Winning coalitions in plurality voting democracies
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Winning coalitions in plurality voting democracies René van den Brink1 · Dinko Dimitrov2 · Agnieszka Rusinowska3 Received: 3 November 2019 / Accepted: 19 September 2020 © The Author(s) 2020
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 deci‑ sive. 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 characteriza‑ tions of the winning coalitions in the corresponding partitions are provided as well.
1 Introduction In parliamentary voting, 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 one party received the majority of
We are grateful to Francis Bloch and Anirban Kar for fruitful discussions, as well as to four anonymous referees for their helpful comments and suggestions. * Dinko Dimitrov [email protected]‑saarland.de René van den Brink [email protected] Agnieszka Rusinowska agnieszka.rusinowska@univ‑paris1.fr 1
Department of Econometrics and OR, and Tinbergen Institute, VU University, De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands
2
Chair of Economic Theory, Saarland University, Campus C3 1, 66123 Saarbrücken, Germany
3
CNRS‑Paris School of Economics, Centre d’Economie de la Sorbonne, 106‑112 Bd de l’Hôpital, 75647 Paris Cedex 13, France
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the votes, then it can be considered the winner of the election. However, very often there is no majority party, and two or more parties need to cooperate and make an agreement to form a (majority) government. Although in such situations there is no party that can guarantee for itself to be in the government, and thus it is not immedi‑ ately clear who is the winner of the election, usually immediately after the elections there is at least one party that claims to be the winner of the election. For instance, the party with the most votes/seats often declares itself as the winner of the elec‑ tion. But if other, smaller, parties form a majority coalition, then the party with the most votes might eventually be the ‘loser’. For example, if after election a right wing party received the most votes, but the left wing parties can form a majority coalition, then it might be that the largest, or most moderate, left wing party can be considered to be the
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