Contents.m

% MatTuGames: A Matlab Game Theory Toolbox 
% Version 1.9.2 (R2024b) 14-Feb-2025
%
%
%  Author:        Holger I. Meinhardt 
%  E-Mail:        holger.meinhardt@partner.kit.edu
%  Institution:   Karlsruhe Institute of Technology (KIT)
%
%
% aux: Some auxiliary files
%--------------------------
% FrameToImage                              - Converts a frame to an image.
% FrameToImage2                             - Converts a frame to an image using MYAA.
% PlayCoreMovie                             - Plays a movie from a collection of frames.
% SaveFrames                                - Saves the frames of a movie to different file formats.
% ToSimplex                                 - Projects data from 3d to 2d.
% ToSimplex3d                               - Projects data from 4d to 3d.
% ginv                                      - Computes a general inverse.
% myaa                                      - MYAA Render figure with antialiasing.
% qrginv                                    - Computes a pseudo-inverse using a QR-method.
% spnull                                    - Returns a sparse orthonormal basis of the null space.
% toSymbols                                 - Converts a vector (vec,n) into digits.
% vtk_export                                - Exports the graphical raw data to VTK legacy format.
%
% 
% bin: Scirpt File
%----------------
% corevert                                  - External bash script to call the cdd library. 
%                                        
%                                        
%                                        
% doc: Document Files                    
%-------------------                     
% MatTuGames_Version_1.9.2.                 - Additions and changes in version 1.9.2
% ReadMe.pdf                                - Installation instruction (PDF)
% ReadMe.md                                 - Installation instruction (Markdown Format)
% getting_started.m                         - Checks the installation
% getting_started.out                       - Reference results of getting_started
% getting_started.md                        - Reference results of getting_started (Markdown Format)   
% manual_mat_tugames.pdf                    - Manual (PDF)
% MatTuGames_References.md                  - Bibliography (Markdown Format)
% MatTuGames_References.pdf                 - Bibliography (PDF)    
% testcase_graphics                         - Checking basic graphic installation.
%
%
% graphics: Graphic Example Files
%-------------------------------
% core_exp.pdf                              - Core plot example 3d.
% core_exp_all.pdf                          - Core plot example 3d.
% core_exp_prk.pdf                          - Core plot example 3d.
% core_exp_prn.pdf                          - Core plot example 3d.
% core_exp_shap.pdf                         - Core plot example 3d.
% core_exp_sol_all.pdf                      - Core plot example 3d.
% core_exp_sol_none.pdf                     - Core plot example 3d.
% core_exp_sol_prk.pdf                      - Core plot example 3d.
% core_exp_sol_prn.pdf                      - Core plot example 3d.
% core_exp_sol_shap.pdf                     - Core plot example 3d.
% manual_exp2_core01.pdf                    - Core plot example 3d.
%
%
%
% mama: Mathematica Symbolic Toolbox Functions to call the Mathematica Package TuGames
%------------------------------------------------------------------------------------
% tug_AdjustedWorthVectors                  - Computes the adjusted worth vectors of k-convex games.   
% tug_AllAntiSurpluses                      - Computes the minimum surpluses.
% tug_AllMaxSurpluses                       - Computes the maximum surpluses.
% tug_AntiPreKernel                         - Computes an anti-pre-kernel point.
% tug_AntiPreKernelQ                        - Checks if an imputation is an anti-pre-kernel point.
% tug_AvConvexQ                             - Checks on average convexity.
% tug_AverageConvexQ                        - Checks on average convexity.
% tug_BalancedCollectionQ                   - Verifies if the induced collections are balanced. Checking Kohlberg's criterion.
% tug_BalancedKSelectionQ                   - Checks if an imputation induces a k-balanced selection.
% tug_BalancedSelectionQ                    - Checks if an imputation induces a balanced selection.
% tug_Bankruptcy                            - Creates a modest bankruptcy game.
% tug_BelongToCoreQ                         - Checks if an imputation belongs to the core.
% tug_BestCoalToMatrix                      - Computes an equivalence matrix. 
% tug_Bsc                                   - Returns the set of most effective coalitions.
% tug_CharacteristicValues                  - Computes the characteristic values.
% tug_Coal2Dec                              - List of proper coalitions in Mathematica order.
% tug_CollectionBalancedQ                   - Checks if a collection is balanced.
% tug_CollectionOfDecreasingExcess          - Creates the collection of decreasing excesses.
% tug_Concession                            - Computes the concession vector.
% tug_ContestedGarment                      - Computes the contested garment. 
% tug_ConvexQ                               - Checks convexity.
% tug_ConvexUnanConditionQ                  - Checks convexity while relying on the unanimity coordinates.
% tug_CoreElementsQ                         - Checks if an imputation belongs to the core.
% tug_CoreQ                                 - Checks if the core is non-empty. 
% tug_CostSavings                           - Creates the cost savings game.
% tug_CriticalVal                           - Computes some critical epsilon values.
% tug_DetQuasiAvConvex                      - Determines a quasi average convex game.
% tug_DetRandCoord                          - Returns random unanimity coordinates.
% tug_DetUCoord                             - Determines the missing unanimity coordinates of size greater than 2. 
% tug_Disagreement                          - Computes the disagreement vector.
% tug_DualGame                              - Creates the dual of a Tu-game.
% tug_EpsCore                               - Computes the least core.
% tug_EqClass                               - Determines the equivalence classes from the set of most effective coalitions.
% tug_EvalSumMinCoord                       - Calculates at most (n-1) inequalities of the unanimity coordinates constraints of nonnegative sums.
% tug_ExcessValues                          - Determines the excesses.
% tug_FindPreKernel                         - Computes a pre-kernel element.
% tug_GameMonotoneQ                         - Checks on monotonicity.
% tug_Gap                                   - Computes the gap function.
% tug_GrandCoalitionLargestValueQ           - Checks if the grand coalition has largest value.
% tug_GreedyBankruptcy                      - Creates the greedy bankruptcy game.
% tug_HarsanyiDividends                     - Creates the unanimity coordinates.
% tug_ImpToVec                              - Converts an imputation to a set of vectors.
% tug_ImputationQ                           - Checks if a payoff vector is an imputation.
% tug_IntersectionOfMaxExcessSets           - Determines if the set of proper coalitions having largest excesses has an empty intersection.
% tug_IntersectionUpperLowerSetQ            - Checks if the intersection of the lower and upper set is non-empty. 
% tug_Kernel                                - Computes a kernel point.
% tug_KernelCalculation                     - Computes a or some kernel element(s).
% tug_KernelImputationQ                     - Checks if an imputation is a kernel point.
% tug_KernelVertices                        - Computes a kernel segment.
% tug_LargestAmount                         - Computes the largest amount.
% tug_LeastCore                             - Determine the least core.
% tug_LexiCenter                            - Computes the lexi center.
% tug_LowerSetIncImputationQ                - Checks if the lower set is included in the imputation set.
% tug_LowerSetQ                             - Checks if an imputation belongs to the lower set.
% tug_MKernel                               - Determines a kernel point.
% tug_MLExtension                           - Computes the multi-linear extension.
% tug_MargValue                             - Determines the marginal contribution vector.
% tug_MaxExcessBalanced                     - Checks if the maximum surpluses are balanced.
% tug_MaxExcessSets                         - Computes the set of proper coalitions having largest excesses.
% tug_MinExcessBalanced                     - Determines if the minimum surpluses are balanced.
% tug_MinUnanimityCoordinates               - Returns the minimum unanimity coordinates. 
% tug_Mnuc                                  - Determines the nucleolus.
% tug_MonotoneQ                             - Checks on monotonicity.
% tug_Nuc                                   - Computes the nucleolus.
% tug_OneNormalization                      - Creates a one normalized game.
% tug_PreKernel                             - Computes a pre-kernel element.
% tug_PreKernelEl                           - Computes a pre-kernel element.
% tug_PreKernelEqualsKernelQ                - Checks if the pre-kernel coincides with the kernel.
% tug_PreKernelQ                            - Checks if an imputation is a pre-kernel element.
% tug_PreNuc                                - Computes the pre-nucleolus.
% tug_ProperAmount                          - Computes the proper amount.
% tug_Quota                                 - Computes the quotas.
% tug_ReasonableOutcome                     - Computes the reasonable outcome.
% tug_ReasonableSet                         - Computes the reasonable set.
% tug_ScrbSolution                          - Determines the Scrb solution.
% tug_SetsToVec                             - Converts the set of most effective coalitions to a set of vectors.
% tug_ShapleyValue                          - Determines the Shapley value.
% tug_ShapleyValueML                        - Determines the Shapley value using multi-linear extension.
% tug_SmallestContribution                  - Determines the smallest contribution vector.
% tug_StrictlyConvexUnanConditionQ          - Examines the sufficient condition of convexity in terms of unanimity coordinates.
% tug_SuperAdditiveQ                        - Checks on super-additivity.
% tug_SymGameSizeK                          - Returns a special type of symmetric game.
% tug_SymGameType2                          - Returns a special type of symmetric game.
% tug_SymGameType3                          - Returns a special type of symmetric game.
% tug_SymGameType4                          - Returns a special type of symmetric game.
% tug_TalmudicRule                          - Computes the Talmudic distribution rule.
% tug_TauValue                              - Determines the Tau value.
% tug_UnanAvConvexQ                         - Checks if the coordinates satisfy the sufficient and necessary condition of average convexity.
% tug_UnanConvexQ                           - Checks if the coordinates satisfy the sufficient and necessary condition of convexity.
% tug_UnanimityCoordinates                  - Determines all unanimity coordinates of the game
% tug_UpperSetIncImputationQ                - Checks if the upper set is included in the imputation set.
% tug_UpperSetQ                             - Checks if an imputation belongs to the upper set.
% tug_UtopiaVector                          - Computes the utopia payoff. 
% tug_ValueExcess                           - Computes an objective function to compute a pre-kernel element.
% tug_VerticesCore                          - Determines the vertices of the core.
% tug_WeaklySuperAdditiveQ                  - Checks if the Tu-game is weakly super-additive.
% tug_WeightedMajority                      - Creates the weighted majority game.
% tug_ZeroMonotoneQ                         - Checks on zero-monotonicity.
% tug_ZeroNormalization                     - Creates the zero normalized game.
% tug_ZeroOneNormalization                  - Creates the zero-one normalized game.
% tug_kCover                                - Determines from the Tu-game the corresponding k-game.
%
%
%
% mat_tugames: Serial Computing
%------------------------------
% ADvalue                                   - Computes the Aumann-Dreze value.
% ANucAirportProb                           - Computes the anti-nucleolus from an airport capital cost problem.
% AP_DummyPlayer_propertyQ                  - Checks if the solution x satisfies the AP-Dummy player property.
% AP_DummyPlayers                           - Returns the player who are AP-Dummy players.
% AP_NullPlayer_propertyQ                   - Checks if the solution x satisfies the AP-Null player property. 
% AP_NullPlayers                            - Returns the players who are AP-Null players. 
% A_DummyPlayer_propertyQ                   - Checks if the solution x satisfies the A-Dummy player property. 
% A_NullPlayer_propertyQ                    - Checks if the solution x satisfies the A-Null player property. 
% A_NullPlayers                             - Returns the players who are A-Null players. 
% AdditiveQ                                 - Checks if the game v is additive.
% AllMarginalContributions                  - Computes all marginal contributions of a Tu game.
% AllMarginalContributionsResToS            - Computes all marginal worth vectors of a TU-game v restricted to coalition S.
% AllSubGames                               - Computes all subgames.
% AlmostAverageConcaveQ                     - Returns true whenever the game v is almost average concave.
% AlmostAverageConvexQ                      - Returns true whenever the game v is almost average convex.
% AlmostConcave_gameQ			    - Returns true whenever the game v is almost concave.
% AlmostConvex_gameQ			    - Returns true whenever the game v is almost convex.
% AntiCoreCoverQ                            - Checks if the anti-core cover is non-empty.
% AntiCorePlot                              - Plots the anti-core.
% AntiCoreVertices                          - Evaluates the vertices of the anti-core.
% AntiImputationVertices                    - Computes all vertices of the anti imputation set.
% AntiReduced_game_propertyQ  		    - Checks whether an imputation x satisfies the anti-reduced game property. 
% AntiUtopiaPayoff                          - Computes the anti-utopia and agreement vector.
% Anti_B0_balancedCollectionQ               - Checks the reversal of weak Kohlberg's criterion.
% Anti_BestCoalitions                       - Computes the set of less effective coalitions.
% Anti_CPCore                               - Computes the closest point of the anti-core to x.
