## Weighted Voting, Threshold Functions, and Zonotopes

## Catherine Stenson (Juniata College)

#### Abstract :

Many voting systems consist of a set of players who can form coalitions; a winning coalition is one that can pass a measure. The winning and losing coalitions can be given by several classes of functions, including switching functions, threshold functions, weighted voting systems, and simple games. Polytopes, and in particular zonotopes, provide a nice perspective on these functions. We define a zonotope $T_n$ related to these functions and describe its coordinates in terms of a measure of voting power. We also give a voting interpretation of the coordinates of the derived zonotope $\bar{T}_n$. This talk does not require any previous knowledge of voting theory or of zonotopes.