[EV] Vanilla Multiball

[callforjudgement]

I missed this thread earlier.

I believe 2:1:1 is mathematically a 3 faction draw if the players' alignments are known; either scum can no-kill to force the other to no-kill (because if the other scum shot, then town would lynch them in response), and town doesn't want to lynch scum and scum don't want to lynch anyone, thus nothing can happen during the day (theoretically, a townie would benefit from being lynched, but the scum won't vote for it and 2 votes are not enough).

1:1:1, on the other hand, is broken. There's some debate as to how it's broken, and it depends a lot on the details of the ruleset. (In particular, in face-to-face play that uses mafiascum.net hammer rules, any attempt by scum to vote for town will cause the game to be decided by the players' reaction times.)

I think it may be best to explicitly rule 1:1:1 and 2:1:1 town wins in multiball in order to avoid the problems with endgame breakage, in which case this will throw off the probabilities somewhat.

Nightless and Vengescum setups don't have these issues, so analysing the EV is more meaningful. (That said, in nightless 1:1:1, the townie has no chance to win and kingmaker ability, which is normally considered broken.)

By the way, although I haven't proven it yet, I suspect that the EV for a vanilla Nightless always stays much the same if you multiply every faction's member count by the same factor. This probably holds at least approximately true even in multiball. As such, you can approximate the various faction EVs via dividing the setup down to a :1 setup (e.g. 3

x

:

x

has a 50% EV because 3:1 does).

[callforjudgement]

In 1:1:1 Nightless, town has no mathematical chance to win. Therefore, it can't be wrong to vote or not vote; whatever they do doesn't affect their own victory chances.

The reaction time argument for 1:1:1 daystart goes like this: town can prevent scum A voting for them by voting for scum A in response (under the assumption that if the townie gets lynched, they lose anyway, and therefore they have no reason not to vote for someone to force them to unvote). If scum A unvotes, the townie then wants to unvote/vote no-lynch; however, if scum B is fast enough, they'll hammer scum A before the townie has a chance to unvote.

I believe that in 1:1:1 daystart with unknown alignments, everyone's best option is to pretend to be town (and thus will vote no-lynch and crossvote a vote on them, both optimal moves as town). Thus, you'd expect no lynch to be hammered very quickly, as not doing so would be a scumclaim. If alignments are known, I can see scum attempting to go for the draw, but doing so is very risky because if the other scum's reactions are faster than either yours or the townie's, you'll end up losing as a result.