Stackelberg-Pareto Synthesis

We study the framework of two-player Stackelberg games played on graphs in which Player 0 announces a strategy and Player 1 responds rationally with a strategy that is an optimal response. While it is usually assumed that Player 1 has a single objective, we consider here the new setting where he has several. In this context, after responding with his strategy, Player 1 gets a payoff in the form of a vector of Booleans corresponding to his satisfied objectives. Rationality of Player 1 is encoded by the fact that his response must produce a Pareto-optimal payoff given the strategy of Player 0. We study for several kinds of ω-regular objectives the Stackelberg-Pareto Synthesis problem which asks whether Player 0 can announce a strategy which satisfies his objective, whatever the rational response of Player 1. We show that this problem is fixed-parameter tractable for games in which objectives are all reachability, safety, Büchi, co-Büchi, Boolean Büchi, parity, Muller, Streett or Rabin objectives. We also show that this problem is \(\mathsf {NEXPTIME} \)-complete except for the cases of Büchi objectives for which it is \(\mathsf {NP} \)-complete and co-Büchi objectives for which it is in \(\mathsf {NEXPTIME} \) and \(\mathsf {NP} \)-hard. The problem is already \(\mathsf {NP} \)-complete in the simple case of reachability objectives and graphs that are trees.