Voting power and proportional representation of voters

Abstract

Our paper provides a justification for the proportional representative (PR) election system for politically diversified societies. We employ the Shapley value concept to measure the political power of parties in a parliament. We prove that for the PR system if parties' size add up to 1 and is uniformly distributed, the expected ratio of a party size to its political power converges to 1, and the variance decreases to 0 as the number of parties increases. The rate of convergence is high. An empirical evidence from the Netherlands elections supports our result. Using the Shapley-Owen index we obtain similar result (this time numerically only) for a voting model that takes into account ideological differences between parties and voters.

On the Democratic Weights of Nations

“…13 In particular, F i secondorder stochastically dominates distribution F j (or F j is a mean-preserving spread of F i ) if n i > n j because the sample median of n i independent draws from G has smaller variance than that of just n j < n i draws; and the respective draw from H adds identical 11 Asymptotic proportionality between weights and voting power has first been investigated by Penrose (1952) in the context of binary alternatives. Related formal results by Neyman (1982), Lindner and Machover (2004), Snyder et al (2005), Jelnov and Tauman (2012) and Theorem 1 below suppose that the relative weight of any given voter becomes negligible as more and more voters are added. The case when relative weights of a few large voters fail to vanish as m → ∞ -giving rise to oceanic games and typically non-proportionality -has been treated by Shapiro and Shapley (1978) and Dubey and Shapley (1979).…”

Worst-case Bounds on Power vs. Proportion in Weighted Voting Games with an Application to False-name Manipulation

“…A third approach focuses on the probabilistic modelling of the WVG weights. For example, Jelnov and Tauman (2014) show that if player weights are sampled uniformly from the unit simplex, the expected Shapley-Shubik power of a player relative to its proportion goes to 1 with rapid convergence in the number of players. Lindner and Machover (2004) study a different model where the ratio of the Shapley-Shubik index to proportional weight in infinite chains of game instances asymptotically approaches 1.…”

On Weights and Quotas for Weighted Majority Voting Games

“…One of the fundamental questions in the analysis of weighted voting games is to determine the relation among weight and power. This question has been addressed for particular classes of random weighted voting games obtained according to a fixed distribution of weights (see for example [36][37][38][39][40]). Having access to a good approximation to the real distribution of the player's weight might allow us to use the techniques on these papers to analyze the relation among power and weight on the complete set of weighted voting games.…”

“…For instance, all examples with 10 and 11 players given by Freixas and Molinero [29]. As an example, consider the following representations [68; 38,31,28,23,11,8,6,5,3,1] and [68; 37,31,28,23,11,8,7,5,3,1], they define the same game, and both are minimum sum and canonical.…”