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Faster Algorithms for the Least-Core value and the Nucleolus in Convex Games

Giacomo Maggiorano, Alessandro Sosso, Gautier Stauffer

Abstract

The nucleolus is a central solution concept in cooperative game theory. While its computation is NP-hard in general, it can be computed in polynomial time for convex games; however, the only published polynomial-time algorithm relies on the ellipsoid method. We develop a combinatorial alternative based on reduced games and iterative least-core value computations. Leveraging submodular function minimization and polyhedral structure in a novel way, we obtain a faster combinatorial algorithm for computing the least-core value, improving the oracle complexity by a factor $n^3$ over previous approaches. As a consequence, we obtain a new strongly polynomial-time and combinatorial algorithm for computing the nucleolus in convex games. Preliminary analysis indicates an improved oracle complexity compared to the ellipsoid-based algorithm.

Faster Algorithms for the Least-Core value and the Nucleolus in Convex Games

Abstract

The nucleolus is a central solution concept in cooperative game theory. While its computation is NP-hard in general, it can be computed in polynomial time for convex games; however, the only published polynomial-time algorithm relies on the ellipsoid method. We develop a combinatorial alternative based on reduced games and iterative least-core value computations. Leveraging submodular function minimization and polyhedral structure in a novel way, we obtain a faster combinatorial algorithm for computing the least-core value, improving the oracle complexity by a factor over previous approaches. As a consequence, we obtain a new strongly polynomial-time and combinatorial algorithm for computing the nucleolus in convex games. Preliminary analysis indicates an improved oracle complexity compared to the ellipsoid-based algorithm.

Paper Structure

This paper contains 16 sections, 20 theorems, 104 equations, 4 algorithms.

Table of Contents

  1. Introduction
  2. Definitions and preliminaries
  3. Remark
  4. A natural reduced game approach for Nucleolus in convex games
  5. The Least Core of Convex Games Combinatorially
  6. Non-Emptiness testing
  7. A simple Farkas' Lemma-driven procedure to search for TEXT
  8. Complexity
  9. Counterexample to Incorrect Theorem \ref{['prop:kuiperswrong']}
  10. Proofs
  11. Testing non-emptiness of base polytopes associated with crossing submodular functions over families closed under complement
  12. Preliminaries
  13. Base Polytope Non-Emptiness Algorithm
  14. Fujishige's Approach
  15. Discrete Sandwich Theorem Approach
  16. ...and 1 more sections

Key Result

Proposition 1

peleg2007introduction Let $(N,v)$ be a game, and $\bar{\eta}$ its prenucleolus. Then, $\forall S \subsetneq N$, $S\neq \emptyset$

Theorems & Definitions (33)

  • Proposition 1
  • Proposition 2
  • Proposition 3
  • Theorem 1
  • Theorem 2
  • Proposition 4
  • Theorem 3
  • proof
  • Proposition 5
  • proof
  • ...and 23 more