Draws in the undirected trapping game on site-percolated lattices

Independently delete each vertex of the nearest-neighbor lattice $\Z^d$ with probability $p$. Two players, knowing the entire resulting board, move a token alternately to an undeleted neighbor that has never been visited; a player with no legal move loses. A draw means that neither player can force a win. We prove that the draw probability is zero in dimension two for every $0<p<1$, whereas in every dimension $d\ge3$ it is positive for some strictly positive deletion probability. The proof expresses the outcome through the effect of distant boundary conditions on maximum independent sets. In the plane, a deterministic boundary bound is incompatible with the fluctuations that a positive draw probability would produce. In higher dimensions, contour shifts and a uniform comparison of permuted deletion configurations retain the influence of the two checkerboard boundary conditions. The comparison is performed for all fixed geometries before a contour is selected from the random board.