The inverse problem for power distributions in committees

Abstract

Several power indices have been introduced in the literature in order to measure the influence of individual committee members on an aggregated decision. Here we ask the inverse question and aim to design voting rules for a committee such that a given desired power distribution is met as closely as possible. We generalize the approach of Alon and Edelman who studied power distributions for the Banzhaf index, where most of the power is concentrated on few coordinates. It turned out that each Banzhaf vector of an n-member committee that is near to such a desired power distribution, also has to be near to the Banzhaf vector of a k-member committee. We show that such Alon-Edelman type results exist for other power indices like e.g. the Public Good index or the Coleman index to prevent actions, while such results are principally impossible to derive for e.g. the Johnston index.

Appendices

Appendix 1: Details for quality function results

In this section we want to prove the missing details for the quality function results announced in Table 1. Since we have started with the proof for the Public Good Index, in Sect. 4.2, we want to continue with the Deegan-Packel index, which arises as the equal division of the ${\text {PGI}}$, see Definition 13.

We can exploit this relation by setting $\widehat{C}^{{\text {DP}}}=\sum _{i=k+1}^n C_i^{{\text {DP}}}$ and denoting the number of minimal winning coalitions that contain at least one member of (k, n] by $\widehat{M}^{{\text {PGI}}}$. With this we have:

Lemma 13

$$\begin{aligned} \widehat{M}^{{\text {PGI}}}\le \widehat{C}^{{\text {DP}}}\cdot (k+1) \end{aligned}$$

Proof

Let S be a minimal winning coalition that contains at least one player from (k, n]. We set $a:=\left| S\cap [1,k]\right|$ and $b:=\left| S\cap (k,n]\right|$, so that $a+b=|S|$, $0\le a\le k$, and $1\le b\le n-k$. Thus, the stated inequality follows from $\sum \nolimits _{i=k+1}^n C_i^{{\text {DP}}}(v,S)=\frac{b}{a+b}\ge \frac{1}{k+1}$, where $v$ denotes the respective game.

Corollary 3

For the Deegan-Packel index we can choose $f_1(k)=(k+1)^2$.

Lemma 14

Let $v$ be a complete simple game with player set $[n]$ and $S\subseteq [n]$ be a coalition. The number of direct left-shifts of S is bounded by $|S|+1$ and $\left\lfloor \frac{n+1}{2}\right\rfloor$.

Proof

W.l.o.g. we assume $1\succeq 2\succeq \cdots \succeq n$. A direct left-shift of S arises either by shifting a player of S one place to the left or by adding the weakest player n. For the other bound we observe that player i can only be shifted to position $i-1$ if $i-1\notin S$ and $i\ne 1$. Thus only $\left\lfloor \frac{n}{2}\right\rfloor$ players can be shifted one position to the left, where equality is only possible if either n is odd or n is even and $n\in S$. Considering the possible addition of player n given at most $\left\lfloor \frac{n+1}{2}\right\rfloor$ cases.

Lemma 15

For the Shift index we can choose $f_1(k)=\frac{k+3}{2}$.

Proof

Let $v=(\mathcal {W},[n])$ be an arbitrary simple game. As defined before, $C_i^{{\text {Shift}}}(v,2^{[n]})$ counts the number of shift-minimal winning coalitions in $v$ containing player i and we have $\overline{C}^{{\text {Shift}}}(v,2^{[n]})=\sum _{i=1}^n C_i^{{\text {Shift}}}(v,2^{[n]})$. For brevity we just write $C_i$ and $\overline{C}$. As further abbreviations we use $v'=(\mathcal {W}',[n])$ for the k-rounding of $v$, $\widehat{C}=\sum _{i=k+1}^n C_i$, i.e. the restriction of $\overline{C}$ to the players in (k, n], and by $\widehat{M}$ we denote the number of shift-minimal winning coalitions in $v$ that contain at least one player from (k, n]. With this we have $\widehat{M}\le \widehat{C}$.

