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. © Springer-Verlag Berlin Heidelberg 2007.