Mathematica Package: TuGames Version 3.1.4

Contents:
 1.  Introduction
 2.  Custom Installation
 3.  Getting Started
 3.1 Basic Example
 3.2 Using a different Solver
 4.  Running the Package in Parallel
 4.1 General Procedure
 4.2 Running the Cddmathlink libraries in Parallel
 5.  MATLink and MATtug (Matlab toolbox MatTuGames)
 6.  Documentation
 7.  Graphics
 8.  Acknowledgment
 9.  License

1. Introduction

TuGames is a Mathematica package to determine and to check some game properties of transferable utility games. It provides more than 200 different functions to calculate, for instance, (pre-)kernel elements, the (pre-)nucleolus, the modiclus, the modified and proper modified pre-kernel, the Shapley value, Lorenz solution, Dutta-Ray solution, excess payoffs, the tau-value, chi-value, Gately point, the vertices of a core, and much more. Moreover, it verifies if the game is convex, average-convex or superadditive just to mention some interesting game properties. It can be used in serial as well as in parallel mode, and in conjunction with MATLink to build up a Matlab connection to invoke the toolbox MatTuGames. This package is exclusively dedicated to Mathematica version 12.x and higher, because TUG-3.0.0 is the first version that is transcribed to the new collection of algorithms for solving convex problems introduced in version 12. For Mathematica versions smaller than 12.0 one should use TUG-2.6.2. This version remains compatible with the most recent Mathematica version as long as ConstrainedMax/ConstrainedMin and LinearProgramming/DualLinearProgramming are supported by Wolfram Research, otherwise one has to switch to TUG-3.0.0 or later. In this respect, the graphical features should run on all platforms. As a highlight, a fast algorithm to seek for a pre-kernel element is implemented in the updated package. This algorithm is described in the book

The Pre-Kernel as a Tractable Solution for Cooperative Games
An Exercise in Algorithmic Game Theory
Series: Theory and Decision Library C, Vol. 45
Meinhardt, Holger Ingmar
2014, XXXIII, 242 p. 8 illus., 4 illus. in color

More information can be found here:

• Book
• DOI

Moreover, we have included a new package called ParaTuGames to literally run the major commands of TuGames in parallel. Notice that this package has not passed its stage of development. Hence, the user should use it with care and for testing only.

In addition, note that these functions/interfaces had to be provided in a Global context to call the parallel commands of Mathematica within a package. This design requires that during a running session with ParaTuGames no other variables of Global context should be defined to avoid name conflicts and wrong results. Though use only the commands that start with the prefix Para, and which returns a description of how to use it. To get an idea of its usage, invoke in the panel of the Documentation Center TUG and click then on hyperlink "ParaTuGames Package" to open the file TUG/Tutorials/ParaExpGamePers12 from the Documentation Center (cf. Section 6). Finally, this package does not require the configuration procedure as under Section 4.

A modified version of the package CooperativeGames that has been developed by M. Carter to run properly with Mathematica 12.x and higher is enclosed. It must be mentioned in this place that some commands of TuGames require routines that have been provided by the package CooperativeGames. A description of the package CooperativeGames can be found in

Hal R. Varian, Economic and Financial Modeling with Mathematica, 
Telos Springer Publisher,1993, Chapter 8. 

Furthermore, if one is interested in computing the vertices of a core the Mathematica package VertexEnum written by K. Fukuda and I. Mizukoshi must also be installed on the computer. But note that this function is very slow in computing all vertices of a core even for modern computers. You can overcome these shortcomings of VertexEnum by installing the C-library Cddmathlink written by the same authors to perform the same computational task more efficiently. It can be found under

Cddlib

for various UNIX, MacOSX and for Window systems. The library is linked via MathLink with the Mathematica Kernel. Under Linux and Mathematica 11.3 we complied successfully the binaries using the following compiler flags

MLFLAGS = -lML64i4 -lpthread -lrt -luuid -ldl 

See also our post at

Vertex Enum

to overcome possible pitfalls, which might occur during a compilation of the Cddmathlink binaries.

For Windows and MacOSX Operating Systems pre-build binaries are available under

Cddlibml Binary

This C-library is required to use the graphical features of the package. Notice that the functions that based on this library are activated. To deactivate them comment in the corresponding lines in the TuGames.m file (cf. Section 2).

