A potential game approach to modelling evolution in a connected society


When studying human behaviour, it is important to understand not just how individuals interact, but also interactions at the level of communities and populations. Most previous modelling of networks has focused on interactions between individual agents. Here we provide a modelling framework to study the evolution of behaviour in connected populations, by regarding subpopulations as the basic unit of interaction and focusing on the population-level connection structure. We find that when the underlying game played by individuals is a potential game, utilizing such a structure greatly simplifies analysis. In addition, according to known general results on the convergence of evolution dynamics to Nash equilibria in a potential game, our formulation provides a tractable model on behavioural dynamics in social networks that needs only conventional techniques from evolutionary game theory.

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Fig. 1: Canonical examples of connected populations.
Fig. 2: Transition from a polarized medium-run equilibrium to an integrated long-run equilibrium through the long-run dynamic.
Fig. 3: Simulation of the combined best-response dynamic in the bilingual game from the medium-run equilibrium \(x_{{\rm{Ao}}}^{{\rm{Lo}}} = m^{{\rm{Lo}}}\), \(x_{{\rm{AB}}}^{{\rm{LR}}} = m^{{\rm{LR}}}\) and \(x_{\rm{oB}}^{{\rm{oR}}} = m^{{\rm{oR}}}\), with \((m^{{\rm{Lo}}},m^{{\rm{LR}}},m^{{\rm{oR}}}) = (0.4,0.4,0.2)\).

Data availability

The simulation results used to produce the figures in this paper and the Supplementary Discussion are available from https://github.com/zusaiEGT/ConnectedEvol.

Code availability

The MATLAB (R2017) code for this simulation is available from https://github.com/zusaiEGT/ConnectedEvol.


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The authors thank I. Obara, R. Sawa and the participants of presentations at the University of Tsukuba, Stony Brook International Conference on Game Theory, East Asian Game Theory Conference and Southern Economic Association annual meetings for helpful suggestions and comments. D.Z. is grateful to the University of Oregon for hospitality during part of this work. The funders had no role in study design, data collection and analysis, decision to publish or preparation of the manuscript.

Author information

J.W. and D.Z. designed the framework of the research, contributed to the design and mathematical analysis of the examples, and jointly wrote the paper. J.W. contributed to surveying the literature and exploring the implications. D.Z. contributed to mathematical formulation of the general model, and wrote the MATLAB code.

Correspondence to Dai Zusai.

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