Large Deviations Theory of Increasing Returns

Abstract

An influential theory of increasing returns has been proposed by the economist W. B. Arthur in the '80s to explain the lock-in phenomenon between two competing commercial products. In the most simplified situation there are two competing products that gain customers according to a majority mechanism: each new customer arrives and asks which product they bought to a certain odd number of previous customers, and then buy the most shared product within this sample. It is known that one of these two companies reaches monopoly almost surely in the limit of infinite customers. Here we consider a generalization [G. Dosi, Y. Ermoliev, Y. Kaniovsky, J. Math. Econom. 23, 1-19 (1994)] where the new customer follows the indication of the sample with some probability, and buy the other product otherwise. Other than economy, this model can be reduced to the urn of Hill, Lane and Sudderth, and includes several models of physical interest as special cases, like the Elephant Random Walk, the Friedman's urn and other generalized urn models. We provide a large deviation analysis of this model at the sample-path level, and give a formula that allows to find the most likely trajectories followed by the market share variable. Interestingly, in the parameter range where the lock-in phase is expected, we observe a whole region of convergence where the entropy cost is sub-linear. We also find a non-linear differential equation for the cumulant generating function of the market share variable, that can be studied with a suitable perturbations theory.

We correct an inverted sign in Eq. (93). 32 pages, 9 figures

References in corpus (15)

• A survey of random processes with reinforcement
• Elephants can always remember: Exact long-range memory effects in a non-Markovian random walk
• Introduction to dynamical large deviations of Markov processes
• Ergodicity and large deviations in physical systems with stochastic dynamics
• Elephant Random Walks and their connection to Pólya-type urns
• Rare behavior of growth processes via umbrella sampling of trajectories
• Giant leaps and long excursions: fluctuation mechanisms in systems with long-range memory
• Large deviations for Generalized Polya Urns with arbitrary urn function
• Large deviations in models of growing clusters with symmetry-breaking transitions
• Similarity of ensembles of trajectories of reversible and irreversible growth processes
• Replica Symmetry Breaking without Replicas
• Random Polymers and Generalized Urn Processes
• Ideal chains with fixed self-intersection rate
• Universal function of the non-equilibrium phase transition of nonlinear Pólya urn
• Energy of the Interacting Self-Avoiding Walk at the $\theta -$point

Cited by in corpus (5)

• Nonequilibrium steady state of trapped active particles
• A simplified Parisi Ansatz II: REM universality
• Universal scaling limits for spin networks via martingale methods
• Elephant random walks with multiple extractions and general reinforcement functions
• Simple model of market share dynamics based on clients' firm-switching decisions