Abstract
We study the issue of assigning weights to players that identify winning coalitions in plurality voting democracies. For this, we consider plurality games which are 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 precisely supportive if it is possible to assign weights to players in such a way that a coalition being winning in a partition implies that the combined weight of its members is maximal over all coalitions in the partition. A plurality game is decisive if in every partition there is exactly one winning coalition. We show that decisive plurality games with at most four players, majority games with an arbitrary number of players, and almost symmetric decisive plurality games with an arbitrary number of players are precisely supportive. Complete characterizations of a partition's winning coalitions 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
Summary
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)
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