We discuss a new approach to data clustering. We find that maximum likelyhood leads naturally to ... more We discuss a new approach to data clustering. We find that maximum likelyhood leads naturally to an Hamiltonian of Potts variables which depends on the correlation matrix and whose low temperature behavior describes the correlation structure of the data. For random, uncorrelated data sets no correlation structure emerges. On the other hand for data sets with a built-in cluster structure, the method is able to detect and recover efficiently that structure. Finally we apply the method to financial time series, where the low temperature behavior reveals a non trivial clustering.
Physica A-statistical Mechanics and Its Applications, 2001
We present and study a Minority Game based model of a financial market where adaptive agents -the... more We present and study a Minority Game based model of a financial market where adaptive agents -the speculators -interact with deterministic agents -called producers. Speculators trade only if they detect predictable patterns which grant them a positive gain. Indeed the average number of active speculators grows with the amount of information that producers inject into the market. Transitions between equilibrium and out of equilibrium behavior are observed when the relative number of speculators to the complexity of information or to the number of producers are changed.
We present a general approach to explain the Zipf's law of city distribution. If the simplest int... more We present a general approach to explain the Zipf's law of city distribution. If the simplest interaction (pairwise) is assumed, individuals tend to form cities in agreement with the well-known statistics PACS numbers: 89.50.+r, 05.20.-y, 05.40.+j Zipf [1], half a century ago, has found that the city sizes obey an astonishingly simple distribution law, which is attributed to the more generic least effort principle of human behavior. Let us denote by q m the number of cities having the population size m, then R(m) =
We study the optimization of the stable marriage problem. All individuals attempt to optimize the... more We study the optimization of the stable marriage problem. All individuals attempt to optimize their own satisfaction, subject to mutually conflicting constraints. We find that the stable solutions are generally not the globally best solution, but reasonably close to it. All the stable solutions form a special sub-set of the meta-stable states, obeying interesting scaling laws. Both numerical and analytical tools are used to derive our results.
Traders in a market typically have widely different, private information on the return of an asse... more Traders in a market typically have widely different, private information on the return of an asset. The equilibrium price of the asset may reflect this information more accurately if the number of traders is large enough compared to the number of the states of the world that determine the return of the asset. We study the transition from markets where prices do not reflect the information accurately into markets where it does. In competitive markets, this transition takes place suddenly, at a critical value of the ratio between number of states and number of traders. The Nash equilibrium market behaves quite differently from a competitive market even in the limit of large economies.
Physica A-statistical Mechanics and Its Applications, 1997
We study the scaling behavior in currency exchange rates. Our results suggest that they satisfy s... more We study the scaling behavior in currency exchange rates. Our results suggest that they satisfy scaling with an exponent close to 0.5, but that it differs qualitatively from that of a simple random walk. Indeed price variations cannot be considered as independent variables and subtle correlations are present. Furthermore, we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity.
PACS. 01.75+m-Science and society. PACS. 02.50Le -Decision theory and game theory. PACS. 05.40+j ... more PACS. 01.75+m-Science and society. PACS. 02.50Le -Decision theory and game theory. PACS. 05.40+j -Fluctuation phenomena, random processes, and Brownian motion.
Physica A-statistical Mechanics and Its Applications, 1997
We investigate the fluctuations induced by irrationality in simple games with a large number of c... more We investigate the fluctuations induced by irrationality in simple games with a large number of competing players. We show that Nash equilibria in such games are "weakly" stable: irrationality propagates and amplifies through players' interactions so that huge fluctuations can results from a small amount of irrationality. In the presence of multiple Nash equilibria, our statistical approach allows to establish which is the globally stable equilibrium. However characteristic times to reach this state can be very large.
Physica A-statistical Mechanics and Its Applications, 2000
We discuss a model of heterogeneous, inductive rational agents inspired by the El Farol Bar probl... more We discuss a model of heterogeneous, inductive rational agents inspired by the El Farol Bar problem and the Minority Game. As in markets, agents interact through a collective aggregate variable -which plays a role similar to price -whose value is fixed by all of them. Agents follow a simple reinforcement-learning dynamics where the reinforcement, for each of their available strategies, is related to the payoff delivered by that strategy. We derive the exact solution of the model in the "thermodynamic" limit of infinitely many agents using tools of statistical physics of disordered systems. Our results show that the impact of agents on the market price plays a key role: even though price has a weak dependence on the behavior of each individual agent, the collective behavior crucially depends on whether agents account for such dependence or not. Remarkably, if the adaptive behavior of agents accounts even "infinitesimally" for this dependence they can, in a whole range of parameters, reduce global fluctuations by a finite amount. Both global efficiency and individual utility improve with respect to a "price taker" behavior if agents account for their market impact.
