sustainability

Article

Sustainable Usability Distribution Mechanisms underMulti-Attribute Sports Management Schemes

En-Cheng Chi 1 and Yu-Hsien Liao 2,*

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Citation: Chi, E.-C.; Liao, Y.-H.

Sustainable Usability Distribution

Mechanisms under Multi-Attribute

Sports Management Schemes.

Sustainability 2021, 13, 1528.

https://doi.org/10.3390/

su13031528

Academic Editor: Andrea Cirà

Received: 30 December 2020

Accepted: 26 January 2021

Published: 1 February 2021

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1 Office of Physical Activities, National Pingtung University, Pingtung County 90003, Taiwan;g731007@mail.nptu.edu.tw

2 Department of Applied Mathematics, National Pingtung University, Pingtung County 900391, Taiwan* Correspondence: twincos@mail.nptu.edu.tw; Tel.: +886-958-631-010

Abstract: Recently, game-theoretical methods have been adopted to analyze the reasonability ofusability distribution mechanisms. On the other hand, sustainability has become a major concep-tion among many fields by focusing on various influences that arose from environmental change,including usability distribution under multi-attribute sports management schemes. In many real-world situations, however, performers and its energy levels (strategies) should be essential factorssimultaneously. Based on maximal-usability among energy level (strategy) vectors, we define anoutput, its efficacious extension and normalization to analyze usability distribution mechanismsunder multi-attribute sports management schemes. We also adopt axiomatic processes to presentthe reasonability for these outputs. By considering reduced scheme and excess mapping, we adoptalternative formulation to offer dynamic processes for the efficacious extension and the normaliza-tion, respectively.

Keywords: multi-attribute sports management scheme; reduced scheme; excess mapping; dy-namic process

1. Introduction

Due to the constant renovation of the trend of real-world environmental change,distribution mechanisms of a combination of different theoretical fields, including sportsusability distribution, have become a major conception in the context of sustainability. Inthe framework of sports usability distribution, the use of several notions could promotethe distribution efficiency no matter for improvement of distribution methods or sportsmanagement techniques of usability. On the other hand, the axiomatic outcomes of game-theoretical distribution mechanisms could be always adopted to analyze various interactionrelationships and related models among agents and coalitions by applying mathematicaloutcomes. In addition to theoretical analysis, game-theoretical outcomes also have beenapplied to offer optimal outcomes or equilibrium situations for many real-word models.Based on the demands for accounting, economics, management sciences and even sportssciences, some more power outputs are introduced as adequate mappings from weights anddecision quota to power. The most extensively applied one is the Banzhaf–Coleman output.The Banzhaf–Coleman output, named after Banzhaf [1], is a power output proposed bythe probability of changing a result of a vote where voting interests are not necessarilydistributed equally among the voters. Plainly speaking, the Banzhaf–Coleman output isa distribution notion that collects each performer’s average marginal contribution fromall participated coalitions. Several related outcomes may be found in, e.g., Banzhaf [1],Owen [2], Dubey and Shapley [3], Moulin [4], Lehrer [5], Haller [6] and van den Brink andvan der Laan [7].

Under traditional schemes, each performer is either entirely participated or not partic-ipated at all in participation with other performers, while under a multi-choice scheme,each performer could be consented to take finite many different energy levels. It is known

Sustainability 2021, 13, 1528. https://doi.org/10.3390/su13031528 https://www.mdpi.com/journal/sustainability

Sustainability 2021, 13, 1528 2 of 16

that outputs on multi-choice schemes could be performed under various issues such aseconomics, management sciences, sports sciences and so on. Hwang and Liao [8,9] andLiao [10] proposed several extensions of the max-reduced scheme and the complement-reduced scheme to characterize several core concepts in the context of multi-choice schemes.Later, Hwang and Liao [11] also considered an extended self-reduced scheme to analyze amulti-choice generalization of the Shapley value.

In different topics, from biomedical engineering, sciences to environment and the man-agement sciences, performers confront an increasing demand to focus on multiple objectiveseffectively in its operational procedures. Related conditions include analyzing allocationtradeoffs, selecting optimal strategy or course designs or an arbitrary other situation whereyou need an effective rule with tradeoffs among several objectives. Under various cases,these real-world effective conditions might be modeled as a mathematical multi-attributeoptimization status. The rules of such conditions require appropriate techniques to presentoptimal outcomes that—unlike traditional notions or viewpoints—apply several propertiesof the objectives into account.

Under the axiomatic processes for outputs under cooperative schemes, consonanceis a crucial property of useful outputs. The notion behind this type of consonance isas follows: for a given scheme, performers might develop prospects of the scheme andmay be willing to consent the computation of its remunerations to be based upon theseprospects. The output concept is consonant if it gives the same remunerations to performersin the original scheme as it does to performers of the imaginary reduced scheme. Thus,consonance is a requisite of the inner “robustness" of compromises. In addition to axiomaticcharacterizations, dynamic processes can be represented that lead the performers to thatoutput, commencing from an arbitrary efficacious remuneration vector. The main base of adynamic notion was raised from Stearns [12].

The above-mentioned existing-outcomes generate one motivation:

• whether the power (usability) outputs could be defined by simultaneously consideringmulti-choice behavior and multi-attribute situations.

Different from the contexts of traditional schemes and multi-choice schemes, we focuson the framework of multi-attribute multi-choice schemes throughout this paper.

1. In Section 2.1, the multi-consideration Banzhaf–Coleman output, the multi-considerationefficient Banzhaf–Coleman output and the multi-consideration normalized Banzhaf–Colemanoutput are further defined by adopting maximal-usability among multi-choice levelvectors on multi-attribute multi-choice schemes.

2. In Section 2.1, some motivating and practical examples are provided to present relatedapplications for sports management.

3. By applying an extended reduction, we propose some axiomatic processes to presentthe reasonability for these outputs in Section 3. In order to analyze the dynamicprocesses of these outputs, we adopt alternative formulations for these outputs interms of excess mappings.

4. In Section 4, specific reductions and excess mapping are applied to present that theseoutputs can be reached by performers who commence from an arbitrary efficaciousremuneration vector.

2. Preliminaries2.1. Definitions and Notations

Let UV be the universe of performers and P ⊆ UV be a set of performers. Lete = (ep)p∈P with ep ∈ N be the vector that shows the amount of energy levels for eachperformer, at which it can operate. For p ∈ UV, we define Ep = {0, 1, . . . , ep} to be the levelrepository of performer p, where 0 means not acting, and E+

p = Ep \ {0}. For P ⊆ UV,P 6= ∅, let EP = ∏p∈P Ep be the product set of the level repositories for performers in P.For every K ⊆ P, we define σK ∈ EP as the vector with σK

p = 1 if p ∈ K, and σKp = 0 if

p ∈ P \ K. Denote 0P to be the zero vector in RP. For m ∈ N, let Γm = {1, . . . , m}.

