Łukasiewicz logics for cooperative games

Abstract

Coalitional resource games (CRGs) provide a natural abstract framework with which to model scenarios in which groups of agents cooperate by pooling resources in order to carry out tasks or achieve individual goals. In this work, we introduce a richer and more general framework, called Łukasiewicz resource games (ŁRG), which is based on many-valued Łukasiewicz logics, whose formulae make it possible to specify the class of piecewise linear polynomial functions with integer and rational coefficients on [0,1]n. The use of Łukasiewicz logics provides a new approach to the representation of the scenario/situations modelled by CRGs. In ŁRGs, each agent is endowed with resources that can be allocated over a set of tasks, where the outcome of a task depends on the profile of resources that are allocated to it. We specify task outcomes using formulae of Łukasiewicz logic. In addition, agents have payoff functions over task outcomes, which are also specified by Łukasiewicz formulae. After motivating and introducing ŁRGs, we formalise notions of coalition structures and the core for ŁRGs and investigate the non-emptiness of the core both from a logical and computational perspective. We prove that ŁRGs are a proper generalisation of CRGs by showing how any CRG can be translated into a ŁRG that is strategically equivalent, in the sense that the former has a non-empty core if and only if so does the latter.