% Anti_ChiValue                             - Computes the anti-chi-value of a TU-game v. 
% Anti_Converse_DGP_Q  			    - Checks whether an imputation x satisfies the anti-converse derived game property.
% Anti_DerivedGame  			    - Computes from (v,x,S) a modified Davis-Maschler anti-derived game vS on S at x for game v.
% Anti_Derived_game_propertyQ  		    - Checks whether an imputation x satisfies a modified anti-derived game property.
% Anti_GenGap                               - Computes the anti-generalized gap function from game v. 
% Anti_Kernel                               - Computes an anti-kernel point.
% Anti_LorenzDom                            - Checks if x anti-Lorenz dominates y in game v.
% Anti_LorenzMinACoreQ                      - Checks if x is Anti_Lorenz minimal in the anti-core of game v, i.e., x is in the Anti_Lorenz set.
% Anti_LorenzSol                            - Determines the anti-Lorenz solution of game v.
% Anti_ModPreKernel  			    - Computes from (v,x) an anti-modified pre-kernel element.
% Anti_ModPreKernel2  			    - Computes from (v,x) an anti-modified pre-kernel element.
% Anti_ModPrekernelQ  			    - Checks whether the imputation x is a modified anti-pre-kernel element of the TU-game v.
% Anti_Modiclus                             - Computes the anti modiclus of a game.
% Anti_Monotonic_Cover                      - Computes the anti-monotonic cover of game v. 
% Anti_Nucl                                 - Computes the anti nucleolus of a game.
% Anti_Nucl_llp                             - Computes the anti nucleolus of a game.
% Anti_PModPreKernel  			    - Computes from (v,x) an anti-proper-modified pre-kernel element.
% Anti_PModPrekernelQ  			    - Checks whether the imputation x is a proper modified anti-pre-kernel element of the TU-game v.
% Anti_PreKernel                            - Computes an anti-prekernel point.
% Anti_PreNucl                              - Computes the anti pre-nucleolus of game v.
% Anti_PreNucl_llp                          - Computes the anti pre-nucleolus of game v.
% Anti_PrekernelQ                           - Checks if an imputation is an anti prekernel point.
% Anti_PropModPreKernel  		    - Checks whether the imputation x is a proper modified anti-pre-kernel element of the TU-game v.	    
% Anti_TauValue                             - Computes the anti-tau-value of a TU-game v.
% Anti_Weak_balancedCollectionQ  	    - Verifies whether the set of induced coalitions is a weak_balanced collection.	    
% Anti_balancedCollectionQ                  - Checks the reversal of Kohlberg's criterion.
% Anti_kernelQ                              - Checks if an imputation is an anti kernel point.
% Anti_modiclusQ                            - Verifies whether the set of induced coalitions is a bi-balanced collection.
% B0_balancedCollectionQ                    - Checking weak Kohlberg's criterion.
% B0_balancedQ                              - Verifies whether the collection of coalitions is weakly balanced.
% BanzhafColeman                            - Computes the Banzhaf/Coleman index of a simple game sv (normalized Banzhaf value by 2^n-1).
% BanzhafOwenValue                          - Computes the Banzhaf-Owen value w.r.t. a priori unions cs.
% BanzhafPenrose                            - Computes the Banzhaf/Penrose and Banzhaf/Coleman index of a simple game sv.
% BaryCenter                                - Computes the barycenter of the core.
% BestCoalitions                            - Computes the set of most effective coalitions.
% COV_propertyQ                             - Verifies if the payoff x satisfies COV property.
% CPCore                                    - Computes the closest point of the core to x.
% CanonicalOrder                            - Orders a set of coalitions by a canonical order.
% CddAntiCoreCoverPlot                      - Plots the anti-core cover set.
% CddAntiCoreCoverVertices                  - Computes all vertices of the anti-core cover set.
% CddAntiCorePlot                           - Plots the anti-core of a game using cddmex. 
% CddAntiCoreQ                              - Checks if the anti-core exists (cddmex).
% CddAntiCoreSimplexPlot                    - Plots the anti-core using simplex projection.
% CddAntiCoreSimplexVertices                - Computes all anti-core vertices using simplex projection.
% CddAntiCoreVertices                       - Computes the vertices of the anti-core (cddmex).
% CddAntiImputationSimplexVertices          - Computes all vertices of the anti-imputation set using simplex projection.
% CddAntiImputationVertices                 - Computes all vertices of the anti-imputation set.
% CddAntiLeastCore                          - Computes the least core of game v using (cddmex).
% CddAntiLeastCoreVertices                  - Computes the vertices of the anti least core of game v (cddmex).
% CddAntiNucl                               - Computes the anti nucleolus of game v (cddmex).
% CddAntiNucl_llp                           - Computes the anti nucleolus of game v (cddmex).
% CddAntiPrenucl                            - Computes the anti pre-nucleolus of game v (cddmex).
% CddAntiPrenucl_llp                        - Computes the anti pre-nucleolus of game v (cddmex).
% CddAnti_WeberSetPlot                      - Plots the anti-Weber set of game v.
% CddAnti_WeberSetSimplexPlot               - Plots the anti-Weber set of game v projected to the simplex.
% CddBelongToLeastCoreQ                     - Checks if a payoff vector belongs to the least-core.
% CddCoreCoverPlot                          - Plots the core cover of a TU game.
% CddCoreCoverSimplexPlot                   - Plots the core cover (simplex projection).
% CddCoreCoverSimplexVertices               - Computes all vertices of the core cover (simplex).
% CddCoreCoverVertices                      - Computes all vertices of the core cover of a TU game.
% CddCoreMovie                              - Creates a movie w.r.t. the strong epsilon-cores.
% CddCorePlot                               - Plots the core of a game using cddmex. 
% CddCoreQ                                  - Checks if the core exists (cddmex).
% CddCoreSimplexMovie                       - Creates a movie w.r.t. the strong epsilon-cores (simplex projection).
% CddCoreSimplexPlot                        - Plots the core (simplex projection).
% CddCoreSimplexVertices                    - Computes the vertices of the core (simplex).
% CddCoreVertices                           - Computes the vertices of the core (cddmex).
% CddExactGame                              - Computes the exact game from v (cddmex).
% CddImputationSimplexVertices              - Computes the vertices of the imputation set (simplex).
% CddImputationVertices                     - Computes the vertices of the imputation set (cddmex).
% CddKernelCatchers                         - Draws some kernel catchers (cddmex).
% CddKernelCatchersSimplex                  - Draws some kernel catchers (simplex).
% CddLeastCore                              - Computes the least core (cddmex).
% CddLeastCoreVertices                      - Computes the least core vertices (cddmex). 
% CddLinear_Production                      - Computes from a production problem (A,mB,p) a linear production game using cddmex. 
% CddLowerSetSimplexVertices                - Computes the vertices of the lower set (simplex). 
% CddLowerSetVertices                       - Computes the vertices of the lower set (cddmex). 
% CddModiclus                               - Computes the modiclus of game v using cddmex.
% CddNucl                                   - Computes the nucleolus using the CDD solver (cddmex).
% CddPreKernel                              - Computes a pre-kernel element (cddmex).
% CddPrenucl                                - Computes the prenucleolus using the CDD solver (cddmex).
% CddPrenucl_llp                            - Computes the prenucleolus using the CDD solver (cddmex).
% CddReasonableSetSimplexVertices           - Computes the vertices of the reasonable set (simplex). 
% CddReasonableSetVertices                  - Computes the vertices of the reasonable set (cddmex). 
% CddStrongCorePlot                         - Plots a strong epsilon core. 
% CddStrongCoreSimplexPlot                  - Plots the strong epsilon core (simplex projection).
% CddTotallyBalancedQ  			    - Checks whether the core of all subgames is non-empty (cddmex).
% CddUpperSetSimplexVertices                - Computes the vertices of the upper set (simplex). 
% CddUpperSetVertices                       - Computes the vertices of the upper set (cddmex). 
% CddWeberSet                               - Computes the vertices of the Weber Set.
% CddWeberSetPlot                           - Plots the Weber set.
% CddWeberSetSimplex                        - Computes the vertices of the Weber Set (simplex).
% CddWeberSetSimplexPlot                    - Plots the Weber set (simplex).
% ChebyCenterCore                           - Computes the Cheby Center of the core of game v using MPT3.
% ChiValue                                  - Computes the chi-value of a TU-game v. This is a generalized Tau value.
% CmpConsistencyQ                           - Checks whether an imputation x satisfies the complement consistency.
% CmpRedGame                                - Computes from (v,x,S) a complement reduced game vS on S at x for game v.
% CoalitionSolidarity                       - Determines the coalition solidarity value.
% ColemanOwenValue                          - Computes the Coleman-Owen value w.r.t. a priori unions cs.
% Complement_Reduced_game                   - Computes from (v,x) all complement reduced games on S at x of game v.
% ComposeMarkets                            - Composite from a parameter set defining number of players involved in each separated market situation a market game (merged markets).
% Composition                               - Composite at least 2 TU games up to 15 to a new extended game.
% ConstantSumQ                              - Checks if the game v has constant-sum.
% Converse_CmpConsistencyQ                  - Checks whether an imputation x satisfies the converse complement consistency property.
% Converse_DGP_Q  			    - Checks whether an imputation x satisfies the converse derived game property.
% Converse_RGP_Q                            - Checks if an imputation satisfies the CRGP.
% CoreCoverQ                                - Checks if the core cover a TU game v is non-empty.
% CorePlot                                  - Plots the core.
% CoreVertices                              - Computes the vertices of the core.
% Cost_Nucl                                 - Computes the nucleolus of a cost game v using the optimization toolbox.
% Cost_PreNucl                              - Computes the pre-nucleolus of a cost game v using the optimization toolbox.
% DCP_propertyQ  			    - Checks whether the solution x satisfies the dual cover property.	    
% DFP_propertyQ  			    - Checks whether the solution x satisfies the dual floor property.	    
% DM_AntiReduced_game  			    - Computes from (v,x) all anti-reduced games on S at x of game v.
% DM_Anti_Derived_game  		    - Computes from (v,x) a modified Davis-Maschler anti-reduced game vS on S at x for game v.
% DM_Derived_game  			    - Computes from (v,x) a modified Davis-Maschler reduced game vS on S at x for game v.
% DM_Reduced_game                           - Computes all Davis-Maschler reduced games.
% DM_TwoPersonGame                          - Computes from (v,x) all reduced two-person games.
% DM_TwoReduced_game  			    - Computes from (v,x) all single and two-person reduced games on S at x of game v.
% DRP_propertyQ  			    - Checks whether the solution x satisfies the dual replication property.	    
% DecomposableQ                             - Checks whether the game v is decomposable w.r.t. the coalition structure cs. 
% DecomposeGame                             - Computes the unique decomposition of a TU-game.
% DecomposeInPositiveGames                  - Decomposes a TU game v into the difference of two positive games (convex games).
% DeeganPackel                              - Computes the Deegan-Packel index from the set of minimal winning coalitions. 
% DeeganPackel_SV                           - Computes the Deegan-Packel index from a simple game to construct the set of minimal winning coalitions. 
% DerivedCostMatrix                         - Computes from a cost matrix and a partition of the player set N the corresponding derived cost matrix. 
% DerivedGame  				    - Computes from (v,x,S) a modified Davis-Maschler derived game vS on S at x for game v.
% Derived_game_propertyQ  		    - Checks whether an imputation x satisfies a modified derived game property.
% DiscShapleyValue                          - Computes the discounted Shapley value.
% DualCover  				    - Computes the maximum characteristic values from the primal or dual game.
% DualFloor  				    - Computes the minimum characteristic values from the primal or dual game.
% Dual_Cover_game  			    - Computes from (v,x) a modified Davis-Maschler reduced game vS on S at x for game v.
% Dual_Cover_propertyQ  		    - Checks whether an imputation x satisfies a modified reduced game property	    
% Dual_Floor_game			    - Computes from (v,x) a modified Davis-Maschler anti-reduced game vS on S at x for game v.
% Dual_Floor_propertyQ			    - Checks whether an imputation x satisfies a modified anti-reduced game property.
% DummyPlayer_propertyQ                     - Checks the dummy player property. 
% DummyPlayers                              - Returns the list of dummy players of game v. 
% DuttaRay  				    - Computes the Dutta-Ray solution for convex games.
% EANSCValue                                - Computes the Equal Allocation of Non-Separable Contribution/Cost Value.
% ECCoverGame  				    - Computes from (v,x) an excess comparability cover of game v.
% ECFloorGame  				    - Computes from (v,x) an excess comparability floor of game v.
