a simplified expression of the shapley function for fuzzy game abstract in this paper a simplified expression of the shapley function for games with fuzzy coalition is proposed which can be regarded as the generalization of shapley functions defined in some particular games with fuzzy coalition the simplified expression of the shapley function is compared with two definitions established by butnariu tsurumi et al a conclusion is drawn that the simplified expression of the shapley function is equivalent to butnariu s definition when characteristic function is a game with proportional values and is equivalent to tsurumi s definition when characteristic function is a game with choquet integral forms furthermore from an angle of interaction between two participation levels the properties of the two games defined by butnariu and tsurumi are respectively studied

fuzzy formulation of auctions and optimal sequencing for multiple auctions abstract current research on auctions is mainly based on the game theory in the first part of this paper we propose a fuzzy set based formulation of auctions we define fuzzy sets to represent the seller and buyers valuations bid possibilities and win possibilities analyzing the properties of these fuzzy sets we study fuzzy versions of discriminating and nondiscriminating bidding strategies for each bidder in a single object auction by using bellman and zadeh s concept of confluence of fuzzy decisions instead of the game theoretic nash equilibrium the monotonically decreasing membership function replaces of the corresponding distribution density function in milgrom and weber s auction theory in the second part of the paper we develop a soft computing approach to maximize the seller s revenue in multiple object auctions through the use of object sequencing the proposed sequencing approach is based on fuzzy rule quantification numerical experience shows the potential of the proposed approach for application in practical auctions

reflective lorentzian lattices of signature 5 1 abstract in this paper we give a complete classification of strongly square free reflective z lattices of signature 5 1 this is done by reducing the classification of lorentzian lattices to those of positive definite lattices the classification of totally reflective genera breaks up into two parts the first part consists of classifying the square free totally reflective primitive genera by calculating strong bounds on the prime factors of the determinant of positive definite quadratic forms lattices with this property we achieve these bounds by combining the minkowski siegel mass formula with the combinatorial classification of reflective lattices accomplished by scharlau blaschke in a second part we use a lattice transformation that goes back to watson to generate all totally reflective primitive genera when starting from the square free case

fuzzy interval cooperative games we introduce a core like concept called f core for games in which the worth of any coalition is given by means of a fuzzy interval such a class of cooperative games was introduced by mares in 1999 different from mares approach the f core generalizes the interval core introduced by alparslan branzei and tijs in 2008 for games where the worth of any coalition is a compact interval of the real line moreover we introduce a balanced like condition and prove that this condition is necessary but not sufficient to guarantee the f core to be non empty

the consistent value of fuzzy games in the framework of fuzzy games we offer an extension of the reduced game introduced by hart and mas colell which we name the self reduced game according to consistency which related to the self reduced game we provide a definition of the consistent value which is a generalization of the shapley value of fuzzy games we adopt three existing concepts from coalitional game theory and reinterpret them in the framework of fuzzy games the first one is that there exists a unique potential function and the resulting payoff vector coincides with the consistent value second based on the properties of balanced contributions and consistency we offer several axiomatizations of the consistent value finally we propose a dynamic process to illustrate that the consistent value can be reached by players who start from an arbitrary efficient payoff vector and make successive adjustments

max min solutions for fuzzy multiobjective matrix games abstract in this paper we consider two person zero sum games with fuzzy multiple payoff matrices we assume that each player has a fuzzy goal for each of the payoffs a degree of attainment of the fuzzy goal is defined and the max min strategy with respect to the degree of attainment of the fuzzy goal is examined if all of the membership functions both for the fuzzy payoffs and for the fuzzy goals are linear the formulated mathematical programming problem which yields the max min strategy can be reduced to the linear programming problem by making use of sakawa s method the variable transformations and the relaxation procedure

games with fuzzy permission structure a conjunctive approach abstract a cooperative game consists of a set of players and a characteristic function which determines the maximal gain or minimal cost that every subset of players can achieve when they decide to cooperate regardless of the actions that the other players take a permission structure over the set of players describes a hierarchical organization where there are players who need permission from certain other players before they are allowed to cooperate with others various assumptions can be made about how a permission structure affects the cooperation possibilities in the conjunctive approach it is assumed that each player needs permission from all his superiors this paper deals with fuzzy permission structures in the conjunctive approach in this model players could depend partially on other players that is they may have certain degree of autonomy first we define a value for games with fuzzy permission structure that only takes into account the direct relations among players and provide a characterization for this value finally we study a value for games with fuzzy permission structure which takes account of the indirect relations among players

a banzhaf function for a fuzzy game in this paper another important solution for fuzzy games called the banzhaf function is investigated first a sim plified expression for the banzhaf function for games with fuzzy coalition is proposed the relationship between this simplified ex pression and two kinds of particular banzhaf functions are dis cussed in detail furthermore in the framework of games with fuzzy coalitions we give a definition of the potential function of fuzzy games and prove that the marginal values of the potential function are the components of the banzhaf function further more a new axiomatic characterization of the banzhaf function for games with fuzzy coalitions is provided

