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Signaling in Data Markets via Free Samples

Nivasini Ananthakrishnan, Alireza Fallah, Michael I. Jordan

Abstract

We study a setting in which a data buyer seeks to estimate an unknown parameter by purchasing samples from one of K data sellers. Each seller has privately known data quality (e.g., high vs. low variance) and a private per-sample cost. We consider a multi-stage game in which the first stage is a free-trial stage in which the sellers have the option of signaling data quality by offering a few samples of data for free. Buyers update their beliefs based on the sample variance of the free data and then run a procurement auction to buy data in a second stage. For the auction stage, we characterize an approximately optimal Bayesian incentive compatible mechanism: the buyer selects a single seller by minimizing a belief-adjusted virtual cost and chooses the purchased sample size as a function of posterior quality and virtual cost. For the free-trial stage, we characterize the equilibrium, taking the above mechanism as the continuation game. Free trials may fail to emerge: for some parameters, all sellers reveal zero samples. However, under sufficiently strong competition (large K), there is an equilibrium in which sellers reveal the maximum allowable number of samples; in fact, it is the unique equilibrium.

Signaling in Data Markets via Free Samples

Abstract

We study a setting in which a data buyer seeks to estimate an unknown parameter by purchasing samples from one of K data sellers. Each seller has privately known data quality (e.g., high vs. low variance) and a private per-sample cost. We consider a multi-stage game in which the first stage is a free-trial stage in which the sellers have the option of signaling data quality by offering a few samples of data for free. Buyers update their beliefs based on the sample variance of the free data and then run a procurement auction to buy data in a second stage. For the auction stage, we characterize an approximately optimal Bayesian incentive compatible mechanism: the buyer selects a single seller by minimizing a belief-adjusted virtual cost and chooses the purchased sample size as a function of posterior quality and virtual cost. For the free-trial stage, we characterize the equilibrium, taking the above mechanism as the continuation game. Free trials may fail to emerge: for some parameters, all sellers reveal zero samples. However, under sufficiently strong competition (large K), there is an equilibrium in which sellers reveal the maximum allowable number of samples; in fact, it is the unique equilibrium.
Paper Structure (28 sections, 10 theorems, 46 equations, 1 figure)

This paper contains 28 sections, 10 theorems, 46 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Theorem 1 (Informal)
  3. Theorem 2 (Informal)
  4. Related Work
  5. Data markets.
  6. Strategic information revelation in data markets.
  7. Information revelation through free samples.
  8. Competition and externalities in data markets.
  9. Persuasion and information design.
  10. Model
  11. Timing and information.
  12. Estimation.
  13. Utility functions.
  14. Solution concept.
  15. The Mechanism Design Subproblem
  16. ...and 13 more sections

Key Result

Proposition 1

Suppose ass:virtual_costs_monotonicity holds. Given the sequence of free samples shared by the sellers $\left (\mathcal{S}^f_i \right )_{i=1}^K$ inducing beliefs $({\pi})_{i=1}^K$ with means $(\bar{\sigma}_i^2)_{i=1}^K$, there is a sample purchasing and payment rule that forms a Bayesian Incentive C

Figures (1)

  • Figure 1: Phase diagram of symmetric equilibria across $(K,\, \sigma_{H}/\sigma_{L})$. Each cell is colored based on the equilibrium detected in the corresponding regime of parameters. Blue: only the informative equilibrium ($m^* = M$). Red: only the uninformative ($m^* = 0$). Green: only an intermediate equilibrium ($m^* = 2$). Purple: coexistence of multiple symmetric equilibria. Gray: no symmetric equilibrium found.

Theorems & Definitions (18)

  • Proposition 1: Approximately Optimal Purchasing and Payment Rules
  • proof : Proof of \ref{['prop:mechanism_soln']}
  • Lemma 1: Optimal weights to minimize variance
  • proof : Proof of \ref{['lem:opt_weights']}
  • Theorem 1: Uninformative Equilibrium
  • Lemma 2: Utility in symmetric data-sharing strategy profiles
  • proof : Proof of \ref{['lem:utility_symmetric_strategy']}
  • Lemma 3: Belief shift from sharing $m$ samples
  • proof : Proof of \ref{['lem:upper_bound_belief_shift']}
  • Theorem 2: Equilibrium is maximally informative when there are many sellers
  • ...and 8 more