Structure and Design of Informational Substitutes

Abstract

We analyze structure and design of informational substitutes and complements, as proposed by Chen and Waggonner (2016). First, we characterize “universal” complements, or information structures such that signals are complements for every decision problem, as precisely variants of the exclusive-or (XOR) of binary signals. This characterization is important because equilibria in the corresponding prediction market games are always the “worst-possible” regardless of design. Second, we show that the problem of designing the market for substitutability is equivalent to solving a linear program, and that for many common information structures, such a linear program can be solved in polynomial time. Third, we extend informational substitutes to predicting continuous distributions and distribution properties, such as mean and median, and show that they sometimes behave unintuitively. In particular, conditionally independent gaussian signals are complements under a wide range of standard decision problems.