By considering diffusion on De Bruijn graphs, we study in details the dynamics of the histories i... more By considering diffusion on De Bruijn graphs, we study in details the dynamics of the histories in the Minority Game, a model of competition between adaptative agents. Such graphs describe the structure of temporal evolution of $M$ bits strings, each node standing for a given string, i.e. a history in the Minority Game. We show that the frequency of visit of each history is not given by $1/2^M$ in the limit of large $M$ when the transition probabilities are biased. Consequently all quantities of the model do significantly depend on whether the histories are real, or uniformly and randomly sampled. We expose a self-consistent theory of the case of real histories, which turns out to be in very good agreement with numerical simulations.
We show that the Minority Game, a model of interacting heterogeneous agents, can be described as ... more We show that the Minority Game, a model of interacting heterogeneous agents, can be described as a spin systems and it displays a phase transition between a symmetric phase and a symmetry broken phase where the games outcome is predicable. As a result a ``spontaneous magnetization'' arises in the spin formalism.
We study analytically a simple game theoretical model of heterogeneous interacting agents. We sho... more We study analytically a simple game theoretical model of heterogeneous interacting agents. We show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz. Such a stationary state differs from the Nash equilibrium where each agent maximizes her own utility. The latter turns out to be characterized by a replica symmetry broken structure. Numerical results fully agree with our analytic findings.
Physica A-statistical Mechanics and Its Applications, 2000
Using the Minority Game model we study a broad spectrum of problems of market mechanism. We study... more Using the Minority Game model we study a broad spectrum of problems of market mechanism. We study the role of different types of agents: producers, speculators as well as noise traders. The central issue here is the information flow : producers feed in the information whereas speculators make it away. How well each agent fares in the common game depends on the market conditions, as well as their sophistication. Sometimes there is much to gain with little effort, sometimes great effort virtually brings no more incremental gain. Market impact is shown to play also an important role, a strategy should be judged when it is actually used in play for its quality. Though the Minority Game is an extremely simplified market model, it allows to ask, analyze and answer many questions which arise in real markets.
Physica A-statistical Mechanics and Its Applications, 2004
We mathematize El Farol bar problem and transform it into a workable model. In general, the avera... more We mathematize El Farol bar problem and transform it into a workable model. In general, the average convergence to optimality at the collective level is trivial and does not even require any intelligence on the side of agents. Secondly, specializing to a particular ensemble of continuous strategies yields a model similar to the Minority Game. Statistical physics of disordered systems allows us to derive a complete understanding of the complex behavior of this model, on the basis of its phase diagram.
We discuss in detail the derivation of stochastic differential equations for the continuum time l... more We discuss in detail the derivation of stochastic differential equations for the continuum time limit of the Minority Game. We show that all properties of the Minority Game can be understood by a careful theoretical analysis of such equations. In particular, i) we confirm that the stationary state properties are given by the ground state configurations of a disordered (soft) spin system; ii) we derive the full stationary state distribution; iii) we characterize the dependence on initial conditions in the symmetric phase and iv) we clarify the behavior of the system as a function of the learning rate. This leaves us with a complete and coherent picture of the collective behavior of the Minority Game. Strikingly we find that the temperature like parameter which is introduced in the choice behavior of individual agents turns out to play the role, at the collective level, of the inverse of a thermodynamic temperature.
Physica A-statistical Mechanics and Its Applications, 1998
We propose and study a simple model of dynamical redistribution of capital in a diversified portf... more We propose and study a simple model of dynamical redistribution of capital in a diversified portfolio. We consider a hypothetical situation of a portfolio composed of N uncorrelated stocks. Each stock price follows a multiplicative random walk with identical drift and dispersion. The rules of our model naturally give rise to power law tails in the distribution of capital fractions invested in different stocks. The exponent of this scale free distribution is calculated in both discrete and continuous time formalism. It is demonstrated that the dynamical redistribution strategy results in a larger typical growth rate of the capital than a static "buy-and-hold" strategy. In the large N limit the typical growth rate is shown to asymptotically approach that of the expectation value of the stock price . The finite dimensional variant of the model is shown to describe the partition function of directed polymers in random media.