a truthful constant approximation for maximizing the minimum load on related machines abstract designing truthful mechanisms for scheduling on related machines is a very important problem in single parameter mechanism design we consider the covering objective that is we are interested in maximizing the minimum completion time of a machine this problem falls into the class of problems where the optimal allocation can be truthfully implemented a major open issue for this class is whether truthfulness affects the polynomial time implementation we provide the first constant factor approximation for deterministic truthful mechanisms in particular we come up with an approximation guarantee of 2 e significantly improving on the previous upper bound of min m 2 e s m s 1

the cg position value for games on fuzzy communication structures abstract a cooperative game for a set of agents establishes a fair allocation of the profit obtained for their cooperation the best known of these allocations is the shapley value a communication structure defines the feasible bilateral communication relationships among the agents in a cooperative situation some solutions incorporating this information have been defined from the shapley value the myerson value the position value etc later fuzzy communication structures were introduced in a fuzzy communication structure the membership of the players and the relations among them are leveled several ways of defining the myerson value for games on fuzzy communication structure were proposed one of them is the choquet by graphs cg version now we study in this work the cg position value and its calculation the cg position value is defined as a solution for games with fuzzy communication structure which considers the bilateral communications as players so the shapley value is applied for a new game the link game over the fuzzy sets of links in the fuzzy communication structure and the profit obtained for each link is allocated between both players in the link as we see in our examples and results the cg position value is more concerned with the graphical position of the players and their communications than the other cg values in this paper we also introduce a procedure to compute exactly the position value avoiding to calculate the characteristic function of the link game for all coalitions this procedure is used to determine the cg position value finally we compare the new value with other cg values in an applied example about the power of the groups in the european parliament

random payoff two person zero sum game with joint chance constraints abstract we study a two person zero sum game where the payoff matrix entries are random and the constraints are satisfied jointly with a given probability we prove that for the general random payoff zero sum game there exists a weak duality between the two formulations i e the optimal value of the minimizing player is an upper bound of the one of the maximizing player under certain assumptions we show that there also exists a strong duality where their optimal values are equal moreover we develop two approximation methods to solve the game problem when the payoff matrix entries are independent and normally distributed finally numerical examples are given to illustrate the performances of the proposed approaches

low dimensional lattices ii subgroups of gl n ℤ the maximal finite irreducible groups of n x n integers for n 4 5 9 11 13 17 19 23 were determined by dade ryskov bulow plesken pohst and plesken as the automorphism groups of certain quadratic forms this paper presents a geometric description of the corresponding n dimensional lattices and gives coordinates which display their symmetries and minimal vectors some very interesting lattices appear

low dimensional lattices i quadratic forms of small determinant in this paper the first of a series dealing with low dimensional lattices and their applications we classify positive definite integral quadratic forms of determinant d 25 in dimensions up to a limit which ranges from 18 for d 1 to for d 25

optimal coalition formation and surplus distribution two sides of one coin abstract a fuzzy coalitional game represents a situation in which players can vary the intensity at which they participate in the coalitions accessible to them as opposed to the treatment as a binary choice in the non fuzzy crisp game building on the property not made use of so far in the literature of fuzzy games that a fuzzy game can be represented as a convex program this paper shows that the optimum of such a program determines the optimal coalitions as well as the optimal rewards for the players two sides of one coin furthermore this program is seen to provide a unifying framework for representing the core the least core and the fuzzy nucleolus among others next we derive conditions for uniqueness of core rewards and to deal with non uniqueness we introduce a family of parametric perturbations of the convex program that encompasses a large number of well known concepts for selection from the core including the dutta ray solution dutta and ray 1989 the equal sacrifice solution yu 1973 the equal division solution selten 1972 and the tau value tijs 1981 we also generalize the concept of the grand coalition of contracting players by allowing for multiple technologies and we specify the conditions for this allocation to be unique and egalitarian finally we show that our formulation offers a natural extension to existing models of production economies with threats and division rules for common surplus

low dimensional lattices iv the mass formula the mass formula expresses the sum of the reciprocals of the group orders of the lattices in a genus in terms of the properties of any of them we restate the formula so as to make it easier to compute in particular we give a simple and reliable way to evaluate the 2 adic contribution our version unlike earlier ones is visibly invariant under scale changes and dualizing we use the formula to check the enumeration of lattices of determinant d 25 given in the first paper in this series we also give tables of the standard mass the l series n m m s m odd and genera of lattices of determinant d 25

a lexicographical solution concept in an n person cooperative fuzzy game abstract in this paper a new solution concept in an n person cooperative fuzzy game with sidepayment is proposed we first define an excess with respect to a player in a fuzzy game with sidepayment usually the concept of an excess is defined for each coalition in contrast however we define an excess of a player by summing up all the excess of a coalition to which he belongs by considering a payoff vector which minimizes the excess of a player in the lexicographical order we proposed a new solution concept in a fuzzy game as well as in a nonfuzzy game then we further consider the problem to extend a nonfuzzy game into a fuzzy game such an extension can be described by the mappings from a characteristic function in a nonfuzzy game into a characteristic function in a fuzzy game an illustrative numerical example of the proposed solution concept to some extended fuzzy games is also presented