In order to install the library, we recommend it to do so in the TUG directory tree. We have provided several folders that should receive the Cddmathlink executables for different Operating System ABIs. To retrieve your system ID execute in a Notebook

$SystemID

We have provided some executables for RHEL 7.5, RHEL 8.2, and MacOSX. Unfortunately, we are not experienced enough under Windows to provide any for those. Thus, we invite the community to fill that gap and to provide some.

In order check for Mathematica versions smaller than 12.1 the paths where you have to install the Cddmathlink library call the command

In[1]:= PacletInformation["TUG"]

Out[1]= {Name -> TUG, Version -> 3.1.4 BuildNumber -> , Qualifier -> , WolframVersion -> 12+,
         SystemID -> All, Description -> A Mathematica Package for Cooperative Game Theory,
	 Category -> , Creator -> Holger Ingmar Meinhardt <holger.meinhardt@partner.kit.edu>,
	 Publisher -> , Support -> , Internal -> False, Location -> /home/kit/xxx/xxxx/.Mathematica/Paclets/Repository/TUG-3.1.4
	 Context -> {TUG`coop`, TUG`vertex`, TUG`}, Enabled -> True, Loading -> Manual}

To get the same information and beyond that under Mathematica version 12.1, it is required to execute

In[1]:= PacletObject["TUG"][All]

Out[1]=  {"Name" -> "TUG", "Version" -> "3.1.4", "WolframVersion" -> "12+", 
          "Qualifier" -> "", "SystemID" -> All, "Description" -> "A Mathematica Package for Cooperative Game Theory",
          "Category" -> Missing["NotAvailable"], "Keywords" -> Missing["NotAvailable"], 
          "UUID" -> Missing["NotAvailable"], 
          "Creator" -> 
          "Holger Ingmar Meinhardt <holger.meinhardt@partner.kit.edu>", 
          "URL" -> "https://github.com/himeinhardt/TuGames", 
          "Internal" -> False, 
          "Context" -> {"TUG`coop`", "TUG`vertex`", "TUG`"}, 
          "Loading" -> Manual, "AutoUpdating" -> False, "Enabled" -> True, 
          "Location" -> "/home/kit/xxx/xxxx/.Mathematica/Paclets/Repository/TUG-3.1.4"}

or alternatively

Information[PacletObject["TUG"]]

Then open the directory for Mathematica versions smaller than 12.1 by

SystemOpen@Lookup[PacletInformation["TUG"], "Location"]

or for Mathematica version 12.1 one opens the directory via

SystemOpen@Lookup[PacletObject["TUG"][All], "Location"]

This returns the root directory of TUG, and shows you the three folders cddmathlink, cddmathlink2 and cddmathlink2gmp that should receive the Cddmathlink executables.

The author has tested the functions extensively under LINUX x86/64, HP-UX and AIX. Furthermore, the package was also be installed and tested successfully under Windows XP and MacOSX. For more recent Windows OS the author has no experience. But the programming language of Mathematica is system independent, thus, there should no problems occur on these operating systems to run the basic functions of the package.

2. Custom Installation

Since Version 2.5.1 the package is distributed as a Paclet, which allows a custom installation. The latest version is TUG-3.1.4, which is only compatible for Mathematica version 12.0 or higher. This version is not any more usable for versions that are smaller than 12.0. Notice that TUG-2.6.2 is the latest version that is compatible with Mathematica versions smaller than 12.0, and will remain compatible with the most recent Mathematica version as long as ConstrainedMax/ConstrainedMin and LinearProgramming/DualLinearProgramming are supported by Wolfram Research, otherwise one has to update to TUG-3.0.0 or later. The installation of the package requires about 90 MB free hard-disk space, and the Mathematica version should not be smaller than 10 while installing Version TUG-2.6.2. For Mathematica versions smaller than Version 10, a manual installation is requested. In that case one has to follow the guidelines related to the associated operating system that can be found from the Mathematica documentation.

The simplest method is to install the Paclet directly from GitHub under Mathematica V12.0 or later while executing

ResourceFunction["GitHubInstall"]["himeinhardt", "TuGames"]

For smaller versions than 12.0 start Mathematica,