Axiomatic and Dynamic Results for Power Indexes under Symmetry

Symmetry exists in a multitude of phenomena in varying forms. The main aim of this article is to analyze the plausibility of the equal allocation non-separable costs, the efficient Banzhaf–Owen index and the efficient Banzhaf–Coleman index from the perspective of symmetry. First, based on the difference between “participation processes” and “allocating results”, different forms of symmetry are proposed. Next, building on these forms of symmetry, axiomatic results are put forth for the three power indexes, whereby the plausibility of the three power indexes is analyzed. Finally, on the basis of these different forms of symmetry and related axiomatic results, this article introduces different dynamic processes to analyze how an initial allocation result approaches the results derived from the three power indexes through dynamically modification.


Introduction
Three power indexes are considered in this article, the equal allocation non-separable costs (EANSC), the efficient Banzhaf-Coleman index (EBCI) and the efficient Banzhaf-Owen index (EBOI). The Banzhaf-Coleman index originally defined in the issue of voting problems by Banzhaf [1]. And later on generalized to arbitrary issues by Owen [2], it is named as Banzhaf-Owen index. On the other hand, Ransmeier [3] introduced the EANSC to analyze the optimum yield for dams operated by the Tennessee Valley Authority. Inspired by the notion of the EANSC, Hwang and Liao [4,5] introduced the EBCI and the EBOI respectively. Under the notion of the EANSC (EBCI, EBOI), participants first receive their marginal utilities (entire marginal utilities, average marginal utilities) and further distribute the remaining utilities equally. Moulin [6] showed that the EANSC is the only power index satisfying efficiency, equal treatment for equal, covariance and consistency. Further, Hwang [7] and Hwang and Liao [4,5] characterized the EBCI and the EBOI by means of the standardness property due to Hart and Mas-Colell [8]. Related researches could be studied in Dubey and Shapley [9], Lehrer [10], Haller [11], Brink and Laan [12], and so on.
Symmetry is defined in different forms and endowed with relative senses in different fields. In many studies on the game theory, the symmetry of allocation is often discussed, analyzed and applied from different perspectives, which involve many aspects, such as symmetry of benefits, resources, and power. For example, the contribution of each department in a company cannot be assessed by one uniform standard, but rather should be evaluated based on the symmetry of the contribution ratio of each department in relation to its business. Maschler and Owen [13], Moulin [6] and Peleg [14] conducted axiomatic characterizations of the Shapley value, the EANSC and the Prekernal respectively by considering this notion of symmetry. An illustrative case is that, in a legislative institution, 1.
In an interactive system or model, disparities may occur among participants and related groups as a result of multiple interactive factors. For example, in a production project of a for-profit organization, its participating members and groups will generate benefits during the participation process, and all members and groups will be remunerated at the end of the project. At this point, discrepancies may arise in the benefits generated and remunerations acquired by the members or groups. Positive and negative differences respectively represent complaints and satisfaction of the members or groups. Unlike past symmetry concepts, this article would like to develop a set of excess concepts for allocation based on the differences between "process involvement" and "allocating outcomes". Then, the excess concepts would be combined with participatory behaviors to define three different symmetry axioms.

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Further, the allocative efficiency of resources would combined with different symmetry axioms to axiomatically characterize three power indexes.

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Finally, this article would used the power indexes, each with a different set of symmetry axioms, to produce three different modification functions and related dynamic processes. To be precise and brief, these dynamic processes apply the functions modification to gradually adjust the randomly efficient outcomes and to approach a state of symmetry. According to the axiomatic results, an efficient power index that approximates one of the symmetrical states implies that it approximates all three power indexes.

Preliminaries
Let UP be a finite and non-empty collection of total participants. A coalition is a non-empty collection of UP. A transferable-utility (TU) game is denoted by (P, C), where P is a coalition and C is a mapping such that C : 2 P −→ R and C(∅) = 0. Denote the class of all TU games by TG. An index on TG is a mapping κ which associates with each TU game (P, C) ∈ TG an element κ(P, C) of R P . An index κ satisfies efficiency (EFF) if for all (P, C) ∈ TG, ∑ i∈P κ i (P, C) = C(P). EFF states that all participants allot whole utility entirely. Definition 1.

