The Math of Mafia

bgrishinko · May 3, 2012

Referencing whether someone has been mafia in recent games as a basis for suspicion has often been a much debated issue. I'm going to lay it to rest with a bit of math. This was bothering me because math is not debatable and this stuff came up in our game, so I put together the binomial distribution of what we were talking about.

This is essentially the coin flip problem. You have a 50/50 chance of heads or tails for every flip. Each time is effectively 50/50. But looking at it on a greater scale, the chances of getting AT LEAST one tails in 5 flips is better than 50/50. Specifically, there is a 97% chance of getting at least one tails in 5 flips.

Apply this to mafia where you have a 1/4 (or 1/3) chance of being anti-town.

p = chance to be anti-town = 25% (approximated, could have used 33% but I wanted to use lower odds)

n = number of games = 6 (arbitarilly chosen... i.e. Person X in the last 6 games)

I'm not going to put the whole formula here... but run the numbers the easy way (Wolfram Alpha) and... results are:

If there is a 25% chance to be anti-town in any given game and someone has been in 6 games recently,

there is a 82% chance they will be anti-town at least once.

Therefore, if Person X hasn't been anti-town in his last 5 games, there is a high chance of being anti-town in a current game.

Razen · May 3, 2012

You still have to remember that it's completely random, though.

As in coin flips, it's completely random how the distribution of mafia roles occurs. And truly random results tend to occur in streaks much more often than you would think. Sure, as you say, it's 97% likely to get one tails in five coin flips. But as each result occurs independently of other results, this means that each time you flip, you have a 50% chance of getting either heads or tails.

In Mafia, all determinations of who ends up with what role occur independently of how they occurred in the last game, as random.org has no clue who ended up being mafia or town in their last game. So going with BG's number of 25% anti-town roles, each player still has exactly a 25% chance of being assigned an anti-town role at random. Meaning that in each game, they have a 25% chance of being anti-town. The odds are still higher that a player gets assigned a town role if you examine each game as its own entity.

Note that I'm not a math major, so I have no idea if this makes sense or not to other people. It does to me.

bgrishinko · May 3, 2012

Right. But we are looking at two different aspects:

Your example puts blinders on the outside world, individual chance is 25%. What your saying is more, what are the chances that on specifically game 6 of 6, what are the chances you will draw scum? 25%.

Mine also says individual chance is still 25%, but the observation isn't on one game, but observing over the course of many games. It isn't an issue of what are your chances this game, but an issue of how often can you repeatedly be the 75% in a row before getting selected. The previous games don't effect the current at all, but statistically speaking, you would need to be pretty lucky to not be chosen as anti-town after 6 games as a trend.

Basel Gill · May 3, 2012

Somehow I knew this would get its own thread.

Blackhoof · May 4, 2012

Right. But we are looking at two different aspects:

Your example puts blinders on the outside world, individual chance is 25%. What your saying is more, what are the chances that on specifically game 6 of 6, what are the chances you will draw scum? 25%.

Mine also says individual chance is still 25%, but the observation isn't on one game, but observing over the course of many games. It isn't an issue of what are your chances this game, but an issue of how often can you repeatedly be the 75% in a row before getting selected. The previous games don't effect the current at all, but statistically speaking, you would need to be pretty lucky to not be chosen as anti-town after 6 games as a trend.

i support putting blinders on the outside world, because Lady Luck has blinders on. all she sees are the odds in front of her, not the amount of times a person has won or lost at thoase odds in the past.while it is UNLIKELY that someoien will be mafia 5/5 times, it is POSSIBLE, and since each toss of the die is independent of the other, you cannot guess reliably when the streak will end. this is called the gamblers fallacy because gamblers on a losing streak will think that there luck will turn soon. maybe it will, maybe it wont, but just because he has a 1/100 chance of winning and he has lost 99 times before, doesnt mean he will win the next time. it doesnt mean for certain that he will win on the 200th or the 1000th time he tries as well. sure it is likely he will have won once, but Lady Luck cares naught for likelyhoods, merely pure, raw odds.your theory is flawed, because, like you said, in each game a person has a 1/4 chance of being mafia. sure we can marvel at how they have been mafia 5 times before, but that doesnt change the odds, so if we lynch someone soley because they have been mafia 5 times before, there is STILL a 3/4 chance that we are wrong.

Alanna · May 4, 2012

I'm so not a math person, but. When I started playing mafia, I was townie about 10 times before I ever got to be mafia lol.

Nolder · May 4, 2012

Songstress · May 4, 2012

I am inclined to agree with Hoof here.

And this concerns me. :wink:

WWWwombat · May 4, 2012

Razen and Hoof are right. BG is committing gambler's fallacy. He is mistaking the Law of Large Numbers for the nonexistent Law of Averages. The roles of each game are independent of each other so what one person is in one game does not affect what they are in another. It is unlikely to be town in 6 given games, but in any one game on its own, the odds of being anti-town are determined by the setup and not by the results of role distribution in previous games.

