# k-Prize Weighted Voting Games

## TL;DR

This paper introduces k-Prize Weighted Voting Games, analyzing stable outcomes, their efficiency, and computational complexity, providing new insights into multi-prize coalition formations among weighted players.

## Contribution

It defines a new class of voting games with multiple prizes, characterizes stable outcomes, and studies their existence, efficiency, and computational complexity.

## Key findings

- Stable outcomes exist under certain conditions.
- Stable outcomes can be Pareto optimal or socially efficient.
- Computational complexity results for finding stable outcomes.

## Abstract

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.