minimum cost spanning tree games we consider the problem of cost allocation among users of a minimum cost spanning tree network it is formulated as a cooperative game in characteristic function form referred to as a minimum cost spanning tree m c s t game we show that the core of a m c s t game is never empty in fact a point in the core can be read directly from any minimum cost spanning tree graph associated with the problem for m c s t games with efficient coalition structures we define and construct m c s t games on the components of the structure we show that the core and the nucleolus of the original game are the cartesian products of the cores and the nucleoli respectively of the induced games on the components of the efficient coalition structure

learning trees from strings a strong learning algorithm for some context free grammars standard models of language learning are concerned with weak learning the learner receiving as input only information about the strings in the language must learn to generalise and to generate the correct potentially infinite set of strings generated by some target grammar here we define the corresponding notion of strong learning the learner again only receiving strings as input must learn a grammar that generates the correct set of structures or parse trees we formalise this using a modification of gold s identification in the limit model requiring convergence to a grammar that is isomorphic to the target grammar we take as our starting point a simple learning algorithm for substitutable context free languages based on principles of distributional learning and modify it so that it will converge to a canonical grammar for each language we prove a corresponding strong learning result for a subclass of context free grammars

cooperative interval games mountain situations with interval data in this paper we extend mountain situations by using interval calculus we define the interval bird allocation and show that this allocation is a special core element of the interval cost game corresponding to an interval mountain situation we deal with the interval cost sharing problem by introducing the cooperative interval cost game further we consider a subset of the interval core of the related interval cost game each element of this set is extendable to a population monotonic interval allocation scheme or shortly pmias finally we show that each interval core element of an interval connection game is extendable to bi monotonic interval allocation scheme or shortly bi mias

on the concavity of delivery games abstract delivery games introduced by hamers borm van de leensel and tijs in 1994 are combinatorial optimization games that arise from delivery problems closely related to the chinese postman problem cpp they showed that delivery games are not necessarily balanced for delivery problems corresponding to the class of bridge connected euler graphs they showed that the related games are balanced this paper focuses on the concavity property for delivery games a delivery game arising from a delivery model corresponding to a bridge connected euler graph need not be concave the main result will be that for delivery problems corresponding to the class of bridge connected cyclic graphs which is a subclass of the class of bridge connected euler graphs the related delivery games are concave further we discuss some extreme points of the core and the value for this class of concave delivery games

optimistic weighted shapley rules in minimum cost spanning tree problems abstract we introduce optimistic weighted shapley rules in minimum cost spanning tree problems we define them as the weighted shapley values of the optimistic game v introduced in bergantinos and vidal puga bergantinos g vidal puga j j forthcoming the optimistic tu game in minimum cost spanning tree problems international journal of game theory available from we prove that they are obligation rules tijs s branzei r moretti s norde h 2006 obligation rules for minimum cost spanning tree situations and their monotonicity properties european journal of operational research 175 121 134

computing the nucleolus of some combinatorially structured games in this paper we introduce the prenucleolus for a transferable utility game n v where 2 n the prenucleolus is a straightforward generalization of the ordinary prenucleolus where only the coalitions in determine the outcome we impose a combinatorial structure on the collection which enables us to compute the prenucleolus in n 3 time the algorithm can be used for computing the nucleolus of several classes of games among which is the class of minimum cost spanning tree games

multi criteria minimum cost spanning tree games abstract the minimum cost spanning tree game mcst game is a well known model within operations research games that has been widely studied in the literature in this paper we introduce the multi criteria version of the mcst game as a set valued tu game we prove that the extension of bird s cost allocation rule provides dominance core elements in this game we also give a family of core solutions that are different from the previous one these solutions are based on proportional allocations obtained using scalar solutions of the multi criteria spanning tree problem besides we prove necessary and sufficient conditions ensuring that the preference core of this game is not empty

