Lion and Man -- Can Both Win?

TL;DR

This paper investigates pursuit and evasion games in metric spaces, revealing conditions under which players have winning strategies, including surprising cases where both can win or neither can win.

Contribution

It demonstrates that in certain metric spaces both players can have winning strategies and constructs examples where neither player has a guaranteed winning strategy.

Findings

In compact metric spaces, at least one player has a winning strategy.

Examples are provided where both players have winning strategies.

A metric space is constructed where neither player has a winning strategy.

Abstract

This paper is concerned with continuous-time pursuit and evasion games. Typically, we have a lion and a man in a metric space: they have the same speed, and the lion wishes to catch the man while the man tries to evade capture. We are interested in questions of the following form: is it the case that exactly one of the man and the lion has a winning strategy? As we shall see, in a compact metric space at least one of the players has a winning strategy. We show that, perhaps surprisingly, there are examples in which both players have winning strategies. We also construct a metric space in which, for the game with two lions versus one man, neither player has a winning strategy. We prove various other (positive and negative) related results, and pose some open problems.