We address the question of market efficiency using the Minority Game (MG) model. First we show th... more We address the question of market efficiency using the Minority Game (MG) model. First we show that removing unrealistic features of the MG leads to models which reproduce a scaling behaviour close to what is observed in real markets. In particular we find that (i) fat tails and clustered volatility arise at the phase transition point and that (ii) the crossover to random walk behaviour of prices is a finite-size effect. This, on one hand, suggests that markets operate close to criticality, where the market is marginally efficient. On the other it allows one to measure the distance from criticality of real markets, using cross-over times. The artificial market described by the MG is then studied as an ecosystem with different species of traders. This clarifies the nature of the interaction and the particular role played by the various populations.
We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the development of financia... more We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the development of financial instruments, with a dynamical picture of an interacting market, in a simple setting. The proliferation of financial instruments apparently provides more means for risk diversification, making the market more efficient and complete. In the simple market of interacting traders discussed here, the proliferation of financial instruments erodes systemic stability and it drives the market to a critical state characterized by large susceptibility, strong fluctuations and enhanced correlations among risks. This suggests that the hypothesis of APT may not be compatible with a stable market dynamics. In this perspective, market stability acquires the properties of a common good, which suggests that appropriate measures should be introduced in derivative markets, to preserve stability.
We study scale free simple graphs with an exponent of the degree distribution $\gamma$ less than ... more We study scale free simple graphs with an exponent of the degree distribution $\gamma$ less than two. Generically one expects such extremely skewed networks -- which occur very frequently in systems of virtually or logically connected units -- to have different properties than those of scale free networks with $\gamma>2$: The number of links grows faster than the number of nodes and they naturally posses the small world property, because the diameter increases by the logarithm of the size of the network and the clustering coefficient is finite. We discuss a simple prototype model of such networks, inspired by real world phenomena, which exhibits these properties and allows for a detailed analytical investigation.
We discuss a new approach to data clustering. We find that maximum likelyhood leads naturally to ... more We discuss a new approach to data clustering. We find that maximum likelyhood leads naturally to an Hamiltonian of Potts variables which depends on the correlation matrix and whose low temperature behavior describes the correlation structure of the data. For random, uncorrelated data sets no correlation structure emerges. On the other hand for data sets with a built-in cluster structure, the method is able to detect and recover efficiently that structure. Finally we apply the method to financial time series, where the low temperature behavior reveals a non trivial clustering.
Physica A-statistical Mechanics and Its Applications, 2001
We present and study a Minority Game based model of a financial market where adaptive agents -the... more We present and study a Minority Game based model of a financial market where adaptive agents -the speculators -interact with deterministic agents -called producers. Speculators trade only if they detect predictable patterns which grant them a positive gain. Indeed the average number of active speculators grows with the amount of information that producers inject into the market. Transitions between equilibrium and out of equilibrium behavior are observed when the relative number of speculators to the complexity of information or to the number of producers are changed.
We present a general approach to explain the Zipf's law of city distribution. If the simplest int... more We present a general approach to explain the Zipf's law of city distribution. If the simplest interaction (pairwise) is assumed, individuals tend to form cities in agreement with the well-known statistics PACS numbers: 89.50.+r, 05.20.-y, 05.40.+j Zipf [1], half a century ago, has found that the city sizes obey an astonishingly simple distribution law, which is attributed to the more generic least effort principle of human behavior. Let us denote by q m the number of cities having the population size m, then R(m) =
We study the optimization of the stable marriage problem. All individuals attempt to optimize the... more We study the optimization of the stable marriage problem. All individuals attempt to optimize their own satisfaction, subject to mutually conflicting constraints. We find that the stable solutions are generally not the globally best solution, but reasonably close to it. All the stable solutions form a special sub-set of the meta-stable states, obeying interesting scaling laws. Both numerical and analytical tools are used to derive our results.
Traders in a market typically have widely different, private information on the return of an asse... more Traders in a market typically have widely different, private information on the return of an asset. The equilibrium price of the asset may reflect this information more accurately if the number of traders is large enough compared to the number of the states of the world that determine the return of the asset. We study the transition from markets where prices do not reflect the information accurately into markets where it does. In competitive markets, this transition takes place suddenly, at a critical value of the ratio between number of states and number of traders. The Nash equilibrium market behaves quite differently from a competitive market even in the limit of large economies.
