Energy Games with Weight Uncertainty

An energy game is played between two players, modeling a resource-bounded system and its environment. The players take turns moving a token along a finite graph. Each edge of the graph is labeled by an integer, describing how its traversal affects the energy level of the system. The system wins if it never runs out of energy. We introduce and study energy games with weight uncertainty (EGWUs), where the exact updates to the energy level are not a-priori known to the system. Instead, an EGWU specifies, for some subsets of edges, upper and lower bounds for their joint weight. EGWUs thus model settings in which there is only an estimation of the effect of some actions or sets of actions on the energy level, for example due to uncertainty about road conditions for an autonomous car or about the location of docking stations for a robot patrolling a warehouse. The system wins an EGWU if it has a strategy to never run out of energy, no matter what the weights are, as long as they respect the bounds. The environment wins if there are weights that respect the bounds with which it can cause the system to run out of energy. Unlike uncertainty about the exact location of the token, which persists during the interaction, the weight of an edge is revealed upon its traversal. The fact the system learns the weights during the interaction makes EGWUs interesting, and we study the memory required to the players, determinacy of the game, and the possibility of coping with uncertainty by a larger initial energy. We give tight complexity bounds to the problems of deciding whether the system or the environment wins, and we study the effect of parameters like the richness of the function estimating the weights, or the distribution of control along the interaction.