# Measuring Voting Power in Convex Policy Spaces

## Abstract

**:**

## 1. Introduction

“Scientists who study power in political and economic institutions seem divided into two disjoint methodological camps. The first one uses non-cooperative game theory to analyze the impact of explicit decision procedures and given preferences over a well-defined—usually Euclidean—policy space. The second one stands in the tradition of cooperative game theory with much more abstractly defined voting bodies: the considered agents have no particular preferences and form winning coalitions which implement unspecified policies. Individual chances of being part of and influencing a winning coalition are then measured by a power index....

“It seems that we are confronted here with a clear-cut case of theory-laden (or theory-biased) observation. Scientists, equipped with a ready-made theoretical conception, “observe” in reality phenomena that fit that conception. And where the phenomena do not quite fit the theory, they are at best consciously ignored, but more often actually misperceived and tweaked into the theoretical mould.”

## 2. Binary Decision Rules

- (1)
- if no voter is in favor of a proposal, reject it;
- (2)
- if all voters are in favor of a proposal, accept it.

**Definition 1.**A

**Boolean game**is a function $v:{2}^{N}\to \{0,1\}$ with $v\left(\varnothing \right)=0$ and $v\left(N\right)=1$. The set of all Boolean games on n players is denoted by ${\mathcal{B}}_{n}$.

**Definition 2.**A

**simple game**is a Boolean game $v:{2}^{N}\to \{0,1\}$ such that $v\left(S\right)\le v\left(T\right)$ for all $\varnothing \subseteq S\subseteq T\subseteq N$. The set of all simple games on n players is denoted by ${\mathcal{S}}_{n}$.

**Definition 3.**Let v be a simple game. By $\mathcal{W}$ we denote the set of all winning and by ${\mathcal{W}}^{m}$ we denote the set of all minimal winning coalitions of v. Similarly, by $\mathcal{L}$ we denote the set of all losing and by ${\mathcal{L}}^{M}$ we denote the set of all maximal losing coalitions of v.

**Definition 4.**A voter that is not contained in any minimal winning coalition is called a

**null**voter.

**Definition 5.**A simple game is called

**proper**if the complement $N\backslash S$ of any winning coalition s is losing. It is called

**strong**if the complement $N\backslash S$ of any losing coalition S is winning. A simple game that is both proper and strong is called

**constant-sum**(or self-dual, or decisive).

**Definition 6.**Given a simple game, characterized by its set of winning coalitions $\mathcal{W}\subseteq {2}^{N}$, we say that voter $i\in N$ is

**more desirable**as voter $j\in N$, denoted by $i\u2ab0j$, if

- (1)
- for all $S\subseteq N\backslash \{i,j\}$ with $S\cup \left\{j\right\}\in \mathcal{W}$, we have $S\cup \left\{i\right\}\in \mathcal{W}$;
- (2)
- for all $S\subseteq N\backslash \{i,j\}$ with $S\cup \left\{i\right\}\in \mathcal{L}$, we have $S\cup \left\{j\right\}\in \mathcal{L}={2}^{N}\backslash \mathcal{W}$.

**Definition 7.**A simple game $(\mathcal{W},N)$ is called

**complete**if for each pair of voters $i,j\in N$ we have $i\u2ab0j$ or $j\u2ab0i$. The set of all complete (simple) games on n voters is denoted by ${\mathcal{C}}_{n}$.

**Definition 8.**Let $(\mathcal{W},N)$ be a complete simple game, where $1\u2ab02\u2ab0\cdots \u2ab0n$, and $S\subseteq N$ be arbitrary. A coalition $T\subseteq N$ is a

**direct left-shift**of S whenever there exists a voter $i\in S$ with $i-1\notin S$ such that $T=S\backslash \left\{i\right\}\cup \{i-1\}$ for $i>1$ or $T=S\cup \left\{n\right\}$ for $n\notin S$. Similarly, a coalition $T\subseteq N$ is a

**direct right-shift**of S whenever there exists a voter $i\in S$ with $i+1\notin S$ such that $T=S\backslash \left\{i\right\}\cup \{i+1\}$ for $i<n$ or $T=S\backslash \left\{n\right\}$ for $n\in S$. A coalition T is a

**left-shift**of S if it arises as a sequence of direct left-shifts. Similarly, it is a

**right-shift**of S if it arises as a sequence of direct right-shifts. A winning coalition S such that all right-shifts of S are losing is called

**shift-minimal**(winning). Similarly, a winning coalition S such that all left-shifts of S are winning is called

**shift-maximal**(losing). By ${\mathcal{W}}^{sm}$ we denote the set of all shift-minimal minimal winning coalitions of $(\mathcal{W},N)$ and by ${\mathcal{L}}^{sM}$ the set of all shift-maximal losing coalitions.

**Definition 9.**A simple game $(\mathcal{W},N)$ is

**weighted**if there exists a quota $q>0$ and weights ${w}_{1},\cdots ,{w}_{n}\ge \phantom{\rule{3.33333pt}{0ex}}0$ such that coalition S is winning if and only if $w\left(S\right)={\sum}_{i\in S}{w}_{i}\ge q$. We denote the corresponding game by $[q;{w}_{1},\cdots ,{w}_{n}]$. The set of all weighted (simple) games on n voters is denoted by ${\mathcal{T}}_{n}$.

