A matching value for cooperative games

Abstract

A cooperative game describes a situation in which a finite set of n players can generate certain profits by their cooperation. How to allocate the profits effectively among each player is very important and it will affect the stability and the sustainable development of their cooperation. In this paper, by analyzing the characteristics of the existing solutions to solve the profit allocation problem, some concepts such as cooperative unit, perfect matching pair are firstly defined, a construction method of matching order is proposed. Then a new solution called matching value is defined, its basic properties are discussed by a series of theorems. Finally, a matching value under total coalition structure is defined and its axiomatization is proved by three axioms.

Keywords

Cooperative games matching value matching order coalition formation coalition structure

1 Introduction

In a general way, game theory studies cooperative and conflict models using mathematical methods. This paper is about the cooperative game theory [1] which is an important theoretical branch of the game theory. As it can deal with many cooperative phenomena in the real life, many theoretical results of it have achieved important applications in some areas, such as economics, politics [2, 3].

Given a cooperative game, the main problem that arises is how to assign the profit to each player in a reasonable way. In some cases, some players constitute a coalition by mutual cooperation and choice, their most concerned problem is that how to allocate the profit effectively and reasonably. In order to solve this problem, it involves the solution concepts of the cooperative games. Now there are many solution concepts which were defined from different point of view, they can roughly be divided into two major categories: dominant solutions and valuation solutions. Dominant solutions are usually a subset of all imputations and they are consisted by infinite multiple elements. Some usually used dominant solutions are introduced as follows: in 1944, the concept of stable set [4] was put forward by Von Neumann and Morgenstern from the dominance point of view, in 1953, the definition of core [5] was given by Gillies, in 1964, the concept of negotiation set [6] was defined by Aumann and Maschler by the possible mutual negotiations between players, in 1965, the kernel [7] was defined by Davis and Maschler based on the beyond value, in 1969, Nucleolus [8] was proposed based on the beyond values and the dictionary sequence. Unlike the dominant solutions, valuation solutions are usually single-value solutions, some important valuation solutions are introduced as follows: in 1953, Shapley value [9] was proposed by Shapley and its existence and uniqueness were proved, in 1965, Banzhaf value [10] was initially introduced in the context of voting games, in 1977, Owen value [11] was presented by Owen and its axiomatic method is proved, in the same year, Myerson value [12] was proposed by graph theory to study some feasible coalitions, in 1981, τ-value was defined by Tijs [13 –15] from the geometry point of view and its relation with the core was discussed. In addition, there are many other solutions of the cooperative games, such as: consensus value [16], the minimum norm solution [17], Raiffa negotiated solution [18], Alexia values [19], procedural value [20], solidarity value [21] and so on. Based on the introduction of the above solutions, it is easy to see that both the dominant solutions and the valuation solutions are gotten on the premise of the grand coalition can be formed, and the relationship between the coalition formation and the profit allocation are totally ignored. Although the final solution is only decided by different coalition values, a contradiction with the existence of the grand coalition may exist for these completely regardless the relationship among the coalition, coalition values and the final allocation. And even some challenges to the rationality and fairness of the existing solutions may appear for their not considering the implied correlation between each coalition.

Therefore, based on the above analysis, in this paper, we have done the following work: 1) The relations between coalition formation and the profit allocation are analyzed, some concepts such as cooperative unit, perfect matching pair are defined, a construction method of matching order is developed, some related theorems are given to prove the existence of the perfect matching pair and many feasible coalition structures are achieved. 2) The structure of the grand coalition is obtained, a matching value is defined and some basic properties of it are discussed by a series of theorems. 3) Based on the form of all coalition structures, a matching value under total coalition structure is proved by three axioms, the effectives of the matching value are verified by a concrete example.

The rest of this paper is structured as follows: Preliminaries are introduced in Section 2. In Section 3, some important concepts are firstly proposed, a construction method of matching order is developed, and then a new solution called matching value is defined, a series of theorems are developed to discuss its properties. In Section 4, based on the forms of all coalition structures, a matching value under total coalition structure is defined, an axiomatization is proved and an example is given to verify the effectives of the matching value. The conclusions and further studies are discussed in Section 5.

2 Preliminaries

2.1 Concepts of classical cooperative games

A classical cooperative game in coalition form is an ordered pair (N, v), where N = {1, 2, …, n} is the set of players, and v:2^N → R is a map, assigning to each coalition S ∈ 2^N a real number v(S), such that v(∅) = 0. This function v is called the characteristic function of the game and v(S) is called the worth (or value) of coalition S.

Throughout this paper, we shall have frequent occasion to consider some basic concepts and properties of (N, v), which are defined as follows (see [22 –26]).

Definition 1. A classical cooperative game (N, v) is supperadditive, if for any R, T⊆ N, T ∩ R = ∅, it holds $v (R) + v (T) \leq v (R \cup T)$ (1)

Definition 2. A classical cooperative game (N, v) is convex, if for any R, T ⊆ N, it holds $v (R) + v (T) \leq v (R \cup T) + v (R \cap T)$ (2)

Following, we write v(i) instead of v({ i}), N ∖ i instead of N ∖ {i} and v(S ∪ i) instead of v(S ∪ { i}) in abbreviation.

Definition 3. A vector x = (x₁, x₂, …, x_n) ∈ Rⁿ is called an imputation for the game (N, v), if x satisfies $x_{i} \geq v (i), i = 1, 2, \dots, n$ (3) $x (N) = v (N)$ (4)

We denote by E (v) the set of all imputations.

Definition 4. The core of (N, v) is the set $C (v) = {x | v (S) - x (S) \leq 0 (\forall S \subseteq N), x (N) = v (N)}$ (5)

In (5), x (S) = ∑x_i, i ∈ S, x (N) = ∑x_i, i ∈ N .

Definition 5. In a game (N, v), if for any S ⊆ N, the sum of the profits which are allocated to all players in the coalition S is not less than the profits coalition S can gain, then (N, v) satisfies the independent test.

Theorem 1.I n a game (N, v), i fC (v)≠ ∅, thenC (v) is the unique solution satisfying the individually rationalty, supperadditivity and converse reduced game property.

Definition 6. Let (N, v) be a cooperative game, Shapley value is defined as follows $φ_{i} (v) = \sum_{S \subseteq N ∖ i} \frac{| S |! (n - | S | - 1)!}{n!} [v (S \cup i) - v (S)]$ (6)

Theorem 2.Shapley value is the unique solution that satisfies the virtuality, anonymity, collective rationality and the additivity.

