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Discrete Choices under Social Influence: Generic Properties
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Abstract
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 j, the ratio of the social influence 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 j 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 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 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 phase diagrams in the space of the model parameters, on which the regions of multiple equilibria are precisely delimited.
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We present a model of opinion dynamics in which agents adjust continuous opinions as a result of random binary encounters whenever their difference in opinion is below a given threshold. High thresholds yield convergence of opinions towards an average opinion, whereas low thresholds result in several opinion clusters: members of the ame cluster share the same opinion but are no longer influenced by members of other clusters.
In this lecture we introduce the concepts and tools of Statistical Mechanics using the Ising model of condensed matter physics, proposed about one century ago to explain magnetism, as a paradigm. The evocative power of this model brought the techniques of Statistical Mechanics to other fields, like brain models in Cognitive Science, Machine Learning Theory, and more recently to the analysis of models in Game Theory, Economy and Sociology.
This Chapter extends some economic models that take advantage of a formalism inspired from statistical mechanics to account for social inuence 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.
In this paper we explore the impact of social influence on the equilibria in the simplest model with binary choices. We investigate the interplay between the social influence and individual decision to adopt with global interactions, both analytically in the case of an infinite number and numerically in the case of a finite number of agents. We derive exactly necessary and sufficient conditions for existence of single or multiple equilibria as a function of the strength of social influence. In particular, this model encompasses the classical downward sloping demand curve as a particular case. The paper also provides a brief introduction to avalanches and hysteresis loops, and includes extensive bibliographical references.
We unveil collective effects induced by imitation and social pressure by analyzing data from three different sources: birth rates, sales of cell phones and the drop of applause in concert halls. We interpret our results within the framework of the Random Field Ising Model, which is a threshold model for collective decisions accounting both for agent heterogeneity and social imitation. Changes of opinion can occur either abruptly or continuously, depending on the importance of herding effects. The main prediction of the model is a scaling relation between the height h of the speed of variation peak and its width w of the form h ∼w -κ , with κ= 2/3 for well connected populations. Our three sets of data are compatible with such a prediction, with κ≈0.62 for birth rates, κ≈0.71 for cell phones and κ≈0.64 for clapping. In this last case, we in fact observe that some clapping samples end discontinuously (w=0), as predicted by the model for strong enough imitation. Copyright EDP Sciences/Società Italiana di Fisica/Springer-Verlag 2005
Risk control and derivative pricing have become of major concern to financial institutions, and there is a real need for adequate statistical tools to measure and anticipate the amplitude of the potential moves of the financial markets. Summarising theoretical developments in the field, this 2003 second edition has been substantially expanded. Additional chapters now cover stochastic processes, Monte-Carlo methods, Black-Scholes theory, the theory of the yield curve, and Minority Game. There are discussions on aspects of data analysis, financial products, non-linear correlations, and herding, feedback and agent based models. This book has become a classic reference for graduate students and researchers working in econophysics and mathematical finance, and for quantitative analysts working on risk management, derivative pricing and quantitative trading strategies.
We here relate the occurrence of extreme market shares, close to either 0 or 100%, in the media industry to a percolation phenomenon across the social network of customers. We further discuss the possibility of observing self-organized criticality when customers and cinema producers adjust their preferences and the quality of the produced films according to previous experience. Comprehensive computer simulations on square lattices do indeed exhibit self-organized criticality towards the usual percolation threshold and related scaling behaviour.
We discuss the influence of information contagion on the dynamics of choices in social networks of heterogeneous buyers. Starting from an inhomogeneous cellular automata model of buyers dynamics, we show that when agents try to adjust their reservation price, the tatonnement process does not converge to equilibrium at some intermediate market share and that large amplitude fluctuations are actually observed. When the tatonnement dynamics is slow with respect to the contagion dynamics, large periodic oscillations reminiscent of business cycles appear.
The role of interindividu al influences in the determination of stock prices
Modem economic theory has shown that there is no incompatibility between the assumption of rational behavior and the possibility of speculative bubbles. Nevertheless the theory of rational bubbles is incomplete insofar as it tells us nothing about the kind of rational exploitation of information that gives lise to a bubble. Our idea is that, at the roots of bubbles, we find a rational use of the information contained in the average opinion of the market. Under certain conditions, this use, which follows the bayesian rules, will conduct to a self-validating mimetic contagion. This process converges to, what we have called, a rational mimetic bubble.
