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Online matching and market imbalance

Benjamin Barrientos, Daniel Freund, Daniela Saban

TL;DR

The paper addresses online bipartite matching with stochastic rewards under supply–demand imbalance by introducing a solvable, parameterized notion of imbalance $\kappa$ and evaluating delayed online algorithms against an offline clairvoyant benchmark $OFF\textrm{-}I$. It provides explicit $\kappa$-dependent competitive fractions for both adversarial and stochastic arrivals, proves these bounds tight, and extends to generalized imbalance scenarios. The work also shows how imbalance affects upstream inventory decisions and supports these insights with numerical experiments on real volunteer-matching data. Collectively, the results offer a principled framework linking market imbalance to algorithmic performance and platform planning, with practical guidance for inventory management and matching policies.

Abstract

Our work introduces the effect of supply/demand imbalances into the literature on online matching with stochastic rewards in bipartite graphs. We provide a parameterized definition that characterizes instances as over- or undersupplied (or balanced), and show that higher competitive ratios against an offline clairvoyant algorithm are achievable, for both adversarial and stochastic arrivals, when instances are more imbalanced. The competitive ratio guarantees we obtain are the best-possible for the class of delayed algorithms we focus on (such algorithms may adapt to the history of arrivals and the algorithm's own decisions, but not to the stochastic realization of each potential match). We then explore the real-world implications of our improved competitive ratios. First, we demonstrate analytically that the improved competitive ratios under imbalanced instances is not a one-way street by showing that a platform that conducts effective supply- and demand management should incorporate the effect of imbalance on its matching performance on its supply planning in order to create imbalanced instances. Second, we empirically study the relationship between achieved competitive ratios and imbalance using the data of a volunteer matching platform.

Online matching and market imbalance

TL;DR

The paper addresses online bipartite matching with stochastic rewards under supply–demand imbalance by introducing a solvable, parameterized notion of imbalance and evaluating delayed online algorithms against an offline clairvoyant benchmark . It provides explicit -dependent competitive fractions for both adversarial and stochastic arrivals, proves these bounds tight, and extends to generalized imbalance scenarios. The work also shows how imbalance affects upstream inventory decisions and supports these insights with numerical experiments on real volunteer-matching data. Collectively, the results offer a principled framework linking market imbalance to algorithmic performance and platform planning, with practical guidance for inventory management and matching policies.

Abstract

Our work introduces the effect of supply/demand imbalances into the literature on online matching with stochastic rewards in bipartite graphs. We provide a parameterized definition that characterizes instances as over- or undersupplied (or balanced), and show that higher competitive ratios against an offline clairvoyant algorithm are achievable, for both adversarial and stochastic arrivals, when instances are more imbalanced. The competitive ratio guarantees we obtain are the best-possible for the class of delayed algorithms we focus on (such algorithms may adapt to the history of arrivals and the algorithm's own decisions, but not to the stochastic realization of each potential match). We then explore the real-world implications of our improved competitive ratios. First, we demonstrate analytically that the improved competitive ratios under imbalanced instances is not a one-way street by showing that a platform that conducts effective supply- and demand management should incorporate the effect of imbalance on its matching performance on its supply planning in order to create imbalanced instances. Second, we empirically study the relationship between achieved competitive ratios and imbalance using the data of a volunteer matching platform.

Paper Structure

This paper contains 39 sections, 30 theorems, 139 equations, 12 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Model and Preliminaries
  3. Definition of imbalance.
  4. Choice of Benchmark.
  5. Overview of Main Results
  6. Adversarial arrivals: lower bound
  7. Stochastic arrivals: lower bound
  8. Numerical experiments
  9. Concluding Remarks
  10. Doubling supply does not double successful matches
  11. Proof of \ref{['lemma: offline is upper bound']} (Section \ref{['sec: model']})
  12. Results on the underage and overage costs (Section \ref{['sec: overview_results']})
  13. Remaining proofs from Section \ref{['sec3']}
  14. Proof of \ref{['claim: AdvOfflineOptBound']}
  15. Proof of \ref{['lemma: DLP lower bounds greedy/off']}
  16. ...and 24 more sections

Key Result

Lemma 2.1

Let $\mathcal{I}$ be an instance and $\mathrm{ALG}$ be any algorithm with $(X_{u,t})_{u \in S, t \in [T]}$ the sequence of matching decisions that where successful. Then, in either setting, $\mathrm{\mathrm{OFF}}\textrm{-}\mathrm{I}(\mathcal{I}) \geq \mathbb E\left[\sum_{ (u,t) \in S \times [T]} X_

Figures (12)

  • Figure 1: Depending on $\mu\in\{1/4,.5,1\}$ the instance is $1/2$-oversupplied, balanced, or $2$-undersupplied.
  • Figure 2: Optimal stocking levels $\eta_A^*$ and $\eta_B^*$ as functions of $(r-c)/r$. The horizontal dotted lines correspond to $\eta\in\{0.736,1.358\}$. The vertical dotted lines correspond to their intersection with the optimal stocking level.
  • Figure 3: Comparison of our tight CRs for adversarial and stochastic settings across $\kappa$. The dotted lines intersect with the $x$-axis at $0.736$, $0.754$, $1.210$, and $1.358$ respectively.
  • Figure 4: Scatter plots for the CR of $\mathrm{GREEDY\textrm{-}D}$: aggregated instances (left) and a particular instance (right).
  • Figure 5: Map/graph of the particular instance (left) and the instance's connected components for different values of $\mu$ (right). Coordinates were randomly perturbed to preserve the privacy of the platform users.
  • ...and 7 more figures

Theorems & Definitions (37)

  • Lemma 2.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Proposition 3.1
  • Proposition 3.2
  • Proposition 4.3
  • ...and 27 more