**Lemma 1.**A weighted game v with normalized weights $w\in {\mathbb{R}}_{\ge 0}^{n}$, i.e., ${\parallel w\parallel}_{1}=1$, and quota $q\in (0,1]$ is proper if and only if there exists a weighted representation with normalized weights ${w}^{\prime}\in {\mathbb{R}}_{\ge 0}^{n}$ and quota ${q}^{\prime}\in \left(\frac{1}{2},1\right]$.

**Lemma 2.**A weighted game v with normalized weights $w\in {\mathbb{R}}_{\ge 0}^{n}$ and quota $q\in (0,1]$ is strong if and only if there exists a weighted representation with normalized weights ${w}^{\prime}\in {\mathbb{R}}_{\ge 0}^{n}$ and quota ${q}^{\prime}\in \left(0,\frac{1}{2}\right]$.

**Lemma 3.**A weighted game v is constant-sum if and only if there exists an $\epsilon >0$ such that for all $q\in \left(\frac{1}{2}-\epsilon ,\frac{1}{2}+\epsilon \right)$ there exists a normalized weighted representation with quota q.

**Definition 10.**Let ${\mathcal{V}}_{n}\subseteq {\mathcal{B}}_{n}$ a class of Boolean games consisting of n voters. A

**power index**(on ${\mathcal{V}}_{n}$) is a mapping $P:{\mathcal{V}}_{n}\to {\mathbb{R}}^{n}$.

**Definition 11.**Let $g:{\mathcal{V}}_{n}\to {\mathbb{R}}^{n}={\left({g}_{i}\right)}_{i\in N}$ be a power index on a class ${\mathcal{V}}_{n}$ of Boolean games. We say that

- (1)
- g is
**symmetric**: if for all $v\in {\mathcal{V}}_{n}$ and any bijection $\tau :N\to N$ we have ${g}_{\tau \left(i\right)}\left(\tau v\right)={g}_{i}\left(v\right)$, where $\tau v\left(S\right)=v\left(\tau \right(S\left)\right)$ for any coalition $S\subseteq N$; - (2)
- g is
**positive**: if for all $v\in {\mathcal{V}}_{n}$ and all $i\in N$ we have ${g}_{i}\left(v\right)\ge 0$ and $g\left(v\right)\ne 0$; - (3)
- g is
**efficient**: if for all $v\in {\mathcal{V}}_{n}$ we have ${\sum}_{i=1}^{n}{g}_{i}\left(v\right)=1$; - (4)
- g satisfies the
**null voter property**: if for all $v\in {\mathcal{V}}_{n}$ and all null voters i of v we have ${g}_{i}\left(v\right)=0$.

**Definition 12.**A

**coalitional game**is a function $v:{2}^{N}\to \mathbb{R}$ with $v\left(\varnothing \right)=0$.

**Definition 13.**Let ${\mathcal{V}}_{n}$ be a subclass of coalitional games consisting of n voters. A

**value**(on ${\mathcal{V}}_{n}$) is a mapping $P:{\mathcal{V}}_{n}\to {\mathbb{R}}^{n}$.

## 3. $(\mathit{j},\mathit{k})$ Simple Games

**Definition 14.**A sequence $S=({S}_{1},\cdots ,{S}_{j})$ of mutually disjoint sets ${S}_{i}\subseteq N$ with ${\cup}_{1\le i\le j}{S}_{i}=N$ is called

**ordered**$\mathbf{j}$

**-partition**.

**Definition 15.**For two ordered j-partitions $S=({S}_{1},\cdots ,{S}_{j})$, $T=({T}_{1},\cdots ,{T}_{j})$ we write $S\stackrel{j}{\subseteq}T$ if

**Definition 16.**A

**Boolean**$(\mathbf{j},\mathbf{k})$

**game**is given by a function $v:{J}^{N}\to K$ with $v\left((\varnothing ,\cdots ,\varnothing ,N)\right)=k$ and $v\left(\right(N,\varnothing ,\cdots ,\varnothing \left)\right)=1$.

**Definition 17.**A $(\mathbf{j},\mathbf{k})$

**simple game**is a Boolean game $v:{J}^{N}\to K$ such that $v\left(S\right)\le v\left(T\right)$ for all ordered j-partitions $S\stackrel{j}{\subseteq}T$.

**Definition 18.**A

**queue**of $N=\{1,\cdots ,n\}$ is a bijection from N to N. The set of all queues is denoted by ${\mathcal{Q}}_{n}$, i.e., $\left|{\mathcal{Q}}_{n}\right|=n!$.

**Definition 19.**Let $v:{J}^{N}\to K$ be a $(j,k)$ simple game, $q\in {\mathcal{Q}}_{n}$ a queue, and $S=({S}_{1},\cdots ,{S}_{j})$ an ordered j-partition. For each $1\le i\le k-1$ the $\mathbf{i}$

**-pivot**is uniquely defined either as

- (1)
- the voter, whose vote in S clinches the aggregated group decision under, at least the output level i, independently of the subsequent voters of i in q, or
- (2)
- the voter, whose vote in S clinches the aggregated group decision under, at most the output level $i+1$, independently of the subsequent voters of i in q.

**Definition 20.**The Shapley-Shubik index of a $(j,k)$ simple game is given by

**Definition 21.**Given a $(j,k)$ simple game v and an ordered j-partition $S\in {J}^{N}$, we denote by ${S}_{i\downarrow}$ the unique ordered j partition which satisfies

- (1)
- ${S}_{h}\backslash \left\{i\right\}={T}_{h}\backslash \left\{i\right\}$ for all $1\le h\le j$ and
- (2)
- $i\in {T}_{max(h+1,j)}$ for the index h with $i\in {S}_{h}$,

**-swing for voter**$\mathbf{i}$ if $1\le l<m\le k$, $v\left(S\right)=l$, and $v\left(T\right)=m$. The number of all $(m,l)$-swings for voter i in v is denoted by ${\eta}_{i}\left(v\right)$.

