Coreness of Cooperative Games with Truncated Submodular Profit Functions

TL;DR

This paper investigates the properties of the core in cooperative games with truncated submodular profit functions, providing polynomial-time algorithms for some cases and hardness results with approximation algorithms for others.

Contribution

It introduces the study of core and related concepts in truncated submodular profit cooperative games, including polynomial-time decision algorithms and hardness results.

Findings

Core emptiness can be decided in polynomial time.

An allocation in the core can be found efficiently when it exists.

Hardness results and approximation algorithms are provided for core-related concepts when core is empty.

Abstract

Coreness represents solution concepts related to core in cooperative games, which captures the stability of players. Motivated by the scale effect in social networks, economics and other scenario, we study the coreness of cooperative game with truncated submodular profit functions. Specifically, the profit function $f(\cdot)$ is defined by a truncation of a submodular function $\sigma(\cdot)$: $f(\cdot)=\sigma(\cdot)$ if $\sigma(\cdot)\geq\eta$ and $f(\cdot)=0$ otherwise, where $\eta$ is a given threshold. In this paper, we study the core and three core-related concepts of truncated submodular profit cooperative game. We first prove that whether core is empty can be decided in polynomial time and an allocation in core also can be found in polynomial time when core is not empty. When core is empty, we show hardness results and approximation algorithms for computing other core-related concepts including relative least-core value, absolute least-core value and least average dissatisfaction value.