coreness of cooperative games with truncated submodular profit functions coreness represents solution concepts related to core in cooperative games which captures the stability of players motivated by the scale effect in social networks economics and other scenario we study the coreness of cooperative game with truncated submodular profit functions specifically the profit function f cdot is defined by a truncation of a submodular function sigma cdot f cdot sigma cdot if sigma cdot ge eta and f cdot 0 otherwise where eta is a given threshold in this paper we study the core and three core related concepts of truncated submodular profit cooperative game we first prove that whether core is empty can be decided in polynomial time and an allocation in core also can be found in polynomial time when core is not empty when core is empty we show hardness results and approximation algorithms for computing other core related concepts including relative least core value absolute least core value and least average dissatisfaction value

on a generalized max type difference equation from automatic control theory abstract this paper studies the boundedness character of positive solutions of the following generalization of a max type difference equation from automatic control theory see mishkis 1977 18 and popov 1966 21 x n 1 max a x n p x n 1 q x n 2 r n n 0 where the parameters a p q and r are positive numbers in the study of the equation we also introduce a new method called oachkatzlschwoif

complementary cooperation minimal winning coalitions and power indices we introduce a new simple game which is referred to as the complementary weighted multiple majority game c wmmg for short c wmmg models a basic cooperation rule the complementary cooperation rule and can be taken as a sister model of the famous weighted majority game wmg for short in c wmmg each player is characterized by a nonnegative vector with a fixed dimension and players in the same coalition cooperate by producing a characteristic vector for this coalition each dimension of this vector equals the maximum of the corresponding dimensions of its members the value of a coalition is 1 if and only if the sum of its characteristic vector is larger than that of its complementary coalition in which case the coalition is called winning otherwise the coalitional value is 0 in this paper we concentrate on the two dimensional c wmmg an interesting property of this case is that there are at most n 1 minimal winning coalitions mwcs for short and they can be enumerated in time o nlogn where n is the number of players this property guarantees that the two dimensional c wmmg is more handleable than wmg in particular we prove that the main power indices i e the shapley shubik index the penrose banzhaf index the holler packel index and the deegan packel index are all polynomially computable to make a comparison with wmg we know that it may have exponentially many mwcs and none of the four power indices is polynomially computable unless p np still for the two dimensional case we show that local monotonicity holds for all of the four power indices in wmg this property is possessed by the shapley shubik index and the penrose banzhaf index but not by the holler packel index or the deegan packel index since our model fits very well the cooperation and competition in team sports we hope that it can be potentially applied in measuring the values of players in team sports say help people give more objective ranking of nba players and select mvps and consequently bring new insights into contest theory and the more general field of sports economics it may also provide some interesting enlightenments into the design of non additive voting mechanisms last but not least the threshold version of c wmmg is a generalization of wmg and natural variants of it are closely related with the famous airport game and the stable marriage roommates problem

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

on a max type and a min type difference equation this note shows that every positive solution to the following third order non autonomous max type difference equationx n 1 maxa nx n x n 2 email protected n 0 when a n n n 0 is a three periodic sequence of positive numbers is periodic with period three the same result was proved for the following min type difference equationx n 1 mina nx n x n 2 email protected n 0

matching games the least core and the nucleolus a matching game is a cooperative game defined by a graph g n e the player set is n and the value of a coalition s n is defined as the size of a maximum matching in the subgraph induced by s we show that the nucleolus of such games can be computed efficiently the result is based on an alternative characterization of the least core which may be of independent interest the general case of weighted matching games remains unsolved

global attractivity of a family of nonautonomous max type difference equations abstract this paper studies a family of nonautonomous max type difference equations with several delays using some transformations we prove global attractivity of positive solutions to these equations under some conditions some of our results considerably extend related ones in the literature two examples are given to illustrate the main results

