Optimization of revenue versus budget in a random market – the law of equal probabilities

Abstract

The main result of this paper is the derivation of a law of equal probabilities for the problem of optimization of revenue versus budget in a random market. It states that the optimal solution is characterized by the condition of equal probabilities across all products in a store. To prove the main result, we solve the optimization problem by the method of Lagrange multipliers, and we prove that the optimal solution is given by the law of equal probabilities. We also prove a universality result that as the number of the products in a store goes to infinity, the revenue versus budget function converges to a non-random limit. Numerical results illustrating both the law of equal probabilities and the universality are provided.

Keywords::

• maximization of revenue versus budget
• Lagrange multipliers
• law of large numbers
• universality

Keywords::

• C300