Complexity of Determining Nonemptiness of the Core

TL;DR

This paper investigates the computational complexity of determining whether the core of a cooperative game is nonempty, showing it is generally NP-complete, but tractable when collaborative possibilities are provided, with implications for multiagent coalition stability.

Contribution

It introduces a concise representation for characteristic form games and analyzes the complexity of core nonemptiness, revealing NP-completeness in general and tractability under certain conditions.

Findings

Determining core nonemptiness is NP-complete with and without transferable utility.

Given collaborative possibilities, checking core membership becomes tractable.

A hybrid utility transfer scenario remains NP-complete even with known collaborative possibilities.

Abstract

Coalition formation is a key problem in automated negotiation among self-interested agents, and other multiagent applications. A coalition of agents can sometimes accomplish things that the individual agents cannot, or can do things more efficiently. However, motivating the agents to abide to a solution requires careful analysis: only some of the solutions are stable in the sense that no group of agents is motivated to break off and form a new coalition. This constraint has been studied extensively in cooperative game theory. However, the computational questions around this constraint have received less attention. When it comes to coalition formation among software agents (that represent real-world parties), these questions become increasingly explicit. In this paper we define a concise general representation for games in characteristic form that relies on superadditivity, and show that it allows for efficient checking of whether a given outcome is in the core. We then show that determining whether the core is nonempty is $\mathcal{NP}$-complete both with and without transferable utility. We demonstrate that what makes the problem hard in both cases is determining the collaborative possibilities (the set of outcomes possible for the grand coalition), by showing that if these are given, the problem becomes tractable in both cases. However, we then demonstrate that for a hybrid version of the problem, where utility transfer is possible only within the grand coalition, the problem remains $\mathcal{NP}$-complete even when the collaborative possibilities are given.