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**=(''N+1-2S'')/(''N+1'')<br> | **=(''N+1-2S'')/(''N+1'')<br> | ||
...which is precisely the original formula, with (''N''+1) swapped in for ''N''. Therefore the formula holds. | ...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''<br> | |||
Nightless games have a 50% win rate for either faction if scum make up a quarter of the initial population. | |||
[[Category:Theory]] | [[Category:Theory]] |
Revision as of 01:44, 19 August 2011
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
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-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.