Games on Random Boards
Riddhipratim Basu, Alexander E. Holroyd, James B. Martin, Johan, W\"astlund

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…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsStochastic processes and statistical mechanics · Markov Chains and Monte Carlo Methods · Theoretical and Computational Physics
