  We discuss the strategy that rational agents can use to maximize their
expected long-term payoff in the co-action minority game. We argue that the
agents will try to get into a cyclic state, where each of the $(2N +1)$ agent
wins exactly $N$ times in any continuous stretch of $(2N+1)$ days. We propose
and analyse a strategy for reaching such a cyclic state quickly, when any
direct communication between agents is not allowed, and only the publicly
available common information is the record of total number of people choosing
the first restaurant in the past. We determine exactly the average time
required to reach the periodic state for this strategy. We show that it varies
as $(N/\ln 2) [1 + \alpha \cos (2 \pi \log_2 N)$], for large $N$, where the
amplitude $\alpha$ of the leading term in the log-periodic oscillations is
found be $\frac{8 \pi^2}{(\ln 2)^2} \exp{(- 2 \pi^2/\ln 2)} \approx
{\color{blue}7 \times 10^{-11}}$.
