Nightless Expectation Rule
The Nightless Expectation Rule is a mathematical rule stemming from The EV Project.
P(Town win) = 1 - 2S/N = (N-2S)/N
An obvious corollary would be:
P(Mafia win) = 2S/N
This proof uses mathematical induction.
- Base case: In a 2 - 1 LyLo, Town will lynch correctly 1/3 of the time at random. This is 1 - 2/3.
- Induction Step: Given that P(Town win with S scum and N players) = (N-2S)/N, consider P(Town win with S scum and N+1 players).
- P(Town win,S,N+1) = (S/(N+1))*P(Town win,S-1,N) + ((N+1-S)/(N+1))*P(Town win,S,N)
- =(S/(N+1))*(N-2(S-1))/N + ((N+1-S)/(N+1))*(N-2S)/N)
...which is precisely the original formula, with (N+1) swapped in for N. Therefore the formula holds.
Letting P(Town win) = 1/2, this means:
- 1/2 = (N-2S)/N
- N/2 = N-2S
- N = 2N-4S
- 4S = N
Nightless games have a 50% win rate for either faction if scum make up a quarter of the initial population.