So far, there are 121 papers I am aware of. If one of yours is not listed here, please email me !


Articles related to the standard Minority Game :

By "standard", I mean Minority Game as first defined: global game, N agents, S strategies, virtual points [inductive agents], global game, global history.



The original paper:


Contains an easy-to-read introduction to the MG:


Numerical evidence that all the interesting quantities only depend on P/N (P=2M). First mention of the existence of a phase transition and first evidence of the presence of "information" in the "non-crowded" region:

A more complete and precise version of their previous paper ; extensive numerical results with qualitative explanations, analytical approach valid for P/N<<1.


Numerical study of the variance in the MG and in the Bar Problem (BP). It is shown that the variance of the BP has also a minimum:


Numerical evidence that the variance does not depend on the dynamics of the histories in the symmetric region (Note from me: this is not true in the asymmetric phase. See here for more details)

A response: the temporal correlation are relevant in the MG for instance in a population with various M is considered.

Self-consistent solution of MG with real histories; the dynamics of histories is non trivial and relevant in most cases. Only in the symmetric phase of the standard MG it is not, at least for the fluctuations.


Extensive numerical study of the symmetric phase and introduction of the majority model:


Introduction of the distance between players and the reduced strategy space, study of Darwinism effects, resource sharing, variance of the attendance, mixed memory population:


A study of the Chaitin algorithmic complexity of the minority sign, as well as the mutual information. In this preprint, N=201:


Intuitive analytical study of the variance of the attendance for the MG in the reduced strategy space and for the BP:


Details of the previous paper:

A try to apply to the original MG the method introduced in the previous preprint of N. F. Johnson  et al.:


Tries to understand the dynamics of a modified MG, in the crowded region, by ergordic invariants:


An adequate mathematical formalism is introduced. Formal connection between the MG and neural networks spin-glasses, finite size determination of the critical point, caracterization of the symmetry breaking and the two phases. Shows that the cause of cooperation is caused by the deterministic behavior of some agents for some histories: if agents have two opposite strategies, no cooperation arises:


A continuous MG (or mixed histories MG ?) ; introduction of exponential learning (Bolzman weights, Logit model) in the use of the strategies. The "temperature" reduces the fluctuations in the symmetric phase, and turns out to be a time scale, or a learning rate in the asymmetric phase. This model is statistically equivalent to MG with binary strategies, hence all exact results for the standard MG with "temperature" hold.


First numerical evidence that the temperature of the above model is a learning rate, that is, a time scale. Also shows that the number of visited histories decreases when P/N increases in the asymmetric phase (finite size effect):


Exact solution to the asymmetric phase for S=2 (agents minimize available information), and of the symmetric for large enough "temperature". Introduction of a corrected inductive dynamics (the agents know their impact on the "market") leading to a Nash equilibrium (minimization of the fluctuations). See cond-mat/9909265 for detailed symmetric replica calculus and cond-mat/0007397 for the symmetry broken replica calculus.

A generalisation of the previous preprint to any S. This preprint is written in the language of economists.

Short review of the two previous works

Review aiming at describing the state of art in econophysics; has a section about the MG:


Analytical approach for the symmetric phase that gives good results for very small P/N:


Numerical evidence that the phase transition is robust under the change of payoff:


A study of the Chaitin algorithmic complexity of the minority sign, as well as the mutual information for the corrected dynamics with continuous parameter:


Aims to study continuous (non rescaled) time in the MG. Also recovers results of spin-glass nature of MG found in preprints cond-mat/9904392, 9908480, and 0004308. Shows numerically that in anti-persistent regions, the system's state depends on the initial condition (see preprint cond-mat/0102257 for another discussion about stochastic continuous time equations).

