The game behind oriented percolation

We characterize the critical parameter of oriented percolation on Z² through the value of a zero-sum game. Specifically, we define a zero-sum game on a percolation configuration of Z², where two players move a token along the non-oriented edges of Z², collecting a cost of 1 for each edge that is open, and 0 otherwise. The total cost is given by the limit superior of the average cost. We demonstrate that the value of this game is deterministic and equals 1 if and only if the percolation parameter exceeds pc, the critical exponent of oriented percolation. Additionally, we establish that the value of the game is continuous at pc. Finally, we show that for p close to 0, the value of the game is equal to 0.