Games with Weighted Multiple Objectives

Games with multiple objectives arise naturally in synthesis of reactive systems. We study games with weighted multiple objectives. The winning objective in such games consists of a set α of underlying objectives, and a weight function w:2^α→IN that maps each subset S of α to a reward earned when exactly all the objectives in S are satisfied. The goal of a player may be to maximize or minimize the reward. As a special case, we obtain games where the goal is to maximize or minimize the number of satisfied objectives, and in particular satisfy them all (a.k.a. generalized conditions). A weight function allows for a much richer reference to the underlying objectives: prioritizing them, referring to desired and less desired combinations, and addressing settings where we cannot expect all sub-specifications to be satisfied together. We focus on settings where the underlying objectives are all Büchi, co-Büchi, reachability, or avoid objectives, and the weight function is non-decreasing (a.k.a. free disposal). For each of the induced classes (that is, type of underlying condition, type of optimization, and type of weight function), we solve the problem of deciding the game and analyze its tight complexity. We also study the tight memory requirements for each of the players. Finally, we consider general weight functions, which make the setting similar to the one of Boolean Muller objectives.