Also, 25% is just a rule of thumb. The actual percentage of anti-town players depends on the roles and the number of anti-town factions. I have had as little as about 15% (my vanilla game) and as many as about 50% (ASoIaF).

cosmicpanda · May 4, 2012

Also, just in case, let's not talk about on-going games.

bgrishinko · May 4, 2012

Razen and Hoof are right. BG is committing gambler's fallacy. He is mistaking the Law of Large Numbers for the nonexistent Law of Averages. The roles of each game are independent of each other so what one person is in one game does not affect what they are in another. It is unlikely to be town in 6 given games, but in any one game on its own, the odds of being anti-town are determined by the setup and not by the results of role distribution in previous games.

Also, 25% is just a rule of thumb. The actual percentage of anti-town players depends on the roles and the number of anti-town factions. I have had as little as about 15% (my vanilla game) and as many as about 50% (ASoIaF).

Well, I was just using the 25% as an average for simplicity sake.I understand the Gambler's Fallacy here, but they are two sides of the same coin. Its kinda hard to explain clearly, but you are observing two different things. Using the gambler as an analogy, each instance the gambler has a 50/50 chance of winning a coin flip. If asked to bet on it, the gambler will have to bet 6 different times a dollar each. Probability dictates he should come out what he started with. Win 3 lose 3. What I'm talking about is placing 1 bet of 6 dollars on a coin landing heads 6 times in a row. That's a much more dangerous bet.

So in my above example Individually, it is always 25% - looking at overall trends, each instance is always 25% still. When looking overall though, you are observing something different. The odds of getting town 6 times in a row are small. You aren't observing individual events, but what are the chances of a certain amount of consecutive hits.

Nynaeve · May 4, 2012

It shouldn't matter that the chances of being mafia in every gave are different, pending on how many players are playing + how many baddies there are,

An example.

Game 1 = 25 players, 5 scum = 1/5 chance of being scum. 4/5 chances of not being scum.

Game 2 = 30 players, 10 scum = 1/3 chances of being scum. 2/3 chances of not being scum.

Now if you ask what are the chances of a person being town in both games it's = 4/5 X 2/3 = 8/15

What are the chances of someone being mafia in both games = 1/5 X 1/3 = 1/15

What are the chances of someone being town in the first and mafia in second = 4/5 X 1/3 = 4/15

What are the chances of someone being scum in the first and town in the second = 1/5 X 2/3 - 2/15.

Aka total chance of being once mafia and once town in these two games is = 7/15.

This example specifically indicates that being scum twice in a row is the rarest of all, odds wise. Being town twice in a row is the most likely outcome. Being once town and once scum is close to the chance of being town twice in a row (unless you wanna get into the chances of being town and then scum or scum and then town... yada yada yada)...... So the players in the second game could have TECHNICALLY claimed that anyone that was scum in the first game is less likely to be scum in this game. This doesn't mean that that it's not possible....there's still a chance of 1/15 of someone being scum twice in a row... but it's less likely :P

This HUGE condition is that the mods use a randomized way to pick people. if they pick people themselves it wouldn't apply (as far as I understand it). And it really doesn't matter who is modding or whatever... it's like saying that the odds to get heads or tails changes pending on who throws the coin :P It's always a 50/50 chance.

Those are my thoughts, anyhow. Could be complete bull LOL... but that's what i understand from a course I took in uni.

Nyn

Blackhoof · May 5, 2012

I understand your calculations, but when someone joins a game there is STILL a 1/4 chance of them being mafia. Until those odds change, the likelihood of them being mafia/town several times in a row is irrelevant.

Nynaeve · May 5, 2012

Why is it irrelevant? I disagree. Lets take the coin toss example. Yes, it's always a 1/2 chance to get heads in ONE toss. But if you ask what are the chances of getting heads twice in a row then the odds change. it's a 1/4 chance. And getting it three times in a row is 1/8. The chance of an individual toss giving you heads still stays at 1/2 but since you're asking what's the chances of it happening several times in a row then you do the initial odd (1/2) power the number of tosses.

And this isn't my personal opinion or anything... it's a formula to calculating odds.

Aka if you ask what are the chances of me getting heads in a coin toss AND getting a 5 in a dice toss?

odds of getting heads = 1/2

odds of getting 5 in a dice toss = 1/6

AND = multiplying.

