Trapping games on random boards

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 analyse 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.

Percolation games, probabilistic cellular automata, and the hard-core model

“…Undirected lattices. The following game is considered in [BHMW16]. Each site of Z d is independently a trap with probability p, and two players alternately move a token.…”

“…A player who cannot move loses. This game is closely related to maximum matchings, and this is used in [BHMW16] to derive results for biased variants in which odd and even sites have different percolation parameters. However, for the unbiased version described above it is unknown whether there exist any p > 0 and d ≥ 2 for which draws occur with positive probability.…”

“…As the size of this region tends to ∞, the probability of a draw starting from the origin converges to the probability of a draw on Z 2 ; the question is whether this limiting probability is positive for any p and q. Related questions are considered in [HM] and [BHMW16], in which the underlying graph is respectively a Galton-Watson tree, and a random subset of the lattice with undirected moves. In our case of a random subset of Z 2 with directed moves, the outcome (first-player win, first-player loss, draw) of the game started from each site can be interpreted in terms of the evolution of a certain one-dimensional discrete-time probabilistic cellular automaton (PCA); the state of the PCA at a given time relates to the outcomes associated to the sites on a given Northwest-Southeast diagonal of Z 2 .…”

Galton–Watson games

“…However, survival is not sufficient for the existence of a draw-intuitively, that requires not just an infinite path, but an infinite path that neither player can profitably deviate from. Indeed, we will find that the draw and escape probabilities D,D, E (1) , E (2) undergo phase transitions as p is varied, but typically not at the same location as the survival phase transition.…”

“…LetÑ,P,D be the analogous probabilities for the misère game. For the escape game, let S (1) (respectively, S (2) ) be the probability that the stopper wins assuming the stopper has the first (respectively, second) move. Similarly let E (1) = 1 − S (2) and E (2) = 1 − S (1) be the win probabilities for the escaper when moving first or second, respectively.…”