This paper presents a game theoretic formulation of a graph traversal problem, with applications to robots moving through hazardous environments in the presence of an adversary, as in military and security applications. The blue team of robots moves in an environment modeled by a time-varying graph, attempting to reach some goal with minimum cost, while the red team controls how the graph changes to maximize the cost. The problem is formulated as a stochastic game, so that Nash equilibrium strategies can be computed numerically. Bounds are provided for the game value, with a guarantee that it solves the original problem. Numerical simulations demonstrate the results and the effectiveness of this method, particularly showing the benefit of mixing actions for both players, as well as beneficial coordinated behavior, where blue robots split up and/or synchronize to traverse risky edges.

In their seminal work, Nayyar et al. (2013) showed that imperfect information can be abstracted away from common-payoff games by having players publicly announce their policies as they play. This insight underpins sound solvers and decision-time planning algorithms for common-payoff games. Unfortunately, a naive application of the same insight to two-player zero-sum games fails because Nash equilibria of the game with public policy announcements may not correspond to Nash equilibria of the original game. As a consequence, existing sound decision-time planning algorithms require complicated additional mechanisms that have unappealing properties. The main contribution of this work is showing that certain regularized equilibria do not possess the aforementioned non-correspondence problem -- thus, computing them can be treated as perfect-information problems. Because these regularized equilibria can be made arbitrarily close to Nash equilibria, our result opens the door to a new perspective to solving two-player zero-sum games and yields a simplified framework for decision-time planning in two-player zero-sum games, void of the unappealing properties that plague existing decision-time planning approaches.

Isolation (or Isola) is a turn-based strategy board game where two players try to confine their opponent on a 7x7 checker-like board. Eventually, they can no longer make a move (thus isolating them). Each player has one piece, which they can move around like a queen in chess -- up-down, left-right, and diagonal. In the above picture, you can see from the black squares that both players have placed their pieces on various parts of the board. But as the game progressed, it shows that the yellow player still has three possible moves (up and to the right, right one square, and right two squares).