  Imitation is simple behavior which uses successful actions of others in order
to deal with one's own problems. Because success of imitation generally depends
on whether profit of an imitating agent coincides with those of other agents or
not, game theory is suitable for specifying situations where imitation can be
successful. One of the concepts describing successfulness of imitation in
repeated two-player symmetric games is unbeatability. For infinitely repeated
two-player symmetric games, a necessary and sufficient condition for some
imitation strategy to be unbeatable was specified. However, situations where
imitation can be unbeatable in multi-player games are still not clear. In order
to analyze successfulness of imitation in multi-player situations, here we
introduce a class of totally symmetric games called unexploitable games, which
is a natural extension of two-player symmetric games without exploitation
cycles. We then prove that, for infinitely repeated unexploitable games, there
exist unbeatable imitation strategies. Furthermore, we also prove that, for
infinitely repeated non-trivial unexploitable games, there exist unbeatable
zero-determinant strategies, which unilaterally enforce some relationships on
payoffs of players. These claims are demonstrated in the public goods game,
which is the simplest unexploitable game. These results show that there are
situations where imitation is unbeatable even in multi-player games.
