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Abstract Different agents need to make a prediction. They observe identical data, but have different models: they predict using different explanatory variables. We study which agent believes they have the best predictive ability—as measured by the smallest subjective posterior mean squared prediction error—and show how it depends on the sample size. With small samples, we present results suggesting it is an agent using a low-dimensional model. With large samples, it is generally an agent with a high-dimensional model, possibly including irrelevant variables, but never excluding relevant ones. We apply our results to characterize the winning model in an auction of productive assets, to argue that entrepreneurs and investors with simple models will be overrepresented in new sectors, and to understand the proliferation of “factors” that explain the cross-sectional variation of expected stock returns in the asset-pricing literature. 

We introduce a simple but general online learning framework in which a learner plays against an adversary in a vector-valued game that changes every round. Even though the learner's objective is not convex-concave (and so the minimax theorem does not apply), we give a simple algorithm that can compete with the setting in which the adversary must announce their action first, with optimally diminishing regret. We demonstrate the power of our framework by using it to (re)derive optimal bounds and efficient algorithms across a variety of domains, ranging from multicalibration to a large set of no-regret algorithms, to a variant of Blackwell's approachability theorem for polytopes with fast convergence rates. As a new application, we show how to (multi)calibeat'' an arbitrary collection of forecasters --- achieving an exponentially improved dependence on the number of models we are competing against, compared to prior work. 

Motivated by markets for “expertise,” we study a bandit model where a principal chooses between a safe and risky arm. A strategic agent controls the risky arm and privately knows whether its type is high or low. Irrespective of type, the agent wants to maximize duration of experimentation with the risky arm. However, only the high type arm can generate value for the principal. Our main insight is that reputational incentives can be exceedingly strong unless both players coordinate on maximally inefficient strategies on path. We discuss implications for online content markets, term limits for politicians, and experts in organizations. 

Motivated by their increasing prevalence, we study outcomes when competing sellers use machine learning algorithms to run real-time dynamic price experiments. These algorithms are often misspecified, ignoring the effect of factors outside their control, for example, competitors’ prices. We show that the long-run prices depend on the informational value (or signal-to-noise ratio) of price experiments: if low, the long-run prices are consistent with the static Nash equilibrium of the corresponding full information setting. However, if high, the long-run prices are supra-competitive—the full information joint monopoly outcome is possible. We show that this occurs via a novel channel: competitors’ algorithms’ prices end up running correlated experiments. Therefore, sellers’ misspecified models overestimate the own price sensitivity, resulting in higher prices. We discuss the implications on competition policy.