Noise Reduction in Coarse Bifurcation Analysis of Stochastic Agent-Based Models: An Example of Consumer Lock-In

We investigate the occurrence of coarse macroscopic states in an agent-based model of consumer lock-in. The system studied here is a modification of an existing model by Garlic and Chli [24] and it serves as a prototypical Ising-type sociological system with binary state variables and spatially-dependent agent parameters. In the regime of globally-coupled agents with independent identically-distributed parameters, we derive an analytic approximate coarse evolution-map for the expectation of the average purchase. Following Barkley et al. [5], we interpret metastable locked-in states as fixed points of this one-dimensional first moment map. We then study the emergence of coarse fronts in the regime of heterogeneous agents with strongly discordant preferences. When agent polarization becomes less pronounced, the front destabilizes and one of the two products prevails, giving rise to inhomogeneous profiles featuring pockets of resistance. Stochastic continuation of the spatially-extended case poses a numerical challenge, as Jacobian-vector products are severely affected by noise. We exploit the non-uniqueness of the lifting step introducing weighted lifting/restriction operators, which result in variance-reduced Jacobian-vector products. We test our numerical strategy and show that weighted operators induce good convergence properties of the Newton-GMRES solver. We then show that macroscopic fronts destabilise at a coarse symmetry-breaking bifurcation.