% ECGValue                                  - Computes the Equal Collective Gains value of a TU-game v. 
% EC_DGP_Q  				    - Checks whether the solution x satisfies excess comparability for each derived game.
% EC_RGP_Q  				    - Checks whether the solution x satisfies excess comparability for each reduced game.
% EC_propertyQ  			    - Checks whether the solution x satisfies excess comparability.	    
% EPSDValue                                 - Computes the egalitarian proportional surplus division value of a individually positive TU-game.
% ESD                                       - Computes the equal surplus division of a TU-game.
% EqDistDividends                           - Computes the equally distributed dividends of coalitions to which players belong, i.e., Shapley-value, of a TU-game v.
% EssentialConstSumQ                        - Checks if v is an essential constant-sum game.
% EssentialQ                                - Checks if the game v is essential.
% ExtShapleyValue                           - Computes the extended Shapley-value.
% FindPartition                             - Tries to find a partition and anti partition w.r.t. payoff vector x.
% FlatQ                                     - Checks if the game v is flat.
% Gap                                       - Determines the gap function.
% GatelyValue				    - Computes the Gately point of an essential game v.
% GenGap                                    - Computes the generalized gap function from game v.
% GetAirPortProb                            - Computes a pseudo-random airport cost allocation problem of size m+1.
% GetMCNetRules                             - Transforms a cell array of the MC-nets representation of a TU game into a structure array.
% GetMarketGame                             - Determines from a random generated game the corresponding market game.
% GetPlayersCharacter                       - Determines from the set of players of the weighted majority game the characters Step, Sum and Null Player.
% GetPlayersCharacter                       - Determines from the set of players of the weighted majority game the characters Step, Sum and Null Player.
% GetProbDist                               - Tries to find for the game v the corresponding probability distribution over the players' orderings w.r.t. pre-imputation x.
% HMS_AntiReduced_game  		    - Computes from (v,x) all Hart/Mas-Colell anti-reduced games on S at x of game v.	    
% HMS_Anti_Derived_game  		    - Computes from (v,x,S) a modified Hart-Mas-Colell anti-reduced game vS on S at x for game v.	    
% HMS_Derived_game  			    - Computes from (v,x,S) a modified Hart-Mas-Colell reduced game vS on S at x for game v.
% HMS_DervGame  			    - Computes from (v,x,S) a modified Hart-Mas-Colell derived game vS on S at x for game v.	    
% HMS_ImputSavingReducedGame                - Computes from (v,x) all Hart/Mas-Colell ISR games.
% HMS_RedGame  				    - Computes from (v,x,S) a Hart-Mas-Colell reduced game vS on S at x for game v.
% HMS_Reduced_game                          - Creates all Hart/Mas-Colell reduced games. 
% HMS_TwoReduced_game  			    - Computes from (v,x) all Hart/Mas-Colell singleton and two-person reduced games on S at x of game v.
% HarsanyiValue                             - Computes a Harsanyi-value of a TU-game v.
% ISRG_propertyQ                            - Checks whether an imputation x satisfies the ISR game property.
% ImpSetEqsLwsQ                             - Checks if the imputation set coincides with the lower set.
% ImputSavingReducedGame                    - Computes from (v,x) all imputation saving reduced games.
% ImputationVertices                        - Computes the vertices of the imputation set.
% InessGame                                 - Computes the inessential game from a payoff vector.
% InteractionSets                           - Determines a system of interaction sets.
% IrredAntiCore                             - Computes from a cost matrix the corresponding extreme points of the irreducible anti-core of the associated m.c.s.t. game. 
% IrredCostMatrix                           - Computes from a cost matrix and a solution tree the irreducible cost matrix. 
% Johnston                                  - Computes the Johnston power index from the set of winning coalitions. 
% Kernel                                    - Computes a kernel point using optimization toolbox.
% KrEqsPrkQ                                 - Checks if the kernel is equal to the pre-kernel.
% LED  					    - Computes the large excess difference w.r.t. the payoff x.
% LED_propertyQ  			    - Checks whether the solution x satisfies large excess difference property.	    
% LS_Nucl                                   - Computes the least square nucleolus of a game.
% LS_PreNucl                                - Computes the least square pre-nucleolus of a game.
% LeastCore                                 - Computes the least core using optimization toolbox.
% LeastCoreVertices                         - Computes the least core vertices. 
% LedcoconsQ  				    - Checks whether an imputation x satisfies large excess difference converse consistency.
% Ledcons_propertyQ  			    - Checks whether an imputation x satisfies the ledcons property 
% LexMaxMinWin                              - Determines the lexicographical maximal coalition from the clm represented as a cell, matrix, or array of integers.
% LexOrder                                  - Sorts a set of coalitions represented as a cell, matrix, or array of integers into the corresponding lexicographical order.
% LorenzDom                                 - Checks if x Lorenz dominates y in game v.
% LorenzMaxCoreQ                            - Checks if x is Lorenz maximal in the core of game v, i.e., x is in the Lorenz set.
% LorenzSet                                 - Determines the Lorenz set of game v.
% LorenzSol  				    - Determines the Lorenz solution of game v.
% MIMC                                      - Computes the vector of minimum increase in players marginal contribution when they leave the grand coalition.
% MLextension                               - Computes the multi-linear extension.
% MMExcess  				    - Computes the minimal and maximal excess vector of game v and its dual.
% MTRCostMatrix                             - Computes from a cost matrix and a solution tree the cost matrix of a minimal spanning tree. 
% MarketGameQ                               - Checks whether the game v is a market game.
% MaxConsistencyQ                           - Checks whether an imputation x satisfies maximal consistency.
% MinimalRep                                - Computes from a simple game v and a threshold th of the minimal representation of an homogeneous weighted majority game. 
% ModDeeganPackel                           - Computes the modified Deegan-Packel index from the set of winning coalitions. 
% ModDeeganPackel_SV                        - Computes the Deegan-Packel index from a simple game to construct the set of minimal winning coalitions. 
% ModHoller  				    - Computes a modified Holler index from the set of winning coalitions.
% ModPGI                                    - Computes the modified public good index from the set of minimal winning coalitions.
% ModPGI_SV                                 - Computes the modified public good index from a simple game to determine the set of minimal winning coalitions. 
% ModPreKernel  			    - Computes from (v,x) a modified pre-kernel element.	    
% ModPrekernelQ  			    - Checks whether the imputation x is a modified pre-kernel element of the TU-game v.	    
% Modiclus                                  - Computes the modiclus of a game.
% MyersonValue                              - Computes the Myerson value of a Tu game.
% NetworkBanzhaf                            - Computes the network Banzhaf power index from the set of winning coalitions of a network E while imposing a threshold of th. 
% NetworkCenterSol                          - Computes the center solution from the minimal representation of a homogeneous network weighted majority game.
% NetworkDeeganPackel                       - Computes the network Deegan-Packel index from the set of winning coalitions of a network E while imposing a threshold of th. 
% NetworkJohnston                           - Computes the network Johnston power index from the set of winning coalitions of a network E while imposing a threshold of th. 
% NetworkMajorityGame                       - Computes a network majority TU game (simple game).
% NetworkMinimalRep                         - Computes from the set of edges, threshold th and the weights w_vec the minimal homogeneous representation of a homogeneous network weighted majority game. 
% NetworkModDeeganPackel                    - Computes the network modified Deegan-Packel index from the set of winning coalitions of a network E while imposing a threshold of th. 
% NetworkModPGI                             - Computes the modified network public good index from the set of minimal winning coalitions of a network.
% NetworkPGI                                - Computes the network public good index from the set of minimal winning coalitions of a network.
% NetworkShapleyShubik                      - Computes the network Shapley-Shubik power index from the set of winning coalitions of a network E while imposing a threshold of th. 
% NucAirportProb                            - Computes the nucleolus from an airport capital cost problem.
% NullPlayer_propertyQ                      - Verifies if x satisfies the null player property.
% NullPlayers                               - Returns the list of null players of game v. 
% One_Normalization                         - Computes from the game v the corresponding one-normalized game.
% OwenValue                                 - Computes the Owen value.
% PDValue                                   - Computes the proportional division value of a individually positive TU-game.
% PGI                                       - Computes the public good index from the set of minimal winning coalitions.
% PGI_SV                                    - Computes the public good index from a simple game to determine the set of minimal winning coalitions. 
% PModPreKernel  			    - Computes from (v,x) a proper modified pre-kernel element.	    
% PModPrekernelQ  			    - Checks whether the imputation x is a proper modified pre-kernel element of the TU-game v.
% PRP_propertyQ  			    - Checks whether the solution x satisfies the primal replication property.	    
% PS_GameBasis                              - Computes the basis for the class of PS games.
% PartitionGameQ                            - Verifies if (th,w_vec) induces a partition game.
% PartitionPlySet                           - Partitions the set of players of the weighted majority game into character Sum, Step, and Null-Player. 
% PartitionSA                               - Computes a partition of S w.r.t. a hypergraph communication situation.
% PartitionSL                               - Computes a partition of S w.r.t. a communication situation.
% PermutationGame                           - Computes from an assignment matrix the permutation game.
% PlayersCharacter                          - Partitions the set of players of the weighted majority game into the character Sum, Step, and Null-Player.
% PlotCostGraph                             - Plots from a cost matrix the associated cost spanning graph. 
% PlyEqFirstStep                            - Returns the last player that is equivalent to the first step.
% PositionValue                             - Computes the position value.
% Potential                                 - Determines the potential of a TU game (recursive).
% PowerSet                                  - Computes all subsets from a set representation.
% PreKernel                                 - Computes a prekernel element.
% PreNucl                                   - Computes the prenucleolus using optimization toolbox.
% PreNucl2                                  - Computes the prenucleolus using optimization toolbox.
% PreNucl_llp                               - Computes the prenucleolus using optimization toolbox.
% PreNucl_mod                               - Computes the pre-nucleolus of game v using the optimization toolbox and a spanning method.
% PreNucl_mod2                              - Computes the pre-nucleolus of game v using the optimization toolbox and an alternative spanning method.
% PreNucl_mod3                              - Computes the pre-nucleolus of game v using the optimization toolbox and an alternative spanning method.
% PreNucl_mod4                              - Computes the pre-nucleolus of game v using the optimization toolbox and an alternative spanning method.
% PrekernelQ                                - Checks if an imputation is a pre-kernel point.
% PrenuclQ                                  - Checks if an imputation is the pre-nucleolus using Kohlberg's criterion.
% PrkEqPnQ                                  - Checks whether the pre-kernel of a game v is a singleton.
% PrkEqsModPrkQ  			    - Checks whether a pre-kernel element is also an element of the modified as well as proper modified pre-kernel	    
% PropModPreKernel  			    - Computes from (v,x) a proper modified pre-kernel element from the dual cover game.
% PropNucl                                  - Computes the proportional nucleolus.
% PropPreNucl                               - Computes the proportional pre-nucleolus.
% ProportionalGame                          - Computes from (q,co) a proportional game.
% REAS_LED_DCGame  			    - Verifies that x is a reasonable vector of game v, then the shifted  ducal cover game satisfies LED w.r.t. the replicated vector (x,x).
% REAS_propertyQ  			    - Checks if the vector x satisfies the reasonableness on both sides
% REC_propertyQ  			    - Checks whether the solution x satisfies reverse excess comparability.	    
% RE_RGP  				    - Checks whether an imputation x is reasonable from both sides for all reduced games.
% ReasSetEqsUpsQ                            - Checks if the reasonable set coincides with the upper set.
% Reconfirmation_propertyQ                  - Checks the RCP.
% RedGame                                   - Creates a Davis-Maschler reduced game.
% Reduced_game_propertyQ                    - Checks the RGP.
% ReverseMCNetsRep                          - Reverse the MC-nets (concise) representation of a cooperative game into a full representation.
% SDCP_propertyQ  			    - Checks whether the solution x satisfies a strong dual cover property.
% SDFP_propertyQ  			    - Checks whether the solution x satisfies a strong dual floor property.
% SD_ShapleyValue  			    - Computes the surplus division Shapley value. 
% SED  					    - Computes the small excess difference w.r.t. the payoff x.
% SED_propertyQ  			    - Checks whether the solution x satisfies small excess difference property.	    
% SedcoconsQ  				    - Checks whether an imputation x satisfies small excess difference converse consistency.
% Sedcons_propertyQ  			    - Checks whether an imputation x satisfies the sedcons property.
% ShapAirPortMod2                           - Computes the Shapley value from an airport capital cost problem (modified).
% ShapleyAirportProb                        - Computes the Shapley value from an airport capital cost problem.