Shows analytically that antipersistent behaviour is caused by neglect of market impact and fast learning rate. This happens for P/N small and small enough "temperature". Exact solution for all parameters of MG with no memory. Shows analytically that in anti-persistent regions, the system's state depends on the initial condition. Derive a critical "temperature":

Extends and completes the analytic solution of MG where agents take into account their impact on the game, by doing a 1-step replica symmetry broken calculus:

Uses exact generating functional techniques a la De Dominicis for a modified MG, which seems to be essentially the same as the original MG; shows how to obtain in principle the exact dynamical solution of the MG in thermodynamic limit. Not only recovers the replica results for the asymmetric phase, but also allows for the first time to address the dynamics of the symmetric phase in anti-persistent regions, i.e. for small T. Able to deal with non-zero initial strategy scores:

Temporal properties of the symmetric phase are investigated numerically:

Sorts out the question of the continuous time limit. Derives stochastic equation in continuous (rescaled) time, which are valid for all parameters of the game; the obtained Fokker-Planck equation is solved. Shows that i) the "temperature" T or learning rate of agents is actually an inverse temperature for the system; ii) only in the limit of infinite "temperature" (i.e. zero), there is a Lyapunov function; iii) for all parameters, the stationary/steady state really corresponds to the minimum of the available information, hence the replica calculus is exact whenever a stationary state is reached; iv) the replica calculus can account for non uniform initial condition in the symmetric phase; v) explains why only the symmetric phase is sensitive to learning rates and to initial conditions v) gives a self-consistent equation for sigma that is also valid for the symmetric phase vi) gives the Hamiltonian for any payoff.

Strategies' scores are kept only during a small time window of T time steps. Exact results for the fluctuations, for a given realisation of the disorder.

Studies the MG where no coin-tossing takes place when two strategies have the same score. Able to produce approximate analytical expressions and to provide an intuitive interpretation of various phenomena in term of a restoring force and a bias.

Introduces a finite memory in the scores of strategies (see also cond-mat/0102257) and finds a phase transition between presence and absence of coordination when the ratio of the learning rate to the oblivion rate is varied (analytical results). Introduces also a new evolving scheme where all the predictions of all strategies of all agents are changed for one piece of information.

Shows that in the symmetric phase, the period two process disappears if random histories are considered

Exact analytical results for MGs with exponential learning and noise not only on the scores (additive noise), but on the decisions themselves (multiplicative noise)


Brings the light of the generating functional technique to the standard MG with random histories. Discusses the difference between the batch and on-line (i.e. standard) MG. Explains a posteriori which are the explicit or implicit approximations of the previous continuous time approaches.


Shows analytically that the broken-ergodicity of MG where agents take into account their impact on the game, is not related to aging, as usual for spin-glasses, but to long term memory

Shows how it is possible to adjust parameters of the game so that one always has cooperation for naive agents (NB: this is done for 'exotic' strategy space, but their conclusions hold for the original MG). This paper has some arguments to predict the value of M. [Note from me : strictly speaking, maximum profit corresponds to Nash equilibria, which are attained by agents taking into account their impact on the game]

Gives bounds on the `complexity' of the MG.


A review of previous papers on the statistical mechanics of the MG

MG and financial markets

The standard MG can be considered as a very crude model of financial markets, because the minority mechanism is found in markets. Quite a lot of the above papers motivate their study of the MG by that of markets. The following papers try to study specifically the relationships between MG and markets, and not only the MG for its own. Several extension have of course to be considered.



Grand canonical MG with dynamic capital and quasi-periodic producers versus non-adaptive agents. Models with a lot of economic details. Shows that speculators reduce fluctuations, and finds a phase transition when the aggressiveness of the speculators increases.

Study of the role of producers, speculators, noise traders, and insiders in a market. Shows that speculators and producers live in symbiosis (exact calculus): they need each other. Detailed replica calculus also valid for a standard MG, generalized to any average correlation amongst speculators' strategies.

Volume and price are produced by heterogeneous agents in this grand-canonical MG, where agents do not trade if their best strategies perform worse than a given threshold.

Extends the above papers towards more realistic models of markets: agents with dynamical capital and reinvestment and more refined grand-canonical mechanism; also study how to hedge with this kind of modified MG.

Dynamical capital and reinvestment is considered. This is extension is enough to obtain stylized facts are obtained near the critical point. Also extends the discussion about producers and speculators of cond-mat/9909265

It is argued that grand-canonical MGs contain a fundamental mechanism for short ranged volatility correlations.

A very simple MG that leads to stylized facts: an agent plays if she believes that she can beat a given benchmark.