critical packing fraction of rectangular particles on the square lattice the random packing of identical and nonoverlapping rectangular particles of size n ifmmode times else texttimes fi m 1l n ml 10 is studied numerically on the square lattice and the corresponding packing fractions p f and percolation probabilities p ensuremath infty are determined we find that for randomly oriented particles there is a critical packing fraction p f c 0 67 ifmmode pm else textpm fi 0 01 such that for all particles sizes n ifmmode times else texttimes fi m for which p f l p f c they do not percolate i e p ensuremath infty ensuremath rightarrow 0 for stackrel ensuremath rightarrow l ensuremath infty while when p f g p f c p ensuremath infty ensuremath rightarrow 1 when stackrel ensuremath rightarrow l ensuremath infty and an infinite cluster exists the value for p f c is found to be consistent with the continuum percolation threshold p c ensuremath cong 0 67 for overlapping particles in two dimensions

capital allocation a la aumann shapley for non differentiable risk measures abstract we study capital allocation rules satisfying suitable properties for convex and quasi convex risk measures by focusing in particular on a family of capital allocation rules based on the dual representation for risk measures and inspired by the aumann shapley allocation principle these rules extend some well known methods of capital allocation for coherent and convex risk measures to the case of non gateaux differentiable risk measures we also analyze the properties of the allocation principles here introduced and discuss their suitability in the quasi convex context

myerson values for games with fuzzy communication structure in 1977 myerson considered cooperative games with communication structure a communication structure is an undirected graph describing the bilateral relationships among the players he introduced the concept of allocation rule for a game as a function obtaining an outcome for each communication structure among the players of the game the myerson value is a specific allocation rule extending the shapley value of the game more recently the authors studied games with fuzzy communication structures using fuzzy graph theoretic ideas now we propose a general framework in order to define fuzzy myerson values players in a coalition need to measure their profit using their real individual and communication capacities at every moment because these attributes are fuzzy when the game is proposed so they look for forming connected coalitions working at the same level the different ways to obtain these partitions by levels determine different myerson values for the game several interesting examples of these ways are studied in the paper following known models in games with fuzzy coalitions the proportional model and the choquet model

average monotonic cooperative fuzzy games in this paper we study average monotonic fuzzy games a subclass of monotonic cooperative fuzzy games and prove that for a continuous average monotonic cooperative fuzzy game which has a crisp maximal coalition of maximum excess at every imputation its aubin core coincides with its fuzzy bargaining set

fuzzy extensions of bargaining sets and their existence in cooperative fuzzy games recently the concept of classical bargaining set given by aumann and maschler in 1964 has been extended to fuzzy bargaining set in this paper we give a modification to correct some weakness of this extension we also extend the concept of the mas colell s bargaining set the other major type of bargaining sets to its corresponding fuzzy bargaining set our main effort is to prove existence theorems for these two types of fuzzy bargaining sets we will also give necessary and sufficient conditions for these bargaining sets to coincide with the aubin core in a continuous superadditive cooperative fuzzy game which has a crisp maximal coalition of maximum excess at each payoff vector we show that both aumann maschler and mas colell fuzzy bargaining sets of a continuous convex cooperative fuzzy game coincide with its aubin core

games with fuzzy authorization structure a cooperative game consists of a set of players and a characteristic function which determines the maximal gain or minimal cost that every subset of players can achieve when they decide to cooperate regardless of the actions that the other players take it is often assumed that the players are free to participate in any coalition but in some situations there are dependency relationships among the players that restrict their capacity to cooperate within some coalitions those relationships must be taken into account if we want to distribute the profits fairly in this respect several models have been proposed in literature in all of them dependency relationships are considered to be complete in the sense that either a player is allowed to fully cooperate within a coalition or they cannot cooperate at all nevertheless in some situations it is possible to consider another option that a player has a degree of freedom to cooperate within a coalition a model for those situations is presented

classification of integral lattices with large class number a detailed exposition of kneser s neighbour method for quadratic lattices over totally real number fields and of the sub procedures needed for its implementation is given using an actual computer program which automatically generates representatives for all isomorphism classes in one genus of rational lattices various results about genera of elementary lattices for small prime level are obtained for instance the class number of 1 2 dimensional 7 elementary even lattices of determinant 7 6 is 395 no extremal lattice in the sense of quebbemann exists the implementation incorporates as essential parts previous programs of w plesken and b souvignier

convex fuzzy games and participation monotonic allocation schemes in this paper the cone of convex cooperative fuzzy games is studied as in the classical case of convex crisp games these games have a large core and the fuzzy shapley value is the barycenter of the core surprisingly the core and the weber set coincide as in the classical case but the coincidence of these sets for a fuzzy game does not imply automatically convexity as in the crisp case participation monotonic allocation schemes pamas are introduced and it turns out that each core element of a convex fuzzy game is pamas extendable