**Definition 22.**The absolute Banzhaf index of a $(j,k)$ simple game is given by

## 4. Continuous Decision Rules

**Definition 23.**A

**continuous Boolean game**is a function $v:{[0,1]}^{n}\to [0,1]$ with $v\left((0,\cdots ,0)\right)=0$ and $v\left((1,\cdots ,1)\right)=1$. The set of all continuous Boolean games on n players is denoted by ${\mathbb{B}}_{n}$.

**Definition 24.**For two vectors $x=({x}_{1},\cdots ,{x}_{n})\in {\mathbb{R}}^{n}$ and $y=({y}_{1},\cdots ,{y}_{n})\in {\mathbb{R}}^{n}$ we write $x\le y$ if ${x}_{i}\le {y}_{i}$ for all $1\le i\le n$.

**Definition 25.**A

**continuous simple game**is a continuous Boolean game $v:{[0,1]}^{n}\to [0,1]$ such that $v\left(S\right)\le v\left(T\right)$ for all real-valued vectors $\mathbf{0}\le S\le T\le \mathbf{1}$, where $\mathbf{0}$ denotes the all-0- and $\mathbf{1}$ the all-1-vector. The set of all continuous simple games on n players is denoted by ${\mathbb{S}}_{n}$.

**Definition 26.**Given a continuous Boolean $v:{[0,1]}^{n}\to [0,1]$, each voter $i\in N$ such that

**null**voter.

**Definition 27.**A continuous simple game $v:{[0,1]}^{n}\to [0,1]$ is called

**proper**if $v\left(x\right)+v(\mathbf{1}-x)\le 1$ for all real-valued vectors $x\in {[0,1]}^{n}$. It is called

**strong**if $v\left(x\right)+v(\mathbf{1}-x)\ge 1$. A continuous simple game that is both proper and strong is called

**constant-sum**(or self-dual, or decisive).

**Definition 28.**Given a continuous simple game $v:{[0,1]}^{n}\to [0,1]$ we say that voter $i\in N$ is more desirable as voter $j\in N$, denoted by $i\u2ab0j$, if

- (1)
- $v\left(\tau \right(x\left)\right)\ge v\left(x\right)$ for all $x\in {[0,1]}^{n}$ with ${x}_{i}\le {x}_{j}$, where τ is equal to the transposition $(i,j)$;
- (2)
- $v\left(\tau \right(x\left)\right)\le v\left(x\right)$ for all $x\in {[0,1]}^{n}$ with ${x}_{i}\ge {x}_{j}$, where τ is equal to the transposition $(i,j)$.

**Definition 29.**A continuous simple game $v:{[0,1]}^{n}\to [0,1]$ is called

**complete**if for each pair of voters $i,j\in N$, we have $i\u2ab0j$ or $j\u2ab0i$. The set of all continuous complete (simple) games on n voters is denoted by ${\mathbb{C}}_{n}$.

**Definition 30.**A continuous simple game $v:{[0,1]}^{n}\to [0,1]$ is

**linearly weighted**, if there exist (normalized) weights ${w}_{1},\cdots ,{w}_{n}\ge 0$ with ${\sum}_{i=1}^{n}{w}_{i}=1$, such that $v\left(({x}_{1},\cdots ,{x}_{n})\right)={\sum}_{i=1}^{n}{w}_{i}{x}_{i}$. The set of all continuous linearly weighted (simple) games on n voters is denoted by ${\mathbb{L}}_{n}$.

**Definition 31.**A continuous simple game $v:{[0,1]}^{n}\to [0,1]$ is called a

**threshold**game if there exists a quota $q\in (0,1]$ and (normalized) weights ${w}_{1},\cdots ,{w}_{n}\ge 0$ with ${\sum}_{i=1}^{n}{w}_{i}=1$ such that $v\left(({x}_{1},\cdots ,{x}_{n})\right)=1$ if ${\sum}_{i=1}^{n}{w}_{i}{x}_{i}\ge q$ and $v\left(({x}_{1},\cdots ,{x}_{n})\right)=0$ otherwise. The set of all continuous threshold games on n voters is denoted by ${\mathbb{T}}_{n}$.

**Definition 32.**A continuous simple game $v:{[0,1]}^{n}\to [0,1]$ is

**weighted**if there exist (normalized) weights ${w}_{1},\cdots ,{w}_{n}\ge 0$ with ${\sum}_{i=1}^{n}{w}_{i}=1$ and a monotonously increasing quota function $q:[0,1]\to [0,1]$ such that $v\left(({x}_{1},\cdots ,{x}_{n})\right)=q\left({\sum}_{i=1}^{n}{w}_{i}{x}_{i}\right)$. The set of all continuous weighted (simple) games on n voters is denoted by ${\mathbb{W}}_{n}$.

**Lemma 4.**The weighted representation of a continuous linearly weighted game is unique.

**Lemma 5.**Let v be a continuous threshold game such that there exists a vector $x\ne \mathbf{1}$ with $v\left(x\right)=1$ and ${x}_{j}<1$ for a non-null voter j. The weighted representation of v, consisting of a quota $q\in (0,1]$ and weights $w\in {[0,1]}^{n}$ with ${\parallel w\parallel}_{1}=1$, is unique.

