On the Complexity of Exchanging

TL;DR

This paper investigates the computational complexity of transforming coalitions in simple games, focusing on the feasibility of rearranging losing coalitions into winning ones and vice versa, across various representations.

Contribution

It provides a complexity analysis of coalition rearrangement problems in simple games, considering different coalition representations and transformation directions.

Findings

Complexity results depend on coalition representation

Transforming losing to winning coalitions is computationally challenging

Results inform the design of algorithms for coalition manipulation

Abstract

We analyze the computational complexity of the problem of deciding whether, for a given simple game, there exists the possibility of rearranging the participants in a set of $j$ given losing coalitions into a set of $j$ winning coalitions. We also look at the problem of turning winning coalitions into losing coalitions. We analyze the problem when the simple game is represented by a list of wining, losing, minimal winning or maximal loosing coalitions.