N} π_{i}^{v} ([U]_{h_{k}}, L)$ = v^L ([U] _{h
_k}) and for any S ⊆ N, $\sum_{i \in S} π_{i}^{v} ([U]_{h_{k}}, L)$ ≥v^L ([U] _{h
_k} ∩ S). So, $\begin{matrix} \tilde{v^{L}} (U_{S}) & = & \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot v^{L} ([U]_{h_{k}} \cap S) \\ \leq & \sum_{k = 1}^{m} [h_{k} - h_{k - 1}] \cdot \sum_{i \in (U_{S})^{cr}} π_{i}^{v} ([U]_{h_{k}}, L) \\ = & \sum_{i \in (U_{S})^{cr}} π_{i}^{{\tilde{r}}^{cp}, v} (U, N, L) . \end{matrix}$

Hence, $π^{{\tilde{r}}^{cp}, v} (U, N, L) \in {\tilde{C}}^{U} (\tilde{v^{L}})$ is proved. □

3.2 On multilinear model

Similar to Choquet model, we consider another level partition over CSF^N, called the multilinear partition, i.e., for any $(W, S, A) \in {GF}^{N}$ , $\begin{matrix} ml (W, S, A) \\ = (\prod_{i \in T} W_{i} \prod_{i \in S ∖ T} (1 - W_{i}), (T, A))_{T \subseteq S} . \end{matrix}$

In the multilinear partition, a probabilistic explanation can be given. It is indeed a level partition, since $\sum_{T \subseteq S} \prod_{i \in T} W_{i} \prod_{i \in S ∖ T} (1 - W_{i}) \cdot (T, A) = (W, S, A)$ .

Further, $\begin{matrix} {\tilde{r}}^{ml} (W, S, A) \\ = \sum_{T \subseteq S} \prod_{i \in T} W_{i} \prod_{i \in S ∖ T} (1 - W_{i}) \cdot r (T, A), \end{matrix}$ and then $\begin{matrix} r^{U} (E) \\ = {\tilde{r}}^{ml} (U, N, E) \\ = \sum_{T \subseteq N} \prod_{i \in T} U_{i} \prod_{i \in N ∖ T} (1 - U_{i}) \cdot r (T, E), \end{matrix}$

Obviously, ${\tilde{r}}^{ml}$ is a profit measure satisfying the Definition 3.1. Then, a specific allocation rule ${\tilde{r}}^{ml}$ -position value is derived, i.e.,

$\begin{matrix} π^{{\tilde{r}}^{ml}, v} (U, N, L) \\ = \sum_{T \subseteq N} \prod_{i \in T} U_{i} \prod_{i \in N ∖ T} (1 - U_{i}) \cdot π^{v} (T, L (T)) . \end{matrix}$ (2)

We observe that ${\tilde{r}}^{ml}$ -position value is a linear combination of crisp position value.

Analogously, we will use multilinear-component efficiency instead of Choquet-component efficiency for an allocation rule and give the characterization of ${\tilde{r}}^{ml}$ -position value by multilinear-component efficiency and f-balanced link contributions.

Multilinear-component efficiency: for any component H ∈ N/L and (U, N, L) ∈CSF^N, $\begin{matrix} \sum_{i \in H} Ψ_{i}^{v} (U, N, L) \\ = \sum_{T \subseteq H} \prod_{i \in T} U_{i} \prod_{i \in H ∖ T} (1 - U_{i}) \cdot v^{L} (T) . \end{matrix}$

Theorem 3.7. There exists one unique allocation rule ${\tilde{r}}^{ml}$ -position value satisfying multilinear-component efficiency and f-balanced link contributions based on multilnear model.

Example 2. Consider the situation in Example 1 with the only change: here U_i is interpreted as a probability that the village i ∈ {1, 2, 3, 4} is willing to jointly develop the river and these probabilities are mutually independent. We calculate the link game based on multilinear model: r^U ({1, 2}) =18, r^U ({2, 3}) =12.5, r^U ({1, 4}) =36, r^U ( {{1, 2} , {2, 3} } ) = 32, r^U ( {{1, 2} , {1, 4} } ) = 43.2, r^U ( {{1, 4} , {2, 3} } ) = 48.5, r^U ( {{1, 2} , {2, 3} , {1, 4} } ) = 60.8. Therefore, the expected payoff vector for theses villages is $π^{{\tilde{r}}^{ml}, v} (U, N, L) = (23 \frac{7}{40}, 14 \frac{1}{2}, 7 \frac{9}{40}, 15 \frac{9}{10}),$ which coincides with that obtained by Equation (2).

Similar results based on multilinear model as Choquet model are presented as follows and we omit the proof.

Theorem 3.8. There exists one unique link potential function with fuzzy coalition P^v for v over CSF^N based on multilinear model. In addition, for any (U, N, L) ∈ CSF^N and i ∈ N, it holds that $D_{i} P^{v} (U, N, L) = π_{i}^{{\tilde{r}}^{ml}, v} (U, N, L)$ .

