Optimal Pricing with Unreliable Signals

Abstract

We study a single-buyer pricing problem with unreliable side information, motivated by the increasing use of AI-assisted decision-making and LLM-based predictions. The seller observes a private sample that may be either accurate (coinciding with the buyer's valuation), or hallucinatory (an independent draw from the prior), without knowing which case has realized. The buyer does not observe the realized signal, yet knows whether it is accurate or hallucinatory. This creates a higher-order informational asymmetry: the seller is uncertain about the reliability of his own side information, while the buyer has private information about that reliability. Adopting a consistency-robustness framework, we characterize the exact Pareto frontier of tradeoffs between consistency (performance under an accurate signal) and robustness (performance under a hallucinatory signal). We show that keeping the unreliable signal private generates substantial value, yielding tradeoffs that strictly dominate any public-signal benchmark. We further show that perfect consistency does not preclude meaningful protection against hallucination: for every prior, there exists a mechanism achieving perfect consistency together with a nontrivial robustness guarantee of $\frac{1}{2}$. Moreover, if the prior has an infinite mean or a mean of at most its monopoly price, we provide a mechanism that is simultaneously 1-consistent and 1-robust. Our results illustrate a new mechanism design paradigm: rather than relying only on information directly possessed by the designer, mechanisms can be built to leverage the other side's knowledge about the reliability of the designer's information.

Figures (6)

• Figure 1: The blue line represents the $R^{\star}(C)$ curve, which illustrates the robustness-consistency tradeoff frontier achieved in this paper. The dotted red line represents the equation $R + C = 1$, indicating the tradeoff available for the public-signal benchmark.
• Figure 2: Illustration of the thresholds $v_L$ and $v_H$.
• Figure 3: Revenue-curve constraints that determine the worst-case distribution.
• Figure 4: Illustration of the first reduction steps for $(\eta,\beta)$. For a given pair $(\eta,\beta)$, which induces $B(v) = \int_{v}^{U} \beta(w) \;\textrm{d}w + \eta$, the figure shows how $\eta$ can be eliminated and $\beta$ can be transformed into $\tilde{\beta}$, which induces $\tilde{B}(v)$, without increasing the objective. The red curve is the revenue curve of the worst-case distribution under the pair $(0, \tilde{\beta})$.
• Figure 5: Step-function reduction for $\beta(\cdot)$.

Theorems & Definitions (21)

• Theorem 1.1: Optimal Consistency--Robustness Tradeoff
• Theorem 1.2
• Theorem 1.3
• proof : Proof of Theorem \ref{['thm:perfect']}
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Lemma 4.1
• proof