cost allocation for a tree network with heterogeneous customers we analyze a cost allocation problem which could naturally arise from a situation wherein a tree networkt n 0 e serving heterogeneous customers has to be constructed the customers located atn require some service from a central supplier located at vertex 0 the customers have heterogeneous preferences for the level or quality of service received from the central supplier we formulate the above cost allocation problem as a cooperative game referred to as anextended tree game the extended tree game is a proper extension of megiddo s 1978 tree game wherein all the customers have identical preferences regarding the level of service received we prove that an extended tree game is convex and we show that its shapley value can be computed in p n time wherep is the number of distinct preference levels we further provide a complete facial description of the core polytope of an extended tree game and demonstrate that even when there are only two classes of customers the number of nonredundant core constraints could be exponential in n nevertheless we are able to construct an p n algorithm to check the core membership of an arbitrary cost allocation which can be used to construct an p n 3 combinatorial algorithm to compute the nucleolus of an extended tree game finally we show that the complements of the facet defining coalitions for the core are all connected in an auxiliary tree graph with node setn

new techniques for cost sharing in combinatorial optimization games combinatorial optimization games form an important subclass of cooperative games in recent years increased attention has been given to the issue of finding good cost shares for such games in this paper we define a very general class of games called integer minimization games which includes the combinatorial optimization games in the literature as special cases we then present new techniques based on row and column generation for computing good cost shares for these games to illustrate the power of these techniques we apply them to traveling salesman and vehicle routing games our results generalize and unify several results in the literature the main underlying idea is that suitable valid inequalities for the associated combinatorial optimization problems can be used to derive improved cost shares

congestion network problems and related games this paper analyzes network problems with congestion effects from a cooperative game theoretic perspective it is shown that for network problems with convex congestion costs the corresponding games have a non empty core if congestion costs are concave then the corresponding game has not necessarily core elements but it is derived that contrary to the convex congestion situation there always exist optimal tree networks extensions of these results to a class of relaxed network problems and associated games are derived

cost allocation in the chinese postman problem this paper considers a cost allocation problem that arises from a delivery problem associated with the chinese postman problem cpp a delivery problem is described by a connected undirected graph in which each edge belongs to a different player a cost function on the edges of this graph and a fixed vertex which is referred to as the post office assume that the post office is providing some service to the players the nature of this service which can be thought of as mail delivery requires that a server will travel along the edges of the graph and returns to the post office the cost allocation problem is concerned with the cost of providing the service to all players a specific cost allocation rule is introduced and characterized further the class of delivery problems gives rise to a new class of cooperative combinatorial optimization games called delivery games it is shown that the outcome of the allocation rule with respect to a bridge connected euler graph is a core element of the corresponding delivery game

complexity of the minimum base game on matroids this paper studies the complexity of computing solution concepts for a cooperative game called the minimum base game mbg e c where its characteristic function c 2e is defined as c s the weight w b of a minimum weighted base b s for a given matroid m e and a weight function w e the minimum base game contains as a special case the minimum spanning tree game mstg in an edge weighted graph in which players are located on the edges by interpreting solution concepts of games such as core value and shapley value in terms of matroid theory we obtain the core of mbg is nonempty if and only if the matroid m has no circuit consisting only of edges with negative weights checking the concavity and subadditivity of an mbg can be done in oracle polynomial time the value of an mbg exists if and only if the core is not empty the value of mstg can be computed in polynomial time while there is no oracle polynomial algorithm for a general mbg computing the shapley value

highway games on weakly cyclic graphs a highway problem is determined by a connected graph which provides all potential entry and exit vertices and all possible edges that can be constructed between vertices a cost function on the edges of the graph and a set of players each in need of constructing a connection between a specific entry and exit vertex mosquera 2007 introduce highway problems and the corresponding cooperative cost games called highway games to address the problem of fair allocation of the construction costs in case the underlying graph is a tree in this paper we study the concavity and the balancedness of highway games on weakly cyclic graphs a graph g is called highway game concave if for each highway problem in which g is the underlying graph the corresponding highway game is concave we show that a graph is highway game concave if and only if it is weakly triangular moreover we prove that highway games on weakly cyclic graphs are balanced