The odds are then = 1/12

Nyn

Red2111 · May 5, 2012

this is more of Statistical Probablity than true math, and tbh even then you can't truely pin a number (usign the 97% chance in 5 times number) on the games even using statistical probablity. not to mention the equation is flawed because it discounts the possability of a Solo(or 3rd) alignment.

the variables constantly change (game type, # of player, type of roles, ect) so any equation used is automatically thrown out. for instance, taking BG's 5 games variations, if all all 5 games have different variables (one is fast, 1 is basic, 1 is kitchen sink, and 2 are advanced) then only the times played int eh Advanced game can be compared togetehr to get an average.

if anything, this is more on the a verage of trying to determine the odds of landing on red versus black in a game of russian roulet. you have a 49/49 chance of landing scum or town and a 2% chance of langing solo (00 green); with mafia, because the scum is always 1/4th or 1/3rd the town's numbers you technically have a 29%/39% landing black (scum) and 69%/59% landing red (town) . and as we know with Russian Roulet, the game is rigid because even though the process random, sometimes the house applies the magnets to make sure the players are at a disadvantage. for instance, if i have a 12 player game, consisting of 3 vets and the rest newbs; and random.org makes all 3 vets on the scum team, i'm altering this outcome because the game starts out unbalanced.

also liek the game of roulet, the ball could land black 99 times out of 100 and on the last it could go green.

point is, it's not possible mathematically to develope an equation to accurately determine (even with a large curve for error) how often a person will flip scum.

Nynaeve · May 5, 2012

Again, I disagree. As long as you can calculate the odds of being scum in each of the individual games separately... you can calculate the overall odds by multiplying all those individual odds together.

And it doesn't matter the nature of the different roles. Aka if it's a group of scum or a solo alignment. You take that into consideration when you calculate the odds of being scum in that game. Aka, you count the total number of scum in the game.

And yes, it's probability, not math.

Nyn

Turin Turambar · May 5, 2012

Individual odds NEVER change. The rest is all window dressing. That is why the "history" display on the roulette wheel is possibly the greatest money maker the Casinos ever discovered. Even better than free cocktails for players. It give a false sense of being able to determine the result of a totally random event by using what has happened in the past on similar events.

No matter how many times a particular player was aligned to Town/Mafia/Solo in the past in a randomly assigned game, the odds they will be of a certain affiliation is based solely the set up of that game.

Nynaeve · May 5, 2012

The way I see it, it only depends on the way the mod picks the players. If he does it randomly, it works. If he doesn't, then it doesn't work. Because this is always based on the assumption that roles are given randomly. As for the rest of the things that relate to setup... llike whether it's basic or advanced or whatever... I still don't see how it applies at all. At the end of the day every game has scum and none scum. *shrugs*

Nyn

Blackhoof · May 6, 2012

Likelihood is irrelevant in individual contests of chance, because no matter how many times you have been mafia in a row, even if you have a LIKELYHOOD of 1/500'000 (found by multiplying the individual chances together) in the next game the odds of being scum are 1/4. That's all there is too it.

Nynaeve · May 6, 2012

That's just because you choose to label them as individual contest of chance where chance is irrelevant lol. If you toss a coin and then I toss a coin you can still calculate the odds of both of us getting a certain outcome. If you toss a dice and I stick my hand in a bag that has 4 green balls and 2 yellow balls, we can calculate the odds of each of you getting outcome number one and me getting outcome number two. It's just amusing that you claim that you can't ask the question 'What are the odds of A happening and then B happening.'

But yeah, whatever. Sort of tired of repeating myself at this point.

Cheers,

Nyn

Blackhoof · May 6, 2012

Assuming I have a fair coin and I flip it 49 times, getting heads every time, does that decrease or increase the chances of getting tails on the 50th?

Neither, since it is a fair, 50/50 coin. Despite the ridiculously improbable trend, it is still a fair coin and still has a 1/2 chance of either heads or tails coming up.

Nynaeve · May 6, 2012

That's because you're asking a different question. You're asking what am I going to get in this specific toss? (regardless of any previous tosses). And you're right, that would be 50/50 regardless of it being your first toss or your 100th toss.

But I'm talking about conditional odds. I'm asking what are the odds of my 3rd toss being heads, if I got heads in my first and second toss. And that, my friend, is NOT 50/50 anymore. Because you have to calculate the odds of you getting Heads three times in a row.

Nyn

Nynaeve · May 6, 2012

Plus you got me to be repetitive. Happy? LOL

Nyn

Nightstrike · May 6, 2012

This thread made me curious to find out what "mafia" was. Looked it up on wikipedia, but I doubt I got all the rules and how to play. There's probably more to it than just rules and the general idea, right? I might have to try it sometime.

As for this thread, I wonder how the roles are assigned. Do you use some sort of "lottery" to give each person their roles, or is it a person who does it? People are probably bad lottery machines. There was this little experiment we did while I was in school that put us to the test to see how random we could be if we tried, and I think we failed, but that was a little different from this situation.

If we don't know whether it's chance or not, then we can use the past results to calculate how likely it is that it was pure chance. On the other hand, if everyone assumes that it is pure chance, and that we know the odds (like in casinos), then the past results can't be used to determine the future.

Is it difficult to start as a beginner with this "mafia" game? Must I know how it's done before I join, or can I learn as I play?

bgrishinko · May 6, 2012

It's fairly simple. Most of the time roles are assigned randomly though for experienced moderators sometimes like to make sure there are new and experienced players on teams, or certain roles are created for specific people.

There is a mafia basics thread somewhere around here too you can look up.

The Math of Mafia

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