Definition 7. For two games (N, u) and (N, v), if for any i ∈ S, T ≠ S, it holds ${v (S) \leq u (S), v (T) = u (T)} \Rightarrow {x_{i} (v) \leq x_{i} (u)}$ (7) then we call it monotonic about coalition S. x_i (v) is an imputation value for player i of (N, v), x_i (u) is an imputation value for player i of (N, u).

It is easy to see that there are some disadvantages of the core. For example, the core is usually an empty set and sometimes the core solution is not unique. In addition, it is not monotonic about coalition S. We also can see that Shapley value does not satisfy the independent test, that is to say, Shapley value is not always contained in the core. What is more, it does not take into account the externality problem of a coalition formation, when a new player joins or quits a coalition, the Shapley value cannot reflect the influence on other players or coalitions. Based on the above analysis, both the core and the Shapley value have their relative deficiencies. Therefore, in the next Section, we will try to define a new solution concept which has some good properties.

3 A matching value of cooperative games

Core and Shapley value are aim to allocate the profits after a coalition is formed. However, the process of coalition formation and profit allocation are mutually influenced. Therefore, in this Section, we will combine these two phases together to propose a new solution concept.

3.1 A construction method of matching order

The formation process of a coalition is very complex, many factors may affect this process. In order to show this process effectively, we firstly give the following Definitions.

Definition 8. Any subset of N is called a cooperative unit, the set of all cooperative units is denoted by $C D$ .

Definition 9. For any two cooperative units A and B, if A chooses B as its optimal cooperative unit, and B chooses A as its optimal cooperative unit, then A and B constitute a perfect matching pair.

Definition 10. We call a process that any two disjoint cooperative units constitute a perfect matching pair a unit link.

If a unit link is completed, a perfect matching pair is formed. Following we describe a possible formation process of the grand coalition {1, 2, 3, 4, 5} by Fig. 1.

Figure 1 only shows a simple form of the possible formation process. In fact, there are many kinds of possible situations. So the discussion on what kinds of coalition can be formed is really important. In this process, the construction of perfect matching pair is the key link. So following we will define an effective order relation between each cooperative units.

Definition 11. For cooperative unit A, B and C, if A cooperates with B, the gained profit of A is not less than the gained profit when A cooperates with C, then we say that A prefers B rather than C, which is denoted by $A : B \underline{≻} C$ (8) we call (8) a matching order. For simplicity, the symbol A : B ≻ C denotes that A strictly prefers B rather than C, the symbol A : B = C denotes that A has the same preference between B and C.

Definition 12. Suppose in a certain phase of coalition formation process, there are some cooperative units R₁, R₂, …, R_m, m ≥ 2, if there exists i, j ∈ {1, 2, …, m}, such that $R_{i} : R_{j} \underline{≻} R_{k}, R_{j} : R_{i} \underline{≻} R_{k}, \forall k \in {1, 2, \dots, m} ∖ {i, j}$ (9) then R_i and R_j constitute a perfect matching pair, we call R_j and R_i the first priority matching unit for each other.

We have known that a unit link involves two cooperative units, sometimes there is a specialty that there is no other difference besides their sizes. So we allocate the net surplus by the standard residual rule, i.e. if there are two units A and B in a cooperation process, unit A will get $v (A) + \frac{| A |}{| A | + | B |} [v (A \cup B) - v (A) - v (B)]$ (10)

Unit B will get $v (B) + \frac{| B |}{| A | + | B |} [v (A \cup B) - v (A) - v (B)]$ (11)

If we set the cooperative unit A and B as the single player, then this allocation result is the same as Shapley value and it is contained in the core.

Theorem 3.For cooperative unitA, BandC $\begin{matrix} A : B \underline{≻} C \Leftrightarrow \frac{1}{| A | + | B |} [v (A \cup B) - v (A) - v (B)] \geq \\ \frac{1}{| A | + | C |} [v (A \cup C) - v (A) - v (C)] \end{matrix}$

Proof. By the Definition of A : B $\underline{≻}$ C, this theorem is clearly true.

Theorem 4.Matching order is a total order. That is, for any cooperative unitA, B and C, it holds

1) Reflexivity $A : B \underline{≻} B, \forall B \in C D;$

2) Antisymmetry $A : B \underline{≻} C, A : C \underline{≻} B \Rightarrow A : B = C, \forall B, C \in C D;$

3) Transitivity $A : B \underline{≻} C, A : C \underline{≻} D \Rightarrow A : B \underline{≻} D, \forall B, C, D \in C D;$ 4) Comparability

$\forall B, C \in C D$ , we have A : B $\underline{≻}$ C or A : C $\underline{≻}$ B.

Theorem 5.Suppose in a certain phase of coalition formation process, the set of all cooperative units are denoted by $C$ , the set of their corresponding first priority matching units are denoted by $F$ . If we have $| C | = | F | = m \geq 2$ , then there must exist a perfect matching pair.

Proof. Let $C = {R_{1}, R_{2}, \dots, R_{m}}$ , since $| C | = | F |$ , so we have $F = {R_{1}, R_{2}, \dots, R_{m}} = C$ . Suppose the first priority matching unit of R₁ is R_p, the first priority matching unit of R_q is R₁, in which p, q ∈ {2, …, m}. If p = q, then R₁ and R_p constitute a perfect matching pair, otherwise we try to find the first priority matching unit of R_p, we set it R_p1, p₁ ∈ {1, …, p - 1, p + 1, …, m}. If p₁ = q, then the proof is complete, otherwise we try to find the first priority matching unit of R_p1, for $| C | = | F |$ , after k (k ≤ m) times, we can get p_k = q. That is to say, there must exist some limited cooperative units such that R₁ in $C$ corresponds to the R₁ in $F$ .We set $\begin{matrix} R_{i} : R_{i + 1} \underline{≻} R_{k}, i = 1, \dots, q - 1, \\ \forall k \in {1, \dots, m} ∖ {i, i + 1} \end{matrix}$ $R_{q} : R_{1} \underline{≻} R_{k}, \forall k \in {1, \dots, m} ∖ {1, q}, q \in {2, \dots, m}$