Physica A-statistical Mechanics and Its Applications, 1997
We study the scaling behavior in currency exchange rates. Our results suggest that they satisfy s... more We study the scaling behavior in currency exchange rates. Our results suggest that they satisfy scaling with an exponent close to 0.5, but that it differs qualitatively from that of a simple random walk. Indeed price variations cannot be considered as independent variables and subtle correlations are present. Furthermore, we introduce a novel statistical analysis for economic data which makes the physical properties of a signal more evident and eliminates the systematic effects of time periodicity.
PACS. 01.75+m-Science and society. PACS. 02.50Le -Decision theory and game theory. PACS. 05.40+j ... more PACS. 01.75+m-Science and society. PACS. 02.50Le -Decision theory and game theory. PACS. 05.40+j -Fluctuation phenomena, random processes, and Brownian motion.
Physica A-statistical Mechanics and Its Applications, 1997
We investigate the fluctuations induced by irrationality in simple games with a large number of c... more We investigate the fluctuations induced by irrationality in simple games with a large number of competing players. We show that Nash equilibria in such games are "weakly" stable: irrationality propagates and amplifies through players' interactions so that huge fluctuations can results from a small amount of irrationality. In the presence of multiple Nash equilibria, our statistical approach allows to establish which is the globally stable equilibrium. However characteristic times to reach this state can be very large.
Physica A-statistical Mechanics and Its Applications, 2000
We discuss a model of heterogeneous, inductive rational agents inspired by the El Farol Bar probl... more We discuss a model of heterogeneous, inductive rational agents inspired by the El Farol Bar problem and the Minority Game. As in markets, agents interact through a collective aggregate variable -which plays a role similar to price -whose value is fixed by all of them. Agents follow a simple reinforcement-learning dynamics where the reinforcement, for each of their available strategies, is related to the payoff delivered by that strategy. We derive the exact solution of the model in the "thermodynamic" limit of infinitely many agents using tools of statistical physics of disordered systems. Our results show that the impact of agents on the market price plays a key role: even though price has a weak dependence on the behavior of each individual agent, the collective behavior crucially depends on whether agents account for such dependence or not. Remarkably, if the adaptive behavior of agents accounts even "infinitesimally" for this dependence they can, in a whole range of parameters, reduce global fluctuations by a finite amount. Both global efficiency and individual utility improve with respect to a "price taker" behavior if agents account for their market impact.
By considering diffusion on De Bruijn graphs, we study in details the dynamics of the histories i... more By considering diffusion on De Bruijn graphs, we study in details the dynamics of the histories in the Minority Game, a model of competition between adaptative agents. Such graphs describe the structure of temporal evolution of $M$ bits strings, each node standing for a given string, i.e. a history in the Minority Game. We show that the frequency of visit of each history is not given by $1/2^M$ in the limit of large $M$ when the transition probabilities are biased. Consequently all quantities of the model do significantly depend on whether the histories are real, or uniformly and randomly sampled. We expose a self-consistent theory of the case of real histories, which turns out to be in very good agreement with numerical simulations.
We show that the Minority Game, a model of interacting heterogeneous agents, can be described as ... more We show that the Minority Game, a model of interacting heterogeneous agents, can be described as a spin systems and it displays a phase transition between a symmetric phase and a symmetry broken phase where the games outcome is predicable. As a result a ``spontaneous magnetization'' arises in the spin formalism.
We study analytically a simple game theoretical model of heterogeneous interacting agents. We sho... more We study analytically a simple game theoretical model of heterogeneous interacting agents. We show that the stationary state of the system is described by the ground state of a disordered spin model which is exactly solvable within the simple replica symmetric ansatz. Such a stationary state differs from the Nash equilibrium where each agent maximizes her own utility. The latter turns out to be characterized by a replica symmetry broken structure. Numerical results fully agree with our analytic findings.
Physica A-statistical Mechanics and Its Applications, 2000
Using the Minority Game model we study a broad spectrum of problems of market mechanism. We study... more Using the Minority Game model we study a broad spectrum of problems of market mechanism. We study the role of different types of agents: producers, speculators as well as noise traders. The central issue here is the information flow : producers feed in the information whereas speculators make it away. How well each agent fares in the common game depends on the market conditions, as well as their sophistication. Sometimes there is much to gain with little effort, sometimes great effort virtually brings no more incremental gain. Market impact is shown to play also an important role, a strategy should be judged when it is actually used in play for its quality. Though the Minority Game is an extremely simplified market model, it allows to ask, analyze and answer many questions which arise in real markets.
