# Consonance, Symmetry and Extended Outputs

## Abstract

**:**

## 1. Introduction

- Could the power indexes be proposed under the multi-choice consideration?

- In Section 2 and Section 5, we extend the Banzhaf–Coleman index, the Banzhaf–Owen index, and the marginal index to multi-choice games, which we name the level-marginal output, the level-accumulated output, the level-average output, the *-level-marginal output, the *-level-accumulated output, and the *-level-average output, respectively. Since these outputs are not efficacious, we also consider the efficacious extensions of these outputs.
- Briefly, the max-reduced game is based on the “maximizing notion”. For the “summing notion” instead of the “maximizing notion”, we consider the sum-reduced game. For the “averaging notion” instead of the “maximizing notion”, we consider the average-reduced game in Section 3.

## 2. Preliminaries

#### 2.1. Definitions and Notation

**Definition**

**1.**

- The level-marginal output,${\mathsf{\Theta}}^{M}$, is the output that associates with$(P,e,h)\in \Lambda $ and each $(p,q)\in {L}^{P,e}$ the value:$${\mathsf{\Theta}}_{p,q}^{M}(P,e,h)=h({e}_{-p},q)-h({e}_{-p},q-1),$$
- The level-accumulated output,${\mathsf{\Theta}}^{AC}$, is the output that associates with$(P,e,h)\in \Lambda $and each$(p,q)\in {L}^{P,e}$the value:$${\mathsf{\Theta}}_{p,q}^{AC}(P,e,h)=\sum _{\stackrel{S\subseteq P}{p\in S}}\left[h\left({({e}_{-p},q)}_{S},{0}_{P\setminus S}\right)-h\left({({e}_{-p},q-1)}_{S},{0}_{P\setminus S}\right)\right],$$
- The level-average output,${\mathsf{\Theta}}^{AV}$, is the output that associates with $(P,e,h)\in \Lambda $and each$(p,q)\in {L}^{P,e}$the value:$${\mathsf{\Theta}}_{p,q}^{AV}(P,e,h)=\frac{1}{{2}^{\left|P\right|-1}}\xb7\sum _{\stackrel{S\subseteq P}{p\in S}}\left[h\left({({e}_{-p},q)}_{S},{0}_{P\setminus S}\right)-h\left({({e}_{-p},q-1)}_{S},{0}_{P\setminus S}\right)\right],$$

- an output $\rho $ conforms to efficacy (EIY) if $\sum _{p\in P}}{\rho}_{p,{e}_{p}}(P,e,h)=h\left(e\right)$ for every $(P,e,h)\in \Lambda $.

**Definition**

**2.**

- The efficacious marginal output$\overline{{\mathsf{\Theta}}^{M}}$is the output that associates with$(P,e,h)\in \Lambda $and each$(p,q)\in {L}^{P,e}$the value:$$\overline{{\mathsf{\Theta}}_{p,q}^{M}}(P,e,h)={\mathsf{\Theta}}_{p,q}^{M}(P,e,h)+\frac{1}{\left|P\right|}\xb7\left[h\left(e\right)-\sum _{k\in P}{\mathsf{\Theta}}_{k,{e}_{k}}^{M}(P,e,h)\right].$$
- The efficacious accumulated output,$\overline{{\mathsf{\Theta}}^{AC}}$, is the output that associates with$(P,e,h)\in \Lambda $and each$(p,q)\in {L}^{P,e}$the value:$$\overline{{\mathsf{\Theta}}_{p,q}^{AC}}(P,e,h)={\mathsf{\Theta}}_{p,q}^{AC}(P,e,h)+\frac{1}{\left|P\right|}\xb7\left[h\left(e\right)-\sum _{k\in P}{\mathsf{\Theta}}_{k,{e}_{k}}^{AC}(P,e,h)\right].$$
- The efficacious average output,$\overline{{\mathsf{\Theta}}^{AV}}$, is the output that associates with$(P,e,h)\in \Lambda $and each$(p,q)\in {L}^{P,e}$the value:$$\overline{{\mathsf{\Theta}}_{p,q}^{AV}}(P,e,h)={\mathsf{\Theta}}_{p,q}^{AV}(P,e,h)+\frac{1}{\left|P\right|}\xb7\left[h\left(e\right)-\sum _{k\in P}{\mathsf{\Theta}}_{k,{e}_{k}}^{AV}(P,e,h)\right].$$

