Procedural Implementation and Axiomatization of the Weighted Nonseparable Cost Value

The paper deﬁnes a new value called the weighted nonseparable cost value (weighted-NSC value), which divides the nonseparable cost on the ground of an exogenous attached weight and compromises egalitarianism and utilitarianism of a value ﬂexibly. First, we construct an optimization model to minimize the deweighted variance of complaint and deﬁne its optimal solution to be the weighted-NSC value. Second, a process is set up to acquire the weighted-NSC value, which enlarges the traditional procedural values. In the process, one player’s marginal contribution is divided up by all participants rather than merely restricted within his precursors. Lastly, adopting the weight in deﬁning a value destructs the classical symmetry. This promotes the deﬁnition of ω -symmetry for the grand-marginal normalized game to defend against the eﬀect of weight and axiomatically sculptures the weighted-NSC value. Dual dummifying player property is also applied to characterize the new deﬁned value.


Introduction
Consider a wireless network with n nodes. e nodes communicate with far-off destinations through the cooperation of the intermediate nodes. Many indexes of the network, such as throughput and lifetime, will be raised dramatically, and the communication cost be reduced greatly through cooperation [1]. e corresponding payoff allocation or the cost distribution problem is of highly interest [2,3]. Cooperative game theory offers a variety of values to address how to fairly distribute the overall payoff in many fields such as economics [4] and networks [5].
e eminent Shapley value [6] allocates to each participant his expected payoff on all possible orders. In the permutations, each newcomer possesses the whole marginal contribution, which is utilitarian. Averaging the total worth is a natural way for allocation problems, which is entirely egalitarian. e corresponding value is thereby named as the equal division value (ED value) in cooperative games. e egalitarian nonseparable cost value (ENSC value, [7]) at first gives everyone his grand marginal contribution and afterwards averages the (positive or negative) nonseparable part to everyone. Sun et al. [8] revealed that the ENSC value embodies utilitarianism in virtue of the player's attitude towards the grand marginal contributions. en, we encounter the issue of equilibrate utilitarianism and egalitarianism during allocation process.
Ruiz et al. [9] initially linked the optimization theory with allocation issues in cooperative games. ey innovatively harnessed the least square value to minimize the variance of all players' complaint. Triggered by Ruiz, Wang et al. [10] proposed the ideal values, targets at narrowing the gap between one's given payoff and expected rewards.
is paper constructs an optimization model considering the deweighted variance of complaint.
rough deriving the unique optimal solution, we define the weighted nonseparable cost value (weighted-NSC value).
Endowing a value with weight is a common path to reconcile the relationship between egalitarianism and utilitarianism. Aristotle's maxim, "Equals should be treated equally, and unequals unequally, in proportion to relevant similarities and differences" stresses the fair allocation should consider players' similarities and differences, which could be reflected by the weight. In addition, the weight coefficient sheds light on participants' importance, power, or negotiation ability. e weighted Shapley value is defined in [11] and be portrayed axiomatically by Kalai and Samet [12]. Yang et al. [13] researched the weighted surplus division value in cooperative games. In this paper, the weighted-NSC value is defined applying the exogenous determined positive weight. Based on the definition, the nonseparable part is assigned to players according to their importance, which can be evaluated by different methods and be adjusted flexibly.
Procedural value [14] is another efficient method to compromise utilitarianism and egalitarianism on an allocation rule in the reason that one's marginal contribution in a given permutation will be shared with his precursor. Sun et al. [8] provided a process to achieve the ENSC value. is article designs an enlarged process to present a more detailed research of the weighted-NSC value. e philosophy idea of our process differs from the process in [14] because the marginal contributions are shared by all players instead of just among the precursor.
Axiomatization presents an effective method to sculpture a value. e weighted-NSC value links each player's payoff with a weight coefficient and logically does not conform to the axiom of symmetry anymore. is evokes us to propose ω-symmetry for the grand-marginal normalized game in order to characterize the weighted-NSC value. e covariance can also depict the weighted-NSC value from a mapping perspective. Ultimately, the dual dummifying player property is proposed to characterize the weighted-NSC value.
We structure the article as follows. Section 2 briefly lists the fundamental notions of cooperative games. Section 3 puts forward the weighted-NSC value and presents an optimization explanation for it through tackling an optimization model. e procedural implementation for this value is provided in Section 4 by means of designing a process. Section 5 engages in axiomatic description of the weighted-NSC value applying several new proposed properties. Section 6 is committed to a general summary of the article and sketching of the potential aspect of further research.