By the arbitrariness of k, we obtain $\begin{matrix} R_{1} : R_{2} \underline{≻} R_{q}, R_{2} : R_{3} \underline{≻} R_{1}, R_{3} : R_{4} \underline{≻} R_{2}, \dots, \\ R_{q - 1} : R_{q} \underline{≻} R_{q - 2}, R_{q} : R_{1} \underline{≻} R_{q - 1} \end{matrix}$

It follows from Definition 12 and R₁ : R₂ $\underline{≻}$ R_q that $\begin{matrix} \frac{1}{| R_{1} | + | R_{2} |} [v (R_{1} \cup R_{2}) - v (R_{1}) - v (R_{2})] \\ \geq \frac{1}{| R_{1} | + | R_{q} |} [v (R_{1} \cup R_{q}) - v (R_{1}) - v (R_{q})] \end{matrix}$

R_q : R₁ $\underline{≻}$ R_q-1 implies that $\begin{matrix} \frac{1}{| R_{1} | + | R_{q} |} [v (R_{1} \cup R_{q}) - v (R_{1}) - v (R_{q})] \\ \geq \frac{1}{| R_{q} | + | R_{q - 1} |} [v (R_{q} \cup R_{q - 1}) - v (R_{q}) - v (R_{q - 1})] \end{matrix}$

Consequently we can get $R_{q - 1} : R_{q} \underline{≻} R_{q - 2}, \dots, R_{2} : R_{3} \underline{≻} R_{1}$ thus we have $\begin{matrix} \frac{1}{| R_{1} | + | R_{2} |} [v (R_{1} \cup R_{2}) - v (R_{1}) - v (R_{2})] \\ \geq \frac{1}{| R_{1} | + | R_{q} |} [v (R_{1} \cup R_{q}) - v (R_{1}) - v (R_{q})] \geq \\ \frac{1}{| R_{q} | + | R_{q - 1} |} [v (R_{q} \cup R_{q - 1}) - v (R_{q}) - v (R_{q - 1})] \\ \geq \frac{1}{| R_{q - 1} + | R_{q - 2} |} [v (R_{q - 1} \cup R_{q - 2}) \\ - v (R_{q - 1}) - v (R_{q - 2})] \geq \\ \dots \geq \frac{1}{| R_{2} | + | R_{3} |} [v (R_{2} \cup R_{3}) - v (R_{2}) - v (R_{3})] \\ \geq \frac{1}{| R_{1} | + | R_{2} |} [v (R_{1} \cup R_{2}) - v (R_{1}) - v (R_{2})] \end{matrix}$

Hence this theorem is true, and in the above inequalities, we should choose the equal sign, so we obtain R₁ : R₂ = R_q $\underline{≻}$ R_k, R_q : R₁ = R_q-1 $\underline{≻}$ R_k, that is to say, R₁ : R_q $\underline{≻}$ R_k, R_q : R₁ $\underline{≻}$ R_k, ∀k ∈ {1, …, m} ∖ {1, q}, so R₁ and R_q constitute a perfect matching pair and the proof of this theorem is complete.

Theorem 6.Suppose in a certain phase of coalition formation process, there are cooperative unitsR₁, …, R_m, m ≥ 2, if these cooperative units try to finish a matching process, then there exists at least one perfect matching.

Proof. Write $C = {R_{1}, \dots, R_{m}}$ , the set of their corresponding first priority matching units are denoted by $F$ . For the first priority matching unit cannot be itself, it holds that $| F | \geq 2$ , if $| C | = | F |$ , by Theorem 5, this theorem is true, if $| C | \neq | F |$ , then $| F | < | C |$ , so there exists R_t ∈ {R₁, …, R_m}, $R_{t} \notin F$ . Write $H = C ∖ F$ , so the elements in $H$ do not appear in $F$ . So the elements in $H$ can not constitute a perfect matching pair in this phase, so this part will be ignored.

Now we consider the set of all cooperative units $C^{'} = C ∖ H$ , we denote by $F^{'}$ the set of their corresponding first priority matching units. If $| C^{'} | = | F^{'} |$ , it follows from Theorem 5 that this Theorem is true, if $| F^{'} | < | C^{'} |$ , we ignore the cooperative units in $H^{'} = C^{'} ∖ F^{'}$ , we can get the new set of cooperative units $C^{″} = C^{'} ∖ H^{'}$ and the set of their corresponding first priority matching units, as it going on, for $| F | \geq 2$ , there must exist at least a perfect matching pair.

Theorem 7.Let (N, ν) be a cooperative game, the grand coalition must be formed.

Proof. By Theorem 6, this theorem holds true.

3.2 Matching value and its properties

By the discussion in Section 3.1, the grand coalition is finally formed. As the players join the coalition with different orders, so the coalition structure also has some differences, if a player joins the coalition earlier, the coalition structure will become more complex, so we can have the following Definition.

Definition 13. Let (N, ν) be a cooperative game, for player i, its related cooperative units are called the structural units of player i, k_i is the number of the structural units of player i, we denote them by S_i1, …, S_{ik
_i} according to their appearing order, S_i1, …, S_{ik
_i} are disjoint and S_i1 ∪ S_i2 ∪ ⋯ ∪ S_{ik
_i} = N ∖ i.

Theorem 8.Let (N, ν) be a cooperative game, for anyi, j ∈ N, ifj ∈ S_{ik
_i}, then

j ≠ i and S_{jk
_j} = N ∖ S_{ik
_i}.

Following we define the final profits of each player from the structural units point of view.

