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An Instrumental Value for Data Production and its Application to Data Pricing
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Proposes an instrumental value for data production processes grounded in Bayesian decision-making, shows it equals information gain (entropy-based) in Bayesian linear regression, and designs pricing mechanisms achieving first-best revenue under perfect customization and log(kappa)-regret under limited customization.
Problem
How to quantify the instrumental (decision-relevant, prior-relative) value of a data production process to a buyer, and how a seller should price such data when the buyer's type (feature direction) is private.
Approach
Define data value as a generalized Bregman divergence of a concave cost-of-uncertainty functional, micro-founded via Bayesian decision-making; in Bayesian linear regression with entropy cost this yields information gain, enabling closed-form valuation without seeing realized data and prior-free mechanism design (IC/IR) for selling data.
Key results
- With perfect customization, a payment t(x)=1/2 log((sigma^2+n)/sigma^2) extracts first-best revenue (0-regret) while remaining IC and IR.
- Under limited customization, no 0-regret mechanism exists for continuous types, but the SVD mechanism has regret at most log(kappa) of the normalized data matrix.
- In the multi-armed bandits special case the seller achieves first-best revenue.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Allow perfect customization of data to the buyer's reported type | Data pricing mechanism · Perfect customization data pricing | regret: first-best revenue; 0-regret | Helped |
| Restrict data customization to derivatives of a pre-collected dataset and use the SVD mechanism | Data pricing mechanism · Limited customization data pricing | regret: at most $\log(\kappa((\sqrt{\Sigma(X)})^{-1}X))$ | Hurt |
| Use isotropic existing data, with all singular values the same | Limited customization data pricing · Limited customization data pricing, isotropic data | regret: 0-regret | Helped |
| Restrict buyer types to standard-basis vectors | Limited customization data pricing · Multi-armed bandits data pricing | regret: 0-regret | Helped |
| Limit the seller to limited customization rather than perfect customization | Data pricing mechanism · Limited customization data pricing | regret: no 0-regret mechanism | Hurt |
Surprising findings
- Perfect customization allows first-degree price discrimination, extracting full surplus, unlike classic item-allocation settings.
- Data Shapley overestimates the marginal value of a second datum in a Gaussian example (log3/4 vs 1/2 log(3/2)).
Limitations
- Results assume publicly known Bayesian linear regression model with Gaussian prior and a single buyer/seller.
- Limited-customization regret is log(kappa), not zero.
Experimental setup
Metrics: revenue regret
Baselines: Data Shapley
Builds on: Data Shapley Bregman divergence Myerson-style payment characterization
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