  We study the rational verification problem which consists in verifying the
correctness of a system executing in an environment that is assumed to behave
rationally. We consider the model of rationality in which the environment only
executes behaviors that are Pareto-optimal with regard to its set of
objectives, given the behavior of the system (which is committed in advance of
any interaction). We examine two ways of specifying this behavior, first by
means of a deterministic Moore machine, and then by lifting its determinism. In
the latter case the machine may embed several different behaviors for the
system, and the universal rational verification problem aims at verifying that
all of them are correct when the environment is rational. For parity
objectives, we prove that the Pareto-rational verification problem is
co-NP-complete and that its universal version is in PSPACE and both NP-hard and
co-NP-hard. For Boolean B\"uchi objectives, the former problem is
$\Pi_2\mathsf{P}$-complete and the latter is PSPACE-complete. We also study the
case where the objectives are expressed using LTL formulas and show that the
first problem is PSPACE-complete, and that the second is 2EXPTIME-complete.
Both problems are also shown to be fixed-parameter tractable (FPT) for parity
and Boolean B\"uchi objectives. Finally, we evaluate two variations of the FPT
algorithm proposed to solve the Pareto-rational verification problem on a
parametric toy example as well as on randomly generated instances.