A MG where the heterogeneity is in the fact that agents have no access to complete information (which can be way too complex): each agent has partial information, which is derived from the complete information by her own filter. The filter contains now the quenched disorder. All qualitative results of the standard MG with producers are reproduced (phase transition, market impact, ...). Exact results from the replica calculus. Emphasis on the economic foundation of the minority mechanism. Allows one to study the strong efficiency hypothesis

Is it possible to identify all parameters of the refined MG of cond-mat/0008387 just by looking at its time series? Yes. Is it possible to predict the direction and amplitude of large movements of such a market model? Yes

The results of the preceding preprint are relevant for real financial markets

Shows that markets are either minority or majority games depending on the ratio between fundamentalists and trend followers in the market.

Agents play with different frequencies. Agents playing often are more likely to use different strategies. In addition, it is shown that in inefficient MG markets, the arbitrage opportunity is proportional to the inverse of its frequency. Application to financial markets

Applies different time series analysis methods to a grand canonical MG and compares with the S&P500

Analyses the cause, duration and amplitude of crashes with help of De Bruijn graphs in a Grand Canonical MG

Continues the analysis of previous preprint and proposes remedies

A different payoff function is proposed, where the gain at time t depends on the action of agents at time t-1 (see also cond-mat/020622. Note that the discussion parallels that of this paper, except that agents have no expectation over future price, hence are neither contrarians or trend-followers in essence.

A much extended grand canonical MG, including market clearing, capital dynamics, etc, with 3 regimes: bubbles/crashes, intermittency, stable prices.

Surveys the merits of various kinds of grand-canonical MG and MG-like models with intertemporal payoffs

Simplifies further the model of cond-mat/0101326: the agents have only one strategy each, and are allowed not to play. In the exact solution, no stylized facts, as P/N is not the correct control parameter for stylized facts. Argues that high volatility regions are due to a signal-to-noise transition, which explains finite size effects in the original MG.

Minority Game with another type of strategies:

A MG where players have an alternative kind of strategies: given a history, each agent i has a probability pi to choose the action which was winning last time the history occurred. Evolutionary means that if an agent has a wealth smaller that d, his pi is changed within a range of R (R=1 here).


Uses the same kind of strategies as above, but modifies the way in which the pi are updated


The asymmetrical MG with the same kind of strategies as the previous paper


Detailed studies of the strange phenomena occurring in the above preprint


Confirms that the memory length is not a crucial parameter for this kind of strategies, and propose some analytical formulas based on random walks


Proceeding that covers preprints cond-mat/9810142 and cond-mat/9905039.


Proposes a mean-field theory for this kind of strategies.


Compares MGs with strategies introduced by D'Hulst and Rodgers (cond-mat/9902001) with MGs studied just above in this section:


Analytical approach (master equation) to the model investigated in the above preprints as well as to the MG introduced by Packzuski and Bassler (cond-mat/9905082)

Finds a global cost function that the behaviour of agents with such strategies minimises. Also adds thermal fluctuations and studies their effect.

Shows that changing the ratio R between points for winning and for losing leads to clustering (the histogram of the pi is peak around 0) to segregation for R>R_c<1.

In this model, a new p is drawn from a uniform distribution [0,1]. The probability of winning is time dependent, with oscillatory behaviour, which means that there is no real stationary state

A nice theory of agent survival in this model, based on first-passage formalism for random walks with time-dependent (oscillating) probabilities. Shows that Rc=1 in the thermodynamic limit.


Points out that the role of R has been studied in their papers, that stochastic behaviour of <p> was also observed, and that drawing a new p from a uniform distribution is an important modification of the original rule.

(Long) reply to the previous comment. The oscillatory behaviour is observed for all rules, but its period and strength are not the identical. Differentiates stochastic and oscillatory behaviours.

Neural Networks playing a MG can cooperate. The relevant parameters are N and eta, the learning rate. Analytic results (without any details)


Gives details of the previous paper. The importance of the strategy parametrisation choice is discussed:


The title says it all. A nice review. Has a section about the two previous works


Extends the previous preprints, in particular to cryptography.


Same principle as the previous preprints, but for more than two alternatives:


Agents have a neural network based on the principle of punishing the errors (see this preprint  from Chialvo and Bak)


Agents have two strategies, and play their worst one with a given probability.


Mixed population of agents with "generalized strategies" and standard agents.