#### 2.2. Motivating and Practical Examples

## 3. Reductions, Consonance, and Characterizations

**Definition**

**3.**

- The complement-reduced game$(S,{e}_{S},{h}_{S,\rho}^{com})$related to ρ and S is defined for every$\omega \in {E}^{S}$by,$${h}_{S,\rho}^{com}\left(\omega \right)=\left\{\begin{array}{cc}0\hfill & if\omega ={0}_{S},\hfill \\ h(\omega ,{e}_{P\setminus S})-{\displaystyle \sum _{p\in P\setminus S}}{\rho}_{p,{e}_{p}}(P,e,h)\hfill & otherwise.\hfill \end{array}\right.$$
- The sum-reduced game$(S,{e}_{S},{h}_{S,\rho}^{sum})$related to ρ and S is defined for every$\omega \in {E}^{S}$by,$${h}_{S,\rho}^{sum}\left(\omega \right)=\left\{\begin{array}{cc}0\hfill & if\omega ={0}_{S},\hfill \\ h\left(e\right)-{\displaystyle \sum _{p\in P\setminus S}}{\rho}_{p,{e}_{p}}(P,e,h)\hfill & if\omega ={e}_{S},\hfill \\ {\displaystyle \sum _{\stackrel{T\subseteq P\setminus S}{T\ne \varnothing}}}\left[h(\omega ,{e}_{T},{0}_{P\setminus (S\cup T)})-{\displaystyle \sum _{p\in T}}{\rho}_{p,{e}_{p}}(P,e,h)\right]\hfill & otherwise.\hfill \end{array}\right.$$
- The average-reduced game$(S,{e}_{S},{h}_{S,\rho}^{ave})$related to ρ and S is defined for every$\omega \in {E}^{S}$by,$${h}_{S,\rho}^{ave}\left(\omega \right)=\left\{\begin{array}{cc}0\hfill & if\omega ={0}_{S},\hfill \\ h\left(e\right)-{\displaystyle \sum _{p\in P\setminus S}}{\rho}_{p,{e}_{p}}(P,e,h)\hfill & if\omega ={e}_{S},\hfill \\ \frac{1}{{2}^{|P\setminus S|}}\xb7{\displaystyle \sum _{\stackrel{T\subseteq P\setminus S}{T\ne \varnothing}}}\left[h(\omega ,{e}_{T},{0}_{P\setminus (S\cup T)})-{\displaystyle \sum _{p\in T}}{\rho}_{p,{e}_{p}}(P,e,h)\right]\hfill & otherwise.\hfill \end{array}\right.$$

