Simple games cover voting systems in which a single alternative, such
as a bill or an amendment, is pitted against the status quo. A simple game
or a yes–no voting system is a set of rules that specifies exactly which
collections of “yea” votes yield passage of the issue at hand, each of these
collections of “yea” voters forms a winning coalition. We are interested in
performing a complexity analysis on problems defined on such families of
games. This analysis as usual depends on the game representation used as
input. We consider four natural explicit representations: winning, losing,
minimal winning, and maximal losing. We first analyze the complexity of
testing whether a game is simple and testing whether a game is weighted.
We show that, for the four types of representations, both problems can be
solved in polynomial time. Finally, we provide results on the complexity
of testing whether a simple game or a weighted game is of a special type.
We analyze strongness, properness, decisiveness and homogeneity, which
are desirable properties to be fulfilled for a simple game. We finalize
with some considerations on the possibility of representing a game in a
more succinct representation showing a natural representation in which
the recognition problem is hard.
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