  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 generalize
Shapley's example by introducing an external parameter. We show that the
periodic behaviour in Shapley's example at some critical parameter value
disintegrates into unpredictable (chaotic) behaviour, with players dithering a
huge number of times between different strategies. At a further critical
parameter the dynamics becomes periodic again and both players aim to play one
ahead of the other.
  We study the dynamics of a two player continuous time bimatrix fictitious
play with the dynamics of a one-parameter family of $3 \times 3$ games that
includes a well-known example of Shapley's as a special case. In this paper we
adopt a geometric (dynamical systems) approach and study the bifurcations of
simple periodic orbits. Here we concentrate on the periodic behaviour, while in
a sequel we shall describe the chaotic behaviour.
