Threshold Policies with Tight Guarantees for Online Selection with Convex Costs

TL;DR

A class of simple threshold policies that are logistically simple and easy to implement but have provable optimality guarantees among all deterministic algorithms are provided, and a lower bound on competitive ratios of randomized algorithms is derived.

Abstract

This paper provides threshold policies with tight guarantees for online selection with convex cost (OSCC). In OSCC, a seller wants to sell some asset to a sequence of buyers with the goal of maximizing her profit. The seller can produce additional units of the asset, but at non-decreasing marginal costs. At each time, a buyer arrives and offers a price. The seller must make an immediate and irrevocable decision in terms of whether to accept the offer and produce/sell one unit of the asset to this buyer. The goal is to develop an online algorithm that selects a subset of buyers to maximize the seller's profit, namely, the total selling revenue minus the total production cost. Our main result is the development of a class of simple threshold policies that are logistically simple and easy to implement, but have provable optimality guarantees among all deterministic algorithms. We also derive a lower bound on competitive ratios of randomized algorithms and prove that the competitive ratio of our threshold policy asymptotically converges to this lower bound when the total production output is sufficiently large. Our results generalize and unify various online search, pricing, and auction problems, and provide a new perspective on the impact of non-decreasing marginal costs on real-world online resource allocation problems.

Figures (6)

• Figure 1: Illustration of $\textsf{CR}_f^*(\rho,k)$, $\textsf{CR}_f^{\textsf{lb}}(\rho,k)$, and $\underline{\textsf{CR}}_f(\rho)$.
• Figure 2: Illustration of admission thresholds with $k = 5$ in three cases. In subfigure (a), the turning point $\tau = 2$ (i.e., $\lambda_0 = \lambda_1 = \lambda_2 = p_{\min}$) and $\underaccent{\bar{}}{k} = \bar{k} = k = 5$; In subfigure (b), the turning point $\tau = 0$ (i.e., $\lambda_0 = p_{\min}$), $\underaccent{\bar{}}{k} = 1$, $\bar{k} =4$, and $k = 5$; In subfigure (c), the turning point $\tau = 3$ (i.e., $\lambda_0 = \lambda_1 = \lambda_2 = \lambda_3 = p_{\min}$), $\underaccent{\bar{}}{k} = 4$, and $\bar{k} = k = 5$. Recall that the sequence of marginal costs are non-decreasing, i.e., $c_{i+1} \geq c_i$ holds for all $i\in [k-1]$.
• Figure 3: Illustration of the three cost functions used in the simulation.
• Figure 4: Comparison of the optimal competitive ratio $\alpha^* = \textsf{CR}_f^*(\rho,k)$ and the average empirical ratio (AER) of $\textsf{TOS}_{\boldsymbol{\lambda}^*}$ under different setups. Subfigure (a): $\alpha^* = \textsf{CR}_f^*(\rho,k)$ under different $f$ (linear, quadratic, and exponential) with $k = 300$. Subfigure (b): AER of $\textsf{TOS}_{\boldsymbol{\lambda}^*}$ under different types of arrival instances (low2high, random, and high2low) with quadratic $f$ and $\rho = 8$. Other than $\alpha^*$, each curve shows the AER of $\textsf{TOS}_{\boldsymbol{\lambda}^*}$ over 1000 arrival instances sampled by their corresponding types. Subfigure (c): AER of $\textsf{TOS}_{\boldsymbol{\lambda}^*}$ under different $f$ (linear, quadratic, and exponential). Each curve shows the AER of $\textsf{TOS}_{\boldsymbol{\lambda}^*}$ over 1000 arrival instances sampled by Type 2: random.
• Figure 5: Empirical ratio (ER) of $\textsf{TOS}_{\boldsymbol{\lambda}^*}$ under different $f$ with $k = 300$. The three real curves in each subfigure show the 25th percentile, 75th percentile, and average of ERs over 1000 arrival instances sampled by Type 2: random. The scatter plots in blue show the minimum and maximum ER among the 1000 samples.

Theorems & Definitions (60)

• Definition 1: Min-Profit
• Definition 2: Min-Production
• Definition 3: Conjugate
• Lemma 1
• Proposition 1
• proof
• Definition 4: Admission Threshold
• Proposition 2
• proof
• Corollary 1: Sufficient Inequalities