Posted Pricing for Online Selection: Limited Price Changes and Risk Sensitivity

TL;DR

The paper addresses online resource allocation via posted pricing under two practical constraints: a cap on price changes (Δ) and tail-risk via CVaRδ of total welfare. It introduces kSelection-(δ,Δ) and a correlated posted-pricing scheme (cPPM-φ) that uses a single random seed to coordinate prices across levels, achieving favorable tail performance while limiting price updates. The authors develop a risk-neutral optimal design for δ=1 across all Δ, and comprehensive risk-sensitive results for Δ=0, Δ=k-1, and general Δ, using a risk-sensitive online primal–dual framework (R-OPD) and, in some cases, delayed differential equations to characterize pricing functions. The framework reveals a clear trade-off between θ (tail risk) and price-change flexibility, with optimal or near-optimal performance in several regimes and a path forward for broader applicability and data-driven extensions.

Abstract

Posted-price mechanisms (PPMs) are a widely adopted strategy for online resource allocation due to their simplicity, intuitive nature, and incentive compatibility. To manage the uncertainty inherent in online settings, PPMs commonly employ dynamically increasing prices. While this adaptive pricing achieves strong performance, it introduces practical challenges: dynamically changing prices can lead to fairness concerns stemming from price discrimination and incur operational costs associated with frequent updates. This paper addresses these issues by investigating posted pricing constrained by a limited, pre-specified number of allowed price changes, denoted by $Δ$. We further extend this framework by incorporating a second critical dimension: risk sensitivity. Instead of evaluating performance based solely on expectation, we utilize a tail-risk objective-specifically, the Conditional Value at Risk (CVaR) of the total social welfare, parameterized by a risk level $δ\in [0, 1]$. We formally introduce a novel problem class kSelection-$(δ,Δ)$ in online adversarial selection and propose a correlated PPM that utilizes a single random seed to correlate posted prices. This correlation scheme is designed to address both the limited price changes and simultaneously enhance the tail performance of the online algorithm. Our subsequent analysis provides performance guarantees under these joint constraints, revealing a clear trade-off between the number of allowed price changes and the algorithm's risk sensitivity. We also establish optimality results for several important special cases of the problem.

Figures (4)

• Figure 1: The performance comparison of the r-static, r-dynamic, and d-dynamic algorithms on a specific instance of the kSelection-$(\delta,\Delta)$ problem, whose formal definition is provided in Section \ref{['sec:problem_setting']}. The algorithms are characterized as follows: (i) the r-static algorithm, developed in sun2024static, employs a single randomized price; (ii) the r-dynamic algorithm, introduced in jazi2025posted, utilizes $k$ independent random seeds to generate $k$ randomized, dynamically increasing prices; (iii) the d-dynamic algorithm, developed in tan2023threshold, uses $k$ deterministic, dynamically changing prices. The left plot illustrates the fluctuation in performance of these randomized algorithms over $10^4$ independent runs on the same instance, whereas the right plot presents the Cumulative Distribution Function (CDF) of their empirical performance across these runs.
• Figure 2: Illustration of cPPM-$\boldsymbol{\phi}$ with $\Delta = 2$ (i.e., dynamic pricing with three total price levels), total units $k = 10$, and reservation vector $\{q_1 = q_2 = 3,\, q_3 = 4\}$. When a random seed $R \sim \mathcal{U}(0,1)$ is sampled, the three prices $p_1$, $p_2$, and $p_3$ are generated according to the pricing functions $\phi_1$, $\phi_2$, and $\phi_3$, respectively. By construction, the pricing functions satisfy $L = \phi_1(0) \leq \phi_1(1) = \phi_2(0) \leq \phi_2(1) = \phi_3(0) \leq \phi_3(1) = U$, which ensures that $p_1 \leq p_2 \leq p_3$ always holds.
• Figure 3: Worst-case $\textsf{CVaR}_{\delta}\textsf{-CR}$ of cPPM-$\boldsymbol{\phi}$ for $\delta \in \{0.2, 0.6, 0.9\}$ with $L=1$, $U=100$, and $k$ ranging from 3 to 100. The pricing functions are designed according to Theorem \ref{['thm:k:cvar:design:phi']}.
• Figure 4: Worst-case $\textsf{CVaR}_{\delta}\textsf{-CR}$ of cPPM-$\boldsymbol{\phi}$ for $\delta \in \{0.2, 0.4, 0.8\}$ with $L=1$, $U=100$, and $k=40$, where the design of the pricing functions are according to Theorem \ref{['thm:general:cvar:design:phi']}, and we have $\max_{i,j} |q_{i}-q_j|\leq 1$.

Theorems & Definitions (17)

• Definition 1
• Theorem 1
• Theorem 2: Risk-Sensitive Fully-Static Pricing
• Theorem 3: Risk-Sensitive Fully-Dynamic Pricing
• Proposition 1
• Theorem 4: Risk-Sensitive $\Delta$-Dynamic Pricing
• proof
• Lemma 1
• Lemma 2
• Proposition 2