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Dynamic Games and Applications

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19-10-2020 | Original Article

Stochastic Value Formation

Author: Jiabin Wu

Published in: Dynamic Games and Applications | Issue 3/2021

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Abstract

We propose a simple model of value formation in two societies (communities). When forming values, individuals face conformity pressure within their own society and they are intolerant toward the other society. When such a value formation process is noisy, the interaction between conformity, intolerance and noise can give rise to interesting dynamic outcomes including two polarized societies and polarization across the two societies.

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We assume that the value space is a closed interval on the real line, which is a common assumption made in the economics literature to study preferences, values, beliefs over a single-dimensional issue. See, for example, Michaeli and Spiro [45, 46] and Cheung and Wu [27].

We use quadratic function because it is commonly adopted in the studies of social norms for tractability. See, for example, Manski and Mayshar [44] and Kuran and Sandholm [40]. Nevertheless, it is the convexity of the intolerance function that leads to our main results.

When we check if a certain state is steady or stable, we drop the time subscript and use \(E_i, E_j\).

The derivation is provided in Anderson et al. [2].

The Fokker–Planck equation is not yet widely used in the economic analysis. See Binmore et al. [10], Friedman and Yellin [30] and Anderson et al. [2, 3] for applications of the Fokker–Planck equation in game theory; and Benhabib et al. [7] for an application of the Fokker–Planck equation to the study of income distributions.

An alternative choice is the strong topology induced by the variation norm. See Bomze [18, 19] and Oechssler and Riedel [47, 48]. The strong topology is more restrictive than the weak topology in the sense that it only allows perturbations from a small fraction of individuals changing their values.

We discretize both the set of values [0,1] (with grid size being 0.0001) and time to make the simulation work. By discretizing time, we use a normal distribution with mean 0 to approximate the noise. In addition, to ensure the dynamic stays in [0, 1], we impose boundary conditions to revert the dynamic back in range every time it escapes. Hence, in the plot we show, a very small portion of the graph is situated outside of [0,1]. The program is available upon request.

The results remain the same if we instead run 100,000 periods.

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Title: Stochastic Value Formation
Author: Jiabin Wu
Publication date: 19-10-2020
Publisher: Springer US
Published in: Dynamic Games and Applications / Issue 3/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI: https://doi.org/10.1007/s13235-020-00370-z

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