The Value of Information in Monotone Decision Problems

A seminal theorem due to Blackwell (1951) shows that every Bayesian decision-maker prefers an informative signal Y to another signal X if and only if Y is statistically sufficient for X. Sufficiency is an unduly strong requirement in most economic problems because it does not incorporate any structure the model might impose. In this paper, we develop a general theory of information that allows us to characterize the information preferences of decision-makers based on how their marginal returns to acting vary with the underlying (unknown) state of the world. Our analysis imposes one central restriction: we consider "monotone decision problems," whereby all decision-makers in the relevant class choose higher actions when higher values of the signal are realized. We show how this restriction can be exploited to characterize information preferences using stochastic dominance orders over the distributions of posterior beliefs generated by different signals. Of particular interest for applied modeling, we identify conditions under which one decision-maker has a higher marginal value of information than another decision-maker, and thus will acquire more information. The results are applied to oligopoly models, labor markets with adverse selection, hiring problems, and a coordination game