Folk solution for simple minimum cost spanning tree problems

Subiza, Begoña; Gimnez-Gmez, Jos-Manuel; Peris, Josep E.

doi:10.1016/j.orl.2016.06.008

Cited by 4 publications

(5 citation statements)

References 5 publications

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“…Let us consider the so-called 2−mcst problems (Estévez-Fernández and Reijnierse 2014; Subiza et al 2016) in which the connection cost between two diferent individuals (houses, villages, ...) can only take one of two possible values (low and high cost). Moreover, we consider problems (N , ) such that…”

Section: −Mcst Problemsmentioning

confidence: 99%

A claims problem approach to the cost allocation of a minimum cost spanning tree

2021

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We propose to allocate the cost of a minimum cost spanning tree by deining a claims problem and using claims rules, then providing easy and intuitive ways to distribute this cost. Depending on the starting point that we consider, we deine two models. On the one hand, the benefit-sharing model considers individuals' costs to the source as the starting point, and then the beneit of building the eicient tree is shared by the agents. On the other hand, the costs-sharing model starts from the individuals' minimum connection costs (the cheapest connection they can use), and the additional cost, if any, is then allocated. As we prove, both approaches provide the same family of allocations for every minimum cost spanning tree problem. These models can be understood as a central planner who decides the best way to connect the agents (the eicient tree) and also establishes the amount each agent has to pay. In so doing, the central planner takes into account the maximum and minimum amount they should pay and some equity criteria given by a particular (claims) rule. We analyze some properties of this family of cost allocations, specially focusing in coalitional stability (core selection), a central concern in the literature on cost allocation.

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Section: −Mcst Problemsmentioning

confidence: 99%

A claims problem approach to the cost allocation of a minimum cost spanning tree

2021

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“….) can only take one of two possible values: low and high cost (see, for instance, [15]; see also [16] where this class has been generalized to the so-called simple mcst problems). Moreover, we assume that c ij = k 1 , i 6 ¼ j, c ii = k 2 , and 0 � k 1 � k 2 .…”

Section: Some Mcst Problems Where α Ceq > Is a Core-selectionmentioning

confidence: 99%

“…E1) Let us consider the so-called 2 − mcst problems in which the connection cost between two different individuals (houses, villages, …) can only take one of two possible values: low and high cost (see, for instance, [ 15 ]; see also [ 16 ] where this class has been generalized to the so-called simple mcst problems).…”

Section: A Core Egalitarian Proposalmentioning

confidence: 99%

“…Proof. From [ 16 ] we know that in this class of problems there is a partition of the set of agents, such that in each subset the Folk solution proposes the same allocation to all individuals in this group ( simple components). To simplify the proof we suppose that there are just two components in the 2 − mcst problem ( N ω , C ) (for more than two components, the reasoning follows an analogous argument): …”

Section: A Core Egalitarian Proposalmentioning

confidence: 99%

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An egalitarian approach for sharing the cost of a spanning tree

2020

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A minimum cost spanning tree problem analyzes the way to efficiently connect individuals to a source. Hence the question is how to fairly allocate the total cost among these agents. Our approach, reinterpreting the spanning tree cost allocation as a claims problem defines a simple way to allocate the optimal cost with two main criteria: (1) each individual only pays attention to a few connection costs (the total cost of the optimal network and the cost of connecting himself to the source); and (2) an egalitarian criteria is used to share costs. Then, using claims rules, we define an egalitarian solution so that the total cost is allocated as equally as possible. We show that this solutions could propose allocations outside the core, a counter-intuitive fact whenever cooperation is necessary. Then we propose a modification to get a core selection, obtaining in this case an alternative interpretation of the Folk solution.

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“…A related result is provided bySubiza et al (2016). They provide a closed-form solution for the folk solution in a class of mcst games that are a subset of clique games in which links between agents have a cost that is either high or low.…”

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confidence: 99%

Clique games: A family of games with coincidence between the nucleolus and the Shapley value

Trudeau

Vidal-Puga

2020

Mathematical Social Sciences

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We introduce a new family of cooperative games for which there is coincidence between the nucleolus and the Shapley value. These socalled clique games are such that players are divided into cliques, with the value created by a coalition linearly increasing with the number of agents belonging to the same clique. Agents can belong to multiple cliques, but for a pair of cliques, at most a single agent belong to their intersection. Finally, if two players do not belong to the same clique, there is at most one way to link the two players through a chain of players, with any two adjacent players in the chain belonging to a common clique. We provide multiple examples for clique games, chief among them minimum cost spanning tree problems. This allows us to obtain new correspondence results between the nucleolus and the Shapley value, as well as other cost sharing methods for the minimum cost spanning tree problem.

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Folk solution for simple minimum cost spanning tree problems

Cited by 4 publications

References 5 publications

A claims problem approach to the cost allocation of a minimum cost spanning tree

A claims problem approach to the cost allocation of a minimum cost spanning tree

An egalitarian approach for sharing the cost of a spanning tree

Clique games: A family of games with coincidence between the nucleolus and the Shapley value

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