Real or real histories: what difference does it make? :

Introduction

At each time step t , all the agents are given the same bit string mu(t) that encode the M last outcomes (0=A won, 1=B won). The quantity M is called the memory of agents. As a dynamical variable, mu(t) moves on a graph called the De Bruijn graph of order M, which looks like this for M=3:

De Bruijn graph of order 3

This dynamics seems to make any analytical approach much more difficult, because mu(t) is not markovian. The idea of Cavagna is to replace the real history mu(t) with a random number, drawn from a uniform distribution. In this case, a random history is better called a common piece of information, and the fact that agents' behavior depends on a completely random information is an illustration of the sun's spot effect.

The question is: do we need real histories at all, i.e. what does it change?

In the symmetric phase, it destroys the antipersistent behavior. This has no influence on simple quantities such as the fluctuations. However, the autocorrelation function for instance changes completely: it loses its periodic behavior.
In the asymmetric phase, it is does, but not much (about 10%):

Fluctuations for real (filled circles) and random (void circles) histories. Continous curves are from the replica calculus. alpha=P/N

Indeed, these values only depend on the histories' frequency distribution function. In the symmetric phase (N/P<0.3347...), the MG's agents manage to produce a uniform distribution, but not in the asymmetric phase (N/P>0.3374). There is a way of finding self-consitently the distribution of the frequency in the asymmetric phase.

For many extension of the MG, the nature of the histories has a big influence even on the outcome of simple quantities. For instance, if agents take even slightly account of their impact, the difference is more pronounced (about 50%):

Fluctuations for agents accounting for their impact (epsilon=0.1)

Even more, if agents account for their exact impact on the game, the difference is of one order of magnitude.

Real histories have a meaning: they represent a quantity which makes sense, and sometimes, this sense is needed for what one wants to modelize. Indeed, consider agents with M=3, and add agents with M=4. In this case, temporal correlations in the real histories do really matter. There are numerous examples of MGs whose properties depend on the nature of histories.

So, yes, random or real histories do really matter; the difference goes beyond several percents, except for time averaged simple macroscopic quantities of the standard MG with naive agents.

In fact whether to take real or random histories depends on what one wants to modelize: if the meaning of the histories is important then consider real histories. If you want to modelize random news arrival, switch to random histories, and your model will be easier to solve.

Papers related to this topic are:

A. Cavagna, Irrelevance of Memory in the Minority Game, PRE (1998), preprint
N. F. Johnson et al., Enhanced winnings in a mixed ability population playing a minority game, (1999) preprint
D. Challet and M. Marsili, Relevance of Memory in Minority Games, (2000) preprint
D. Zheng , B.-H. Wang, Statistical Properties of the Attendance Time Series in the Minority Game (2001), preprint
Ch.-Y. Lee, Is Memory in the Minority Game Relevant?, PRE 64,015105R (2001)
R. Metzler, Antipersistent Binary Time Series (2001), preprint