ChangeLog

   :Title: TuGames.m
   :History:
   Version 1.0: without options
   Version 1.1: with options
   Version 1.3:
     Minor code revision.

     Some robustness problems with the function family ModifiedNucleolus[]
     have been fixed, but it is not completely solved. For convex games or inessential games
     the solution computed might still be wrong.

     Some new functions to compute the pre-kernel, anti-pre-kernel
     and some related functions were incorporated. The function names
     are PreKernelSolution[] and AntiPreKernelSolution[].
     These functions are based on an algorithm that relies on the indirect function
     approach to compute pre-kernel solutions. In order to check if the solution is
     an anti-pre-kernel, the functions MinExcessBalanced[], AntiPreKernelQ[]
     and AllAntiSurpluses[] have been added.

   Version 1.5:
     Some code revision.

     The functions DefineGame[], MaxExcessBalanced[], MinExcessBalanced[],
      KernelImputationQ[], SuperAdditiveQ[] and AvConvex[] have been completely rewritten.
     For this reason the functions
             AvSumCheck[]
             ContributionSum[]
      have been deprecated.

    The function AverageConvexQ[] is the same as AvConvexQ[], but it allows issuing an option.

    The functions PreKernelQ[] and AntiPreKernelQ[] have been modified.

     Some new functions related to unanimity coordinates have been added.
     For instance
     DetUCoord[] to get the unanimity coordinates from a set of positive coordinates.
     HarsanyiDividends[] provides an alternative approach to compute the
              unanimity coordinates (Harsanyi dividends) of the game.
     CharacteristicValues[] computes the characteristic values of the game from
              the unanimity coordinates.

     The functions
     UnanConvexQ[]
     UnanAvConvexQ[]
     provide alternative approaches to check convexity and average-convexity.

    Version 1.6:
     We added some new functions related to the prenucleolus, like BalancedSelectionQ[].

     Bug fixes:

     AverageConvexQ[]        - Fix incorrect computation and a precision problem.
     DetUCoord[]             - Fix incorrect computation.
     UnanAvConvexIffCondQ[]  - Fix incorrect computation and a precision problem.
     AdjustedWorthVectors[]  - Fix incorrect computation.

     Modification:

     Some general code revision.

     First code adjustment to use this package in parallel.

     Furthermore, some basic functions can now be called within a running Matlab session while
     using our Matlab toolbox MatTuGames, which is available from the following URL:
     http://www.mathworks.cn/matlabcentral/fileexchange/35933?product=EL

    Version 1.7:

     Bug fixes:

     PreKernelElement[]        - Fix a precision problem.
     KernelVertices[]          - Fix a precision problem.
     DirectionOfImprovement[]  - Fix a precision problem.
     MaxExcessBalanced[]       - Fix a precision problem.
     MinExcessBalanced[]       - Fix a precision problem.
     KernelImputationQ[]       - Fix a precision problem.
     KernelImputationListQ[]   - Fix a precision problem.
     ModifiedNucleolus[]       - Fix incorrect computation with games having empty core.
     ModifiedKernel[]          - Fix incorrect computation with games having empty core.
     LexiCenter[]              - Fix incorrect computation with games having empty core.
     PreNucleolus[]            - Fix incorrect computation with games having empty core.

     Modification:

     The parallel mode has been optimized. For more details see the README file and/or
     the notebook file TuGamesParallel.nb in the Documentation folder.

     Tested up to 32 SubKernels. For running even the cddmathlink libraries in parallel see
     the explanation in the README file and/or for an example the notebook file
     TuGamesMovieParaModeV6.nb in the Documentation directory.

     To keep track our package with the new graphic concept of Mathematica 6.x and higher,
     the following functions have been added or modified to depict solutions for three-person games:

     AnimationKernelProperty2d[]  - new function, dedicated for Mathematica 6.x and higher only!
     StrongEpsCore2d[]            - new function, dedicated for Mathematica 6.x and higher only!
     FilledCoreV6[]               - is an upgrade of FilledCore[] for version 6.x and higher.

     The graphical functions to plot game solutions for four-person games reside in the package
     IOTuGamesV6.m. To use the old graphic concept one has to load the package IOTuGames.m
     in connection with << Version5`Graphics`.
     All graphical extensions require the cddmathlink libraries.

    Version 1.8:

     ModifiedNucleolus[]       - Complete code revision to get rid of large round-off errors.
                                 We use now a fast simplex method instead of an interior point
                                 method to increase its reliability.
     ModifiedKernel[]          - See ModifiedNucleolus[].
     LexiCenter[]              - See ModifiedNucleolus[].
     PreNucleolus[]            - See ModifiedNucleolus[].

     The behavior of these functions have changed. Options are obsolete now. ModifiedNucleolus[] does
     not compute anymore the pre-nucleolus for inessential games, it returns the empty set instead.


    Version 2.2:
       Transcription of the old option rules to the new ones invented by Mathematica 8.x.
       This package is now exclusively dedicated to Mathematica version 8.x and higher.