Preliminaries
A player set N and a characteristic function v: 2 N ⟶ R altogether are contributing factors to characterize a cooperative game, written as 〈N, v〉. Without confusion, we abbreviate it as a game and correspondingly denote as v.
Here, 2 N denotes the power set of N. Given any coalition S ∈ 2 N \ ϕ , the characteristic function designates S the worth v(S), trivially with v(ϕ) � 0. e cardinal of the coalition S is labelled by s. e set of all games on N is written as G N .
For v 1 , v 2 ∈ G N and real number p ∈ R, the games v 1 + v 2 and p · v 1 are, respectively, given by , for S ⊆ N\ k, j , we call them symmetric players.
We view a payoff vector (or payoff) as an n-dimensional vector x � (x k ) k∈N ∈ R n and incidentally define x(S) � k∈S x k for S ⊆ N. If a payoff x possesses x(N) � v(N), we name it efficient and denote the preimputation set by e component ϕ k (v) invariably can be regarded as an evaluation of player k's benefit when taking part in the game v.
e well-known Shapley value [6] distributes to each participant the expected marginal contribution, which is defined by the formula as follows: (1) e mean value of v(N) provides the concept of the equal division value (ED value): e egalitarian nonseparable cost value (ENSC value [7]) allocates the nonseparable part equally after giving each one his grand marginal contribution. Given any v ∈ G N , Take a coefficient 0 ≤ α ≤ 1, and the α-ENSC value [15] is portrayed by convex combination, αENSC(v) + (1 − α) ED(v), which is equivalent to the following formula: Let ϕ: G N ⟶ R n be a value, and we enumerate the wellknown axioms:

The Weighted Nonseparable Cost Value and
Its Optimization Implementation e grand marginal contribution of player i, b v i , is testified to be the supremum of the core [16]. Consequently, all players in a game will aspire to b v i and prefer the allocation Invariably, we could not decrease everyone's complaint simultaneously. According to the egalitarian philosophy, we strive to find out an allocation to balance each one's complaint instead of merely push down the highest complaint. e variance of the complaint is applicable to evaluate fluctuation degree of players' complaint.
Furthermore, we might confront the situation that every player has different importance and be treated differently.
is evokes us that the weight coefficient needs to be taken into account during inspecting the variance of the complaint. Given the weight system Δ n : � ω ∈ R n ++ | i∈N ω i � 1}, we modify the traditional variance and define the deweighted variance as i∈N ( to equilibrate the complaint of each player ("deweighted" here clarifies removing the effect of the weight). e detailed study of the deweighted variance is unfolded via an optimization model as follows. Given any game v ∈ G N and weight vector ω ∈ Δ n , model I : min Explicitly, equation (5) presents a convex optimization problem. Its optimal solution is either unattainable or exists uniquely. To seek model I's optimum solution, we formulate its Lagrange function as follows: en, calculate the partial derivative of L(x, λ) with x i , i ∈ N, and we generate Lagrange conditions: Furthermore, we demonstrate the constraint of efficiency through computing the partial derivative of equation (6) with λ: Combining equations (7) and (8), model I's unique optimal solution is Consequently, a new value of allocating the payoff emerges in our mind. Definition 1. For any game v ∈ G N and weight vector ω ∈ Δ n , the weighted nonseparable cost value (weighted-NSC value) is defined as where b v i represents player i's grand marginal contribution. We name it the weighted nonseparable cost value be- respectively, called the separable cost and nonseparable cost [7]. After giving player i his separable cost, the weighed-NSC value divides the remaining nonseparable part among all players based on a weight vector ω ∈ Δ n , while the ENSC value distributes this part equally to all players. e weight reflects a player's bargaining ability or accounts for a player's probability of forming coalitions with others. In practical application, the weight can be altered flexibly using different methods. Model I offers an implementation process for the weighted-NSC value. Without more ado, we get the following conclusion based on the derivation process of model I.

