  We propose an analytically tractable variation of the minority game in which
rational agents use probabilistic strategies. In our model, $N$ agents choose
between two alternatives repeatedly, and those who are in the minority get a
pay-off 1, others zero. The agents optimize the expectation value of their
discounted future pay-off, the discount parameter being $\lambda$. We propose
an alternative to the standard Nash equilibrium, called co-action equilibrium,
which gives higher expected pay-off for all agents. The optimal choice of
probabilities of different actions are determined exactly in terms of simple
self -consistent equations. The optimal strategy is characterized by $N$ real
parameters, which are non-analytic functions of $\lambda$, even for a finite
number of agents. The solution for $N \leq 7$ is worked out explicitly
indicating the structure of the solution for larger $N$. For large enough
future time horizon, the optimal strategy switches from random choice to a
win-stay lose-shift strategy, with the shift probability depending on the
current state and $\lambda$.
