Simple and Robust Quality Disclosure: The Power of Quantile Partition

TL;DR

This work studies robust quality disclosure under uncertainty by restricting platforms to simple signaling rules and evaluating performance via the minimax competitive ratio against Bayesian-optimal revenue. It proves that K-quantile partition disclosures are minimax optimal within a broad class, with the optimal robust ratio Γ_K^* characterized as the unique solution to Λ_K(Γ)=1 and thresholds obtained by a backward recursion Q_{r-1}^* = 1 - λ_{Γ_K^*}(1 - Q_r^*); for any fixed K, the robust ratio of a K-quantile partition has a clean max-over-bins form, and uniform percentile buckets yield the intuitive 1 + 1/K bound. The analysis further shows a universal 2-approximation barrier for finite-signal monotone (quality-threshold) partitions, highlighting the advantage of rank-based partitioning. A central methodological contribution is a tight reduction from robust quality disclosure to a robust disclosure design framework based on convex-indirect-revenue functions h, enabling extremal hinge-function representations that drive the tractable, exact characterization. Overall, the results justify percentile-based disclosures in practice and provide a principled design framework for robust signaling and pricing in markets with uncertain quality.

Abstract

Quality information on online platforms is often conveyed through simple, percentile-based badges and tiers that remain stable across different market environments. Motivated by this empirical evidence, we study robust quality disclosure in a market where a platform commits to a public disclosure policy mapping the seller's product quality into a signal, and the seller subsequently sets a downstream monopoly price. Buyers have heterogeneous private types and valuations that are linear in quality. We evaluate a disclosure policy via a minimax competitive ratio: its worst-case revenue relative to the Bayesian-optimal disclosure-and-pricing benchmark, uniformly over all prior quality distributions, type distributions, and admissible valuations. Our main results provide a sharp theoretical justification for quantile-partition disclosure. For K-quantile partition policies, we fully characterize the robust optimum: the optimal worst-case ratio is pinned down by a one-dimensional fixed-point equation and the optimal thresholds follow a backward recursion. We also give an explicit formula for the robust ratio of any quantile partition as a simple "max-over-bins" expression, which explains why the robust-optimal partition allocates finer resolution to upper quantiles and yields tight guarantees such as 1 + 1/K for uniform percentile buckets. In contrast, we show a robustness limit for finite-signal monotone (quality-threshold) partitions, which cannot beat a factor-2 approximation. Technically, our analysis reduces the robust quality disclosure to a robust disclosure design program by establishing a tight functional characterization of all feasible indirect revenue functions.

Figures (4)

• Figure 1: An illustration of $K$-Quantile Partition disclosure. Here $K = 4$ and $Q_0 = 0, Q_1 = 0.25, Q_2 = 0.5, Q_3 = 0.75, Q_4 = 1$. The $y$-axis is cumulative probability mass. Left: uniform discrete prior with $\mathsf{supp}(F)=\{0,0.5,1\}$. Right: uniform continuous prior with $\mathsf{supp}(F)=[0,1]$. With the same $K$-Quantile Partition, the induced posterior-mean distributions can differ across priors.
• Figure 2: The blue dashed lines are upper boundary (i.e., $h(x) \equiv 1$) and lower boundary (i.e., $h(x) = x$) of the functional space defined in \ref{['thm:rdd:optimal robust policy']} (here we normalize $h(1) = 1$ for the presentation simplicity). The red solid lines in \ref{['fig:left tight bound']} are some feasible convex function $h \in \mathcal{H}$ in this functional space $\mathcal{H}$, while the red solid lines in \ref{['fig:right tight bound']} are some extremal convex functions (i.e., $h(x) = \max\{x, c\}$ that we establish in \ref{['lem:stoploss-representation']} for some $c\in[0, 1]$) in this functional space $\mathcal{H}$.
• Figure 3: The blue dashed lines are upper boundary (i.e., $h(x) = x + c$) and lower boundary (i.e., $h(x) = \max\{x, c\}$) of the functional space $\mathcal{H}_{\textsc{sand}}$ when we fix a particular value of $c$. The red solid line in \ref{['fig:left sandwich']} is a feasible convex function $h \in\mathcal{H}_{\textsc{sand}}$ in this functional space, while the red solid line in \ref{['fig:right sandwich']} is the constructed convex function $h$ in this functional space such that it leads to worst-case competitive ratio.
• Figure 4: An illustration of $K$-Quality Partition disclosure. Here $K = 4$ and $\omega_0 = 0, \omega_1 = 0.25, \omega_2 = 0.5, \omega_3 = 0.75, \omega_4 = 1, \xi_0 = 0, \xi_r = 1$ for all $r\in[4]$. The $y$-axis is cumulative probability mass. Left: uniform discrete prior with $\mathsf{supp}(F)=\{0,0.5,1\}$. Right: uniform continuous prior with $\mathsf{supp}(F)=[0,1]$. With the same $K$-Quality Partition, the induced posterior-mean distributions can differ across priors. In particular, for the uniform discrete prior on the left, one induced signal is degenerate because there is no prior probability mass between the quality thresholds $\omega_2 = 0.5$ and $\omega_3 = 0.75$.

Theorems & Definitions (51)

• Definition 2.1: Disclosure policy
• Definition 2.2: Robust quality disclosure
• Theorem 3.1: Robust optimal quantile partition
• Proposition 3.2: Monotonicity of $\Gamma_K^*$
• Proposition 3.3: Decreasing margin
• Theorem 3.4: Robust competitive ratio
• Corollary 3.5
• Corollary 3.6
• Definition 4.1: Robust disclosure design
• Definition 4.2: Optimal posted-price revenue function