We propose a generic model for multiple choice situations in the presence of herding and compare it with recent empirical results from a Web-based music market experiment. The model predicts a phase transition between a weak imitation phase and a strong imitation, `fashion' phase, where choices are driven by peer pressure and the ranking of individual preferences is strongly distorted at the aggregate level. The model can be calibrated to reproduce the main experimental results of Salganik et al. (Science, 311, pp. 854-856 (2006)); we show in particular that the value of the social influence parameter can be estimated from the data. In one of the experimental situation, this value is found to be close to the critical value of the model.
Evidence relating unemployment to health is found at every level of social science analysis from national population rates to individual psychophysiological stress response. At the population level of analysis, increase in the unemployment rate indicates recession and/or structural economic decline. At the individual level, unemployment is interpreted as a stressful life event. In both cases, inverse associations are found between measures of unemployment and indicators of health. We identify social science literatures associating health indicators with each of the following: economic growth, socioeconomic status, sociocultural change, economic instability, the status of being unemployed, social stress and work stress. Outstanding research issues include the requirements to identify and measure the effects of conditional factors and control variables in multivariate analysis and to examine a broader range of both severity of unemployment and severity of health outcomes. A research agenda proposes studies at the macro, meso and micro levels of analysis. We urge such research for its potential contribution both to analytic social science and to economic and social policy.
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 other agents make the same decision. We consider two special cases of heterogeneity in the individuals' decision rules, corresponding in the literature to the Random Utility Models of Thurstone, and of McFadden and Manski. In the first one the heterogeneity fluctuates with time, leading to a standard model in Physics: the Ising model at finite temperature (known as annealed disorder) in a uniform external field. In the second approach the heterogeneity among agents is fixed; in Physics this is a particular case of the quenched disorder model known as a random field Ising model, at zero temperature. We study analytically the equilibrium properties of the market in the limiting case where each agent is influenced by all the others (the mean field limit), and we illustrate some dynamic properties of these models making use of numerical simulations in an Agent based Computational Economics approach. Considering the optimization of the profit by the seller within the case of fixed heterogeneity with global externality, we exhibit a new regime where, if the mean willingness to pay increases and/or the production costs decrease, the seller's optimal strategy jumps from a solution with a high price and a small number of buyers, to another one with a low price and a large number of buyers. This regime, usually modelled with ad hoc bimodal distributions of the idiosyncratic heterogeneity, arises here for general monomodal distributions if the social influence is strong enough.
The paper extends the Brock-Durlauf social interactions model to richer social structures modelled with arbitrary interaction topologies and examines in detail the star, the wheel and the path. It explores Nash equilibria when agents act on the basis of expectations over, and, alternatively, actual knowledge their neighbors’ decisions. It links social interactions with econometrics of systems of simultaneous equations. The local dynamics near steady states combine spectral properties of the adjacency matrix and of the nonlinearities of reaction functions. For regular interaction topologies, adjustment exhibit relative persistence. Cyclical interaction is associated with endogenous spatial oscillations and islands of conformity. Copyright Springer-Verlag Berlin/Heidelberg 2006
This important study shows that an understanding of product differentiation is crucial to understanding how modern market economies function and that differentiated markets can be analyzed using discrete choice models of consumer behavior.
Product differentiation - in quality, packaging, design, color, and style - has an important impact on consumer choice. It also provides a rich source of data that has been largely unexplored because there has been no generally accepted way to model the information available. This important study shows that an understanding of product differentiation is crucial to understanding how modern market economies function and that differentiated markets can be analyzed using discrete choice models of consumer behavior. It provides a valuable synthesis of existing, often highly technical work in both differentiated markets and discrete choice models and extends this work to establish a coherent theoretical underpinning for research in imperfect competition.The discrete choice approach provides an ideal framework for describing the demands for differentiated products and can be used for studying most product differentiation models in the literature. By introducing extra dimensions of product heterogeneity, the framework also provides richer models of firm location. Discrete Choice Theory of Product Differentiation introduces students and researchers to the field, starting at the beginning and moving through to frontier research. The first four chapters detail the consumer-theoretic foundations underlying choice probability systems (including an overview of the main models used in the psychological theory of choice), while the next four chapters apply the probabilistic choice approach to oligopoly models of product differentiation, product selection, and location choice. The final chapter suggests various extensions of the models presented as well topics for further research.
We investigate models of opinion formation which are based on the social impact theory. The following approaches are discussed: (i) general mean field theory of social impact, (ii) a social impact model with learning, (iii) a model of a finite group with a strong leader, (iv) a social impact model with dynamically changing social temperature, (v) a model with individuals treated as active Brownian particles interacting via a communication field.