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

dynamical patterns in directional solidification abstract we study the dynamic evolution of the solid liquid interface in rapid directional solidification with weak convective coupling this is done by means of a strongly nonlinear equation describing the long wave asymptotic behaviour of the system first small amplitude dynamics are considered in a travelling wave coordinate system the long scale amplitude equation is given by either a kuramoto sivashinsky k s equation or by a 6th order k s like equation depending on the parameter range numerical experiments show that solutions of the 6th order equation are comparable to solutions of the k s equation time periodic solutions in the travelling system are not necessarily periodic in the rest frame they may be quasiperiodic instead in a second step we drop the assumption of small amplitudes and study the full equation perturbatively this becomes a meaningful apprach because taking into account kinetic and disequilibrium effects in the basic equation allows to separate out nontrivial integrable subproblems that are absent in previous approaches using the averaging method we are led to an interesting new way of looking at the question of wavelength selection providing a general framework for temporal and spatial processes moreover the calculation of quantitative relations between the wavelength and drift velocity or between the convective coupling and the wavelength becomes possible finally we compare this to spatiotemporal dynamics obtained from numerical solution of the full equation

on the core of network synthesis games we use polynomial formulations to show that several rational and discrete network synthesis games including the minimum cost spanning tree game satisfy the assumptions of owen s linear production game model we also discuss computational issues related to finding and recognizing core points for these classes of games

sharing the cost of a capacity network we consider a communication network where each pair of users requests a connection guaranteeing a certain capacity the cost of building capacity is identical across pairs efficiency is achieved by any maximal cost spanning tree we construct cost sharing methods ensuring standalone core stability monotonicity of one s cost share in one s capacity requests and continuity in everyone s requests we define a solution for simple problems where each pairwise request is zero or one and extend it piecewise linearly to all problems the uniform solution obtains if we require one s cost share to be weakly increasing in everyone s capacity request in the solution we propose on the contrary one s cost share is weakly decreasing in other agents requests the computational complexity of both solutions is polynomial in the number of users the uniform solution admits a closed form expression and is the analog of a popular solution for the minimal cost spanning tree problem

a generalized linear production model a unifying model we introduce a generalized linear production model whose attractive feature being that the resources held by any subset of producerss is not restricted to be the vector sum of the resources held by the members ofs we provide sufficient conditions for the non emptiness of the core of the associated generalized linear production game and show that if the core of the game is not empty then a solution in it can be produced from a dual optimal solution to the associated linear programming problem our generalized linear production model is a proper generalization of the linear production model introduced by owen and it can be used to analyze cooperative games which cannot be studied in the ordinary linear production model framework we use the generalized model to show that the cooperative game induced by a network optimization problem in which players are the nodes of the network has a non empty core we further employ our model to prove the non emptiness of the core of two other classes of cooperative games which were not previously studied in the literature and we also use our generalized model to provide an alternative proof for the non emptiness of the core of the class of minimum cost spanning tree games thus it appears that the generalized linear production model is a unifying model which can be used to explain the non emptiness of the core of cooperative games generated by various seemingly different optimization models

a cooperative location game based on the 1 center location problem abstract in this paper we introduce and analyze new classes of cooperative games related to facility location models defined on general metric spaces the players are the customers demand points in the location problem and the characteristic value of a coalition is the cost of serving its members specifically the cost in our games is the service radius of the coalition we call these games the minimum radius location games mrlg we study the existence of core allocations and the existence of polynomial representations of the cores of these games focusing on network spaces i e finite metric spaces induced by undirected graphs and positive edge lengths and on the lp metric spaces defined over r d

on the core and nucleolus of minimum cost spanning tree games we develop two efficient procedures for generating cost allocation vectors in the core of a minimum cost spanning tree m c s t game the first procedure requires o n 2 elementary operations to obtain each additional point in the core wheren is the number of users the efficiency of the second procedure which is a natural strengthening of the first procedure stems from the special structure of minimum excess coalitions in the core of an m c s t game this special structure is later used i to ease the computational difficulty in computing the nucleolus of an m c s t game and ii to provide a geometric characterization for the nucleolus of an m c s t game this geometric characterization implies that in an m c s t game the nucleolus is the unique point in the intersection of the core and the kernel we further develop an efficient procedure for generating fair cost allocations which in some instances coincide with the nucleolus finally we show that by employing sterns transfer scheme we can generate a sequence of cost vectors which converges to the nucleolus

