Learning and Pricing with Models That Do Not Explicitly Incorporate Competition

Abstract

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 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 show that it is possible for sellers to be better off when using a monopoly model than at the Nash equilibrium.

Recent Developments in Dynamic Pricing Research: Multiple Products, Competition, and Limited Demand Information

“…Maximum likelihood estimation is also quite commonly used (Broder and Rusmevichientong 2012, Carvalho and Puterman 2005, den , Boer and Zwart 2011, 2014. Other approaches used include linear least squares estimation (Bertsimas and Perakis 2006, Besbes and Zeevi 2014, Cooper et al 2013, Kachani et al 2007, Keskin and Zeevi 2013, 2014, and simple empirical estimation which is quite different from all other approaches and is described in detail below when we review the corresponding papers (Besbes and Zeevi 2009, 2011, Chen and Farias 2013, Wang et al 2014. Unlike the Bayesian approach, these approaches do not require prior knowledge of an unknown parameter.…”

Dynamic Pricing and Learning: Historical Origins, Current Research, and New Directions

“…Lai and Wei (1982) study how the speed of convergence of least-squares linear regression estimates depend on the amount of dispersion in the explanatory variables. Their results are applied in several dynamic pricing problems with linear demand functions, such as Le Guen (2008) and Cooper et al (2012). Similarly, results on the convergence rate of maximum-likelihood estimators, as in Borovkov (1998) and den Boer and Zwart (2011a), are crucial in the analysis of pricing policies by Zeevi (2011), Broder andRusmevichientong (2012) and den Boer and Zwart (2010).…”

Dynamic pricing and learning with competition: insights from the dynamic pricing challenge at the 2017 INFORMS RM & pricing conference

“…A main take-away from this strand of literature is the importance of having the 'right' amount of price experimentation. The importance of incorporating competition into these learning-and-earning models, and the potential detrimental effect of ignoring competition, has been demonstrated by Schinkel et al (2002), Tuinstra (2004), Bischi et al (2004Bischi et al ( , 2007, Isler and Imhof (2008), Cooper et al (2014), and Anufriev et al (2013), building forth on earlier work by Kirman (Kirman, 1975, 1983, Brousseau and Kirman, 1992.…”