On the transition to efficiency in Minority Games

It is now well known that Minority Games (MGs) can display two types of phase transitions separating ergodic phases from a non-ergodic regimes [1][2][3]. One type of transition is characterized by a diverging susceptibility signalling the existence of an informationally efficient phase with vanishing predictability of the bid time-series [4,5]. The second type of transition (referred to as memory-onset transition) occurs instead at finite integrated response and is marked by a de Almeida-Thouless-instability and a replica-symmetry broken phase at non-zero predictability, similar to phase transitions observed in models of spin glasses [6,7]. From a physical viewpoint, the divergence of static and dynamic susceptibilities signals a sensitivity of the MG-dynamics to perturbations in the stationary state. A geometric interpretation of this phenomenon has been devised in [8], and rests on the observation that the microscopic N -dimensional state vector describing the system of N interacting agents evolves in an αN -dimensional space spanned by the quenched disorder of the problem. Here α is the key control parameter of the model. The analysis of the MG shows that the effective dimension of phase space is reduced to (1 -φ)N as a fraction φ ≡ φ(α) of agents 'freezes' during the course of the dynamics: frozen agents are those who use just one of their strategies in the long run, so that their degrees of freedom are effectively removed from the dynamics. The breakdown of ergodicity is observed to occur at some α = α c satisfying the condition [1 -φ(α c )]N = α c N , i.e. it occurs when the dimensionality of the reduced phase space becomes equal to that of the space in which the effective dynamics is defined.

The aim of the present paper is to test variations of the MG, in which the above geometric picture is systematically modified, for the existence or otherwise of phase transitions at diverging integrated response and for the presence of efficient phases associated with this type of transition. Such alterations of the model occur naturally when an additional degree of heterogeneity (besides the quenched randomness of the strategies) is added to the agents’ learning rules, and appear to be a key ingredient of more realistic models of the learning of agents. They are indeed very much in the spirit of David Sherrington’s approach to Minority Games. David suggested the addition of heterogeneity and complexity, and to study their effects on the phase behaviour of the MG in numerous discussions as well as in earlier joint articles with the authors of the present paper [9,10], and it is a pleasure to submit work along this line to the special issue in honour of David’s 65th birthday. Specifically, we will here consider MGs with agent-dependent impact correction, grandcanonical MGs [11] with heterogeneous incentives to trade, and El-Farol type games with heterogeneous comfort levels [12,13]. We demonstrate that in order to observe a transition to an efficient phase, it is for a large class of MGs necessary, and in absence of memory onset transitions also sufficient, that the above geometric interpretation holds for a finite fraction of the agents. In turn, no efficient phase occurs when no such group of agents exists.

Batch MGs [5] are discrete zero-temperature dynamical systems describing the coupled time-evolution of N agents, labelled by i = 1, . . . , N in the following. Each agent, in the simplest setup, is described by one continuous dynamical variable q i (t) which evolves according to the following rules:

α is here a finite positive control parameter while {ξ µ i , ω µ i } are quenched random variables, usually drawn independently from the set {-1, 0, 1} with weights 1/4, 1/2, 1/4, satisfying ω µ i ξ µ i = 0. h i (t) is an external perturbation field, used to measure the response of the system, and will be set to zero eventually. These equations can be derived from a finance-inspired setup that has been discussed at length in the literature (see e.g. [2]) and we shall not repeat it here in detail. In a nutshell, agents’ choices are encoded in the Ising spins s i (t), whose possible values represent the two (heterogeneous) trading strategies of which every agent disposes. q i (t) is then a ‘valuation’ by which agent i assesses the performance of his strategies, so that if q i (t) → ±∞ asymptotically, then agent i will stick to one of his strategies (s i (t) → ±1) when t → ∞. Otherwise, he will keep switching strategies forever. These two types of agents are referred to as ‘frozen’ and ‘fickle’, respectively. Note that frozen agents are insensitive to (small) perturbations of the dynamics in the steady state. µ denotes the state of the world and may take on P = αN values. The quantity A µ represents in turn the bid imbalance (’excess demand’) in state µ. Efficient states are characterized by zero bid imbalance (on average) and zero predictability. In an efficient phase no statistical forecast of a bid imbalance i

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