Theorem 1.
Given any game v ∈ G N and weight vector ω ∈ Δ n , model I' s optimal solution exists uniquely and equals to the weighted-NSC value: Setting aside the weight coefficient, we derive the following corollaries. Given any v ∈ G N , model II : min Corollary 1. Given any game v ∈ G N , model II' s optimal solution exists uniquely and equals to the ENSC value: Identically, take v ∈ G N and 0 ≤ α ≤ 1: Mathematical Problems in Engineering Corollary 2. Given any game v ∈ G N , model III's optimal solution exists uniquely and equals to the α-ENSC value, i.e., Take v ∈ G N , ω ∈ Δ n , and x ∈ I * (v), and we link each e(i, x) with ω i and put forward player i's deweighted complaint as Let ϕ: G N ⟶ R n ; we propose the corresponding axiom as follows: (i) Equal deweighted complaint property (EC): for all v ∈ G N and ω ∈ Δ n , the value ϕ verifies that Based on equation (10), it is easy to assert the following for any v ∈ G N and ω ∈ Δ n .

The Procedural Explanation
Procedural values [14] are hinged on a process which settles how to divide a player's marginal contribution among his predecessors in a given permutation. When a participant intends to collaborate with existing players, a fraction of his marginal contribution is compelled to be handed over and be shared with these people. Motivated by the philosophy of egalitarianism, this paper modifies the conventional process creatively and designs an extended process to obtain the weighted-NSC value.
For any v ∈ G N is denoted T(N) as all possible orders of N. Given an order τ ∈ T(N), τ(k) is the player who is at position k in this order. Player i's precursor (including i) under τ is defined as follows:

Process 1
Step 1: the participants set up permutations at random, and each τ in T(N) is equal possible Step 2: for player i(1 ≤ i ≤ n), his precursor set is established as referred in equation (18), and the value v(p τ is produced as the marginal contribution Step 3: player i claims his grand marginal contribution b v i Step 4: the surplus (or deficit) after player i' claim in Step 5: compute i' expected payoff on all orders in T(N) In step 4 of Process 1, the surplus (or deficit) is placed into a pool after b v i is taken away. In contrast to [14], our process not only broadens the allocation object from p τ i to N but also links the weight with benefit allocation. e philosophy idea in Process 1 is analogous to the redistribution of social welfare, i.e., after each person gets his or her own salary, the excess part will be taxed according to the given rate by Tax Bureau. In this sense, this process offers a significant tool for social welfare allocation. We illustrate Process 1 by an example. Example 1. Given players' weight vector ω � ω 1 , ω 2 , ω 3 and the entrance order τ � (2, 3, 1) ∈ T(N).
(1) Firstly, player 2 joins the game. He charges his grand marginal contribution b v 2 , and then, all players, 2 on account of the weight ω. (2) en, there comes player 3 right after player 2, He also claims his own grand marginal contribution b v 3 , and the surplus is distributed by all players rather than just his predecessor based on the weight ω.
Similarly, the nonseparable cost is allocated with the weight ω after player 1 asks for τ � (2, 3, 1). Table 1 demonstrates the allocation result in every step and the total payoff to players under the permutation τ � (2, 3, 1).
Obviously, each one's payoff is irrelevant to permutations; therefore, the expected payoff in all orders of the arrival is the same as in Table 1, which is From this example, we automatically deduce the following conclusion and attach a detailed precise proof for it. Theorem 3. Given any game v ∈ G N and weight system ω ∈ Δ n , the expected payoff ϕ ω (v) calculated by Process 1 equals to the weighted-NSC value.
Proof. For any v ∈ G N and ω ∈ Δ n , randomly take an order τ ∈ T(N), and player i' payoff ϕ ω i,τ (v) in the above process is determined as (20) erefore, the expect value ϕ ω (v) in step 5 is Based on equality (20), the following formula can be derived: 4 Mathematical Problems in Engineering In essence, the right side of equality (20) is independent of the permutation; hence, is theorem offers a procedural explanation for the weighted-NSC value and incidentally gives rise to the following remarks evidently. □ Remark 1. Given any v ∈ G N , if step 4 in Process 1 is executed presuming that players are of equal importance, i.e., ω i � 1/n, the expected value in step 5 is exactly the ENSC value. 1]) and step 4 is executed presuming that players are of equal importance, i.e., ω i � 1/n, the expected value in step 5 is exactly the α-ENSC value.