       FilledCore[] is deprecated now. Using the old graphic concept Version 1.8 is required.

    Version 2.3:
       Modification:
       Change protected command SubsetQ[] to SubSetQ[] from the VertexEnum package. Order is
       reversed to SubsetQ[] which is new in Mathematica version 10.x.

       Some minor code revision.

       New Functions:
        MLExtension[]           - Computes the multilinear extension of the game.
        ShapleyValueML[]        - Computes the Shapley value from the multilinear extension of the game.
        PreKernel[]             - Computes a pre-kernel point by Algorithm 8.2.1 of Meinhardt (2013).

       Bug fixes:
        FindKernelSolution[] convergence process to the kernel should be now more robust.

    Version 2.4:

       Some minor code revision.

    Version 2.5:

       Adding the function ApproxNuc[] to compute the (p, k)-nucleolus which is an approximation of
       the nucleolus by a non-linear optimization approach, i.e., minimizes a p-norm. The function
       NonLinNuc[] is based on this function to compute the nucleolus. We extended this idea to the
       pre-nucleolus through the commands ApproxPreNuc[] and NonLinPreNuc[]. In addition, we added
       the least square computation of the (pre-)nucleolus by the functions LSNuc[] and LSPreNuc[].

       Functions to compute the barycenter of the extreme points of the core, dual cover game,
       dual extension, primal extension, modiclus, a proper modified pre-kernel element,
       potential of a game, the Lorenz solution, and Dutta-Ray solution for convex games
       have been added.

       Bug fixes:
        An insufficient coloring of the function FilledCoreV6[] caused by an incomplete delaunay
        triangulation of the Mathematica built-in function DelaunayTriangulation[] has been fixed.



    Version 2.5.1:
	Installation procedure has changed. The package is now distributed as a Paclet. The documentation
	was revised and extended. About 230 pages were added to the documentation. The error
	handling of functions was improved.

	The functions to compute the modiclus and related solutions have been parallelized, and are part
	of the ParaTuGames package.

	Some minor code revision and bug fixes.

    Version 2.5.2:

	We have revised the Installation procedure of the Cddmathlink library, which makes it not any more necessary
	to explicitly formulate some conditions for all operating systems. Moreover, some default binaries for RHEL 7.5
	and Mathematica 11.3 ship now with the package. Alternatively, we also added some for RHEL systems for
	Mathematica 10 and later. These binaries must be copied into the pre-specified folder.


	Some minor code revision and bug fixes.

    Version 2.5.3:

        Adding some binaries for Mathematica 10.0 or later on OS X 10.9 or later. We are very grateful
        to Szabolcs Horvát for providing these to the community.

        Change of the License to the MIT License terms.


    Version 2.5.4:

        Functions to compute and to verify the simplified modified pre-kernel/nucleolus are added. They are called
        SMPrenucleolus[],IsSMPrenuceolusQ[],SMPreKernel[],IsSMPreKernelQ[]. For the last two commands we implemented
        parallel counterparts called ParaSMPreKernel[], and ParaIsSMPreKernelQ[] respectively.

        Adding the function BalancedCollectionQ[] that should replace in the future the function BalancedSelectionQ[].
        For n=>4 the function returns incorrect results, probably caused by a bug of the DualLinearProgramming[] function.

        Example of incorrect results:

        Consider the collection of sets given by

        cS4={{1}, {2}, {3}, {1, 2}, {1, 3}, {2, 4}, {3, 4}, {2, 3, 4}};

        then the return value is false

         In[29]:= First[BalancedSystemQ[cS4, Range[4]]]

         Out[29]= False

        However, the system is balanced, since the balancing weights are given by

        whgs = {3,1,1,1,1,2,2,1}/5

        Related to this context we provide the function BalancedInequalityQ[] to check whether a balanced system
        satisfies a balanced inequality of a TU-game. Notice that if all balanced systems satisfying this property
        non-emptiness of the core is guaranteed. Recall that for n=4 we need to check 9 equivalence classes, however,
        for n=6 we have to check 158 classes.


     Version 2.6.0:

       Adding functions to compute the Chi-Value, EPSD-Value, Gately-Value, PD-Value, and the nucleolus by the Leng and Parlar (2010) formulae for three-person
       zero-normalized and super-additive games. Changing the package extension from *.m to *.wl.

       Performing some code maintenance and minor bug fixes.

     Version 2.6.1:

       Code revision and optimization. The (anti-)pre-kernel computation is now faster up to a factor 3 in serial as well as in parallel.

    Version 2.6.2:

       Adding functions:
            -  AlmostConvexQ[] to check if the game is almost convex.
            -  AlmostConcaveQ[] to check if the game is almost concave.
            -  ADMCGameQ[] to check the property if the game satisfies almost diminishing marginal contributions.
            -  AIMCGameQ[] to check the property if the game satisfies almost increasing marginal contributions.
            -  kConvexity[] to check if the game is k-convex.
            -  EANSCValue[] to compute the Equal Allocation of Non-Separable Contribution/Cost value.

       Revised the functions:
              ConvexQ[], ConcaveQ[], Nuc1convex[].