linear production games with fuzzy control the aim of this paper is to analyse linear production with committee control situations arising when controllers face the possibility of graduating their options in order to model these situations we consider several kinds of fuzzy controls which can be modelled as different simple fuzzy games an lp game which is an extension of lp games with committee control introduced in curiel et al on balanced games and games with committee control or spectrum 11 1989 83 88 is obtained and its core is studied

a new shapley value for games with fuzzy coalitions abstract this paper deals with cooperative games over fuzzy coalitions in these situations there is a continuous set of fuzzy coalitions instead of a finite set of them as in the classical case the unit square in an n dimensional space there exist in the literature two different extensions of the known shapley value for crisp games to games with fuzzy coalitions the crisp shapley value and the diagonal value the first value only uses a finite information in the set of fuzzy coalitions the vertices of the square while the second one uses a neighbourhood of the diagonal of the square we propose a new extension of the shapley value improving the crisp shapley value for games with fuzzy coalitions this new version uses the faces of the square namely an infinity quantity of information we analyze several properties of the new value we endow it with an axiomatization and we study the behavior when it is applied to known fuzziness of crisp games

group schemes and local densities the subject matter of this paper is an old one with a rich history beginning with the work of gauss and eisenstein maturing at the hands of smith and minkowski and culminating in the fundamental results of siegel more precisely if l is a lattice over z for simplicity equipped with an integral quadratic form q the celebrated smith minkowski siegel mass formula expresses the total mass of l q which is a weighted class number of the genus of l q as a product of local factors these local factors are known as the local densities of l q subsequent work of kneser tamagawa and weil resulted in an elegant formulation of the subject in terms of tamagawa measures in particular the local density at a non archimedean place p can be expressed as the integral of a certain volume form ld over autzp l q which is an open compact subgroup of autqp l q the question that remains is whether one can find an explicit formula for the local density through the work of pall for p 6 2 and watson for p 2 such an explicit formula for the local density is in fact known for an arbitrary lattice over zp see p and wa the formula is obviously structured though cs seems to be the first to comment on this unfortunately the known proof as given in p and k does not explain this structure and involves complicated recursions on the other hand conway and sloane cs 13 have given a heuristic explanation of the formula in this paper we will give a simple and conceptual proof of the local density formula for p 6 2 the view point taken here is similar to that of our earlier work ghy and the proof is based on the observation that there exists a smooth affine group scheme g over zp with generic fiber autqp l q which satisfies g zp autzp l q this follows from general results of smoothening blr as we explain in section 3 for the purpose of obtaining an explicit formula it is necessary to have an explicit construction of g the main contribution of this paper is to give such an explicit construction of g in section 5 and to determine its special fiber in section 6 finally by comparing ld and the canonical volume form can of g we obtain the explicit formula for the local density in section 7 the smooth group schemes constructed in this paper should also be of independent interest

linear fuzzy game with coalition interaction and its coincident solutions abstract in this study we consider the solution concepts for fuzzy coalition games i e cooperative games with fuzzy coalition under a certain participation level in general cooperative games with fuzzy coalition are based on the assumption that all the fuzzy coalition values for different fuzzy coalitions must be represented by the same formula which may omit the coalition interaction under different participation ratios for players considering these conditions we propose the coincident fuzziness form for games with fuzzy coalition which are represented by a mapping from the characteristic function of the crisp game to that of the fuzzy coalition game the proposed fuzzy coalition games admit the differences in coalition interactions for different fuzzy coalitions where the coalition interactions are represented by the fuzzy coalition values in different ways or formulas for a fixed fuzzy coalition the maximum fuzzy coalition game is proven to be the choquet integral form on the condition that the associate crisp game is convex in order to seek appropriate solutions for the proposed games based on a certain participation level the fuzzy shapley axioms are defined and the explicit shapley value is represented by the shapley value of the associated crisp games moreover the fuzzy core of this proposed fuzzy coalition game is the stable solution set for the given fuzzy coalition which is also denoted by the crisp cores furthermore we study the relationship between the fuzzy shapley value and the fuzzy core by adding restrictions on the fuzzy core we propose a strong fuzzy core which is a more stable solution than the fuzzy core in addition the fuzzy core is equivalent to the strong fuzzy core under some condition

a generalization of the aumann shapley value for risk capital allocation problems abstract the paper proposes a new method to allocate risk capital to divisions or lines of business within a firm existing literature advocates an allocation rule that in game theoretic terms is equivalent to using the aumann shapley value as allocation mechanism the aumann shapley value however is only well defined if a specific differentiability condition is satisfied the rule that we propose is characterized as the limit of an average of path based allocation rules with grid size converging to zero the corresponding allocation rule is equal to the aumann shapley value if it exists if the aumann shapley value does not exist the allocation rule is equal to the weighted average of the aumann shapley values of nearby capital allocation problems