Now we want to study the changes in the $C_i$ by going from $v$ to $v'$. At first we consider the cases where coalition S is a SMWC (shift-minimal winning coalition) in $v$ but not in $v'$.

• (1) If $S\cap (k,n]\ne \emptyset$ then removing S from $v$ results in a decrease of 1 for player i and there is at least one player in (k, n]. Thus the negative change of the $C_i$ of that type is bounded by $\widehat{M}\le \widehat{C}$.

• (2) If $S\subseteq [1,k]$, then S is winning in $v'$ too. Since S is not a SMWC in $v'$ (by assumption), there must be a direct right-shift $S'=S{\backslash } j\cup h$ of S, where $j\in S$, $h\in [1,k]{\backslash } S$ or $h=\emptyset$, such that $S'$ is winning in $v'$. According to the rounding procedure, there is a subset $\emptyset \ne T\subseteq 2^{(k,n]}$ such that $S'\cup T$ is a SMWC in $v$. Given $S'$ there are at most $\frac{k+1}{2}$ choices for S, see Lemma 14. Thus, we have that the negative change is bounded by $\frac{k+1}{2}\cdot \widehat{M}\le \frac{k+1}{2}\cdot \widehat{C}$.

Next, we consider the cases where coalition S is not a SMWC in $v$ but in $v'$. Since the players in (k, n] are null players, we can deduce $S\in 2^{[1,k]}$.

• (3) Assume that S is losing in $v$ but winning in $v'$. According to the rounding procedure there exists coalition $\emptyset \ne T\in 2^{(k,n]}$ such that $S\cup T$ is a SMWC in $v$. Thus, the total increase for $C_i$ is bounded by $\widehat{M}\le \widehat{C}$.

• (4) If S is winning in $v$, then there must be a direct right-shift $S'$ such that $S'$ is winning in $v$ but losing in $v'$. We proceed similarly as in case (2) and deduce an upper bound of $\frac{k+1}{2}\cdot \widehat{M}$ for the change of $C_i$.

By proving the bound $\widehat{M}^{{\text {Shift}}}\le \widehat{C}^{{\text {SDP}}}\cdot (k+1)$, similar to the proof of Lemma 13, we can conclude $f_1(k)=\frac{k+3}{2}$.

Corollary 4

For the Shift Deegan-Packel index we can choose $f_1(k)=\frac{(k+1)(k+3)}{2}$.

Lemma 16

For ${\text {ColPrev}}$ we can choose $f_1(k)=2$.

Proof

Let us write $\eta _i(v,2^{[n]})=C_i^{{\text {swing}}}(v,2^{[n]})$ for the number of swings for player i and $\overline{\eta }(v,2^{[n]})=\overline{C}^{{\text {swing}}}(v,2^{[n]})$. By $v'$ we denote ${\varGamma }(v,k)$, where ${\varGamma }$ is the k-rounding function. From Lemmas 8 and 5 we conclude

$$\begin{aligned} \left| \eta _i(v',2^{[n]})-\eta _i(v,2^{[n]})\right| \le \varepsilon \cdot \overline{\eta }(v,2^{[n]}) \end{aligned}$$

(1)

for all $1\le i\le k$. By $\mathcal {W}(v)$ and $\mathcal {W}(v')$ we denote the set of winning coalitions of $v$ and $v'$, respectively. From the proof of Lemma 8 we obtain

$$\begin{aligned} \Big \vert |\mathcal {W}(v')|-|\mathcal {W}(v)|\Big \vert \le \varepsilon \cdot \overline{\eta }(v,2^{[n]}). \end{aligned}$$

(2)