Physica A-statistical Mechanics and Its Applications, 2004
We mathematize El Farol bar problem and transform it into a workable model. In general, the avera... more We mathematize El Farol bar problem and transform it into a workable model. In general, the average convergence to optimality at the collective level is trivial and does not even require any intelligence on the side of agents. Secondly, specializing to a particular ensemble of continuous strategies yields a model similar to the Minority Game. Statistical physics of disordered systems allows us to derive a complete understanding of the complex behavior of this model, on the basis of its phase diagram.
We discuss in detail the derivation of stochastic differential equations for the continuum time l... more We discuss in detail the derivation of stochastic differential equations for the continuum time limit of the Minority Game. We show that all properties of the Minority Game can be understood by a careful theoretical analysis of such equations. In particular, i) we confirm that the stationary state properties are given by the ground state configurations of a disordered (soft) spin system; ii) we derive the full stationary state distribution; iii) we characterize the dependence on initial conditions in the symmetric phase and iv) we clarify the behavior of the system as a function of the learning rate. This leaves us with a complete and coherent picture of the collective behavior of the Minority Game. Strikingly we find that the temperature like parameter which is introduced in the choice behavior of individual agents turns out to play the role, at the collective level, of the inverse of a thermodynamic temperature.
Physica A-statistical Mechanics and Its Applications, 1998
We propose and study a simple model of dynamical redistribution of capital in a diversified portf... more We propose and study a simple model of dynamical redistribution of capital in a diversified portfolio. We consider a hypothetical situation of a portfolio composed of N uncorrelated stocks. Each stock price follows a multiplicative random walk with identical drift and dispersion. The rules of our model naturally give rise to power law tails in the distribution of capital fractions invested in different stocks. The exponent of this scale free distribution is calculated in both discrete and continuous time formalism. It is demonstrated that the dynamical redistribution strategy results in a larger typical growth rate of the capital than a static "buy-and-hold" strategy. In the large N limit the typical growth rate is shown to asymptotically approach that of the expectation value of the stock price . The finite dimensional variant of the model is shown to describe the partition function of directed polymers in random media.
We address the question of market efficiency using the Minority Game (MG) model. First we show th... more We address the question of market efficiency using the Minority Game (MG) model. First we show that removing unrealistic features of the MG leads to models which reproduce a scaling behaviour close to what is observed in real markets. In particular we find that (i) fat tails and clustered volatility arise at the phase transition point and that (ii) the crossover to random walk behaviour of prices is a finite-size effect. This, on one hand, suggests that markets operate close to criticality, where the market is marginally efficient. On the other it allows one to measure the distance from criticality of real markets, using cross-over times. The artificial market described by the MG is then studied as an ecosystem with different species of traders. This clarifies the nature of the interaction and the particular role played by the various populations.
We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the development of financia... more We contrast Arbitrage Pricing Theory (APT), the theoretical basis for the development of financial instruments, with a dynamical picture of an interacting market, in a simple setting. The proliferation of financial instruments apparently provides more means for risk diversification, making the market more efficient and complete. In the simple market of interacting traders discussed here, the proliferation of financial instruments erodes systemic stability and it drives the market to a critical state characterized by large susceptibility, strong fluctuations and enhanced correlations among risks. This suggests that the hypothesis of APT may not be compatible with a stable market dynamics. In this perspective, market stability acquires the properties of a common good, which suggests that appropriate measures should be introduced in derivative markets, to preserve stability.
We study scale free simple graphs with an exponent of the degree distribution $\gamma$ less than ... more We study scale free simple graphs with an exponent of the degree distribution $\gamma$ less than two. Generically one expects such extremely skewed networks -- which occur very frequently in systems of virtually or logically connected units -- to have different properties than those of scale free networks with $\gamma>2$: The number of links grows faster than the number of nodes and they naturally posses the small world property, because the diameter increases by the logarithm of the size of the network and the clustering coefficient is finite. We discuss a simple prototype model of such networks, inspired by real world phenomena, which exhibits these properties and allows for a detailed analytical investigation.
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Papers by Matteo Marsili