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A multi-choice scheme is denoted by (P, e, h), where P 6= ∅ is a finite collection ofperformers, e = (ep)p∈P ∈ EP is the vector that appears the amount of energy levels foreach performer and h : EP → R is a map which apportions to each ω = (ωp)p∈P ∈ EP

the usability that the performers can accept when each performer p takes energy levelωp ∈ Ep with h(0P) = 0. A scheme (P, e, h) will be denoted by the map h if there is noconfusion. Given a multi-choice scheme (P, e, h) and ω ∈ EP, we denote (P, ω, h) to bethe multi-choice subscheme defined by restricting h to {χ ∈ EP | χp ≤ ωp ∀p ∈ P}. Amulti-attribute multi-choice scheme is a triple (P, e, Hm), where m ∈ N, Hm = (ht)t∈Γm

and (P, e, ht) is a multi-choice scheme for every t ∈ Γm. Let ∆ be the collection of allmulti-attribute multi-choice schemes.

Given (P, e, h) ∈ ∆ and ω ∈ RP, we define that S(ω) = {p ∈ P| ωp 6= 0} and ωK to bethe restriction of ω at K for each K ⊆ P. Further, we define h∗(K) = maxω∈EP{h(ω)|S(ω) =K} to be the maximal-usability (Here we apply bounded multi-choice schemes, defined as(P, e, h) such that, there exists Eh ∈ R such that h(ω) ≤ Eh for every ω ∈ EP. We apply it toguarantee that h∗(K) is well-defined.) among all vectors ω with S(ω) = K. A remunerationvector of (P, e, Hm) ∈ ∆ is a vector x = (xt)t∈Γm and xt = (xt

p)p∈P ∈ RP, where xtp denotes

the remuneration to performer p in (P, e, ht) for every t ∈ Γm and for every p ∈ P. Aremuneration vector x of (P, e, Hm) is multi-attribute efficacious if ∑p∈P xt

p = ht∗(

P)

forevery t ∈ Γm. The collection of all multi-attribute efficacious vectors of (P, e, Hm) is denotedby ME(P, e, Hm).

An output is a map ρ apportioning to each (P, e, Hm) ∈ ∆, an element

ρ(

P, e, Hm) = (ρt(P, e, Hm))t∈Γm

,

where ρt(P, e, Hm) = (ρtp(

P, e, Hm))p∈P ∈ RP and ρt

p(

P, e, Hm) is the remuneration of the

performer p apportioned by ρ in(

P, e, ht).Next, we introduce three outputs under the multi-attribute multi-choice situation.

Definition 1. The multi-consideration Banzhaf–Coleman output (MBCO), Θ, is defined to be forevery (P, e, Hm) ∈ ∆, for every t ∈ Γm and for every p ∈ P,

Θtp(P, e, Hm) =

12|P|−1 ∑

S⊆Pp∈S

[ht∗(S)− ht

∗(S \ {p})].

Under the output Θ, all performers receive an average marginal contribution of maximal-usability in each S ⊆ P.

An output ρ conforms multi-attribute efficacy (MEIY) if for every (P, e, Hm) ∈ ∆ andfor every t ∈ Γm, ∑p∈P ρp(P, e, Hm) = ht

∗(P). MEIY presents that all performers completelyallocate all the usability. Clearly, the MBCO violates MEIY. In the following, we consideran efficacious extension and an normalization.

Definition 2.

• The multi-consideration efficacious Banzhaf–Coleman output (MEBCO), Θ, is defined forevery (P, e, Hm) ∈ ∆, for every t ∈ Γm and for every p ∈ P,

Θtp(P, e, Hm) = Θt

p(P, e, Hm) +1|P| ·

[ht∗(P)− ∑

k∈PΘt

k(P, e, Hm)].

Under the output Θ, all performers first receive its MBCO, and further allocate the remainingusability equally.

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• The multi-consideration normalized Banzhaf–Coleman output (MNBCO), Φ, is defined forevery (P, e, Hm) ∈ ∆∗, for every t ∈ Γm and for every p ∈ P,

Φtp(P, e, Hm) =

ht∗(P)

∑k∈P

Θtk(P, e, Hm)

·Θtp(P, e, Hm),

where ∆∗ = {(P, e, Hm) ∈ ∆ | ∑p∈P

Θtp(P, e, Hm) 6= 0 for every t ∈ Γm}. Under the notion

of Φ, all performers allocate the maximal-usability of the grand coalition proportionally byapplying the MBCO of all performers.

Lemma 1. The MEBCO and the MNBCO conform MEIY on ∆ and ∆∗, respectively.

Proof. For every (P, e, Hm) ∈ ∆ and for every t ∈ Γm,

∑p∈P

Θtp(P, e, Hm) = ∑

p∈P

[Θt

p(P, e, Hm) + 1|P| ·

[ht∗(P)− ∑

k∈PΘt

k(P, e, Hm)]]

= ∑p∈P

Θtp(P, e, Hm) + |P|

|P| ·[ht∗(P)− ∑

k∈PΘt

k(P, e, Hm)]

= ∑p∈P

Θtp(P, e, Hm) + ht

∗(P)− ∑k∈P

Θtk(P, e, Hm)

= ht∗(P).

Thus, the MEBCO conforms MEIY on ∆. For every (P, e, Hm) ∈ ∆∗ and for everyt ∈ Γm,

∑p∈P

Φtp(P, e, Hm) = ∑

p∈P

[ht∗(P)

∑k∈P

Θtk(P,e,Hm)

·Θtp(P, e, Hm)

]= ht

∗(P)∑

k∈PΘt

k(P,e,Hm)·[

∑p∈P

Θtp(P, e, Hm)

]= ht

∗(P).

Thus, the MNBCO conforms MEIY on ∆∗.

2.2. Motivating and Practical Examples

As we mentioned in the introduction, each performer might be allowed to participatewith different levels in real situations respectively. On the other hand, multi-attributeanalysis is a notion of multiple criterion analysis that is concerned with situations involvingsimultaneously more than one objective to be optimized. Multi-attribute analysis has beenadopted in many issues, including biomedical engineering, economics, politics, sportsmanagement sciences, logistics, where efficacious decisions need to be adopted in thepresence of trade-offs among several objectives. For instance, minimizing cost whilemaximizing comfort while marketing a central air conditioning system, and maximizingefficacy whilst minimizing energy consumption and emission of pollutants are examplesof multi-attribute efficacious problems involving respectively two and three objectives.Under various cases, there might be more than three objectives. Hence, we focus on theframework of multi-attribute multi-choice schemes throughout this paper. The advantagesof our methods are that these outputs of the multi-attribute multi-choice scheme alwaysexist and result in a kind of global outcome for a specific performer by summarizing theresult under all its energy levels.

Here we provide a brief motivating example of multi-choice schemes in the settingof “sports management”. Let P = {1, 2, . . . , p} be a set of all performers of a sportsmanagement system (P, e, h). The function h could be treated as a usability function whichassigns to each level vector α = (αp)p∈P ∈ EP the worth that the performers can obtainwhen each performer p participates at operation strategy αp ∈ Ep in (P, e, h). Modeled inthis way, the sports management system (P, e, h) could be considered as a multi-choice

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scheme, with h being each characteristic function and Ep being the set of all operationstrategies of the performer p.

