# Discrete Choices under Social Influence: Generic Properties

## Author Info

• Mirta B. Gordon

()
(TIMC-IMAG - Techniques de l'Ingénierie Médicale et de la Complexité - Informatique, Mathématiques et Applications, Grenoble - CNRS : UMR5525 - Université Joseph Fourier - Grenoble I)

(CAMS - Centre d'analyse et de mathématique sociale - CNRS : UMR8557 - École des Hautes Études en Sciences Sociales (EHESS), LPS - Laboratoire de Physique Statistique de l'ENS - CNRS : UMR8550 - Université Pierre et Marie Curie (UPMC) - Paris VI - Université Paris VII - Paris Diderot - École normale supérieure [ENS] - Paris)

• Denis Phan

(CREM - Centre de Recherche en Economie et Management - CNRS : UMR6211 - Université de Rennes 1 - Université de Caen Basse-Normandie, GEMAS - Groupe d'étude des méthodes de l'analyse sociologique - CNRS : UMR8598 - Université Paris IV - Paris Sorbonne)

• Viktoriya Semeshenko

(TIMC-IMAG - Techniques de l'Ingénierie Médicale et de la Complexité - Informatique, Mathématiques et Applications, Grenoble - CNRS : UMR5525 - Université Joseph Fourier - Grenoble I)

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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 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.

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## Bibliographic Info

Paper provided by HAL in its series Working Papers with number halshs-00135405.

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Handle: RePEc:hal:wpaper:halshs-00135405

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## Related research

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

## References

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## Citations

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Cited by:
1. D. Sornette, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models," Papers 1404.0243, arXiv.org.
2. 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, Elsevier, vol. 392(4), pages 785-796.
3. 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.
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, Springer, vol. 71(4), pages 597-610, October.
5. Jean-Philippe Bouchaud, 2011. "Panel Statement: The endogenous dynamics of markets: price impact and feedback loops," Chapters, European Central Bank, European Central Bank.
6. Jean-Philippe Bouchaud, 2012. "Crises and collective socio-economic phenomena: simple models and challenges," Papers 1209.0453, arXiv.org, revised Dec 2012.
7. 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.
8. 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, Springer, vol. 68(1), pages 133-144, March.

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