On Non-Ergodic Phases in Minority Games

Abstract

In the first part of this review, we survey the known analytic results regarding the non-ergodic phase of standard Minority Games. Such phases are characterized by the fact that, at odds with ergodic regimes, the steady state properties of the game (e.g. the volatility) depend both on the initial conditions chosen for the agents' learning process as well as on the learning rate. Secondly, we present a discussion of the effects of finite-memory learning, which lifts the non-ergodicity, in the context of spherical Minority Games.

Statistical mechanics of competitive resource allocation using agent-based models

“…After some lighter computations, one recovers all the equations of the replica calculus; in addition, one also can discuss in greater rigor the validity of simple expressions for σ 2 0 . The symmetric phase still resists full analysis [131], which prompted the introduction of a further simplified MG, of the spherical kind [65] (see Sec. II D).…”

“…It does however solve the problem of non-ergodicity of the symmetric phase since initial score valuations are gradually forgotten [41]. Unfortunately, it also has a great influence on analytical results, since an infinitesimal λ has so far prevented from obtaining any mathematical insight about the stationary state from generating functionals: they still yield the exact effective agent dynamics but nobody has found a way to extract information about the stationary state because there are no more frozen agents [41,131]. The spherical MG with payoff discounting is of course exactly solvable with this method [13,131].…”

“…Unfortunately, it also has a great influence on analytical results, since an infinitesimal λ has so far prevented from obtaining any mathematical insight about the stationary state from generating functionals: they still yield the exact effective agent dynamics but nobody has found a way to extract information about the stationary state because there are no more frozen agents [41,131]. The spherical MG with payoff discounting is of course exactly solvable with this method [13,131]. Replicas can be applied in some cases: Marsili et al [126] study an MG with impact and discounting; the quantity minimized by the dynamics is now…”

Spherical grand-canonical minority games with and without score discounting

“…Spherical MGs might therefore serve as a fully solvable starting point to which non-linear effects might be added in a perturbative fashion, maintaining analytical tractability as far as possible [14]. Spherical MGs with memory-loss have briefly been considered in [15], but to our knowledge no systematic study of the detailed phase behaviour and ergodicity properties of spherical grand-canonical MGs has been carried out. The paper is based on [16] and structured as follows: we will first introduce the spherical GCMG in Sec.…”