Prophet Upper Bounds for Online Matching and Auctions

TL;DR

New and improved upper bounds on the competitiveness achievable by an algorithm for the general online 2-bounded auction and the (single-minded) prophet matching problems are provided.

Abstract

In the online 2-bounded auction problem, we have a collection of items represented as nodes in a graph and bundles of size two represented by edges. Agents are presented sequentially, each with a random weight function over the bundles. The goal of the decision-maker is to find an allocation of bundles to agents of maximum weight so that every item is assigned at most once, i.e., the solution is a matching in the graph. When the agents are single-minded (i.e., put all the weight in a single bundle), we recover the maximum weight prophet matching problem under edge arrivals (a.k.a. prophet matching). In this work, we provide new and improved upper bounds on the competitiveness achievable by an algorithm for the general online 2-bounded auction and the (single-minded) prophet matching problems. For adversarial arrival order of the agents, we show that no algorithm for the online 2-bounded auction problem achieves a competitiveness larger than $4/11$, while no algorithm for prophet matching achieves a competitiveness larger than $\approx 0.4189$. Using a continuous-time analysis, we also improve the known bounds for online 2-bounded auctions for random order arrivals to $\approx 0.5968$ in the general case, a bound of $\approx 0.6867$ in the IID model, and $\approx 0.6714$ in prophet-secretary model.

Figures (5)

• Figure 1: Agent 2 selects exactly one edge marked with a star at random and shows it with a weight equal to 4. Agent 3 puts a weight of $4/\varepsilon$ on the diagonal edge with probability $\varepsilon$.
• Figure 2: Each edge of the graph is labeled with the pair defining its weight and distribution.
• Figure 3: Rolling particle representation for the $\sqrt{3}-1$ upper bound in the prophet-secretary single-choice problem. When the algorithm sees agent $A$ at time $x=T(A)$, it has to decide whether to stop or continue. The expected earned weight of stopping is $\lambda x+1$, and the expected weight for not stopping is $\lambda$. If $x>T^{\star}$, the algorithm decides to stop.
• Figure 4: Instances in the prophet-secretary model for matching (left) and general 2-bounded auctions (right). The expected weight recovered by stopping on the first lateral agent at time $x$ is $\lambda x+2$ for matching and $\lambda x + 3/2$ for auctions. The expected weight recovered by stopping on the second lateral agent at time $y$ is $\lambda y+1$, and for not stopping is $\lambda$.
• Figure 5: The number of important agents is distributed according to a Poisson with parameter $\theta$, and the expected waiting time between important agents distributes as an exponential with rate $\theta$. If the algorithm stops at an important agent at time $t$, it gets an expected profit of $\lambda t + 2-\exp(-\theta t/2)$.

Theorems & Definitions (9)

• Theorem 1
• Theorem 2
• proof : Proof of Theorem \ref{['thm:upper-bounds-matching-adversarial']}\ref{['ub-matching-a']}
• proof : Proof of Theorem \ref{['thm:upper-bounds-matching-adversarial']}\ref{['ub-matching-b']}
• proof : Proof of Theorem \ref{['thm:upper-bounds-matching']}\ref{['ro-ub-matching-a']}
• proof : Proof of Theorem \ref{['thm:upper-bounds-matching']}\ref{['ro-ub-matching-b']}
• Lemma 1
• proof
• proof : Proof of Theorem \ref{['thm:upper-bounds-matching']}\ref{['ro-ub-matching-c']}