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Stable Task Allocation in Multi-Agent Systems with Lexicographic Preferences

Spyros Reveliotis, Eva Robillard

TL;DR

This work addresses stable task allocation in multi-agent systems where agents have lexicographic (ordinal) preferences over feasible subsets of tasks, formalized as a bipartite one-to-many matching with downward-closed feasibility sets $F(a)$ and a lexicographic-tree representation $\mathcal{D}(a)$ for agent preferences. The authors develop an integer-programming framework that encodes stable matchings via binary variables $X_{an}$ associated with tree nodes, enforces one allocation per agent, at most one agent per task, and stability against blocking pairs, and they propose relaxations using blocking variables $Y_{ta}$ to cope with infeasibility. They establish a tree-based criterion for substitutability, relate it to known stability results, and show how substitutability can imply the existence of stable allocations and enable GS-type algorithms under certain conditions. The work provides a rigorous foundation for optimizing stable task allocations in complex preference structures and outlines a scalable path (column generation) to solve the resulting IPs in large-scale settings, with potential applications in centralized multi-agent service platforms.

Abstract

Motivated by the increasing interest in the explicit representation and handling of various "preference" structures arising in modern digital economy, this work introduces a new class of "one-to-many stable-matching" problems where a set of atomic tasks must be stably allocated to a set of agents. An important characteristic of these stable-matching problems is the very arbitrary specification of the task subsets constituting "feasible" allocations for each agent. It is shown that as long as the agents rank their feasible task allocations lexicographically with respect to their stated preferences for each atomic task, matching stability reduces to the absence of blocking agent-task pairs. This result, together with a pertinent graphical representation of feasible allocations, enable (i) the representation of the space of stable matchings as a set of linear constraints with binary variables, and (ii) the specification and handling of certain notions of optimality within this space of stable matchings. The last part of the paper also addresses the notion of "substitutability" in the considered problem context.

Stable Task Allocation in Multi-Agent Systems with Lexicographic Preferences

TL;DR

This work addresses stable task allocation in multi-agent systems where agents have lexicographic (ordinal) preferences over feasible subsets of tasks, formalized as a bipartite one-to-many matching with downward-closed feasibility sets and a lexicographic-tree representation for agent preferences. The authors develop an integer-programming framework that encodes stable matchings via binary variables associated with tree nodes, enforces one allocation per agent, at most one agent per task, and stability against blocking pairs, and they propose relaxations using blocking variables to cope with infeasibility. They establish a tree-based criterion for substitutability, relate it to known stability results, and show how substitutability can imply the existence of stable allocations and enable GS-type algorithms under certain conditions. The work provides a rigorous foundation for optimizing stable task allocations in complex preference structures and outlines a scalable path (column generation) to solve the resulting IPs in large-scale settings, with potential applications in centralized multi-agent service platforms.

Abstract

Motivated by the increasing interest in the explicit representation and handling of various "preference" structures arising in modern digital economy, this work introduces a new class of "one-to-many stable-matching" problems where a set of atomic tasks must be stably allocated to a set of agents. An important characteristic of these stable-matching problems is the very arbitrary specification of the task subsets constituting "feasible" allocations for each agent. It is shown that as long as the agents rank their feasible task allocations lexicographically with respect to their stated preferences for each atomic task, matching stability reduces to the absence of blocking agent-task pairs. This result, together with a pertinent graphical representation of feasible allocations, enable (i) the representation of the space of stable matchings as a set of linear constraints with binary variables, and (ii) the specification and handling of certain notions of optimality within this space of stable matchings. The last part of the paper also addresses the notion of "substitutability" in the considered problem context.

Paper Structure

This paper contains 7 sections, 4 theorems, 49 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Literature review of the stable-matching problem
  3. The considered matching problem
  4. Matching Stability
  5. An IP-based representation of matching stability and optimal stable matchings
  6. Substitutability
  7. Conclusions

Key Result

Proposition 1

Every matching ${\mathcal{M}} \in \mathbb{M}$ is individually rational.

Figures (3)

  • Figure 1: The lexicographic trees of Example \ref{['ex:l-t']}.
  • Figure 2: The lexicographic trees $\mathcal{D}(a_1)$ and $\mathcal{D}(a_2)$ in Example \ref{['ex:empty']}.
  • Figure 3: The lexicographic trees $\mathcal{D}(a_1)$ and $\mathcal{D}(a_2)$ in Example \ref{['ex:uneq']}.

Theorems & Definitions (17)

  • Definition 1
  • Remark 1
  • Definition 2
  • Example 1
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 1
  • Definition 6
  • Definition 7
  • ...and 7 more