Mafia 1 - Poly

[DrSpike]

I'll vote for him if you like.

[rah]

The simplest thing to do is that once the villager's count is down to 4, the mafia wins. As far as I'm concerned, if they last that long, they've beaten the odds and deserve to win.

[Jamski]

The simplest thing to do is that once the villager's count is down to 4, the mafia wins.

Nah, that's the most exciting vote of all... shame we have to wait so long for SPACED COWBOY!

-Jam

[rah]

Originally posted by vovan
Yes, it would be great if you could change it, rah.

Done

[Adagio]

Originally posted by vovan
Yes, it would be great if you could change it, rah.

I read it as "(...) change it to rah"

[Jamski]

Vovanism ----> vovan.

Ummm, OK

-Jam

[DrSpike]

Originally posted by rah
The simplest thing to do is that once the villager's count is down to 4, the mafia wins. As far as I'm concerned, if they last that long, they've beaten the odds and deserve to win.

It is intuitive that the odds of surviving that long must be quite low for the mafia, but it turns out to be a mistake. Put aside your intuitions and have a try at calculating it, then see what you think.

Besides, stopping the game at 4 is against the spirit, since the game ends at the most interesting point. True if the game goes on for the final vote at 3 *then* the odds of the overall game are against the mafia, but again, not by as much as you might think. Overall the case for allowing the vote at 3 which decides matters is a strong one IMO.

[Spaced Cowboy]

Sorry guys, too many early moring meets this AM, my vote is for Skanky, just because I think that if it was Jamski, he would have knocked me off already.

[rah]

Originally posted by DrSpike

It is intuitive that the odds of surviving that long must be quite low for the mafia, but it turns out to be a mistake. Put aside your intuitions and have a try at calculating it, then see what you think.

What's the calc. I figured it about 50/50 that the villagers would nail the mafia by around 7 villagers.

[DrSpike]

Well it's hard to pin down, but my back of the envelope suggests not much worse that 50/50 for the mafia that he survives to the last 4. Under my proposed scheme where the vote continues the mafia has a 20ish% of winning.

That probably is better odds than you might think at first glance. Take the last 3 as an illustrative case. With 3 people the mafia's vote changes the chance he'll be killed from a a higher probability to 1/4, since the other 2 players' combinations are:

MAFIA MAFIA
Player 1 MAFIA
MAFIA Player 2
Player 1 Player 2

With the Mafia's vote deciding in any of the cases except MAFIA MAFIA.

Overall under my scheme I'd be surprised if in many repititions of the game it'd be much worse to be a mafia than an individual player. Under the scheme proposed where the mafia wins at 4 his chances are a lot greater than the other players.

Plus it's more fun to vote it out!

[rah]

Yes, it's more fun to vote them out.

Assuming randon chance of identifying mafia.
assume that each round the odds of identifying mafia is
1/16
1/15
1/14
1/13
1/12
1/11
1/10
1/9
1/8
1/7
1/6
1/5
1/4
1/3
the sum of the odds would be
1/16 0.06
1/15 0.13
1/14 0.20
1/13 0.28
1/12 0.36
1/11 0.39
1/10 0.55
1/9 0.66
1/8 0.79
1/7 0.93
1/6 1.10
1/5 1.30
1/4 1.55
1/3 1.88

So after 6 villagers have voted the odds are in favor of the mafia being identified.

But since the mafia will never vote for themselves, (I'm not sure how to factor that in) I'm sure that reduces it (less at the higher levels and more so near the bottom). SO it's probably closer to after their are five left.

[DrSpike]

Um, the mafia only has a chance of dying on the rounds the villagers vote! He isn't going to kill himself. For instance in the first round the probability is not 1/16, it's 0.

Think about it...........your method suggests a chance that the mafia will die in excess of 100%.

Now, if you add the randomised probabilities for the correct rounds together it would come out at just over 100% after the voting with 3 left (the reasons aren't important except for mathematicians). Of course the trick is working out a 'correction' for the mafia's vote. It can be derived I think (it may need to be numerically simulated), but I cheated to yield the estimate above. My back of the envelope calculations are accurate to +/-5% I'd say.

[rah]

HAHAHAHHA, oppps forgot about the first round so recalced and the alternating rounds. I'll recalc it.
SUM
1/15 0.07
1/13 0.14
1/11 0.23
1/9 0.35
1/7 0.49
1/5 0.62
1/3 1.02

Yes, you are quite correct. I doesn't swing in the villagers favor till the bitter end and not even then if you add in mafia not voting for themselves. Yes, quite counter intuitive.

[DrSpike]

The mafia doesn't vote every even round. He is too busy killing someone.

If you recalc it for the odd rounds you'll get what I suggested in the last post, around about 100% chance of death after the voting for the last 3. But of course that is not a prediction, it's just true by definition. The prediction comes from scaling the probabilities for the effect of the mafia's vote.

[DrSpike]

Ah, I see you recalculated it. It's quite interesting really. I always try and use things like this as examples when teaching, because the underlying principles can seem sooooooo boring in the abstract. My latest idea is "Optimal game-theoretic strategies in The Weakest Link"