Let (N, ν) be a cooperative game, k_i the number of structural units of player i, the set of all structural units is denoted by $M_{i1} = {S_{i 1}, \dots, S_{{ik}_{i1}}}$ . Player i firstly chooses S_i1, the profit of player i is $ψ (i) = v (i) + \frac{1}{| S_{i 1} | + 1} [v (S_{i 1} \cup i) - v (S_{i 1}) - v (i)]$ then the new formed cooperative unit S_i1 ∪ i will choose S_i2, the profit of the cooperative unit S_i1 ∪ i is $\begin{matrix} v^{'} (S_{i 1} \cup i) = v (S_{i 1} \cup i) + \\ \frac{| S_{i 1} | + 1}{| S_{i 2} | + | S_{i 1} | + 1} [v (S_{i 2} \cup S_{i 1} \cup i) - v (S_{i 2}) - v (S_{i 1} \cup i)] \end{matrix}$

Thus the profit of player i is $\begin{matrix} ψ (i) = v (i) + \frac{1}{| S_{i 1} | + 1} [v^{'} (S_{i 1} \cup i) - v (S_{i 1}) - v (i)] \\ = v (i) + \frac{v (S_{i 1} \cup i) - v (S_{i 1}) - v (i)}{| S_{i 1} | + 1} \\ + \frac{v (S_{i 2} \cup S_{i 1} \cup i) - v (S_{i 2}) - v (S_{i 1} \cup i)}{| S_{i 2} | + | S_{i 1} | + 1} \end{matrix}$

Then S_i2 ∪ S_i1 ∪ {i} will choose S_i3 to constitute a perfect matching pair, the profit of player i is $\begin{matrix} ψ (i) = v (i) + \frac{v (S_{i 1} \cup i) - v (S_{i 1}) - v (i)}{| S_{i 1} | + 1} \\ + \frac{v (S_{i 2} \cup S_{i 1} \cup i) - v (S_{i 2}) - v (S_{i 1} \cup i)}{| S_{i 2} | + | S_{i 1} | + 1} \\ + \frac{v (S_{i 3} \cup S_{i 2} \cup S_{i 1} \cup i) - v (S_{i 3}) - v (S_{i 2} \cup S_{i 1} \cup i)}{| S_{i 3} | + | S_{i 2} | + | S_{i 1} | + 1} \end{matrix}$ it is not stop until the grand coalition is formed. The profit of player i is $\begin{matrix} ψ (i) = v (i) + \frac{v (S_{i 1} \cup i) - v (S_{i 1}) - v (i)}{| S_{i 1} | + 1} + \\ \frac{v (S_{i 2} \cup S_{i 1} \cup i) - v (S_{i 2}) - v (S_{i 1} \cup i)}{| S_{i 2} | + | S_{i 1} | + 1} + \dots + \\ \frac{v (S_{{ik}_{i}} \cup \dots \cup S_{i 1} \cup i) - v (S_{{ik}_{i}}) - v (S_{i (k_{i} - 1)} \cup \dots \cup S_{i 1} \cup i)}{| S_{{ik}_{i}} | + \dots + | S_{i 2} | + | S_{i 1} | + 1} \end{matrix}$

Therefore, we can give the formula of the structure matching value shown as follows.

Definition 14. Let (N, ν) be a cooperative game, for any i ∈ N = {1, 2, …, n}, we have

$\begin{matrix} ψ_{i} = v (i) \\ + \sum_{j = 1}^{k_{i}} \frac{v (\cup_{t = 1}^{j} S_{it} \cup i) - v (S_{ij}) - v (\cup_{t = 1}^{j - 1} S_{it} \cup i)}{1 + \sum_{t = 1}^{j} | S_{it} |} \end{matrix}$ (12)

$S_{ij} \in M_{i} = {S_{i 1}, \dots, S_{{ik}_{i}}}$ is called the structural units of player i, k_i is the number of the structural units, we call ψ_i a structural matching value of player i.

In Section 3.1, we know that the structure of the grand coalition is not unique. Usually we have the following specialty, for example, in the process of coalition formation, if A : B = C, B : A $\underline{≻}$ C and C : A $\underline{≻}$ B, so A and B or A and C can constitute the perfect matching. Under this circumstance, we need to average all the possible structure matching value. So we have the following Definition.

Definition 15. Let (N, ν) be a cooperative game, suppose the possible structure number of the grand coalition is u, the structure matching value of player i is denoted as $ψ_{i}^{1}, \dots, ψ_{i}^{u}$ , so the matching value of player i is $ψ_{i} (v) = \frac{ψ_{i}^{1} + ψ_{i}^{2} + \dots + ψ_{i}^{u}}{u}, \forall i \in N$ (13) so the matching value is ψ (v) = (ψ₁ (v) , …, ψ_n (v)).

Generally speaking, u is usually equal to 1 of cooperative game (N, ν), there may exists many possible grand coalitions only if some special cases appear, in this case, we need to average them.

By the above Definitions and Theorems, We can easily get the following properties of (N, ν).

Property 1.Matching valueψ (v) satisfies the individually rationality, collective rationality and it is unique.

Property 2.Let (N, ν) be a cooperative game, for any two playersiandj, if when $i, j \notin S \Rightarrow v (S \cup i) = v (S \cup j), \forall S \subseteq N ∖ i, j$ we have $ψ_{i} (v) = ψ_{j} (v)$ then matching value ψ (v) satisfies symmetry.

Property 3.If playeriis lagging behind, then its matching value will be small.

Theorem 9.Let (N, ν) be a cooperative game, we denote by $L$ the set of the structure units of all players. For a certain grand coalition structure, the structure matching value satisfies the independence test under $L$ , that is $\sum_{i \in S} ψ_{i} \geq v (S), \forall S \in L$

Proof. If S = N, by Property 1, this theorem is clearly true. If S ≠ N, when S does not constitute a perfect matching pair with some cooperative unit, the sum of the profits in coalition S is v (S), for S can constitute a perfect pair with some cooperative unit, then the sum of the profits is v′ (S), by the superaddidivity, we obtain v′ (S) ≥ v (S), by the definition of the structure matching value, we have ∑_i∈Sψ_i ≥ v (S) and this completes the proof of the theorem.

Definition 16. For any two cooperative games (N, u) and (N, ν) with the same grand coalition structure, for any i ∈ S, if {v (S) ≤ u (S) , v (T) = u (T) , ∀ T ≠ S} ⇒ {x_i (v) ≤ x_i (u)}. Then we call this the structure coalition monotoniciy.

Theorem 10.Matching valueψ (v) satisfies the structure coalition monotonicity.

From the above Theorems, we can see that matching value has good properties, following we will discuss its relation with other solutions, we firstly give the following lemma.

Lemma 1.Let (N, ν) be a cooperative game, if all the grand coalition structures may exist, when |B| = |C|, we haveA : B = C, ∀ A, B, C ∈ 2^N.

Proof. For all the grand coalition structures may exist, so all possible unit links may exist, thus i : j = t, that is, in the first phase of the matching process, each player is equally important relative to other players, so after the cooperative unit is formed, if |B| = |C| $A : B = C, \forall A, B, C \in 2^{N}$

So Lemma 1 holds good.

