Minimum Cost Spanning Tree Games Research Articles

Two-stage Minimum Cost Spanning Tree Game under Fuzzy Optimistic Coalition

This paper discusses the problem of cost allocation when players have different levels of optimism based on the two-stage minimum spanning tree game, and uses Choquet integral to calculate the characteristic function of fuzzy optimistic coalition and fuzzy pessimistic coalition. It is proved that the subgame of the two-stage clear optimistic coalition minimum cost spanning tree game is also a convex game. Finally, an example is used to prove that the two-stage fuzzy pessimistic coalition minimum cost spanning tree game has a dynamical instability solution.

Nonempty Core of Minimum Cost Spanning Tree Games with Revenues

Nonempty Core of Minimum Cost Spanning Tree Games with Revenues

Some new results on generalized additive games

A Generalized Additive Game (GAG) (Cesari et al. in Int J Game Theory 46(4):919–939, 2017) is a Transferable Utility (TU) game (N, v), where each player in N is provided with an individual value, and the worth v(S) of a coalition \(S \subseteq N\) is obtained as the sum of the individual values of players in another subset \(\mathcal {M}(S)\subseteq N\). Based on conditions on the map \(\mathcal {M}\) (which associates to each coalition S a set of beneficial players \(\mathcal {M}(S)\) not necessarily included in S), in this paper we characterize classes of GAGs that satisfy properties like monotonicity, superadditivity, (total) balancedness, PMAS-admissibility and supermodularity, for all nonnegative vectors of individual values. We also illustrate the application of such conditions on \(\mathcal {M}\) over particular GAGs studied in the literature (e.g., glove games (Shapley and Shubik in Int Econ Rev 10:337–362, 1969), generalized airport games (Norde et al. in Eur J Oper Res 136(3):635–654, 2002), fixed tree games (Bjørndal et al. in Math Methods Oper Res 59(2):249–270, 2004), link-connection games (Moretti in Multi-agent systems and agreement technologies, vol 10767. Springer, Cham, 2008; Nagamochi et al. in Math Oper Res 22(1):146–164, 1997), simple minimum cost spanning tree games (Norde et al. in Eur J Oper Res 154(1):84–97, 2004; Tijs et al. in Eur J Oper Res 175(1):121–134, 2006) and graph coloring games (Deng et al. in Math Program 87(3):441–452, 2000; Hamers et al. in Math Program 145(1–2):509–529, 2014)).

Network Connectivity Game

We investigate the cost allocation strategy associated with the problem of providing service /communication between all pairs of network nodes. There is a cost associated with each link and the communication between any pair of nodes can be delivered via paths connecting those nodes. The example of a cost efficient solution which could provide service for all node pairs is a (non-rooted) minimum cost spanning tree. The cost of such a solution should be distributed among users who might have conflicting interests. The objective of this paper is to formulate the above cost allocation problem as a cooperative game, to be referred to as a Network Connectivity (NC) game, and develop a stable and efficient cost allocation scheme. The NC game is related to the Minimum Cost Spanning Tree games and to the Shortest Path games. The profound difference is that in those games the service is delivered from some common source node to the rest of the network, while in the NC game there is no source and the service is established through the two-way interaction among all pairs of participating nodes. We formulate Network Connectivity (NC) game and construct an efficient cost allocation algorithm which finds some points in the core of the NC game. Finally, we discuss the Egalitarian Network Cost Allocation (ENCA) rule and demonstrate that it finds an additional core point.

Minimum-cost spanning tree games: Bird rule revisited

Minimum-cost spanning tree (MCST) game is a classical model in the theory of cooperative games. The MCST game has been drawing continuous attention since it was raised by Claus and Kleitman in 1973. The MCST game is closely related to the MCST problem, a fundamental model in graph theory and combinatorial optimization, and has been widely applied in cost allocations for constructions of water networks, power networks, road networks and railway networks, etc. Due to its simplicity and intuitiveness, the Bird rule is one of the most famous solutions in the MCST game and has been extensively studied in the literature. Based on the linear programming formulation of the MCST problem by Edmonds, we provide a new equivalent closed-form formula for the Bird rule. We also study a generalization of the MCST game with weighted nodes. We show that the core is nonempty in the generalized model, and in particular, a modified Bird rule is proved to be still applicable.