joint migration inversion simultaneous determination of velocity fields and depth images using all orders of scattering the future in seismic exploration and seismic monitoring is the inclusion of all orders of scattering in the imaging and inversion algorithms using the full wavefield in an inversion process allows us to surpass imprints provided by incomplete acquisition and provide more accurate information on the subsurface as noise becomes signal we describe an inversion process that does not use the standard parameters from finite difference type modeling being local velocity and density but that describes the seismic data in terms of elastic reflectivity operators and propagation operators with these operators our full wavefield modeling process builds the complete two way seismic response a major advantage of our alternative to the traditional full waveform inversion methodology is that reflectivity operators are traveltime free and propagation operators are scattering free making the inversion problem significantly more linear a major advantage of our algorithm is 1 that multiples are utilized to impro

global behavior of the max type difference equation x n max 1 x n m a n x n r in this paper we study the global behavior of the following max type difference equation x n max 1 x n m a n x n r n 0 1 2 where a n n 0 is a sequence of positive numbers with a n 0 1 for every n 0 and sup a n 1 and m r 1 2 3 and the initial values x d x d 1 x 1 0 with d max m r we show that 1 if x n n d is a positive solution of this equation then for every 0 k 2 m 1 x 2 mn k n 0 is eventually monotone 2 if a n n 0 is a periodic sequence then every positive solution of this equation is eventually periodic with period 2 m

inversion of seismic reflection data in the acoustic approximation the nonlinear inverse problem for seismic reflection data is solved in the acoustic approximation the method is based on the generalized least squares criterion and it can handle errors in the data set and a priori information on the model multiply reflected energy is naturally taken into account as well as refracted energy or surface waves the inverse problem can be solved using an iterative algorithm which gives at each iteration updated values of bulk modulus density and time source function each step of the iterative algorithm essentially consists of a forward propagation of the actual sources in the current model and a forward propagation backward in time of the data residuals the correlation at each point of the space of the two fields thus obtained yields the corrections of the bulk modulus and density models this shows in particular that the general solution of the inverse problem can be attained by methods strongly related to the methods of migration of unstacked data and commerc

core stability of minimum coloring games in cooperative game theory a characterization of games with stable cores is known as one of the most notorious open problems we study this problem for a special case of the minimum coloring games introduced by deng et al 1999 which arise from a cost allocation problem when the players are involved in conflict in this paper we show that the minimum coloring game on a perfect graph has a stable core if and only if every vertex of the graph belongs to a maximum clique we also consider the problem on the core largeness the extendability and the exactness of minimum coloring games as a consequence we show that it is conp complete to decide whether a given game has a large core is extendable or is exact

on some systems of difference equations abstract in this note we show that the following systems of difference equations u n 1 v n 1 v n v n 1 u n 1 u n u n 1 v n 1 u n v n 1 u n 1 v n u n 1 u n 1 v n v n 1 v n 1 u n with n n 0 and complex initial values u 0 and v 0 are solvable explicitly by means of riccati equations and based on the formulae for the general solutions we present the behaviour of their solutions as n while the result is natural for the first system it is a bit surprising for the other two systems

solving coalitional resource games coalitional resource games crgs are a form of non transferable utility ntu game which provide a natural formal framework for modelling scenarios in which agents must pool scarce resources in order to achieve mutually satisfying sets of goals although a number of computational questions surrounding crgs have been studied there has to date been no attempt to develop solution concepts for crgs or techniques for constructing solutions in this paper we rectify this omission following a review of the crg framework and a discussion of related work we formalise notions of coalition structures and the core for crgs and investigate the complexity of questions such as determining nonemptiness of the core we show that while such questions are in general computationally hard it is possible to check the stability of a coalition structure in time exponential in the number of goals in the system but polynomial in the number of agents and resources as a consequence checking stability is feasible for systems with small or bounded numbers of goals we then consider constructive approaches to generating coalition structures we present a negotiation protocol for crgs give an associated negotiation strategy and prove that this strategy forms a subgame perfect equilibrium we then show that coalition structures produced by the protocol satisfy several desirable properties pareto optimality dummy player and pseudo symmetry