cooperation among agents with a proximity relation abstract a cooperative game consists of a set of players and a characteristic function determining the maximal gain or minimal cost that every subset of players can achieve when they decide to cooperate regardless of the actions that the other players take the relationships of closeness among the players should modify the bargaining among them and therefore their payoffs the first models that have studied this closeness used a priori unions or undirected graphs in the a priori union model a partition of the big coalition is supposed each element of the partition represents a group of players with the same interests the groups negotiate among them to form the grand coalition and later inside each one players bargain among them now we propose to use proximity relations to represent leveled closeness of the interests among the players and extending the a priori unions model

characterization of the equilibrium strategy of the bimatrix game with fuzzy payoff in this paper we consider fuzzy bimatrix games which mean bimatrix games with fuzzy payoff matrices we define two types of concepts of equilibrium strategy and investigate their relationships moreover we prove that these nash equilibrium strategies exist in any fuzzy bimatrix games

low dimensional lattices iii perfect forms the perfect lattices or quadratic forms in dimensions n leqslant 7 found by barnes stacey and others are studied and their automorphism groups orbits of minimal vectors and eutactic coefficients are determined it is shown that just 30 of the 33 known seven dimensional perfect lattices are extreme a simple and self contained proof is given of the classification of perfect lattices in dimensions n leqslant 4 and hence of the determination of the densest lattice packings in these dimensions

cores and weber sets for fuzzy extensions of cooperative games this paper investigates fuzzy extensions of cooperative games and the coincidence of the solutions for fuzzy and crisp games we show that an exact game has an exact fuzzy extension such that its fuzzy core coincides with the core for games with empty cores we exploit lovasz extensions to establish the coincidence of weber sets for fuzzy and crisp games

a note on the core minimal conversely consistent enlargement abstract the moulin reduction is extended by reducing the number of the players to the framework of fuzzy games based on this extended moulin reduction the axiomatic result of the core in tadenuma s 25 case does not hold in the context of fuzzy games and the core also violates the converse consistency therefore we propose the minimal core which possesses the converse consistency and is also the smallest solution satisfying the core

the nucleolus is well posed the lexicographic order is not representable by a real valued function contrary to many other orders or preorders so standard tools and results for well posed minimum problems cannot be used we prove that under suitable hypotheses it is however possible to guarantee the well posedness of a lexicographic minimum over a compact or convex set this result allows us to prove that some game theoretical solution concepts based on lexicographic order are well posed in particular this is true for the nucleolus

genericity and robustness of full surplus extraction we study whether priors that admit full surplus extraction fse are generic an issue that becomes a gauge to evaluate the validity of the current mechanism design paradigm we consider the space of priors on the universal type space and thereby relax the assumption of a fixed finite number of types made by cremer and mclean 1988 we show that fse priors are topologically generic contrary to the result of heifetz and neeman 2006 that fse is generically impossible both geometrically and measure theoretically instead of using the bdp approach or convex combinations of priors adopted in heifetz and neeman 2006 we prove our genericity results by showing a robustness property of cremer mclean mechanisms

measuring the interactions among variables of functions over the unit hypercube abstract by considering a least squares approximation of a given square integrable function f 0 1 n r by a multilinear polynomial of a specified degree we define an index which measures the overall interaction among variables of f this definition extends the concept of banzhaf interaction index introduced in cooperative game theory our approach is partly inspired from multilinear regression analysis where interactions among the independent variables are taken into consideration we show that this interaction index has appealing properties which naturally generalize several properties of the banzhaf interaction index in particular we interpret this index as an expected value of the difference quotients of f or under certain natural conditions on f as an expected value of the derivatives of f finally we discuss a few applications of the interaction index in aggregation function theory

a unified approach to truthful scheduling on related machines we present a unified framework for designing deterministic monotone polynomial time approximation schemes ptass for a wide class of scheduling problems on uniformly related machines this class includes among others minimizing the makespan maximizing the minimum load and minimizing the p norm of the machine loads vector previously this kind of result was only known for the makespan objective monotone algorithms have the property that an increase in the speed of a machine cannot decrease the amount of work assigned to it our results imply the existence of a truthful mechanism that can be implemented in polynomial time where the social goal is approximated within arbitrary precision

percolation of dimers on square lattices abstract a theoretical approach based on exact calculations of configurations on finite rectangular cells is applied to study the percolation of homonuclear dimers on square lattices an efficient algorithm allows us to calculate the detailed structure of the configuration space for m l x l y cells with m varying from 16 to 36 the percolation process has been monitored by following the percolation function defined as the ratio between the number of percolating configurations and the total number of available configurations for a given cell size and concentration of occupied sites the percolation threshold has been calculated by means of two complementary methods one based on well known renormalization techniques and the other based on determining the inflection point of the percolation function curves a comparison of the results obtained by these two methods has been performed the study includes the use of finite size scaling theory to extrapolate numerical results towards the thermodynamic limit the effect of jamming due to dimers is also established finally the critical exponents and have been obtained and values compared with numerical results and expected theoretical estimations the present results show agreement and even improvement in the case of with respect to some numeric values available in the literature

