Is Selling Complete Information (Approximately) Optimal?

TL;DR

This paper investigates the effectiveness of selling only fully informative experiments in complex Bayesian decision environments, showing it is approximately optimal in some cases and highlighting limitations in others.

Contribution

It provides approximation bounds for selling only fully informative experiments in multi-action environments and characterizes when this approach is optimal in multi-dimensional settings.

Findings

Selling only fully informative experiments approximates optimal revenue within a factor of O(m) for binary state and multiple actions.

The size of the optimal menu must grow at least linearly with the number of actions.

In uniform distributions, selling only the fully informative experiment is proven to be optimal.

Abstract

We study the problem of selling information to a data-buyer who faces a decision problem under uncertainty. We consider the classic Bayesian decision-theoretic model pioneered by [Blackwell, 1951, 1953]. Initially, the data buyer has only partial information about the payoff-relevant state of the world. A data seller offers additional information about the state of the world. The information is revealed through signaling schemes, also referred to as experiments. In the single-agent setting, any mechanism can be represented as a menu of experiments. [Bergemann et al., 2018] present a complete characterization of the revenue-optimal mechanism in a binary state and binary action environment. By contrast, no characterization is known for the case with more actions. In this paper, we consider more general environments and study arguably the simplest mechanism, which only sells the fully informative experiment. In the environment with binary state and $m\geq 3$ actions, we provide an $O(m)$-approximation to the optimal revenue by selling only the fully informative experiment and show that the approximation ratio is tight up to an absolute constant factor. An important corollary of our lower bound is that the size of the optimal menu must grow at least linearly in the number of available actions, so no universal upper bound exists for the size of the optimal menu in the general single-dimensional setting. For multi-dimensional environments, we prove that even in arguably the simplest matching utility environment with 3 states and 3 actions, the ratio between the optimal revenue and the revenue by selling only the fully informative experiment can grow immediately to a polynomial of the number of agent types. Nonetheless, if the distribution is uniform, we show that selling only the fully informative experiment is indeed the optimal mechanism.