[EV] Oracular Mafia

[callforjudgement]

Oracular Mafia

• 8 Town 1-shot Day Oracles (can ask the moderator a true-or-false question about which players have which roles, will get a truthful answer)
  
• 5 Mafia 1-shot Day Oracles (note: the action isn't useful as scum know the whole setup already, it just lets them figure out when the moderator's reply would arrive)

This is almost certainly broken, but I'm interested in how.

Can town get a 100% win rate?
What's town's best strategy in trying to get a

perfect

win (i.e. no mislynches)?

[callforjudgement]

They can choose what questions they ask. Mostly about who's scum.

There's surely got to be some set of 13 questions you can assign the 13 players, and win based on the answers even if scum lie about what some of the answers were.

[callforjudgement]

Right.

[callforjudgement]

It's pretty funny but like dethy, it's more of a logic game than a mafia game

I don't think it'd be super fun to play

Right, which is why I marked this as an EV thread ("solve the puzzle"), not as a setup thread ("I'm planning to run this").

As mith points out, you have to try to rely on the information coming from scum somehow (I used similar reasoning to pick the numbers in the first place). It's also a test of my theory that "WIFOM tends to cancel out", i.e. that scum normally do about as well going along with a breaking strategy pretending that they're town, and intentionally violating it and likely getting caught as a result; it's not necessarily shedding doubt on the theory (as I have no idea whether scum should lie or not), but rather pointing out something that I missed (that because town have to allow for both strategies, they get less useful information than if everyone was town

even if

scum tell the truth).

For what it's worth, the solution to this sort of puzzle is often to catch one scum at a time (because this allows you both to disregard their result, and to gain information from the scum nightkill after you've lynched the one caught scum). I can't see an obvious way to apply that principle here, though, especially as you effectively have to use all the Oracle shots Day 1 (to stop scum killing a player with an unused shot).

[callforjudgement]

OK, I think I've figured out how to analyse this sort of setup.

The trick is to look at how many scumteams are

ruled out

by each answer. If a player claims a particular Oracle result, then we know that either a) the claimer is scum, or b) the result is correct; so all the scumteams where the claimer is not scum and the result would be wrong are impossible. For example, suppose player A goes first in terms of asking a question; there are 792 scumteams that don't include player A, so with an optimal question, half of them (396) will be ruled out. So there are now only (1287-396 = 891) possibilities left for the scumteam.

You then take the player on the fewest of the remaining scumteams, and ask them to go next (as their information will have the most potential use); this won't be player A, who is on as many scumteams as possible. Each of the 891 remaining scumteams contains 5 scum, each player is on (891*5/13) = ~347.2 scumteams on average. That means that there must be some player who's on at most 347 possible scumteams (regardless of what question was asked, assuming it splits the scumteams into two equal halves). Call this player B. There are at least 544 possible scumteams which do not contain player B, and player B's question will eliminate 272 of them. That brings us down to (891-272 = 619) possible scumteams.

Unfortunately I'm not sure if the same sort of analysis can be continued, as I see no reason why it's necessarily possible to keep crafting questions such that the player on the fewest possible scumteams is necessarily a player who hasn't asked yet. If the same rate of scumteam reduction continues, though (which is far from certain!), we'd end up eliminating about 30% of the possible scumteams at each stage. That'd leave us with approximately 12 potential scumteams after all the questions have been asked. That's a large enough number that I'd now be surprised if a 100% perfect win strategy existed, unless it could somehow leave everyone as potential scum (so that the scum's nightkills necessarily eliminated some possible teams without the need to ask any more questions). It might also be possible to design the questions such that no matter the answer, some specific player's odds of being scum were drastically reduced (thus making that player's question more valuable), but that seems somewhat at odds with trying to split the remaining possibilities in half.

[callforjudgement]

Well, you can do something like putting all the setups where B is scum (and A is not scum) on the same half of the split. That gives you a good advantage in scumteam elimination if A says that the other half of the split is correct (because now you know that either A is scum or B is town). It's less useful if the split goes the other way, though, and there's limits to how much you can do this for two players B and C using A's question (as they could be scum together). Perhaps you want all the scumteams where B is scum and C isn't on one side of the split, all the scumteams where C is scum and B isn't on the other side, and to split the both-scum and neither-scum cases equally.

[callforjudgement]

Huh, that's actually much lower than I expected, even on the condition that the groups have to be symmetrical.

I imagine in an actual game you'd use scumhunting to help aim the questions, so maybe this setup is less broken / more interesting than I thought.

[callforjudgement]

I think you'd have to start by figuring out how many questions (asked by different players) you need to confirm one player as town (which is also an interesting question in its own right). That seems like the first step in your strategy there.

The answer can't possibly be less than 5 (the number of scum), as if 4 scum asked questions and just lied about the results you'd have no way to know which of the remaining 9 players was the fifth scum. My guess is that it's probably 6, although I'm far from certain on this.

[callforjudgement]

Right, you can use it as an anti-gambit mechanism but I don't see why town would be gambitting in this setup anyway.