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Heterogeneous Society in Binary Choices with Externalities

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Abstract

We study a two-strategy model with negative externalities proposed by Schelling, in a dynamical setting where a society consists of two interacting populations with different behaviors derived from experiments with human participants. The resulting dynamics is a three-dimensional piecewise smooth map with one discontinuity, which inherits some of the characteristics of each homogeneous population dynamics, while others are lost and new ones emerge. We propose a technique to represent the dynamics on a bidimensional space and prove that the heterogeneous society dynamics can be obtained as a linear combination of the dynamics of the two homogeneous populations. As expected, complexity arises with respect to some aspects. Firstly, the number of equilibria expands to infinity and we were able to determine possible focal equilibria in the sense of Schelling. Secondly, when heterogeneity is introduced, the period adding structure of cycles is replaced by a period incrementing structure. Thirdly, the phenomenon of overreaction and cyclic oscillations can be mitigated even if it never completely disappears. We also derive the orbits of cycles of period two and provide numerical evidences of coexistence of cycles with different periods. It is worth noticing that with the heterogeneous society, the dynamics does not depend on the society aggregate choices only, rather on each population choice; neglecting it will make impossible to determine the future evolution of the system. The implications are important as heterogeneity makes the system path-dependent and a policy maker, considering aggregate society choice only, would be unable to make the proper decisions, unless further information is considered.

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Notes

  1. Thomas C. Schelling–Facts”. Nobelprize.org. Nobel Media AB 2014. Web. 10 Nov 2016. http://www.nobelprize.org/nobel_prizes/economic-sciences/laureates/2005/schelling-facts.html.

  2. In this paper we use the same terminology as in the Population Games literature, see [37, p. 3].

  3. The careful reader will notice that in [8] the choices are labeled A and B, respectively, and that regions are determined by the payoff intersection point d which coincides with \(z^{*}\) as defined in (3).

  4. For the sake of brevity in the following proposition we omit subscript t.

  5. The small difference in terms of notation has to be attributed to the fact that in [8] the dynamics is studied by considering agents choosing L instead of R.

  6. Both these diagrams and Fig. 7 diagrams are in the parameters space \(\left( \delta _L,\delta _R\right) \). However, in Fig. 7 colors represent the period k of cycles, while in Fig. 12 they represent the number of coexistent cycles.

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Acknowledgements

The authors are grateful to Gian-Italo Bischi, Laura Gardini, Amnon Rapoport, the participants to the June, 8th, 2011 seminar at Centro de Modelamiento Matemático, Universidad de Chile and the participants to the 46th Annual Meeting of the Society of Mathematical Psychology in Potsdam, for helpful suggestions and constructive discussions. Usual caveats apply. This work has been performed within the activity of the PRIN project “Local interactions and global dynamics in economics and finance: models and tools”, MIUR, Italy, and under the auspices of COST Action IS1104 “The EU in the new complex geography of economic systems: models, tools and policy evaluation”.

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Authors and Affiliations

  1. Department of Economics, University of Molise, Via Francesco De Sanctis, snc, 86100, Campobasso, Italy

    Arianna Dal Forno

  2. Department of Psychology, Center for Logic, Language, and Cognition, University of Torino, Via Verdi 10, 10124, Torino, Italy

    Ugo Merlone

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  1. Arianna Dal Forno
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Correspondence to Ugo Merlone.

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Dal Forno, A., Merlone, U. Heterogeneous Society in Binary Choices with Externalities. Dyn Games Appl 9, 433–457 (2019). https://doi.org/10.1007/s13235-018-0270-x

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