Distributed Learning Dynamics Converging to the Core of $B$-Matchings

TL;DR

This work presents two learning dynamics that converge to the core of the bipartite B-matching problems, centralized dynamics in the nature of the Hungarian method and distributed dynamics, which converge to the core with probability one.

Abstract

$B$-Matching is a special case of matching problems where nodes can join multiple matchings with the degree of each node constrained by an upper bound, the node's $B$-value. The core solution of a bipartite $B$-matching is both a matching between the nodes respecting the upper bound constraint and an allocation of the weights of the edges among the nodes such that no group of nodes can deviate and collectively gain higher allocation. We present two learning dynamics that converge to the core of the bipartite $B$-matching problems. The first dynamics are centralized dynamics in the nature of the Hungarian method, which converge to the core in a polynomial time. The second dynamics are distributed dynamics, which converge to the core with probability one. For the distributed dynamics, a node maintains only a state consisting of (i) the aspiration levels for all of its possible matches and (ii) the matches, if any, to which it belongs. The node does not keep track of its history nor is it aware of the environment state. In each stage, a randomly activated node proposes to form a new match and changes its aspiration based on the success or failure of its proposal. At this stage, the proposing node inquires about the aspiration of the node it wants to match with to calculate the feasibility of the match. The environment matching structure changes whenever a proposal succeeds. A state is absorbing for the distributed dynamics if and only if it is in the core of the $B$-matching.

Figures (5)

• Figure 1: Sample run of the multi-robot task assignment setting. On the right, the optimal value 'OPT' is plotted in purple and the total feasible aspirations of the agents, as the $B$-Matching Proposals dynamics evolve, 'PROP' is plotted in blue. On the left, an illustration of the final $B$-copies matching between the robots and the tasks is shown.
• Figure 2: The average of the relative total feasible aspirations achieved by the $B$-Matching Proposals dynamics across $100$ runs.
• Figure 3: The average relative welfare across $50$ configurations when using $B_1, B_2,$ and $B_3$ as the $B$-value functions. The runs that use $B_1$ are represented by the blue line, those using $B_2$ are represented by the red line, and those using $B_3$ are represented by the yellow line.
• Figure 4: The average relative welfare across $50$ configurations when using different values for the discretization width.
• Figure 5: The average relative welfare across 50 configurations, and the dynamics performance as we gradually remove nodes and their associated edges from the graph.

Theorems & Definitions (21)

• Definition 1
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Proposition 1
• proof
• Definition 7
• Definition 8