- $\rho $ conforms to complement-consonance (ComCSA) if for every $(P,e,h)\in \Lambda $, for every $S\subseteq P$, and for every $(p,q)\in {L}^{S,{e}_{S}}$, ${\rho}_{p,q}(P,e,h)={\rho}_{p,q}(S,{e}_{S},{h}_{S,\rho}^{com})$.
- $\rho $ conforms to sum-consonance (SumCSA) if for every $(P,e,h)\in \Lambda $, for every $S\subseteq P$, and for every $(p,q)\in {L}^{S,{e}_{S}}$, ${\rho}_{p,q}(P,e,h)={\rho}_{p,q}(S,{e}_{S},{h}_{S,\rho}^{sum})$.
- $\rho $ conforms to average-consonance (AveCSA) if for every $(P,e,h)\in \Lambda $, for every $S\subseteq P$, and for every $(p,q)\in {L}^{S,{e}_{S}}$, ${\rho}_{p,q}(P,e,h)={\rho}_{p,q}(S,{e}_{S},{h}_{S,\rho}^{ave})$.
- $\rho $ conforms to bilateral complement-consonance (BilComCSA) if for every $(P,e,h)\in \Lambda $, for every $S\subseteq P$ with $\left|S\right|=2$, and for every $(p,q)\in {L}^{S,{e}_{S}}$, ${\rho}_{p,q}(P,e,h)={\rho}_{p,q}(S,{e}_{S},{h}_{S,\rho}^{com})$.
- $\rho $ conforms to bilateral sum-consonance (BilSumCSA) if for every $(P,e,h)\in \Lambda $, for every $S\subseteq P$ with $\left|S\right|=2$, and for every $(p,q)\in {L}^{S,{e}_{S}}$, ${\rho}_{p,q}(P,e,h)={\rho}_{p,q}(S,{e}_{S},{h}_{S,\rho}^{sum})$.
- $\rho $ conforms to bilateral average-consonance (BilAveCSA) if for every $(P,e,h)\in \Lambda $, for every $S\subseteq P$ with $\left|S\right|=2$, and for every $(p,q)\in {L}^{S,{e}_{S}}$, ${\rho}_{p,q}(P,e,h)={\rho}_{p,q}(S,{e}_{S},{h}_{S,\rho}^{ave})$.

**Lemma**

**1.**

- 1.
- The output$\overline{{\mathsf{\Theta}}^{M}}$conforms to BilComCSA.
- 2.
- The output$\overline{{\mathsf{\Theta}}^{AC}}$conforms to BilSumCSA.
- 3.
- The output$\overline{{\mathsf{\Theta}}^{AV}}$conforms to BilAveCSA.

**Proof**

**of Lemma 1.**

- Marginal-criterion for games (MCG): for every $(P,e,h)\in \Lambda $ with $\left|P\right|\le 2$, $\rho (P,e,h)=\overline{{\mathsf{\Theta}}^{M}}(P,e,h)$.
- Accumulated-criterion for games (ACCG): for every $(P,e,h)\in \Lambda $ with $\left|P\right|\le 2$, $\rho (P,e,h)=\overline{{\mathsf{\Theta}}^{AC}}(P,e,h)$.
- Average-criterion for games (AVCG): for every $(P,e,h)\in \Lambda $ with $\left|P\right|\le 2$, $\rho (P,e,h)=\overline{{\mathsf{\Theta}}^{AV}}(P,e,h)$.

**Remark**

**1.**

**Lemma**

**2.**

- 1.
- If ρ conforms to MCG and BilComCSA, then it also conforms to EIY.
- 2.
- If ρ conforms to ACCG and BilSumCSA, then it also conforms to EIY.
- 3.
- If ρ conforms to ACCG and BilAveCSA, then it also conforms to EIY.

**Proof**

**of Lemma 2.**

**Theorem**

**1.**

- 1.
- ρ conforms to MCG and BilComCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{M}}$.
- 2.
- ρ conforms to ACCG and BilSumCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{AC}}$.
- 3.
- ρ conforms to ACCG and BilAveCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{AV}}$.

