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# Nightless Expectation Rule

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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 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*) - =((
*SN-2S*)+(^{2}+2S*N*))/(^{2}-2SN+N-2S-SN+2S^{2}*N*(N+1)*) - =(
*N*+^{2}*N*-2*SN*)/(*N**(*N*+1)) - =(
*N+1-2S*)/(*N+1*)

- P(Town win,

...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.