In the following, we also provide a practical application of power distribution ina sports association, such as NBA, MLB and so on. Let P = {1, 2, . . . , p} be a set of allperformers of a sports association. In the sports association, all the performers are electedby voting or recommendation by sports parties (or teams). All performers have the powerto propose, discuss, establish and veto all bills (or rules). They dedicate different levels ofattention and participation to different bills depending on their academic expertise and thepublic opinion they represent. The level of involvement is also closely associated with thealliance strategy formed for the interests of different sports parties. For the aforementionedreasons, strategies adopted by each performer of the parliament show distinct levels ofparticipation and certain amounts of ambiguity. The function h could be treated as apower function which assigns to each level vector α = (αp)p∈P ∈ EP the power that theperformers can dedicate when each performer p participates at operation strategy αp ∈ Ep.Modeled in this way, the sports association operational system (P, e, h) could be consideredas a multi-choice scheme, with h being each characteristic function and Ep being the set ofall operation strategies of the performer p. To evaluate the influence of each performer inthe sports association, using the power indicators we proposed, we first assess the influenceeach sports association performer has arisen over previous bill meetings based on variousstrategies, which are the outputs mentioned in Definitions 1 and 2.

Here we provide an application with real data as follows. Let (P, e, Hm) ∈ ∆ withP = {i, j, k}, m = 2 and e = (2, 1, 1). Thus, (P, e, Hm) =

((P, e, h1), (P, e, h1)

). Further, let

h1(2, 1, 1) = 5, h1(1, 1, 1) = 7, h1(2, 1, 0) = 3, h1(2, 0, 1) = 2, h1(2, 0, 0) = 9, h1(1, 1, 0) =3, h1(1, 0, 1) = −4, h1(0, 1, 1) = 5, h1(1, 0, 0) = −1, h1(0, 1, 0) = 2, h1(0, 0, 1) = −3,h2(2, 1, 1) = 9, h2(1, 1, 1) = 5, h2(2, 1, 0) = 7, h2(2, 0, 1) = 3, h2(2, 0, 0) = 2, h2(1, 1, 0) = 9,h2(1, 0, 1) = 3, h2(0, 1, 1) = −4, h2(1, 0, 0) = 5, h2(0, 1, 0) = −2, h2(0, 0, 1) = 5 andh1(0, 0, 0) = 0 = h2(0, 0, 0). Therefore, we have that h1

∗({i, j, k}) = 7, h1∗({i, j}) = 3,

h1∗({i, k}) = 2, h1

∗({j, k}) = 5, h1∗({i}) = 9, h1

∗({j}) = 2, h1∗({k}) = −3, h2

∗({i, j, k}) = 9,h2∗({i, j}) = 9, h2

∗({i, k}) = 3, h2∗({j, k}) = −4, h2

∗({i}) = 5, h2∗({j}) = −2, h2

∗({k}) = 5and h1

∗(∅) = 0 = h2∗(∅). By Definitions 1 and 2,

Θ1i (P, e, Hm) = 18

4 , Θ1j (P, e, Hm) = 15

4 , Θ1k(P, e, Hm) = −2

4 ,

Θ1i (P, e, Hm) = 55

12 , Θ1j (P, e, Hm) = 46

12 , Θ1k(P, e, Hm) = −5

12 ,

Φ1i (P, e, Hm) = 144

31 , Φ1j (P, e, Hm) = 120

31 , Φ1k(P, e, Hm) = −16

31 ,Θ2

i (P, e, Hm) = 274 , Θ2

j (P, e, Hm) = 0, Θ2k(P, e, Hm) = −3

4 ,

Θ2i (P, e, Hm) = 31

4 , Θ2j (P, e, Hm) = 1, Θ2

k(P, e, Hm) = 14 ,

Φ2i (P, e, Hm) = 81

8 , Φ2j (P, e, Hm) = 0, Φ2

k(P, e, Hm) = −98 .

In the following sections, we would like to demonstrate that the MBCO, the MEBCOand the MNBCO could present “optimal distribution mechanisms” among all performers, inthe sense that this organization can receive remuneration from each combination of opera-tional levels of all performers under multi-choice behavior and multi-attribute situations.

3. Axiomatic Outcomes

Here, we demonstrate that there exists corresponding reduced schemes that could beapplied to axiomatize the MBCO, the MEBCO and the MNBCO.

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Let ρ be an output, (P, e, Hm) ∈ ∆ and S ⊆ P. The 1-reduced scheme (S, eS, H1,mS,ρ ) is

defined by H1,mS,ρ = (h1,t

S,ρ)t∈Γm and

h1,tS,ρ(ω) =

0 , ω = 0S,

12|P\S| ∑

Q⊆P\S

[maxτ∈EQ{h(ω, τ, 0P\{S∪Q}

)|S(τ) = Q}

− ∑p∈Q

ρp(P, e, h)]

, otherwise

for every ω ∈ ES. An output ρ conforms 1-consonance (1CSE) if ρtp(S, eS, H1,m

S,ρ ) =

ρtp(P, e, Hm) for every (P, e, Hm) ∈ ∆, for every S ⊆ P with |S| = 2, for every t ∈ Γm and for

every p ∈ S. Further, ρ conforms 1-criterion for schemes (1CS) if ρ(P, e, Hm) = Θ(P, e, Hm)for every (P, e, Hm) ∈ ∆ with |P| ≤ 2. In the following, we characterize the MBCO byadopting 1CSE and 1CS.

Lemma 2.

1. The MBCO conforms 1CSE on ∆.2. On ∆, the MBCO is the only output conforming 1CS and 1CSE.

Proof. To demonstrate result 1, let (P, e, Hm) ∈ ∆ and S ⊆ P. The proof is finished if|P| = 1. Suppose that |P| ≥ 2 and S = {p, q} for some p, q ∈ P, for every t ∈ Γm and forevery p ∈ S,

Θtp(S, eS, H1,m

S,Θ) = 12|S|−1 · ∑

T⊆Sp∈T

[(h1,t

S,Θ)∗(T)− (h1,tS,Θ)∗(T \ {p})

]= 1

2|S|−1·2|P\S| · ∑T⊆Sp∈T

∑Q⊆P\S

[ht∗(T ∪Q)− ht

∗(T \ {p} ∪Q)]

= 12|P|−1 · ∑

K⊆Pp∈K

[h(K)− h(K \ {p})

]= Θt

p(P, e, Hm).

