  Concurrent multi-player mean-payoff games are important models for systems of
agents with individual, non-dichotomous preferences. Whilst these games have
been extensively studied in terms of their equilibria in non-cooperative
settings, this paper explores an alternative solution concept: the core from
cooperative game theory. This concept is particularly relevant for cooperative
AI systems, as it enables the modelling of cooperation among agents, even when
their goals are not fully aligned. Our contribution is twofold. First, we
provide a characterisation of the core using discrete geometry techniques and
establish a necessary and sufficient condition for its non-emptiness. We then
use the characterisation to prove the existence of polynomial witnesses in the
core. Second, we use the existence of such witnesses to solve key decision
problems in rational verification and provide tight complexity bounds for the
problem of checking whether some/every equilibrium in a game satisfies a given
LTL or GR(1) specification. Our approach is general and can be adapted to
handle other specifications expressed in various fragments of LTL without
incurring additional computational costs.
