  We introduce a natural variant of weighted voting games, which we refer to as
k-Prize Weighted Voting Games. Such games consist of n players with weights,
and k prizes, of possibly differing values. The players form coalitions, and
the i-th largest coalition (by the sum of weights of its members) wins the i-th
largest prize, which is then shared among its members. We present four solution
concepts to analyse the games in this class, and characterise the existence of
stable outcomes in games with three players and two prizes, and in games with
uniform prizes. We then explore the efficiency of stable outcomes in terms of
Pareto optimality and utilitarian social welfare. Finally, we study the
computational complexity of finding stable outcomes.
