So I was recently playing a game of Mafia (sometimes known as Werewolf) and I happened upon a situation where I tried to choose the best option by calculating the probability of a who a person was. Since Mafia is such a heavily modified game with a diverse set of rules and role types, I'll go through the rules and roles I was using when I played it (sorry for the long read).
The basic rules are this:
Some players are Mafia, some are townspeople (some townspeople have special abilities, which I'll cover later). Every "night" (i.e. the beginning of each round), the Mafia choose one townsperson to kill (i.e. eliminate from the game). Then in the "morning" (i.e. the main part of each round), the group (both the Mafia and the townspeople) have to vote on who they think the Mafia is and lynch them (i.e. eliminating them from the game). The game is won (from the perspective of the town) when no more Mafia are alive.
They way we played, four of five possible roles dealt out, one to each of the four people playing. The five roles were:
- Mafia (generic mafia bad-guy),
- Crazy Person (not Mafia or town. Essentially a bad-guy. Wins if they can convince the town to kill them),
- Doctor (part of the town. Can cure two people during the night [but not themselves]),
- Detective (part of the town. Ability is irrelevant to question), and
- Dentist (part of the town. Ability is irrelevant to question).
Note that there are two "bad-guy" types so that there will always be a villain in the game. We made it so that if the town thought that three was no mafia and that a certain player was the crazy person, they could vote to send him to an asylum and win the game.
With all of that out of the way, here's the situation: I (let's say, player D or Dan) was the doctor. During the night I healed players A and B (let's say Aidan and Ben). Only player C (let's say Charlie) and myself were left open for killing. When the night was over, no one had been killed. Some conversation happened and we started saying who we were. Aidan stated that he was the Mafia. This led us to believe he was either the Crazy Person (wanting us to kill him, or triple bluffing etc.) or Mafia (wanting us to think he was crazy person and not kill him etc.). There was no situation we could think of where he could have been the town.
For various reasons, we were fairly sure that Ben, Charlie and myself were all the town, leaving us only with how to respond to Aidan. We weren't sure either way about if he was Mafia or crazy person so I decided it was a 50-50 chance between the two.
I had healed Aidan and Ben, so since no one had died (and he couldn't have killed himself as Mafia), he either had to have been Mafia and tried to kill Ben, or been the Crazy person (in which case no one would die anyway).
Assuming he selected victims randomly, without any sort of predictable strategy, it would be 1/3 chance of him selecting Ben. Since it was 50-50 he was Mafia, the chance he was Mafia and killed Ben was $\frac{1}{2}\times\frac{1}{3}=\frac{1}{6}\approx 16.7\%$ and therefore the chance he was a crazy person is $1-\frac{1}{6}=\frac{5}{6} \approx 83.3\%$.
In the end, we went with this logic and voted him into the asylum and it turned out he was indeed the Crazy Person.
Despite the fact that it worked (whether or not it was right I won, so who cares right?), I'm quite dubious of the way I got to the answer. Like, how can it be a 50% chance he's a Crazy person, but actually an 80% chance. Was the answer I came to correct and if not, what is the right answer?
If you're still reading at this point, thanks for putting up with such a long winded question (I say as I continue typing).
Best Answer
Great question :-)
Let $N$ be the event that no-one was killed, and $C$ the event that Aidan is crazy. Then you want the conditional probability
\begin{align} \textsf{Pr}(C\mid N) &=\frac{\textsf{Pr}(C\cap N)}{\textsf{Pr}(N)} \\ &=\frac{\textsf{Pr}(C\cap N)}{\textsf{Pr}(C\cap N)+\textsf{Pr}(\overline{C}\cap N)}\\ &=\frac{\frac12\cdot1}{\frac12\cdot1+\frac12\cdot\frac13}\\ &=\frac34\;. \end{align}
So the chance wasn't quite as high as you thought, but it was still the right choice.
You calculated
\begin{align} 1-\textsf{Pr}(\overline{C}\cap N)&=\textsf{Pr}((\overline C\cap\overline N)\cup(C\cap N)\cup(C\cap\overline N))\\ &=\textsf{Pr}((\overline C\cap\overline N)\cup C)\;, \end{align}
the unconditional probability that either Aidan is Mafia and someone was killed or he's crazy (and hence no-one was killed). I don't think this corresponds to any conditional probability given your knowledge of $N$.
Regarding your question "how can it be a $50\%$ chance he's a crazy person, but actually an $80\%$ chance" (in fact $75\%$), the former is the unconditional probability for him to be a crazy person and the latter is the conditional probability, given your knowledge of $N$.
(By "unconditional", I mean conditional only on the facts that you regarded as definitely settled: That the three of you were all townspeople and Aidan was either Mafia or crazy. Of course the original unconditional probability for Aidan to be crazy was only $\frac15$.)