Influence versus utility in the evaluation of voting rules: a new look at the Penrose formula

Abstract

In this paper, we present a contribution to the analysis of the relationship between influence/power measurement and utility measurement, the two most popular social objective criteria used when evaluating voting mechanisms. For one particular probabilistic model describing the preferences of the electorate, the so-called impartial culture (IC) model used by Banzhaf, the Penrose formula shows that the two objectives coincide. The IC probabilistic model assumes that voter preferences are independent and neutral. In this article, we prove a general version of the Penrose formula, allowing for preference correlations and biases in the electorate. We use that formula to illustrate, for a spectrum of well-known probabilistic models, how the divergence between the two social objectives impacts the ranking and performances of the voting mechanisms.

Appendix

Let us consider $n=mr+1$ and $q=lr+1$ with $0<l<m$. We prove that if f is continuous at $\theta \equiv \frac{l}{m}$, then:

$$\begin{aligned} \underset{r\rightarrow \infty }{\lim }\,n\times Influence\,\left( i,SE_{F},{\mathfrak {Maj}}_{q}\right) =f(\theta ). \end{aligned}$$

Let $A=\left\{ p\in \left] \theta -\epsilon ,\theta +\epsilon \right[ \right\}$ and $B=\left\{ p\in \left[ 0,\theta -\epsilon \right] \cup \left[ \theta +\epsilon ,1\right] \right\}$ for some $\epsilon >0$. By Formula (12):

$$\begin{aligned}& n\times Influence\left( i,SE_{F},{\mathfrak {Maj}}_{q}\right) \\&=\left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{0}^{1}p^{lr}(1-p)^{(mr-lr)}f(p)dp\\&=\left[ \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{(mr-lr)} f(p)dp\right] \\ &+\left[ \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{B} p^{lr}(1-p)^{(mr-lr)}f(p)dp\right] . \end{aligned}$$

Let us first consider $\left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{B} p^{lr}(1-p)^{(mr-lr)}f(p)dp$. By using two times the lower bound and one time the upper bound of the following general inequality³²:

$$\begin{aligned} \sqrt{2\pi k}\left( \frac{k}{e}\right) ^{k}\le k!\le \sqrt{2\pi k}\left( \frac{k}{e}\right) ^{k}\exp \left( \frac{1}{12k}\right) , \end{aligned}$$

we get that

$$\begin{aligned} \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \le \left( \frac{mr}{lr}\right) ^{lr}\left( \frac{mr}{mr-lr}\right) ^{mr-lr}\sqrt{\frac{mr}{2\pi lr(mr-lr)}}\exp \left( \frac{1}{12mr}\right) . \end{aligned}$$

Therefore, rearranging terms, one gets that:

$$\begin{aligned} 0&\le \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{B}p^{lr}(1-p)^{(mr-lr)} f(p)dp\\&\le \left( 1+\frac{1}{mr}\right) \sqrt{\frac{mr}{2\pi \theta (1-\theta )} }\exp \left( \frac{1}{12mr}\right) \int _{B}\left[ \left( \frac{p}{\theta }\right) ^{\theta }\left( \frac{1-p}{1-\theta }\right) ^{(1-\theta )}\right] ^{mr}f(p)dp. \end{aligned}$$

The logarithm of the expression $\left[ \left( \frac{p}{\theta }\right) ^{\theta }\left( \frac{1-p}{1-\theta }\right) ^{(1-\theta )}\right] ^{mr}$ writes:

$$\begin{aligned} mr\left( \theta \ln \left( \frac{p}{\theta }\right) +(1-\theta )\ln \left( \frac{1-p}{1-\theta }\right) \right) . \end{aligned}$$

The function $\theta \ln \left( \frac{p}{\theta }\right) +(1-\theta )\ln \left( \frac{1-p}{1-\theta }\right)$ is strictly concave (as a function of p) on $\left] 0,1\right[$. It is uniquely maximized when $p=\theta$ and takes the value 0. Then there exists $c>0$ such that $\theta \ln \left( \frac{p}{\theta }\right) +(1-\theta )\ln \left( \frac{1-p}{1-\theta }\right) <-c$ for all $p\in B$ and therefore:

$$\begin{aligned} \int _{B}\left[ \left( \frac{p}{\theta }\right) ^{\theta }\left( \frac{1-p}{1-\theta }\right) ^{(1-\theta )}\right] ^{mr}f(p)dp<\exp (-mrc), \end{aligned}$$

yielding that

$$\begin{aligned} 0&\le \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{B}p^{lr}(1-p)^{(mr-lr)} f(p)dp \\&\le \left( 1+\frac{1}{mr}\right) \sqrt{\frac{mr}{2\pi \theta (1-\theta )} }\exp \left( \frac{1}{12mr}-cmr\right) \text {.} \end{aligned}$$