**Proof**

**of Theorem 1.**

**Example**

**1.**

**Example**

**2.**

**Example**

**3.**

**Example**

**4.**

## 4. Symmetry, Accordance, and Characterizations

- $\rho $ conforms to marginal symmetry (MASYM) if for every $(P,e,h)\in \Lambda $ with ${\mathsf{\Theta}}_{p,{k}_{p}}^{M}(P,e,h)={\mathsf{\Theta}}_{q,{k}_{q}}^{M}(P,e,h)$ for some $(p,{k}_{p}),(q,{k}_{q})\in {L}^{P,e}$ and for every $\omega \in {E}^{P\setminus \{p,q\}}$, ${\rho}_{p,{k}_{p}}(P,e,h)={\rho}_{q,{k}_{q}}(P,e,h)$.
- $\rho $ conforms to accumulated symmetry (ACSYM) if for every $(P,e,h)\in \Lambda $ with ${\mathsf{\Theta}}_{p,{k}_{p}}^{AC}(P,e,h)={\mathsf{\Theta}}_{q,{k}_{q}}^{AC}(P,e,h)$ for some $(p,{k}_{p}),(q,{k}_{q})\in {L}^{P,e}$ and for every $\omega \in {E}^{P\setminus \{p,q\}}$, ${\rho}_{p,{k}_{p}}(P,e,h)={\rho}_{q,{k}_{q}}(P,e,h)$.
- $\rho $ conforms to average symmetry (AVSYM) if for every $(P,e,h)\in \Lambda $ with ${\mathsf{\Theta}}_{p,{k}_{p}}^{AV}(P,e,h)={\mathsf{\Theta}}_{q,{k}_{q}}^{AV}(P,e,h)$ for some $(p,{k}_{p}),(q,{k}_{q})\in {L}^{P,e}$ and for every $\omega \in {E}^{P\setminus \{p,q\}}$, ${\rho}_{p,{k}_{p}}(P,e,h)={\rho}_{q,{k}_{q}}(P,e,h)$.
- $\rho $ conforms to accordance (ADE) if for every $(P,e,h),(P,e,\zeta )\in \Lambda $ with $h\left(\omega \right)=\zeta \left(\omega \right)+{\sum}_{p\in S\left(\omega \right)}{b}_{p,{\omega}_{p}}$ for some $b\in {\mathbb{R}}^{{L}^{P,e}}$ and for every $\omega \in {E}^{P}$, $\rho (P,e,h)=\rho (P,e,\zeta )+b$.

**Lemma**

**3.**

- 1.
- If an output ρ conforms to EIY, MASYM, and ADE, then ρ conforms to MCG.
- 2.
- If an output ρ conforms to EIY, ACSYM, and ADE, then ρ conforms to ACCG.
- 3.
- If an output ρ conforms to EIY, AVSYM, and ADE, then ρ conforms to AVCG.

**Proof**

**of Lemma 3.**

**Theorem**

**2.**

- 1.
- An output ρ conforms to EIY, MASYM, ADE, and BilComCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{M}}$.
- 2.
- An output ρ conforms to EIY, ACSYM, ADE, and BilSumCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{AC}}$.
- 3.
- An output ρ conforms to EIY, AVSYM, ADE, and BilAveCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{AV}}$.

**Proof**

**of Theorem 2.**

## 5. Different Generalizations and Related Results

**Definition**

**4.**

- The *-efficacious marginal output,$\overline{{\mathsf{\Theta}}^{\ast M}}$, is the output on Λ that associates with$P,e,h\in \Lambda $and all$(p,q)\in {L}^{P,e}$the value:$$\overline{{\mathsf{\Theta}}_{p,q}^{\ast M}}(P,e,h)={\mathsf{\Theta}}_{p,q}^{\ast M}(P,e,h)+\frac{1}{\left|P\right|}\xb7\left[h\left(e\right)-{\displaystyle \sum _{k\in P}}{\mathsf{\Theta}}_{k,{e}_{k}}^{M}(P,e,h)\right],$$
- The *-efficacious accumulated output, $\overline{{\mathsf{\Theta}}^{\ast AC}}$, is the output on Λ that associates with $P,e,h\in \Lambda $ and all $(p,q)\in {L}^{P,e}$ the value:$$\overline{{\mathsf{\Theta}}_{p,q}^{\ast AC}}(P,e,h)={\mathsf{\Theta}}_{p,q}^{\ast AC}(P,e,h)+\frac{1}{\left|P\right|}\xb7\left[h\left(e\right)-{\displaystyle \sum _{k\in P}}{\mathsf{\Theta}}_{k,{e}_{k}}^{AC}(P,e,h)\right],$$
- The *-efficacious average output,$\overline{{\mathsf{\Theta}}^{\ast AV}}$, is the output on Λ that associates with $P,e,h\in \Lambda $ and all $(p,q)\in {L}^{P,e}$ the value:$$\overline{{\mathsf{\Theta}}_{p,q}^{\ast AV}}(P,e,h)={\mathsf{\Theta}}_{p,q}^{\ast AV}(P,e,h)+\frac{1}{\left|P\right|}\xb7\left[h\left(e\right)-{\displaystyle \sum _{k\in P}}{\mathsf{\Theta}}_{k,{e}_{k}}^{AV}(P,e,h)\right],$$