% ShapleyQ                                  - Checks if the imputation x is a Shapley value of game v.
% ShapleyValue                              - Computes the Shapley value (potential).
% ShapleyValueLB                            - Computes the Shapley value from the linear basis.
% ShapleyValueM                             - Computes the Shapley value based on all marginal contributions.
% ShapleyValueML                            - Computes the Shapley value using multi-linear extension. 
% ShapleyValuePot                           - Computes the Shapley value and potential.
% SolidarityPGI                             - Computes the solidarity Holler index w.r.t. a priori unions cs. 
% SolidarityShapleyValue                    - Determines the solidarity Shapley value. 
% SolidarityValue                           - Determines the solidarity value. 
% SortMg                                    - Sorts a sub/power set w.r.t. its cardinality.
% SplitSimpleGame                           - Splits a simple game with winning players into sub-games with and without winning players.
% StandardSolution                          - Determines the standard solution.
% StrConverse_DGP_Q  			    - Checks whether an imputation x satisfies the strong converse derived game property.
% StrConverse_RGP_Q                         - Checks the strong RGP.
% StrLedcoconsQ  			    - Checks whether an imputation x satisfies satisfies strong large excess difference converse consistency.	    
% StrSedcoconsQ  			    - Checks whether an imputation x satisfies satisfies strong small excess difference converse consistency (SEDCOCONS).	    
% SubCoalitions                             - Computes the power set (subsets) from an array.
% SubDual                                   - Determines the dual of a subgame.
% SubGame                                   - Creates a subgame.
% SubSets                                   - Creates all subsets of super set.
% Sum_Marg_Contributions                    - Returns 1 whenever for a coalition the sum of marginal contributions is positive.
% SuperAddSolQ                              - Checks if the vector x is an element of a super additive solution of the game v.
% SuperSets                                 - Computes the super-sets of set S. 
% Talmudic_Rule                             - Computes the Talmudic rule.
% TauValAirportProb                         - Computes the tau value from an airport capital cost problem.
% TauValue                                  - Computes the Tau value.
% UnionStableBasis                          - Determines a basis of a union stable system.
% UpperPayoff				    - Computes the upper and minimum claim vector of game v.
% WSysKernelQ                               - Checks the correctness of a kernel vector w.r.t. a weight system.
% WSysNuclQ                                 - Checks the correctness of the nucleolus w.r.t. a weight system.
% WSysPreKernelQ                            - Checks the correctness of a pre-kernel vector w.r.t. a weight system.
% WSysPreNuclQ                              - Checks the correctness of the pre-nucleolus w.r.t. a weight system.
% WeakReduced_game_propertyQ  		    - Checks whether an imputation satisfies the weak reduced game property.
% Weak_balancedCollectionQ                  - Checking weak Kohlberg's criterion.
% Weak_balancedCollectionQ                  - Checking weak Kohlberg's criterion.
% WedgeProdGame                             - Computes from an MC-nets (concise) representation a cooperative production game (wedge production game).
% ZeroOne_Normalization                     - Creates a zero-one normalized game.
% additive_game                             - Creates an additive game.
% admissibleGame                            - Computes a symmetric compromise admissible game.
% airport_costgame                          - Computes from an airport problem the associated airport cost game.
% airport_game                              - Computes from an airport problem the associated savings game.
% airport_profit                            - Computes from a cost and benefit vector the associated surplus game.
% alphaVector                               - Computes recursively an alpha vector from the positive core.
% anti_coreQ  				    - Checks the existence of the anti-core of game v.
% anti_partition                            - Computes from a partition its anti partition.
% apex_game                                 - Creates an apex game.
% apu_PGI                                   - Computes the Holler index w.r.t. a priori unions cs. 
% apu_SolidarityValue                       - Determines the solidarity value w.r.t. a priori unions.
% assignment_game                           - Creates an assignment game.
% average_concaveQ                          - Returns true whenever the game v is average-concave.
% average_convexQ                           - Checks the Tu-game on average convexity.
% average_excess                            - Computes the average excess of game v.
% balancedCollectionQ                       - Checking Kohlberg's criterion.
% balancedCoverQ                            - Checks whether the characteristic function v is equal to a balanced cover.
% balancedQ                                 - Verifies whether the collection of coalitions is balanced.
% bankruptcy_airport                        - Computes from a bankruptcy problem the airport surplus game.
% bankruptcy_game                           - Creates a bankruptcy game.
% banzhaf                                   - Computes the Banzhaf value.
% basis_coordinates                         - Determines the basis coordinates of a Tu game.
% basis_game                                - Determines bases games.
% belongToAllSubCores                       - Checks whether all projections of an imputation x are a member of an associated core of a subgame.
% belongToAntiCoreQ                         - Checks if a payoff vector belongs to the anti-core.
% belongToCoreQ                             - Checks if a payoff vector belongs to the core.
% belongToImputationSetQ                    - Checks if a payoff vector belongs to imputation set.
% belongToLeastCoreQ                        - Checks if a payoff vector belongs to the least core.
% belongToLowerSetQ                         - Checks if a payoff vector belongs to lower set.
% belongToUpperSetQ                         - Checks if a payoff vector belongs to upper set.
% belongToWeberSetQ                         - Checks if the imputation x belongs to the Weber set using MPT3.
% bidding_collusion_game                    - Determines a bidding collusion game from a bid vector.
% bint_AssignmentGame                       - Creates an assignment game (bintprog).
% bs_PreKernel                              - Computes from (v,x) a pre-kernel element using MATLAB's backslash instead of pinv.
% cardinality_game                          - Assigns zero to a coalition of size<=k<n, otherwise its cardinality.
% cardinality_game2                         - Assigns a zero to a coalition of size<=k<n otherwise its cardinality times 100.
% center_solution                           - Computes the center solution from the minimal representation of a homogeneous weighted majority game.
% cfr_Anti_Kernel                           - Computes an anti kernel element with coalition formation restrictions.
% cfr_Anti_Nucl                             - Computes the anti nucleolus of game v with coalition formation restrictions.
% cfr_Anti_PreKernel                        - Computes an anti-pre-kernel element with coalition formation restrictions.
% cfr_Anti_PreNucl                          - Computes the anti pre-nucleolus of game v with coalition formation restrictions.
% cfr_Anti_Weak_balancedCollectionQ  	    - Verifies whether the set of induced coalitions is a weak_balanced collection with coalition formation restrictions.
% cfr_Anti_balancedCollectionQ  	    - Verifies whether the set of induced coalitions is an anti balanced collection with coalition formation restrictions.
% cfr_Kernel                                - Computes a Kernel element with coalition formation restrictions.
% cfr_PreKernel                             - Computes a Pre-Kernel element with coalition formation restrictions.
% cfr_PreNucl                               - Computes the pre-nucleolus of game v with coalition formation restrictions.
% cfr_balancedCollectionQ  		    - Verifies whether the set of induced coalitions is a balanced collection with coalition formation restrictions.
% cfr_nucl                                  - Computes the nucleolus of game v with coalition formation restrictions.
% cfr_weak_balancedCollectionQ  	    - Verifies whether the set of induced coalitions is a weakly_balanced collection with coalition formation restriction.
% clToMatlab                                - Computes the unique integer representation of coalitions.
% clp_kernel                                - Computes a kernel point using the CLP solver.
% clp_weightedKernel                        - Computes a weighted kernel point using the CLP solver.
% cls_kernel                                - Computes a kernel point using the CLS solver.
% cls_weightedKernel                        - Computes a weighted kernel point using the CLS solver.
% coeff_linearbasis                         - Determines the coefficients (dividends) of a linear basis from a TU game.
% complementary_basis                       - Computes the complementary basis of the n-person TU game space.
% complementary_dividends                   - Computes the complementary dividends.
% complementary_game                        - Generates a producer and buyer game.
% compromiseAdmissibleQ                     - Checks if the core cover a TU game v is non-empty.
% compromiseAntiAdmissibleQ                 - Checks if the anti-core cover a TU game v is non-empty.
% compromiseStableQ                         - Checks if the game is compromise stable.
% concave_gameQ                             - Checks the concavity of a Tu-game.
% contentment                               - Computes the contentment vector of game v w.r.t. x.
% convex_gameQ                              - Checks the convexity of a Tu-game.
% coreQ                                     - Checks the non-emptiness of the core.
% cp_kernel                                 - Computes a kernel element using the cone programming solver of MATLAB's Optimization Toolbox.
% cp_prekernel                              - Computes a pre-kernel element using the cone programming solver of MATLAB's Optimization Toolbox.
% cplex_AntiNucl                            - Computes the anti nucleolus of game v using the CPLEX solver.
% cplex_AntiNucl_llp                        - Computes the anti nucleolus of game v using the CPLEX solver.
% cplex_AntiPreNucl                         - Computes the anti prenucleolus using the CPLEX solver.
% cplex_AntiPreNucl_llp                     - Computes the anti prenucleolus using the CPLEX solver.
% cplex_AntiPreNucl_mod                     - Computes the anti pre-nucleolus of game v using cplexmex (fast).
% cplex_AntiPreNucl_mod2                    - Computes the anti pre-nucleolus of game v using cplexmex (fast/method 2).
% cplex_AntiPreNucl_mod3                    - Computes the anti pre-nucleolus of game v using cplexmex (fast/method 3).
% cplex_AntiPreNucl_mod4                    - Computes the anti pre-nucleolus of game v using cplexmex (fast/method 4).
% cplex_AssignmentGame                      - Creates an assignment game using the CPLEX solver.
% cplex_LeastCore                           - Computes the least core using cplexmex.
% cplex_alphaVector  			    - Computes recursively an alpha vector from the positive core of game v using cplexmex.
% cplex_cfr_nucl                            - Computes the nucleolus of game v with coalition formation restrictions using the CPLEX solver.
% cplex_cs_nucl                             - Computes the nucleolus of game v w.r.t. coalition structure cs using the CPLEX solver.
% cplex_exact_game                          - Computes the exact game from v using the CPLEX solver.
% cplex_flow_game                           - Computes from a flow problem a TU flow game (CPLEX).
% cplex_kernel                              - Computes a kernel point using the CPLEX solver.
% cplex_linprog_game                        - Computes from a production matrix (A,H,mB,mD,p) a linear programming game using cplexmex.
% cplex_modiclus                            - Computes the modiclus of game v using cplexmex.
% cplex_nucl                                - Computes the nucleolus using the CPLEX solver.
% cplex_nucl_llp                            - Computes the nucleolus using the CPLEX solver.
% cplex_prekernel                           - Computes a prekernel point using the CPLEX solver.
% cplex_prenucl                             - Computes the prenucleolus using the CPLEX solver.
% cplex_prenucl_llp                         - Computes the prenucleolus using the CPLEX solver.
% cplex_prenucl_mod                         - Computes the pre-nucleolus of game v using cplexmex (fast).
% cplex_prenucl_mod2                        - Computes the pre-nucleolus of game v using cplexmex (fast/method 2). 
% cplex_prenucl_mod3                        - Computes the pre-nucleolus of game v using cplexmex (fast method 3).
% cplex_prenucl_mod4                        - Computes the pre-nucleolus of game v using cplexmex (fast/method 4). 
% cplex_weightedKernel                      - Computes a weighted kernel point using the CPLEX solver.
% cplex_weightedNucl                        - Computes a weighted nucleolus using the CPLEX solver.
% cplex_weightedNucl_llp                    - Computes a weighted nucleolus using the CPLEX solver.
% cplex_weightedPreKernel                   - Computes a weighted prekernel point using the CPLEX solver.
% cplex_weightedPreNucl                     - Computes a weighted prenucleolus using the CPLEX solver.
% cplex_weightedPreNucl_llp                 - Computes a weighted prenucleolus using the CPLEX solver.
% critical_value1                           - Computes the biggest gain of any group of players.
% critical_value2                           - Computes a critical value w.r.t. the strong epsilon-core.
% critical_value_star                       - Computes a critical value which contains the intersection of the imputation and reasonable set
% cs_Anti_Kernel                            - Computes an anti-kernel element from a coalition structure.
% cs_Anti_Nucl                              - Computes the anti nucleolus of game v w.r.t. coalition structure cs.
% cs_Anti_PreKernel                         - Computes an anti-pre-kernel element from a coalition structure.
% cs_Anti_PreNucl                           - Computes the anti pre-nucleolus of game v w.r.t. coalition structure cs.
% cs_Anti_Weak_balancedCollectionQ          - Verifies whether the set of induced coalitions is a weak_balanced collection w.r.t. the coalition structure cs. Checking Kohlberg's criterion.
