Linking the Kar and folk solutions through a problem separation property

Abstract

Minimum cost spanning tree problems connect agents e¢ ciently to a source with the cost of using an edge …xed. We revisit the dispute between the Kar and folk solutions, two solution concepts to divide the common cost of connection based on the Shapley value. We introduce a property called Weak Problem Separation that allows, under conditions, to divide the problem in two: connecting an agent to the source and connecting agents to each other. It allows us to characterize the set of all a¢ ne combinations of the Kar and folk solutions.

A review of cooperative rules and their associated algorithms for minimum-cost spanning tree problems

“…2. (Bergantiños and Vidal-Puga 2007a) A comparative of the Shapley value in the irreducible game (folk solution) and in the private game (Kar solution) can be found in Trudeau (2014b). Ando and Kato (2010) prove that the Shapley value in the irreducible game, as well as the egalitarian solution and the nucleolus, can be computed in time O(|N | 2 ).…”

A new stable and more responsive cost sharing solution for minimum cost spanning tree problems

“…All common cost sharing solutions satisfy a property called Independence of Irrelevant Edges that says that no cost share depends on the cost of an irrelevant edge (Trudeau, 2010).…”