computing the least core and nucleolus for threshold cardinality matching games cooperative games provide a framework for fair and stable profit allocation in multi agent systems core least core and nucleolus are such solution concepts that characterize stability of cooperation in this paper we study the algorithmic issues of the least core and nucleolus of threshold cardinality matching games tcmg a tcmg is defined on a graph g v e and a threshold t in which the player set is v and the profit of a coalition s v is 1 if the size of a maximum matching in g s meets or exceeds t and 0 otherwise we first show that for a tcmg the problems of computing least core value finding and verifying least core payoff are all polynomial time solvable we also provide a general characterization of the least core for a large class of tcmg cf theorem 2 next based on gallai edmonds decomposition in matching theory we establish a concise formulation of the nucleolus for a special case of tcmg when the threshold t equals 1 for arbitrary t we prove that the nucleolus of tcmg can be obtained in polynomial time for bipartite graphs and graphs with a perfect matching for a tcmg the problems of computing least core value finding and verifying least core payoff are polynomial time solvable we provide a general characterization of the least core for a large class of tcmg we establish a concise formulation of the nucleolus for a typical case of tcmg when the threshold t equals 1 for an arbitrary threshold t the nucleolus of tcmg can be obtained in polynomial time for graphs with a perfect matching for an arbitrary threshold t the nucleolus of tcmg can be obtained in polynomial time for bipartite graphs

global stability of a difference equation with maximum we prove that every positive solution to the difference equationx n maxa 1x n 1 a 1 a 2x n 2 a 2 a kx n k a k email protected n 0 where email protected n a i 0 a email protected 0 1 i 1 k converges to the following quantity maxa 1 1 a 1 1 a k 1 a k 1 confirming a quite recent conjecture of interest we also prove another result on global convergence which concerns some cases when not all a i i 1 k belong to the interval 0 1

combining full wavefield migration and full waveform inversion a glance into the future of seismic imaging the next generation seismic migration and inversion technology considers multiple scattering as vital information allowing the industry to derive significantly better reservoir models with more detail and less uncertainty while requiring a minimum of user intervention three new insights have been uncovered with respect to this fundamental transition unblended or blended multiple scattering can be included in the seismic migration process and it has been proposed to formulate the imaging principle as a minimization problem the resulting process yields angle dependent reflectivity and is referred to as recursive full wavefield migration wfm the full waveform inversion process for velocity estimation can be extended to a recursive optionally blended anisotropic multiple scattering algorithm the resulting process yields angle dependent velocity and is referred to as recursive full waveform inversion wfi the mathematical equations of wfm and wfi have an identical structure but the physical meaning behind the expressions is fundamentally different in wfm the reflection process is central and the aim is to estimate reflection operators of the subsurface using the up and downgoing incident wavefields including the codas in each gridpoint in wfi however the propagation process is central and the aim is to estimate velocity operators of the subsurface using the total incident wavefield sum of up and downgoing in each gridpoint angle dependent reflectivity in wfm corresponds with angle dependent velocity anisotropy in wfi the algorithms of wfm and wfi could be joined into one automated joint migration inversion process in the resulting hybrid algorithm being referred to as recursive joint migration inversion jmi the elaborate volume integral solution was replaced by an efficient alternative wfm and wfi are alternately applied at each depth level where wfm extrapolates the incident wavefields and wfi updates the velocities without any user interaction the output of the jmi process offers an integrated picture of the subsurface in terms of angle dependent reflectivity as well as anisotropic velocity this two fold output reflectivity image and velocity model offers new opportunities to extract accurate rock and pore properties at a fine reservoir scale

eventual periodicity of some systems of max type difference equations the periodicity of the next system of max type difference equations xn 1 max1 i1 m1f1i1xn ki1 1 1 1 xn ki1 2 1 2 xn ki1 l 1 l n xn t1s 1 xn 2 max1 i2 m2f2i2xn ki2 1 2 1 xn ki2 2 2 2 xn ki2 l 2 l n xn t2s 2 xn l max1 il mlflilxn kil 1 l 1 xn kil 2 l 2 xn kil l l l n xn tls l n n0n n0 where s l mj tj kij h j n j h 1 l j h 1 l 1 2 l 1 2 l is a permutation of 1 2 l 1 2 l and fjij 0 l n0 0 fjij 0 l n0 0 j 1 l j 1 l ij 1 mj ij 1 mj is studied it is shown that under some conditions posed on functions fjijfjij all positive solutions of the system are eventually periodic with period ts for some t nt n

