% MatTuGames: A Matlab Game Theory Toolbox
% Version 1.9 (R2021a) 25-Jun-2021
%
%
% Author: Holger I. Meinhardt
% E-Mail: Holger.Meinhardt@wiwi.uni-karlsruhe.de
% Institution: University of Karlsruhe (KIT)
%
%
% aux: Some auxiliary files
%--------------------------
% FrameToImage - Converts a frame to an image.
% PlayCoreMovie - Plays a movie from a collection of frames.
% ginv - Computes a general inverse.
% myaa - MYAA Render figure with anti-aliasing.
% qrginv - Computes a pseudo-inverse using a QR-method.
% vtk_export - Exportes 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.m - Additions and changes in version 1.9
% 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_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.
% 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.
% AlmostConcave_gameQ - Returns true whenever the game v is almost concave.
% AlmostConvex_gameQ - Returns true whenever the game v is almost convex.
% AdditiveQ - Checks if the game v is additive.
% AllMarginalContributions - Computes all marginal contributions of a Tu game.
% AllSubGames - Computes all subgames.
% 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_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_ModPreKernel - 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_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_Weak_balancedCollectionQ - Checking reverse weak Kohlberg's criterion.
% Anti_balancedCollectionQ - Verifies whether the set of induced coalitions is an anti balanced collection.
% Anti_balancedCollectionQ - Checks the reversal of Kohlberg's criterion.
% Anti_kernelQ - Checks if an imputation is an anti kernel point.
% B0_balancedCollectionQ - Checking weak Kohlberg's criterion.
% B0_balancedQ - Verifies whether the collection of coalitions is weakly balanced.
% 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.
% 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).
% 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).
% 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.
% Complement_Reduced_game - Computes from (v,x) all complement reduced games on S at x of game v.
% 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.
% 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.
% 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.
% EssentialConstSumQ - Checks if v is an essential constant-sum game.
% EssentialQ - Checks if the game v is essential.
% ExtShapleyValue - Computes the extended Shapley-value.
% 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.
% 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.
% 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
% LorenzDom - Checks if x Lorenz dominates y in game v.
% 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.
% 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.
% 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.
% 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.
% 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.
% 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.
% PrekernelQ - Checks if an imputation is a pre-kernel point.
% PrenuclQ - Checks if an imputation is the pre-nucleolus using Kohlberg's criterion.
% 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.
% 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.
% 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.
% 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.
% 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.
% 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.
% Weak_balancedCollectionQ - Checking weak Kohlberg's criterion.
% ZeroOne_Normalization - Creates a zero-one normalized game.
% additive_game - Creates an additive game.
% admissibleGame - Computes a symmetric compromise admissible 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.
% 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.
% 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.
% 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.
% bint_AssignmentGame - Creates an assignment game (bintprog).
% 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.
% 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_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.
% 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_AssignmentGame - Creates an assignment game using the CPLEX solver.
% cplex_LeastCore - Computes the least core using cplexmex.
% cplex_exact_game - Computes the exact game from v using the CPLEX solver.
% cplex_kernel - Computes a kernel point using the CPLEX solver.
% 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_mod4 - Computes the pre-nucleolus of game v using cplexmex (fast/method 4).
% 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
% cvx_kernel - Computes a kernel point using the CVX solver.
% cvx_prekernel - Computes a prekernel point using the CVX solver.
% 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.
% 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_exact_game - Computes the exact game from v using the GLPK solver.
% glpk_kernel - Computes a kernel point using the GLPK solver.
% 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.
% 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_kernel - Computes a kernel point using the GUROBI solver.
% 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.
% 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.
% hypergraphQ - Checks whether the system is a hypergraph communication situation.
% interest_game - Computes from an interest problem the corresponding game.
% ipopt_kernel - Computes a kernel point using the IPOPT solver.
% ipopt_prekernel - Computes a 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.
% lin_prekernel - Computes a 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.
% lowersetQ - Checks the existence of the lower set.
% 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_exact_game - Computes the exact game from v using the MOSEK solver.
% 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_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.
% near_ringQ - Checks if a collection of coalitions is a near ring.
% nucl - Computes the nucleolus using optimization toolbox.
% 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.
% 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.
% potential - Determines the potential of a TU game (basis).
% 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.
% qpc_kernel - Computes a kernel point using the QPC solver.
% qpc_prekernel - Computes a prekernel point using the QPC solver.
% qrg_prekernel - Computes a 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.
% savings_game - Creates a saving game from a cost game.
% scrb_solution - Computes separable costs-remaining benefits allocation.
% 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.
% 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.
% 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.
% weighted_majority - Creates a 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).
% 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_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_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_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_DualFloor - The minimum characteristic values from the primal or dual game.
% 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_GenGap - Computes the generalized gap function from game v.
% 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_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_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_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_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_bint_AssignmentGame - Creates an assignment game (bintprog).
% p_clp_kernel - Computes a kernel point using the CLP solver.
% p_cls_kernel - Computes a 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_cvx_kernel - Computes a kernel point using the CVX solver.
% p_cvx_prekernel - Computes a 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_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_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_ipopt_kernel - Computes a kernel point using the IPOPT solver.
% p_ipopt_prekernel - Computes a 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_lin_prekernel - Computes a 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_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_ols_prekernel - Computes a prekernel point using optimization toolbox.
% 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_qpc_kernel - Computes a kernel point using the QPC solver.
% p_qpc_prekernel - Computes a prekernel point using the QPC solver.
% p_qrg_prekernel - Computes a 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_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_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_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.