[EV] 8p Multiball

[Ircher]

The problem is town has veto power, but otherwise have practically lost from the start.

Town does not control a majority of votes, so town's best option D1 is to no lynch.

That gives: (Assuming RB is used N1 and cannot RB and kill at the same time):

Chance of getting RB'd: 1/6 - (1/6 * 1/6) = 5/36 --> Subtract the probability that scum!A RBs the killer for scum!B, but scum!B RBs the roleblocker for scum!A.

Mafia Chance of Shooting a Townie : 2/3 * 31/36 = 62/108 = 31/54 (Subtract the chance of being RB'd from 1)
Mafia Chance of Shooting a Werewolf (Doesn't matter which) : 1/3 * 31/36 = 31/108
Mafia Chance of Kill being Blocked : 5/36 = 15/108

Same chances for Werewolves. ^

Chance of Cross-Kill = 31/108 * 31/108 = 961/11664 ~ 0.0824 = 8.24% Chance.
Chance of Mafia killing Werewolf and Werewolf shooting Townie = 31/108 * 31/54 ~ 0.1648 = 16.48% Chance
Chance of Mafia killing Werewolf and Werewolf being blocked = 31/108 * 5/36 ~ 0.0399 = 3.99% Chance
Chance of both being blocked = 5/36 * 5/36 ~ 0.0193 = 1.93% Chance
Chance of 2 Townies (Possibly the same) being shot : 31/54 * 31/54 ~ 0.3296 = 32.96% Chance
Chance of Mafia shooting Townie and Werewolf Blocked = 31/54 * 5/36 ~ 0.0797 = 7.97% Chance

That gives us:

8% Chance of 4:1:1 with 4 BPs.
8% Chance of 4:2:1 with 4 BPs.
33% Chance of 4:2:1 with 3 BPs.
2% Chance of 4:2:2 with 4 BPs.
16% Chance of 4:2:2 with 3 BPs.
25% Chance of 4:2:2 with 2 BPs.
8% Chance of 3:2:2 with 3 BPs.

For 4:2:2 with 2 BPs:

Chance townie was BP : 1/2
Chance townie was not BP : 1/2

Mafia Chance of hitting BP Townie = 2/3 * 1/2 = 1/3
Mafia Chance of hitting Non-BP Townie = 2/3 * 1/2 = 1/3
Mafia Chance of hitting a Werewolf = 1/3
(And Vice-Versa)

Chance of Cross-Kill : 1/3 * 1/3 = 1/9
Chance of killing 2 Townies : 1/3 - (1/3 * 1/3) = 2/9 (Subtract probability of hitting same target)
Chance of killing 1 Townie and no one else : 1/3 * 1/3 = 1/9
Chance of killing 0 Townies and no one else : 1/3 * 1/3 = 1/9
Chance of killing 1 Werewolf and no one else : 1/3 * 1/3 = 1/9
Chance of killing 1 Werewolf and 1 Townie : 1/3 * 1/3 = 1/9

Double-Check: 2/9 + 1/9 + 1/9 + 2*1/9 + 2*1/9 + 1/9 = 9/9

So:

2/9 Chance 2:2:2 with 2 BPs (Town should try to force a lynch in this case probably)
1/9 Chance 3:2:2 with 1 BP (I'd still no lynch as town)
1/9 Chance 4:2:2 Vanilla (Is this chance stored on the wiki?)
2/9 Chance 4:2:1 with 1 BP (Town should lynch)
2/9 Chance 3:2:1 with 2 BPs (Probably no lynch)
1/9 Chance 4:1:1 with 2 BPs (Town should lynch)

For 3:2:2 with 3 BPs:

Mafia Chance of hitting Townie : 3/5
Mafia Chance of hitting Werewolf : 2/5

(Same for Werewolves)

Chance of Cross-Kill : 2/5 * 2/5 = 4/25
Chance of 1 Werewolf Death : 2/5 * 3/5 = 6/25
Chance of 1 BP dying : 3/5 * 1/5 = 3/25
Chance of 2 separate Townies hit : 3/5 * 2/5 = 6/25

Check : 4/25 + 2*6/25 + 3/25 + 6/25 = 25/25

So:

4/25 Chance of 3:1:1 with 3 BPs.
12/25 Chance of 3:2:1 with 2 BPs.
3/25 Chance of 2:2:2 with 2 BPs.
6/25 Chance of 3:2:2 with 1 BP.

Someone else can figure the rest out.

[Ircher]

Even though no lynch is probably the best strategy for town, I will calculate some stuff if a lynch is achieved D1, but do keep in mind that town needs scum's help to lynch scum, so the actual values for actual play would probably be lower that what the EV reflects.

Both RB's will be used N1 if available.

Chance Goon is lynched = 1/4
Chance RB is lynched = 1/4
Chance BP is lynched = 1/2

If BP is lynched:

3:2:2 with 3 BPs going into the night.

Chance of getting RB'd = 1/5 - (1/5 * 1/5) = 4/25

Mafia shoot Townie : 3/5 * 21/25 = 63/125
Mafia shoot Werewolf : 2/5 * 21/25 = 42/125
Mafia Blocked : 4/25 = 20/125

Cross-Kill : 42/125 * 42/125 ~ 0.1129 = 11.29%
Townie + Werewolf Shot : 42/125 * 63/125 ~ 0.1693 = 16.93%
Both Blocked : 4/25 * 4/25 = 0.0256 = 2.56%

Meh.... Too many possibilities....

[Ircher]

Woah, that's impressive. Can't this sort of thing be computerized or something?
Thanks for the analysis, I'll be super happy if someone made it full!

Yes, but I'm too lazy to figure that out.

[Ircher]

So Ircher. You assume that town will always nolynch. Why?
Town can definitely make lynches go through. scum want to lynch scum too, remember?

I think it's generally in town's advantage to no lynch each day until town outnumber scum (or it gets so ridiculousky scum-sided, that randomly lynching will help town's chances).

We would have to calculate the statistics for both though, and seeing it play out in an actual game might help.

[Ircher]

I would like to see an 8p run before deciding on an optimal strategy.

[Ircher]

I will /pre-in for your next test run btw.

[Ircher]

Not really tbh. The number of options to explore do decrease by one degree, but 7p results may vary too much from 8p to be practical since the setup shifts to even numbers from odd numbers at start.