Games on Random Boards

TL;DR

This paper analyzes a two-player graph game on percolation clusters in Euclidean lattices, establishing conditions for the existence of winning strategies and the impact of boundary conditions on matchings.

Contribution

It provides new results on game outcomes on percolation clusters, including conditions for no draws and critical windows for advantage, linking game theory with percolation and matching theory.

Findings

No draws occur when one parity's sites are sufficiently rare.

A critical window for advantage depends on the size and percolation parameter.

Boundary conditions significantly influence maximum matchings.

Abstract

We consider the following two-player game on a graph. A token is located at a vertex, and the players take turns to move it along an edge to a vertex that has not been visited before. A player who cannot move loses. We analyze outcomes with optimal play on percolation clusters of Euclidean lattices. On Z^2 with two different percolation parameters for odd and even sites, we prove that the game has no draws provided closed sites of one parity are sufficiently rare compared with those of the other parity (thus favoring one player). We prove this also for certain d-dimensional lattices with d>=3. It is an open question whether draws can occur when the two parameters are equal. On a finite ball of Z^2, with only odd sites closed but with the external boundary consisting of even sites, we identify up to logarithmic factors a critical window for the trade-off between the size of the ball and the percolation parameter. Outside this window, one or other player has a decisive advantage. Our analysis of the game is intimately tied to the effect of boundary conditions on maximum-cardinality matchings.