Hence, Θ conforms 1CSE.By result 1, Θ conforms 1CSE on ∆. Clearly, Θ conforms 1CS on ∆. To demonstrate

uniqueness of result 2, suppose that ρ conforms 1CS and 1CSE. Let (P, e, h) ∈ ∆. If |P| ≤ 2,then ρ(P, e, h) = Θ(P, e, h) by 1CS. The condition |P| > 2: Let p ∈ P and t ∈ Γm, and letS ⊂ P with |S| = 2 and p ∈ S. Therefore,

ρtp(P, e, Hm) = ρt

p(S, eS, H1,mS,ρ ) (by 1CSE)

= Θtp(S, eS, H1,m

S,ρ ) (by 1CS)

= 12|S|−1 · ∑

T⊆Sp∈T

[(h1,t

S,ρ)∗(T)− (h1,tS,ρ)∗(T \ {p})

]= 1

2|S|−1·2|P\S| ∑T⊆Sp∈T

∑Q⊆P\S

[ht∗(T ∪Q)− ht

∗((T \ {p}) ∪Q

)]

= 12|P|−1 · ∑

K⊆Pp∈K

ht∗(K)− ht

∗(K \ {p})

= Θtp(P, e, Hm).

Thus, ρ(P, e, Hm) = Θ(P, e, Hm).

In the following we take some examples to manifest that each of the properties appliedin Lemma 2 are independent of the rest of properties.

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Example 1. Consider an output ρ by for every (P, e, Hm) ∈ ∆, for every t ∈ Γm and for everyp ∈ P, ρt

p(P, e, Hm) = 0. Clearly, ρ conforms 1CSE, but it does not conform 1CS.

Example 2. Consider an output ρ for every (P, e, Hm) ∈ ∆, for every t ∈ Γm and for every p ∈ P,

ρtp(P, e, Hm) =

{Θt

p(P, e, Hm) if |P| ≤ 2,0 otherwise.

On ∆, ρ conforms 1CS, but it does not conform 1CSE.

It is easy to demonstrate that the output Θ and Φ violates 1CSE. In the following, weconsider the 2-reduced scheme. Let ρ be an output, (P, e, Hm) ∈ ∆ and S ⊆ P. The 2-reducedscheme (S, eS, H2,m

S,ρ ) is defined by H2,mS,ρ = (h2,t

S,ρ)t∈Γm and

h2,tS,ρ(ω) =

0 , ω = 0S,

maxτ∈EP\S

{h(ω, τ)|S(τ) = P \ S} − ∑p∈P\S

ρp(P, e, h) , S(ω) = S,

12|P\S| ∑

Q⊆P\S

[maxτ∈EQ{h(ω, τ, 0P\{S∪Q}

)|S(τ) = Q}

− ∑p∈Q

ρp(P, e, h)]

, otherwise

for every ω ∈ ES. An output ρ conforms 2-consonance (2CSE) if ρtp(S, eS, H2,m

S,ρ ) =

ρtp(P, e, Hm) for every (P, e, Hm) ∈ ∆, for every S ⊆ P with |S| = 2, for every t ∈ Γm

and for every p ∈ S. Furthermore, ρ conforms 2-criterion for schemes (2CS) if ρ(P, e, Hm) =Θ(P, e, Hm) for every (P, e, Hm) ∈ ∆ with |P| ≤ 2.

Clearly, (S, eS, HmS,ρ) does not exist if ∑p∈S Θt

p(P, e, Hm) = 0. Thus, we consider the3-consonance as follows. An output ρ conforms 3-consonance (3CSE) if (S, eS, Hm

S,ρ) ∈ ∆∗

for some (P, e, Hm) ∈ ∆ and for some S ⊆ P with |S| = 2, it holds that ρtp(S, eS, Hm

S,ρ) =

ρtp(P, e, Hm) for every t ∈ Γm and for every p ∈ S. Further, ρ conforms 3-criterion for

schemes (3CS) if ρ(P, e, Hm) = Φ(P, e, Hm) for every (P, e, Hm) ∈ ∆ with |P| ≤ 2.Subsequently, we characterize the MEBCO and the MNBCO by respectively adopting

2CSE, 3CSE, 2CS and 3CS. In order to establish consonance of the MEBCO and the MNBCO,it will be useful to introduce alternative formulation for the MEBCO and the MNBCO interms of excess. Let (P, e, Hm) ∈ ∆, S ⊆ P and x be a remuneration vector in (P, e, Hm).Define that xt(S) = ∑p∈S xt

p for every t ∈ Γm. The excess of a coalition S ⊆ P at x isconsidered to be

Ex(S, Hm, x) = (Ex(S, ht, xt))t∈Γm and Ex(S, ht, xt) = ht∗(S)− xt(S). (1)

The value Ex(S, ht, xt) can be regarded as the complaint of coalition S when allperformers in S receive its remunerations from xt in (P, e, ht).

Lemma 3. For every (P, e, Hm) ∈ ∆, for every x ∈ ME(P, e, Hm), for every p, q ∈ P and forevery t ∈ Γm,

∑S⊆P\{p,q}

Ex(S ∪ {p}, ht, xt) = ∑S⊆P\{p,q}

Ex(S ∪ {q}, ht, xt)

⇐⇒ x = Θ(P, e, Hm).

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Proof. Let (P, e, Hm) ∈ ∆ and x ∈ ME(P, e, Hm). For every t ∈ Γm and for every p, q ∈ P,

∑S⊆P\{p,q}

Ex(S ∪ {p}, ht, xt) = ∑S⊆P\{p,q}

Ex(S ∪ {q}, ht, xt)

⇐⇒ ∑S⊆P\{p,q}

[ht∗(S ∪ {i})− xt(S ∪ {i})

]= ∑

S⊆P\{p,q}

[ht∗(S ∪ {j})− xt(S ∪ {j})

]⇐⇒

[∑

S⊆P\{p,q}ht∗(S ∪ {i})

]− 2|P|−2 · xt

p =

[∑

S⊆P\{p,q}ht∗(S ∪ {j})

]− 2|P|−2 · xt

q

⇐⇒ xtp − xt

q = 12|P|−2 · ∑

S⊆P\{p,q}

[ht∗(S ∪ {i})− ht

∗(S ∪ {j})].

(2)By definition of Θ,

Θtp(P, e, Hm)−Θt

q(P, e, Hm) = 12|P|−2 · ∑

S⊆P\{p,q}

[ht∗(S ∪ {p})− ht

∗(S ∪ {q})]. (3)

By Equations (2) and (3), for every p, q ∈ P,

xtp − xt

q = Θtp(P, e, Hm)−Θt

q(P, e, Hm).

Hence,

∑q∈P

[xt

p − xtq

]= ∑

q∈P

[Θt

p(P, e, Hm)−Θtq(P, e, Hm)

].

That is, |P| · xtp−∑q∈P xt

q = |P| ·Θtp(P, e, Hm)−∑q∈P Θt

q(P, e, Hm). Since x ∈ ME(P, e, Hm)

and Θ conforms MEIY, |P| · xtp− ht

∗(P) = |P| ·Θtp(P, e, Hm)− ht

∗(P). Hence, xtp = Θt

p(P, e, Hm)

for every t ∈ Γm and for every p ∈ P, i.e., x = Θ(P, e, Hm).

