       Re:mathematica and CGT

• To: mathgroup at smc.vnet.net
• Subject: [mg92770] Re:[mg92741] mathematica and CGT
• From: Holger Meinhardt <Holger.Meinhardt at wiwi.uni-karlsruhe.de>
• Date: Mon, 13 Oct 2008 06:16:21 -0400 (EDT)

```
Dear Franciso,

I checked your small problem with my Mathematica package
``TuGames'' and I can confirm your result. I get the following resutls:

In:=
ExpGame:=(T={1,2,3};
Clear[v];
v[{}]=0;v[{1}]=0;v[{2}]=2;v[{3}]=2;
v[{1,2}]=2;v[{1,3}]=2;v[{2,3}]=2;
v[T]=4;)

In:=
ConvexQ[ExpGame]

Out=
False

In:=
AvConvexQ[ExpGame]

Out=
False

In:=
CoreQ[ExpGame]

Out=
True

In:=
CddGmpVerticesCore[ExpGame]

Out=
{{0,2,2}}

In:=
VerticesCore[ExpGame]

Out=
{{{0,2,2}},{{7,10,8,4,1,6,5}}}

In:=
cq={{0,2,2},{1,2,1},{2,2,0} ,{2,0,2}};

In:=
BelongToCoreQ[ExpGame,cq]

Out=
{True,False,False,False}

In:=
Kernel[ExpGame]

From In:=
Game has nonempty core

Out=
{0,2,2}

In:=
perm=Permutations[{4,0,0}];

In:=
PreKernelSolution[ExpGame,perm,SolutionExact\[Rule] True]

Out=
{{0,2,2}}

An old version of my package can be found on:

http://library.wolfram.com/infocenter/MathSource/5709/

To get some rough idea what can be done with the package
consult also the following web-site

http://members.wri.com/jeffb/visualization/gametheory.shtml

Upon request I can also provide you with a new version
of my package.

With best regards,
Holger

> Dear Friends:
> This is a question indirectly (but importantly for me) related to
> Mathematica. >I am trying to make Mathemtica code to represent
> cooperative game theory >concepts (of course, expecting it to work
> only for very simple games, because >making algorithms for
> practically all these concepts is an NP-Hard problem).

> Of course, one would want to check the code with simple numerical
> examples. I >come across the following one in the book by Milan Mares
> (Fuzzy cooperative >games, p. 21).

> The example is the following: Game (N,v), N={1,2,3}, v(empty set)=0
> v(1)=0
> v(2)=v(3)=v(12)=v(13)=v(23)=2
> v(N)=4

> The problem: find the core of this game.
> Now, Mares claims that this core has many components: (0,2,2),
> (1,2,1), >(2,2,0) and (2,0,2).
> My Mathematica code finds only one core element (0,2,2), as both
> players 2 and >3 can guarantee for themselves at least 2 without
> joining any coalition.
> I think this is the correct answer, but I may by fatally wrong...
> Can anybody out there help me?
> Thanks,
> Francisco

```

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