Theorem 11.Let (N, ν) be a cooperative game, if all the grand coalition structures may exist, then $v (S) = \sum_{i \in S} v (i) + \sum_{i = 1}^{| S | - 1} ɛ_{i}, \forall S \in 2^{N}, | S | \geq 2, ɛ_{i} \geq 0$

Proof. By Lemma 1 and the superadditivity, theorem 11 holds.

Theorem 12.Let (N, ν) be a cooperative game, if all of the grand coalition structures may exist, then $φ_{i} (v) = ψ_{i} (v)$

Proof. By Theorem 11, $v (S) = \sum_{i \in S} v (i) + \sum_{i = 1}^{| S | - 1} ɛ_{i}, \forall S \in 2^{N}, | S | \geq 2, ɛ_{i} \geq 0$

the Shapley value of player i is $φ_{i} (v) = v (i) + \frac{ɛ_{1} + ɛ_{2} + \dots + ɛ_{n - 1}}{n}, \forall i \in N$

For simplicity, we denote the above φ_i (v) as (*). It is clearly that when n = 2, (*) holds true, when n = 3, all possible grand coalition structures are shown in Fig. 2.

We can calculate the matching value under the above three circumstances shown as follows. $\begin{matrix} ψ_{1}^{1} = v (1) + 1 / - 2 * [v (1, 2) - v (1) - v (2)] \\ + 1 / - 3 * [v (1, 2, 3) - v (1, 2) - v (3)] \\ ψ_{1}^{2} = v (1) + 1 / - 2 [v (1, 3) - v (1) - v (3)] + \\ 1 / - 3 [v (1, 2, 3) - v (1, 3) - v (2)] \\ ψ_{1}^{3} = v (1) + 1 / - 3 [v (1, 2, 3) - v (1) - v (2, 3)] \end{matrix}$

Thus $ψ_{1} (v) = 1 / - 3 (ψ_{1}^{1} + ψ_{1}^{2} + ψ_{1}^{3}) = v (1) + 1 / - 3$ (ɛ₁ + ɛ₂) then we can calculate ψ₂ (v) and ψ₃ (v) ψ_i (v) = v (i) + 1/ - 3 (ɛ₁ + ɛ₂), ∀i ∈ N = {1, 2, 3}

Consequently when n = 3, (*) holds good.

Now we suppose that when n = k, k ≥ 3, (*) holds. When n = k + 1, it means that one more player k + 1 is added, when n = k, we set {k + 1} = N_k+1 ∖ N_k. Henceforth v (N_k+1) - v (N_k) = v (k + 1) + ɛ_k. Then we get $v (N_{k + 1}) - v (N_{k + 1} ∖ i) - v (i) = ɛ_{k}, \forall i \in N_{k + 1}$

Therefore, ɛ_k is exceeded of the net surplus for each player. Since the appearing probabilities of all the grand coalition structures are the same, so we can average all the net surplus. Then we obtain $ψ_{i} (v) = v (i) + \frac{ɛ_{1} + ɛ_{2} + \dots + ɛ_{k}}{k + 1}, \forall i \in N_{k + 1}$

In summary, (*) holds good and the proof of this theorem is complete.

Theorem 13.Let (N, ν) be a three-person convex cooperative game, then $φ_{i} (v) \in C (v)$

Proof. The same proof method as the Theorem 12, this theorem holds true.

4 The axiomatization of the matching value under total coalition structures

By Section 3, we can know that if we remove the constraint condition of the perfect matching, then any two cooperative units can constitute a unit link. Then we can get the following conclusion.

Theorem 14.Let (N, ν) be a cooperative game, the number of coalition structures under imperfect matching is $C_{n}^{2} * C_{(n - 1)}^{2} * \dots * C_{2}^{2}$

Definition 17. We call the set of all structure matching values as the matching value under total coalition structure, which is denoted as M (v). $\begin{matrix} M (v) = \cup_{u = 1}^{C_{n}^{2} * C_{(n - 1)}^{2} * \dots * C_{2}^{2}} (ψ_{1}^{u}, \dots, ψ_{n}^{u}) \\ ψ_{i}^{u} = v (i) + \sum_{j = 1}^{k_{i}} \frac{v (\cup_{t = 1}^{j} S_{it}^{u} \cup i) - v (S_{ij}^{u}) - v (\cup_{t = 1}^{j - 1} S_{it}^{u} \cup i)}{1 + \sum_{t = 1}^{j} | S_{it}^{u} |} \end{matrix}$

Example 1. Let (N, ν) be a three-person cooperative game, we know that v(1) = v(2) = v(3) = 0, v(1,2) = 1, v(1,3) = 2, v(2,3) = 3, v(1,2,3)= 4 .

By Theorem 14, the total number of all the coalition structures is $C_{3}^{2} * C_{2}^{2} = 3$ , we can obtain the structure units of player 1 is:

$M_{1}^{1} = {{2}, {3}}$ , $M_{1}^{2} = {{3}, {2}}$ , $M_{1}^{3} =$ {{2, 3}}, thus we get $ψ_{1}^{1} = 3 / - 2, ψ_{1}^{2} = 5 / - 3, ψ_{1}^{3} =$ 1/ - 3, then we obtain $\begin{matrix} M (v) \\ = {(\frac{3}{2}, \frac{3}{2}, 1), (\frac{5}{3}, \frac{2}{3}, \frac{5}{3}), (\frac{1}{3}, \frac{11}{6}, \frac{11}{6})} \end{matrix}$

Theorem 15.M (v) satisfies the collective rationality, that is, for ∀x ∈ M (v), we havex (N) = v (N).

Proof. By Definition 17, this theorem is clearly true.

Definition 18. Let (N, ν) be a cooperative game, its reduction game is defined as (R, v_R,x), in which $v_{R, x} (T) = {\begin{matrix} 0, & T = \emptyset, \\ v (T), & T \subset R, R \subset N, \\ v (N) - x (N ∖ R), & T = R, R \subset N . \end{matrix}$

Definition 19. For a solution σ of (N, v), if for any x ∈ σ, there exists R ⊂ N such that $\begin{matrix} x_{R} \in σ (R, v_{R, x}), x_{N ∖ R} \in σ (N ∖ R, v_{N ∖ R, x}) \\ \frac{x (R) - v (R)}{| R |} = \frac{x (N ∖ R) - v (N ∖ R)}{| N ∖ R |} \end{matrix}$ then σ satisfies the resolution balance fixed reduction properties(we call it RBFR in short).