Remark 1. Clearly,

∑S⊆P\{p,q}

Ex(S \ {p}, Hm, Θ(P, e, Hm)) = ∑S⊆P\{p,q}

Ex(S \ {q}, Hm, Θ(P, e, Hm))

for every (P, e, Hm) ∈ ∆ and for every p, q ∈ P.

Theorem 1.

1. The MEBCO conforms 2CSE on ∆.2. If ρ conforms 2CS and 2CSE, then it also conforms MEIY.3. On ∆, the MEBCO is the only output conforming 2CS and 2CSE.

Proof. To demonstrate result 1, let (P, e, Hm) ∈ ∆ and S ⊆ P. The proof is finished if|P| = 1. Suppose that |P| ≥ 2, x = Θ(P, e, Hm) and S = {p, q} for some p, q ∈ P. For everyt ∈ Γm and for every l ∈ S,

Ex({l}, h2,t

S,x, xtS)

= h2,tS,x({l})− xt

l

=(

12|P\S|

· ∑Q⊆P\S

[ht∗({l} ∪Q)− ∑

k∈Qxt

k])− xt

l

= 12|P\S|

· ∑Q⊆P\S

(ht∗({l} ∪Q)− ∑

k∈Qxt

k − xtl

)= 1

2|P\S|· ∑

Q⊆P\S

[ht∗({l} ∪Q

)− ∑

k∈{l}∪Qxt

k

]= 1

2|P\S|· ∑

Q⊆P\{p,q}Ex(Q ∪ {l}, ht, xt).

(4)

Sustainability 2021, 13, 1528 9 of 16

Since Θ conforms MEIY, xtS ∈ Ex(S, eS, h2,t

S,x) by definition of 2-reduced scheme. Fur-ther, by Lemma 3 and Equation (4),

Ex({p}, h2,t

S,x, xtS)

= 12|P\S|

· ∑Q⊆P\{p,q}

Ex(Q ∪ {p}, ht, xt) (by Equation (4))

= 12|P\S|

· ∑Q⊆P\{p,q}

Ex(Q ∪ {q}, ht, xt) (by Lemma 3)

= Ex({q}, h2,t

S,x, xtS). (by Equation (4))

Therefore, xS = Θ(S, eS, h2,mS,Θ

). That is, Θ conforms 2CSE.To demonstrate result 2, suppose ρ conforms 2CS and 2CSE. Let (P, e, Hm) ∈ ∆ and

t ∈ Γm. By 2CS, ρ conforms MEIY if |P| ≤ 2. The condition |P| > 2: Suppose, onthe contrary, that there exists (P, e, Hm) ∈ ∆ such that ∑p∈P ρt

p(P, e, Hm) 6= ht∗(P). This

presents that there exists p ∈ P and q ∈ P such that [ht∗(P)− ∑k∈P\{p,q} ρt

k(P, e, Hm)] 6=[ρt

p(P, e, Hm) + ρtq(P, e, Hm)]. By 2CSE, ρ conforms MEIY for two-person schemes and this

contradicts with

ρtp(P, e, Hm) + ρt

q(P, e, Hm) = ρtp({p, q}, e{p,q}, h2,t

{p,q},ρ) + ρtq({p, q}, e{p,q}, h2,t

{p,q},ρ)

= ht∗(P)− ∑

k∈P\{p,q}ρt

k(P, e, Hm).

Hence, ρ conforms MEIY.To demonstrate result 3, Θ conforms 2CSE by result 1. Clearly, Θ conforms 2CS. To

demonstrate uniqueness, suppose ρ conforms 2CS and 2CSE; hence, by result 2, ρ alsoconforms MEIY. Let (P, e, Hm) ∈ ∆. By 2CS, ρ(P, e, Hm) = Θ(P, e, Hm) if |P| ≤ 2. Thecondition |P| > 2: Let p ∈ P, t ∈ Γm and S = {p, q} for some q ∈ P \ {p}. Then,

ρtp(P, e, Hm)−Θt

p(P, e, Hm)

= ρtp(S, eS, h2,m

S,ρ )−Θtp(S, eS, h2,m

S,Θ) (by 2CSE of ρ, Θ)

= Θtp(S, eS, h2,m

S,ρ )−Θtp(S, eS, h2,m

S,Θ) (by 2CS of ρ, Θ)

= 12 ·[(h2,t

S,ρ)∗(S) + (h2,tS,ρ)∗({p})− (h2,t

S,ρ)∗({q})]

− 12 ·[(h2,t

S,Θ)∗(S) + (h2,t

S,Θ)∗({p})− (h2,t

S,Θ)∗({q})

].

(5)

By definitions of h2,tS,ρ and h2,t

S,Θ,

(h2,tS,ρ)∗({p})− (h2,t

S,ρ)∗({q}) = 12|P\S|

· ∑Q⊆P\S

[ht∗({p} ∪Q)− ht

∗({q} ∪Q)]

= (h2,tS,Θ

)∗({p})− (h2,tS,Θ

)∗({q}).(6)

By Equation (6), Equation (5) becomes

ρtp(P, e, Hm)−Θt

p(P, e, Hm)

= 12 ·((h2,t

S,ρ)∗(S)− (h2,tS,Θt)∗(S)

)= 1

2 ·(

ρtp(P, e, Hm) + ρt

q(P, e, Hm)−Θtp(P, e, Hm) + Θt

q(P, e, Hm))

.

That is, for every p, q ∈ P,

ρtp(P, e, Hm)− ρt

q(P, e, Hm) = Θtp(P, e, Hm)−Θt

q(P, e, Hm).

By MEIY of ρ and Θ,

|P| · ρtp(P, e, Hm)− ht

∗(P) = |P| ·Θtp(P, e, Hm)− ht

∗(P).

Sustainability 2021, 13, 1528 10 of 16

Thus, ρtp(P, e, Hm) = Θt

p(P, e, Hm) for every p ∈ P, i.e., ρ(P, e, Hm) = Θ(P, e, Hm).

In the following we take some examples to show that each of the properties applied inTheorem 1 are independent of the rest of properties.

Example 3. Consider an output ρ for every (P, e, Hm) ∈ ∆, for every t ∈ Γm and for every p ∈ P,ρt

p(P, e, Hm) = 0. Clearly, ρ conforms 2CSE, but it does not conform 2CS.

Example 4. Consider an output ρ for every (P, e, Hm) ∈ ∆, for every t ∈ Γm and for every p ∈ P,

ρtp(P, e, Hm) =

{Θt

p(P, e, Hm) if |P| ≤ 2,0 otherwise.

On ∆, ρ conforms 2CS, but it does not conform 2CSE.

Lemma 4. Let (P, e, Hm) ∈ ∆∗ and x ∈ ME(P, e, Hm). Then, for every p, q ∈ P and t ∈ Γm, ∑S⊆P\{p,q}

Ex(S ∪ {p}, ht,xt

at ) = ∑S⊆P\{p,q}

Ex(S ∪ {q}, ht,xt

at )

⇐⇒ x = Φ(P, e, Hm),

where at = ht(P)∑

k∈PΘt

k(P,e,Hm).

