## [EV] Purgatory

This forum is for discussion of individual Open Setups, including theoretical balance.
Awoo
Best New Mish Mash Player

Joined: September 01, 2017
Pronoun: He
I don't think mith did this one yet!!

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.

================================
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!!!)

================================

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

================================
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

================================

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!

================================

9 mafia dips below again. Appears to converge to 1/32.

11 mafia also behaves as expected. 1/64

13 mafia also behaves as expected. 1/128

Looks 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.

mith
Godfather

Joined: March 27, 2002
Location: Plano, TX
Pronoun: He
Heh.

So, the reason this happens for 1 Mafia is that there are effectively P slots for the Mafia to end up with, and as P grows larger the ratio between hell (win) and heaven (loss) slots goes to 1/2.

At first glance, I would have expected this to be similar for more Mafia. The obvious difference, though, is that you can reach a final judgement which is not a simple "put Mafia in one slot or the other" choice - that is, if you hit 3:3, you have a 1/2 chance of losing immediately, without getting to try to put both the other scum in the right place, while deciding correctly is only worth 1 "point".

All that said, at 3:4 I still get an EV of 17/35, and I'd be really surprised if 3:6 drops that far. Are you sure your calculations are correct?

implosion
Polymath

Joined: September 09, 2010
Location: zoraster's wine cellar
Pronoun: He
How are you calculating this? Programmatically? I could very easily be missing something but the asymptotic numbers look wrong to me.

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.

I think this (and in fact any fixed number of mafia with an increasing number of town) should converge to 50%. As t grows large, the probability that all of the mafia are sorted into either hell or heaven by the time that the game reaches a judgment day grows to 1, because like mith says they're effectively randomly being put into "slots" and the number of slots that could be alive in a judgment day stays fixed. But then effectively all 3 mafia are being randomly sorted into heaven and hell, which should give a 50% chance that the majority wind up in heaven and a 50% chance that the majority wind up in hell.

That said, if this is some programming error, I am interested to see what the setup that these actually are the EVs for is :p

Sukima
Watcher

Joined: October 03, 2014
Pronoun: He
You can have an even number of mafia though, because once one side hits N of 2N mafia, game is over.

Sukima
Watcher

Joined: October 03, 2014
Pronoun: He
From just manually cranking out scenarios in a spreadsheet, I'm getting for 3:6....

17/84+1/14+3/21+3/28 = 17+6+12+9 / 84 = 11/21 EV for town, ~52.38%?

Did you maybe forget that judged people always go to hell regardless of if it would have been a heaven phase next, and the next phase is always hell? Since that allows for situations like Townie Hell, Judgement day Mafia Hell, Mafia Hell again Town wins.

Also getting 17/35 for the 3:4 set.

Awoo
Best New Mish Mash Player

Joined: September 01, 2017
Pronoun: He
uhhhhhhhhhhhhhhhhhhhhh

i think thers aa

problemm in my programmmmmmmmmmmmmmmm

but i dont knoww hat it is

Sukima
Watcher

Joined: October 03, 2014
Pronoun: He
Did you make sure that judgement days go into hell days? Since that favors town pretty greatly where if they get a correct judgement day, they immediately get another chance to lynch mafia.

[ + ]