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# Discrete Choices under Social Influence: Generic Properties

## Author

Listed:
• Mirta B. Gordon

() (AMA - Analyse de données, Modélisation et Apprentissage automatique [Grenoble] - LIG - Laboratoire d'Informatique de Grenoble - UPMF - Université Pierre Mendès France - Grenoble 2 - UJF - Université Joseph Fourier - Grenoble 1 - INPG - Institut National Polytechnique de Grenoble - CNRS - Centre National de la Recherche Scientifique - UGA - Université Grenoble Alpes)

() (CAMS - Centre d'analyse et de mathématique sociale - EHESS - École des hautes études en sciences sociales - CNRS - Centre National de la Recherche Scientifique, LPS - Laboratoire de Physique Statistique de l'ENS - FRDPENS - Fédération de recherche du Département de physique de l'Ecole Normale Supérieure - ENS Paris - CNRS - Centre National de la Recherche Scientifique - ENS Paris - École normale supérieure - Paris - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique)

• Denis Phan

(CREM - Centre de recherche en économie et management - UNICAEN - Université de Caen Normandie - NU - Normandie Université - UR1 - Université de Rennes 1 - UNIV-RENNES - Université de Rennes - CNRS - Centre National de la Recherche Scientifique, GEMAS - Groupe d'étude des méthodes de l'analyse sociologique - UP4 - Université Paris-Sorbonne - CNRS - Centre National de la Recherche Scientifique)

• Viktoriya Semeshenko

(TIMC-IMAG - Techniques de l'Ingénierie Médicale et de la Complexité - Informatique, Mathématiques et Applications, Grenoble - UJF - Université Joseph Fourier - Grenoble 1 - Grenoble INP - Institut polytechnique de Grenoble - Grenoble Institute of Technology - IMAG - Institut d'Enseignement Supérieur et de Recherche en Alimentation, Santé Animale, Sciences Agronomiques et de l'Environnement - CNRS - Centre National de la Recherche Scientifique - UGA - Université Grenoble Alpes)

## 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 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 $j$ 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 {\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.

## Suggested Citation

• Mirta B. Gordon & Jean-Pierre Nadal & Denis Phan & Viktoriya Semeshenko, 2007. "Discrete Choices under Social Influence: Generic Properties," Working Papers halshs-00135405, HAL.
• Handle: RePEc:hal:wpaper:halshs-00135405
Note: View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-00135405
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File URL: https://halshs.archives-ouvertes.fr/halshs-00135405/document

## Citations

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Cited by:

1. Paolo Dai Pra & Fulvio Fontini & Elena Sartori & Marco Tolotti, 2011. "Endogenous equilibria in liquid markets with frictions and boundedly rational agents," Working Papers 7, Department of Management, Università Ca' Foscari Venezia.
2. Lo Schiavo, Mauro & Prinari, Barbara & Gronski, Jessica A. & Serio, Angelo V., 2015. "An artificial neural network approach for modeling the ward atmosphere in a medical unit," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 116(C), pages 44-58.
3. Mirta B. Gordon & Jean-Pierre Nadal & Denis Phan & Viktoriya Semeshenko, 2012. "Entanglement between Demand and Supply in Markets with Bandwagon Goods," Papers 1209.1321, arXiv.org, revised Dec 2012.
4. Y. P. Ma & S. Gonçalves & S. Mignot & J.-P. Nadal & M. B. Gordon, 2009. "Cycles of cooperation and free-riding in social systems," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 71(4), pages 597-610, October.
5. D. Sornette, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models," Papers 1404.0243, arXiv.org.
6. M. B. Gordon & J. R. Iglesias & V. Semeshenko & J. P. Nadal, 2009. "Crime and punishment: the economic burden of impunity," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 68(1), pages 133-144, March.
7. Jean-Philippe Bouchaud, 2012. "Crises and collective socio-economic phenomena: simple models and challenges," Papers 1209.0453, arXiv.org, revised Dec 2012.
8. Kindler, A. & Solomon, S. & Stauffer, D., 2013. "Peer-to-peer and mass communication effect on opinion shifts," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(4), pages 785-796.

### Keywords

social influence; externalities; discrete choice; heterogeneous agents; socioeconomic behavior;

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