Proof. Let (P, e, Hm) ∈ ∆∗ and x ∈ ME(P, e, Hm). For every p, q ∈ P and t ∈ Γm,

∑S⊆P\{p,q}

Ex(S ∪ {p}, ht, xt

at ) = ∑S⊆P\{p,q}

Ex(S ∪ {q}, ht, xt

at )

⇐⇒ ∑S⊆P\{p,q}

[ht(S ∪ {p})− xt(S∪{p})

at

]= ∑

S⊆P\{p,q}

[ht(S ∪ {q})− xt(S∪{q})

at

]⇐⇒

[∑

S⊆P\{p,q}ht(S ∪ {p})

]− 2|P|−2·xt

pat =

[∑

S⊆P\{p,q}ht(S ∪ {q})

]− 2|P|−2·xt

qat

⇐⇒ xtp − xt

q = at

2|P|−2 · ∑S⊆P\{p,q}

[ht(S ∪ {p})− ht(S ∪ {q})

].

(7)

By definition of Φ,

Φtp(P, e, Hm)−Φt

q(P, e, Hm) = at

2|P|−2 · ∑S⊆P\{p,q}

[ht(S ∪ {p})− ht(S ∪ {q})

](8)

By Equations (7) and (8), for every pair {p, q} ⊆ P and t ∈ Γm,

xtp − xt

q = Φtp(P, e, Hm)−Φt

q(P, e, Hm).

Hence,

∑q 6=p

[xt

p − xtq

]= ∑

q 6=p

[Φt

p(P, e, Hm)−Φtq(P, e, Hm)

].

That is,(|P| − 1

)· xt

p − ∑q 6=p xtq =

(|P| − 1

)· Φt

p(P, e, Hm) − ∑q 6=p Φtq(P, e, Hm).

Since x ∈ ME(P, e, Hm) and Φ conforms MEIY, |P| · xtp − ht

∗(P) = |P| · Φtp(P, e, Hm) −

ht∗(P). Hence, xt

p = Φtp(P, e, Hm) for every (P, e, Hm) ∈ ∆∗, for every p ∈ P and for every

t ∈ Γm.

Theorem 2.

1. The MNBCO conforms 3CSE on ∆∗.2. If ρ conforms 3CS and 3CSE, then it also conforms MEIY.

Sustainability 2021, 13, 1528 11 of 16

3. On ∆∗, the MNBCO is the only output conforming 3CS and 3CSE.

Proof. The proofs of outcomes 1 and 2 are similar to the proofs of outcomes 1 and 2 ofTheorem 1, so we omit them. Next, we demonstrate result 3. By result 1, the MNBCOconforms 3CSE. Clearly, the MNBCO conforms 3CS. To demonstrate uniqueness, supposeρ conforms 3CSE and 3CS on ∆∗. By result 2, ρ conforms MEIY on ∆∗. Let (P, e, Hm) ∈ ∆∗.The proof could be finished by induction on |P|. By 3CS, it is trivial that ρ(P, e, Hm) =Θ(P, e, Hm) if |P| ≤ 2. Suppose that it holds if |P| ≤ l − 1, l ≥ 3. The condition |P| = l: Let

p, q ∈ P with p 6= q and t ∈ Γm. By Definition 2, Φtk(P, e, Hm) = ht

∗(P)∑

h∈PΦt

h(P,e,Hm)·Φt

k(P, e, Hm)

for every k ∈ P. Assume that ωtk =

Φtk(P,e,Hm)

∑h∈P

Φth(P,e,Hm)

for every k ∈ P and for every t ∈ Γm.

Therefore,

ρtp(P, e, Hm)

= ρtp(

P \ {q}, eP\{q}, h2,tP\{q},ρ

)(by 3CSE of ρ)

= Φtp(

P \ {q}, eP\{q}, h2,tP\{q},ρ

)(by 3CS of ρ)

=

(h2,t

P\{q},ρ

)∗

(P\{q}

)∑

k∈P\{q}Φt

k

(P\{q},eP\{q} ,h

2,tP\{q},ρ

) ·Φtp

(P \ {q}, eP\{q}, h2,t

P\{q},ρ

)=

ht∗(P)−ρt

p(P,e,Hm)

∑k∈P\{q}

Φtk(P,e,Hm)

·Φtp(P, e, Hm) (by Equation (8))

=ht∗(P)−ρt

p(P,e,Hm)

−Φtq(P,e,Hm)+ ∑

k∈PΦt

k(P,e,Hm)·Φt

p(P, e, Hm).

(9)

By Equation (9),

ρtp(P, e, Hm) · [1−ωt

q] = [ht∗(P)− ρt

q(P, e, Hm)] ·ωtq

=⇒ ∑p∈P

ρtp(P, e, Hm) · [1−ωt

q] = [ht∗(P)− ρt

q(P, e, Hm)] · ∑p∈P

ωtq

=⇒ ht∗(P) · [1−ωt

q] = [ht∗(P)− ρt

q(P, e, Hm)] · 1 (by MEIY of ρ)=⇒ ht

∗(P)− ht∗(P) ·ωt

q = ht∗(P)− ρt

q(P, e, Hm)

=⇒ Φtq(P, e, Hm) = ρt

q(P, e, Hm).

The proof is finished.

In the following we take some examples to manifest that each of the properties appliedin Theorem 1 are independent of the rest of the properties.

Example 5. Consider an output ρ by for every (P, e, Hm) ∈ ∆∗, for every t ∈ Γm and for everyp ∈ P, ρt

p(P, e, Hm) = 0. On ∆∗, ρ conforms 3CSE, but it does not conform 3CS.

Example 6. Consider an output ρ by for every (P, e, Hm) ∈ ∆∗, for every t ∈ Γm and for everyp ∈ P,

ρtp(P, e, Hm) =

{Φt

p(P, e, Hm) if |P| ≤ 2,0 otherwise.

On ∆∗, ρ conforms 3CS, but it does not conform 3CSE.

4. Dynamic Processes

Here, we adopt excess mappings and reductions to offer dynamic outcomes for theMEBCO and the MNBCO.

In order to present the dynamic processes of the MEBCO and the MNBCO, we firstlydefine switch mappings by means of excess mappings. The switch mappings are basedon the notion that each performer shortens the complaint relating to its own and others’

Sustainability 2021, 13, 1528 12 of 16

non-participation, and uses these regulations to revise the original remuneration. In thefollowing, we firstly introduce the dynamic processes for the MEBCO.

Definition 3. Let (P, e, Hm) ∈ ∆ and p ∈ P. The switch mapping is considered to be f =( f t)t∈Γm , where f t = ( f t

p)p∈P and f tp : ME(P, e, Hm)→ R is defined by

f tp(x) = xt

p + α ∑q∈P\{p}

12|P|−2 ∑

Q⊆P\{p,q}

[Ex(Q \ {q}, ht, xt)− Ex(Q \ {p}, ht, xt)

]Define [x]0 = x, [x]1 = f ([x]0), . . . , [x]n = f ([x]n−1) for every n ∈ N.