**Lemma 6.**The quota function q of a continuous weighted game $v:{[0,1]}^{n}\to [0,1]$ is unique. If q is monotone and continuous, then also the weights ${w}_{i}$ are unique.

**Lemma 7.**All continuous linearly weighted games are proper, strong, and constant-sum.

**Lemma 8.**A continuous threshold game v with (normalized) weights w and quota $q\in (0,1]$ is proper if and only if $q>\frac{1}{2}$.

**Lemma 9.**A continuous threshold game v with (normalized) weights w and quota $q\in (0,1]$ is strong if and only if $q\le \frac{1}{2}$.

**Corollary 1.**No continuous threshold game can be constant-sum.

**Lemma 10.**A continuous weighted game v with (normalized) weights w and quota function $q:[0,1]\to [0,1]$ is proper if and only if $q\left(y\right)+q(1-y)\le 1$ for all $y\in [0,1]$.

**Lemma 11.**A continuous weighted game v with (normalized) weights w and quota function $q:[0,1]\to [0,1]$ is strong if and only if $q\left(y\right)+q(1-y)\ge 1$ for all $y\in [0,1]$.

**Corollary 2.**A continuous weighted game v with (normalized) weights w and quota function $q:[0,1]\to [0,1]$ is constant-sum if and only if $q\left(y\right)+q(1-y)=1$ for all $y\in [0,1]$.

**Definition 33.**Let ${\mathbb{V}}_{n}\subseteq {\mathbb{B}}_{n}$ a class of continuous Boolean games consisting of n voters. A power index (on ${\mathbb{V}}_{n}$) is a mapping $P:{\mathbb{V}}_{n}\to {\mathbb{R}}^{n}$.

## 5. Examples of Continuous Games

## 6. Generalizing Three Power Indices

#### 6.1. Shapley-Shubik Index

- (1)
- According to the veil of ignorance, the set of vote vectors has no structure, i.e., votes are independent and each of the ${2}^{n}$$\{0,1\}$-vectors occurs with equal probability.
- (2)
- Assume that the voters are arranged in a sequence and called one by one. After the ith voter in the current sequence has expressed his vote, an output alternative may be excluded independently from the votes of the subsequent voters. Here, all sequences are equally probable and the exclusion of an output alternative is counted just once, i.e., it is counted for the first player who excludes it.

**Definition 34.**

- $\overline{\tau}:{[0,1]}^{n}\times \{1,\cdots ,n\}\to {[0,1]}^{n}$, $(x,i)\mapsto ({y}_{1},\cdots ,{y}_{n})$, where ${y}_{j}={x}_{j}$ for all $1\le j\le i$ and ${y}_{j}=1$ otherwise;
- $\underline{\tau}:{[0,1]}^{n}\times \{1,\cdots ,n\}\to {[0,1]}^{n}$, $(x,i)\mapsto ({y}_{1},\cdots ,{y}_{n})$, where ${y}_{j}={x}_{j}$ for all $1\le j\le i$ and ${y}_{j}=0$ otherwise.

**Definition 35.**Let $v:{[0,1]}^{n}\to [0,1]$ be a continuous simple game. The Shapley-Shubik index ${SSI}_{i}\left(v\right)$ of voter i in v is given by

**Lemma 12.**Let $P:{\mathcal{S}}_{n}\to {\mathbb{R}}^{n}$ be a power index. If P satisfies symmetry, efficiency, the null voter property, and the transfer axiom, then P coincides with the Shapley-Shubik index.

**Definition 36.**For two Boolean games ${v}_{1},{v}_{2}\in {\mathcal{S}}_{n}$ we define ${v}_{1}\vee {v}_{2}$ by $\left({v}_{1}\vee {v}_{2}\right)\left(S\right)=max\left({v}_{1}\left(S\right),{v}_{2}\left(S\right)\right)$ for all $S\subseteq N$. Similarly, we define ${v}_{1}\wedge {v}_{2}$ by $\left({v}_{1}\wedge {v}_{2}\right)\left(S\right)=min\left({v}_{1}\left(S\right),{v}_{2}\left(S\right)\right)$.

**Definition 37.**A power index $P:{\mathcal{V}}_{n}\to {\mathbb{R}}^{n}$ satisfies the

**transfer axiom**, if

**Definition 38.**For two continuous Boolean games ${v}_{1},{v}_{2}$ we define ${v}_{1}\vee {v}_{2}$ by $\left({v}_{1}\vee {v}_{2}\right)\left(x\right)=max\left({v}_{1}\left(x\right),{v}_{2}\left(x\right)\right)$ for all $x\in {[0,1]}^{n}$. Similarly, we define ${v}_{1}\wedge {v}_{2}$ by $\left({v}_{1}\wedge {v}_{2}\right)\left(x\right)=min\left({v}_{1}\left(x\right),{v}_{2}\left(x\right)\right)$.

**Lemma 13.**The Shapley-Shubik index SSI is symmetric, positive, and satisfies both the null voter property and the transfer axiom on ${\mathbb{S}}_{n}$.

**Conjecture 1.**The Shapley-Shubik index for continuous simple games is efficient, i.e., ${\sum}_{i=1}^{n}{SSI}_{i}\left(v\right)=1$ for all $v\in {\mathbb{S}}_{n}$.

**Conjecture 2.**Let $P:{\mathbb{S}}_{n}\to {\mathbb{R}}^{n}$ be a power index. If P satisfies symmetry, efficiency, the null voter property, and the transfer axiom, then P coincides with the Shapley-Shubik index according to Definition 35.

