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We present a spherical version of the grand-canonical minority game (GCMG),and solve its dynamics in the stationary state. The model displays severaltypes of transitions between multiple ergodic phases and one non-ergodic phase.We derive analytical solutions, including exact expressions for the volatility,throughout all ergodic phases, and compute the phase behaviour of the system.In contrast to conventional GCMGs, where the introduction of memory-lossprecludes analytical approaches, the spherical model can be solved also whenexponential discounting is taken into account. For the case of homogeneousincentives to trade epsilon and memory loss rates rho, an efficient phase isfound only if rho=epsilon=0. Allowing for heterogeneous memory-loss rates wefind that efficiency can be achieved as long as there is any finite fraction ofagents which is not subject to memory loss.

Spherical grand-canonical minority games with and without score discounting