The Value of Information in Resource-Constrained Pricing

Abstract

Firms that price perishable resources -- airline seats, hotel rooms, seasonal inventory -- now routinely use demand predictions, but these predictions vary widely in quality. Under hard capacity constraints, acting on an inaccurate prediction can irreversibly deplete inventory needed for future periods. We study how prediction uncertainty propagates into dynamic pricing decisions with linear demand, stochastic noise, and finite capacity. A certified demand forecast with known error bound~$ε^0$ specifies where the system should operate: it shifts regret from $O(\sqrt{T})$ to $O(\log T)$ when $ε^0 \lesssim T^{-1/4}$, and we prove this threshold is tight. A misspecified surrogate model -- biased but correlated with true demand -- cannot set prices directly but reduces learning variance by a factor of $(1-ρ^2)$ through control variates. The two mechanisms compose: the forecast determines the regret regime; the surrogate tightens estimation within it. All algorithms rest on a boundary attraction mechanism that stabilizes pricing near degenerate capacity boundaries without requiring non-degeneracy assumptions. Experiments confirm the phase transition threshold, the variance reduction from surrogates, and robustness across problem instances.

Figures (12)

• Figure 1: Overview of the unified framework. boundary attraction underlies all settings. Without offline information, the system learns online at the minimax $O(\sqrt{T})$ rate. A certified informed price $(\bm{p}^0,\bm{d}^0)$ can improve the regime to near-logarithmic regret when $\epsilon^0 \lesssim T^{-1/4}$, while correlated surrogate signals reduce learning variance and improve constants. The combined policy compounds both benefits.
• Figure 2: Regret scaling across time horizons $T$ confirms theoretical predictions: (a) full information grows as $O(\log T)$, (b) no information scales as $O(\sqrt{T})$, and (c) informed price with $\epsilon^0 = T^{-1/2}$ recovers near-logarithmic regret.
• Figure 3: Regret trajectories for the five algorithms on Scale 2 ($m=1$, $n=4$, $\sigma=2.2$, 500 replications). Shaded regions show $\pm 1$ standard deviation. Combining both information channels (Surrogate+Informed) yields the lowest regret; the separation between algorithms grows with $T$.
• Figure 4: Phase transition in informed-price regret (300 replications per point; error bars: $\pm 1$ SE). Below $\epsilon^0 \approx T^{-1/4}$ (dashed vertical), regret stays near the $O(\log T)$ floor; above it, regret rises to the $O(\sqrt{T})$ baseline (dashed horizontal). The transition sharpens with $T$, consistent with Theorem \ref{['thm:incumbent']}.
• Figure 5: Schematic of the boundary attraction mechanism in Algorithm \ref{['alg:resolve']}. The red shaded regions represent the danger zone where resource degeneracy causes instability: when a demand component falls below threshold $\zeta(T-t+1)^{-1/2}$, dual variables can diverge. Fluid solutions in this region are rounded to the boundary (zero demand), ensuring dual variables remain bounded and the re-solve policy maintains stability.

Theorems & Definitions (23)

• Proposition 1: gallego1994optimal
• Theorem 2
• Theorem 3
• Remark 1: Optimality and special cases
• Lemma 4: keskin2014dynamic
• Proposition 5
• Theorem 6
• Proposition 7
• Remark 2: How much correlation suffices?
• Theorem 8: Regret with Surrogate Assistance