[SETUP/EV] Connect

[BNL]

2 Mafia Goon
2 Vanilla Townies

There is a Day 1 discussion period, then the game moves into Night 1 where the townies try to connect by privately guessing who the other townie is.

If the townies connect, they win
If none of the townies find each other, they lose.

Otherwise there’s a second discussion period on Day 2, and the townies are given a second chance to connect on Night 2. If they manage to do so, they win, otherwise scum win.

I think this might be runnable as a tiny Marathon, but I want to make sure there isn’t a broken strategy. Can anyone figure out the optimal strategies and the EV?

As a bonus question, what’s the EV if there is no discussion on Day 1?

[Michael Scott]

Hi; I'm still pretty much new to Mafia so I'm doing this as an exercise.
1/9+8/9*1/8 should be the EV with zero discussion overall. Which equates to 2/9, really low chances.
Is a townie who successfully connected, informed at the start of D2? That would change things.

[Michael Scott]

In that case, the EV would be:
1/9 + (4/9)*(1/2) = 33%. Still pretty low.

[Michael Scott]

Don't think there are any breaking strategies.
D1 discussion, apart from reads, is only meaningful functionally to govern pairings. In the case of each VT being paired with a goon, both goons simply say they didn't connect, making their other scum. From a VT's perspective who didn't connect, it's impossible to distinguish between both of the other pair when they both claim they didn't connect, thus D2 discussion doesn't even matter.

If it's unknown pairing, scum simply each fake declare a town as "not paired".

[callforjudgement]

If town are told whether they connected or not, there's a 67% EV strategy: each player targets the next player on the playerlist D1, leading to a 33% chance of a scum autowin (which happens if the scum are not adjacent in the playerlist), but if scum don't win, come day 2 both townies know for certain who the other townie is (either because they connected or by elimination if they didn't), making winning trivial.

If town aren't told about the success rate of their connections, I can't see a strategy above 33% EV, although there are several different strategies which all hit 33%. (The simplest is likely pair-targetting N1, i.e. the top two players on the playerlist target each other, so do the bottom two players. After doing this, D2 is irrelevant; you either win instantly or lose instantly.)

[BBmolla]

Similar to Magic Train.

What happens if in Day 2 mafia have to select one of their own to die?

Edit: Literally nothing lmfao. Lemme think.

[mith]

I agree with the 1/3 or 2/3 depending on whether town is told they were successful. I don’t think the 1/3 would be hard to prove just by exhaustive search of the day 1 strats. (Here’s a less obvious one: pick one player; everyone else targets him, he targets someone at random. There is a 1/2 chance of losing - you picked scum - but if you don’t the player is confirmed town, so do the same thing again. 1/2 chance of picking town and 2/3 chance of that player targetting the other town in two tries, so 1/3. In practice this one would be hard to actually do withiut provable randomness, since scum could block ever picking town; at some point, either someone has to switch or the players voting together just target each other. Pretty much every other strat is more practical. )

[Gamma Emerald]

maybe edit setup size? might have to tweak mechanics for that

[mith]

3 Mafia, 2 Town might be interesting, actually. With a playerlist type strategy, it's 50-50 whether town are consecutive, and if town are told they were successful you could have some interesting games with scum claims. Running through all the possible strategies is a little more complicated though, and no time for that now...

[edit]I was thinking this, but managed to not type it: obviously in this version you would need to remove the scum day 1 win condition.[/edit]

[BNL]

What do you think of this if townies survive to Day 2 no matter what, but the number of connected townies is revealed?

[callforjudgement]

At 2:2, there's a strategy that gives a 5 in 6 chance of a town win (A and B target each other, C targets A, D targets B; day 2, A and D target each other and B and C target each other if the night result was 0 (50% chance) for a 2 in 3 victory chance, otherwise (50% chance) both townies can deduce the other's identity for a guaranteed win). I can't see a way to bring it up to 100%, though.

At 3 scum 2 town, things are more complex, and a good EV analysis would likely involve brute-forcing through all the possible day 1 strategies.

[BuJaber]

What do you think of this if townies survive to Day 2 no matter what, but the number of connected townies is revealed?

Reveal the number of connected players publicly without telling anyone if they connected?
That would make things very interesting.

[the worst]

It's about 40°C and I'm trying to get home so can't think about EVs rn. Playing 4p for two cycles seems interesting if townies aren't informed if they connected and it takes mutual connection for town to win.

Assuming during discussion #1 players always forced two dichotomies (i.e. players {1,2} + {3,4} will pair each other) it's only a 50% chance a townie will be the one who takes charge d1 and then only a 1/6 chance of a town win d1.

Going into d2 both townies have an effective guilty which brings d2 into similar EV to a 3p LyLo I think?

if I'm being obtuse and someone's already addressed this or if I'm playing thru it sub-optimally in my head please let me know

[callforjudgement]

The problem is that the players don't both have guilties; you can be correct without knowing you're correct, if the other player didn't connect to you.

Now I'm wondering if it's best to inform one of the townies whether they were correct, but not the other. (My guess is no, because that introduces an asymmetry that leads to weird claiming strategies D2.)

[BNL]

Another idea I have which may be closer to balanced:

3 Mafia, 2 Town. Townies try to connect twice. Number of connected townies is revealed after first attempt.

[the worst]

The problem is that the players don't both have guilties; you can be correct without knowing you're correct, if the other player didn't connect to you.

Now I'm wondering if it's best to inform one of the townies whether they were correct, but not the other. (My guess is no, because that introduces an asymmetry that leads to weird claiming strategies D2.)

Assuming though:

scum is A & D
Discussion 1: Player A dictates that A+B will pair & C+D will pair.
Night 1: B pairs with A; C pairs with D. action fails.
Discussion 2: both townies know that their actions failed; there is no reason for town to lie in this setup which means B knows for a fact A is scum and C knows for a fact D is scum. if either of them weren't the case, they would have mutually paired the night before.

I know it's a soft guilty but unless town outright lies I think it's effectively an accurate one.. I think.

[BNL]

I just remembered a 50% EV strategy for the original game, with 2 Mafia 2 Town, 2 Guesses, and instant loss with no connection:

Night 1: A -> B <-> C <- D
Night 2: A <-> B C <-> D

If B/C are town town win N1. If A/B or C/D are town they survive N1 and win N2. All other pairings lose N1.

[callforjudgement]

Oh, that's clever. So it is possible to beat 33%.

[the worst]

That's very clever.