Theorem 16.M (v) satisfies RBFR.

Proof. For x ∈ M (v), there exists at least one coalition structure, we set this coalition structure as L. The final grand coalition N must be constituted by {R} and {N ∖ R}, in the corresponding coalition structure of x, the final two cooperative units which constitute the grand coalition are the aim we try to find, we choose the arbitrary one as R to prove.

We get two cooperative models from (N, v) which are denoted by (R, v_R,x) and (N ∖ R, v_N∖R,x) respectively, in (R, v_R,x), for the constructed mode of coalition structure, there exists a coalition structure L_R, such that ∀i ∈ R, we have $M_{i}^{L R} = M_{i}^{L} ∖ {{N ∖ R}}$ . Thus $\begin{matrix} ψ_{i}^{L} = v (i) \\ + \sum_{j = 1}^{k_{i} - 1} \frac{v (\cup_{t = 1}^{j} S_{it}^{L} \cup i) - v (S_{ij}^{L}) - v (\cup_{t = 1}^{j - 1} S_{it}^{L} \cup i)}{1 + \sum_{t = 1}^{j} | S_{it}^{L} |} \\ + \frac{v (N) - v (N ∖ R) - v (R)}{| N |} \\ = v (i) + \frac{v (N) - v (N ∖ R) - v (R)}{| N |} + \\ \sum_{j = 1}^{k_{i} - 2} \frac{v (\cup_{t = 1}^{j} S_{it}^{L} \cup i) - v (S_{ij}^{L}) - v (\cup_{t = 1}^{j - 1} S_{it}^{L} \cup i)}{1 + \sum_{t = 1}^{j} | S_{it}^{L} |} \end{matrix}$

Since v(R) = x (R) - |R| * $+ \frac{v (N) - v (R) - v (R)}{| N |}$ , therefore $\begin{matrix} \frac{v (R) - v (S_{i (k_{i} - 1)}^{L}) - v (\cup_{t = 1}^{k_{i} - 2} S_{it}^{L} \cup i)}{| R |} \\ + \frac{v (N) - v (N ∖ R) - v (R)}{| N |} \\ = \frac{x (R) - v (S_{i (k_{i} - 1)}^{L}) - v (\cup_{t = 1}^{k_{i} - 2} S_{it}^{L} \cup i)}{| R |} \end{matrix}$

So $\begin{matrix} ψ_{i}^{L} = v (i) + \frac{x (R) - v (S_{i (k_{i} - 1)}^{L}) - v (\cup_{t = 1}^{k_{i} - 2} S_{it}^{L} \cup i)}{| R |} + \\ \sum_{j = 1}^{k_{i} - 2} \frac{v (\cup_{t = 1}^{j} S_{it}^{L} \cup i) - v (S_{ij}^{L}) - v (\cup_{t = 1}^{j - 1} S_{it}^{L} \cup i)}{1 + \sum_{t = 1}^{j} | S_{it}^{L} |} = \\ v (i) + \frac{v_{R, x} (R) - v (S_{i (k_{i} - 1)}^{L}) - v (\cup_{t = 1}^{k_{i} - 2} S_{it}^{L} \cup i)}{| R |} + \\ \sum_{j = 1}^{k_{i} - 2} \frac{v (\cup_{t = 1}^{j} S_{it}^{L} \cup i) - v (S_{ij}^{L}) - v (\cup_{t = 1}^{j - 1} S_{it}^{L} \cup i)}{1 + \sum_{t = 1}^{j} | S_{it}^{L} |} = ψ_{i}^{L_{R}} \end{matrix}$

Therefore, we can have

∀i ∈ N ∖ R, $M_{i}^{L N ∖ R} = M_{i}^{L} ∖ {{R}}$ , and $ψ_{i}^{L} =$ $ψ_{i}^{L_{N ∖ R}}$ . By Definition 17, we obtain

x_R ∈ M (R, v_R,x) and x_N∖R ∈ M (N ∖ R, v_N∖R,x)

So we can have $\begin{matrix} \frac{x (R) - v (R)}{| R |} = \frac{x (N ∖ R) - v (N ∖ R)}{| N ∖ R |} \\ = \frac{v (N) - v (R) - v (N ∖ R)}{| N |} \end{matrix}$ and the proof of this theorem is complete.

Definition 20. For a solution σ of the cooperative game (N, v), ∀x ∈ E (v), if there exists R ⊂ N such that $\begin{matrix} x_{R} \in σ (R, v_{R, x}), x_{N ∖ R} \in σ (N ∖ R, v_{N ∖ R, x}) \\ \frac{x (R) - v (R)}{| R |} = \frac{x (N ∖ R) - v (N ∖ R)}{| N ∖ R |} \end{matrix}$

We get x ∈ σ (N, v), then σ satisfies the association balance fixed reduction properties (we call it ABFR for short).

Theorem 17.M (v) satisfies the ABFR.

Proof. For ∀x ∈ E (v), there exists R ⊂ N, such that x_R ∈ M (R, v_R,x), x_N∖R ∈ M (N ∖ R, v_N∖R,x). By Theorem 16, it follows that x (R) = v_R,x (R) = v (N) - x (N ∖ R), therefore x (N) = v (N). Since $\frac{x (R) - v (R)}{| R |} = \frac{x (N ∖ R) - v (N ∖ R)}{| N ∖ R |}$

So $\frac{x (R) - v (R)}{| R |} = \frac{v (N) - x (R) - v (N ∖ R)}{| N ∖ R |}$ then we can conclude $\begin{matrix} x (R) = v (R) + | R | * \frac{v (N) - v (R) - v (N ∖ R)}{| N |} \\ x (N ∖ R) = v (N) - x (R) = v (N ∖ R) \\ + | N ∖ R | * \frac{v (N) - v (R) - v (N ∖ R)}{| N |} \end{matrix}$

We set w = v_R,x, y = x_R, so y ∈ M (R, w). By Theorem 17, there exist Q ⊂ R such that $\begin{matrix} y_{Q} \in M (R, w_{Q, y}), y_{R ∖ Q} \in M (R, w_{R ∖ Q, y}) \\ \frac{y (Q) - v (Q)}{| Q |} = \frac{y (R ∖ Q) - v (R ∖ Q)}{| R ∖ Q |} \end{matrix}$