% cs_Anti_balancedCollectionQ               - Verifies whether the set of induced coalitions is an anti balanced collection w.r.t. the coalition structure cs.  Checking Kohlberg's criterion.
% cs_Banzhaf                                - Computes the Banzhaf value w.r.t. a communication situation.
% cs_GetPrk                                 - Computes a pre-kernel element from each possible partition of N.
% cs_Kernel                                 - Computes a kernel element from a coalition structure.
% cs_PreKernel  			    - Computes from (v,x) a pre-kernel element w.r.t. the coalition structure cs.
% cs_PreNucl                                - Computes the pre-nucleolus of game v w.r.t. coalition structure cs.
% cs_Weak_balancedCollectionQ               - Verifies whether the set of induced coalitions is a weakly_balanced collection w.r.t. the coalition structure cs. Checking Kohlberg's criterion.
% cs_balancedCollectionQ                    - Verifies whether the set of induced coalitions is a balanced collection w.r.t. the coalition structure cs. Checking Kohlberg's criterion.
% cs_nucl                                   - Computes the nucleolus of game v w.r.t. coalition structure cs.
% cvx_kernel                                - Computes a kernel point using the CVX solver. 
% cvx_prekernel                             - Computes a prekernel point using the CVX solver. 
% cvx_weightedKernel                        - Computes a weighted kernel point using the CVX solver.
% cvx_weightedPreKernel                     - Computes a weighted prekernel point using the CVX solver.
% diffOperator                              - Computes the difference operator of the game w.r.t. coalition T.
% disagreement                              - Computes the disagreement vector of game v.
% dual_game                                 - Creates the dual of a Tu-game.
% equal_treatmentQ                          - Checks if a vector x satisfies ETP.
% essentialSet                              - Computes the set of essential coalitions.
% exact_game                                - Computes the exact game from v using Matlab's Optimization toolbox.
% exact_gameQ  				    - Checks whether game v is an exact game using Matlab's Optimization toolbox.
% excess                                    - Determines the excesses w.r.t. a payoff vector. 
% feasible_dividends                        - Computes a collection of feasible dividends.
% flow_game                                 - Computes from a flow problem a TU flow game using the optimization toolbox.
% flow_probMinCut                           - Computes from a flow problem a minimal cut.
% formatPowerSet  			    - Formats the Matlab cell output that contains the representation of coalitions into matrix form.
% gameToMama                                - Converts a TU-game into Mathematica representation.
% gameToMatlab                              - Converts a Tu-game into Matlab representation.
% game_two                                  - Constructs a 2-game from the coalition size 2 and number of players.
% game_Wsys                                 - Creates a set of games from an asymmetric weight system (all types).
% game_basis                                - Computes a game basis of the n-person TU game space.
% game_space                                - Computes the game space which replicates a payoff as a pre-kernel element.
% genUnionStable                            - Creates a union stable system.
% getCOV                                    - Computes from a sample of observations obs and for n-assets the covariance matrix V of a portfolio with indefinite risk-return relationship. 
% getCOV2                                   - Computes from a sample of observations obs and for n-assets the covariance matrix V of a portfolio with negative risk-return relationship.          
% getCOV3                                   - Computes from a sample of observations obs and for n-assets the covariance matrix V of a portfolio with positive risk-return relationship. 
% getMinimalWinning                         - Computes from a simple game the minimal winning coalitions.
% getPSgame                                 - Computes a PS game from the PS game basis.
% getSymCostMatrix                          - Computes a symmetric cost matrix from the cardinality of the player set and a upper bound value to specify the range from which the random number are drawn. 
% getgame                                   - Creates a Tu-game from the unanimity coordinates.
% glpk_AntiNucl                             - Computes the anti nucleolus of game v using the GLPK solver.
% glpk_AntiNucl_llp                         - Computes the anti nucleolus of game v using the GLPK solver.
% glpk_AntiPreNucl                          - Computes the anti pre-nucleolus using the GLPK solver.
% glpk_AntiPreNucl_llp                      - Computes the anti pre-nucleolus using the GLPK solver.
% glpk_alphaVector                          - Computes recursively an alpha vector from the positive core.
% glpk_cfr_nucl                             - Computes the nucleolus of game v with coalition formation restrictions using the GLPK solver.
% glpk_cs_nucl                              - Computes the nucleolus of game v w.r.t. coalition structure cs using the GLPK solver.
% glpk_exact_game                           - Computes the exact game from v using the GLPK solver.
% glpk_flow_game                            - Computes from a flow problem a TU flow game (GLPK).
% glpk_kernel                               - Computes a kernel point using the GLPK solver. 
% glpk_linprog_game                         - Computes from a production matrix (A,H,mB,mD,p) a linear programming game using glpkmex.
% glpk_modiclus                             - Computes the modiclus of game v using glpkmex.
% glpk_nucl                                 - Computes the nucleolus using the GLPK solver.
% glpk_nucl_llp                             - Computes the nucleolus using the GLPK solver.
% glpk_prekernel                            - Computes a prekernel point using the GLPK solver. 
% glpk_prenucl                              - Computes the prenucleolus using the GLPK solver.
% glpk_prenucl_llp                          - Computes the prenucleolus using the GLPK solver.
% glpk_weightedKernel                       - Computes a weighted kernel point using the GLPK solver.
% glpk_weightedNucl                         - Computes a weighted nucleolus using the GLPK solver.
% glpk_weightedNucl_llp                     - Computes a weighted nucleolus using the GLPK solver.
% glpk_weightedPreKernel                    - Computes a weighted prekernel point using the GLPK solver.
% glpk_weightedPreNucl                      - Computes a weighted prenucleolus using the GLPK solver.
% glpk_weightedPreNucl_llp                  - Computes a weighted prenucleolus using the GLPK solver.
% grMaxFlowGame                             - Computes from a flow problem a TU flow game.
% greedy_bankruptcy                         - Creates the greedy bankruptcy game.
% gurobi_AntiNucl                           - Computes the anti nucleolus of game v using the GUROBI solver.
% gurobi_AntiNucl_llp                       - Computes the anti nucleolus of game v using the GUROBI solver.
% gurobi_AntiPreNucl                        - Computes the anti prenucleolus using the GUROBI solver.
% gurobi_AntiPreNucl_llp                    - Computes the anti prenucleolus using the GUROBI solver.
% gurobi_AssignmentGame                     - Creates an assignment game using the GUROBI solver.
% gurobi_alphaVector                        - Computes recursively an alpha vector from the positive core.
% gurobi_cfr_nucl                           - Computes the nucleolus of game v with coalition formation restrictions using the GUROBI solver.
% gurobi_cs_nucl                            - Computes the nucleolus of game v w.r.t. coalition structure cs using the GUROBI solver.
% gurobi_exact_game                         - Computes the exact game from v using the GUROBI solver.
% gurobi_flow_game                          - Computes from a flow problem a TU flow game (GUROBI).
% gurobi_kernel                             - Computes a kernel point using the GUROBI solver. 
% gurobi_linprog_game                       - Computes from a production matrix (A,H,mB,mD,p) a linear programming game using gurobimex.
% gurobi_modiclus                           - Computes the modiclus of game v using the GUROBI.
% gurobi_nucl                               - Computes the nucleolus using the GUROBI solver.
% gurobi_nucl_llp                           - Computes the nucleolus using the GUROBI solver.
% gurobi_prekernel                          - Computes a prekernel point using the GUROBI solver. 
% gurobi_prenucl                            - Computes the prenucleolus using the GUROBI solver.
% gurobi_prenucl_llp                        - Computes the prenucleolus using the GUROBI solver.
% gurobi_weightedKernel                     - Computes a weighted kernel point using the GUROBI solver.
% gurobi_weightedNucl                       - Computes a weighted nucleolus using the GUROBI solver.
% gurobi_weightedNucl_llp                   - Computes a weighted nucleolus using the GUROBI solver.
% gurobi_weightedPreKernel                  - Computes a weighted prekernel point using the GUROBI solver.
% gurobi_weightedPreNucl                    - Computes a weighted prenucleolus using the GUROBI solver.
% gurobi_weightedPreNucl_llp                - Computes a weighted prenucleolus using the GUROBI solver.
% harsanyi_dividends                        - Determines the the unanimity coordinates.
% holler                                    - Computes the Holler index.
% homogeneous_representationQ               - Checks if the weighted majority game possesses a homogeneous representation.
% hsl_prekernel                             - Computes a prekernel point using HSL solvers. 
% hsl_weightedPreKernel                     - Computes a weighted prekernel point using HSL solvers.
% hypergraphQ                               - Checks whether the system is a hypergraph communication situation.
% interest_game                             - Computes from an interest problem the corresponding game.
% intersection_basis                        - Computes the intersection basis of the n-person TU game space.
% ipopt_kernel                              - Computes a kernel point using the IPOPT solver. 
% ipopt_prekernel                           - Computes a prekernel point using the IPOPT solver.
% ipopt_weightedKernel                      - Computes a weighted kernel point using the IPOPT solver.
% ipopt_weightedPreKernel                   - Computes a weighted prekernel point using the IPOPT solver.
% ireffQ                                    - Checks if a payoff satisfies IR as well as the Eff property.
% jury_game                                 - Computes from a quota and the number of jurors a simple game.
% k_Converse_RGP_Q                          - Checks if an imputation satisfies the k-CRGP.
% k_Reconfirmation_propertyQ                - Checks the k-RCP.
% k_Reduced_game_propertyQ                  - Checks the k-RGP.
% k_StrConverse_RGP_Q                       - Checks the strong k-CRGP. 
% k_anticover                               - Determines from the Tu-game the corresponding anti k-game.
% k_concaveQ                                - Checks k-concavity of the Tu-game.
% k_convexQ                                 - Checks k-convexity of the Tu-game.
% k_cover                                   - Determines from the Tu-game the corresponding k-game.
% kernelQ                                   - Checks if an imputation is a kernel point.
% landlord                                  - Computes a production game arising from l-landlords and t-tenants.
% lin_prekernel                             - Computes a prekernel point using optimization toolbox.
% lin_weightedPreKernel                     - Computes a weighted prekernel point using optimization toolbox.
% linear_basis                              - Determines the linear basis of the n-person TU game space.
% linear_production                         - Computes from a production problem (A,mB,p) a linear production game. 
% linprog_game                              - Computes from a production matrix (A,H,mB,mD,p) a linear programming game.
% lowersetQ                                 - Checks the existence of the lower set. 
% ma57_prekernel                            - Computes from (v,x) a pre-kernel element using HSL MA57.
% ma86_prekernel                            - Computes from (v,x) a pre-kernel element using HSL MA86.
% ma87_prekernel                            - Computes from (v,x) a pre-kernel element using HSL MA87.
% ma97_prekernel                            - Computes from (v,x) a pre-kernel element using HSL MA97.
% market2_game                              - Determines from two disjoint sets a market game.
% market_game                               - Determines from two disjoint sets a market game.
% mcst_game                                 - Computes from a cost matrix the corresponding mcst game.
% mex_coalitions                            - Computes the set of coalitions with maximum excesses
% minNoBlockPayoff                          - Computes the minimum no blocking payoff from game v.
% min_aspiration                            - Computes the minimum aspiration level of players of game v.
% min_epsshift                              - Computes for an almost-convex game the min epsilon shift to construct a convex game. 
% min_game                                  - Generates a minimum game.
% min_homogrep                              - Computes from the threshold th and the weights w_vec the minimal homogeneous representation of an homogeneous weighted majority game. 
% minimal_representation                    - Computes from the threshold th and the weights w_vec the minimal representation of an homogeneous weighted majority game.
% minimal_winning                           - Computes the minimal winning coalitions.
% modiclusQ                                 - Verifies whether the set of induced coalitions is a bi-balanced collection.
% monotone_gameQ                            - Checks monotonicity of the TU game.
% monotonic_cover                           - Determines the monotonic cover from a TU game.
% msk_AntiNucl                              - Computes the anti nucleolus of game v using the MOSEK solver.
% msk_AntiNucl_llp                          - Computes the anti nucleolus of game v using the MOSEK solver.
% msk_AntiPreNucl                           - Computes the anti prenucleolus using the MOSEK solver.
% msk_AntiPreNucl_llp                       - Computes the anti prenucleolus using the MOSEK solver.
% msk_AssignmentGame                        - Creates an assignment game using the MOSEK solver.
% msk_alphaVector                           - Computes recursively an alpha vector from the positive core.
% msk_cfr_nucl                              - Computes the nucleolus of game v with coalition formation restrictions using the MOSEK solver.
