Synthesis with Rational Environments

Synthesis is the automated construction of a system from its specification. The system has to satisfy its specification in all possible environments. The environment often consists of agents that have objectives of their own. Thus, it makes sense to soften the universal quantification on the behavior of the environment and take the objectives of its underlying agents into an account. Fisman et al. introduced rational synthesis: the problem of synthesis in the context of rational agents. The input to the problem consists of temporal-logic formulas specifying the objectives of the system and the agents that constitute the environment, and a solution concept (e.g., Nash equilibrium). The output is a profile of strategies, for the system and the agents, such that the objective of the system is satisfied in the computation that is the outcome of the strategies, and the profile is stable according to the solution concept; that is, the agents that constitute the environment have no incentive to deviate from the strategies suggested to them.

In this paper we continue to study rational synthesis. First, we suggest an alternative definition to rational synthesis, in which the agents are rational but not cooperative. In the non-cooperative setting, one cannot assume that the agents that constitute the environment take into account the strategies suggested to them. Accordingly, the output is a strategy for the system only, and the objective of the system has to be satisfied in all the compositions that are the outcome of a stable profile in which the system follows this strategy. We show that rational synthesis in this setting is 2ExpTime-complete, thus it is not more complex than traditional synthesis or cooperative rational synthesis. Second, we study a richer specification formalism, where the objectives of the system and the agents are not Boolean but quantitative. In this setting, the goal of the system and the agents is to maximize their outcome. The quantitative setting significantly extends the scope of rational synthesis, making the game-theoretic approach much more relevant.