- *-marginal-criterion for games (*MCG): for every $(P,e,h)\in \Lambda $ with $\left|P\right|\le 2$, $\rho (P,e,h)=\overline{{\mathsf{\Theta}}^{\ast M}}(P,e,h)$.
- *-accumulated-criterion for games (*ACCG): for every $(P,e,h)\in \Lambda $ with $\left|P\right|\le 2$, $\rho (P,e,h)=\overline{{\mathsf{\Theta}}^{\ast AC}}(P,e,h)$.
- *-average-criterion for games (*AVCG): for every $(P,e,h)\in \Lambda $ with $\left|P\right|\le 2$, $\rho (P,e,h)=\overline{{\mathsf{\Theta}}^{\ast AV}}(P,e,h)$.

**Remark**

**2.**

**Lemma**

**4.**

- 1.
- The output$\overline{{\mathsf{\Theta}}^{\ast M}}$conforms to BilComCSA.
- 2.
- The output$\overline{{\mathsf{\Theta}}^{\ast AC}}$conforms to BilSumCSA.
- 3.
- The output$\overline{{\mathsf{\Theta}}^{\ast AV}}$conforms to BilAveCSA.

**Proof**

**of Lemma 4.**

**Lemma**

**5.**

- 1.
- If ρ conforms to *MCG and BilComCSA, then it also conforms to EIY.
- 2.
- If ρ conforms to *ACCG and BilSumCSA, then it also conforms to EIY.
- 3.
- If ρ conforms to *AVCG and BilAveCSA, then it also conforms to EIY.

**Proof**

**of Lemma 5.**

**Theorem**

**3.**

- 1.
- ρ conforms to *MCG and BilComCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{\ast M}}$.
- 2.
- ρ conforms to *ACCG and BilSumCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{\ast AC}}$.
- 3.
- ρ conforms to *AVCG and BilAveCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{\ast AV}}$.

**Proof**

**of Theorem 3.**

**Example**

**5.**

**Example**

**6.**

**Example**

**7.**

**Example**

**8.**

- $\rho $ conforms to *-marginal symmetry (*MASYM) if for every $(P,e,h)\in \Lambda $ with ${\mathsf{\Theta}}_{p,{k}_{p}}^{\ast M}(P,e,h)={\mathsf{\Theta}}_{q,{k}_{q}}^{\ast M}(P,e,h)$ for some $(p,{k}_{p}),(q,{k}_{q})\in {L}^{P,e}$ and for every $\omega \in {E}^{P\setminus \{p,q\}}$, ${\rho}_{p,{k}_{p}}(P,e,h)={\rho}_{q,{k}_{q}}(P,e,h)$.
- $\rho $ conforms to *-accumulated symmetry (*ACSYM) if for every $(P,e,h)\in \Lambda $ with ${\mathsf{\Theta}}_{p,{k}_{p}}^{\ast AC}(P,e,h)={\mathsf{\Theta}}_{q,{k}_{q}}^{\ast AC}(P,e,h)$ for some $(p,{k}_{p}),(q,{k}_{q})\in {L}^{P,e}$ and for every $\omega \in {E}^{P\setminus \{p,q\}}$, ${\rho}_{p,{k}_{p}}(P,e,h)={\rho}_{q,{k}_{q}}(P,e,h)$.
- $\rho $ conforms to *-average symmetry (*AVSYM) if for every $(P,e,h)\in \Lambda $ with ${\mathsf{\Theta}}_{p,{k}_{p}}^{\ast AV}(P,e,h)={\mathsf{\Theta}}_{q,{k}_{q}}^{\ast AV}(P,e,h)$ for some $(p,{k}_{p}),(q,{k}_{q})\in {L}^{P,e}$ and for every $\omega \in {E}^{P\setminus \{p,q\}}$, ${\rho}_{p,{k}_{p}}(P,e,h)={\rho}_{q,{k}_{q}}(P,e,h)$.

