In revenue management research and practice, demand models are used that describe how demand for a seller's products depends on the decisions, such as prices, of that seller. Even in settings where the demand for a seller's products also depends on decisions of other sellers, the models often do not explicitly account for such decisions. It has been conjectured in the revenue management literature that such monopoly models may incorporate the effects of competition, because the parameter estimates of the monopoly models are based on data collected in the presence of competition. In this paper we take a closer look at such a setting to investigate the behavior of parameter estimates and decisions if monopoly models are used in the presence of 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. We analyze the behavior of the sellers' parameter estimates and prices under various assumptions regarding the sellers' knowledge and estimation procedures, and 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 have many potential limit points that are neither the Nash equilibrium nor the cooperative solution and that depend on the initial conditions. We compare the sellers' revenues at potential limit prices with their revenues at the Nash equilibrium and the cooperative solution, and we show that it is possible for sellers to be better off when using a monopoly model than at the Nash equilibrium.