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Network Structures and Poverty Traps

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Abstract

We build an evolutionary network game of economic agents that choose actions of being either a high-profile or a low-profile economic agent. Those economic agents reside in the vertices of an undirected graph or network given by their types, and their strategic interaction is driven by imitative behavior. Then, the share of types of economic agents forms networks described by a mean field formalism which depends on agents’ payoff functions, as well as on the current state of the economic network. We show the fact that, in this context of networks, a neighbor is imitated if her strategy outperformed the focal individual’s in the previous iterations. The main result is that there are three equilibria (each with a non-degenerate basin of attraction), one completely made up of high-profile individuals, one made up of low-profile individuals (i.e., the poverty trap), and a mixture. The main parameters from being in one or the other equilibrium are: (i) the degree of node, (ii) cost of being high-profile, and (iii) initial distribution of types. We conclude with simple numerical examples to show that outcome depends on network structures and on both the education costs, c, and the value of \(\beta \) which is the incentive to choose the high-profile action.

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Notes

  1. Jackson and Watts [25] and Jackson [26] pointed out that a network g is stochastically stable if its steady state probability \(\mu (g, \varepsilon )\) is bounded below as the error rate \(\varepsilon \) aspires to 0 that is \(\mu (g, \varepsilon ) \rightarrow _\varepsilon a>0\), \(a\in [0,1]\).

  2. All experiments have been extended to the greater number of nodes to verify and estimate the stability of received results. We have reproduced the experiments on graphs with higher numbers of nodes (up to 100), and it turned out that incrementation of the number of nodes does not affect the overall result.

  3. In simulation analysis, we consider two rules to describe the imitation process in populated networks over a period of time. Rule 1: Mean field dynamics by imitation of success. Each agent changes her behavior if at least one neighbor has better payoff. In this case the network remains the structure but the ratio of agents will be changed. There are two ways for the switching rule [23, 35]: (i) Initial distribution of agents is nonuniform. If agent i receives an opportunity to revise her strategy, then she estimates her neighbors as one homogeneous player with aggregated payoff function. This payoff function is equal to mean value of payoffs of players, considered as one homogeneous player. If payoff function of homogeneous player is better then agent i changes her strategy to the strategy of her more popular neighbor. (ii) Initial distribution of agents is uniform. In this case agent i keeps her own strategy. Rule 2: Mean field dynamics by imitation under dissatisfaction. Dynamic process occurs according to the rule in formula (3). In this case, if agent i has a payoff which is less or equal a payoff of her neighbor, then she remains with her strategy; if agent i has the same strategy as her neighbor, then she also keeps her strategy even if the neighbor’s payoff is bigger; if neighbor’s payoff is bigger and corresponds to another/different strategy, then agent i changes her strategy for copying the strategy of the neighbor.

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Acknowledgements

We thank the editor, Georges Zaccour, and the anonymous reviewers for their constructive comments, which helped us to improve the manuscript. We benefitted from discussions and comments from Costas Azariadis, Nicola Dimitri, Sebastian Ille, William Olvera-Lopez, Joss E. Sanchez-Perez and Laura Policardo. Thanks to the Research Workgroup on Seminario de Investigacion y Estudio en Teoria Economica at the Department of Mathematical Economics, University of San Luis Potosi, and SNI-CONACYT Mexico.

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Authors and Affiliations

  1. Department of Economics, Society, and Politics, The University of Urbino Carlo Bo, Urbino, Italy

    Edgar J. Sánchez Carrera

  2. Faculty of Applied Mathematics and Control Processes, St. Petersburg State University, Saint Petersburg, Russia

    Elena Gubar

  3. AGIMA Ltd., Saint Petersburg, Russia

    Andrey F. Oleynik

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  1. Edgar J. Sánchez Carrera
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  2. Elena Gubar
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Correspondence to Edgar J. Sánchez Carrera.

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Carrera, E.J.S., Gubar, E. & Oleynik, A.F. Network Structures and Poverty Traps. Dyn Games Appl 9, 236–253 (2019). https://doi.org/10.1007/s13235-018-0256-8

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