Proposition 3.9. Let (U, N, L) ∈ CSF^N, $\tilde{r} = {\tilde{r}}^{ml}$ , the following statements hold

If (N, L) is cycle-free, then $r^{U} (E) = r^{\bar{v}} (E)$ for any E ⊆ L.

$\sum_{i \in H} π_{i}^{{\tilde{r}}^{ml}, v} (U, N, L) = \sum_{i \in H} π_{i}^{\bar{v}} (N, L)$ for any H ∈ N/L.

If the game v is superadditive, then $C (r^{\bar{v}}) \subseteq C (r^{U}) .$

If game v is convex, and (N, L) has no cycles, then r^U is also convex and $π^{{\tilde{r}}^{ml}, v} (U, N, L) \in {\tilde{C}}^{U} (\tilde{v^{L}})$ .

4 Another characterization of

\tilde{r}

-position value

Inspired by Kamijo [6], in this section we will provide another characterization of the $\tilde{r}$ -position value which is described by effort function. The effort function is an evaluation of the effort of the player to the surplus generated from essential links for communication structures with fuzzy coalition. Here our aim is to find an allocation rule for the game v over CSF^N in view of the effort function.

Given any communication structure with fuzzy coalition (U, N, L). Let e_i (A, L) denote the effort player i makes to the dividend generated by essential link set A while the formed links are L. Assume that e_i (∅ , L) =0 for any i ∈ N, e (A, L) = (e_i (A, L) ) _(i∈N) is said to be an effort function if for any A ⊆ L, A≠ ∅, it satisfies that

e_i (A, L) ≥0 for all i ∈ N,

∑_i∈Ne_i (A, L) >0.

With the effort function defined, we are dedicated to researching how to divide the dividend of any link set among players. We only consider the setting in which the share of each player in the dividend of any link set is proportional to his effort, that is, each player i accounts for $\frac{e_{i} (A, L)}{\sum_{i \in N} e_{i} (A, L)}$ of the dividend generated by essential links A.

Let ${\tilde{α}}_{A} = \sum_{A^{'} \subseteq A} (- 1)^{| A | - | A^{'} |} r^{U} (A^{'})$ be the dividend of link set A in (U, N, L) by v, the resulting rule has the following form, i.e., for all i ∈ N, $Ψ_{i}^{v} (U, N, L) = \sum_{A \subseteq L} {\tilde{α}}_{A} \cdot \frac{e_{i} (A, L)}{\sum_{j \in N} e_{j} (A, L)},$ which is an allocation rule since $\sum_{i \in N} Ψ_{i}^{v} (U, N, L) = r^{U} (L)$ and called the fuzzy linear proportional effort (FLPE) allocation rule.

It is a easily observed result that the $\tilde{r}$ -position value is a FLPE allocation rule with the effort function $e_{i} (A, L) = {\begin{matrix} c_{0} | A_{i} |, & if A_{i} \neq \emptyset, \\ 0, & otherwise \end{matrix}$ (c₀ > 0) imposed.

Now, we provide a characterization of the $\tilde{r}$ -position value using the effort function following Kamijo [6], in which we suppose that A ⊆ L.

Null effort For all i ∈ N, e_i (A, L) =0 if A_i =∅.

Null effort implies that if a player has no bilateral relationship with others in the set of communication links A, this player makes no effort in the generation of the dividend of A.

Independence from L For all i ∈ N, e_i (A, L′) = e_i (A, A) for any A ⊆ L′ ⊆ L.

Independence from L says that the effort each player makes is only relevant with the essential links, but not the formed links.

Balanced link contributions by effort For all i, j ∈ N, $\begin{matrix} \sum_{l \in A_{j}} [e_{i} (A, L) - e_{i} (A ∖ {l}, L)] \\ = \sum_{l \in A_{i}} [e_{j} (A, L) - e_{j} (A ∖ {l}, L)] \end{matrix}$

This expression can be interpreted as: for any i, j ∈ N, the sum of the marginal contribution of each link in A_j to the effort of player i is the same as the sum of the marginal contribution of each link in A_i to the effort of player j.

Effort efficiency ∑_i∈Ne_i (A, L) = c|A| where c is a non-negative constant.

Effort efficiency requires that the sum of the effort of each player among N is equal to constant times the number of edges in the essential links.

Theorem 4.1. A FLPE allocation rule Ψ^v is the $\tilde{r}$ -position value if and only if the effort function satisfies null effort, independence from L, balanced link contributions by effort and effort efficiency.

Proof. (1) (necessity) The $\tilde{r}$ -position value is a FLPE allocation rule for which can be written as $π_{i}^{\tilde{r}, v} (U, N, L) = \sum_{A \subseteq L} {\tilde{α}}_{A} \cdot \frac{e_{i} (A, L)}{\sum_{i \in N} e_{i} (A, L)}$ with $e_{i} (A, L) = {\begin{matrix} c_{0} | A_{i} |, & if A_{i} \neq \emptyset, \\ 0, & otherwise \end{matrix}$ (c₀ > 0) for each i ∈ N. We can quickly verify that this effort function satisfies null effort, independence from L, balanced link contributions by effort and effort efficiency. The detailed proof is omitted.

