Learning Underspecified Models

This paper examines whether one can learn to play an optimal action while only knowing part of true specification of the environment. We choose the optimal pricing problem as our laboratory, where the monopolist is endowed with an underspecified model of the market demand, but can observe market outcomes. In contrast to conventional learning models where the model specification is complete and exogenously fixed, the monopolist has to learn the specification and the parameters of the demand curve from the data. We formulate the learning dynamics as an algorithm that forecast the optimal price based on the data, following the machine learning literature (Shalev-Shwartz and Ben-David (2014)). Inspired by PAC learnability, we develop a new notion of learnability by requiring that the algorithm must produce an accurate forecast with a reasonable amount of data uniformly over the class of models consistent with the part of the true specification. In addition, we assume that the monopolist has a lexicographic preference over the payoff and the complexity cost of the algorithm, seeking an algorithm with a minimum number of parameters subject to PAC-guaranteeing the optimal solution (Rubinstein (1986)). We show that for the set of demand curves with strictly decreasing uniformly Lipschitz continuous marginal revenue curve, the optimal algorithm recursively estimates the slope and the intercept of the linear demand curve, even if the actual demand curve is not linear. The monopolist chooses a misspecified model to save computational cost, while learning the true optimal decision uniformly over the set of underspecified demand curves.