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Approximate Stability of Subadditive Games and Traveling Salesman Games

Nicolas Besson-Niebles, Sylvain Bouveret, Nadia Brauner, Nicolas Brulard

TL;DR

The paper studies how to achieve fair and stable cost allocations for traveling salesman problems in collaborative transportation within Short Food Supply Chains, especially when the core is empty. It extends semicore concepts to subadditive TU games, derives exact formulas for the cost of semicore stability and the optimal strong ε-semicore, and connects these with established approximate stability notions. A general bounding framework is proposed to bound stability costs via proxy games with nonempty cores, alongside simple polynomial bounds tailored to TSGs. These results offer actionable guidance for policymakers and practitioners evaluating subsidies or incentives to ensure coalition stability in SFSC transportation.

Abstract

The core of Transferable Utility (T.U.) games is a well-known solution concept from cooperative game theory yielding a cost allocation among n agents (called players) forming a coalition that is stable (i.e. no subset of players has an interest to deviate). In this paper, inspired by a practical application in the context of a decision support system for collaborative transportation in a Short Food Supply Chain (SFSC), we mainly focus on Traveling Salesman Games (TSGs), where the objective is to allocate the cost of a Traveling Salesman Problem (TSP) with n locations and 1 depot to n players, each linked to exactly one of the locations. Given the computational complexity of computing an element of the core and the cost of a TSP, we study semicore allocations: a relaxation of the core that only requires that the subsets of size n -1 and of size 1 do not wish to deviate from the coalition. In the literature, instances of TSGs with empty cores and semicores are found. Hence, this paper first surveys the methods to approximate stability whenever the core is empty, such as the cost of stability (computing the minimum amount of money to subsidize the coalition with to attain stability) and the $ε$-core (which is a set of allocations that allow subsets of players to exceed their actual cost, but at most of a value of $ε$). We prove that these two solution

Approximate Stability of Subadditive Games and Traveling Salesman Games

TL;DR

The paper studies how to achieve fair and stable cost allocations for traveling salesman problems in collaborative transportation within Short Food Supply Chains, especially when the core is empty. It extends semicore concepts to subadditive TU games, derives exact formulas for the cost of semicore stability and the optimal strong ε-semicore, and connects these with established approximate stability notions. A general bounding framework is proposed to bound stability costs via proxy games with nonempty cores, alongside simple polynomial bounds tailored to TSGs. These results offer actionable guidance for policymakers and practitioners evaluating subsidies or incentives to ensure coalition stability in SFSC transportation.

Abstract

The core of Transferable Utility (T.U.) games is a well-known solution concept from cooperative game theory yielding a cost allocation among n agents (called players) forming a coalition that is stable (i.e. no subset of players has an interest to deviate). In this paper, inspired by a practical application in the context of a decision support system for collaborative transportation in a Short Food Supply Chain (SFSC), we mainly focus on Traveling Salesman Games (TSGs), where the objective is to allocate the cost of a Traveling Salesman Problem (TSP) with n locations and 1 depot to n players, each linked to exactly one of the locations. Given the computational complexity of computing an element of the core and the cost of a TSP, we study semicore allocations: a relaxation of the core that only requires that the subsets of size n -1 and of size 1 do not wish to deviate from the coalition. In the literature, instances of TSGs with empty cores and semicores are found. Hence, this paper first surveys the methods to approximate stability whenever the core is empty, such as the cost of stability (computing the minimum amount of money to subsidize the coalition with to attain stability) and the -core (which is a set of allocations that allow subsets of players to exceed their actual cost, but at most of a value of ). We prove that these two solution

Paper Structure

This paper contains 9 sections, 7 theorems, 21 equations, 1 table.

Table of Contents

  1. Cooperative game theory concepts for collaborative transportation in Short Food Supply Chains
  2. Definitions from cooperative game theory
  3. Semicore allocations and their importance in the context of SFSC
  4. Approximately fair allocations
  5. Computation of Approximate Semicore Stability
  6. Bounds of the cost of (semi)core stability
  7. A general method
  8. Polynomial bounds for the cost of semicore stability of TSGs
  9. Conclusion

Key Result

Proposition 1

Let $(\mathcal{N}, c)$ be a cost function game such that $Core(\mathcal{N}, c) = \emptyset$. We have: $CoS(\mathcal{N}, c) = n \cdot wOeC(\mathcal{N}, c)$.

Theorems & Definitions (14)

  • Proposition 1
  • proof
  • Lemma 1
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Theorem 3
  • proof
  • ...and 4 more