% msk_cs_nucl                               - Computes the nucleolus of game v w.r.t. coalition structure cs using the MOSEK solver.
% msk_exact_game                            - Computes the exact game from v using the MOSEK solver.
% msk_flow_game                             - Computes from a flow problem a TU flow game (MOSEK).
% msk_kernel                                - Computes a kernel point using the MOSEK solver.
% msk_linear_production                     - Computes from a production problem (A,mB,p) a linear production game using mosekmex. 
% msk_linprog_game                          - Computes from a production matrix (A,H,mB,mD,p) a linear programming game using mosekmex.
% msk_modiclus                              - Computes the modiclus of game v using the MOSEK solver.
% msk_nucl                                  - Computes the nucleolus using the MOSEK solver.
% msk_nucl_llp                              - Computes the nucleolus using the MOSEK solver.
% msk_prekernel                             - Computes a prekernel point using the MOSEK solver.
% msk_prenucl                               - Computes the prenucleolus using the MOSEK solver.
% msk_prenucl_llp                           - Computes the prenucleolus using the MOSEK solver.
% msk_prenucl_mod                           - Computes the pre-nucleolus of game v using mosekmex and a spanning method.
% msk_prenucl_mod2                          - Computes the pre-nucleolus of game v using mosekmex and an alternative spanning method.
% msk_prenucl_mod3                          - Computes the pre-nucleolus of game v using mosekmex and an alternative spanning method.
% msk_prenucl_mod4                          - Computes the pre-nucleolus of game v using mosekmex and an alternative spanning method.
% msk_weightedKernel                        - Computes a weighted kernel point using the MOSEK solver.
% msk_weightedNucl                          - Computes a weighted nucleolus using the MOSEK solver.
% msk_weightedNucl_llp                      - Computes a weighted nucleolus using the MOSEK solver.
% msk_weightedPreKernel                     - Computes a weighted prekernel point using the MOSEK solver.
% msk_weightedPreNucl                       - Computes a weighted prenucleolus using the MOSEK solver.
% msk_weightedPreNucl_llp                   - Computes a weighted prenucleolus using the MOSEK solver.
% near_ringQ                                - Checks if a collection of coalitions is a near ring. 
% nucl                                      - Computes the nucleolus using optimization toolbox.
% nucl_formula                              - Computes the nucleolus of a three-person super-additive game v from a formula.
% nucl_llp                                  - Computes the nucleolus using optimization toolbox.
% nullShapley                               - Determines a basis of the null space for the Shapley-value for n-persons.
% nullShapleyLB                             - Determines a counting basis of the null space for the Shapley-value for n-persons.
% oases_kernel                              - Computes a kernel point using the OASES solver.
% oases_prekernel                           - Computes a prekernel point using the OASES solver.
% oases_weightedKernel                      - Computes a weighted kernel point using the OASES solver.
% oases_weightedPreKernel                   - Computes a weighted prekernel point using the OASES solver.
% oddeven_game                              - Assigns |S|-1 if S is odd and |S|+1 if S is even.
% ols_prekernel                             - Computes a prekernel point using optimization toolbox.
% ols_weightedPreKernel                     - Computes a weighted prekernel point using optimization toolbox.
% one_concaveQ                              - Checks whether the game v is 1-concave.
% one_convexQ                               - Checks whether the game v is 1-convex.
% positive_gameQ                            - Returns true (1) if all Harsanyi dividends are non-negative, otherwise false (zero).
% potential                                 - Determines the potential of a TU game (basis).
% probability_game                          - Computes from a positive vector x the corresponding probability game.
% product_game                              - Computes form a vector x the corresponding product game.
% production_game                           - Creates an affine production game.
% production_game2                          - Creates an affine production game.
% production_game_sq                        - Creates a quadratic production game. 
% profit_matrix                             - Creates the profit matrix of an assignment game. 
% proper_amount                             - Computes the largest amount players contribute to a proper coalition.
% ps_gameQ                                  - Checks whether a game is a PS game.
% psstar_gameQ                              - Checks whether a game is a PS* game.
% pure_overhead                             - Creates the matrix of pure overhead games.
% qpBB_kernel                               - Computes a kernel point using the QPBB solver.
% qpBB_weightedKernel                       - Computes a weighted kernel point using the QPBB solver.
% qpc_kernel                                - Computes a kernel point using the QPC solver.
% qpc_prekernel                             - Computes a prekernel point using the QPC solver.
% qpc_weightedKernel                        - Computes a weighted kernel point using the QPC solver.
% qpc_weightedPreKernel                     - Computes a weighted prekernel point using the QPC solver.
% qrg_prekernel                             - Computes a prekernel point using qrginv instead of pinv.
% qrg_weightedPreKernel                     - Computes a weighted prekernel point using qrginv instead of pinv.
% quotas                                    - Determines the quotas of a game.
% reasonable_outcome                        - Determines the reasonable outcome.
% replicate_Shapley                         - Replicates the Shapley value for a game space. 
% replicate_prk                             - Replicates a pre-kernel solution as a pre-kernel of a game space. 
% root_game                                 - Computes from game v its associated root game.
% rootedQ                                   - Checks whether the game v is rooted.
% satisfaction                              - Computes the satisfaction of a partition and of its anti partition.
% savings_game                              - Creates a saving game from a cost game.
% scrb_solution                             - Computes separable costs-remaining benefits allocation. 
% secDiffOperatorQ                          - Checks whether the second order differences are all positive which indicates convexity.
% select_starting_pt                        - Selects a starting point for the pre-kernel computation.
% semi_concaveQ                             - Checks semi-concavity.
% semi_convexQ                              - Checks semi-convexity.
% separable_cost_allocation                 - Computes the separable cost allocation.
% separating_collectionQ                    - Verifies if a collection is separating.
% shiftGame  				    - Computes from the game v the t-shift game of v.
% simple_game                               - Creates a simple game.
% sm_Kernel                                 - Computes an element of the simplified Kernel of a game.
% sm_PreKernel  			    - Computes an element of the simplified pre-kernel of game v.	    
% sm_PreNucl                                - Computes the simplified pre-nucleolus of a game.
% sm_nucl                                   - Computes the simplified nucleolus of a game.
% smallest_amount                           - Computes the smallest amount vector.
% sortsets                                  - Sorts a sub/power set w.r.t. its cardinality.
% streps_value                              - Determines the strong epsilon-game.
% strictconvex_gameQ                        - Returns 1 whenever the game v is strictly convex.
% sub_additiveQ				    - Returns true whenever the game v is sub-additive.
% substitutes                               - Establishes which pair of players are substitutes.
% super_additiveQ                           - Checks the Tu-game on super additivity.
% superadditive_cover                       - Computes from game v its superadditive cover.
% surplus_game                              - Computes from a cost game c the corresponding surplus game v. 
% symmetricQ                                - Checks if the game v is symmetric.
% totallyAntiBalancedQ                      - Checks whether the anti-core of all subgames is non-empty. 
% totallyBalancedCoverQ                     - Checks whether the characteristic function v is equal to a totally balanced cover.
% totallyBalancedQ  			    - Checks whether the core of all subgames is non-empty.
% tricameral_assembly                       - Computes from a set of parameters a simple game. 
% unanimity_games                           - Computes the unanimity coordinates.
% union_stableQ                             - Checks whether a system is union stable.
% uppersetQ                                 - Checks the existence of the upper set.
% value_matrix  			    - Computes from an assignment matrix the corresponding value matrix for a permutation game.	    
% vclToMatlab                               - Computes a Tu-game and the corresponding unique integer representation of coalitions
% veto_players                              - Determines the veto players of a simple game. 
% veto_rich_players                         - Returns a list of veto players for the TU-game v.
% weakly_sub_additiveQ			    - Returns true whenever the game v is weakly sub-additive.
% weakly_super_additiveQ                    - Checks the Tu-game on weakly super additivity.
% weightedAntiBalancedCollectionQ           - Checks the reversal of weighted Kohlberg's criterion.
% weightedAnti_B0_balancedCollectionQ       - Checks the reversal of weighted weak Kohlberg's criterion.
% weightedAnti_Kernel                       - Computes a weighted anti-kernel point.
% weightedAnti_Nucl                         - Computes a weighted anti nucleolus of game.
% weightedAnti_Nucl_llp                     - Computes a weighted anti nucleolus of game.
% weightedAnti_PreKernel                    - Computes a weighted anti-prekernel point.
% weightedAnti_PreNucl                      - Computes a weighted anti pre-nucleolus of game.
% weightedAnti_PreNucl_llp                  - Computes a weighted anti pre-nucleolus of game.
% weightedB0_balancedCollectionQ            - Checking a weighted weak Kohlberg's criterion.
% weightedBalancedCollectionQ               - Checking a weighted Kohlberg's criterion.
% weightedCsB0_balancedCollectionQ  	    - Verifies whether the set of induced coalitions is a B0_balanced collection w.r.t. the coalition structure cs.
% weightedCsBalancedCollectionQ             - Verifies whether the set of induced coalitions is a weighted balanced collection w.r.t. the coalition structure cs.
% weightedCsKernel                          - Computes a weighted kernel element w.r.t. coalition structure cs.
% weightedCsKernelQ                         - Checks whether the imputation x is a weighted kernel element of the TU-game v w.r.t. coalition structure cs.
% weightedCsNucl                            - Computes a weighted nucleolus of game v w.r.t. coalition structure cs.
% weightedCsPreKernel                       - Computes a weighted pre-kernel element w.r.t. a coalition structure cs.
% weightedCsPreNucl                         - Computes a weighted pre-nucleolus of game v w.r.t. coalition structure cs.
% weightedGraphGame                         - Computes from a weighted graph problem a weighted graph TU game.
% weightedKernel                            - Computes a weighted kernel point using optimization toolbox.
% weightedKernelQ                           - Checks if an imputation is a weighted kernel point.
% weightedNucl                              - Computes a weighted nucleolus using optimization toolbox.
% weightedNucl_llp                          - Computes a weighted nucleolus using optimization toolbox.
% weightedPreKernel                         - Computes a weighted prekernel element.
% weightedPreNucl                           - Computes a weighted prenucleolus using optimization toolbox.
% weightedPreNucl_llp                       - Computes a weighted prenucleolus using optimization toolbox.
% weighted_cfr_Kernel                       - Computes a weighted Kernel element with coalition formation restrictions.
% weighted_cfr_PreKernel                    - Computes a weighted Pre-Kernel element with coalition formation restrictions.
% weighted_cfr_PreNucl                      - Computes the weighted pre-nucleolus of game v with coalition formation restrictions.
% weighted_cfr_balancedCollectionQ          - Verifies whether the set of induced coalitions is a balanced collection with coalition formation restrictions.
% weighted_cfr_nucl                         - Computes the weighted nucleolus of game v with coalition formation restrictions.
% weighted_cfr_weak_balancedCollectionQ     - Verifies whether the set of induced coalitions is a weakly_balanced collection with coalition formation restriction.
% weighted_majority                         - Creates a weighted majority game.
% weighted_truncated                        - Computes from the threshold th and the weights w_vec a truncated weighted majority game.
% winning_coalitions                        - Determines the whole set of winning coalitions.
% winning_players                           - Computes from a pre-defined set of winning coalitions (e.g. minimal winning coalitions) the set of winning players. 
% zero_monotonicQ                           - Checks zero monotonicity.
% zero_normalization                        - Creates a zero normalized game. 
%
%
% Class Objects
% -------------
% TuACore                                   - subclass object of TuSol (anti-core plot).
% TuAPrn                                    - subclass object of TuSol (anti pre-nucleolus from various solvers).
% TuASol                                    - subclass object of TuGame (game solutions).
% TuAVert                                   - subclass object of TuSol (anti-core vertices).
% TuCons                                    - subclass object of TuSol (consistency).
% TuCore                                    - subclass object of TuSol (core plot).
% TuGame                                    - to perform several computations for retrieving and modifying game data.
% TuKcons                                   - subclass object of TuSol (generalized consistency).
% TuKrn                                     - subclass object of TuSol (kernel solutions from various solvers).
% TuMCnets                                  - subclass object of TUGAME from getting an MC-nets representation game data.
% TuNuc                                     - subclass object of TuSol (nucleolus from various solvers).
% TuPrk                                     - subclass object of TuSol (pre-kernel solutions from various solvers).
% TuPrn                                     - subclass object of TuSol (pre-nucleolus from various solvers).
% TuProp                                    - subclass object of TuGame (game properties).
% TuRep                                     - subclass object of TuSol (prk replication).
% TuShRep                                   - subclass object of TuSol (Shapley value replication).
% TuSol                                     - subclass object of TuGame (game solutions).