**Lemma 14.**The Shapley-Shubik index of the weighted median aggregation rule, according to Definition 35 is given by the Shapley-Shubik index of the weighted game $\left[{\parallel w\parallel}_{1}/2;{w}_{1},\cdots ,{w}_{n}\right]$.

#### 6.2. Banzhaf Index

- (1)
- According to the veil of ignorance, the set of vote vectors has no structure, i.e., votes are independent and each of the ${j}^{n}$J-vectors occurs with equal probability.
- (2)
- Relevant for the measurement of influence is only the number of $(m,l)$-swings (or swings for simple games) for voter i arising if voter i shifts his chosen alternative by one.

**Definition 39.**Let $v:{[0,1]}^{n}\to [0,1]$ be a continuous simple game. The (absolute) Banzhaf index ${BZI}_{i}\left(v\right)$ of voter i in v is given by

**Lemma 15.**Let $P:{\mathcal{S}}_{n}\to {\mathbb{R}}^{n}$ be a power index. If P satisfies symmetry, the null voter property, the transfer axiom, and the Banzhaf total power, then P coincides with the Banzhaf index.

**Definition 40.**A power index $P:{\mathcal{V}}_{n}\to {\mathbb{R}}^{n}$ satisfies

**Banzhaf total power**, if

**Definition 41.**A power index $P:{\mathbb{V}}_{n}\to {\mathbb{R}}^{n}$ satisfies

**Banzhaf total power**, if ${\parallel BZI\left(v\right)\parallel}_{1}$ coincides with

**Lemma 16.**The Banzhaf index BZI is symmetric, positive, and satisfies the null voter property, the transfer axiom, and Banzhaf total power on ${\mathbb{S}}_{n}$.

**Conjecture 3.**Let $P:{\mathbb{S}}_{n}\to {\mathbb{R}}^{n}$ be a power index. If P satisfies symmetry, the null voter property, the transfer axiom, and the Banzhaf total power, then P coincides with the Banzhaf index according to Definition 39.

#### 6.3. Nucleolus

**Definition 42.**Given a continuous simple game $v:{[0,1]}^{n}\to [0,1]$ and a vector $w\in {[0,1]}^{n}$ with ${\parallel w\parallel}_{1}=1$, the

**excess**of a coalition $x\in {[0,1]}^{n}$ is given by ${e}^{v}(x,w)=v\left(x\right)-{w}^{T}x\in [-1,1]$.

**Definition 43.**Given a continuous simple game $v:{[0,1]}^{n}\to [0,1]$ and a vector $w\in {[0,1]}^{n}$ with ${\parallel w\parallel}_{1}=1$, the

**excess function**is given by ${E}_{w}^{v}:[-1,1]\to [0,1]$,

**Definition 44.**For two integrable functions ${f}_{1}:[-1,1]\to [0,1]$ and ${f}_{2}:[-1,1]\to [0,1]$, we write ${f}_{1}\le {f}_{2}$ if there exists a constant $c\in [-1,1]$ such that ${f}_{1}\left(y\right)\le {f}_{2}\left(y\right)$ for all $y\in [c,1]$ and ${\int}_{c}^{1}{f}_{1}\left(y\right)\phantom{\rule{0.166667em}{0ex}}dy<{\int}_{c}^{1}{f}_{2}\left(y\right)\phantom{\rule{0.166667em}{0ex}}dy$.

**Definition 45.**For a continuous simple game $v:{[0,1]}^{n}\to [0,1]$ the

**nucleolus**$Nuc\left(\mathbf{v}\right)$ is given by

**Conjecture 4.**Under mild technical assumptions for a continuous simple game $v:{[0,1]}^{n}\to [0,1]$, we have $|Nuc(v\left)\right|\le 1$.

## 7. Power Indices When Votes Are Not Equiprobable

**Definition 46.**Let $v:{[0,1]}^{n}\to [0,1]$ be a continuous simple game. The density Shapley-Shubik index ${SSI}_{i}^{f}\left(v\right)$ of voter i in v is given by

**Definition 47.**Let $v:{[0,1]}^{n}\to [0,1]$ be a continuous simple game. The density (absolute) Banzhaf index ${BZI}_{i}^{f}\left(v\right)$ of voter i in v is given by

## 8. Conclusions

## Conflicts of Interest

## A. Determining the SSI for Two Continuous Simple Games

**Table A1.**${SSI}_{1}\left(\widehat{v}\right)$ for $\widehat{v}({x}_{1},{x}_{2},{x}_{3})=\frac{1}{6}\xb7(1{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2})$.

$\pi \in {\mathcal{S}}_{3}$ | 3-fold integral |
---|---|

$(1,2,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{6}{6}-\frac{{x}_{1}^{2}+5}{6}+\frac{{x}_{1}^{2}}{6}-\frac{0}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

$(1,3,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{6}{6}-\frac{{x}_{1}^{2}+5}{6}+\frac{{x}_{1}^{2}}{6}-\frac{0}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

$(2,1,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{2{x}_{2}^{2}+4}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3}{6}+\frac{{x}_{1}^{2}+2{x}_{2}^{2}}{6}-\frac{2{x}_{2}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

$(2,3,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{2{x}_{2}^{2}+3{x}_{3}^{2}+1}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}+\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}-\frac{2{x}_{2}^{2}+3{x}_{3}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

$(3,1,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{3{x}_{3}^{2}+3}{6}-\frac{{x}_{1}^{2}+3{x}_{3}^{2}+2}{6}+\frac{{x}_{1}^{2}+3{x}_{3}^{2}}{6}-\frac{3{x}_{3}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

$(3,2,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{2{x}_{2}^{2}+3{x}_{3}^{2}+1}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}+\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}-\frac{2{x}_{2}^{2}+3{x}_{3}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

**Table A2.**${SSI}_{2}\left(\widehat{v}\right)$ for $\widehat{v}({x}_{1},{x}_{2},{x}_{3})=\frac{1}{6}\xb7(1{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2})$.