**Lemma**

**6.**

- 1.
- If an output ρ conforms to EIY, *MASYM, and ADE, then ρ conforms to *MCG.
- 2.
- If an output ρ conforms to EIY, *ACSYM, and ADE, then ρ conforms to *ACCG.
- 3.
- If an output ρ conforms to EIY, *AVSYM, and ADE, then ρ conforms to *ACCG.

**Proof**

**of Lemma 6.**

**Theorem**

**4.**

- 1.
- An output ρ conforms to EIY, *MASYM, ADE, and BilComCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{\ast M}}$.
- 2.
- An output ρ conforms to EIY, *ACSYM, ADE, and BilSumCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{\ast AC}}$.
- 3.
- An output ρ conforms to EIY, *AVSYM, ADE, and BilAveCSA if and only if$\rho =\overline{{\mathsf{\Theta}}^{\ast AV}}$.

**Proof**

**of Theorem 4.**

## 6. Conclusions

- Based on the multi-choice consideration, we propose several extensions of the Banzhaf–Coleman index, the Banzhaf–Owen index, and the marginal index on multi-choice games. Since these extensions are not efficacious, we also consider the efficacious extensions of these outputs.
- In order to present the rationality of these efficacious outputs, we adopt the complement-reduction, the sum-reduction, the average-reduction and related consonance property to characterize these efficacious outputs.
- By applying the related properties of symmetry and accordance, alternative axiomatic results of these efficacious outputs are also proposed.

- Liao [16] defined the maximal EANSC to compute a kind of global value for a specific performer by adopting the maximal marginal values of performers among total levels. Differing from the extended EANSC of Liao [16], we also focus on the Banzhaf–Coleman index and the Banzhaf–Owen index and consider the outputs, the reduction, and several properties by simultaneously analyzing the performers and the energy levels. The other major dissimilarity is the fact that we offer the related properties of symmetry and accordance to characterize the outputs introduced in this paper. The related properties of symmetry and accordance were not present in Liao [16].
- Liao [17] proposed the duplicate EANSC to compute a kind of global value for a specific performer by considering the replicated behavior of performers among total levels. Differing from the extended EANSC of Liao [17], we also focus on the Banzhaf–Coleman index and the Banzhaf–Owen index and consider the outputs, the reduction, and several properties by simultaneously analyzing the performers and the energy levels. The other major dissimilarity is the fact that we offer the related properties of symmetry and accordance to characterize the outputs introduced in this paper. The related properties of symmetry and accordance were not present in Liao [17].

## Author Contributions

## Funding

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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Liao, Y.-H.
Consonance, Symmetry and Extended Outputs. *Symmetry* **2021**, *13*, 72.
https://doi.org/10.3390/sym13010072

**AMA Style**

Liao Y-H.
Consonance, Symmetry and Extended Outputs. *Symmetry*. 2021; 13(1):72.
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**Chicago/Turabian Style**

Liao, Yu-Hsien.
2021. "Consonance, Symmetry and Extended Outputs" *Symmetry* 13, no. 1: 72.
https://doi.org/10.3390/sym13010072