Obligation rules for minimum cost spanning tree situations and their monotonicity properties

Abstract

Abstract:We introduce the class of Obligation rules for minimum cost spanning tree situations. The main result of this paper is that such rules are cost monotonic and induce also population monotonic allocation schemes. Another characteristic of Obligation rules is that they assign to a minimum cost spanning tree situation a vector of cost contributions which can be obtained as product of a double stochastic matrix with the cost vector of edges in the optimal tree provided by the Kruskal algorithm. It turns out that the Potters value (P -value) is an element of this class.

A characterization of Kruskal sharing rules for minimum cost spanning tree problems

“…The literature devoted to the analysis of this property in various models is surveyed in Thomson (1995). The main result of this paper is not only important for the characterization itself, but it also provides us with an easy way to obtain the sharing functions, formerly called obligation functions in Tijs et al (2006), associated with the rules. We also prove that a family of weighted Shapley rules belongs to the family of Kruskal sharing rules and we calculate the associated sharing functions.…”

“…Feltkamp et al (1994) introduced the equal remaining obligation rule, which was studied by Bergantiños and Vidal-Puga (2004, 2007a, and 2007b. This rule belongs to a wide family of rules, introduced by Tijs et al (2006), the family of obligation rules. These rules are defined through Kruskal's algorithm and the philosophy of "construct and charge" (Moretti et al, 2005), i.e., the minimal tree is built arc by arc and the cost of each arc is paid, after its construction, by all the agents who benefit from it.…”

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AbstractIn Tijs et al (2006) a new family of cost allocation rules is introduced. In this paper we provide the first characterization of this family by means of population monotonicity and a property of additivity in the context of cost spanning tree problems.

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Generalized minimum spanning tree games

“…For the MSTG, some heuristic methods have been proposed to calculate a cost allocation in the core such as the Bird rule (Bird 1976), the obligation rule (Tijs et al 2006) and Folk solution (Bogomolnaia and Moulin 2010). The Bird rule of a MSTG on the graph G is described as follows.…”

Minimum cost connection networks: Truth-telling and implementation

“…Every agent wants to be connected to the source making the MCST model a special case of the CN model. For the MCST model it has been shown (Hougaard and Tvede, 2012) that the set of allocation rules satisfying reductionism (a strong form of UII) and monotonicity (a strong form of NI) contains fixed relative cost share rules such as the equal split rule, the folk rule discussed in Bogomolnaia & Moulin (2010) and the rest of the family of obligation rules introduced and analyzed in Tijs et al (2006). The family of obligation rules are Stand Alone core stable where Stand alone core stable means that no coalition of agents pays more than the cost of building a network satisfying the connection demands of its members.…”