Price of anarchy in a Bertrand oligopoly market

Abstract

The price of anarchy quantifies the inefficiency that occurs in the total system objective in the user optimization as compared to the system optimization setting. It is well known that this inefficiency occurs due to lack of coordination among the competitors in the system. In this thesis, we study the price of anarchy in a Bertrand oligopoly market by comparing the total profits in the two settings. The main contribution of this thesis is a lower and an upper bound for the price of anarchy that only depends on the price sensitivity matrix characterizing the demand sellers face. We first derive these bounds for a symmetric affine demand model. Using the same approach, we also provide a lower bound for asymmetric affine demand as well as a lower and an upper bound for nonlinear demand. These bounds are easy to compute. In addition, we illustrate that the worst-case price of anarchy value occurs for a uniform demand model when quality differences do not exist among sellers. This implies that in many real-world instances where quality differences exist, the performance under the user optimization may in fact be close to what is achieved under system optimization. We illustrate several insights on the bounds we present through simulations.