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This is a lot more complicated than what I had in mind.  The original idea
is natural extensible...  Say you, me, and Joe.  I have two games with Joe,
selected by the method I described.  You have two games with Joe,
independantly selected by the same mathod.  I have two games with you,
again selected independantly...   Now, granted, in even this 3-person case,
it could result in 6 different games.  If four people play, it could be 12
different games.  If five people play, it could be 20 different games... 
But I don't think that this is really a problem.  For one thing, if 10
people play, each player is not likely to have 9 variants, and each
opponent wanting a different one.  And also, if there are lots of different
games played, I'm not sure that's a problem anyway.