So far, there are 158 papers listed here. If one of yours is missing
here, please email me
!
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Articles related to the standard Minority
Game :
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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:
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D. Challet and Y.-C. Zhang, Emergence of Cooperation and
Organization in an Evolutionary Game, Physica A 246, 407
(1997), preprint
An easy-to-read introduction to the MG:
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Y.-C. Zhang, Modeling Market Mechanism with Evolutionary Games,
Europhys. News 29, 51 (1998), preprint
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:
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R. Savit et al., Adaptive
Competition, Market Efficiency, Phase Transitions and Spin-Glasses
(1997), preprint
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R. Savit et al., Adaptive Competition, Market Efficiency, Phase
Transition, PRL, 82(10), 2203 (1999) (a more attractive version
of the above preprint)
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). The variance of the BP has also a
minimum:
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N. F. Johnson et al., Volatility and Agent Adaptability in a
Self Organizing Market, Physica A, 256, 230 (1998), preprint
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 herefor
more details)
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A. Cavagna, Irrelevance of Memory in
the Minority Game, PRE (1998), preprint
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A. Cavagna, Comment on "Adaptive
Competition, Market Efficiency, and Phase Transitions", Phys. Rev.
Lett. 84 5, 1058 (2000)
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R. Savit, Reply to Comment on "Adaptive Competition, Market
Efficiency, and Phase Transitions"Phys. Rev. Lett. 84, 1059 (2000)
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:
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M. A. R. de Cara et al., Competition, efficiency and collective
behavior in the "El Farol" bar problem (1998), preprint
Introduction of the distance between players and the
reduced strategy space, study of Darwinism effects, resource sharing,
variance of the attendance, mixed memory population:
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D. Challet and Y.-C. Zhang, On the Minority Game: Analytical and
Numerical Studies, Physica A 256, 514 (1998), preprint
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:
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M. Hart, P. Jefferies, N. F. Johnson
and P. M. Hui, Crowd-Anticrowd model of the Minority Game (2000), preprint
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M. Hart, P. Jefferies, N. F. Johnson
and P. M. Hui, Crowd-anticrowd theory of the Minority Game (2000), preprint
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M. Hart, P. Jefferies, P.M. Hui, N.F. Johnson, Crowd-Anticrowd
Theory of Multi-Agent Market Games (2000), preprint
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.
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A. Cavagna, J.P. Garrahan, I.
Giardina and D. Sherrington, A Thermal Model for Adaptive Competition
in a Market, , Phys. Rev. Lett. 83, 4429 (1999). preprint
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D. Challet, M. Marsili and R.
Zecchina, Comment on "A Thermal Model for Adaptive Competition in a
Market", submitted (2000), preprint
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A. Cavagna, J.P. Garrahan, I. Giardina and D. Sherrington, Reply
to Comment on ``Thermal Model for Adaptive Competition in a Market'', preprint
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):
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G. Bottazzi, G. Devetag G. Dosi (SASAS Pisa, Italy), Learning
and Emergent Coordination in Speculative Markets: Some Properties of
"Minority Game" Dynamics, preprint
(1999)
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.
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D. Challet, M. Marsili and R. Zecchina, Statistical Mechanics of
Heterogeneous Agents (1999), preprint
A generalisation of the previous preprint to any S. This
preprint is written in the language of economists.
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M. Marsili, D. Challet and R. Zecchina, Exact Solution of a
Modified El Farol's Bar Problem (1999), preprint
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:
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R. D'hulst, G. J. Rodgers and M. Ausloss, Strategy Selection in
the Minority Game(1999), Physica A 278, 579 (2000), preprint
Numerical evidence that the phase transition is robust
under the change of payoff:
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Y. Li, A. VanDeeme, R. Savit, The Minority Game with Variable
Payoffs (2000), preprint
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).
