Mirta B. Gordon

Social Interactions in Economic Theory: An Insight from Statistical Mechanics

This Chapter extends some economic models that take advantage of a formalism inspired from statistical mechanics to account for social influence in individual decisions. Starting with a framework suggested by Durlauf, Blume and Brock, we introduce three classes of models shifting progressively from rational towards adaptive expectations. We discuss the risk and opportunity of transposing the tools, methods and concepts from statistical mechanics to economics. We also analyze some issues seldom addressed, such as a comparison between two models of heterogeneous idiosyncratic preferences, corresponding to cases with quenched and annealed disorder in statistical physics, respectively.

DISCRETE CHOICES UNDER SOCIAL INFLUENCE: GENERIC PROPERTIES

We consider a model of socially interacting individuals that make a binary choice in a context of positive additive endogenous externalities. It encompasses as particular cases several models from the sociology and economics literature. We extend previous results to the case of a general distribution of idiosyncratic preferences, called here Idiosyncratic Willingnesses to Pay (IWP).Positive additive externalities yield a family of inverse demand curves that include the classical downward sloping ones but also new ones with non constant convexity. When

Characterization of the Sonar Signals Benchmark

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A random walk in the literature on criminality: A partial and critical view on some statistical analyses and modelling approaches

, the ratio of the social influene strength to the standard deviation of the IWP distribution, is small enough, the inverse demand is a classical monotonic (decreasing) function of the adoption rate. Even if the IWP distribution is mono-modal, there is a critical value of

MDLChunker: A MDL-Based Cognitive Model of Inductive Learning

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Inducing High-Level Behaviors from Problem-Solving Traces Using Machine-Learning Tools

above which the inverse demand is non monotonic, decreasing for small and high adoption rates, but increasing within some intermediate range. Depending on the price there are thus either one or two equilibria.Beyond this first result, we exhibit the {\em generic} properties of the boundaries limiting the regions where the system presents different types of equilibria (unique or multiple). These properties are shown to depend {\em only} on qualitative features of the IWP distribution: modality (number of maxima), smoothness and type of support (compact or infinite).The main results are summarized as {\em phase diagrams} in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited.

Collective states in social systems with interacting learning agents

We study the classification of sonar targets first introduced by Gorman & Sejnowski (1988). We discovered that not only the training set and the test set of this benchmark are both linearly separable, although by different hyperplanes, but that the complete set of patterns, training and test patterns together, is also linearly separable. The distances of the patterns to the separating hyperplane determined by learning with the training set alone, and to the one determined by learning the complete data set, are presented.

Efficient adaptive learning for classification tasks with binary units

We are interested in the possible contributions of mathematical modelling of crime. We refer to numerous and quite recent papers that analyse and discuss empirical data in an attempt to discover stylised trends worthy of being understood through simple models. We summarise part of this literature and try to understand the reasons of important discrepancies in their conclusions. Then we present some recent modelling attempts that may help to understand the large variance in the statistical-based conclusions.

Chapter 8 Choice under Social Influence: Effects of Learning Behaviours on the Collective Dynamics

This paper presents a computational model of the way humans inductively identify and aggregate concepts from the low-level stimuli they are exposed to. Based on the idea that humans tend to select the simplest structures, it implements a dynamic hierarchical chunking mechanism in which the decision whether to create a new chunk is based on an information-theoretic criterion, the Minimum Description Length (MDL) principle. We present theoretical justifications for this approach together with results of an experiment in which participants, exposed to meaningless symbols, have been implicitly encouraged to create high-level concepts by grouping them. Results show that the designed model, called hereafter MDLChunker, makes precise quantitative predictions both on the kind of chunks created by the participants and also on the moment at which these creations occur. They suggest that the simplicity principle used to design MDLChunker is particularly efficient to model chunking mechanisms. The main interest of this model over existing ones is that it does not require any adjustable parameter.

PHASE TRANSITIONS IN OPTIMAL UNSUPERVISED LEARNING

Many researchers consider interactive learning environments to be interesting solutions for overcoming the limits of classical one-to-many teaching methods. However, these environments should incorporate accurate representations of student knowledge to provide relevant guidance. In a problem-solving environment, one way to build and update this student model is model tracing, or using a detailed representation of cognitive skills to precisely follow what the student is doing. Some model-tracing tutors such as PAT (Personal Algebra Tutor) contain rules that the system can use to solve the problem and assess the students solution.