Abstract

We consider plurality voting games being simple games in partition function form such that in every partition there is at least one winning coalition. Such a game is said to be weighted if it is possible to assign weights to the players in such a way that a winning coalition in a partition is always one for which the sum of the weights of its members is maximal over all coalitions in the partition. A plurality game is called decisive if in every partition there is exactly one winning coalition. We show that in general, plurality games need not be weighted, even not when they are decisive. After that, we prove that (i) decisive plurality games with at most four players, (ii) majority games with an arbitrary number of players, and (iii) decisive plurality games that exhibit some kind of symmetry, are weighted. Complete characterizations of the winning coalitions in the corresponding partitions are provided as well.

Highlights

  • A plurality voting system is a system where each voter can vote for one party, and the seats in parliament are allocated to the parties depend‐ ent on the cast votes

  • If after election a right wing party received the most votes, but the left wing parties can form a majority coalition, it might be that the largest, or most moderate, left wing party can be considered to be the winner of the election

  • We introduce plurality games as a special type of simple games in partition func‐ tion form

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Summary

Introduction

A plurality voting system is a system where each voter can vote for one party, and the seats in parliament are allocated to the parties depend‐ ent on (usually proportional to) the cast votes. If after election a right wing party received the most votes, but the left wing parties can form a majority coalition, it might be that the largest, or most moderate, left wing party can be considered to be the winner of the election In such situations, whether a coalition is winning or losing might depend on the way how players outside the coalition are organized into coalitions. We assume that (i) a winning coalition cannot become losing when it grows, and (ii) there are negative externalities of other coalitions growing in the sense that bigger outside coalitions give ‘more resistance’ and outside coali‐ tions becoming bigger cannot turn a losing coalition into a winning one.1 Within this model of (monotonic) plurality games, we study the possibility of assigning weights to players (parties) such that a winning coalition in a partition is always one that has the maximal sum of its players’ weights over all coalitions in the partition. All proofs are collected in Appendix A (proofs from Sect. 3), Appendix B (proofs from Sect. 4), and Appendix C (proofs from Sect. 5)

Plurality games
Games with at most four players
Majority games and symmetric players
Related literature

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