concave measures and the fuzzy core of exchange economies with heterogeneous divisible commodities the main purpose of this paper is to prove the existence of the fuzzy core of an exchange economy with a heterogeneous divisible commodity in which preferences of individuals are given by nonadditive utility functions defined on a s algebra of admissible pieces of the total endowment of the commodity the problem is formulated as the partitioning of a measurable space among finitely many individuals applying the yosida hewitt decomposition theorem we also demonstrate that partitions in the fuzzy core are supportable by prices in l 1

on characterization of equilibrium strategy of two person zero sum games with fuzzy payoffs in this paper we consider fuzzy matrix games namely two person zero sum games with fuzzy payoffs based on fuzzy max order for such games we define three kinds of concepts of minimax equilibrium strategies and investigate their properties first we shall show that these equilibrium strategies are characterized as nash equilibrium strategies of a family of parametric bi matrix games with crisp payoffs second we investigate properties of values of fuzzy matrix games by means of possibility and necessity measures in addition we give a numerical example to illustrate utility of our approaches

a banzhaf value for games with fuzzy communication structure computing the power of the political groups in the european parliament abstract in 2013 jimenez losada et al introduced several extensions of the myerson value for games with fuzzy communication structure in a fuzzy communication structure the membership of the players and the relations among them are leveled now we study a banzhaf value for these situations the myerson model is followed to define the fuzzy graph banzhaf value taking as base point the choquet integral we propose an axiomatization for this value introducing leveled amalgamation of players an algorithm to calculate this value is provided and its complexity is studied finally we show an applied example computing by this fuzzy value the power of the groups in the european parliament

low dimensional lattices v integral coordinates for integral lattices we say that an n dimensional classically integral lattice lambda is s integrable for an integer s if it can be described by vectors s frac 1 2 x 1 x k with all x i in mathbb z in a euclidean space of dimension k geqslant n equivalently lambda is s integrable if and only if any quadratic form f x corresponding to lambda can be written as s 1 times a sum of k squares of linear forms with integral coefficients or again if and only if the dual lattice lambda contains a eutactic star of scale s this paper gives many techniques for s integrating low dimensional lattices such as e 8 and the leech lattice a particular result is that any one dimensional lattice can be 1 integrated with k 4 this is lagrange s four squares theorem let phi s be the smallest dimension n in which there is an integral lattice that is not s integrable in 1937 ko and mordell showed that phi 1 6 we prove that phi 2 12 phi 3 14 21 leqslant phi 4 leqslant 25 16 leqslant phi 5 leqslant 22 phi s 4 leqslant 4s 2 s odd phi s leqslant 2 pi es 1 o 1 s even and phi s geqslant 2 ln ln s ln ln ln s 1 o 1

max consistency complement consistency and the core of fuzzy games recently hwang introduced two extensions of the max reduced game to fuzzy games by reducing the number of the players based on these reduced games he offered analogous axiomatizations of the core related to serrano and volij in the context of fuzzy games here we continue and develop hwang s work on similar topics however we turn to a different definition of the reduced game by simultaneously reducing both the number of the players and the participation levels we propose such extensions of the complement reduced game and of the max reduced game to fuzzy games and define the related properties of consistency and converse consistency based on these properties we offer several analogous characterizations of the core related to peleg serrano and volij and tadenuma respectively

existence of an aumann maschler fuzzy bargaining set and fuzzy kernels in tu fuzzy games abstract in this paper we first give fuzzy extensions of the well known solution concepts of kernel and prekernel for a transferable utility tu game and establish the existence of fuzzy kernels in tu fuzzy games which generalizes the corresponding result for classical tu games by davis and maschler 5 we then prove that the fuzzy kernel is a subset of the fuzzy bargaining set in a tu fuzzy game which gives the fuzzy counterpart of the corresponding fact for classical tu games by davis and maschler as a consequence we establish the existence of fuzzy aumann maschler fuzzy bargaining sets for tu fuzzy games which generalizes the well known existence theorem for bargaining sets by aumann and maschler 1 and peleg 11

games with incomplete information played by bayesian players i iii this article originally appeared in management science november 1967 volume 14 number 3 pp 159 182 published by the institute of management sciences the paper develops a new theory for the analysis of games with incomplete information where the players are uncertain about some important parameters of the game situation such as the payoff functions the strategies available to various players the information other players have about the game etc however each player has a subjective probability distribution over the alternative possibilities in most of the paper it is assumed that these probability distributions entertained by the different players are mutually consistent in the sense that they can be regarded as conditional probability distributions derived from a certain basic probability distribution over the parameters unknown to the various players but later the theory is extended also to cases where the different players subjective probability distributions fail to satisfy this consistency assumption in cases where the consistency assumption holds the original game can be replaced by a game where nature first conducts a lottery in accordance with the basic probability distribution and the outcome of this lottery will decide which particular subgame will be played i e what the actual values of the relevant parameters will be in the game yet each player will receive only partial information about the outcome of the lottery and about the values of these parameters however every player will know the basic probability distribution governing the lottery thus technically the resulting game will be a game with complete information it is called the bayes equivalent of the original game part i of the paper describes the basic model and discusses various intuitive interpretations for the latter part ii shows that the nash equilibrium points of the bayes equivalent game yield bayesian equilibrium points for the original game finally part iii considers the main properties of the basic probability distribution

