[SETUP] Amber Flag

[callforjudgement]

Amber Flag

• 12 Vanilla Townies
  
• 3 Mafia Goons
  
• Mafia win if at least two of them are alive on N6. Town win if two scum die by N6.
  
• The Mafia nightkill is
  not
  compulsive.

This setup was originally a thought experiment into the value of the scum nightkill. The maximum EV strategy is of course to never nightkill at all, but I have a feeling that the maximum winrate strategy may be different. I'm mostly interested in what the best scum strategy is here, because it's likely to give a lot of information as to how games with unusual kill patterns should be balanced.

The setup also looks surprisingly playable for a thought experiment, so I might as well post it as a [SETUP] (especially because the [EV] is trivial to calculate and possibly not that interesting).

[callforjudgement]

I'd run it if there were interest (I agree it's a bit large, but it might make sense in something like Team Mafia which ran a pretty similar setup last time). The thought experiment was the main reason to post it, though, as getting enough interest seems difficult.

I like Irrelephant's comparison. (Perhaps you could even add the gambling mechanic to Amber Flag: you can vote to flip someone, but that also makes them lynchproof, although prevents them posting if they're scum. The white-flag ending means that an agreement to never lynch someone wouldn't necessarily lose the game if they were scum.) That said, Gambling Mountainous has a fairly well-known winrate without the special mechanic (i.e. almost unwinnable for town), but Amber Flag's scum win rate in the absence of kills is much less clear. (I'm guessing it's pretty townsided if scum never kills, but who knows? It's basically 15p Black Flag Nightless where the "losing numbers" are 7:1, which I'm guessing is rather more scumsided than the 10p version. That said, the 10p version has IIRC never been lost by town, so…)

[callforjudgement]

Making WIFOM kills would only be correct if town actually attempted to analyse nightkills, and even then it

still

wouldn't be correct most of the time unless town were fatally over-reliant on nightkill analysis

and

didn't give enouugh consideration to the WIFOM possibility.

In general, the WIFOM approach (i.e. doing something intentionally suboptimal because people believe you'd never do it as scum) is something that mathematically comes up as part of "unbreakable strategies" for scum, but the mathematical strategies typically suggest doing it only a small proportion of the time if town are playing perfectly. You do it more if town are giving a lot of weight to your actions but not considering WIFOM, and less if town aren't giving much weight to your actions or are considering WIFOM too much. "Town aren't giving much weight to your actions" is basically the meta we have at the moment.

[callforjudgement]

Why is it called Amber Flag?

It reminded me of the other Flag setups (White Flag and Black Flag are the extreme versions of the setup, and Red Flag is related in that scum have a choice about what to do overnight). Amber seemed like a good colour for the flag to be.

[callforjudgement]

If scum never nightkill, then town get 6 tries to find 2 out of 3 scum in 15 players; it doesn't matter what order they lynch in because they get no additional information from nightkills (there aren't any), and under EV assumptions, no additional information from flips either (because EV assumptions include "no associative tells"). So we can simply count the possibilities:

• Number of possible arrangements of scum on the playerlist: (3 choose 15) = (15×14×13)÷(3×2×1) = 455
  
• Number of arrangements in which exactly 3 scum are in the first 6 lynches: (3 choose 6) = (6×5×4)÷(3×2×1) = 20
  
• Number of arrangements in which exactly 2 scum are in the first 6 lynches, thus 1 scum in the other 9 players: (2 choose 6) × (1 choose 9) = (6×5)÷(2×1)×(9)÷(1) = 135
  
• Total number of arrangements in which town win: 155 out of 455
  
• Total town EV: 155 ÷ 455 or approximately 34.066%.

If scum always nightkill a townie, the EV is equivalent to that of 12:3 White Flag. Mathdino did the calculations for that a while back, producing an EV of 53.6%.

Other nightkill strategies for town will have EVs that are somewhere in between.

[callforjudgement]

I don't think 34% is a scumsided EV for a Nightless (which is what you get if scum never kill).

The usually mentioned numbers for a Mini Nightless are 9:4, which has an EV of 1 - (8/13) = 5/13 or around 38.5%. So 34% isn't far from there.

That said, my guess is that scum's best strategy is to kill any players who are really strong as town, then stop killing, which is likely to really hurt town's chances without increasing the EV too much.