Motivated by the prevalence of prediction problems in the economy, we study markets in which firms sell models to a consumer to help improve their prediction. Firms decide whether to enter, choose models to train on their data, and set prices. The consumer can purchase multiple models and use a weighted average of the models bought. Market outcomes can be expressed in terms of the bias-variance decompositions of the models that firms sell. We show that market structure can depend in subtle and nonmonotonic ways on the statistical properties of available models. Moreover, firms may choose inefficiently biased models to deter entry by competitors or to obtain larger profits. Keywords: prediction, models, competition, mean squared error, bias-variance decomposition.

Although you'd never guess it from mainstream media, the most significant antitrust case in more than 20 years is under way in Washington. In it, the US justice department, alongside the attorneys general of eight states, is suing Google for abusively monopolising digital advertising technologies, thereby subverting competition through "serial acquisitions" and anti-competitive auction manipulation. Or, to put it more prosaically, arguing that Google – which has between 90% and 95% of the search market – has maintained its monopoly not by making a better product, but by locking down almost every avenue through which consumers might find a different search engine and making sure they only see Google wherever they look. Basically, because the US government has been asleep at the wheel for almost a quarter of a century and has finally woken up to its democratic responsibilities. The last time it stirred itself to take on an aggressive monopolist was in 2001, when it sued Microsoft for illegally tying its Internet Explorer browser to Windows as part of a (successful) campaign to destroy Netscape, maker of the first distinctive commercial web browser, which Bill Gates and co perceived as a potentially lethal competitive threat.

We study the profit-maximization problem of a monopolistic market-maker who sets two-sided prices in an asset market. The sequential decision problem is hard to solve because the state space is a function. We demonstrate that the belief state is well approximated by a Gaussian distribution. We prove a key monotonicity property of the Gaussian state update which makes the problem tractable, yielding the first optimal sequential market-making algorithm in an established model. The algorithm leads to a surprising insight: an optimal monopolist can provide more liquidity than perfectly competitive market-makers in periods of extreme uncertainty, because a monopolist is willing to absorb initial losses in order to learn a new valuation rapidly so she can extract higher profits later.

I first spoke with Cory Doctorow two years ago. I was trying to get a handle on the sci-fi genre known as cyberpunk, most famously associated with the work of William Gibson. Doctorow, who is often described as a post-cyberpunk writer, is both a theorist-practitioner of science fiction and a vigorous commentator on technology and policymaking; his answers to my questions were long, thoughtful, and full of examples. And so, after that first talk, I made plans to speak with him again, not for research purposes but as the basis for the interview below. Doctorow, who is fifty-one, grew up in Toronto, the descendant of Jewish immigrants from what are now Poland, Russia, and Ukraine.

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.

The issue of how the explicit introduction of information impacts traditional economic analysis of, for example, competitive equilibrium or monopolistic behaviour has been investigated fruitfully within the paradigm of asymmetric information. The success of explaining non-perfectly competitive outcomes has however led to a neglect of the issue of how information itself can be considered valuable. A single Value of Information (VoI) literature does not exits and the issue spans over diverse economic fields and even other disciplines that aim to discriminate signal from noise. Of course any investigation of a Value of Information needs a specific reference frame in which such a "value" may occur.

We study the profit-maximization problem of a monopolistic market-maker who sets two-sided prices in an asset market. The sequential decision problem is hard to solve because the state space is a function. We demonstrate that the belief state is well approximated by a Gaussian distribution. We prove a key monotonicity property of the Gaussian state update which makes the problem tractable, yielding the first optimal sequential market-making algorithm in an established model. The algorithm leads to a surprising insight: an optimal monopolist can provide more liquidity than perfectly competitive market-makers in periods of extreme uncertainty, because a monopolist is willing to absorb initial losses in order to learn a new valuation rapidly so she can extract higher profits later.

The increasing use of group discounts has provided opportunities for buying groups with diverse preferences to coordinate their behavior in order to exploit the best offers from multiple vendors. We analyze this problem from the viewpoint of the vendors, asking under what conditions a vendor should adopt a volume-based price schedule rather than posting a fixed price, either as a monopolist or when competing with other vendors. When vendors have uncertainty about buyers' valuations specified by a known distribution, we show that a vendor is always better off posting a fixed price, provided that buyers' types are i.i.d. and that other vendors also use fixed prices. We also show that these assumptions cannot be relaxed: if buyers are not i.i.d., or other vendors post discount schedules, then posting a schedule may yield higher profit for the vendor. We provide similar results under a distribution-free uncertainty model, where vendors minimize their maximum regret over all type realizations.

We study the profit-maximization problem of a monopolistic market-maker who sets two-sided prices in an asset market. The sequential decision problem is hard to solve because the state space is a function. We demonstrate that the belief state is well approximated by a Gaussian distribution. We prove a key monotonicity property of the Gaussian state update which makes the problem tractable, yielding the first optimal sequential market-making algorithm in an established model. The algorithm leads to a surprising insight: an optimal monopolist can provide more liquidity than perfectly competitive market-makers in periods of extreme uncertainty, because a monopolist is willing to absorb initial losses in order to learn a new valuation rapidly so she can extract higher profits later.