With this we conclude

$$\begin{aligned}&\left| \frac{\eta _i(v',2^{[n]})}{|\mathcal {W}(v')|}-\frac{\eta _i(v,2^{[n]})}{|\mathcal {W}(v)|}\right| \\&\quad \le \left| \frac{\eta _i(v',2^{[n]})}{|\mathcal {W}(v)|}-\frac{\eta _i(v,2^{[n]})}{|\mathcal {W}(v)|}\right| +\left| \frac{\eta _i(v',2^{[n]})}{|\mathcal {W}(v')|}-\frac{\eta _i(v',2^{[n]})}{|\mathcal {W}(v)|}\right| \\&\quad = \frac{\left| \eta _i(v',2^{[n]})-\eta _i(v,2^{[n]})\right| }{|\mathcal {W}(v)|} +\frac{\eta _i(v',2^{[n]})}{|\mathcal {W}(v')|}\cdot \frac{\Big \vert |\mathcal {W}(v')|-|\mathcal {W}(v)|\Big \vert }{|\mathcal {W}(v)|}\\&\quad \le \frac{\varepsilon \cdot \overline{\eta }(v,2^{[n]})}{|\mathcal {W}(v)|}+ \frac{\eta _i(v',2^{[n]})}{|\mathcal {W}(v')|}\cdot \frac{\varepsilon \cdot \overline{\eta }(v,2^{[n]})}{|\mathcal {W}(v)|}\\&\quad \le 2\cdot \frac{\varepsilon \cdot \overline{\eta }(v,2^{[n]})}{|\mathcal {W}(v)|} \end{aligned}$$

where we have used the triangle inequality, Inequality (1), Inequality (2), and the fact that the number of swings for player i cannot be larger than the number of winning coalitions. Thus we can choose $f_1(k)=2$.

Lemma 17

For ${\text {ColIni}}$ we can choose $f_1(k)=2$.

Proof

As pointed out in e.g. Dubey et al. (1981) ${\text {ColIni}}(v)$ equals ${\text {ColPrev}}(v^\star )$, where $v^\star$ denotes the dual game of $v$, see e.g. Taylor and Zwicker (1999) for a definition. Since the class of simple games is closed under taking the dual and $\widehat{{\text {ColIni}}}=\widehat{{\text {ColPrev}}}$, we can chose the same quality function as in Lemma 16.

Appendix 2: Technical details for the (absolute) Johnston index

In this section we provide the delayed proof from Sect. 4.3. Before going into the details, we provide a useful bound for binomial coefficients:

Lemma 18

For integers $1\le m\le n$ we have ${{n-1}\atopwithdelims (){m-1}}\le \frac{2^{n-1}}{\sqrt{n}}$.

Proof

Since the binomial coefficients attain their maximum at the center, it suffices to prove the proposed inequality for $m-1=\left\lfloor \frac{n-1}{2}\right\rfloor$. For $t\ge 1$ we have ${{2t}\atopwithdelims ()t}\le \frac{2^{2t}}{\sqrt{3t+1}}$. From ${{2t-1}\atopwithdelims (){t-1}}+{{2t-1}\atopwithdelims (){t}}={{2t}\atopwithdelims (){t}}$ and ${{2t-1}\atopwithdelims (){t-1}}={{2t-1}\atopwithdelims (){t}}$, we conclude ${{2t-1}\atopwithdelims (){t-1}}\le \frac{2^{2t-1}}{\sqrt{3t+1}}$ for all $t\ge 1$. Thus, we have ${{n-1}\atopwithdelims (){m-1}}\le \frac{2^{n-1}}{\sqrt{n}}$ for all $n\ge 1$.