(2) (sufficiency) Next we consider the uniqueness part. Assume that Ψ^v is a FLPE allocation rule with the effort function e (A, L) satisfying the four properties mentioned in this theorem. We will prove that the result $e_{i} (A, L) = {\begin{matrix} c_{0} | A_{i} |, & {if \end{matrix} A_{i} \neq \emptyset, 0, & {otherwise$ (c₀ > 0) is true by induction on the number of elements in the link set A.

If |A|=1, let A = {{i, j} }. By the null effort, effort efficiency and independence from L successively, we get that $\begin{matrix} \sum_{k \in N} e_{k} ({{i, j}}, L) \\ = \sum_{k \in N; {{i, j}}_{k} \neq \emptyset} e_{k} ({{i, j}}, L) \\ = e_{i} ({{i, j}}, L) + e_{j} ({{i, j}}, L) \\ = c | {{i, j}} | = c \end{matrix}$ Meanwhile, from the balanced link contributions by effort, we get the equation $e_{i} ({{i, j}}, L) - e_{j} ({{i, j}}, L) = 0 .$

Therefore, we can obtain that if k ∉ {i, j}, $e_{i} ({{i, j}}, L) = e_{j} ({{i, j}}, L) = \frac{c}{2}$ and e_k ( {{i, j} } , L ) = 0. Put $c_{0} = \frac{c}{2}$ , the result holds.

Let $z \in ℤ, z > 1$ , we suppose the result is true when |A| < z.

If |A| = z, it follows from the null effort that e_i (A, L) =0 when A_i =∅. Thus, let T = {i|A_i ≠ ∅} = {1, 2, …, t}, according to the balanced link contributions by effort, we have |A_j|e_i (A, L) - |A_i|e_j (A, L) =0 for all i, j ∈ T. Applying effort efficiency, the following equations system is obvious, ${\begin{matrix} | A_{2} | e_{1} (A, L) - | A_{1} | e_{2} (A, L) & = 0, \\ | A_{3} | e_{1} (A, L) - | A_{1} | e_{3} (A, L) & = 0, \\ ⋮ \\ | A_{t} | e_{1} (A, L) - | A_{1} | e_{t} (A, L) & = 0, \\ e_{1} (A, L) + e_{2} (A, L) + \dots + e_{t} (A, L) & = c | A | . \end{matrix}$

We notice that this system of equations has a unique solution for its determinant is non-zero. Moreover, the equations hold if we take $e_{i} (A, L) = \frac{c}{2} | A_{i} | = c_{0} | A_{i} |$ when A_i≠ ∅. In a summary, $e_{i} (A, L) = {\begin{matrix} c_{0} | A_{i} |, & if A_{i} \neq \emptyset, \\ 0, & otherwise \end{matrix}$ and then the result proves to be true if |A| = z.

The theorem proof is completed. □

Now we show the logical independence of the axioms in the characterization.

An effort function e¹ (A, L) given by $e_{i}^{1} (A, L) = {\begin{matrix} 1, & if | A | = 1 and A_{i} = \emptyset, \\ | A_{i} |, & if A_{i} \neq \emptyset, \\ 0, & if A_{i} = \emptyset and | A | \neq 1 \end{matrix}$ for all i ∈ N satisfies independence from L, balanced link contributions by effort and effort efficiency, but does not satisfy null effort.

An effort function e² (A, L) given by $\begin{matrix} e_{i}^{2} (A, L) \\ = {\begin{matrix} 0, & if A_{i} = \emptyset, \\ \frac{| A |}{| {i | A_{i} \neq \emptyset} |} + | A_{i} |, & if A = L, A_{i} \neq \emptyset, \\ | A_{i} |, & if A ⫋ L, A_{i} \neq \emptyset \end{matrix} \end{matrix}$ for all i ∈ N satisfies null effort, balanced link contributions by effort and effort efficiency, but does not satisfy independence from L.

An effort function e³ (A, L) given by $e_{i}^{3} (A, L) = {\begin{matrix} 0, & if A_{i} = \emptyset, \\ \frac{| A |}{| {i | A_{i} \neq \emptyset} |}, & if A_{i} \neq \emptyset \end{matrix}$ for all i ∈ N satisfies null effort, independence from L and effort efficiency, but does not satisfy balanced link contributions by effort.

An effort function e⁴ (A, L) given by $e_{i}^{4} (A, L) = 0$ for all i ∈ N satisfies null effort, independence from L and balanced link contributions by effort, but does not satisfy effort efficiency.

Footnotes

Acknowledgments

The research has been supported by the National Natural Science Foundation of China (Grant Nos. 71571143, 71601156, 71671140 and71271171).

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