Lemma 5. f (x) ∈ ME(P, e, Hm) for every (P, e, Hm) ∈ ∆ and for every x ∈ ME(P, e, Hm).

Proof. Let (P, e, Hm) ∈ ∆, t ∈ Γm, p, q ∈ P and x ∈ ME(P, e, Hm). Similar to Equation (2),

∑q∈P\{p}

12|P|−2 ∑

Q⊆P\{p,q}

[Ex(Q \ {q}, ht, xt)− Ex(Q \ {p}, ht, xt)

]= ∑

q∈P\{p}

[(1

2|P|−2 ∑Q⊆P\{p,q}

[ht(Q \ {q})− ht(Q \ {p})

])− xt

p + xtq

].

(10)

By definition of Θ,

Θtp(P, e, Hm)−Θt

q(P, e, Hm) = 12|P|−2 ∑

Q⊆P\{p,q}

[ht(Q \ {q})− ht(Q \ {p})

]. (11)

By Equations (10) and (11),

∑q∈P\{p}

12|P|−2 ∑

Q⊆P\{p,q}

[Ex(Q \ {q}, ht, xt)− Ex(Q \ {p}, ht, xt)

]= ∑

q∈P\{p}

(Θt

p(P, e, Hm)−Θtq(P, e, Hm)− xt

p + xtq

)=

((|P| − 1)

(Θt

p(P, e, Hm)− xtp)− ∑

q∈P\{p}Θt

q(P, e, Hm) + ∑q∈P\{p}

xtq

)=

(|P|(Θt

p(P, e, Hm)− xtp)− ht∗(P) + ht

∗(P))

(by MEIY of Θ, x ∈ ME(P, e, Hm)

)= |P| ·

(Θt

p(P, e, Hm)− xtp

).

(12)

Moreover,

∑p∈P

∑q∈P\{p}

12|P|−2 ∑

Q⊆P\{p,q}

[Ex(Q \ {q}, ht, xt)− Ex(Q \ {p}, ht, xt)

]= ∑

p∈P|P| ·

(Θt

p(P, e, Hm)− xtp

)= |P| ·

(∑

p∈PΘt

p(P, e, Hm)− ∑p∈P

xtp

)= |P| ·

(ht∗(P)− ht

∗(P)) (

by MEIY of Θ, x ∈ ME(P, e, Hm))

= 0.

(13)

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Therefore, we have that

∑p∈P

f tp(x)

= ∑p∈P

[xt

p + α ∑q∈P\{p}

12|P|−2 ∑

Q⊆P\{p,q}

[Ex(Q \ {q}, ht, xt)− Ex(Q \ {p}, ht, xt)

]]= ∑

p∈Pxt

p + α ∑p∈P

∑q∈P\{p}

12|P|−2 ∑

Q⊆P\{p,q}

[Ex(Q \ {q}, ht, xt)− Ex(Q \ {p}, ht, xt)

]= ht

∗(P) + 0(by Equation (13) and x ∈ ME(P, e, Hm)

)= ht

∗(P).

Hence, f (x) ∈ ME(P, e, Hm) if x ∈ ME(P, e, Hm).

Theorem 3. Let (P, e, Hm) ∈ ∆. If 0 < α < 2|P| , then {[x]n}∞

n=1 converges to Θ(P, e, Hm) foreach x ∈ ME(P, e, Hm).

Proof. Let (P, e, Hm) ∈ ∆, t ∈ Γm, p ∈ P and x ∈ ME(P, e, Hm). By Equation (12) and thedefinition of f ,

f tp(x)− xt

p = α ∑q∈P\{p}

12|P|−2 ∑

Q⊆P\{p,q}

[Ex(Q \ {q}, ht, xt)− Ex(Q \ {p}, ht, xt)

]= α · |P| ·

(Θt

p(P, e, Hm)− xtp

).

Hence,

Θtp(P, e, Hm)− f t

p(x) = Θtp(P, e, Hm)− xt

p + xtp − f t

p(x)= Θt

p(P, e, Hm)− xtp − α · |P| · (Θt

p(P, e, Hm)− xtp)

=(

1− α · |P|)[

Θtp(P, e, Hm)− xt

p

].

Therefore, for every n ∈ N,

Θ(P, e, Hm)− [x]P =(

1− α · |P|)n[

Θ(P, e, Hm)− x].

If 0 < α < 2|P| , then−1 <

(1− α · |P|

)< 1 and {[x]n}∞

n=1 converges to Θ(P, e, Hm).

Similar to the work of Maschler and Owen [13], a dynamic outcome could be providedunder reductions as follows.

Definition 4. Let ρ be an output, (P, e, Hm) ∈ ∆, S ⊆ P and x ∈ ME(P, e, Hm). The (x, ρ)-reduced scheme (S, eS, Hr,m

ρ,S,x) is given by Hr,mρ,S,x = (hr,t

ρ,S,x)t∈Γm and for every T ⊆ S,

hr,tρ,S,x(ω) =

maxτ∈EP\S

{h(ω, τ)|S(τ) = P \ S} − ∑p∈P\S

xtp , S(ω) = S,

h2,tS,ρ(ω) , otherwise.

Similar to the work of Maschler and Owen [13], we also introduce related switchmapping as follows. The R-switch mapping is g = (gt)t∈Γm , where gt = (gt

p)p∈P andgt

p : ME(P, e, Hm)→ R is defined by

gtp(x) = xt

p + α ∑k∈P\{p}

(Θt

p({p, k}, hr,t

Θ,{p,k},x)− xt

p

).

Define [θ]0 = x, [θ]1 = g([θ]0), . . . , [θ]n = g([θ]n−1) for every n ∈ N.

Lemma 6. g(

x)∈ ME(P, e, Hm) for every (P, e, Hm) ∈ ∆ and for every x ∈ ME(P, e, Hm).

Sustainability 2021, 13, 1528 14 of 16

Proof. Let (P, e, Hm) ∈ ∆, t ∈ Γm, p, k ∈ P and x ∈ ME(P, e, Hm). Let S = {p, k}, by MEIYof Θ and Definition 4,

Θtp(S, eS, Hr,m

Θ,S,x) + Θt

k(S, eS, Hr,mΘ,S,x

) = xtp + xt

k.

By 2CSE and 2CS of Θ,

Θtp(S, eS, Hr,m

Θ,S,x)−Θt

k(S, eS, Hr,mΘ,S,x

) = (hr,tΘ,S,x

)∗({p})− (hr,tΘ,S,x

)∗({k})= (h2,t

S,Θ)∗({p})− (h2,t

S,Θ)∗({k})

= Θtp(S, eS, H2,m

S,Θ)−Θt

k(S, eS, H2,mS,Θ

)

= Θtp(P, e, Hm)−Θt

k(P, e, Hm).