It follows that $y (Q) = w (Q) + | Q | * \frac{w (R) - w (Q) - w (R ∖ Q)}{| R |}$

For $\begin{matrix} w (R) & = & v_{R, x} (R) = v (N) - x (N ∖ R) w (Q) \\ = & v_{R, x} (Q) = v (Q), w (R ∖ Q) = v_{R, x} (R ∖ Q) \\ = & v (R ∖ Q) \end{matrix}$

Hence $\begin{matrix} y (Q) = v (Q) + | Q | * \frac{v (R) - v (Q) - v (R ∖ Q)}{| R |} + \\ | Q | * \frac{v (N) - v (R) - v (N ∖ R)}{| N |} = x (Q) \end{matrix}$ then we can obtain $\begin{matrix} x (R ∖ Q) \\ = v (R ∖ Q) + | R ∖ Q | * \frac{v (R) - v (Q) - v (R ∖ Q)}{| R |} + \\ | R ∖ Q | * \frac{v (N) - v (R) - v (N ∖ R)}{| N |} \end{matrix}$

For R, we can get Q and R ∖ Q, for Q, we have P ⊂ Q, such that $\begin{matrix} x (P) = v (P) + | P | * \frac{v (Q) - v (P) - v (Q ∖ P)}{| Q |} \\ + | P | * \frac{v (R) - v (Q) - v (R ∖ Q)}{| R |} \\ + | P | * \frac{v (N) - v (R) - v (N ∖ R)}{| N |} \end{matrix}$

then we get $\begin{matrix} x (i) = v (i) + | i | * \frac{v (S_{i 1} \cup i) - v (S_{i 1}) - v (i)}{| S_{i 1} \cup i |} + \\ | i | * \frac{v (S_{i 2} \cup S_{i 1} \cup i) - v (S_{i 2}) - v (S_{i 1} \cup i)}{| S_{i 2} \cup S_{i 1} \cup i |} + \dots + | i | * \\ \frac{v (S_{{ik}_{i}} \cup \dots \cup S_{i 1} \cup i) - v (S_{{ik}_{i}}) - v (S_{i (k_{i} - 1)} \cup \dots \cup S_{i 1} \cup i)}{| S_{{ik}_{i}} \cup \dots \cup S_{i 1} \cup i |} \\ = v (i) + \sum_{j = 1}^{k_{i}} \frac{v (\cup_{t = 1}^{j} S_{it} \cup i) - v (S_{ij}) - v (\cup_{t = 1}^{j - 1} S_{it} \cup i)}{1 + \sum_{t = 1}^{j} | S_{it} |} = 1 pt 1 pt ψ_{i} \\ S_{i 1} \cup S_{i 2} \cup \dots \cup S_{{ik}_{i}} = N ∖ i, i \in R . \end{matrix}$ In the same way, for any i ∈ N ∖ R, we can get x (i) = ψ_i.

So for i ∈ N, we get x (i) = ψ_i. Therefore x ∈ M (N, v) and this completes the proof.

Prior to giving the axiomatization theorem of the total coalition structure matching value, we will give an important lemma.

Lemma 2.For a two-person cooperative game, if its solutionσsatisfies collective rationality and RBFR, thenσ = M (v).

Proof. For σ satisfies collective rationality, so any x = (x₁, x₂) ∈ σ, we have x₁ + x₂ = v (1, 2) = v (N).

For σ satisfies RBFR, there exists R ⊂ N, such that $\frac{x (R) - v (R)}{| R |} = \frac{x (N ∖ R) - v (N ∖ R)}{| N ∖ R |}$

For N = {1, 2}, then x₁ - v (1) = x₂ - v (2), we obtain $\begin{matrix} x_{1} = v (1) + 0.5 \times [v (1, 2) - v (1) - v (2)], \\ x_{2} = v (2) + 0.5 \times [v (1, 2) - v (1) - v (2)], \end{matrix}$ thus σ ={(x₁, x₂)}. By the Definition of M (v), we get $M (v) = {(v (1) + \frac{v (1, 2) - v (1) - v (2)}{2}, v (2) + \frac{v (1, 2) - v (1) - v (2)}{2})}$ so the proof of this theorem is complete.

Then we give an axiomatization theorem of the total coalition structure matching value.

Theorem 18.The total coalition structure matching value is the unique solution satisfying collective rationality, RBFR and ABFR.

Proof. Existence By Theorem 15, Theorem 16 and Theorem 17, we can know that M (v) satisfies collective rationality, RBFR and ABFR. So the existence is proved.

Uniqueness Suppose a solution σ of (N, v) satisfies collective rationality, RBFR and ABFR. Then we only need to prove σ = M (v). Obviously when n = 1, we have σ = M (v), when n = 2, by Lemma 2, we get that σ = M (v) is clearly true. Following we will prove that when n ≥ 3, x_N∖R ∈ σ (N ∖ R, v_N∖R,x). For σ satisfies collective rationality and RBFR, so ∀x ∈ σ, there exists R ⊂ N, such that ${\begin{matrix} \frac{x (R) - v (R)}{| R |} = \frac{x (N ∖ R) - v (N ∖ R)}{| N ∖ R |} \\ x (R) + x (N ∖ R) = v (N) \end{matrix}$

So we can have $x (R) = v (R) + | R | * \frac{v (N) - v (R) - v (N ∖ R)}{| N |}$ for x_R ∈ σ (R, v_R,x), the same as the proof of Theorem 17, we can get that ∀i ∈ N, we have x (i) = ψ_i, so x ∈ M (v). Thus we have σ ⊆ M (v). For ∀x ∈ M (v), there must exist at least a coalition structure, in the formation process of coalition structure, there must exist two cooperative units to finish the unit link, we denote the cooperative unit as S₁ = R and |S₁|=2, by the properties of M (v), we can get x_{S
₁} ∈ M (S₁, v_S₁,x), for |S₁|=2, by Lemma 2, we obtain $M (S_{1}, v_{S_{1}, x}) = σ (S_{1}, v_{S_{1}, x})$ so x_{S
₁} ∈ σ (S₁, v_S₁,x). Suppose S₂ constitutes the perfect matching pair with S₁, if |S₂|=1, we set S₂ = P, then it is clearly that x_{S
₂} ∈ σ (S₂, v_S₂,x) $\begin{matrix} \frac{x (S_{1}) - v (S_{1})}{| S_{1} |} = \frac{x (S_{2}) - v (S_{2})}{| S_{2} |} = \\ \frac{v (S_{1} \cup S_{2}) - v (S_{1}) - v (S_{2})}{| S_{1} \cup S_{2} |} + C \end{matrix}$