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Juan P. Garrahan, Esteban Moro,
David Sherrington, Continuous time dynamics of the Thermal Minority
Game (2000), preprint
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David Sherrington, Juan P. Garrahan, Esteban Moro, Statistical
Physics of Adaptive Correlation of Agents in a Market (2000), preprint
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:
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J. A. F. Heimel, A. C. C. Coolen, Generating Functional Analysis
of the Dynamics of the Batch Minority Game with Random External
Information (2000), preprint
Temporal properties of the symmetric phase are investigated
numerically:
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D. Zheng , B.-H. Wang, Statistical Properties of the Attendance
Time Series in the Minority Game (2001), preprint
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.
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M. Marsili, D. Challet, On the continuous time limit and
stationary states of the Minority Game (2001), preprint
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.
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M. L. Hart, P. Jefferies, N. F. Johnson, Dynamics of the Time
Horizon Minority Game (2001), preprint
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.
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P. Jefferies, M.L. Hart, N.F. Johnson, Deterministic Dynamics in
the Minority Game (2001), preprint
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.
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M. Marsili, R. Mulet, F. Ricci-Tersenghi and R. Zecchina,
Learning to Coordinate in a Complex and Non-Stationary World, preprint
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)
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J. A. F. Heimel, A. C. C. Coolen, D. Sherrington, Dynamics of
the Batch Minority Game with Inhomogeneous Decision noise (2001), preprint
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
The Lempel-Ziv complexity is measured in MG history bit-strings for
real and random histories (and intermediary cases): it has a maximum at
the transition point for real histories but not random ones, for which
it constant.
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Milan Rajkovic, Zoran Mihailovic, Quantifying Complexity
in the Minority Game,Physica A, vol 325, issues 1-2, (2003) 40-47, preprint
The Gini
index (measure of wealth inequality amonts the players) is
investigated: it is maximal at the phase transition point
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K. H. Ho, F. K. Chow, H. F. Chau, Cooperation Comes With
Exploitation -- A Study Of The Wealth Inequality In The Minority Game
(2004) preprint
A closed-form exact mathematical theory of MG with real market histories using generating functionals.
Shows numerically that the fluctuations of the attendance in the
symmetrical phase are depend on the nature of the market history. This
only shows up for large enough systems.
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K. H. Ho, W. C. Man, F. K. Chow, and H. F. Chau, Memory Is Relevant In The Symmetric Phase Of The Minority Game (2004), preprint
Proposes a mathematical theory of the variance of the attendance as
a function of the bias in initial strategy valuations for small m for
real market histories. Fraction of frozen agents, convergence time and
wealth distribution are also treated.
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K. Y. Michael Wong, S. W. Lim, and Zhuo Gaon, Effects of diversity on multi-agents systems: Minority Games (2005), preprint
Studies populations where each agent can have strategies with various market history length
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K. E. Mitman, S. C. Choe, N. F. Johnson, Competitive advantage for multiple-memory strategies in an artificial market (2005), preprint
Spherical means that the agents can play linear combinations of
their two strategies, with a geometrical constraint. Generating
functional give exact expressions for the volatility, but the
phenomenology is significantly different from that of the standard MG
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T. Galla, A.C.C. Coolen, D. Sherrington, Dynamics of a spherical minority game (2003), preprint
The Spherical MG redux: the spherical condition leaves less liberty
to the player, and makes the phenomenology of the model much closer to
that of the standard MG, while still giving exact expressions for the
volatility
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T. Galla, D. Sherrington, Stationary states of a spherical Minority Game with ergodicity breaking (2005), preprint
Dynamical analytical solution of batch MG with S>1. Instead of
frozen agents, one should consider the whole distribution of strategy
use frequency. This makes the closure of the set of effective equations
more complex.
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N. Shayeghi, A. C. C. Coolen, Generating functional analysis of batch minority game with arbitrary strategy number (2006), preprint
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.
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D. Challet, M. Marsili and Y.-C. Zhang, Modelling Market
Mechanism with Minority Game (1999), preprint
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.
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P. Jefferies, M.L. Hart, P.M. Hui, N.F. Johnson, From market
games to real-world markets (2000),preprint
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
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D. Challet, A. Chessa, M. Marsili, Y.-C. Zhang, From Minority
Games to real markets, Quantitative Finance (2001), preprint
It is argued that grand-canonical MGs contain a fundamental
mechanism for short ranged volatility correlations.
