Cooperative Games with Overlapping Coalitions: Charting the Tractability Frontier

TL;DR

This paper investigates the computational complexity of the core in overlapping coalition formation games, introduces Linear Bottleneck Games, and identifies conditions for polynomial-time solutions, advancing understanding of stability in multiagent resource allocation.

Contribution

It provides the first complexity analysis of the core in OCF games, introduces Linear Bottleneck Games, and establishes conditions for polynomial-time algorithms.

Findings

Deciding core non-emptiness is computationally challenging.

Linear Bottleneck Games always have a non-empty core.

Certain conditions enable polynomial-time core computations.

Abstract

In many multiagent scenarios, agents distribute resources, such as time or energy, among several tasks. Having completed their tasks and generated profits, task payoffs must be divided among the agents in some reasonable manner. Cooperative games with overlapping coalitions (OCF games) are a recent framework proposed by Chalkiadakis et al. (2010), generalizing classic cooperative games to the case where agents may belong to more than one coalition. Having formed overlapping coalitions and divided profits, some agents may feel dissatisfied with their share of the profits, and would like to deviate from the given outcome. However, deviation in OCF games is a complicated matter: agents may decide to withdraw only some of their weight from some of the coalitions they belong to; that is, even after deviation, it is possible that agents will still be involved in tasks with non-deviators. This means that the desirability of a deviation, and the stability of formed coalitions, is to a great extent determined by the reaction of non-deviators. In this work, we explore algorithmic aspects of OCF games, focusing on the core in OCF games. We study the problem of deciding if the core of an OCF game is not empty, and whether a core payoff division can be found in polynomial time; moreover, we identify conditions that ensure that the problem admits polynomial time algorithms. Finally, we introduce and study a natural class of OCF games, Linear Bottleneck Games. Interestingly, we show that such games always have a non-empty core, even assuming a highly lenient reaction to deviations.