The dynamic Nash bargaining solution for 2-stage cost sharing game

The problem of constructing the Dynamic Nash Bargaining Solution in a 2-stage game is studied. In each stage, a minimum cost spanning tree game is played, all players select strategy profiles to construct graphs in the stage game. At the second stage, players may change the graph using strategy profiles with transition probabilities, which decided by players in the first stage. The players' cooperative behavior is considered. As solution the Dynamic Nash Bargaining Solution is proposed. A theorem is proved to allow the Dynamic Nash Bargaining Solution to be time-consistent.

The degree and cost adjusted folk solution for minimum cost spanning tree games

In this paper two cost sharing solutions for minimum cost spanning tree problems are introduced, the degree adjusted folk solution and the cost adjusted folk solution. These solutions overcome the problem of the classical reductionist folk solution as they have considerable strict ranking power, without breaking established axioms. As such they provide an affirmative answer to an open question, put forward in Bogomolnaia and Moulin (2010).

Efficient algorithms for subdominant cycle-complete cost functions and cycle-complete solutions

The cycle-complete solution introduced by Trudeau (2012) is a solution concept for minimum cost spanning tree games and was proved to have desirable properties such as core-selection and sensitivity to change of the cost function. The cycle-complete solution is defined as the Shapley value of the minimum cost spanning tree game associated with the subdominant cycle-complete cost function of a given cost function. In this study, we characterize subdominant cycle-complete cost functions and provide an O(n2logn) time algorithm for computing such functions, where n is the number of players. This algorithm leads to a new algorithm for computing the cycle-complete solution of a minimum cost spanning tree game with an O(n2logn) time bound.

Social enterprise tree network games

We investigate the cost allocation strategy associated with the problem of providing service among network users residing at nodes of a social network. We assume that the social network platform is established in a symmetric complete network. There is a cost associated with each link and the service between any pair of nodes can be delivered via a directed path. The example of a cost efficient solution for such network is a (non-rooted) minimum cost directed spanning tree. The network cost should be distributed among users who might have conflicting interests. The objective of this paper is to formulate the above cost allocation problem as a cooperative game, to be referred to as a Social Enterprise Tree Network (SETN) game, and develop a “fair” and efficient cost allocation scheme. The SETN game is related to the minimum cost spanning tree game. The profound difference is that in the minimum cost spanning tree game the service is delivered from some common source node to the rest of the network, while under the social network platform there is no source and the service is established through the interaction among all participating nodes. The input to our cost allocation problem is the optimal non-rooted directed spanning tree SETN. We formulate several associated SETN games in characteristic function form. Then we construct a couple of efficient cost allocation algorithms that find some points in the core of those SETN games and thus result in subsidy-free cost allocations.

On the Core of Cost-Revenue Games: Minimum Cost Spanning Tree Games with Revenues

In this paper, we analyze cost sharing problems arising from a general service by explicitly taking into account the generated revenues. To this cost-revenue sharing problem, we associate a cooperative game with transferable utility, called cost-revenue game. By considering cooperation among the agents using the general service, the value of a coalition is defined as the maximum net profit that the coalition may obtain by means of cooperation. As a result, a coalition may profit from not allowing all its members to get the service that generates the revenues. We focus on the study of the core of cost-revenue games. Under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members, it is shown that a cost-revenue game has a non-empty core for any vector of revenues if, and only if, the dual game of the cost game has a large core. Using this result, we investigate minimum cost spanning tree games with revenues. We show that if every connection cost can take only two values (low or high cost), then, the corresponding minimum cost spanning tree game with revenues has a non-empty core. Furthermore, we provide an example of a minimum cost spanning tree game with revenues with an empty core where every connection cost can take only one of three values (low, medium, or high cost).

On the core and nucleolus of directed acyclic graph games

We introduce directed acyclic graph (DAG) games, a generalization of standard tree games, to study cost sharing on networks. This structure has not been previously analyzed from a cooperative game theoretic perspective. Every monotonic and subadditive cost game--including monotonic minimum cost spanning tree games--can be modeled as a DAG-game. We provide an efficiently verifiable condition satisfied by a large class of directed acyclic graphs that is sufficient for the balancedness of the associated DAG-game. We introduce a network canonization process and prove various structural results for the core of canonized DAG-games. In particular, we characterize classes of coalitions that have a constant payoff in the core. In addition, we identify a subset of the coalitions that is sufficient to determine the core. This result also guarantees that the nucleolus can be found in polynomial time for a large class of DAG-games.