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J.-P. Bouchaud, I. Giardina, M.
Mezard, On a universal mechanism for long ranged volatility
correlations (2000), preprint
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I. Giardina, J.-P. Bouchaud, M. Mézard, Microscopic
Models for Long Ranged Volatility Correlations (2001), preprint (a few
more details)
A very simple MG that leads to stylized facts: an agent plays
if she believes that she can beat a given benchmark.
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D. Challet, M. Marsili and Y.-C.
Zhang, Stylized Facts of Financial Markets and Market Crashes in
Minority Games (2001), preprint
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D. Challet, M. Marsili and Y.-C. Zhang, Minority Games and
Stylized Facts (2001), preprint
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
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J. Berg, M. Marsili, A. Rustichini, R. Zecchina, Statistical
mechanics of asset markets with private information (2001), preprint
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
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S. Lamper, S. Howison, N. F. Johnson, Prediction of Large Future
Changes in a competitive evolving population (2001), preprint
The results of the preceding preprint are relevant for real
financial markets
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N. F. Johnson, D. Lamper, P. Jefferies, M. L. Hart, S. Howison,
Application of Multi-Agent Games to the Prediction of Financial Time
Series, preprint
Shows that markets are either minority or majority games
depending on the ratio between fundamentalists and trend followers in
the market.
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M. Marsili, Market mechanism and expectations in minority and
majority games, Physica A 299, pp. 93-103 (2001), paper
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
The same model with several types of decision noise, analyzed
with generating functionals
Applies different time series analysis methods to a grand
canonical MG and compares with the S&P500
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F. F. Ferreira, G. Francisco, B. S. Machado, P. Muruganandam,
Time Series Analysis for Minority Game Simulations of Financial Markets
(2002), preprint
Analyses the cause, duration and amplitude of crashes with help
of De Bruijn graphs in a Grand Canonical MG
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P. Jefferies, D. Lamper, N. F. Johnson, Anatomy of extreme
events in a complex adaptive system (2002), preprint
Continues the analysis of previous preprint and proposes
remedies
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M. L. Hart, D. Lamper, N. F. Johnson, Crash Avoidance in a
Complex System (2002), preprint
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 thispaper,
except that agents have no expectation over future price,hence are
neither contrarians nor trend-followers in essence.
A much extended grand canonical MG, including market clearing,
capital dynamics, inter-temporal payoff, etc, with 3 regimes: bubbles/crashes,
intermittency, stable prices.
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I. Giardina, J.-Ph. Bouchaud, Bubbles, Crashes and Intermittency
in Agent Based Market Models (2002), preprint
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 correctcontrol
parameter for stylized facts. Argues that high volatilityregions are
due to a signal-to-noise transition, which explainsfinite size effects
in the original MG.
The payoff is modified so that the game is a majority game for
small A, and a minority game for large A. Large fluctuations arise near
the point where the nature of the game changes from minority to
minority game
- A. De Martino, I. Giardina, M. Marsili, A. Tedeschi, Generalized
minority games with adaptive trend-followers and contrarians (2004) preprint
More detailled study of the previous model, showing in particular intermittency
- A. Tedeschi, A. De Martino, I. Giardina, Coordination, intermittency and trends in generalized Minority Games (2005) preprint
Shows that the structure of bit string histories and binary strategies
is the cause of periods of predictability and unpredictability, found
in a previous preprint. Applies this to Nasdaq price change amplitude prediction with agent-based models.
- Jorgen Vitting Andersen, Didier Sornette, A Mechanism for Pockets of Predictability in Complex Adaptive Systems (2004) preprint
Develops theoretical tools for fitting agent-based models to data.
- N. Gupta, R. Hauser, N. F. Johnson, Forecasting time-series using artificial market models (2005), preprint
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Minority Game with another type of
strategies:
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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).
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N. F. Johnson et al., Self Organized Segregation within an
Evolving Population, PRL 82, 3360 (1999), preprint
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
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E. Burgos, Horacio Ceva, R.P.J.
