Learning and Pricing with Models That Do Not Explicitly Incorporate Competition

We consider repeated pricing games in which two competing sellers use mathematical models to choose the prices of their products. Over the sequence of games, each seller attempts to estimate the values of the parameters of a demand model that expresses demand as a function only of its own price using data comprised only of its own past prices and demand realizations. Thus, as is often the case in practice, the sellers’ models do not explicitly account for other sellers. We study the behavior of the sellers’ prices and parameter estimates under various assumptions regarding the sellers’ knowledge and estimation procedures. We identify situations in which (a) the sellers’ prices converge to the Nash equilibrium associated with knowledge of the correct demand model, (b) the sellers’ prices converge to the cooperative solution, and (c) the sellers’ prices converge to other values that are neither the Nash equilibrium nor the cooperative solution and that depend on the initial prices.