Therefore,

2 ·[Θt

p(S, eS, Hr,mΘ,S,x

)− xtp

]= Θt

p(P, e, Hm)−Θtk(P, e, Hm)− xt

p + xtk. (14)

By definition of g and Equation (14),

gtp(x) = xt

p +α2 ·[

∑k∈P\{p}

Θtp(P, e, Hm)− ∑

k∈P\{p}xt

p

− ∑k∈P\{p}

Θtk(P, e, Hm) + ∑

k∈P\{p}xt

k

]= xt

p +α2 ·[

∑k∈P\{p}

Θtp(P, e, Hm)−

(|P| − 1

)xt

p

− ∑k∈P\{p}

Θtk(P, e, h) +

(ht∗(P)− xt

p)]

= xtp +

α2 ·[(|P| − 1

)Θt

p(P, e, Hm)−(|P| − 1

)xt

p

−(ht∗(P)−Θt

p(P, e, Hm))+(ht∗(P)− xt

p)]

= xtp +

|P|·α2 ·

[Θt

p(P, e, Hm)− xtp

].

(15)

Therefore, we have that

∑p∈P

gtp(x) = ∑

p∈P

[xt

p +|P|·α

2 ·[Θt

p(P, e, Hm)− xtp]]

= ∑p∈P

xtp +

|P|·α2 ·

[∑

p∈PΘt

p(P, e, Hm)− ∑p∈P

xtp]

= ht∗(P) + |P|·α

2 ·[ht∗(P)− ht

∗(P)]

= ht∗(P).

Thus, g(

x)∈ ME(P, e, Hm) for every x ∈ ME(P, e, Hm).

Theorem 4. Let (P, e, Hm) ∈ ∆. If 0 < α < 4|P| , then {[θ]n}∞

n=1 converges to Θ(P, e, Hm) foreach x ∈ ME(P, e, Hm).

Proof. Let (P, e, Hm) ∈ ∆, t ∈ Γm and x ∈ ME(P, e, Hm). By Equation (15), gtp(x) =

xtp +

|P|·α2 ·

[Θt

p(P, e, Hm)− xtp

]for every p ∈ P. Therefore,

(1− |P| · α

2)·[Θt

p(P, e, Hm)− xtp]=[Θt

p(P, e, Hm)− gtp(x)

]Therefore, for every n ∈ N,

Θ(P, e, Hm)− [θ]P =(

1− |P| · α2

)n[Θ(P, e, Hm)− x

].

Sustainability 2021, 13, 1528 15 of 16

If 0 < α < 4|P| , then −1 <

(1− |P|·α2

)< 1 and {[θ]n}∞

n=1 converges to Θ(P, e, h) for

every (P, e, Hm) ∈ ∆, for every t ∈ Γm and for every p ∈ P.

Subsequently, we present the dynamic outcomes for the MNBCO.

Definition 5.

• Let (P, e, Hm) ∈ ∆∗ and p ∈ P. The N-switch mapping is considered to be f N = ( f N,t)t∈Γm ,where f N,t = ( f N,t

p )p∈P and f N,tp : ME(P, e, Hm)→ R is considered by

f N,tp (x) = xt

p + α ∑q∈P\{p}

[at

2|P|−2 ∑S⊆P\{p,q}

[Ex(S ∪ {p}, ht, xt

at )− Ex(S ∪ {q}, ht, xt

at )]]

,

where at = ht∗(P)

∑k∈P

Φtk(P,e,Hm)

. Let [λ]0 = x, [λ]1 = f N,t([λ]0), . . . , [λ]n = f N,t([λ]n−1) for

every n ∈ N.• The N-R-switch mapping is gN = (gN,t)t∈Γm , where gN,t = (gN,t

p )p∈P and gN,tp :

ME(P, e, Hm)→ R is considered by

gN,tp (x) = xt

p + α ∑k∈P\{p}

(Φt

p({p, k}, hr,t

Φ,{p,k},x)− xt

p

).

Let [χ]0 = x, [χ]1 = gN,t([χ]0), . . . , [χ]n = gN,t([χ]n−1) for every n ∈ N.

Theorem 5.

1. f N(x) ∈ ME(P, e, Hm) and gN(x) ∈ ME(P, e, Hm) for every (P, e, Hm) ∈ ∆∗ and forevery x ∈ ME(P, e, Hm).

2. Let (P, e, Hm) ∈ ∆∗. If 0 < α < 2|P| , then {[λ]n}∞

n=1 converges to Φ(P, e, Hm) for eachx ∈ ME(P, e, Hm).

3. Let (P, e, Hm) ∈ ∆∗. If 0 < α < 4|P| , then {[χ]n}∞

n=1 converges to Φ(P, e, Hm) for eachx ∈ ME(P, e, Hm).

Proof. The proofs of this theorem are similar to Lemmas 5, 6 and Theorems 3 and 4.

5. Conclusions

In this article, we investigate the multi-consideration Banzhaf–Coleman output, themulti-consideration efficacious Banzhaf–Coleman output and the multi-considerationnormalized Banzhaf–Coleman output. Based on the reduced scheme, several axiomaticoutcomes for these outputs are proposed to analyze the reasonability. By adopting reduc-tions and excess mappings, we also present alternative formulations and related dynamicoutcomes for these outputs. One should compare related existing outcomes with theoutcomes proposed in this article.

• The multi-consideration Banzhaf–Coleman output, the multi-consideration efficaciousBanzhaf–Coleman output and the multi-consideration normalized Banzhaf–Colemanoutput are presented initially in the contexts of traditional schemes and multi-attributemulti-choice schemes.

• The notion of our switch mappings in Definitions 3–5 and related dynamic outcomesare based on that of Maschler and Owen’s [13] dynamic outcomes for the Shapleyvalue [14]. The major difference is that our switch mappings in Definitions 3 and 5are based on “excess mappings”, and the Maschler and Owen [13] switch mapping isbased on “reduced schemes”.

The outcomes proposed in this article raise one motivation:

• Whether there exist other generalizations and related outcomes for some more outputsin the context of multi-attribute multi-choice schemes.

Sustainability 2021, 13, 1528 16 of 16

To our knowledge, these issues are still open questions.

Author Contributions: Conceptualization, Y.-H.L.; methodology, Y.-H.L.; software, E.-C.C. andY.-H.L.; validation, Y.-H.L.; formal analysis, Y.-H.L.; investigation, Y.-H.L.; resources, E.-C.C. andY.-H.L.; data curation, E.-C.C. and Y.-H.L.; writing—original draft preparation, E.-C.C. and Y.-H.L.;writing—review and editing, E.-C.C. and Y.-H.L.; visualization, E.-C.C. and Y.-H.L.; supervision,E.-C.C. and Y.-H.L.; project administration, E.-C.C. and Y.-H.L.; funding acquisition, none. All authorshave read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Conflicts of Interest: The authors declare no conflict of interest.

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