Perazzo, Quenching and Annealing in the Minority Game (2000), preprint
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E. Burgos, Horacio Ceva, R.P.J. Perazzo, Dynamical quenching and
annealing in self-organization multiagent models (2000), 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
Studies in details this kind of strategies with and without memory;
finds that the system performs better without memory. Computes the
autocorrelation of the memory. Finally, allows the player not to play
(grand canonical game).
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Roland Kay and Neil F. Johnson, Memory and
self-induced shocks in an evolutionary population competing for limited
resources (2003),
, preprint
Proceeding that covers preprints cond-mat/9810142 and
cond-mat/9905039.
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P. M. Hui, T. S. Lo, N. F. Johnson, Segregation in a competing
and evolving population(2000), preprint
Proposes a mean-field theory for this kind of strategies.
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T. S. Lo, P. M. Hui, N. F. Johnson, Theory of Evolutionary
Minority Game (2000), preprint
Compares MGs with strategies introduced by D'Hulst and
Rodgers (cond-mat/9902001) with MGs studied just above in this
section:
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T. S. Lo, S. W. Lim, P. M. Hui, N. F. Johnson, Evolutionary
minority game with heterogeneous strategy distribution (2000), preprint
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.
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E. Burgos, Horacio Ceva, R.P.J. Perazzo, Thermal Treatment of
the Minority Game (2001), preprint
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.
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S. Hod, E. Nakar, Segregation vs. Clustering in the Evolutionary
Minority Game, (2002), preprint
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
-
S. Hod, E. Nakar, Semianalytical approach to the Evolutionary
Minority Game, (2002), preprint
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.
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E. Burgos, Horacio Ceva, and R.P.J. Perazzo, Comment on
Self-Segregation versus Clustering in the Evolutionary Minority Game
(2003) preprint
(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.
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S. Hod, E. Nakar, Strategy updating rules and strategy
distributions in dynamical multiagent systems (2003), preprint
Shows that the critical ratio R_c depends on d and N. Introduces
a simplified model where p can take only 3 values (called "Three-Group
Evolutinary MG") and finds N_c=(d/cst(1-R))2
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K. Chen, B.-H. Wang, and B. Yuan, Adiabatic
theory for the population distribution in the evolutionary minority
game, Phys. Rev. E 69, 025102 (2004)
preprint
Derives an intermittency correction value to N_c for the Three-Group EMG.
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K. Chen, B.-H. Wang, and B. Yuan, Theory of the Three-Group Evolutionary Minority Game (2004)
preprint
Neural Networks playing a MG can cooperate. The relevant
parameters are N and eta, the learning rate. Analytic results
(without any details)
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W. Kinzel, R. Metzler, I. Kanter, Dynamics of Interacting Neural
Networks, J. Phys. A 33 (2000), L141-L147, preprint
Gives details of the previous paper. The importance of the
strategy parametrisation choice is discussed:
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W. Kinzel, R. Metzler, I. Kanter, Interacting Neural Networks
(2000), preprint
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:
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Liat Ein-Dor, Richard Metzler, Ido Kanter, Wolfgang Kinzel,
Multi-Choices Minority Game (2000), preprint
Agents have a neural network based on the principle of
punishing the errors (see this preprint from
Chialvo and Bak)
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J. Wakeling, P. Bak, Intelligent Systems in the Context of
Surrounding Environment, PRE 64, 051920 (2001), preprint
Agents have two strategies, and play their worst one with a
given probability.
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P. Jefferies, M. Hart, N.F. Johnson,
P.M. Hui, Generalized strategies in the Minority Game (2000), preprint
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M. Hart, P. Jefferies, N. F. Johnson, P. M. Hui, Stochastic
strategies in the Minority Game (2000)preprint
Mixed population of agents with "generalized
strategies" and standard agents.
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P. Jefferies, M. Hart, N.F. Johnson, P.M. Hui, Mixed population
Minority Game with generalized strategies (2000), preprint
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.
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G. Reents, R. Metzler, W. Kinzel, A New Stochastic Strategy for
the Minority Game (2000), preprint
Strategies are drawn from a set whose size does not depend
on the system size.
