Non-equilibrium time-dependent solution to discrete choice with social interactions

TL;DR

This paper presents a fast, non-equilibrium, time-dependent solution to the binary decision model with social interactions, revealing path-dependent behaviors, metastable effects, and enabling model calibration from data.

Contribution

It introduces a novel resolvent-based method for solving the model out of equilibrium, applicable to various binary decision models and providing a likelihood function for calibration.

Findings

Metastable effects occur above a critical rationality threshold.

Altruistic agents are more prone to metastable regimes despite higher short-term utility.

The method can be extended to other models like Kirman's ant model and Ising model.

Abstract

We solve the binary decision model of Brock and Durlauf in time using a method reliant on the resolvent of the master operator of the stochastic process. Our solution is valid when not at equilibrium and can be used to exemplify path-dependent behaviours of the binary decision model. The solution is computationally fast and is indistinguishable from Monte Carlo simulation. Well-known metastable effects are observed in regions of the model's parameter space where agent rationality is above a critical value, and we calculate the time scale at which equilibrium is reached from first passage time theory to a much greater approximation than has been previously conducted. In addition to considering selfish agents, who only care to maximise their own utility, we consider altruistic agents who make decisions on the basis of maximising global utility. Curiously, we find that although altruistic agents coalesce more strongly on a particular decision, thereby increasing their utility in the short-term, they are also more prone to being subject to non-optimal metastable regimes as compared to selfish agents. The method used for this solution can be easily extended to other binary decision models, including Kirman's ant model, and under reinterpretation also provides a time-dependent solution to the mean-field Ising model. Finally, we use our time-dependent solution to construct a likelihood function that can be used on non-equilibrium data for model calibration. This is a rare finding, since often calibration in economic agent based models must be done without an explicit likelihood function. From simulated data, we show that even with a well-defined likelihood function, model calibration is difficult unless one has access to data representative of the underlying model.