The setup is purgatory https://wiki.mafiascum.net/index.php?title=Purgatory designed by whoever has the credit on that wiki page. Go read the rules there.
What you need to know is that I restricted the setup to always have an odd number of players (since the game needs to start with a hell phase according to the rules), and there also needs to be an odd number of mafia (since based on the win conditions, x mafia in heaven x mafia in hell, 0 mafia left in game is not a win or a loss for either faction :S)
1 Mafia, 2t townies:
2 : 1 evaluates to 66% winrate.
1/3 chance to hit the mafia in hell phase 1 and win. 2/3 chance to go to a coinflip judgement day = 1/3 + 1/3 = 2/3. But the problem with this is that there are no townies in heaven to actually MAKE the judgement day descision.
So let's add an extra rule to mend this hold in the original setup: If there were no townies in heaven on judgement day, the townies in hell do the choosing! :evil face:
But as you add more townies... the setup gets worse for town.
4 : 1 evaluates to 60% winrate.
And as you add more and more townies, the EV approaches 50% from above, as shown by this chart.
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Let's try the values in the original setup.
6 : 3 evaluates to 32% EV. looks bad
10 : 5 evaluates to 13% EV. looks even worse!!!! (lets remember things work out a little differently in practice then on paper due to the human element of using logic and stuff!!!)
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Let's try 3 mafia, 2t townies
This appears to converge to 25% EV from above as the number of town grows arbitrarily large, just like the last setup.s
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Now lets try 5 mafia 2t town.
Wait wtf? This one starts at a 16% EV, dips to a minimum of 11.56% at 22 townies, then starts increasing asymptotically towards 12.5% EV (or 1/8).
So it would seem like the pattern so far is that when # of mafia = m = 2n + 1, n in naturals including 0, town's winrate as the number of townies approaches infinity approaches 1 / 2^(n+1).
But what about this weird behavior about dipping below the convergence value then going back up to it? lets get more results and observe
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7 mafia
behaves the same as 5 mafia, dipping below the convergence value again.Appears to converge to 1/ 16 = 0.06%, as expected. It sure is weird to think that a game with INFINITE townies gets exponentially more threatened by a mafia team with 2 extra members!
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9 mafia
dips below again. Appears to converge to 1/32.11 mafia
also behaves as expected. 1/6413 mafia
also behaves as expected. 1/128Looks like we found the pattern. Thats super weird. Not sure why the graph starts behaving the way it does once we get to 5 mafia.