       Minor Bug fixes and code revision.

    Version 3.0.0:

       Not any more backward compatible to Mathematica versions smaller than 12 due to the port
       to the new collection of algorithms for solving convex problems introduced in version 12.

       ConstrainedMax/ConstrainedMin and LinearProgramming/DualLinearProgramming are replaced by LinearOptimization.

       This replacement has been conducted for the following functions:

       Nucleolus                  -- Part of CooperativeGames. Function originally written by M. Carter.
       LeastCoreAux               -- Part of CooperativeGames. Function originally written by M. Carter.
       ModifiedNucleolus          -- Part of TuGames.
       ModifiedKernel             -- Part of TuGames.
       PreNucleolus               -- Part of TuGames.
       LexiCenter                 -- Part of TuGames.
       Modiclus                   -- Part of TuGames.
       IsModiclusQ                -- Part of TuGames.
       Kernel                     -- Part of TuGames.
       KernelCalculation          -- Part of TuGames.
       BalancedInequalityQ        -- Part of TuGames.
       BalancedCollectionQ        -- Part of TuGames.
       EpsCore                    -- Part of TuGames.
       FirstCriticalVal           -- Part of TuGames.
       DeltaLP                    -- Part of TuGames.
       FeasibleConstraints        -- Part of TuGames.
       KernelVertices             -- Part of TuGames.
       SolvePrimal                -- Part of TuGamesAux.
       SolveDual                  -- Part of TuGamesAux.
       BalancedSystemQ            -- Part of TuGamesAux.
       ParaModiclus               -- Part of ParaTuGames.
       ParaIsModiclus             -- Part of ParaTuGames.

      New added functions are:
      WeaklyBalancedCollectionQ   -- Part of TuGames.
      WeaklyBalancedSystemQ       -- Part of TuGamesAux.


     The function WeaklyBalancedCollectionQ replaces WeaklyBalancedSelectionQ, the latter will be deprecated in a future version.

   Version 3.0.1:

      Improved exception handling for functions using LinearOptimization. Performance improving of revised functions in Version 3.

      Some minor bug fixes.

   Version 3.0.2:

     Adjusting the functions

       ApproxPreNuc                -- Part of TuGames,
       ApproxNuc                   -- Part of TuGames,
       DuttaRay                    -- Part of TuGames,
       LorenzSolution              -- Part of TuGames,
       NonLinPreNuc                -- Part of TuGames,
       NonLinNuc                   -- Part of TuGames,
       PreKernelSolution           -- Part of TuGames

    to the new set of optimization algorithms.

    Adding options to the functions PreKernelQ, MaxExcessBalanced, AntiPreKernelQ, MinExcessBalanced, ParaPreKernelQ, ParaMaxExcessBalanced, ParaAntiPreKernelQ, ParaMinExcessBalanced.

    Improving the performance of PreKernelSolution.

    Some minor bug fixes and code revision.



    Version 3.0.3:

    1. Revision
       Functions usage documentation.

    2. Modification

    2.1 Adding the new functions:
       -  VetoRichPlayers[] determining the set of veto players.
       -  DecomposeInPositiveGames[] decomposes a TU game into positive games, i.e., convex games.
       -  AvConcaveQ[] checking if the game is average-concave.
       -  AverageConcaveQ[] checking if the game is average-concave.
       -  AlmostAverageConvexQ[] checking if the game is almost average-convex.
       -  AlmostAverageConcaveQ[] checking if the game is almost average-concave.
       -  PositiveGameQ[] checking if the game is positive, i.e., convex.
       -  SemiConvexQ[] checking if the game is semi-convex.



    3. Bug Fixes

       Fixing Bug in AnimationKernelProperty2d with the FigureSize option in Manipulate.

       Fixing an issue in ModifiedNucleolus and ModifiedKernel w.r.t. the boundary Infinity, which was replaced by 1000*v[T].

       Some minor bug fixes and code revision.

   Version 3.1.0:

    1. Revision

      Update of the Documentation references pages and adjustment to the paclet system of Mathematica Version 13.0.

    Version 3.1.1:

    1. Modification

    Adding the new functions:

    ProductGame, ProbabilityGame, HarsanyiValue, ShapAirProb, TauValAirProb, NucAirProb.

    Version 3.1.2:

    1. Revision

      Update of the Documentation references pages.

    2. Modification

    Adding the new functions:

    AirportProblem.

    3. Bug Fixes

    Fixing Bug related to the Harsanyi dividends in ProductGame and KernelCalculation related to missing output.

    Some minor bug fixes and code revision.

    Version 3.1.4:

    1. Revision

      Update of the Documentation references pages.
      Code Revision for FilledCoreV6 to avoid overlapping labeling.

    2. Modification

    Adding the new functions:

    FilledWeberSetV6, PlotWeberSet3dV6, CddGmpPlotWeberSet, EqDistDividends, CostLocationGame, DualProbabilityGame, PQNorm.

    3. Bug Fixes

    Some minor bug fixes and code revision.