C is a constant. For σ satisfies ABFR, we can get $x_{S_{1} \cup S_{2}} \in σ (S_{1} \cup S_{2}, v_{S_{1} \cup S_{2}, x})$

If |S₂|=2, we set S₂ = Q, by Lemma 2, we get x_{S
₂} ∈ σ (S₂, v_S₂,x), then we obtain $\begin{matrix} \frac{x (S_{1}) - v (S_{1})}{| S_{1} |} = \frac{x (S_{2}) - v (S_{2})}{| S_{2} |} = \\ \frac{v (S_{1} \cup S_{2}) - v (S_{1}) - v (S_{2})}{| S_{1} \cup S_{2} |} + C^{'} \end{matrix}$

C′ is a constant, then we have $x_{S_{1} \cup S_{2}} \in σ (S_{1} \cup S_{2}, v_{S_{1} \cup S_{2}, x})$

If |S₂|≥3, then the coalition structure of S₂ must be constituted by some cooperative units just like P and Q, for however many P and Q constitute the perfect matching pair with S₁, a new cooperative unit T is gotten and x_T ∈ σ (T, v_T,x). Thus x_S₁∪S₂ ∈ σ (S₁ ∪ S₂, v_{S₁∪S₂,x}) is true. So we do not stop to finish the unit link for S₁ ∪ S₂ until S₁ ∪ S₂ ∪ ⋯ ∪ S_k = N, by the above analysis, we get $\begin{matrix} x_{S_{1} \cup S_{2} \cup \dots \cup S_{k - 1}} \in σ (S_{1} \cup S_{2} \cup \dots \cup S_{k - 1}, \\ v_{S_{1} \cup S_{2} \cup \dots \cup S_{k - 1}, x}) x_{S_{k}} \in σ (S_{k}, v_{S_{k}, x}) \\ \frac{x (S_{1} \cup S_{2} \cup \dots \cup S_{k - 1}) - v (S_{1} \cup S_{2} \cup \dots \cup S_{k - 1})}{| S_{1} \cup S_{2} \cup \dots \cup S_{k - 1} |} \\ = \frac{x (S_{k}) - v (S_{k})}{| S_{k} |} \\ = \frac{v (N) - v (S_{1} \cup S_{2} \cup \dots \cup S_{k - 1}) - v (S_{k})}{| N |} \end{matrix}$

Because σ satisfies ABFR, we conclude that x (N) = x_{S₁∪S₂∪⋯∪S_k-1∪S_k} ∈ σ (N, v_N,x) = σ (N, v), that is, x ∈ σ, thus M (v) ⊆ σ, for σ ⊆ M (v), so σ = M (v), the uniqueness is proved and the proof of this theorem is complete.

In order to verify the effectiveness of the matching value defined in this paper, we will give the following Example.

Example 2. Let (N, ν) be a three-person cooperative game, v (1) =60, v (2) =96, v (3) =84, v (1, 2) =204, v (1, 3) =156, v (2, 3) =192, v (1, 2, 3) =291.

1) We use the core to give the final allocation result. $\begin{matrix} C (v) = {(x_{1}, x_{2}, x_{3}) | 60 \leq x_{1} \leq 99, 96 \leq x_{2} \leq 135, \\ 84 \leq x_{3} \leq 87, x_{1} + x_{2} + x_{3} = 291} \end{matrix}$ We can easily see that the core solution is not unique.

2) We use the Shapley value to give the finalallocation result. Suppose player {1} and {2} arefirstly constitute a coalition R = {1, 2}, we get φ₁ (v) = 84, φ₂ (v) = 120, then {3} joins their coalition to constitute the grand coalition {1, 2, 3}, we get the final allocation results by Shapley value $φ_{1}^{'} (v) = 83, φ_{2}^{'} (v) = 119, φ_{3} (v) = 89$ . Since φ₃ (v) = 89 > v (3), $φ_{1}^{'} (v) = 83 < φ_{1} (v)$ , $φ_{2}^{'} (v) = 119 < φ_{2} (v)$ , so {1, 2} will not agree with {3} to constitute the grand coalition N = {1, 2, 3}. So the final allocation result is x = (84, 120, 84). However, for this allocation result, we can easily find that v (N) =291 > v (1, 2) + v (3), thus the collective rationality is not satisfied.

3) We use the matching value to give the final allocation result. We get that {1} and {2} constitute a coalition {1, 2}, ψ₁ (v) = 84, ψ₂ (v) = 120, when {3} joins {1, 2}, $ψ_{1}^{'} (v) = 85, ψ_{2}^{'} (v) = 121, ψ_{3} (v) = 85$ . We have $ψ_{1}^{'} (v) > φ_{1} (v), ψ_{2}^{'} (v) > φ_{2} (v), ψ_{3}^{'} (v) > v (3)$ So the coalition {1, 2} must agree with {3} to form the grand coalition, the final solution is x^* = (85, 121, 85).

We can easily verify that it belongs to the core. By this example, we can get the conclusion that the matching value can maximize the group profits. Under the incomplete cooperation circumstances, the grand coalition is gotten, compared with the Shapley value, it is more close to the reality, because not all the coalitions can be formed in reality. For each player in this coalition, player 1 and 2 can get their satisfactory profits, player 3 also can get his ideal profits on the premise that he joins the grand coalition.

5 Conclusions and further study

In this paper, the formation process of the grand coalition is firstly discussed, some concepts such as perfect matching pair, matching order are defined, then a new solution called matching value is proposed, its basic properties are discussed by a series of theorems, and then based on the forms of all coalition structures, a matching value under total coalition structure is proved by three axioms, the characteristics and performance of the matching value are shown by an example. Thus the discussion on the applications of the matching value will be our next step of work.

Footnotes

Acknowledgments

This work was supported by the National Natural Science Foundation of China and the Specialized Research Fund for the Doctoral Program of Higher Education (No. 71071018, 71371030, 20111101110036, 71271029).

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