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The equal allocation non-separable costs (EANSC), θ M , is the index on TG which associates with (P, C) ∈ TG and all participants i ∈ P the value where θ M i (P, C) = C(P) − C(P \ {i}) is the marginal index of i. The marginal index of i could be treated as the marginal contribution of i in the grand coalition P. Under the index θ M , all participants receives their marginal indexes respectively, and allocate the rest of utility simultaneously.

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The efficient Banzhaf-Owen index (EBOI), θ BO , is the index on TG which associates with (P, C) ∈ TG and all participants i ∈ P the value The efficient Banzhaf-Coleman index (EBCI), θ BC , is the index on TG which associates with (P, C) ∈ TG and all participants i ∈ P the value where The Banzhaf-Coleman index of i could be treated as the average marginal contribution of i among all participated coalition. Since the Banzhaf-Coleman index violates EFF, Hwang and Liao [4] proposed the efficient Banzhaf-Coleman index θ BC . Under the index θ BC , all participants receives their Banzhaf-Coleman indexes respectively, and allocate the rest of utility simultaneously.
It is shown that these indexes satisfies EFF simultaneously.

Example 1.
In the following, a practical example is given below to illustrate the three indexes and related applications.

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In a legislative institution, some power appraisal bodies conduct evaluations based on differences in the influence of legislative representatives as to their participation in the overall sitting, known as the marginal index by Definition 1. After the influence of each legislative representative is evaluated, the remaining power is allocated to all the representatives, that is, the EANSC in Definition 1.

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Some power appraisal bodies conduct evaluations based on differences in the overall influence of legislative representatives as to their participation in a bill committee in relation to their professionalism, known as the Banzhaf-Owen index by Definition 1. After the overall influence of each legislative representative is evaluated, the remaining power is allocated to all the representatives, that is, the EBOI in Definition 1.

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Some power appraisal bodies conduct evaluations based on differences in the average influence of legislative representatives as to their participation in a bill committee in relation to their professionalism, known as the Banzhaf-Coleman index by Definition 1. After the average influence of each legislative representative is evaluated, the remaining power is allocated to all the representatives, that is, the EBCI in Definition 1.
Subsequently, some existing axioms and related axiomatic results for these three indexes are provided. For each of these efficient indexes, there is a corresponding definition of reduced game as follows. Given an index κ, (P, C) ∈ TG and S ⊆ P.
For each of these reduced games, there is a corresponding consistency as follows. Let κ be an index.
. Hwang [7] and Hwang and Liao [4] introduced three extended analogues of the standard for games due to Hart and Mas-Colell [8]. Let κ be an index.
Hwang [7] and Hwang and Liao [4] characterized these power indexes by adopting related properties of consistency and standardness as follows. • The index θ M is the only index satisfying COMS and CCON.
• The index θ BO is the only index satisfying EBOS and ESCON.
• The index θ BC is the only index satisfying EBCS and EACON.

Symmetry and Axiomatic Results
In this section, several types of symmetry axiom would be introduced to characterize these indexes. In the following, symmetry refers to the difference between the "participating process" and "allocating results" perceived by participants and related groups.
Let (P, C) ∈ TG, S ⊆ P and κ be an index. Define that κ S (P, C) for the restriction of κ(P, C) to S and |κ(P, C)| S = ∑ i∈S κ i (P, C). The excess of S ⊆ P at κ is Based on the consideration of balancing the differences among all members and groups with respect to different participation behaviors, one could identify three different types of symmetry axiom and provides specific definitions for each type. Let κ be an index.
The complement symmetry under excess states that when two participants do not participate in the largest possible environment, the excesses derived from the power index should be identical. Based on Example 1, in a legislative institution, this type of symmetry focuses on the need to balance the difference between any two legislative representatives as to their respective participation in the overall sitting. The sum symmetry under excess states that when two participants drop out of all participating environments respectively, the sum of the excesses derived from the allocation method should be identical. Based on Example 1, in a legislative institution, this type of symmetry focuses on the need to balance the total difference between any two legislative representatives as to their respective participation in all professional committee bills. • κ satisfies average symmetry under excess (ASE) if for all (P, C) ∈ TG and for all i, j ∈ P, The average symmetry under excess states that when two participants drop out of all participating environments respectively, the average excesses derived from the allocation method should be identical. Based on Example 1, in a legislative institution, this type of symmetry focuses on the need to balance the average difference between any two legislative representatives as to their respective participation in all professional committee bills.
The following results present that the three index satisfy three different types of symmetry axiom respectively. Lemma 1.
The following theorem presents that the three indexes could be characterized by means of corresponding symmetry respectively. Theorem 1.

