On Reward-Penalty-Selection Games

TL;DR

This paper introduces a cooperative game based on the Reward-Penalty-Selection Problem, proving its convexity, superadditivity, total balancedness, and providing efficient methods to compute the Shapley value and core elements.

Contribution

It defines a new cooperative game model for RPSP, proves key structural properties, and offers polynomial-time algorithms for computing solution concepts.

Findings

RPS games are convex, superadditive, and totally balanced.

Shapley value can be computed in polynomial time.

Core elements correspond to feasible flows in a network graph.

Abstract

The Reward-Penalty-Selection Problem (RPSP) can be seen as a combination of the Set Cover Problem (SCP) and the Hitting Set Problem (HSP). Given a set of elements, a set of reward sets, and a set of penalty sets, one tries to find a subset of elements such that as many reward sets as possible are covered, i.e. all elements are contained in the subset, and at the same time as few penalty sets as possible are hit, i.e. the intersection of the subset with the penalty set is non-empty. In this paper we define a cooperative game based on the RPSP where the elements of the RPSP are the players. We prove structural results and show that RPS games are convex, superadditive and totally balanced. Furthermore, the Shapley value can be computed in polynomial time. In addition to that, we provide a characterization of the core elements as a feasible flow in a network graph depending on the instance of the underlying RPSP. By using this characterization, a core element can be computed efficiently.