$\pi \in {\mathcal{S}}_{3}$ | 3-fold integral |
---|---|

$(2,1,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{6}{6}-\frac{{x}_{2}^{2}+4}{6}+\frac{{x}_{2}^{2}}{6}-\frac{0}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{2}{6}$ |

$(2,3,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{6}{6}-\frac{{x}_{2}^{2}+4}{6}+\frac{{x}_{2}^{2}}{6}-\frac{0}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{2}{6}$ |

$(1,2,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{{x}_{1}^{2}+5}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3}{6}+\frac{{x}_{1}^{2}+2{x}_{2}^{2}}{6}-\frac{{x}_{1}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{2}{6}$ |

$(3,2,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{3{x}_{3}^{2}+3}{6}-\frac{2{x}_{2}^{2}+3{x}_{3}^{2}+1}{6}+\frac{2{x}_{2}^{2}+3{x}_{3}^{2}}{6}-\frac{3{x}_{3}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{2}{6}$ |

$(1,3,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{{x}_{1}^{2}+3{x}_{3}^{2}+2}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}+\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}-\frac{{x}_{1}^{2}+3{x}_{3}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{2}{6}$ |

$(3,1,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{{x}_{1}^{2}+3{x}_{3}^{2}+2}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}+\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}-\frac{{x}_{1}^{2}+3{x}_{3}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{2}{6}$ |

**Table A3.**${SSI}_{3}\left(\widehat{v}\right)$ for $\widehat{v}({x}_{1},{x}_{2},{x}_{3})=\frac{1}{6}\xb7(1{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2})$.

$\pi \in {\mathcal{S}}_{3}$ | 3-fold integral |
---|---|

$(3,1,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{6}{6}-\frac{3{x}_{3}^{2}+3}{6}+\frac{3{x}_{3}^{2}}{6}-\frac{0}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{6}$ |

$(3,2,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{6}{6}-\frac{3{x}_{3}^{2}+3}{6}+\frac{3{x}_{3}^{2}}{6}-\frac{0}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{6}$ |

$(1,3,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{{x}_{1}^{2}+5}{6}-\frac{{x}_{1}^{2}+3{x}_{3}^{2}+2}{6}+\frac{{x}_{1}^{2}+3{x}_{3}^{2}}{6}-\frac{{x}_{1}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{6}$ |

$(2,3,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{2{x}_{2}^{2}+4}{6}-\frac{2{x}_{2}^{2}+3{x}_{3}^{2}+1}{6}+\frac{2{x}_{2}^{2}+3{x}_{3}^{2}}{6}-\frac{2{x}_{2}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{6}$ |

$(1,2,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}+\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{6}$ |

$(2,1,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}+\frac{{x}_{1}^{2}+2{x}_{2}^{2}+3{x}_{3}^{2}}{6}-\frac{{x}_{1}^{2}+2{x}_{2}^{2}}{6}\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{6}$ |

**Table A4.**${SSI}_{1}\left(\tilde{v}\right)$ for $\tilde{v}({x}_{1},{x}_{2},{x}_{3})={x}_{1}{x}_{2}^{2}{x}_{3}^{3}$.

$\pi \in {\mathcal{S}}_{3}$ | 3-fold integral |
---|---|

$(1,2,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(1-{x}_{1}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{2}$ |

$(1,3,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(1-{x}_{1}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{2}$ |

$(2,1,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{2}^{2}-{x}_{1}{x}_{2}^{2}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

$(2,3,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{3}^{3}-{x}_{1}{x}_{3}^{3}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{8}$ |

$(3,1,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{2}^{2}{x}_{3}^{3}-{x}_{1}{x}_{2}^{2}{x}_{3}^{3}+{x}_{1}{x}_{2}^{2}{x}_{3}^{3}-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{12}$ |

$(3,2,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{2}^{2}{x}_{3}^{3}-{x}_{1}{x}_{2}^{2}{x}_{3}^{3}+{x}_{1}{x}_{2}^{2}{x}_{3}^{3}-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{12}$ |

**Table A5.**${SSI}_{2}\left(\tilde{v}\right)$ for $\tilde{v}({x}_{1},{x}_{2},{x}_{3})={x}_{1}{x}_{2}^{2}{x}_{3}^{3}$.

$\pi \in {\mathcal{S}}_{3}$ | 3-fold integral |
---|---|

$(1,2,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{1}-{x}_{1}{x}_{2}^{2}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{3}$ |

$(1,3,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{1}{x}_{3}^{3}-{x}_{1}{x}_{2}^{2}{x}_{3}^{3}+{x}_{1}{x}_{2}^{2}{x}_{3}^{3}-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{8}$ |

$(2,1,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(1-{x}_{2}^{2}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{2}{3}$ |

$(2,3,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(1-{x}_{2}^{2}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{2}{3}$ |

$(3,1,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{1}{x}_{3}^{3}-{x}_{1}{x}_{2}^{2}{x}_{3}^{3}+{x}_{1}{x}_{2}^{2}{x}_{3}^{3}-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{8}$ |

