The Full Strategy Minority Game

The Full Strategy Minority Game ( F S M G ) is an instance of the Minority Game ( M G ) which includes a single copy of every potential agent. In this work, we explicitly solve the F S M G thanks to certain symmetries of this game. Furthermore, by considering the M G as a statistical sample of the F S M G , we compute approximated values of the key variable σ 2 / N in the symmetric phase for different versions of the M G . As another application we prove that our results can be easily modified in order to handle certain kinds of initial biased strategy scores, in particular when the bias is introduced at the agents’ level. We also show that the F S M G verifies a strict period two dynamics (i.e., period two dynamics satisfied with probability 1) giving, to the best of our knowledge, the first example of an instance of the M G for which this feature can be analytically proved. Thanks to this property, it is possible to compute in a simple way the probability that a general instance of the M G breaks the period two dynamics for the first time in a given simulation.