Mirta Gordon | Centre National de la Recherche Scientifique / French National Centre for Scientific Research (original) (raw)

Papers by Mirta Gordon

RePEc: Research Papers in Economics, 2005

International audienceIn this paper, we consider a discrete choice model where heterogeneous agen... more International audienceIn this paper, we consider a discrete choice model where heterogeneous agents are subject to mutual influences. We explore some consequences on the market's behaviour, in the simplest case of a uniform willingness to pay distribution. We exhibit a first-order phase transition in the profit optimization by the monopolist: if the social influence is strong enough, there is a regime where, if the mean willingness to pay increases, or if the production costs decrease, the optimal solution for the monopolist jumps from a solution with a high price and a small number of buyers, to a solution with a low price and a large number of buyers. Depending on the path of prices adjustments by the monopolist, simulations show hysteretic effects on the fraction of buyers

We study social organizations with possible coexistence at equilibrium of cooperating individuals... more We study social organizations with possible coexistence at equilibrium of cooperating individuals and pure consumers (free-riders). We investigate this polymorphic equilibrium using a game-theoretic approach and a statistical physics analysis of a simple model. The agents face a binary decision problem: whether to contribute or not to the public good, through the maximization of an additive utility that has two competing terms, a fixed cost for cooperating and an idiosyncratic moral cost for free-riding proportional to the fraction of cooperators. We study the equilibria regimes of this model. We show that there is a fraction of expected cooperators below which cooperation fails to emerge. Besides the homogeneous stable equilibria (everybody cooperates or everybody free-rides), it exists a solution in which cooperators coexist with free-riders. This polymorphic equilibrium is a consequence of the heterogeneous (idiosyncratic) perceptions of the social reproval by the different indiv...

Cornell University - arXiv, Mar 14, 2002

Typical learning curves for Soft Margin Classifiers (SMCs) learning both realizable and unrealiza... more Typical learning curves for Soft Margin Classifiers (SMCs) learning both realizable and unrealizable tasks are determined using the tools of Statistical Mechanics. We derive the analytical behaviour of the learning curves in the regimes of small and large training sets. The generalization errors present different decay laws towards the asymptotic values as a function of the training set size, depending on general geometrical characteristics of the rule to be learned. Optimal generalization curves are deduced through a fine tuning of the hyperparameter controlling the trade-off between the error and the regularization terms in the cost function. Even if the task is realizable, the optimal performance of the SMC is better than that of a hard margin Support Vector Machine (SVM) learning the same rule, and is very close to that of the Bayesian classifier.

In this article we study the effects of introducing structure in the input distribution of the da... more In this article we study the effects of introducing structure in the input distribution of the data to be learnt by a simple perceptron. We determine the learning curves within the framework of Statistical Mechanics. Stepwise generalization occurs as a function of the number of examples when the distribution of patterns is highly anisotropic. Although extremely simple, the model seems to capture the relevant features of a class of Support Vector Machines which was recently shown to present this behavior.

Journal de Physique, 1987

Nous présentons un modèle de mémoire à long terme : apprentissage avec bornes irréversibles. Les ... more Nous présentons un modèle de mémoire à long terme : apprentissage avec bornes irréversibles. Les meilleures valeurs des bornes et la capacité de mémoire sont déterminés numériquement. Nous montrons qu'il est possible en général de calculer analytiquement la capacité de mémoire si l'on résout le problème de marche aléatoire associé à chaque règle d'apprentissage. Nos estimations 2014 faites pour plusieurs règles d'apprentissage 2014 sont en excellent accord avec les résultats numériques et de mécanique statistique. Abstract. 2014 We present a model of long term memory : learning within irreversible bounds. The best bound values and memory capacity are determined numerically. We show that it is possible in general to calculate analytically the memory capacity by solving the random walk problem associated to a given learning rule. Our estimations 2014 done for several learning rules 2014 are in excellent agreement with numerical and analytical statistical mechanics results.

Journal de Physique I, 1993

Physical Review E, 2001

We study the typical learning properties of the recently introduced Soft Margin Classifiers (SMCs... more We study the typical learning properties of the recently introduced Soft Margin Classifiers (SMCs), learning realizable and unrealizable tasks, with the tools of Statistical Mechanics. We derive analytically the behaviour of the learning curves in the regime of very large training sets. We obtain exponential and power laws for the decay of the generalization error towards the asymptotic value, depending on the task and on general characteristics of the distribution of stabilities of the patterns to be learned. The optimal learning curves of the SMCs, which give the minimal generalization error, are obtained by tuning the coefficient controlling the trade-off between the error and the regularization terms in the cost function. If the task is realizable by the SMC, the optimal performance is better than that of a hard margin Support Vector Machine and is very close to that of a Bayesian classifier.

Journal of Physics A: Mathematical and General, 2000

Upper and lower bounds for the typical storage capacity of a constructive algorithm, the Tilingli... more Upper and lower bounds for the typical storage capacity of a constructive algorithm, the Tilinglike Learning Algorithm for the Parity Machine [M. Biehl and M. Opper, Phys. Rev. A 44 6888 (1991)], are determined in the asymptotic limit of large training set sizes. The properties of a perceptron with threshold, learning a training set of patterns having a biased distribution of targets, needed as an intermediate step in the capacity calculation, are determined analytically. The lower bound for the capacity, determined with a cavity method, is proportional to the number of hidden units. The upper bound, obtained with the hypothesis of replica symmetry, is close to the one predicted by Mitchinson and Durbin [Biol. Cyber. 60 345 (1989)].

