Stackelberg-Pareto Synthesis (Extended Version)

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 $\omega$-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, B\"uchi, co-B\"uchi, Boolean B\"uchi, parity, Muller, Streett or Rabin objectives. We also show that this problem is NEXPTIME-complete except for the cases of B\"uchi objectives for which it is NP-complete and co-B\"uchi objectives for which it is in NEXPTIME and NP-hard. The problem is already NP-complete in the simple case of reachability objectives and graphs that are trees.