This paper is about binary choices with externalities. These are either-or situations,not choices of degree or quantity. An "externality" is present when you care about my choice or my choice affects yours. You may not care, but need to know-whether to pass on left or right when we meet. You may not need to know, but care---you will drive whether or not I drive--- but prefer that I keep off the road.You may both care and need to know. The literature of externalities has mostly to do with how much of a good or a bad should be produced,consumed,or allowed.HereI consider only the interdependence of choices to do or not to do, to join or not to join, to stay or to leave, to vote yes or no, to conform or not to conform to some agreement or rule or restriction. Players will accept hockey helmets (or not) by individual choice for several reasons. Chicago star Bobby Hull cites the simplest factor: "vanity." But many players honestly believe that helmets will cut their efficiency and put them at a disadvantage, and others fear the ridicule of opponents. The use of helmets will spread only through fear caused by injuries like Green's-or through a rule making them mandatory. . . One player summed up the feelings of many: "It's foolish not to wear a helmet. But I don't because the other guys don't. I know that' silly, but most of the players feel the same way. If the league made us do it, though, we'd all wear them and nobody would mind." Shortly after Teddy Green of the Bruins took a hockey stick in his brain, the player, Don Awrey, commented to a Newsweek(1969) reporter, "WhenI saw the way Teddy looked, it was an awful feeling . . . I'm going to start wearing a helmet now, and I don't care what anybody says." A voluntary helmet may be seen as cowardly,but nobody thinks a baseball player timid when he dons the batting helmet without which the league will not let him bat. Motorcycle helmets are not only worn regularly,but probably worn more gladly, in states that require them. When ever ascribed motives matter, the way a choice is organized or constrained will itself be a part of the "outcome" and affect the payoffs.
Why are the social problems of ghettos so bad? This article proposes that ghettos are communities that have experienced epidemics of social problems. One important implication of this theory is that the pattern of neighborhood effects on social problems should be nonlinear in large cities. As neighborhood quality decreases, there should be a sharp increase in the probability that an individual will develop a social problem. The jump should occur somewhere near the bottom of the distribution of neighborhood quality. This hypothesis is tested by analyzing the pattern of neighborhood effects on dropping out and teenage childbearing. The analysis strongly supports the hypothesis, with exceptions for certain subgroups. Even after controlling for individual characteristics, black and white adolescents are exposed to sharp increases in the risk of dropping out and having a child in the worst neighborhoods in large cities.
I. The nature of the problem, 183. — II. Functional and nonfunctional demand, 188. — III. The bandwagon effect, 190. — IV. The snob effect, 199. — V. The Veblen effect, 202. — VI. Mixed effects, 205. — VII. Conclusion, 206.
One of the core assumptions in the study of complex systems is that there exist ‘universal’ features analogous to those that characterize the notion of universality in statistical physics. That is to say, sometimes the details do not matter: certain aspects of complex behaviour transcend the particularities of a given system, and are to be anticipated in any system of a multitude of simultaneously interacting components. There can be no tougher test of this idea than that posed by the nature of human social systems. Can there really be any similarities between, say, a collection of inanimate particles in a fluid interacting via simple, mathematically defined forces of attraction and repulsion, and communities of people each of whom is governed by an unfathomable wealth of psychological complexity? The traditional approach to the social sciences has tended to view these psychological factors as irreducible components of human social interactions. But attempts to model society using the methods and tools of statistical physics have now provided ample reason to suppose that, in many situations, the behaviour of large groups of people can be understood on the basis of very simple interaction rules, so that individuals act essentially as automata responding to a few key stimuli in their environment. This is clearly a challenging, perhaps even disturbing, idea. I will review some of the evidence in its favour. I will show, with examples ranging from pedestrian dynamics to social choice theory, economics, demography, and the formation of businesses and alliances, that modelling the behaviour of individuals and social and political institutions according to the viewpoint of statistical physics does seem capable of capturing some of the important features of social systems. These models reveal many of the characteristic elements displayed by other complex systems: collective dynamics that changes via abrupt shifts (phase transitions), metastability, critical phenomena and scale-free statistical variations. I will discuss what this implies for the notions of human free will and determinism.
