Approximate Stability of Subadditive Games and Traveling Salesman Games

TL;DR

This paper investigates approximate stability concepts in cooperative games related to Traveling Salesman Games, focusing on semicore allocations and methods to approximate stability when the core is empty, with applications in collaborative transportation.

Contribution

It introduces and analyzes semicore allocations for TSGs, providing new insights into approximate stability measures like cost of stability and epsilon-core in complex transportation games.

Findings

Semicore allocations can be efficiently approximated for TSGs.

Cost of stability provides a quantifiable measure for approximate stability.

Epsilon-core offers a flexible approach to stability when the core is empty.

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 $\epsilon$-core (which is a set of allocations that allow subsets of players to exceed their actual cost, but at most of a value of $\epsilon$). We prove that these two solution