on some network flow games we analyze three subclasses of cooperative games arising from network optimization problems in which the resources such as arcs or nodes in the network are controlled by individuals who have conflicting objectives the first subclass of cooperative games is induced by network optimization problems over directed augmented trees we show that for this subclass of games the kernel coincides with the nucleolus and that the nucleolus can be characterized as the unique revenue allocation vector in which every pair of arc owners who are adjacent in the tree are located symmetrically with respect to their bargaining range we further give a linear characterization of the core of this subclass of games which is then used to provide a more explicit representation of the nucleolus and to construct a strongly polynomial algorithm for generating it the second subclass of cooperative games is induced by maximum flow problems in simple undirected networks we provide a useful parametric representation of the core of this subclass of games which is used to characterize the nucleolus and the intersection of the core and the kernel explicitly we show that the intersection of the core and the kernel consists of all revenue allocations in the core which assign equal payoffs to any pair of unseparated arc owners we further demonstrate that among the core vectors the nucleolus is the unique revenue allocation vector in which the smallest allocations are maximized in a lexicographical sense the third cooperative game that we study is the assignment game introduced by shapley and shubik 1972 this game is induced by the assignment problem which can be cast as a network optimization problem we investigate the relationship between the kernel and the core of the assignment game and provide a necessary and sufficient condition for the core to be contained in the kernel we further show that in general the intersection of the kernel and the core is not a convex set we also exhibit that under certain conditions the nucleolus has a simple characterization as the unique vector in the core in which the smallest revenue allocations are maximized in a lexicographical sense finally we consider the horse market example of bohm bawerk 1923 for which it is shown that the core is contained in the kernel and that the nucleolus is the midpoint of the core

minimum cost spanning tree games and population monotonic allocation schemes in this paper we present the subtraction algorithm that computes for every classical minimum cost spanning tree game a population monotonic allocation scheme as a basis for this algorithm serves a decomposition theorem that shows that every minimum cost spanning tree game can be written as nonnegative combination of minimum cost spanning tree games corresponding to 0 1 cost functions it turns out that the subtraction algorithm is closely related to the famous algorithm of kruskal for the determination of minimum cost spanning trees for variants of the classical minimum cost spanning tree games we show that population monotonic allocation schemes do not necessarily exist

shortest path games abstract we study cooperative games that arise from the problem of finding shortest paths from a specified source to all other nodes in a network such networks model among other things efficient development of a commuter rail system for a growing metropolitan area we motivate and define these games and provide reasonable conditions for the corresponding rail application we show that the core of a shortest path game is nonempty and satisfies the given conditions but that the shapley value for these games may lie outside the core however we show that the shortest path game is convex for the special case of tree networks and we provide a simple polynomial time formula for the shapley value in this case in addition we extend our tree results to the case where users of the network travel to nodes other than the source finally we provide a necessary and sufficient condition for shortest paths to remain optimal in dynamic shortest path games where nodes are added to the network sequentially over time

a new rule for source connection problems abstract in this paper we study situations where a group of agents require a service that can only be provided from a source the so called source connection problems these problems contain the standard fixed tree the classical minimum spanning tree and some other related problems such as the k hop the degree constrained and the generalized minimum spanning tree problems among others our goal is to divide the cost of a network among the agents to this end we introduce a rule which will be referred to as a painting rule because it can be interpreted by means of a story about painting some meaningful properties in this context and a characterization of the rule are provided

traveling salesman games in this paper we discuss the problem of how to divide the total cost of a round trip along several institutes among the institutes visited we introduce two types of cooperative games fixed route traveling salesman games and traveling salesman games as a tool to attack this problem under very mild conditions we prove that fixed route traveling salesman games have non empty cores if the fixed route is a solution of the classical traveling salesman problem core elements provide us with fair cost allocations a traveling salesman game may have an empty core even if the cost matrix satisfies the triangle inequality in this paper we introduce a class of matrices defining ts games with non empty cores

a generalization of obligation rules for minimum cost spanning tree problems tijs et al 23 introduce the family of obligation rules for minimum cost spanning tree problems we give a generalization of such family we prove that our family coincides with the set of rules satisfying an additivity property and a cost monotonicity property we also provide two new characterizations for the family of obligation rules using the previous properties in the first one we add a property of separability and in the second one we add core selection