Axiomatizations
Axiomatization is a fundamental method to depict a solution in cooperative games. e axiom of symmetry is applied to describe the ENSC value in [15]. e weight coefficient in the definition of NSC ω (v) contributes to asymmetry. If players still pursue equitable treatment after deleting the influence of weight, it is necessary to generate a new symmetry property which is restricted in a special kind of game.
(ω-SYM): the value ϕ: G N ⟶ R n verifies that where v ∈ G N is a grand-marginal normalized game and ω ∈ Δ n . is axiom indicates that players will be awarded equally in the grand-marginal normalized games if we do not consider the factor of weight. Apparently, the weighted-NSC value does not abide by the classical symmetry anymore while complies with ω-SYM, and we dedicate ω-SYM to describe the weighted-NSC value. Given any v ∈ G N and ω ∈ Δ n , the following can be obtained.

Proof
(1) Take ω ∈ Δ n ; it is trivial to testify that the weighted- Applying the property (ω-SYM), the following equation is obtained: Summing up all j ∈ N in equation (26), we can get Combined with the efficiency, we have Let v: � v − v 0 , which is obviously an inessential game. Owing to the property (IN), it is easy to derive □ Remark 3. In eorem 4, the axiom (L) is enough to substitute for (A) according to the above proof. Different from the axioms of (A) and (L), covariance researches a value through the angle of mapping: Alternatively, we can apply this property to depict the weighted-NSC value.
. is states that also ϕ possesses the axiom (IN) and then ϕ coincides with the weighted-NSC value combined with eorem 4.
One is dummifying when the coalition (including this player) worth equals to the cumulative of each member's individual worth. Dummifying player's presence in a coalition hinders cooperation to some extent. Giving a dummifying player his individual worth same to be a plausible distribution. e player who brings dummifying influence to coalitions will be researched in a dual way. Relevant axiom is put forward as follows: (iii) Dual dummifying player property (DD): Generally, the grand marginal contributions cannot be guaranteed. However, this property insures this ideal payoff for the dual dummifying player. We depict the weighted-NSC value with the dual dummifying player property as follows. Given any v ∈ G N and ω ∈ Δ n , the following can be obtained.

Proof
(1) Given any ω ∈ Δ n , the weighted-NSC value explicitly satisfies (E), (L), and (ω-SYM), and we now check (DD). Given any v ∈ G N , take a dual dummifying player i ∈ N randomly, and we attain v(N) � j∈N b v j owing to Definition 3. Consequently, we compute is exactly verifies DD. (2) Proof of uniqueness: let ϕ: G N ⟶ R n be a value with (E), (L), (DD), and (ω-SYM). For anyv ∈ G N , c ∈ R, and d ∈ R n and according to the property (L) of ϕ, where d(S) � i∈S d i , for S ⊆ N. Obviously, en, erefore, each i ∈ N is a dual dummifying player in d. Since ϕ i (d) � d i holds due to (DD), we deduce that which shows that ϕ satisfies (COV). With Corollary 3, we can easily verify uniqueness. □ □ Example 2. Assume a partial wireless network is composed of 3 nodes as in Figure 1. Node n 1 contributes nothing to any coalition, and node n 1 and node n 3 communicate through node n 2 . If a node is located within the communication range of another one, there is a link between them.
e situation could be formulated in terms of a cooperative game with characteristic function. Presume that each node's value is as follows:   Evidently, the weighted-NSC value provides a rational payoff distribution rule which can flexibly balance the relationship between utilitarianism and egalitarianism during allocation process according to actual situations.

Conclusions
e weighted-NSC value defined in this article compromises the principle of equalitarianism and utilitarianism with given exogenous weight vector in the allocation of payoff. In contrast to the optimization model of the ENSC value and the α-ENSC value, we define the deweighted variance of complaint and set up the corresponding model to attain the weighted-NSC value. Furthermore, we devise a detailed process to obtain the weighted-NSC value, which is similar but different to the traditional procedural values in terms of allocation objects of the marginal contribution. In our process, players' marginal contribution is divided by all the participants of the game. In addition, we portray the weighted-NSC value axiomatically applying new defined axioms.
Inspired by the geometrical study of cooperative games in [17], the weighted-NSC value can be introduced into graph-restricted games and help to research the topological structure of wireless networks, which will probably improve the connectivity performance of wireless networks. In addition, applying the weighted value in cooperative games to the sustainable supply chain management [18] is worthy of further research.

Data Availability
All data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.