cooperation through social influence abstract we consider a simple and altruistic multiagent system in which the agents are eager to perform a collective task but where their real engagement depends on the willingness to perform the task of other influential agents we model this scenario by an influence game a cooperative simple game in which a team or coalition of players succeeds if it is able to convince enough agents to participate in the task to vote in favor of a decision we take the linear threshold model as the influence model we show first the expressiveness of influence games showing that they capture the class of simple games then we characterize the computational complexity of various problems on influence games including measures length and width values shapley shubik and banzhaf and properties of teams and players finally we analyze those problems for some particular extremal cases with respect to the propagation of influence showing tighter complexity characterizations

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

boundedness character of a class of difference equations abstract a complete picture regarding the boundedness character of positive solutions to the following difference equation x n max a x n 1 p x n k p n n 0 where k 2 and the parameters a and p are positive real numbers is given in particular for the case p k 1 0 k k k 1 k 1 we prove that all solutions to the equation are bounded we also present corresponding results concerning the following closely related difference equation x n a x n 1 p x n k p n n 0

solutions of a max type system of difference equations we show in an elegant way that the general solution to the following max type system of difference equations xn 1 maxaxn ynxn yn 1 maxayn xnyn n n0 for the case y0 x0 a 0 y0 x0 max a 1 a is given by xn afk n 1 n x0fk n y0fk n 1 n n n y1 x0 y0 and yn y0fk n 1 1afk n 1 n 1x0fk n 1 1 1 n n 2 where k n n for n even k n n 1 for n odd n n0 n n mod2 for n n0 and where the sequence fn n n is the fibonacci sequence correcting and giving a proof of a wrong result in the paper d simsek b demir c cinar on the solutions of the system of difference equations xn 1 max a xn yn xn yn 1 max a yn xn yn discrete dyn nat soc 2009 2009 11 pp article id 325296

on the complexity of cooperative solution concepts we study from a complexity theoretic standpoint the various solution concepts arising in cooperative game theory we use as a vehicle for this study a game in which the players are nodes of a graph with weights on the edges and the value of a coalition is determined by the total weight of the edges contained in it the shapley value is always easy to compute the core is easy to characterize when the game is convex and is intractable np complete otherwise similar results are shown for the kernel the nucleolus the e core and the bargaining set as for the von neumann morgenstern solution we point out that its existence may not even be decidable many of these results generalize to the case in which the game is presented by a hypergraph with edges of size k 2

on a symmetric system of max type difference equations we study behavior of positive solutions of the following max type system of difference equationsx n 1 maxc y n px n 1 p y n 1 maxc x n py n 1 p email protected n 0 where p email protected 0 extending some results in the literature among other results we prove that if p email protected 0 1 then every positive solution of the system converges to 1 1

on some periodic systems of max type difference equations we show that all positive solutions to the system of max type difference equations xn 1 max1 i m1f1ixn ki 1 1 1 xn ki 2 1 2 xn ki l 1 l n xn s 1 xn 2 max1 i m2f2ixn ki 1 2 1 xn ki 2 2 2 xn ki l 2 l n xn s 2 xn l max1 i mlflixn ki 1 l 1 xn ki 2 l 2 xn ki l l l n xn s l n n0 where s l mj ki t j n j t 1 l and for a fixed j i 1 mj and where the functions fji 0 l n0 0 j 1 l i 1 mj satisfy some conditions are eventually periodic with not necessarily prime period s a related result for the corresponding system of min type difference equations is also proved

simultaneous inversion of the background velocity and the perturbation in full waveform inversion abstractthe gradient of standard full waveform inversion fwi attempts to map the residuals in the data to perturbations in the model such perturbations may include smooth background updates from the transmission components and high wavenumber updates from the reflection components however if we fix the reflection components using imaging the gradient of what is referred to as reflected waveform inversion rwi admits mainly transmission background type updates the drawback of existing rwi methods is that they lack an optimal image capable of producing reflections within the convex region of the optimization because the influence of velocity on the data was given mainly by its background propagator and perturbed reflectivity components we have optimized both components simultaneously using a modified objective function specifically we used an objective function that combined the data generated from a source using the background velocity and that by the perturbed velocity through born modeli