Generalized minimum spanning tree games

The minimum-cost spanning tree game is a special class of cooperative games defined on a graph with a set of vertices and a set of edges, where each player owns a vertex. Solutions of the game represent ways to distribute the total cost of a minimum-cost spanning tree among all the players. When the graph is partitioned into clusters, the generalized minimum spanning tree problem is to determine a minimum-cost tree including exactly one vertex from each cluster. This paper introduces the generalized minimum spanning tree game and studies some properties of this game. The paper also describes a constraint generation algorithm to calculate a stable payoff distribution and presents computational results obtained using the proposed algorithm.

On the core of cost-revenue games: Minimum cost spanning tree games with revenues

In this paper, we analyze cost sharing problems arising from a general service by explicitly taking into account the generated revenues. To this cost-revenue sharing problem, we associate a cooperative game with transferable utility, called cost-revenue game. By considering cooperation among the agents using the general service, the value of a coalition is defined as the maximum net revenues that the coalition may obtain by means of cooperation. As a result, a coalition may profit from not allowing all its members to get the service that generates the revenues. We focus on the study of the core of cost-revenue games. Under the assumption that cooperation among the members of the grand coalition grants the use of the service under consideration to all its members, it is shown that a cost-revenue game has a nonempty core for any vector of revenues if, and only if, the dual game of the cost game has a large core. Using this result, we investigate minimum cost spanning tree games with revenues. We show that if every connection cost can take only two values (low or high cost), then, the corresponding minimum cost spanning tree game with revenues has a nonempty core. Furthermore, we provide an example of a minimum cost spanning tree game with revenues with an empty core where every connection cost can take only one of three values (low, medium, or high cost).

Submodularity of minimum-cost spanning tree games

We give a necessary condition and a sufficient condition for a minimum-cost spanning tree game introduced by Bird to be submodular (or convex). When the cost is restricted to two values, we give a characterization of submodular minimum-cost spanning tree games. We also discuss algorithmic issues.Copyright © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 63(3), 231–238 2014

The Degree and Cost Adjusted Folk Solution for Minimum Cost Spanning Tree Games

In this paper two cost sharing solutions for minimum cost spanning tree problems are introduced, the degree adjusted folk solution and the cost adjusted folk solution. These solutions overcome the problem of the classical reductionist folk solution as they have considerable strict ranking power, without breaking established axioms. As such they provide affirmative answers to open questions, put forward in Bogomolnaia and Moulin (2010) and Bogomolnaia et al. (2010).

Computation of the Shapley value of minimum cost spanning tree games: #P-hardness and polynomial cases

We show that computing the Shapley value of minimum cost spanning tree games is #P-hard even if the cost functions are restricted to be {0,1}-valued. The proof is by a reduction from counting the number of minimum 2-terminal vertex cuts of an undirected graph, which is #P-complete. We also investigate minimum cost spanning tree games whose Shapley values can be computed in polynomial time. We show that if the cost function of the given network is a subtree distance, which is a generalization of a tree metric, then the Shapley value of the associated minimum cost spanning tree game can be computed in O(n4) time, where n is the number of players.

Minimum cost spanning tree games and spillover stability

This article discusses interactive minimum cost spanning tree problems and argues that the standard approach of using a transferable utility game to come up with a fair allocation of the total costs has some flaws. A new model of spillover games is presented, in which each player’s decision whether or not to cooperate is properly taken into account.

Minimum cost arborescences

In this paper, we analyze the cost allocation problem when a group of agents or nodes have to be connected to a source, and where the cost matrix describing the cost of connecting each pair of agents is not necessarily symmetric, thus extending the well-studied problem of minimum cost spanning tree games, where the costs are assumed to be symmetric. The focus is on rules which satisfy axioms representing incentive and fairness properties. We show that while some results are similar, there are also significant differences between the frameworks corresponding to symmetric and asymmetric cost matrices.

REDUCTION OF ULTRAMETRIC MINIMUM COST SPANNING TREE GAMES TO COST ALLOCATION GAMES ON ROOTED TREES

A minimum cost spanning tree game is called ultrametric if the cost function on the edges of the underlying network is an ultrametric. We show that every ultrametric minimum cost spanning tree game is reduced to a cost allocation game on a rooted tree. It follows that there exist O(n 2 ) time algorithms for computing the Shapley value, the nucleolus and the egalitarian allocation of the ultrametric minimum cost spanning tree games, where n is the number of players.

Connection Situations Under Uncertainty

This paper deals with cost allocation problems arising from connection situations where edge costs are closed intervals of real numbers. To solve such problems, we extend classical solutions from the theory of minimum cost spanning tree games. We study the properties of such solutions and suggest how to use them under uncertainty conditions.