For pt.II see ibid., vol.14, p.3603 (1981). Uniaxially anisotropic magnetic systems with both a random and a uniform magnetic field along the easy axis exhibit rather exotic phase diagrams as function of the anisotropy (a), temperature(T), random field (H0) and uniform field (H). For fixed strong anisotropies, below the tricritical temperature. the coexistence surface (H=0) bifurcates for H not=0 into a pair of symmetrical 'wings' of first-order transitions, bounded by two critical loci which meet at the tricritical point at H=0. On the other side, for fixed weak anisotropies the spin-flop phase spreads out into two symmetrical 'horns' (for H>0 and H<0) containing two tricritical lines which meet at the bicritical point at H=0. At intermediate fixed anisotropies, for large uniform fields, the upper parts of the 'horns' overlap the 'wings'. The global four-dimensional phase diagram has been explicitly constructed using mean-field theory. In the vicinity of the new multicritical point, there is a large variety of critical (random Ising-like, pure xy-like), normal and special bicritical (random Ising-like), tricritical (random Ising-like, pure xy-like), fourth-order (random Ising-like) behaviours, as well as crossovers between these behaviours. The results are applicable to the description of dilute antiferromagnets in a uniform field.
Mean-field and renormalisation group theories are used to study the diagrams of anisotropic n-component d-dimensional magnetic systems with a uniaxially random magnetic field along the easy axis. For strong anisotropy only the longitudinal phase exists. The transition into this phase exhibits a tricritical point. When the anisotropy is decreased, a 'transverse' or 'spin-flopped' phase appears for low temperatures and large values of the random field. This phase is separated from the longitudinal one by a spin-flop first-order line, ending at a critical end-point. For very weak anisotropy, a bicritical point appears in the phase diagram. For a certain value of the anisotropy the critical end-point, the bicritical point and the tricritical point all coincide defining a new multicritical point.
This paper examines the spatial patterns of unemployment in Chicago between 1980 and 1990. We study unemployment clustering with respect to different social and economic distance metrics that reflect the structure of agents' social networks. Specifically, we use physical distance, travel time, and differences in ethnic and occupational distribution between locations. Our goal is to determine whether our estimates of spatial dependence are consistent with models in which agents' employment status is affected by information exchanged locally within their social networks. We present non-parametric estimates of correlation across Census tracts as a function of each distance metric as well as pairs of metrics, both for unemployment rate itself and after conditioning on a set of tract characteristics. Our results indicate that there is a strong positive and statistically significant degree of spatial dependence in the distribution of raw unemployment rates, for all our metrics. However, once we condition on a set of covariates, most of the spatial autocorrelation is eliminated, with the exception of physical and occupational distance. Racial and ethnic composition variables are the single most important factor in explaining the observed correlation patterns. Copyright © 2002 John Wiley & Sons, Ltd.
Recent work in economics has begun to integrate sociological ideas into the modeling of individual behavior. In particular, this new approach emphasizes how social context and social interdependences influence the ways in which individuals make choices. This paper provides an overview of an approach to integrating theoretical and empirical analysis of such environments. The analysis is based on a framework due to Brock and Durlauf (2001, forthcoming). Empirical evidence on behalf of this perspective is assessed and some policy implications are explored.
I am grateful to Joseph B. Kadane for numerous constructive suggestions offered during discussions of this research. The financial sponsorship of the U.S. Department of Transportation through grant DOT-OS-4006 is also acknowledged. The opinions and conclusions expressed herein are solely those of the author.
We investigate the use of standard statistical models for quantal choice in a game theoretic setting. Players choose strategies based on relative expected utility and assume other players do so as well. We define a quantal response equilibrium (ORE) as a fixed point of this process and establish existence. For a logit specification of the error structure, we show that as the error goes to zero, QRE approaches a subset of Nash equilibria and also implies a unique selection from the set of Nash equilibria in generic games. We fit the model to a variety of experimental data sets by using maximum likelihood estimation. Journal of Economic Literature Classification Numbers: C19, C44, C72, C92.
The influence of peer behavior on an individual's choices has received renewed interest in recent years. However, accurate measures of this influence are difficult to obtain. Standard reduced-form methods lead to upwardly biased estimates due to simultaneity, common shocks, and nonrandom peer group selection. This paper describes a structural econometric model of peer effects in binary choice, as well as a simulated maximum likelihood estimator for its parameters. The model is nonparametrically identified under plausible restrictions, and can place informative bounds on parameter values under much weaker restrictions. Monte Carlo results indicate that this estimator performs better than a reduced form approach in a wide variety of settings. A brief application to youth smoking demonstrates the method and suggests that previous studies dramatically overstate peer influence.
In 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.
A general concept is presented which allows of setting up mathematical models for stochastic and quasi deterministic dynamic processes in social systems.