Journal of Physics A: Mathematical and General, 1994

We study the dynamics of a stepped crystal surface during evaporation, using the classical model ... more We study the dynamics of a stepped crystal surface during evaporation, using the classical model of Burton, Cabrera and Frank, in which the dynamics of the surface is represented as a motion of parallel, monoatomic steps. The validity of the continuum approximation treated by Frank is checked against numerical calculations and simple, qualitative arguments. The continuum approximation is found to suffer from limitations related, in particular, to the existence of angular points. These limitations are often related to an adatom detachment rate of adatoms which is higher on the lower side of each step than on the upper side ("Schwoebel effect").

Bulletin of Mathematical Biology, 2001

We present a simple model in order to discuss the interaction of the genetic and behavioral syste... more We present a simple model in order to discuss the interaction of the genetic and behavioral systems throughout evolution. This considers a set of adaptive perceptrons in which some of their synapses can be updated through a learning process. This framework provides an extension of the well-known Hinton and Nowlan model by blending together some learning capability and other (rigid) genetic effects that contribute to the fitness. We find a halting effect in the evolutionary dynamics, in which the transcription of environmental data into genetic information is hindered by learning, instead of stimulated as is usually understood by the so-called Baldwin effect. The present results are discussed and compared with those reported in the literature. An interpretation is provided of the halting effect.

We study the mean field dynamics of a model introduced by Phan et al [Wehia, 2005] of a polymorph... more We study the mean field dynamics of a model introduced by Phan et al [Wehia, 2005] of a polymorphic social community. The individuals may choose between three strategies: either not to join the community or, in the case of joining it, to cooperate or to behave as a free-rider. Individuals' preferences have an idiosyncratic component and a social component. Cooperators bear a fixed cost whereas free-riders support a cost proportional to the number of cooperators. We study the dynamics of this model analytically in the mean field approximation for both parallel and sequential updating. As we vary one of the parameters while keeping the other parameters fixed, the phase diagram experiences a rich class of bifurcations. Noticeably, a limit cycle is shown to exist in both parallel and sequential updating, under certain parameter settings. A comparison of the analytical predictions with computer simulations is also included.

summary – STATISTICAL PHYSICS OF COLLECTIVE PHENOMENA IN SOCIAL ANDECONOMIC SCIENCESThis article ... more summary – STATISTICAL PHYSICS OF COLLECTIVE PHENOMENA IN SOCIAL ANDECONOMIC SCIENCESThis article shows how statistical physics may contribute to the modelling of collective phenomenain economics and social science. The main topic here is the study of the global (aggregate) behaviorof a large population, when the agents make choices under social inﬂuence. We present severalexamples, starting from pioneering works in economics and sociology, such as those of T. Schellingwhose approach has all the ﬂavour of physicists’ approaches. keywords – Collective phenomena, Discrete choices with externalities, Emergence, Socialinﬂuence, Schelling, Statistical physics. 1. INTRODUCTIONCela a-t-il un sens d’´etablir une analogie entre le comportement d’un mat´eriau ma-gn´etique comme un aimant et celui d’un ensemble d’individus rationnels devant faireun choix ou prendre une d´ecision? S’il y en a une, elle ne saute pas aux yeux, ousi on la devine elle semble a priori na¨ive. Pourtant, vers la ﬁn des ann´ees 70, Tho-mas Schelling, ´economiste, sociologue et politologue, utilisait des analogies avec lesph´enom`enes physiques pour mod´eliser des faits stylis´es en sciences sociales. Dansson livre [Schelling, 1978], dont le titre en francais est La tyrannie des petites d´eci-sions, il montre comment certains ph´enom`enes, dits de masse critique, peuvent ˆetremod´elis´es avec des approches emprunt´ees a la Physique. Ainsi, il montre comment

We review the problem of extending the applicability of support vector machines (SVM) to graph da... more We review the problem of extending the applicability of support vector machines (SVM) to graph data. Many similarity measures, generally called kernels, on graph data have been proposed in the last decade. Yet some of them, like the optimum assignment kernel (15), are not positive semidefinite, which limits their application in SVM. In this paper we recall the necessary conditions for using SVM. While the Mercer theorem gives necessary and sufficient conditions for vectorial data, we show that for graph data an embedding in a Hilbert space has to be defined explicitly, and that weaker conditions do not suffice. For several kernels proposed in the literature we demonstrate that an underlying Hilbert space does exist by specifying the corresponding basis.Our findings are illustrated with small examples from the graph kernel literature.

... Chaque agent a des préférences Hi qui lui sont propres (idiosyncratiques) ; plus Hi est grand... more ... Chaque agent a des préférences Hi qui lui sont propres (idiosyncratiques) ; plus Hi est grand, plus la volonté de faire le choix ωi = 1 est grande. Dans le cas du marché, c'est le prix de réserve de l'agent : l'agent ach`ete si le prix P est au plus égal `a Hi. ...

We explore the effects of social influence in a simple market model in which a large number of ag... more We explore the effects of social influence in a simple market model in which a large number of agents face a binary choice: ’to buy/not to buy ’ a single unit of a product at a price posted by a single seller (monopoly market). We consider the case of positive externalities: an agent is more willing to buy if the other agents with whom he/she interacts make the same decision. We compare two special cases known in the economics literature as the Thurstone and the McFadden approaches. We show that they correspond to modeling the heterogenity in individual decision rules with, respectively, annealed and quenched disorder. More precisely the first case leads to a standard Ising model at finite temperature in a uniform external field, and the second case to a random field Ising model (RFIM) at zero temperature. We illustrate some dynamic properties of these models, making use of numerical simulations in an ACE (Agent based Computational Economics) approach, and we study analytically the ...