True Love Expectation Rule
In this case, the Lover pairs can be generalized to "Town" or "Scum" pairs based on whether they have scum in the pairing or not.
Then it follows that in a True Love-style game with S scum pairings and Pa total pairings,
P(Town win) = 1 - S/Pa = (Pa-S)/Pa
P(Mafia win) = S/Pa
Simply put, the theoretical scum win rate is precisely the proportion of the pairings they are in.
Note that the chance of scum dying on any given lynch is S/Pa.
Relationship to Nightless Expectation Rule
If you consider that Pa = N/2 (where N is the number of players in the game), then this rule exactly matches up with the Nightless Expectation Rule.
- Base Case: Given a 1 - 1 LyLo (two pairs of Lovers, one of which has scum in it), the chance that Town will win is the chance of lynching a member of the scum pair, or 1/2.
This is 1 - 1/2 = 1/2.
- Induction step: Given that the rule holds for S scum and PA players, it is necessary to show that it holds for S scum and PA+1.
- P(Town win,S,PA+1) = (S/(Pa+1))(P(Town win,S-1,Pa)) + ((Pa+1-S)/(Pa+1))(P(Town win,S,Pa))
- =(S/(Pa+1))(Pa-(S-1))/(Pa) + ((Pa+1-S)/(Pa+1))((Pa-S)/Pa)
- =(SPa - S2 + S + Pa2 - SPa + Pa - S - SPa + S2)/(Pa(Pa+1))
- =(Pa2 + Pa - SPa)/(Pa(Pa+1))
...which is precisely the original formula, with (Pa+1) swapped in for Pa. Thus completes the proof.
Letting P(Town win) = 1/2), this means:
- 1/2 = (Pa-S)/Pa
- Pa/2 = Pa - S
- S = Pa/2
Thus, True Love has a theoretical 50% win rate when scum control half of the initial pairings.