The basis of this concept is the master equation for the probability distribution over appropriately chosen personal and material macrovariables of the society. The probabilistic transition rates depend on motivation potentials governing the decisions and actions of the social agents. The transition from the probability distribution to quasi-meanvalues leads to in general nonlinear coupled differential equations for the macrovariables of the chosen social sector. Up to now several models about population dynamics, collective political opinion formation, dynamics of economic processes and the formation of settlements have been published.
The partition function of a two-dimensional "ferromagnetic" with scalar
"spins" (Ising model) is computed rigorously for the case of vanishing
field. The eigenwert problem involved in the corresponding computation
for a long strip crystal of finite width (n atoms), joined straight
to itself around a cylinder, is solved by direct product decomposition;
in the special case n=â an integral replaces a sum. The choice of
different interaction energies (±J,±Jâ²) in the (0 1) and (1 0) directions
does not complicate the problem. The two-way infinite crystal has
an order-disorder transition at a temperature T=Tc given by the condition
sinh(2J / kTc) sinh(2Jâ² / kTc)=1. The energy is a continuous function
of T; but the specific heat becomes infinite as -log |T-Tc|. For
strips of finite width, the maximum of the specific heat increases
linearly with log n. The order-converting dual transformation invented
by Kramers and Wannier effects a simple automorphism of the basis
of the quaternion algebra which is natural to the problem in hand.
In addition to the thermodynamic properties of the massive crystal,
the free energy of a (0 1) boundary between areas of opposite order
is computed; on this basis the mean ordered length of a strip crystal
is (exp (2J / kT) tanh(2Jâ² / kT))n.
L'article consiste en une discussion de modèles élaborés dans les années 1970 pour rendre compte de la diffusion du service téléphonique. Ces modèles font jouer un rôle essentiel à l'externalité positive de consommation qui est associée à la croissance d'un réseau de communication: le réseau étant d'autant plus utile qu'il est plus étendu, il croît par effet " boule de neige ", attirant en son sein des individus dont le revenu est de plus en plus faible. L'étude dynamique et statique de ce processus de diffusion montre l'existence d'un nombre impair de tailles d'équilibre pour le parc téléphonique, alternativement stables et instables. Les différents modèles originellement apparus dans la littérature sont d'abord commentés en détail, puis on développe à partir de cette matière les fondements micro-économiques d'une théorie intégrée de la demande d'accès à un réseau de télécommunication. /// In this paper, we discuss a number of models which were proposed in the 1970 s in order to explain the diffusion of the telephone service. These models are essentially based on the presence of a positive consumption externality associated with the growth of a communication network: since the network brings all the more utility as it is large, it grows according to a " snowball " effect by attracting individuals with decreasing disposable income. The dynamic and static study of that diffusion process discloses the existence of an odd number of equilibrium sizes for the network, alternatively stable and unstable. We first comment in detail the papers where the original models were introduced. Then, using this literature as a basis, we develop microeconomic fundations for an integrated theory of the demand of access to a telecommunication network.
This note tries to explain why many successful restaurants, plays, sporting events, and other activities do not raise prices even with persistent excess demand. The authors approach assumes that demand by a typical consumer is positively related to quantities demanded by other consumers. This can explain not only the puzzle about prices, but also why consumer demand is often fickle, why it is much easier to go from being "in" to being "out" than from "out" to "in," and why supply does not increase to reduce the excess demand. Copyright 1991 by University of Chicago Press.
This paper analyzes a model in which individuals care about both consumption (intrinsic utility) and social status. Status depends on public perceptions about an individual's predispositions rather than on the individual's actions. However, since predispositions are unobservable, actions signal predispositions and therefore affect status. When status is sufficiently important relative to intrinsic utility, many individuals conform to a rigid standard of behavior, despite heterogeneous intrinsic preferences. When status is relatively unimportant, no conformity emerges. The model produces both customs and fads, and it suggests an explanation for the development of multiple subcultures with distinct norms. Copyright 1994 by University of Chicago Press.
This paper presents a theoretical basis for the argument that large exchange rate shocks—such as the 1980s dollar cycle—may
have persistent effects on trade flows and the equilibrium exchange rate itself. We begin with a partial-equilibrium model
in which large exchange rate fluctuations lead to entry or exit decisions that are not reversed when the currency returns
to its previous level. Then we develop a simple model of the feedback from hysteresis in trade to the exchange rate itself.
Here we see that a large capital inflow, which leads to an initial appreciation, can result in a persistent reduction in the
exchange rate consistent with trade balance.