tomographic full waveform inversion tfwi by combining fwi and wave equation migration velocity analysis convergence of full waveform inversion can be improved by extending the velocity model along either the subsurface offset axis or the time lag axis the extension of the velocity model along the time lag axis enables us to linearly model large time shifts caused by velocity perturbations the extension is based on a new linearization of the scalar wave equation where the extended velocity perturbation is convolved in time with the laplacian of the background wavefield this linearization is accurate for both reflected events and transmitted events and in particular for diving waves recorded at large offsets the modeling capabilities of the proposed linearization enable the simultaneous inversion of reflections and diving waves even when the starting velocity model is far from being accurate we solve the resulting optimization problem with a nested algorithm the inner iterations are based on the proposed linearization and on a mixing of scales between the short and long wavelength components of the v

estimating primaries by sparse inversion and application to near offset data reconstruction accurate removal of surface related multiples remains a challenge in many cases to overcome typical inaccuracies in current multiple removal techniques we have developed a new primary estimation method estimation of primaries by sparse inversion epsi epsi is based on the same primary multiple model as surface related multiple elimination srme and also requires no subsurface model unlike srme epsi estimates the primaries as unknowns in a multidimensional inversion process rather than in a subtraction process furthermore it does not depend on interpolated missing near offset data because it can reconstruct missing data simultaneously sparseness plays a key role in the new primary estimation procedure the method was tested on 2d synthetic data

on the system of difference equations xn cnyn 3 an bnyn 1xn 2y n 3 yn γnxn 3 αn βnxn 1yn 2 xn 3 we show that the system of difference equationsx n c ny n 3a n b ny n 1x n 2y n 3 y email protected nx n email protected email protected nx n 1y n 2x n 3 email protected n 0 where the sequences a n b n c n a n b n c n email protected n 0 and initial values x i y i email protected 1 2 3 are real numbers such that c n 0 c n 0 email protected n 0 can be solved in closed form and for the case when all the sequences a n b n c n a n b n and c n are constant we describe the asymptotic behavior of well defined solutions of the system

on the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t with periodic parameters we investigate the boundedness of positive solutions of the reciprocal max type difference equation xn max a1n 1 xn 1 a2n 1 xn 2 atn 1 xn t where for each value of i the sequence ain n 0 of positive numbers is periodic with period pi we give both sufficient conditions on the pi s for the boundedness of all solutions and sufficient conditions for all solutions to be unbounded this work essentially complements the work by bidwell and franke who showed that as long as every positive solution of our equation is bounded then every positive solution is eventually periodic thereby leaving open the question as to when solutions are bounded

on positive solutions of some classes of max type systems of difference equations global attractivity results on positive solutions of some special cases of the next max type system of difference equations xn i max1 xn l1 i i p1 i j1 1m1 i xn j1 i1 sj1 i1 qj1 i1 xn lri i i pri i jri 1mri i xn jri iri sjri iri qjri iri n n0 where ri n i 1 k k n sjt it 1 k qjt it 0 i 1 k t 1 ri and pj i 0 1 mj i n i 1 k j 1 ri are presented we also present some results on the behavior of positive solutions of the difference equation xn max p1xn 1 pkxn k xn max p1xn 1 pkxn k where pj j 1 k are nonnegative numbers

finite rationality and interpersonal complexity in repeated games a measure of complexity for repeated game strategies is studied this measure facilitates the investigation of some issues regarding finite rationality and the structure of subgame perfect equilibria of repeated games with discounting specifically the complexity of a strategy in a given repeated game is defined to be the cardinality of the induced strategy set i e the number of distinct strategies induced by the original strategy in all possible subgames we observe that this cardinality is equal to the size cardinality of the state set of the smallest automaton which can implement the strategy thus in a sense complexity is measured on the basis of the amount of computing power inherent in the strategy a measure of strategic memory is also studied the following results are obtained 1 combining two notions of bounded rationality epsilon equilibrium and finite complexity we find that every subgame perfect equilibrium of the repeated game can be approximated with regard to payoffs by a subgame perfect epsilon equilibrium of finite complexity 2 for a generic class of normal form stage games at every discount robust subgame perfect drsp equilibrium there are necessary relationships among the complexities and memories of the players strategies in the two player case strategies must be equally complex and must have equal memories 3 for a second class of two pla yer stage games we show that the payoff vectors for all drsp equilibria are obtainable via equilibria in which the players strategies are equally complex and have equal memoiies

