# Selling Information Through Consulting

## TL;DR

This paper develops simple, optimal mechanisms for a monopolist selling correlated private information to a budget-constrained decision maker, simplifying previous complex approaches and providing practical computation methods.

## Contribution

It introduces a unified, simple format for optimal information selling mechanisms under budget constraints, applicable across various correlation settings, with efficient polynomial-size linear programs.

## Key findings

- Mechanisms act as consultants recommending actions with tailored payments.
- Optimal mechanisms are computationally efficient, solvable via polynomial-size LPs.
- The approach simplifies previous exponential LP methods for budget-constrained settings.

## Abstract

We consider a monopoly information holder selling information to a budget-constrained decision maker, who may benefit from the seller's information. The decision maker has a utility function that depends on his action and an uncertain state of the world. The seller and the buyer each observe a private signal regarding the state of the world, which may be correlated with each other. The seller's goal is to sell her private information to the buyer and extract maximum possible revenue, subject to the buyer's budget constraints. We consider three different settings with increasing generality, i.e., the seller's signal and the buyer's signal can be independent, correlated, or follow a general distribution accessed through a black-box sampling oracle. For each setting, we design information selling mechanisms which are both optimal and simple in the sense that they can be naturally interpreted, have succinct representations, and can be efficiently computed. Notably, though the optimal mechanism exhibits slightly increasing complexity as the setting becomes more general, all our mechanisms share the same format of acting as a consultant who recommends the best action to the buyer but uses different and carefully designed payment rules for different settings. Each of our optimal mechanisms can be easily computed by solving a single polynomial-size linear program. This significantly simplifies exponential-size LPs solved by the Ellipsoid method in the previous work, which computes the optimal mechanisms in the same setting but without budget limit. Such simplification is enabled by our new characterizations of the optimal mechanism in the (more realistic) budget-constrained setting.