Criticality in networked markets: profit-seeking behavior destabilizes evolutionarily stable states

Abstract

Oscillation phenomenon in the dynamics of network-marketing firms is empirically observed. Its origin is explored using a population-dynamic model of firm membership. Members, who are classified into two types distinguished by socio-economic status (SES), are driven by the search for profit. A perturbation expansion reveals that the fluctuations ⟨ξ²⟩ of the fraction of the high-SES type diverges as ⟨ξ²⟩∼ |μ − μ_c|⁻¹ as μ → μ⁻_c, where μ is the relative per capita rate of recruiting the high-SES type and μ_c is the critical value coinciding with a bifurcation of the population dynamics. Diverging fluctuations destabilize the evolution-
arily stable state (ESS) of an evolutionary game between the types consequently generating oscillations in the number of members of the firm. Empirical data from publicly-listed network-marketing firms reveal Hurst exponents that are greater than 0.5, which confirm the presence of criticality as predicted by the model.