How Helen spent her afternoon... (Partner Stats)

I was curious to find out the likelihood that the talismans would be given out, so I toyed around with some numbers, and worked out the following table...


http://i50.tinypic.com/70baip.png to see it un-cropped

The columns tell us how many people are left in the game. The rows tell us how many partnerships (as 2-person groups exclusively, for the purposes of this analysis) are still intact.

The "input" used to kickoff the system was "20 players, 10 partnerships."

Based on this, we can see that the bottom-left triangle is impossible to reach, since from (20.10), there's no way to get the number of partnerships to drop that fast. The top-right triangle is also impossible, because, based on the number of players remaining at any point, we can never have less players than 2x the number of intact partnerships.

We can figure out the likelihood of talismans having been given out at a certain point by picking a "Final X players" column and adding together the contents of rows labeled with "1" or "0" partnerships remaining.

Thus, we can see that (ignoring social factors entirely and looking entirely at statistics) at the beginning of the game, there was a 95% talismans would get dealt out by the final five. Conclusion: Brenda, your season was RIDICULOUS. (Not really a conclusion, since we already know that... damn 5%.)

So...

• If Marty leaves tonight, that leaves us 5 pairs in F12.
  If we change the "input" to that, we would see there being a 90% chance talismans get handed out by F5.
• If Marty survives tonight, that leaves us 4 pairs in F12.
  If we change the "input" to that, we would see there being a 94% chance talismans get handed out by F5.

Brook/Jimmy T and Jimmy J/Erica really screwed the pooch on this one, but the odds should still be strongly in our favor to see these immunities come into play.

I would like to see how you would tackle attempting to add social factors into this. Taking that more partnerships will try to work together than not early game - especially with a reward for doing so - and this then gradually turning to multiple partnerships teaming up or turning on each other.

Numbers get interesting to me at this point; We had three intact at F8. but the one decimated before that was at F14. Despite being on opposite sides, there was definitely an element of let's keep three partnerships alive between us - because none of us wanted to get into the situation where there was two only. That is giving the other remaining partnership an excuse to take you out. We were only comfortable doing that once two sets had struck up a deal to ride the twist out to F5, and even then never all that comfortable. I know I was the one to break it but still :P I was super paranoid every round that this was the time when they were going to do it to us.

Was our Season then the Nash Equilibrium from Greece on a much more concentrated scale?