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[5]:=
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[6]:=
ConvexQ[ExpGame]
Out[6]=
False
In[7]:=
AvConvexQ[ExpGame]
Out[7]=
False
In[8]:=
CoreQ[ExpGame]
Out[8]=
True
In[9]:=
CddGmpVerticesCore[ExpGame]
Out[9]=
{{0,2,2}}
In[25]:=
VerticesCore[ExpGame]
Out[25]=
{{{0,2,2}},{{7,10,8,4,1,6,5}}}
In[23]:=
cq={{0,2,2},{1,2,1},{2,2,0} ,{2,0,2}};
In[24]:=
BelongToCoreQ[ExpGame,cq]
Out[24]=
{True,False,False,False}
In[11]:=
Kernel[ExpGame]
From In[11]:=
Game has nonempty core
Out[11]=
{0,2,2}
In[12]:=
perm=Permutations[{4,0,0}];
In[13]:=
PreKernelSolution[ExpGame,perm,SolutionExact\[Rule] True]
Out[13]=
{{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