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A. Bazzan, R. Bordini, G. Andrioti, R. Vicari and J. Wahle,
Wayward Agents in a Commuting Scenario (Personalities in the Minority
Game) (2000), Proc. 4th Int. Conf. on MultiAgent Systems (ICMAS-2000), preprint
A quantum MG !
Shows numerically that
Q learning (a kind of Reinforcement learning procedure) yields a
stationary state close to a Nash equilibrium:
-
M. Andrecut and M. K.
Ali, Q learning in the Minority Game, PRE 64, 067103 (2001), link
Extended
classifier sytems are used and their performance is analyzed:
Zero-th
level classifiers this time:
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Minority Game on networks:
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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).
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M. Paczuski, K. E. Bassler and A. Corral, Self-organized
Networks of Competing Boolean Agents (1999), preprint
Same kind of personal
history as the previous paper, but the agents placed on a circle.
Agents can cooperate:
- T. Kalinowski, H.-J. Schulz and M. Briese, Cooperation in the
Minority Game with Local Information, Physica A 277 (3-4), 502-508
(2000)
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:
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E. Burgos, H. Ceva, R. P. J. Perazzo, A Local Minority Game
(2002), preprint
A generalized MG with N_c choices, and mixed local and global
information
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H. F. Chau, F. K. Chow, K. H. Ho, Minority Game With Peer
Pressure (2003), preprint
A global MG where the agents are linked to neighbours with a
random Poisonnian graph, and compare their performance with that of
their neighbours, and imitate them when outperformed. Scale-free
network of influence and better global coordination emerge
-
M. Anghel, Zoltán Toroczkai, Kevin E. Bassler, G.
Korniss, Competition in Social Networks: Emergence of a Scale-free
Leadership Structure and Collective E ciency (2004), Phys. Rev. Lett. preprint
A crow/anti-crowd approach to networked population as introduced in the previous paper.
- Neil F. Johnson,Pak Ming Hui, Crowd-Anticrowd Theory of
Collective Dynamics in Competitive, Multi-Agent Populations and
Networks (2003), preprint
A more refined approach than the previous paper.
- T. S. Lo, H. Y. Chan, P. M. Hui, and N. F. Johnson, Theory
of Networked Minority Games based on Strategy Pattern Dynamics (2004) preprint
MG where the personal history depends on the actions of neighbours at the previous time step (a la Paczuski). The adaptability as a function of the number of neighbours K is investigated.
Same as previous paper. Several kinds of network, including
growing networks are investigated, as well as the role of Darwinian
evolution.
-
Baosheng Yuan, Bing-Hong Wang, and Kan Chen, Evolutionary Dynamics in Complex Networks of Competing Boolean Agents (2004), preprint
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Various extensions to the Minority Game:
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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:
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Frantisek Slanina, Yi-Cheng Zhang, Dynamical spin-glass-like
behavior in an evolutionary game, preprint
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
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K. F. Yip, P. M. Hui, T. S. Lo, N. F. Johnson, Efficient
resource distribution in a minority game with a biased pool of
strategies, Physica A 321, 318-324 (2003)
Several evolutionary schemes for the standard MG:
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Y. Li et al., Evolution in Minority
Games I. Games with a Fixed Strategy Space (1999), preprint
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Y. Li et al., Evolution in Minority Games II. Games with
Variable Strategy Spaces (1999), preprint
Genetic algorithms are used in order to make agents evolve. In
3 words: usually less fluctuations:
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M. Sysi-Aho, A. Chakraborti and K.
Kaski, Intelligent Minority Games with Genetic-Crossover Strategies
(2002), preprint
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M. Sysi-Aho, A. Chakraborti and K.
Kaski, Hybridized genetic strategies in game theory (2002), preprint
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M. Sysi-Aho, A. Chakraborti and K.
Kaski, Biology Helps You to Win a Game (2003), preprint
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."
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H.J. Quan , B.H. Wang , P.M. Hui , X.S. Luo , Cooperation in the
mixed population minority game with imitation , Chinese Physics
Letters, 18 (9): 1156-1158 SEP 2001
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]
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R. D'hulst, G. J. Rodgers, Percolation and Depinning Transition
in Cut-and-Paste Models of Adaptation (2001), preprint
The MG is modified in order to model the emergence of
colonies of birds in presence of predation.
