On the transition to efficiency in Minority Games

The existence of a phase transition with diverging susceptibility in batch Minority Games (MGs) is the mark of informationally efficient regimes and is linked to the specifics of the agents learning rules. Here we study how the standard scenario is affected in a mixed population game in which agents with the `optimal learning rule (i.e. the one leading to efficiency) coexist with ones whose adaptive dynamics is sub-optimal. Our generic finding is that any non-vanishing intensive fraction of optimal agents guarantees the existence of an efficient phase. Specifically, we calculate the dependence of the critical point on the fraction $q$ of `optimal agents focusing our analysis on three cases: MGs with market impact correction, grand-canonical MGs and MGs with heterogeneous comfort levels.