On Some Evacuation Games with Random Walks

Abstract

We consider a single-player game where a particle on a board has to be steered to evacuation cells. The actor has no direct control over this particle but may indirectly influence the movement of the particle by blockades. We examine optimal blocking strategies and the recurrence property experimentally and conclude that the random walk of our game is recurrent. Furthermore, we are interested in the average time in which an evacuation cell is reached.

Appendix

1.1 Method of Monotonous Iterations

Given an equation system (E) of the form $x =f(x)$ and a starting vector $x^{(0)}$ we compute $x^{(i+1)}$ by

$$\begin{aligned} x^{(i+1)}=f(x^{(i)}). \end{aligned}$$

This method converges towards a solution, for instance, if the following conditions hold

• 1. (E) has a unique solution, and

• 2. the sequence $x^{(i)}$ is monotonically increasing in all coordinates and has an upper bound in each coordinate.

The first condition holds for the problems outlined in this paper.

The second condition depends on the starting vector $x^{(0)}$. Only a good starting vector will lead to the solution of (E). A good solution for our problems is $x^{(0)}=(0,0,\dots ,0)$ for which the second property is fulfilled.