Nightless Expectation Rule

The Nightless Expectation Rule is a mathematical rule stemming from The EV Project.

In Nightless games with no power roles, S scum alive, and N total players alive,

P(Town win) = 1 - 2S/N = (N-2S)/N

An obvious corollary would be:

P(Mafia win) = 2S/N

Proof

This proof uses mathematical induction.

• Base case: In a 2 - 1 ELo, Town will eliminate 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)
  • =((SN-2S²+2S)+(N²-2SN+N-2S-SN+2S²))/(N*(N+1))
  • =(N²+N-2SN)/(N*(N+1))
  • =(N+1-2S)/(N+1)
    

...which is precisely the original formula, with (N+1) swapped in for N. Therefore the formula holds.

Balance

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.