% TuVal                                     - subclass object of TuGame (fairness and related values).
% TuVert                                    - subclass object of TuSol (core vertices).
%
%
% pct_tugames: Parallel Computing
%-------------------------------
% p_ADvalue                                 - Computes the Aumann-Dreze value.
% p_AP_DummyPlayer_propertyQ                - Checks if the solution x satisfies the AP-Dummy player property.
% p_AP_DummyPlayers                         - Returns the player who are AP-Dummy players. 
% p_AP_NullPlayer_propertyQ                 - Checks if the solution x satisfies the AP-Null player property. 
% p_AP_NullPlayers                          - Returns the players who are AP-Null players. 
% p_A_DummyPlayer_propertyQ                 - Checks if the solution x satisfies the A-Dummy player property. 
% p_A_NullPlayer_propertyQ                  - Checks if the solution x satisfies the A-Null player property.
% p_A_NullPlayers                           - Returns the players who are A-Null players. 
% p_AllMarginalContributions                - Computes all marginal contributions of a Tu-game.
% p_AlmostConcave_gameQ			    - Returns true whenever the game v is almost concave.
% p_AlmostConvex_gameQ			    - Returns true whenever the game v is almost convex.
% p_AntiB0_balancedCollectionQ              - Checks the reversal of weighted Kohlberg's criterion.
% p_AntiReduced_game_propertyQ  	    - Checks whether an imputation x satisfies the anti-reduced game property.
% p_Anti_ChiValue                           - Computes the anti-chi-value of a TU-game v. 
% p_Anti_Converse_DGP_Q  		    - Checks whether an imputation x satisfies the anti-converse derived game property.
% p_Anti_Derived_game_propertyQ  	    - Checks whether an imputation x satisfies a modified anti-derived game property.
% p_Anti_Gap                                - Computes the anti-gap function from game v. 
% p_Anti_GenGap                             - Computes the anti-generalized gap function from game v. 
% p_Anti_ModPreKernel  			    - Computes from (v,x) a modified pre-kernel element.
% p_Anti_ModPrekernelQ  		    - Checks whether the imputation x is a modified anti-pre-kernel element of the TU-game v.
% p_Anti_PModPreKernel  		    - Computes from (v,x) a proper modified anti-pre-kernel element.
% p_Anti_PModPrekernelQ  		    - Checks whether the imputation x is a proper modified anti-pre-kernel element of the TU-game v.
% p_Anti_PreKernel                          - Computes an anti-pre-kernel element.
% p_Anti_PrekernelQ                         - Checks if an imputation is an anti prekernel point.
% p_Anti_PropModPreKernel                   - Computes from (v,x) a proper modified anti-pre-kernel element.
% p_Anti_TauValue                           - Computes the anti-tau-value of a TU-game v. 
% p_B0_balancedCollectionQ                  - Checking weak Kohlberg's criterion.
% p_BanzhafColeman                          - Computes the Banzhaf/Coleman index of a simple game sv using MATLAB's PCT.
% p_BanzhafOwenValue                        - Computes the Banzhaf-Owen value w.r.t. a priori unions cs using Matlab's PCT.
% p_BanzhafPenrose                          - Computes the Banzhaf/Penrose and Banzhaf/Coleman index of a simple game sv using MATLAB's PCT.
% p_BestCoalitions                          - Computes  the set of most effective coalitions.
% p_COV_propertyQ                           - Verifies if the payoff x satisfies COV property.
% p_CddTotallyBalancedQ  		    - Checks whether the core of all subgames is non-empty (cddmex).
% p_ChiValue				    - Computes the chi-value of a TU-game v.
% p_CmpConsistencyQ			    - Checks whether an imputation x satisfies the complement consistency.
% p_CmpRedGame				    - Computes from (v,x,S) a complement reduced game vS on S at x for game v.
% p_CoalitionSolidarity                     - Determines the coalition solidarity value.
% p_ColemanOwenValue                        - Computes the Coleman-Owen value w.r.t. a priori unions cs using Matlab's PCT. Is not normalized to one.
% p_Complement_Reduced_game		    - Computes from (v,x) all complement reduced games on S at x of game v.
% p_Converse_CmpConsistencyQ		    - Checks whether an imputation x satisfies the converse complement consistency property.
% p_Converse_DGP_Q  			    - Checks whether an imputation x satisfies the converse derived game property.
% p_Converse_RGP_Q                          - Checks if an imputation satisfies the CRGP.
% p_DM_AntiReduced_game  		    - Computes from (v,x) all anti-reduced games on S at x of game v.
% p_DM_Anti_Derived_game  		    - Computes from (v,x) a modified Davis-Maschler anti-reduced game vS on S at x for game v.
% p_DM_Derived_game  			    - Computes from (v,x) a modified Davis-Maschler reduced game vS on S at x for game v.
% p_DM_Reduced_game                         - Computes all Davis-Maschler reduced games.
% p_DM_TwoReduced_game                      - Computes from (v,x) all single and two-person reduced games on S at x of game v using MATLAB's PCT.
% p_DecomposeGame                           - Computes the unique decomposition of a TU-game.
% p_DeeganPackel                            - Computes the Deegan-Packel index from the set of minimal winning coalitions. 
% p_DeeganPackel_SV                         - Computes the Deegan-Packel index from a simple game to construct the set of minimal winning coalitions. 
% p_Derived_game_propertyQ  		    - Checks whether an imputation x satisfies a modified derived game property.
% p_DualCover  				    - The maximum characteristic values from the primal or dual game.
% p_DualEssentialSet                        - Computes the set of dually essential coalitions.
% p_DualFloor  				    - The minimum characteristic values from the primal or dual game.
% p_DuallyEssentialSet                      - Computes the set of dually essential coalitions.
% p_DummyPlayer_propertyQ                   - Verifies if x satisfies the dummy player property. 
% p_DummyPlayers                            - Returns the list of dummy players of game v. 
% p_ECCoverGame  			    - Computes from (v,x) an excess comparability cover of game v.
% p_ECFloorGame  			    - Computes from (v,x) an excess comparability floor of game v.
% p_ECGValue                                - Computes the Equal Collective Gains value of a TU-game v. 
% p_EC_DGP_Q  				    - Checks whether the solution x satisfies excess comparability for each derived game.
% p_EC_RGP_Q  				    - Checks whether the solution x satisfies excess comparability for each reduced game.
% p_EC_propertyQ  			    - Checks whether the solution x satisfies excess comparability.
% p_Gap                                     - Determines the gap function.
% p_GenGap				    - Computes the generalized gap function from game v.
% p_GetMarketGame                           - Determines from a random generated game the corresponding market game.
% p_HMS_AntiReduced_game  		    - Computes from (v,x) all Hart/Mas-Colell anti-reduced games on S at x of game v.
% p_HMS_Anti_Derived_game  		    - Computes from (v,x,S) a modified Hart-Mas-Colell anti-reduced game vS on S at x for game v.
% p_HMS_Derived_game  			    - Computes from (v,x,S) a modified Hart-Mas-Colell reduced game vS on S at x for game v.
% p_HMS_ImputSavingReducedGame              - Computes from (v,x) all Hart/Mas-Colell ISR games.
% p_HMS_Reduced_game                        - Creates all Hart/Mas-Colell reduced games. 
% p_HMS_TwoReduced_game                     - Computes from (v,x) all Hart/Mas-Colell singleton and two-person reduced games on S at x of game v.
% p_ISRG_propertyQ                          - Checks whether an imputation x satisfies the ISR game property.
% p_ImputSavingReducedGame                  - Computes from (v,x) all imputation saving reduced games.
% p_InessGame                               - Computes the inessential game from a payoff vector.
% p_Johnston                                - Computes the Johnston power index from the set of winning coalitions. 
% p_Kernel                                  - Computes a kernel point using the optimization toolbox.
% p_LED_RGPvsDGP  			    - Determines the shift of reduced games of the ECC game or reduced game of the dual cover restricted to N w.r.t. the derived games.
% p_LS_Nucl                                 - Computes the least square nucleolus of a game.
% p_LS_PreNucl                              - Computes the least square pre-nucleolus of a game.
% p_LedcoconsQ  			    - Checks whether an imputation x satisfies large excess difference converse consistency. 
% p_Ledcons_propertyQ  			    - Checks whether an imputation x satisfies the ledcons property.
% p_MarketGameQ                             - Checks whether the game v is a market game.
% p_MaxConsistencyQ			    - Checks whether an imputation x satisfies maximal consistency.
% p_ModDeeganPackel                         - Computes the modified Deegan-Packel index from the set of winning coalitions. 
% p_ModDeeganPackel_SV                      - Computes the Deegan-Packel index from a simple game to construct the set of minimal winning coalitions. 
% p_ModHoller  				    - Computes the modified Holler index from the set of winning coalitions.
% p_ModPGI				    - Computes the modified public good index from the set of winning coalitions.
% p_ModPGI_SV                               - Computes the modified public good index from a simple game to determine the set of minimal winning coalitions. 
% p_ModPreKernel  			    - Computes from (v,x) a modified pre-kernel element.
% p_ModPrekernelQ  			    - Checks whether the imputation x is a modified pre-kernel element.
% p_MyersonValue                            - Computes the Myerson value of a Tu game.
% p_NetworkBanzhaf                          - Computes the network Banzhaf power index from the set of winning coalitions of a network E while imposing a threshold of th. 
% p_NetworkDeeganPackel                     - Computes the network Deegan-Packel index from the set of winning coalitions of a network E while imposing a threshold of th. 
% p_NetworkJohnston                         - Computes the network Johnston power index from the set of winning coalitions of a network E while imposing a threshold of th. 
% p_NetworkMajorityGame			    - Computes from a network problem (E,c,th) a network majority TU game (simple game).
% p_NetworkModDeeganPackel                  - Computes the network modified Deegan-Packel index from the set of winning coalitions of a network E while imposing a threshold of th. 
% p_NetworkModPGI			    - Computes the network modified public good index from the set of winning coalitions of a network E while imposing a threshold.
% p_NetworkPGI				    - Computes the network public good index from the set of minimal winning coalitions of a network E while imposing a threshold. 
% p_NetworkShapleyShubik                    - Computes the network Shapley-Shubik power index from the set of winning coalitions of a network E while imposing a threshold of th. 
% p_NullPlayer_propertyQ                    - Verifies if x satisfies the null player property.
% p_NullPlayers                             - Returns the list of null players of game v. 
% p_OwenValue                               - Computes the Owen value.
% p_PGI					    - Computes the public good index from the set of minimal winning coalitions.
% p_PGI_SV                                  - Computes the public good index from a simple game to determine the set of minimal winning coalitions. 
% p_PModPreKernel  			    - Computes from (v,x) a proper modified pre-kernel element.
% p_PModPrekernelQ  			    - Checks whether the imputation x is a proper modified pre-kernel element.
% p_PermutationGame                         - Computes from an assignment matrix the permutation game.
% p_PositionValue                           - Computes the position value.
% p_PreKernel                               - Computes a pre-kernel element.
% p_PrekernelQ                              - Checks if an imputation is a pre-kernel point.
% p_PropModPreKernel  			    - Computes from (v,x) a proper modified pre-kernel element.
% p_REAS_propertyQ  			    - Checks if the vector x satisfies the reasonableness on both sides.
% p_REC_propertyQ  			    - Checks whether the solution x satisfies reverse excess comparability.
% p_Reconfirmation_propertyQ                - Checks the RCP.
% p_RedGame                                 - Creates a Davis-Maschler reduced game.
% p_Reduced_game_propertyQ                  - Checks the RGP.
% p_SD_ShapleyValue  			    - Computes the surplus division Shapley value.
% p_SedcoconsQ  			    - Checks whether an imputation x satisfies small excess difference converse consistency.
% p_Sedcons_propertyQ  			    - Checks whether an imputation x satisfies the sedcons property.
% p_ShapleyValue                            - Computes the Shapley value (potential).
% p_ShapleyValueLB                          - Computes the Shapley value from the linear basis.
% p_ShapleyValueM                           - Computes the Shapley value while relying on all marginal contributions.
% p_SolidarityShapleyValue                  - Determines the solidarity Shapley value. 
% p_SolidarityValue                         - Determines the solidarity value. 
% p_StrConverse_DGP_Q  			    - Checks whether an imputation x satisfies the strong converse derived game property.
% p_StrConverse_RGP_Q  			    - Checks whether an imputation x satisfies the strong CRGP.
% p_StrLedcoconsQ  			    - Checks whether an imputation x satisfies satisfies strong large excess difference converse consistency.
