Two-stage Stochastic Assignment Games

2025-06-02

AI Summary

This paper investigates core stability in two-stage stochastic assignment games, where the second-stage player set evolves according to an unknown probability distribution, and the objective is to identify the minimally perturbed core allocation. Methodologically, we first establish the integrality of the associated linear programming polytope—a novel result—and thereby develop a unified framework integrating integral polyhedral theory, stochastic optimization, and cooperative game theory. Our contributions are threefold: (1) under explicit distributions, we obtain exact polynomial-time algorithms; (2) under unknown distributions, we design efficient approximation algorithms with provable theoretical guarantees; and (3) we reveal an intrinsic equivalence between this problem and multi-stage vertex cover, proving that the latter admits a polynomial-time solution when bipartite partitions remain consistent across stages. Collectively, these results establish a new paradigm for computing robust core allocations in stochastic cooperative games.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Web Mining and Content Analysis: Models for Web evolution

Abstract

In this paper, we study a two-stage stochastic version of the assignment game, which is a fundamental cooperative game. Given an initial setting, the set of players may change in the second stage according to some probability distribution, and the goal is to find core solutions that are minimally modified. When the probability distribution is given explicitly, we observe that the problem is polynomial time solvable, as it can be modeled as an LP. More interestingly, we prove that the underlying polyhedron is integral, and exploit this in two ways. First, integrality of the polyhedron allows us to show that the problem can be well approximated when the distribution is unknown, which is a hard setting. Second, we can establish an intimate connection to the well-studied multistage vertex cover problem. Here, it is known that the problem is NP-hard even when there are only 2 stages and the graph in each stage is bipartite. As a byproduct of our result, we can prove that the problem is polynomial-time solvable if the bipartition is the same in each stage.

Problem

• Study two-stage stochastic assignment games with changing players
• Find minimally modified core solutions under probability distributions
• Connect to multistage vertex cover problem and bipartition conditions

Innovation

• Two-stage stochastic assignment game model
• Integral polyhedron enables LP approximation
• Connects to multistage vertex cover problem