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J. L. Tella, M. A. R. de Cara, O. Pla, F. Guinea, A Model
for Predation Pressure in Colonial Birds (2001), preprint
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.
-
F. Földy, Z. Somogyvári, P. Erdi, Hierarchically
Organized Minority Games (2003), preprint
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 transitions when the
resource level is varied. Has strong analogies with market entry games
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R. Savit, S. A.
Brueckner, H. Van Dyke Parunak, J. Sauter, Phase Structure of Resource
Allocation Games (2003), preprint
The MAJORITY game is solved exactly for naive and sophisticated
agents. Very similar to a Hopfield model. A transition between
spin-glass and retrieval phases found. Although all the agents freeze
in both cases, the stationary state reached by the naive agents is not
a Nash equilibrium.
- P. Kozlowski and M. Marsili, Statistical Mechanics of the
Majority Game 2003, preprint
A continuous time MG situation where users send requests to one
of two computers, namely to the one that was the least busy some time
ago. If the information about the state of the computers is wrong with
some probability, the efficiency of the system improves upto a point.
- Richard Metzler, Mark Klein, Yaneer Bar-Yam, Efficiency
through disinformation, preprint
MG played between human beings. Finds that the lengthy of the
bit string history given to the agents is irrelevant to cooperation,
and that humans do cooperate. Interesting discussion about the game
theoretical aspect of the game.
- G. Bottazi and G. Devetag, Coordination and Self-Ogranization
in Minority Games: Experimental Evidence (2002) preprint
MG played between human beings. Main result: the length of the
bit string history is not as much relevant as its nature: the players
cannot cooperate if fed with random histories
- T. Platkowski and M. Ramsza, Playing minority game, Physica
A 323, 726 (2003),
MG played between human beings. Finds interesting oscillating
fluctuations as a function of time. The simplest the behaviour, the
better the player. Argues that human beings have a memory of 3 time
steps.
- R. Savit, K. Koelle, W. Treynor and R. Gonzalez, Man and
Superman: Human limitations, Innovation and Emergence in Resource
Competition, in Proceedings of Collectives and the design of complex
systems (2003)
MG played by one human being against artificial players. Data
suggests that human beings have a memory of 3 time steps. Human players
perform poorly in the asymmetric phase, and well in the symmetric phase.
- P. Laureti, P. Ruch, J. Wakeling, Y.-Ch. Zhang, The
Interactive Minority Game: a Web-based investigation of human market
interactions, Physica A 331 (2004) 651-659, preprint
- P. Ruch, J. Wakeling, Y.-Ch. Zhang, The Interactive Minority
Game: Instructions for Experts, preprint
Human players are given either information that makes them naive
or sophisticated. Resulting fluctuations are accordingly high or low.
- T. Chmura, T. Pitz, Minority Game - an experimental investigation and simulations (2005)preprint
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
anti-persistence is large enough (i.e. alpha small enough).
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Ph.D Theses related to the MG
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-
D. Challet, Modelling Market
Dynamics: Minority Games and beyond, Fribourg (CH), July 2000, psfile
-
A. de Martino, Replica Symmetry
Breaking and Long Term Memory in Large Games with Heterogeneous
Players, SISSA-Trieste (IT) (2001), psfile
-
J. A. F Heimel, Dynamics of Learning by Neurons and Agents:
Generating Functionals for Disordered Systems (2002), pdffile
-
R. Metzler, Neural Networks, Game Theory and Time Series
Generation (2002), psfile
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M. A. R. de Cara, Métodos estadísticos en Ecología Colonialidad y aprendizaje en aves (2003), pdffile
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 remedies
- 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
Review of the application of statistical mechanics to various heterogeneous agent-based models, including MG
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A. De Martino, M. Marsili, Statistical mechanics of socio-economic systems with heterogeneous agents (2006), preprint
Review of current use of the MG as a model of financial markets
- T. Galla, G. Mosetti, Y.-C. Zhang, Anomalous fluctuations in
Minority Games and related multi-agent models of financial markets
(2006), preprint
;)
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