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Stability in Online Assignment Games

Emile Martinez, Felipe Garrido-Lucero, Umberto Grandi

TL;DR

This work addresses stability in online assignment games, showing that stable allocations must derive from maximum social welfare matchings and introducing three non-binary measures—the stability index, the κ-approximate core, and the optimality ratio—that link stability to welfare. It develops theoretical bounds tying these metrics together, demonstrates that stability guarantees can imply near-optimal welfare, and analyzes randomized online algorithms under edge- and vertex-arrival models. The paper provides both general online guarantees (through Half pricing and stability-vs-optimality relations) and model-specific results (vertex-weighted and edge-weighted with free disposal), with tight lower bounds illustrating intrinsic limits. Overall, it lays foundational theory for stability-conscious algorithm design in online matching with transfers, highlighting practical implications for housing markets and similar markets where arrivals are sequential and future data are uncertain.

Abstract

The assignment game models a housing market where buyers and sellers are matched, and transaction prices are set so that the resulting allocation is stable. Shapley and Shubik showed that every stable allocation is necessarily built on a maximum social welfare matching. In practice, however, stable allocations are rarely attainable, as matchings are often sub-optimal, particularly in online settings where eagents arrive sequentially to the market. In this paper, we introduce and compare two complementary measures of instability for allocations with sub-optimal matchings, establish their connections to the optimality ratio of the underlying matching, and use this framework to study the stability performances of randomized algorithms in online assignment games.

Stability in Online Assignment Games

TL;DR

This work addresses stability in online assignment games, showing that stable allocations must derive from maximum social welfare matchings and introducing three non-binary measures—the stability index, the κ-approximate core, and the optimality ratio—that link stability to welfare. It develops theoretical bounds tying these metrics together, demonstrates that stability guarantees can imply near-optimal welfare, and analyzes randomized online algorithms under edge- and vertex-arrival models. The paper provides both general online guarantees (through Half pricing and stability-vs-optimality relations) and model-specific results (vertex-weighted and edge-weighted with free disposal), with tight lower bounds illustrating intrinsic limits. Overall, it lays foundational theory for stability-conscious algorithm design in online matching with transfers, highlighting practical implications for housing markets and similar markets where arrivals are sequential and future data are uncertain.

Abstract

The assignment game models a housing market where buyers and sellers are matched, and transaction prices are set so that the resulting allocation is stable. Shapley and Shubik showed that every stable allocation is necessarily built on a maximum social welfare matching. In practice, however, stable allocations are rarely attainable, as matchings are often sub-optimal, particularly in online settings where eagents arrive sequentially to the market. In this paper, we introduce and compare two complementary measures of instability for allocations with sub-optimal matchings, establish their connections to the optimality ratio of the underlying matching, and use this framework to study the stability performances of randomized algorithms in online assignment games.

Paper Structure

This paper contains 11 sections, 15 theorems, 67 equations, 8 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. The Assignment Game
  3. Sub-Optimality and Stability
  4. Stability Index
  5. Approximated Core
  6. Online Stable Allocations
  7. Stability Under Uncertainty
  8. Equal Pricing and Edge Arrival Model
  9. Vertex Arrival Model
  10. Conclusions
  11. Missing proofs

Key Result

Theorem 2.8

Let $\Gamma$ be an assignment game and $(x,\alpha,\beta)$ be solutions of the pair primal-dual linear programs such that $x$ is integral. It follows that $(\alpha,\beta)\in C(\Gamma)$ and there exists a price vector $\boldsymbol{p}$ such that $(x,\boldsymbol{p})$ is a stable allocation, with $u_i(x,

Figures (8)

  • Figure 1: An assignment game instance. Buyers' valuations are denoted over the edges while sellers' valuations are denoted under their houses.
  • Figure 2: Two possible allocations in our assignment game example. The first allocation is not individually rational, while the second allocation has a blocking pair.
  • Figure 3: An allocation with $\mathop{\mathrm{\mathcal{J}}}\nolimits(\mathop{\mathrm{\mu,\boldsymbol{p}}}\nolimits) = \frac{5}{9}$.
  • Figure 4: Two online matching markets models.
  • Figure 5: Two edge arrival instances
  • ...and 3 more figures

Theorems & Definitions (40)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Remark 2.6
  • Definition 2.7
  • Theorem 2.8
  • Example 2.9
  • Definition 3.1
  • ...and 30 more