Strategy switches and co-action equilibria in a minority game

TL;DR

This paper introduces an analytically solvable variation of the minority game where rational agents optimize probabilistic strategies, revealing a strategy shift from random to win-stay lose-shift as the future horizon increases.

Contribution

It develops a new co-action equilibrium concept and derives exact solutions for optimal strategies in a probabilistic minority game model.

Findings

Optimal strategies are characterized by self-consistent equations.

For small N, explicit solutions are provided.

Strategies switch from random to win-stay lose-shift with longer horizons.

Abstract

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 $\lambda$.