  In the 60's Shapley provided an example of a two player fictitious game with
periodic behaviour. In this game, player $A$ aims to copy $B$'s behaviour and
player $B$ aims to play one ahead of player $A$. In this paper we continue to
study a family of games which generalize Shapley's example by introducing an
external parameter, and prove that there exists an abundance of periodic and
chaotic behavior with players dithering between different strategies. The
reason for all this, is that there exists a periodic orbit (consisting of
playing mixed strategies) which is of {\em `jitter type'}: such an orbit is
neither attracting, repelling or of saddle type as nearby orbits jitter closer
and further away from it in a manner which is reminiscent of a random walk
motion. We prove that this behaviour holds for an open set of games.