A repeated MG with no memory; only losers at the last time step change their decision with probability p. Typical fluctuations are of order 1. Exactly solved. Note that this make the game a MG without memory: inductive agents without memory taking into account their impact on the game are also able to create fluctuations of order 1, see cond-mat/0004376.


Strategies are drawn from a set whose size does not depend on the system size.


A quantum MG !


Shows numerically that Q learning (a kind of Reinforcement learning procedure) yields a stationary state close to a Nash equilibrium:



Extended classifier sytems are used and their performance is analyzed:



Zero-th level classifiers this time:



Minority Game with local or personal histories:

Introduction of personal histories (my personal history = "what I have done for the M last time steps") instead of global histories: cooperation still arises.


Another kind of personal histories: it consists of the previous actions of M random neighbours. The connection with the Kauffman networks is then obvious (M=K).


Same kind of personal history as the previous paper, but the agents placed on a circle. Agents can cooperate:


Agents play local MGs with local information (there are as many MGs as agents) on square lattices (1-d, 2-d, ...). In this model, there are situations where all agents win at the same time. The effective disorder is annealed.

Same as above, but with strategies à la Johnson:



Various extensions to the Minority Game:

First numerical study of the asymmetric MG:


Another kind of asymmetric MG: 1=in the game, 0=out of the game, and agents have incentive to participate, even if the game is risky:

Numerical study of the asymmetric MG with time varying resource level with a setup of ref preprint (boolean networks). Best results for K=2.

Shows numerically that the fluctuations are reduced by an increase of the asymmetry


Several evolutionary schemes for the standard MG:

Genetic algorithms are used in order to make agents evolve. In 3 words: usually less fluctuations:



Agents are placed on a circle and imitate their left neighbour with probability p if the latter gains more than themselves


NOTE: I have no access to the paper itself, hence, just reproduce the abstract as it:
"After studying the effects of imitation on the mixed population of adaptive agents with different memories competing in a minority game, we have found that when the pure population lies in a crowded regime, the introduction of imitation can considerably improve cooperation among agents in a money market."

Two different models of minority games with three alternatives are considered:


A MG where agent have to chose between K rooms; the ones in the less chosen room win. Reproduces statistical features of the standard MG (attendance fluctuations, phase transition, ...)


A MG where agents' changing decisions are replaced by a global cut-and-paste process. [Note from me: in the original MG, the fluctuations are minimal at the point where H goes to zero, in the thermodynamic limit]


The MG is modified in order to model the emergence of colonies of birds in presence of predation.


A game where the agents have K choices, and are rewarded if their choice at time t is chosen by more people at time t+1. This is in essence the same idea as in this preprint

Several minority games are coupled: each individual game plays the sign of its outcome. The strength of interaction is shown to have a measurable effect at all levels.



A game where the two alternatives are two suppliers that have each a given quantity of resource. Being in the minority is not enough. As this amounts to consider a different kind of payoff in the MG, fluctuations are similar in essence to those of the MG. Phase transition when the resource level is varied.



Relevant papers:

Properties of anti-persistent time series are studied on a De Bruijn Graph. Relevant for MGs with populations of various memory length (cond-mat/9903164 and cond-mat/9909265). Explains why players with a larger memory have an edge only if the anti-persistence is large enough (i.e. alpha small enough).

Ph.D Theses related to the MG


  1. D. Challet, Modelling Market Dynamics: Minority Games and beyond, Fribourg (CH), July 2000, psfile

  2. A. de Martino, Replica Symmetry Breaking and Long Term Memory in Large Games with Heterogeneous Players, SISSA-Trieste (IT) (2001), psfile

  3. J. A. F Heimel, Dynamics of Learning by Neurons and Agents: Generating Functionals for Disordered Systems (2002), psfile

  4. R. Metzler, Neural Networks, Game Theory and Time Series Generation (2002), psfile

Reviews

On analytical results on the MG, and on the relationship between the MG and markets

    M. Marsili, Toy models of markets with heterogeneous interacting agents (2002), psfile

On MG-inspired market models, crash prediction, and vaccines

    D. lamper et al., Managing catastrophic changes in a collective, (2002), preprint

On the relationship between payoffs, learning, and efficiency. And on what payoff to give to the agents so that they minimize a given quantity

    D. Challet, Competition between adaptive agents: from learning to collective efficiency and back (2002), preprint




Econophysics Forum