How to Buy Advice∗

A decision maker, whose payoff is influenced by an unknown stochastic process, seeks the advice of an advisor, who may be informed about the process. We identify a sufficient condition on the correlation between the advisor’s information and the true stochastic process, called conservativeness, for which there exists a strategy D of the decision maker that will yield him an almost first-best payoff in every period. We also demonstrate that without conservativeness no strategy can approximate the first-best payoff. The belief-free strategy D satisfies various desirable properties. It only requires a fixed budget – regardless of the realizations of the stochastic process and whether or not the advisor is actually informed about it, the total payoff to the decision maker will never fall below a fixed threshold. Moreover, perperiod compensation to the advisor is independent of the present realization of the process, and depends solely on the expected value of the advice as reported by the advisor. ∗We thank Ehud Kalai, Wojciech Olszewski, and Rakesh Vohra for fruitful discussions. †Kellogg School of Management, Northwestern University, Evanston, IL 60208, USA. E-mails: [email protected] and [email protected].