What is the EV of 2:1 mountainous?

[MURDERCAT]

This seems like a silly question but I was thinking about it.

I know it's "33%" but does it actually play out like that with how voting works?

If scum votes first, the player that was voted knows they are scum when there is no hammer and cross votes. Player 3 has a 50% shot at the 1v1.

If town votes first, they have a 50% chance of hitting town, scum wins.
If they hit scum, scum cross votes and the other player has a 50% shot at the 1v1.

So this all adds up to (1/3*1/2) + (2/3*1/2*1/2) = 1/3

But scum should never vote first, because you then purposefully head down the 50% path instead of the 25% path.
So if scum never votes first, does that mean the EV is 25%?

Well not really, because if scum never votes first then the first person who votes is conf town. So if the game doesn't end when they vote, the second 50/50 is no longer a 50/50, the third player votes with the conf town, leading to an EV of 50%.

Of course if you treat the player who voted first as conf town then scum would always vote first. So there is this interesting balancing that happens. I'm curious what the true percentage of scum-first votes in 2:1 is. Stay tuned for the next marathon weekend where I will be running 500 2:1 mountainous games.

[Ircher]

This is a complex question, and honestly, there is no single "correct" answer. Ultimately, it comes down to how we want to model this. You might be interested in this thread which talks about Nash Equilibrium and some of the things that you are talking about here.

[MURDERCAT]

This will be a fun thread to read, thanks Ircher

[MURDERCAT]

Like with most things worth thinking about, someone else has thought about this long ago and done a better job of it

[Jake The Wolfie]

Of course if you treat the player who voted first as conf town then scum would always vote first. So there is this interesting balancing that happens. I'm curious what the true percentage of scum-first votes in 2:1 is. Stay tuned for the next marathon weekend where I will be running 500 2:1 mountainous games.

I would be interested in playing scum in 500 games, yes.

[Awoo]

EV is theoretical. You're asking for what the results look like in practice. It has been known for a long time that EV is not nessecarily equal to practical EV - most famously that town overperforms EV in setups where mafia has no less nightkill power, and underperforms EV in setups like mountainous with a nightkill.

The EV of 2:1 mountainous is 1/3 for town. Someone (probably mith) did a longer form analysis like you did with the votes and everything and it still works out to 1/3.

[MURDERCAT]

I guess I view EV as the expected outcome if all players play optimally but have completely random reads. The voting behavior here I think still falls under what I consider to be EV. Similarly if there was an open setup with a bulletproof doc and a cop, I would consider the cop claiming on D1 to be part of optimal play (and therefore EV), but I would then assume the cop is choosing targets randomly. This maybe different from how other people think about EV though.

[implosion]

highly relevant thread: viewtopic.php?f=5&t=75419

[implosion]

i'm dumb! ircher already linked that! ignore me.

[implosion]

But yeah from a not-EV standpoint there's also a very significant difference between a 2:1 mountainous and an entire game that has boiled down to a 2:1 final day. GreyICE's comments in the other thread get at that pretty well, namely that in the latter case you have to consider that if someone townish is still alive, it might be conspicuous. You could model that by, say, assigning each player in the list a number from 1 to N representing how townish they are, possibly weighting the scum to have scummier numbers by an amount based on how strong you think the town is, and then discussing optimal town voting strategy and scum nightkilling strategy based on those numbers. Which could be quite interesting.

Just 2:1 mountainous avoids all that confounding stuff, though.

[Ythan]

This subject is closely aligned with my game design interests, therefore ego.

[MURDERCAT]

This is partly directed towards implosion, but also a general question. Do you think people are good at intuitively understanding and finding the Nash equilibrium in these situations if they've played mafia for a while (i.e. not just 3p ELo but across various situations that occur in mafia)? I haven't had that many scum games go to ELo but my intuition is I would never want to vote first. Because the games are played by people who actually want to win that specific game and not necessarily balance site meta as a whole, I imagine there are potentially large deviations. I know there is a bit of data in that linked thread, but that's not nearly enough for conclusions to be drawn IMO (though interesting nonetheless).

[Infinity 324]

I think the point is that people are not necessarily trying to change site meta, but they would notice that scum tend not to vote first very often and TR the player who votes first as a result. Not sure that people actually do this, but maybe they should

[Awoo]

Yeah, implosion made the thread I was thinking of. Read that. It answers this topic.

[implosion]

In a real game that goes to 3 players, there are other factors at play, most of all pre-existing reads. In general I think people might weigh this kind of meta somewhat, but ultimately people are going to trust their own reads. That's why I phrased the practical advice in the other thread as saying you should bias yourself - you shouldn't ignore the idea but you also shouldn't just vote based on it and nothing else of course.

I think there's also a phenomenon of what you said where as scum you naturally would never vote first, but sometimes you can discern that one of the other players is definitely going to vote for you and you can preempt them and "get credit" for voting first, even if they haven't explicitly said so yet.

One other thing that I would ask you please do not do but that is amusing is that to preempt this, one thing players can do is mandate that players claim who they want to vote simultaneously. But of course on a forum this is impossible. But there is one way to bypass this - namely, robust steganography. You can steganographically hide the info of who you're planning to vote for, and then everyone can reveal the decrypted versions. This is why robust steganography is banned in a lot of rule sets, or at least one reason (it can also be used similarly for a simultaneous massclaim).

[MURDERCAT]

I am not good enough to worry about incorporating strategies like that into my game

[yessiree]

As for who should vote first, speaking from a purely theoretical standpoint, scum should always match what town does, and town should always do the opposite. Here's a basically a summary of implosion's thread using a nash equilibria payoff matrix

town \ scum	always vote first	never vote first
always vote first	1/4 \ 1/2	1 \ 0
never vote first	1 \ 0	1/4 \ 1/2

[callforjudgement]

Under EV assumptions, 2:1 mountainous is has an EV of 1/3.

In practice, the win rates if players are prevented from scumhunting will also probably converge to 1/3. A simple way to get closer to the 1/3 figure is for the playerlist to vote on who should vote first (in order to avoid the day timing out without a decision being made). You can vote on who should vote first in

that

vote if you fear a secondary deadlock, and so on, in order to get as close as desired to an EV of 1/3 (after a few iterations, the votes become so meaningless that there's no particular reason why voting first in them should give a measurable disadvantage).

It is worth noting that (given that we're assuming that there's no scumhunting, players are just trying to maximise theoretical win rates) scum can always guarantee a 2/3 win rate by acting identically to how they would act as town. This implies that town can't guarantee more than a 1/3 win rate, so can pick any arbitrary strategy that would achieve this (such as everyone agreeing to vote for a provably random player, or if that strategy is banned, using the strategy I suggested above). This means that the setup's EV is 1/3 even when properly broken.

[MURDERCAT]

Voting on who to vote first is a really cool way of thinking about this