Minimal Representations for Majority Games

Abstract

This paper presents some new results about majority games. Isbell (1959) was the first to find a majority game without a minimum normalized integer representation; he needed 12 voters to construct such a game. Since then, it has been an open problem to find the minimum number of voters of a majority game without a minimum normalized integer representation. Our main new results are:1. All majority games with less than 9 voters have a minimum integer representation.2. For 9 voters, there are 14 majority games without a minimum integer representation, but all these games admit a minimum normalized integer representation.3. For 10 voters, there exist majority games with neither a minimum integer representation nor a minimum normalized integer representation.KeywordsSimple gamesWeighted gamesMajority gamesMinimum and minimal weighted representations/realizationsComputing games