  We give an example of a three-person deterministic graphical game that has no
Nash equilibrium in pure stationary strategies. The game has seven positions,
four outcomes (a unique cycle and three terminal positions), and its normal
form is of size 2 x 2 x 4 only. Thus, our example strengthens significantly the
one obtained in 2014 by Gurvich and Oudalov; the latter has four players, five
terminals, and a 2 x 4 x 6 x 8 normal form. Furthermore, our example is minimal
with respect to the number of players. Both examples are tight but not
Nash-solvable. Such examples were known since 1975, but they were not related
to deterministic graphical games. Moreover, due to the small size of our
example, we can strengthen it further by showing that it has no Nash
equilibrium not only in pure but also in independently mixed strategies, for
both Markovian and a priori evaluations.
