# Positional Voting and Doubly Stochastic Matrices

## TL;DR

This paper explores the mathematical structure of positional voting systems using doubly stochastic matrices, providing elementary proofs, explicit constructions of paradoxical profiles, and methods to select weights for desired outcomes.

## Contribution

It introduces a simplified, matrix-based framework for analyzing positional voting systems, including explicit constructions and weight selection techniques.

## Key findings

- Elementary proofs for key results in positional voting theory
- Explicit construction of paradoxical voting profiles
- Method to choose weights for desired election outcomes

## Abstract

We provide elementary proofs of several results concerning the possible outcomes arising from a fixed profile within the class of positional voting systems. Our arguments enable a simple and explicit construction of paradoxical profiles, and we also demonstrate how to choose weights that realize desirable results from a given profile. The analysis ultimately boils down to thinking about positional voting systems in terms of doubly stochastic matrices.