(17)

Let us now consider $\left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A} p^{lr}(1-p)^{(mr-lr)}f(p)dp$. Since f is continuous at $\theta$, for any $\delta >0$, we can choose $\epsilon$ small enough that $-\delta \le f(p)-f(\theta )\le \delta$ for all $p\in A$. Therefore:

$$\begin{aligned}&\left( f(\theta )-\delta \right) \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr}dp \\&\le \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr} f(p)dp \\&\le \left( mr+1\right) \left( f(\theta )+\delta \right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr}dp. \end{aligned}$$

(18)

Adding inequalities (17) and (18), we obtain:

$$\begin{aligned}&\left( f(\theta )-\delta \right) \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr}dp\\&\le \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{0}^{1}p^{lr}(1-p)^{mr-lr} f(p)dp\\&\le \left( mr+1\right) \left( f(\theta )+\delta \right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr}dp+\left( 1+\frac{1}{mr}\right) \sqrt{\frac{mr}{2\pi \theta (1-\theta )}}\exp \left( \frac{1}{12mr}-cmr\right) . \end{aligned}$$

Let us now show that:

$$\begin{aligned} \underset{r\rightarrow \infty }{\lim }\left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr}dp=1. \end{aligned}$$

From

$$\begin{aligned} \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr}dp\le \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{0}^{1}p^{lr}(1-p)^{mr-lr}dp, \end{aligned}$$

and the identity in footnote 25, we get the following inequality:

$$\begin{aligned} \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr}dp\le 1. \end{aligned}$$

(19)

From inequality (17) applied to the case where f is constant (i.e. the uniform distribution) we obtain:

$$\begin{aligned} \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{B}p^{lr}(1-p)^{(mr-lr)}dp\le \left( 1+\frac{1}{mr}\right) \sqrt{\frac{mr}{2\pi \theta (1-\theta )}}\exp \left( \frac{1}{12mr}-cmr\right) \end{aligned}$$

and therefore

$$\begin{aligned} \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{0}^{1}p^{lr}(1-p)^{mr-lr}dp&\le \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{(mr-lr)}dp\\&+\left( 1+\frac{1}{mr}\right) \sqrt{\frac{mr}{2\pi \theta (1-\theta )}} \exp \left( \frac{1}{12mr}-cmr\right) . \end{aligned}$$

We conclude that:

$$\begin{aligned} 1-\left( 1+\frac{1}{mr}\right) \sqrt{\frac{mr}{2\pi \theta (1-\theta )}} \exp \left( \frac{1}{12mr}-cmr\right) \le \left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{(mr-lr)}dp. \end{aligned}$$

(20)

Collecting inequalities (19) and (20), and taking the limit when r goes to infinity, we deduce that:

$$\begin{aligned} \underset{r\rightarrow \infty }{\lim }\left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{A}p^{lr}(1-p)^{mr-lr}dp=1. \end{aligned}$$

Hence:

$$\begin{aligned} f(\theta )-\delta&\le \underset{r\rightarrow \infty }{LimInf}\left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{0}^{1}p^{lr}(1-p)^{mr-lr}f(p)dp\\ &\underset{r\rightarrow \infty }{\le LimSup}\left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{0}^{1}p^{lr}(1-p)^{mr-lr}f(p)dp\\ &\le f(\theta )+\delta {\text{ for any }}\delta >0, \end{aligned}$$

and therefore:

$$\begin{aligned} \underset{r\rightarrow \infty }{\lim }\left( mr+1\right) \left( {\begin{array}{c}mr\\ lr\end{array}}\right) \int _{0}^{1}p^{lr}(1-p)^{mr-lr}f(p)dp{\text { exists and is equal to }}f(\theta ). \end{aligned}$$