# Evasive maneuver game

First of all, I want to state that I'm not an expert in the game theory and searching the references for the game I just made up. Solving this game by itself seems like a decent project for strong highschool / undergrad students and I would like to suggest it to some people, but I'm not sure whether anything is known about it already and need to familiarize with the topic.

The game is a zero-sum probabilistic game with two players: runner and catcher.

Runner moves in some space, say a line or a plane with limited speed $$v$$ and arbitrary direction.

Catcher knows the exact position of a runner. Once catcher can declare some circle of radius $$R$$, and after the time $$t$$ passes, if the runner is in the circle, the catcher wins. Otherwise, runner wins.

Runner doesn't know the declaration moment.

We also state that if for $$T >> t$$ catcher does not declare anything, runner wins automatically (but I think that runner's behavior should actually have some limit for $$T \rightarrow \infty$$).

I've tried to tackle the problem for the following discrete model: the time is discrete, runner moves 1 step left or right each turn, catcher declares 1 possible position, $$t = 2$$ turns. Then, there is a strategy for a runner which allows to escape with probability $$\frac{\sqrt{5} - 1}{2} \approx 0.62$$ independent on a catcher behavior. I believe it is optimal but didn't prove it fully. Note that escape with probability $$> \frac{2}{3} \approx 0.66$$ is not possible due to the following catcher strategy: choose $$3$$ possible positions of a runner on a second turn with equal probability.

Note also that the symmetric random walk allows the escape with probability $$\frac{1}{2}$$ which is much worse.

The discrete approximation problem you mention was proposed by Isaacs, and solved independently by Dubins [1] and Karlin [2]. These papers show that the runner has a unique optimal strategy which depends on the previous move only and yields escape probability $$p=\frac{\sqrt{5} - 1}{2}$$. Conversely, for every $$\epsilon>0$$, the catcher has a strategy that succeeds with probability at least $$1-p-\epsilon=(3-\sqrt{5})/2 -\epsilon$$. However, the catcher does not have an optimal strategy. A generalization is described in [3]. This problem is related to Pursuit-Evasion type game on graph ("Flyswatter game") Isaacs' original motivation involved continuum games of the type you describe, which are considered in the theory of differential games.