Strategy switches and co-action equilibria in a minority game

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 λ. 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 λ, even for a finite number of agents. The solution for N≤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 λ.