A Generalization of the Minisum and Minimax Voting Methods

TL;DR

This paper introduces a flexible family of approval voting schemes based on p-norms, generalizing existing methods and offering a balanced approach to approvals and disapprovals, with practical advantages over traditional methods.

Contribution

It proposes a new family of voting methods using p-norms, unifying and extending minisum and minimax approaches, and explores their properties and extensions.

Findings

Small p values like 2 or 3 balance approvals and disapprovals effectively.

Large finite p values maximize voter coverage, outperforming minimax.

The methods can be extended to ternary voting systems.

Abstract

In this paper, we propose a family of approval voting-schemes for electing committees based on the preferences of voters. In our schemes, we calculate the vector of distances of the possible committees from each of the ballots and, for a given $ p $-norm, choose the one that minimizes the magnitude of the distance vector under that norm. The minisum and minimax methods suggested by previous authors and analyzed extensively in the literature naturally appear as special cases corresponding to $ p = 1 $ and $ p = \infty, $ respectively. Supported by examples, we suggest that using a small value of $ p, $ such as 2 or 3, provides a good compromise between the minisum and minimax voting methods with regard to the weightage given to approvals and disapprovals. For large but finite $ p, $ our method reduces to finding the committee that covers the maximum number of voters, and this is far superior to the minimax method which is prone to ties. We also discuss extensions of our methods to ternary voting.