We previously mentionned that the sellers should adjust the quantities brought to the market every morning to take into account the expected number of customers, including eventually fluctuations. In order to optimise the next day's profit, a seller with a perfect knowledge of , the probability distribution of the number of visitors, would bring to the market a quantity n'q where n' is given by the following equation:
The above expression is optimal for one day, but does not take into account future gains that could be obtained by systematically bringing extras to make unexpected customers loyal.
Anyway, we did not suppose for the simulations that sellers have a perfect knowledge of the probability distribution of visitors, but that they use a simple routine to add extra whenever they observe fluctuations in the number of visits. The extra at time t is computed according to
where is small and is the variance of the number of buyers computed from the beginning of the simulation. The initial value of is non zero at the beginning of the simulation. We checked by several numerical simulations with different choices of initial and of that the only observable changes were variations of , the critical threshold for order, in the ten percent range . The existence of two dynamical regimes persists.
Another possible refinement would consist in improving the predictive ability of the seller with respect to the number of customers. We tried a moving average prediction rather than the prediction based only on the preceeding day but this only downgraded performance ( increases).