From bocce to positivity: some probabilistic linear algebra

TL;DR

This paper explores probabilistic questions in linear algebra and game theory inspired by bocce, providing methods to compute the likelihood of positive solutions and game outcomes under certain assumptions.

Contribution

It introduces a novel connection between geometric probability in bocce and probabilistic analysis of linear systems and zero-sum games, deriving explicit probability formulas.

Findings

Derived probabilities for positive solutions of linear systems

Calculated likelihood of row player advantage in zero-sum games

Established assumptions under which these probabilities hold

Abstract

A question in geometric probability about the location of the balls in a game of bocce leads to related questions about the probability that a system of linear equations has a positive solution and the probability that a random zero-sum game favors the row player. Under reasonable assumptions we are able to find these probabilities.