Mafia 1 - Poly

[DrSpike]

Originally posted by Sheep
Umm then the Mafia wins, because all the villagers are dead.

Some more thought needs to be given to the rules. Earlier when you said both die if there is a tie in the townspersons' votes I thought it sounded strange. And clearly if all get one vote then all die, including the Mafia!

Presumably a Reservoir Dogs standoff.

This has to be the last chance the villagers have, and if they get it wrong then the mafia just kills the last remaining villager and wins. So in that case the Mafia does have to survive to last to win.

But you need to decide why the mafia wins if the vote is tied. And, what is more, your proposed system of breaking ties at any stage ruins things, since the even number of players at the start ensures that the townspeople are voting at 3 persons left, not the mafia killing. This is the way it has to be, so the earlier proposed method makes no sense.

[DrSpike]

I think it should be as follows:

The mafia votes last, and will win unless the villagers both vote for him, since he will just vote for the villager that was voted for if it is required.

For ties earlier on I don't know..........you could get the other voters to break the tie, or you could use past votes.

[Jamski]

Sorry Spikie, I thought it was you, me and Skanky left for some reason.

How about.... if the vote is a tie NOONE is killed by the townspeople that day, as they have no clear marjority. The next night the Mafia kills again, and the people vote again.

-Jam

[DrSpike]

You can't do that because for consistency you always want the townpeople to vote when there is an odd number, and the mafia to kill when there is an even number. I know I'm dead, but my posts still show up right?

[Jamski]

I have the solution for this problem lined up for "Mafia 2" - but you'll all have to wait untill Spaced Cowboy has voted to see what it is. Its simple, but elegant.

-Jam

[Sheep]

BTW. Next round I will be asking you all to play again. Except for the few people whom have not said enough. We can again make this a 16 villager game.

I am pming Spaced Cowboy.

[Jamski]

OK, but You should use my "Mafia 2 rules" that avoid the tied-vote problem.

Mafia 2 Rules :

One person is elected Sheriff by the village at the start of the game. In the event of a tie when voting the mafia off, the sheriff can choose between the tied candidates.
In the event of the sheriff's death, he should appoint a deputy either before he dies, or by "leaving a will"
If the only two players left are the sherrif and the Mafia, then the mafia kills the sheriff first It is possible that the Mafia is elected as sheriff, and then they are set up really Although it will be suspicious if the mafia DON'T kill the sheriff.

Tied game problem solved.

Optional rule :

Each player has 3 units of "bribe money". All bribes must be in the public forum, and consist of player A giving player B x units of bribe money to vote for player C. In the event that the village wins and the mafia is defeated, then the villager with the most pocketed bribe money is the winner

Comments?

-Jam

[Sheep]

This really comlicated things.

[Adagio]

I don't like the bribe thing... but the sherif thingy sounds good

[Jamski]

Well the sheriff makes things easier. The bribe money's just an idea, but who WOULDN'T want to bribe people to thier point of view (while at the same time getting poorer)

-Jam

[Jamski]

Or we could use bribe money, but those who are against it get no bribe money to spend, and may not be bribed. They could be "incorruptable"

-Jam

[DrSpike]

I don't really see the point of the sheriff thing. Better IMO is reacting to ties by putting forward the tied villagers and asking villagers who voted for neither to switch to one or the other. Since there is always an odd number of voters this is fine. When the game gets to 3 players it is addressed as I suggest above.

[Jamski]

Because having a sheriff is FUN! Even with a odd number of voters its possible to have a tie in every round (because there can be more than 2 candidates)

-Jam

[DrSpike]

Originally posted by Jamski
Because having a sheriff is FUN!

You're just trying to live out those cowboy fantasies you had as a child.

Your friends always wanted you to be the indian I bet.

[Jamski]

I didn't have any friends

Actually I was ALWAYS the sheriff

So... is Spaced Cowboy EVER going to vote?

-Jam

[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"