set valued cooperative games with fuzzy payoffs the fuzzy assignment game abstract in this paper we study cooperative games with fuzzy payoffs the main advantage of the approach presented is the incorporation into the analysis of the problem of ambiguity inherent in many real world collective decision situations we propose extensions of core concepts which maintain the fuzzy nature of allocations and lead to a more satisfactory study of the problem within the fuzzy context finally we illustrate the extended core concepts and the approach to obtain the corresponding allocations through the analysis of assignment games with uncertain profits

monte carlo simulation of the percolation process caused by the random sequential adsorption of k mers on heterogeneous triangular lattices abstract mixed site bond percolation is studied for a random sequential adsorption process of k mers on heterogeneous lattices with variable connectivity by means of monte carlo simulation the percolation phase diagrams are built to render evidence about complex structures critical exponents are also calculated to show that the universality class corresponding to ordinary percolation in two dimensions is preserved

the fuzzy core in games with fuzzy coalitions in this paper the fuzzy core of games with fuzzy coalition is proposed which can be regarded as the generalization of crisp core the fuzzy core is based on the assumption that the total worth of a fuzzy coalition will be allocated to the players whose participation rate is larger than zero the nonempty condition of the fuzzy core is given based on the fuzzy convexity three kinds of special fuzzy cores in games with fuzzy coalition are studied and the explicit fuzzy core represented by the crisp core is also given because the fuzzy shapley value had been proposed as a kind of solution for the fuzzy games the relationship between fuzzy core and the fuzzy shapley function is also shown surprisingly the relationship between fuzzy core and the fuzzy shapley value does coincide as in the classical case

cooperative fuzzy games we derive versions of known results dealing with core equilibria and shapley values of cooperative games in the case of cooperative fuzzy games i e games defined on fuzzy subsets of the set of n players a fuzzy coalition is an n vector i associating with each player i his rate of participation i 0 1 in the fuzzy coalition and the real number v is the coalition worth assumed to be positively homogeneous if also v is superadditive or equivalently concave the fuzzy game with side payments has a nonempty core associated with the coalition worth function for any ordinary game with side payments is a fuzzy extension thereof viz the fuzzy game whose coalition worth function is the least positively homogeneous superadditive function majorizing the coalition worth function of the original game the fuzzy extension always has a nonempty core moreover if the original game has a nonempty core it coincides with that of its fuzzy extension analogous results are established for games without side payments an axiomatization of values of fuzzy games with side payments is also given the results are applied to show that the set of walras equilibria coincides with the fuzzy core of an economy

cores of fuzzy games and their convexity in this paper the proper core and dominance core of a fuzzy game are introduced to guarantee equality between the proper core and the dominance core some mild conditions are provided another important issue for the convexity of the dominance core is also addressed

games on fuzzy communication structures with choquet players myerson 1977 used graph theoretic ideas to analyze cooperation structures in games in his model he considered the players in a cooperative game as vertices of a graph which undirected edges defined their communication possibilities he modified the initial games taking into account the graph and he established a fair allocation rule based on applying the shapley value to the modified game now we consider a fuzzy graph to introduce leveled communications in this paper players play in a particular cooperative way they are always interested first in the biggest feasible coalition and second in the greatest level choquet players we propose a modified game for this situation and a rule of the myerson kind

nontransferable utility games with fuzzy coalition restrictions abstract a value for nontransferable utility ntu games with fuzzy coalition restrictions is introduced and characterized in a similar way as the shapley value for transferable utility tu games has been extended in the literature to study games with restricted cooperation we extend the shapley ntu value to deal with ntu games in situations in which there are fuzzy dependency relationships among the players

existence of fuzzy prekernels and mas colell bargaining sets in tu games abstract in this paper we first prove the existence of fuzzy prekernels and fuzzy nucleolus in tu fuzzy games which generalize the corresponding results for classical tu games as an application we apply the existence of fuzzy prekernels to derive an existence theorem for mas colell fuzzy bargaining sets and a stronger existence theorem for mas colell bargaining sets than the one by vohra 1991 18 we also provide an axiomatization theorem for fuzzy prekernels

the equalizer and the lexicographical solutions for cooperative fuzzy games characterization and properties in this paper we analyze the lexicographical solution for fuzzy tu games we study its properties and obtain a characterization the lexicographical solution was introduced by sakawa and nishizaki fuzzy sets and systems 61 1994 265 275 as a solution for crisp tu games and then extended as a value for fuzzy tu games we approach the problem by means of the close relationship that exists between the lexicographical solution for crisp tu games and the least square nucleolus a crisp value defined by ruiz et al internat j game theory 25 1996 113 134 previously and also based on this relationship we axiomatically characterize the equalizer solution for fuzzy tu games both values the equalizer and the lexicographical solutions are based on the consideration of a measure of dissatisfaction of players rather than coalitions

