Selling supplemental information

1. Problem Statement

The paper addresses the mechanism design problem faced by a data broker selling information to a decision maker (DM) who already possesses private information.

• Context: Decision makers (e.g., firms, lenders, physicians) often hold partial, noisy, or incomplete private signals about the state of the world. They seek to purchase supplemental information from a broker to refine their decisions.
• The Challenge: Unlike standard screening problems where types are scalar and ordered, the DM's "type" is a signal structure (a partition of the state space). This creates a multidimensional screening problem with two specific complications:
  1. Non-ordered Type Space: Signals are not totally ordered (some are horizontally differentiated, others vertically).
  2. Endogenous Type: The DM's payoff depends on the combination of her private signal and the purchased signal. As she acquires information, her interim beliefs and optimal actions change, altering her valuation.
• Objective: Characterize the data broker's optimal selling mechanism (menu of signals and prices) to maximize revenue, specifically investigating conditions under which the broker can extract the efficient surplus (i.e., extract the full value of the information, leaving the DM with zero rent).

2. Methodology and Model Framework

A. Information Structure

The paper adopts the Gentzkow and Kamenica (2017) framework, modeling information as a signal (a partition of the extended state space $\Omega \times [0,1]$).

• Signal Join ($\pi \vee \pi'$): The core algebraic operation is the "join" of two signals. If a DM owns signal $\pi$ and buys $\pi'$, her effective information is $\pi \vee \pi'$. This allows for a rigorous analysis of how supplemental data combines with existing data.
• Valuation: The value of a supplemental signal $\sigma$ to a type $\pi$ is the increase in expected utility:
  $$V(\sigma | \pi) = U(\sigma \vee \pi) - U(\pi)$$
  where $U(\cdot)$ is the expected utility of the optimal action given the signal.

B. Mechanism Design Setup

• Timing: The broker posts a menu $\{(\sigma(\pi), t(\pi))\}$ ex-ante (before the DM's private signal realization). The DM selects a signal and pays a price before observing her private realization.
• Constraints: The mechanism must satisfy:
  • Incentive Compatibility (IC): Truth-telling is optimal.
  • Individual Rationality (IR): Participation yields non-negative utility.
• Goal: Maximize expected revenue $\int t(\pi) dF(\pi)$.

3. Key Contributions and Results

The paper provides a complete characterization of the optimal menu across different environments, distinguishing between binary-action and multi-action settings.

A. The Binary-Action Setting ($|A|=2$)

Result: The data broker can always extract the efficient surplus, regardless of the distribution of DM types.

• Mechanism: The broker offers a distinct binary signal to each type $\pi$.
• The "Minimal Complementary Signal" ($\pi^c$):
  • For any private signal $\pi$, the broker constructs a signal $\pi^c$ that acts as an action recommendation.
  • Logic: If the action induced by $\pi$ is optimal, $\pi^c$ says "do not switch." If the action is suboptimal, $\pi^c$ says "switch."
  • Key Property (Lemma 3): The incremental value of this specific signal $\pi^c$ is maximized when the base signal is exactly $\pi$.
  • Implication: A DM of type $\pi'$ cannot mimic type $\pi$ to gain more value. If $\pi'$ obeys the recommendation, they switch actions only when $\pi$ would, but $\pi'$ might switch from a suboptimal action to a wrong one (or vice versa), reducing their net gain compared to $\pi$. If they disobey, they effectively switch their default action, incurring a loss.
• Outcome: The broker charges $t(\pi) = U - U(\pi)$ (the full value of the information), leaving zero rent.

B. General Environments (Many Actions and States)

Result: Efficient surplus extraction generally fails because the marginal value of correction depends on both the state and the specific action the DM would take otherwise.

• Failure Case: In a general payoff matrix, a "switch" recommendation might be highly valuable to one type (correcting a bad action) but less valuable to another (correcting a mediocre action to a good one, but not the best). This breaks the single-crossing property required for full extraction.
• Sufficient Conditions for Success (Proposition 3): The paper identifies a specific class of payoff structures where efficient extraction is still possible: Diagonal Payoffs (State-Action Matching).
  • Structure: $u(a_i, \omega_j) = u_i$ if $i=j$, and $0$ otherwise.
  • Mechanism: The broker offers an "Error-Correcting Refinement" ($\hat{\pi}$).
    • If $\pi$ induces the optimal action, $\hat{\pi}$ sends a confirmation message.
    • If $\pi$ induces a suboptimal action, $\hat{\pi}$ reveals the true state.
  • Why it works: Under diagonal payoffs, the marginal benefit of being corrected depends only on the state, not on the specific suboptimal action the DM was taking. This restores the property that the signal is most valuable to the type for whom it was designed, allowing full surplus extraction.

C. Counter-Example

The paper constructs a specific example with an upper-triangular payoff matrix where efficient extraction fails.

• In this scenario, a partially informed DM (who knows one state perfectly) and an uninformed DM have different valuations for the same supplemental signal.
• The uninformed DM benefits from the signal even when the informed DM does not (because the informed DM already knows the state), but the structure of the payoffs prevents the broker from designing a menu that separates them without leaving rent.

4. Significance and Implications

1. Theoretical Advancement:

   • The paper solves a complex multidimensional screening problem where the type space is a set of information structures, not a scalar.
   • It demonstrates that the algebraic structure of signals (specifically the join operation) is crucial for solving these problems, distinguishing it from literature using Blackwell experiments or market segmentation.
   • It challenges the standard assumption that efficient surplus extraction requires single-crossing properties, showing that in binary-action settings, the specific geometry of the signal space allows for full extraction without it.

2. Practical Relevance:

   • Data Brokers: The results explain why data brokers can often extract high revenues by selling "data append" services. By offering signals that are perfectly complementary to a firm's existing data (e.g., "switch" recommendations), they can capture the full value of the decision improvement.
   • Information Design: The paper highlights that the correlation between the sold signal and the buyer's private signal is a powerful tool. By selling signals that are correlated with the buyer's private data, the broker can induce full information revelation in the combined signal, thereby maximizing efficiency and revenue.

3. Policy Implications:

   • The findings suggest that in markets for data (e.g., consumer profiling, credit scoring), the ability to sell correlated supplemental information leads to efficient outcomes (perfect price discrimination or optimal medical diagnosis) but eliminates consumer surplus. This raises questions about the welfare effects of data markets and the potential need for regulation to prevent excessive rent extraction by data brokers.

Summary of Main Theorems

• Proposition 1: In a two-type setting, efficient extraction is possible iff the value of the minimal complementary signal for the low type is greater than or equal to its value for the high type.
• Proposition 2: In any binary-action setting, the broker always extracts the efficient surplus using minimal complementary signals.
• Proposition 3: In multi-action settings with diagonal payoffs, the broker always extracts the efficient surplus using error-correcting refinements.
• General Case: Without diagonal payoffs, efficient extraction generally fails due to the dependence of marginal correction value on the specific action taken.