$(3,2,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{3}^{3}-{x}_{2}^{2}{x}_{3}^{3}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

**Table A6.**${SSI}_{3}\left(\tilde{v}\right)$ for $\tilde{v}({x}_{1},{x}_{2},{x}_{3})={x}_{1}{x}_{2}^{2}{x}_{3}^{3}$.

$\pi \in {\mathcal{S}}_{3}$ | 3-fold integral |
---|---|

$(1,2,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{1}{x}_{2}^{2}-{x}_{1}{x}_{2}^{2}{x}_{3}^{3}+{x}_{1}{x}_{2}^{2}{x}_{3}^{3}-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

$(1,3,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{1}-{x}_{1}{x}_{3}^{3}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{8}$ |

$(2,1,3)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{1}{x}_{2}^{2}-{x}_{1}{x}_{2}^{2}{x}_{3}^{3}+{x}_{1}{x}_{2}^{2}{x}_{3}^{3}-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{6}$ |

$(2,3,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left({x}_{2}^{2}-{x}_{2}^{2}{x}_{3}^{3}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{1}{4}$ |

$(3,1,2)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(1-{x}_{3}^{3}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{4}$ |

$(3,2,1)$ | $\underset{x\in {[0,1]}^{3}}{\int}\left(1-{x}_{3}^{3}+0-0\right)\phantom{\rule{0.166667em}{0ex}}dx=\frac{3}{4}$ |

## B. Determining the SSI for a Weighted Median Aggregation Rule

#### B.1. Shapley-Shubik Power for Voter 2

#### 2.1.1. $\pi \in \left\{(2,1,3,4),(2,1,4,3),(2,3,1,4),(2,3,4,1),(2,4,1,3),(2,4,3,1)\right\}$

#### 2.1.2. $\pi \in \left\{(3,2,1,4),(3,2,4,1),(4,2,1,3),(4,2,3,1)\right\}$

#### 2.1.3. $\pi =(1,2,3,4)$, $\pi =(1,2,4,3)$

#### 2.1.4. $\pi \in \left\{(1,3,2,4),(1,4,2,3),(3,1,2,4),(4,1,2,3)\right\}$

Ordering | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}v\left(\overline{\tau}(x,{\pi}^{-1}\left(2\right)-1)\right)\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}$ | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}v\left(\overline{\tau}(x,{\pi}^{-1}\left(2\right))\right)\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}$ | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}v\left(\underline{\tau}(x,{\pi}^{-1}\left(2\right))\right)\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}$ | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}v\left(\underline{\tau}(x,{\pi}^{-1}\left(2\right)-1)\right)\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}$ |
---|---|---|---|---|

${x}_{1}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{2}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{3}$ | ${x}_{3}$ | ${x}_{2}$ | ${x}_{1}$ | ${x}_{1}$ |

${x}_{1}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{3}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{2}$ | ${x}_{3}$ | ${x}_{3}$ | ${x}_{1}$ | ${x}_{1}$ |

${x}_{2}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{1}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{3}$ | ${x}_{3}$ | ${x}_{1}$ | ${x}_{1}$ | ${x}_{1}$ |

${x}_{2}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{3}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{1}$ | ${x}_{1}$ | ${x}_{1}$ | ${x}_{3}$ | ${x}_{3}$ |

${x}_{3}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{1}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{2}$ | ${x}_{1}$ | ${x}_{1}$ | ${x}_{1}$ | ${x}_{3}$ |

${x}_{3}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{2}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}<\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}{x}_{1}$ | ${x}_{1}$ | ${x}_{1}$ | ${x}_{2}$ | ${x}_{3}$ |

Ordering | ${\int}_{({x}_{1},{x}_{3},{x}_{4})}\u2606$ |
---|---|

${x}_{1}<{x}_{2}<{x}_{3}$ | ${\int}_{0}^{{x}_{2}}{\int}_{{x}_{2}}^{1}\left({x}_{3}-{x}_{2}+{x}_{1}-{x}_{1}\right)d{x}_{3}\phantom{\rule{0.166667em}{0ex}}d{x}_{1}=\frac{{x}_{2}^{3}-2{x}_{2}^{2}+{x}_{2}}{2}$ |

${x}_{1}<{x}_{3}<{x}_{2}$ | ${\int}_{0}^{{x}_{2}}{\int}_{{x}_{1}}^{{x}_{2}}\left({x}_{3}-{x}_{3}+{x}_{1}-{x}_{1}\right)d{x}_{3}\phantom{\rule{0.166667em}{0ex}}d{x}_{1}=0$ |

${x}_{2}<{x}_{1}<{x}_{3}$ | ${\int}_{{x}_{2}}^{1}{\int}_{{x}_{1}}^{1}\left({x}_{3}-{x}_{1}+{x}_{1}-{x}_{1}\right)d{x}_{3}\phantom{\rule{0.166667em}{0ex}}d{x}_{1}=\frac{-{x}_{2}^{3}+3{x}_{2}^{2}-3{x}_{2}+1}{6}$ |

${x}_{2}<{x}_{3}<{x}_{1}$ | ${\int}_{{x}_{2}}^{1}{\int}_{{x}_{3}}^{1}\left({x}_{1}-{x}_{1}+{x}_{3}-{x}_{3}\right)d{x}_{1}\phantom{\rule{0.166667em}{0ex}}d{x}_{3}=0$ |

