  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.