global stability of a max type difference equation we show that every positive solution to the difference equationx n maxa 1x n p 1 a 1 a 2x n p 2 a 2 a kx n p k a k n n 0 where p i i 1 k are natural numbers such that 1 0 a i 1 1 i 1 k converges to max 1 i ka i 1 a i 1 this result improves and complements the main result in our recent note s stevic global stability of a difference equation with maximum appl math comput 210 2009 525 529 since it also considers the case when a i 1 0 i 1 k

on the max type equation xn 1 max axn xn 2 abstract we show that every well defined solution of the difference equation x n 1 max a x n x n 2 n n 0 where a r is eventually periodic with period three

on the computational complexity of coalitional resource games we study coalitional resource games crgs a variation of qualitative coalitional games qcgs in which each agent is endowed with a set of resources and the ability of a coalition to bring about a set of goals depends on whether they are collectively endowed with the necessary resources we investigate and classify the computational complexity of a number of natural decision problems for crgs over and above those previously investigated for qcgs in general for example we show that the complexity of determining whether conflict is inevitable between two coalitions with respect to some stated resource bound i e a limit value for every resource is co np complete we then investigate the relationship between crgs and qcgs and in particular the extent to which it is possible to translate between the two models we first characterise the complexity of determining equivalence between crgs and qcgs we then show that it is always possible to translate any given crg into a succinct equivalent qcg and that it is not always possible to translate a qcg into an equivalent crg we establish some necessary and some sufficient conditions for a translation from qcgs to crgs to be possible and show that even where an equivalent crg exists it may have size exponential in the number of goals and agents of its source qcg

periodicity of a class of nonautonomous max type difference equations abstract in this note we prove several general and easily applicable results regarding the periodicity of positive solutions of the following difference equations x n max f 1 x n k 1 n f m x n k m n x n s n n 0 and x n min f 1 x n k 1 n f m x n k m n x n s n n 0 where 1 k 1 k m m s n and where the functions f i 0 n 0 0 1 i m satisfy some additional conditions these results considerably extend and unify numerous results on max type difference equations existing in the literature

on the solutions of a system of difference equations with maximum in this paper we study the following max type system of difference equations x n max 1 x n m min 1 a y n r y n max 1 y n m min 1 b x n t n n 0 where a b 0 m r t 1 2 with r m and t m we show that every solution of this system with the initial values x d y d x d 1 y d 1 x 1 y 1 0 is eventually periodic with period 2m where d max m r t

reasoning about coalitional games we develop investigate and compare two logic based knowledge representation formalisms for reasoning about coalitional games the main constructs of coalitional game logic cgl are expressions for representing the ability of coalitions which may be combined with expressions for representing the preferences that agents have over outcomes modal coalitional game logic mcgl is a normal modal logic in which the main construct is a modality for expressing the preferences of groups of agents for both frameworks we give complete axiomatisations and show how they can be used to characterise solution concepts for coalitional games we show that while cgl is more expressive than mcgl the former can only be used to reason about coalitional games with finitely many outcomes while mcgl can be used to reason also about games with infinitely many outcomes and is in addition more succinct we characterise the computational complexity of satisfiability for cgl and give a tableaux based decision procedure

long term behavior of positive solutions of a system of max type difference equations abstract some sufficient conditions which guaranty the attractivity of all positive solutions of the next system of difference equations x n max a y n 1 p j 2 k x n j q j y n max a x n 1 p j 2 k y n j q j n n 0 where a p 0 and q j j 2 k are nonnegative numbers are given the boundedness character of positive solutions of the system is also studied