${x}_{3}<{x}_{1}<{x}_{2}$ | ${\int}_{0}^{{x}_{2}}{\int}_{{x}_{3}}^{{x}_{2}}\left({x}_{1}-{x}_{1}+{x}_{1}-{x}_{3}\right)d{x}_{1}\phantom{\rule{0.166667em}{0ex}}d{x}_{3}=\frac{{x}_{2}^{3}}{6}$ |

${x}_{3}<{x}_{2}<{x}_{1}$ | ${\int}_{0}^{{x}_{2}}{\int}_{{x}_{2}}^{1}\left({x}_{1}-{x}_{1}+{x}_{2}-{x}_{3}\right)d{x}_{1}\phantom{\rule{0.166667em}{0ex}}d{x}_{3}=\frac{{x}_{2}^{2}-{x}_{2}^{3}}{2}$ |

#### 2.1.5. $\pi =(3,4,2,1)$, $\pi =(4,3,2,1)$

#### 2.1.6. $\pi \in \left\{(1,3,4,2),(1,4,3,2),(3,1,4,2),(4,1,3,2),(3,4,1,2),(4,3,1,2)\right\}$

$max{x}^{\prime}$ | $min{x}^{\prime}$ | $v\left(\overline{\tau}(x,{\pi}^{-1}\left(2\right)-1)\right)$ | $v\left(\overline{\tau}(x,{\pi}^{-1}\left(2\right))\right)$ |
---|---|---|---|

${x}_{1}$ | $\ne {x}_{1}$ | ${x}_{1}$ | $\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\left\{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\begin{array}{ccc}\hfill max({x}_{2},{x}_{3},{x}_{4})& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& {x}_{2}\le {x}_{1}\hfill \\ \hfill {x}_{1}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& {x}_{2}\ge {x}_{1}\hfill \end{array}\right.$ |

$\ne {x}_{1}$ | ${x}_{1}$ | $min({x}_{3},{x}_{4})$ | $min({x}_{2},{x}_{3},{x}_{4})$ |

$\ne {x}_{1}$ | $\ne {x}_{1}$ | ${x}_{1}$ | ${x}_{1}$ |

$max{x}^{\prime}$ | $min{x}^{\prime}$ | $v\left(\underline{\tau}(x,{\pi}^{-1}\left(2\right))\right)$ | $v\left(\underline{\tau}(x,{\pi}^{-1}\left(2\right)-1)\right)$ |

${x}_{1}$ | $\ne {x}_{1}$ | $max({x}_{2},{x}_{3},{x}_{4})$ | $max({x}_{3},{x}_{4})$ |

$\ne {x}_{1}$ | ${x}_{1}$ | $\phantom{\rule{-0.166667em}{0ex}}\left\{\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\begin{array}{ccc}\hfill min({x}_{2},{x}_{3},{x}_{4})& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& {x}_{2}\ge {x}_{1}\hfill \\ \hfill {x}_{1}& \phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}:\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}\phantom{\rule{-0.166667em}{0ex}}& {x}_{2}\le {x}_{1}\hfill \end{array}\right.$ | ${x}_{1}$ |

$\ne {x}_{1}$ | $\ne {x}_{1}$ | ${x}_{1}$ | ${x}_{1}$ |

#### 2.1.7. Summarizing the 24 Permutations for ${SSI}_{2}\left(v\right)$

#### B.2. Shapley-Shubik Power for Voter 3 and Voter 4

#### B.3. Shapley-Shubik Power for Voter 1

#### 2.3.1. $\pi \in \left\{(1,2,3,4),(1,2,4,3),(1,3,2,4),(1,3,4,2),(1,4,2,3),(1,4,3,2)\right\}$

#### 2.3.2. $\pi \in \left\{(2,1,3,4),(2,1,4,3),(3,1,2,4),(3,1,4,2),(4,1,2,3),(4,1,3,2)\right\}$

#### 2.3.3. $\pi \in \left\{(2,3,1,4),(3,2,1,4),(2,4,1,3),(4,2,1,3),(3,4,1,2),(4,3,1,2)\right\}$

#### 2.3.4. $\pi \in \left\{(2,3,4,1),(3,2,4,1),(2,4,3,1),(4,2,3,1),(3,4,2,1),(4,3,2,1)\right\}$

#### 2.3.5. Summarizing the 24 Permutations for ${SSI}_{1}\left(v\right)$

## C. Determining the Nuc for Two Continuous Simple Games

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^{1.}We have ${\mathcal{S}}_{n}\subseteq {\mathcal{C}}_{n}\subseteq {\mathcal{T}}_{n}$, ${\mathcal{S}}_{n}\ne {\mathcal{C}}_{n}$ for $n\ge 4$, and ${\mathcal{C}}_{n}\ne {\mathcal{T}}_{n}$ for $n\ge 6$.^{2.}In some applications one may consider the mentioned condition as a natural requirement. As an example consider the players’ input x as their most preferred alternative. If all players unanimously prefer x to all other alternatives, then x clearly should be implemented.

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Kurz, S. Measuring Voting Power in Convex Policy Spaces. *Economies* **2014**, *2*, 45-77.
https://doi.org/10.3390/economies2010045

**AMA Style**

Kurz S. Measuring Voting Power in Convex Policy Spaces. *Economies*. 2014; 2(1):45-77.
https://doi.org/10.3390/economies2010045

**Chicago/Turabian Style**

Kurz, Sascha. 2014. "Measuring Voting Power in Convex Policy Spaces" *Economies* 2, no. 1: 45-77.
https://doi.org/10.3390/economies2010045