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The index θ M is the only index satisfying EFF and CSE.

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The index θ BO is the only index satisfying EFF and ASE.

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The index θ BC is the only index satisfying EFF and SSE.

Dynamic Processes
Based on corresponding symmetry, this section proposes dynamic processes that lead the participants to these indexes, starting from a payoff vector generated from arbitrary efficient indexes.
A dynamic process mainly refers to a modification process in which unsatisfactory outcomes gradually trend towards balanced outcomes. Subsequently, the power indexes that produce balanced outcomes are generally assessed by using axiomatic characterization to validate their uniqueness and verify that they meet fair, just, and widely accepted axioms. According to the outcomes of a literature review and the axiomatic characterization results of this study, these power indexes were the solution concept that were individually fair, just, and widely accepted. Stearns [17] was the first to introduce the concept of dynamic processes. Maschler and Owen [13] adopted the reduction concept of "reallocate unsatisfactory outcomes" proposed by Hart and incorporated the variance derived from modification and reallocation into the dynamic process of the Shapley value. Hwang [7] and Hwang and Liao [4,5] combined the differences between process involvement and allocation outcomes with different participatory behaviors to create a set of excess concepts for a compensation vector. The concepts were then applied to develop different dynamic processes for EANSC, EBCI, and EBOI.
Based on the aforementioned outcomes of dynamic processes, this article first determines the differences between process involvement and outcomes and adopted different participatory behaviors to develop a set of excess concepts related to power index. Then, this article examines different symmetry properties and developed corresponding modification functions to gradually adjust the asymmetrical outcomes to achieve symmetry. The axiomatic characterization results indicated that the efficient outcomes that achieve symmetry are consistent with the outcomes derived from EANSC, EBCI, and EBOI.
Let (P, C) ∈ TG and κ be an efficient index. Define the modifying functions M M = (M M i ) i∈P , M BO = (M BO i ) i∈P , M BC = (M BC i ) i∈P as follows. For all i ∈ P, x = κ(P, C) and Based on the results of Theorem 1, it is obvious that The modifying functions measure the sum of the differences of these different excesses between any participant and any other participant respectively. A coefficient is further adopted as the modifier to adjust the original outcomes.
The following results show that these modifying functions are well-defined, i.e., if ∑ i∈P κ i (P, C) = C(P), then ∑ i∈P C i (κ(P, C)) = C(P), where C = M M , M BO , M BC . These results play key roles to prove the necessary convergence results.

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The modifying functions used in this article were inspired by the correction function proposed by Maschler and Owen [13]. However, the main differences between the modifying functions used in this article and those proposed by Maschler and Owen [13] are as follows.

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The correction function proposed by Maschler and Owen [13] is based on reduction.

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The modifying functions used in this article are based on excess.
• The excess concepts used to develop the modifying functions are inspired by the excess notions due to Hwang [7] and Hwang and Liao [4,5]. However, the main differences between the functions used in this article and those proposed by Hwang [7] and Hwang and Liao [4,5] are as follows.
-Hwang [7] and Hwang and Liao [4,5] adopted payoff vectors when defining the differences between process involvement and outcomes.

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This article determines the differences between process involvement and outcomes and adopted different participatory behaviors to develop a set of excess concepts related to power indexes.
Through the axiomatic and dynamic processes of this article, one could identify another motivation: • Can other notions be used along with the three power indexes to propose different axiomatic characterizations and dynamic processes?
We believe this is a noteworthy topic of research. Data Availability Statement: Data available on request due to privacy restrictions. The data presented in this study are available on request from the corresponding author.