Lemma 19

For $l=1$ and integers k, m, n satisfying the restrictions from Definition 18 and $1\le m\le n$, we have

$$\begin{aligned} \begin{array}{rll} \displaystyle {\text {JS}}_i(v_{n,m}^{k,l}) &{}= \displaystyle \sum _{j=0}^n {n \atopwithdelims ()j}\cdot 1 +\sum _{j=0}^{m-1}{n \atopwithdelims ()j}\cdot 1 =2^n+\sum _{j=0}^{m-1}{n \atopwithdelims ()j} &{}\quad \forall i\in [1,k) \\ \displaystyle {\text {JS}}_k(v_{n,m}^{k,l}) &{}= \displaystyle \sum _{j=m+1}^{n}{n \atopwithdelims ()j}\cdot 1\,+\,{n \atopwithdelims ()m}\cdot \frac{1}{m+1}&{}\,\\ \displaystyle {\text {JS}}_i(v_{n,m}^{k,l}) &{}= \displaystyle {{n-1} \atopwithdelims (){m-1}}\cdot \frac{1}{m+1}&{} \quad \forall i\in (k,k+n] \\ \displaystyle {\text {JS}}_i(v_{n,n+1}^{k,l}) &{}= \displaystyle \sum _{j=0}^n {n \atopwithdelims ()j}\cdot 1 =2^n&{} \quad \forall i\in [1,k] \\ {\text {JS}}_i(v_{n,n+1}^{k,l}) &{}= 0&{} \quad \forall i\in (k,k+n] \\ \displaystyle {\text {JS}}_i(v_{n,0}^{k,l}) &{}= \displaystyle \sum _{j=0}^{n+1} {n \atopwithdelims ()j}\cdot 1 =2^{n+1}&{} \quad \forall i\in [1,k) \\ {\text {JS}}_i(v_{n,0}^{k,l}) &{}= 0&{} \quad \forall i\in [k,k+n] \\ \xi ^{{\text {JS}}} &{}= \displaystyle n\cdot {{n-1} \atopwithdelims (){m-1}}\cdot \frac{1}{m+1}.&{}\, \end{array} \end{aligned}$$

Proof

We note that all swing coalitions are of the form $\{i\}\cup T$, $\{i,k\}\cup T$, and $\{k\}\cup T$, where $1\le i\le k-1$ and $\emptyset \subseteq T\subseteq (k,n+k]$.

Proof

(of Lemma 12) From Lemmas 18 and 19 we conclude ${\text {JS}}_i(v_{n,m}^{k,l}) = 3\cdot 4^{\tilde{n}}\quad \forall i\in [1,k)$, ${\text {JS}}_k(v_{n,m}^{k,l}) \ge 4^{\tilde{n}} -\frac{\sqrt{2}\cdot 4^{\tilde{n}}}{\sqrt{\tilde{n}+1}}$, ${\text {JS}}_k(v_{n,m}^{k,l}) \le 4^{\tilde{n}}$, and $\xi ^{{\text {JS}}} \le \frac{\sqrt{2}\cdot 4^{\tilde{n}}}{\sqrt{\tilde{n}}}$. For $\tilde{n}\ge 1$ we then have

$$\begin{aligned} \Vert {\text {JS}}(v_{n,m}^{k,l})-{\text {JS}}(v_{n,n+1}^{k,l})\Vert _1\ge & {} k\cdot 4^{\tilde{n}}+\xi ^{{\text {JS}}} \ge \left( k\cdot \frac{\sqrt{\tilde{n}}}{\sqrt{2}}+1\right) \xi ^{{\text {JS}}},\\ \Vert {\text {JS}}(v_{n,m}^{k,l})-{\text {JS}}(v_{n,0}^{k,l})\Vert _1\ge & {} k\cdot 4^{\tilde{n}} -\frac{\sqrt{2}\cdot 4^{\tilde{n}}}{\sqrt{\tilde{n}+1}}+\xi ^{{\text {JS}}} \ge (k-1)4^{\tilde{n}}+\xi ^{{\text {JS}}}\\\ge & {} \left( (k-1)\cdot \frac{\sqrt{\tilde{n}}}{\sqrt{2}}+1\right) \xi ^{{\text {JS}}},\text { and }\lim \nolimits _{\tilde{n}\rightarrow \infty } \frac{\xi ^{{\text {JS}}}}{\sum \limits _{i=1}^{n+k} {\text {JS}}_i(v_{n,m}^{k,l})}= 0. \end{aligned}$$