% p_StrSedcoconsQ  			    - Checks whether an imputation x satisfies satisfies strong small excess difference converse consistency.
% p_StrategicEquivalentPrK                  - Computes the pre-kernel of game v from a strategic equivalent game.
% p_SubSets                                 - Creates all subsets of super set.
% p_TauValue                                - Computes the Tau value.
% p_UpperPayoff				    - Computes the utopia and minimum claim vector of game v.
% p_UtopiaPayoff			    - Computes the utopia and minimum claim vector of game v.
% p_WSysBestCoalitions                      - Computes  the set of most effective coalitions w.r.t. a weight system.
% p_WSys_game_space                         - Computes a game space w.r.t. a weight system which replicates a payoff as a weighted pre-kernel element.
% p_WSys_game_space_red                     - Computes a game space w.r.t. a weight system which replicates a payoff as a weighted pre-kernel element.
% p_WSys_replicate_prk                      - Replicates a weighted pre-kernel point of a game space w.r.t. a weight system. 
% p_WeakReduced_game_propertyQ              - Checks whether an imputation x satisfies the weak reduced game property (consistency)
% p_airport_profit                          - Computes from a cost and benefit vector the associated surplus game.
% p_apu_SolidarityValue                     - Determines the solidarity value w.r.t. a priori unions.
% p_assignment_game                         - Creates  an assignment game.
% p_average_concaveQ			    - Returns true whenever the game v is average-concave.
% p_average_convexQ                         - Checks on average convexity.
% p_balancedSetQ                            - Verifies whether the set of induced coalitions is a balanced collection. 
% p_banzhaf                                 - Computes  the Banzhaf value.
% p_basis_coordinates                       - Determines the basis coordinates of a Tu game.
% p_basis_game                              - Determines bases games.
% p_belongToAllSubCores                     - Checks whether all projections of an imputation x are a member of an associated core of a subgame.
% p_bint_AssignmentGame                     - Creates an assignment game (bintprog).
% p_bs_PreKernel                            - Computes from (v,x) a pre-kernel element using Matlab's PCT and backslash operation.
% p_clp_kernel                              - Computes a kernel point using the CLP solver.
% p_clp_weightedKernel                      - Computes a weighted kernel point using the CLP solver.
% p_cls_kernel                              - Computes a kernel point using the CLS solver.
% p_cls_weightedKernel                      - Computes a weighted kernel point using the CLS solver.
% p_coeff_linearbasis                       - Determines the coefficients (dividends) of a linear basis from a TU game.
% p_convex_gameQ                            - Checks on convexity.
% p_coreQ                                   - Checks the non-emptiness of the core.
% p_cplex_AssignmentGame                    - Creates an assignment game using the CPLEX solver.
% p_cplex_exact_game                        - Computes the exact game from v using the CPLEX solver.
% p_cplex_kernel                            - Computes a kernel point using the CPLEX solver.
% p_cplex_prekernel                         - Computes a prekernel point using the CPLEX solver.
% p_cplex_weightedKernel                    - Computes a weighted kernel point using the CPLEX solver.
% p_cplex_weightedPreKernel                 - Computes a weighted prekernel point using the CPLEX solver.
% p_cvx_kernel                              - Computes a kernel point using the CVX solver. 
% p_cvx_prekernel                           - Computes a prekernel point using the CVX solver. 
% p_cvx_weightedKernel                      - Computes a weighted kernel point using the CVX solver.
% p_cvx_weightedPreKernel                   - Computes a weighted prekernel point using the CVX solver.
% p_disagreement			    - Computes the disagreement vector of game v.
% p_equal_treatmentQ                        - Checks if a vector x satisfies ETP.
% p_essentialSet                            - Computes the set of essential coalitions.
% p_exact_game                              - Computes the exact game from v using Matlab's Optimization toolbox.
% p_excess                                  - Computes the excesses.
% p_flow_game                               - Computes from a flow problem a TU flow game using the optimization toolbox.
% p_game_basis                              - Computes a game basis of the n-person TU-game space.
% p_game_space                              - Computes the game space which replicates a payoff as a pre-kernel element.
% p_game_space_red                          - Computes the game space which replicates a payoff as a pre-kernel element.
% p_genUnionStable                          - Creates a union stable system.
% p_getMinimalWinning                       - Computes from a simple game the minimal winning coalitions. 
% p_getgame                                 - Creates a Tu-game from the unanimity coordinates. 
% p_glpk_exact_game                         - Computes the exact game from v using the GLPK solver.
% p_glpk_kernel                             - Computes a kernel point using the GLPK solver. 
% p_glpk_prekernel                          - Computes a prekernel point using the GLPK solver. 
% p_glpk_weightedKernel                     - Computes a weighted kernel point using the GLPK solver.
% p_glpk_weightedPreKernel                  - Computes a weighted prekernel point using the GLPK solver.
% p_grMaxFlowGame                           - Computes from a flow problem a TU flow game.
% p_gurobi_AssignmentGame                   - Creates an assignment game using the GUROBI solver.
% p_gurobi_exact_game                       - Computes the exact game from v using the GUROBI solver.
% p_gurobi_flow_game                        - Computes from a flow problem a TU flow game (GUROBI).
% p_gurobi_kernel                           - Computes a kernel point using the GUROBI solver. 
% p_gurobi_prekernel                        - Computes a prekernel point using the GUROBI solver. 
% p_gurobi_weightedKernel                   - Computes a weighted kernel point using the GUROBI solver.
% p_gurobi_weightedPreKernel                - Computes a weighted prekernel point using the GUROBI solver.
% p_harsanyi_dividends                      - Determines the the unanimity coordinates.
% p_holler                                  - Computes the Holler index.
% p_homogeneous_representationQ             - Checks if the weighted majority game possesses a homogeneous representation.
% p_hsl_prekernel                           - Computes a prekernel point using HSL solvers. 
% p_hsl_weightedPreKernel                   - Computes a weighted prekernel point using HSL solvers.
% p_ipopt_kernel                            - Computes a kernel point using the IPOPT solver.
% p_ipopt_prekernel                         - Computes a prekernel point using the IPOPT solver.
% p_ipopt_weightedKernel                    - Computes a weighted kernel point using the IPOPT solver.
% p_ipopt_weightedPreKernel                 - Computes a weighted prekernel point using the IPOPT solver.
% p_k_Converse_RGP_Q                        - Checks if an imputation satisfies the k-CRGP.
% p_k_Reconfirmation_propertyQ              - Checks the k-RCP.
% p_k_Reduced_game_propertyQ                - Checks the k-RGP.
% p_k_StrConverse_RGP_Q                     - Checks the strong k-CRGP. 
% p_k_convexQ                               - Checks k-convexity of the Tu-game.
% p_k_cover                                 - Determines from the Tu-game the corresponding k-game.
% p_landlord                                - Computes a production game arising from l-landlords and t-tenants.
% p_lin_prekernel                           - Computes a prekernel point using optimization toolbox.
% p_lin_weightedPreKernel                   - Computes a weighted prekernel point using optimization toolbox.
% p_linear_basis                            - Determines the linear basis of the n-person TU game space.
% p_mcst_game                               - Computes from a cost matrix the corresponding mcst game.
% p_min_aspiration			    - Computes the minimum aspiration level of players of game v.
% p_minimal_representation  		    - Computes from the threshold th and the weights w_vec the minimal representation of an homogeneous weighted majority game.
% p_minimal_winning                         - Computes the minimal winning coalitions.
% p_monotone_gameQ                          - Checks monotonicity of the Tu-game.
% p_msk_AssignmentGame                      - Creates an assignment game using the MOSEK solver.
% p_msk_bintAssignmentGame		    - Computes from an assignment problem the corresponding symmetric assignment game.
% p_msk_exact_game                          - Computes the exact game from v using the MOSEK solver.
% p_msk_kernel                              - Computes a kernel point using the MOSEK solver.
% p_msk_prekernel                           - Computes a prekernel point using the MOSEK solver.
% p_msk_weightedKernel                      - Computes a weighted prekernel point using the MOSEK solver.
% p_msk_weightedPreKernel                   - Computes a weighted kernel point using the MOSEK solver.
% p_nullShapley                             - Determines a basis of the null space for the Shapley-value for n-persons.
% p_oases_kernel                            - Computes a kernel point using the OASES solver.
% p_oases_prekernel                         - Computes a prekernel point using the OASES solver.
% p_oases_weightedKernel                    - Computes a weighted kernel point using the OASES solver.
% p_oases_weightedPreKernel                 - Computes a weighted prekernel point using the OASES solver.
% p_ols_prekernel                           - Computes a prekernel point using optimization toolbox.
% p_ols_weightedPreKernel                   - Computes a weighted prekernel point using optimization toolbox.
% p_one_concaveQ                            - Checks whether the game v is 1-concave.
% p_one_convexQ                             - Checks whether the game v is 1-convex.
% p_parity_basis                            - Computes a basis of the n-person TU game space.
% p_parity_coeff                            - Computes the parity transform of the TU-game v.
% p_potential                               - Determines the potential of a TU game (basis).
% p_proper_amount			    - Computes the largest amount players can contribute to a proper coalition.
% p_pure_overhead                           - Creates the matrix of pure overhead games.
% p_qpBB_kernel                             - Computes a kernel point using the QPBB solver.
% p_qpBB_weightedKernel                     - Computes a weighted kernel point using the QPBB solver.
% p_qpc_kernel                              - Computes a kernel point using the QPC solver.
% p_qpc_prekernel                           - Computes a prekernel point using the QPC solver.
% p_qpc_weightedKernel                      - Computes a weighted prekernel point using the QPC solver.
% p_qpc_weightedPreKernel                   - Computes a weighted kernel point using the QPC solver.
% p_qrg_prekernel                           - Computes a prekernel point using qrginv instead of pinv.
% p_qrg_weightedPreKernel                   - Computes a weighted prekernel point using qrginv instead of pinv.
% p_reasonable_outcome                      - Determines the reasonable outcome.
% p_replicate_Shapley                       - Replicates the Shapley value for a game space. 
% p_replicate_prk                           - Replicates a pre-kernel solution as a pre-kernel of a game space. 
% p_secDiffOperatorQ                        - Checks whether the second order differences are all positive which indicates convexity.
% p_select_starting_pt                      - Selects a starting point for the pre-kernel computation.
% p_semi_convexQ                            - Checks semi-convexity.
% p_smallest_amount			    - Computes the smallest amount vector of the game.
% p_sub_additiveQ			    - Returns true whenever the game v is sub additive.
% p_substitutes                             - Establishes which pair of players are substitutes.
% p_super_additiveQ                         - Checks the Tu-game on super additivity.
% p_superadditive_cover                     - Computes from game v its superadditive cover.
% p_totallyBalancedCoverQ                   - Checks whether the characteristic function v is equal to a totally balanced cover.
% p_totallyBalancedQ  			    - Checks whether the core of all subgames is non-empty.
% p_tricameral_assembly                     - Computes from a set of parameters a simple game. 
% p_unanimity_games                         - Computes the unanimity coordinates.
% p_union_stableQ                           - Checks whether a system is union stable.
% p_veto_rich_players			    - Returns a list of veto players for the TU-game v.
% p_weightedAnti_PreKernel                  - Computes a weighted anti-prekernel point.
% p_weightedKernel                          - Computes a weighted kernel point using the optimization toolbox.
% p_weightedPreKernel                       - Computes a weighted pre-kernel element.
% p_weighted_DualEssentialSet               - Computes the set of weighted dually essential coalitions.
% p_weighted_DuallyEssentialSet             - Computes the set of weighted dually essential coalitions.
% p_weighted_EssentialSet                   - Computes the set of weighted essential coalitions using Matlab's PCT.
% p_zero_monotonicQ                         - Checks zero monotonicity. 
%
%
% Class Objects
% -------------
% p_TuCons                                  - subclass object of p_TuSol (consistency).
% p_TuKcons                                 - subclass object of p_TuSol (generalized consistency).
% p_TuKrn                                   - subclass object of p_TuSol (kernel solutions from various solvers).
% p_TuPrk                                   - subclass object of p_TuSol (pre-kernel solutions from various solvers).
% p_TuProp                                  - subclass object of TuGame (game properties).
% p_TuRep                                   - subclass object of p_TuSol (prk replication).
% p_TuShRep                                 - subclass object of p_TuSol (Shapley value replication).
% p_TuSol                                   - subclass object of TuGame (game solutions).
% p_TuVal                                   - subclass object of TuGame (fairness and related values).
%
%
%
%
% tools: Sed File
%---------------
% sed_core                                  - Converts cdd file format into Matlab format.