InterJournal Complex Systems, 307
Status: Submitted
Manuscript Number: [307]
Submission Date: 990911
Comment on manuscript revision number 14517
Referee report for MS 251
Author(s): David A. Meyer

Subject(s): CX.14, CX.44

Category: Brief Article


I enjoyed reading this well written and interesting paper. In it the authors describe and analyze some simulations of the Minority Game, in which an odd number N of agents repeatedly select between two alternatives, seeking to be in the minority at each timestep. The agents make their choices by comparing each of a set of private strategies (maps from sequences of m minority outcomes to an m+1st predicted outcome) to the historical record and following the strategy which (naively) would have performed the best.

The authors explain the game carefully and take the time to motivate it as a nontrivial, but still analyzable, example of a complex adaptive system. This is important, since without analyzing concrete models it is all too easy to lapse into essentially poetic discourse.

The results the authors describe include two "phases", according to the value of the parameter 2m/N: when this is much smaller than some critical value of order 1, the agents exhibit "herding behavior" which results in relatively poor utilization of the possible resources; when it is much larger than the critical value, the agents are well coordinated, but do no better than they would choosing randomly; only near the critical point is the system efficacious.

I believe that this paper should be accepted for InterJournal and also for publication in the ICCS2 Proceedings. Let me make a few relatively minor comments the authors should consider as they revise it into the format required for the Proceedings (It is not currently in that format, and may also be over the recommended length.). In order of roughly decreasing importance:

I described the procedure used by the agents to select their strategy at each timestep as naive; by this I mean that they evaluate how well each strategy would have done historically without considering their own "market impact", i.e., how their use of that strategy would have changed the sequence of outcomes. Of course, that information is not recorded in the sequence of bits encoding the history, but it is important to point out that this is what is going on in the game as I believe that use of the more sophisticated procedure would change the results significantly.

I'm curious to what extent the so-called phase transition is a phase transition in the the technical sense. Specifically, presumably the sharpness of the transition should increase with the number of agents. Perhaps the authors could comment on potential approaches toward proving that there is a phase transition in this system.

The references should be updated to reflect publication of several of the preprints listed and to include the relevant among the many recent papers on this subject.

In the first paragraph of Section 4 there is a typo: it should be "low m phase", not "low mc phase".

I'm not sure the quantity plotted in Figure 5 should be called the "information"; this is usually reserved for the quantity involving the sum of the products of the individual probabilities and their logs.

Finally, let me reiterate that the authors have described interesting work and have discussed it thoughtfully. This is particularly apparent in the the concluding section, in which they emphasize the dynamic nature of the Minority Game (which puts their work in the minority of nonequilibrium analyses of game- and decision-theoretic problems!), draw the connection with spin-glass models, and comment on implications for the epistemology of complex systems. I am pleased to have had the opportunity to review this paper.

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