the equal allocation of nonseparable costs an extension under fuzzy behavior by considering the players and the activity levels simultaneously we propose a consistent solution which is a generalization of the equal allocation of nonseparable costs eansc on fuzzy games we also adopt two pre existing concepts from traditional game theory and reinterpret them in the framework of fuzzy games first based on the consistency which related to an extended reduced game we offer two axiomatizations of the extended eansc second by applying the excess function we propose a dynamic process to illustrate that the extended eansc can be reached by players who start from an arbitrary efficient solution and make consecutive amendments

the general prenucleolus of n person cooperative fuzzy games abstract in this paper we define the concept of the general prenucleolus minimizing the player complaint vector in the lexicographic order over the preimputation set for the cooperative games with fuzzy coalitions for the general prenucleolus we get a sufficient condition that the complaints of all players are in equal amount under the case of linear complaint functions concluding the proposed player excess sakawa and nishizaki 1994 15 as a result we can obtain the proposed fuzzy solutions with the corresponding linear complaint functions such as the equalizer solution molina and tejada 2002 5 and shapley function etc we prove that the least square general prenucleolus minimizing the variance of the resulting player complaints is the general prenucleolus thus an optimal solution considered from two aspects of the lexicographic order and the least square criterion is obtained in addition several general prenucleoli are proposed in terms of specific situations

fuzzy harsanyi solution for a kind of fuzzy coalition games abstract the fuzzy linear game in ref 32 is a class of fuzzy cooperative games with fuzzy coalition fuzzy coalition game for short which admits the difference of interactions for different fuzzy coalitions the fuzzy shapley value for fuzzy linear game has been studied as shapley value is just one point in the harsanyi solution it is interesting to study the explicit form of harsanyi solution in the fuzzy coalition game this paper defines the fuzzy harsanyi solution which distributes the harsanyi dividends such that the dividend shares of players in each fuzzy coalition are proportional to the corresponding players participation index when the fuzzy coalition payoff is average distributed between players in the fuzzy coalition the fuzzy harsanyi solution also coincides with the fuzzy shapley value just as their relationship in a crisp cooperative game we provide two axiomatic characterizations for the fuzzy harsanyi solution one uses axioms and the other uses distribution matrices the fuzzy harsanyi solution is a unique value on the fuzzy linear game that satisfies fuzzy efficiency fuzzy null player and fuzzy additivity property meanwhile based on probability distribution fuzzy harsanyi solution is seen as an expected marginal contribution of each player in the fuzzy coalition which provides more choices for players in the fuzzy coalition

solutions for fuzzy matrix games in this paper we deal with two person zero sum games with fuzzy payoffs and fuzzy goals we have presented two models for studying two person zero sum matrix games with fuzzy payoffs and fuzzy goals we assume that each player has a fuzzy goal for each of the payoffs we obtained that the fuzzy relation approach and the max min solution are equivalent

optimal auction design this paper considers the problem faced by a seller who has a single object to sell to one of several possible buyers when the seller has imperfect information about how much the buyers might be willing to pay for the object the seller s problem is to design an auction game which has a nash equilibrium giving him the highest possible expected utility optimal auctions are derived in this paper for a wide class of auction design problems

linear quadratic uncertain differential game with application to resource extraction problem uncertain differential game investigates interactive decision making of players over time and the system dynamics is described by an uncertain differential equation this paper goes further to study the two player zero sum uncertain differential game in order to guarantee the saddle point nash equilibrium a max min theorem is provided furthermore when the system dynamics is described by a linear uncertain differential equation and the performance index function is quadratic the existence of saddle point nash equilibrium is obtained via the solvability of a corresponding riccati equation finally a resource extraction problem is analyzed by using the theory proposed in this paper

a mass formula for unimodular lattices with no roots we derive a mass formula for n dimensional unimodular lattices having any prescribed root system we use katsurada s formula for the fourier coefficients of siegel eisenstein series to compute these masses for all root systems of even unimodular 32 dimensional lattices and odd unimodular lattices of dimension n 30 in particular we find the mass of even unimodular 32 dimensional lattices with no roots and the mass of odd unimodular lattices with no roots in dimension n 30 verifying bacher and venkov s enumerations in dimensions 27 and 28 we also compute better lower bounds on the number of inequivalent unimodular lattices in dimensions 26 to 30 than those afforded by the minkowski siegel mass constants

fuzzy games a characterization of the core in this paper we extend to the cooperative fuzzy games case the reduced game introduced by davis and maschler the kernel of a cooperative game naval res logist quart 12 1965 223 259 we shall introduce two types of such a reduced game to cooperative fuzzy games and define the related properties of consistency and converse consistency we further offer two axiomatizations of the core on the domain of all cooperative fuzzy games by means of consistency and its converse