Privacy Can Arise Endogenously in an Economic System with Learning Agents

TL;DR

The paper develops a game-theoretic model of privacy in price discrimination with learning agents, showing that privacy can emerge endogenously as buyers obfuscate valuations (buyer-induced privacy) or as sellers credibly commit to ignore signals (seller-induced privacy). It derives a PBNE for the one-shot game, highlighting a key threshold $\alpha^* = c_B/\Delta\theta$ that governs signaling and pricing, and then extends to repeated interactions where reputation dynamics produce privacy even without commitment. The authors analyze no-regret and no-policy-regret learning for the seller, showing that regret can impede achieving the single-shot optimum $\mathbb{U}^*_1$, while policy-regret-minimizing strategies can attain $\mathbb{U}^*_1$ asymptotically, and reputation-based privacy can arise from observed histories. Through simulations, they demonstrate convergence of utilities, estimator consistency, and alignment of seller actions with equilibrium predictions, while also exploring the impact of biased estimates. The results offer policy-relevant insights: credible privacy commitments or auditing can improve seller utility by stabilizing privacy-enhanced equilibria, bridging formal privacy guarantees with practical privacy considerations in economic settings.

Abstract

We study price-discrimination games between buyers and a seller where privacy arises endogenously--that is, utility maximization yields equilibrium strategies where privacy occurs naturally. In this game, buyers with a high valuation for a good have an incentive to keep their valuation private, lest the seller charge them a higher price. This yields an equilibrium where some buyers will send a signal that misrepresents their type with some probability; we refer to this as buyer-induced privacy. When the seller is able to publicly commit to providing a certain privacy level, we find that their equilibrium response is to commit to ignore buyers' signals with some positive probability; we refer to this as seller-induced privacy. We then turn our attention to a repeated interaction setting where the game parameters are unknown and the seller cannot credibly commit to a level of seller-induced privacy. In this setting, players must learn strategies based on information revealed in past rounds. We find that, even without commitment ability, seller-induced privacy arises as a result of reputation building. We characterize the resulting seller-induced privacy and seller's utility under no-regret and no-policy-regret learning algorithms and verify these results through simulations.

Figures (5)

• Figure 1: Plots of the $\overline{\theta}$-buyer and seller utilities as a function of $\alpha$ in the $\underline{\theta} \geq \mu \overline{\theta}$ setting (left) and the $\underline{\theta} < \mu \overline{\theta}$ setting (right).
• Figure 2: Convergence of seller and buyer utilities for various algorithms. $\underline{\theta} < \mu\overline{\theta}$ with our experiment parameters, so the buyer's $(\alpha=0)$-PD and $(\alpha=\alpha^*)$-PD utilities are the same (see Corollary \ref{['cor:order of utilities']}).
• Figure 3: $\hat{\alpha}_t$ and $\bar{\hat{\alpha}}_t$ over time when seller is playing Exp3, CExp3, or $\alpha^*$-PD. In all cases, $\hat{\alpha}$ is a consistent estimator of the seller's true probability of price discrimination.
• Figure 4: Relative frequency of actions for the seller playing Exp3, CExp3 and $\alpha^*$-PD. The number of PD and reversePD actions for the Exp3 seller are both $0$, as is expected.
• Figure 5: Cumulative average utility of the seller playing against CBER-buyers using biased $\hat{\alpha}$'s.

Theorems & Definitions (35)

• Definition 1: PD game
• Theorem 2.1
• Remark 1: Buyer-induced privacy
• Corollary 1
• Corollary 2
• proof
• Remark 2
• Definition 2: Consistent belief based equilibrium responding (CBER) buyers
• Definition 3: Consistent sequence
• Lemma 3.1