## TuGames

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 8.x and higher. In this respect, the graphical features should run on all platforms.

## Getting Started

The most simplest method is to install the Paclet directly from GitHub under Mathematica V11.3 or later while executing

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

for versions smaller than Mathematica V11.3, please consult the project home at GitHub

TuGames

Having installed the paclet properly, we load the package by

Needs["TUG"] 

to be able to consider a small three-person TU-game. For this purpose, define first the player set

In[2]:= T={1,2,3};

and then the characteristic values through

In[3]:= clv={0,0,0,0,90,100,120,220};

These data are necessary to define the TU-game, which can be accomplished by

In[4]:= ExpGame = DefineGame[T,clv];

After that we are in position to impose some basic operations like

In[5]:= ConvexQ[ExpGame]
Out[5]= True

In[6]:= CoreQ[ExpGame]
Out[6]= True

In[7]:= AverageConvexQ[ExpGame]
Out[7]= True

In[8]:= GameMonotoneQ[ExpGame]
Out[8]= True

In[9]:= ker=Kernel[ExpGame]
170  230  260
Out[9]= {---, ---, ---}
3    3    3

In[10]:= shv=ShapleyValue[ExpGame]
Out[10]= {65, 75, 80}

In[11]:= nc=Nucleolus[ExpGame]
170  230  260
Out[11]= {---, ---, ---}
3    3    3

In[12]:= prn=PreNucleolus[ExpGame]
170  230  260
Out[12]= {---, ---, ---}
3    3    3

For more getting information about its usage and features, open the Mathematica Documentation Center, and enter into the panel just TUG or guide/InstalledAddOns, or scroll down, and click at the end of the page at the right corner on the field Add-ons and Packages`. This directs you to the Documentation of the package from which several examples can be called up or more detailed information can be retrieved. From there, one has access to about 230 pages of documentation.