What are the flaws in this setup?

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Post Post #5 (isolation #0) » Wed Feb 21, 2018 11:32 am

Post by mith »

EV for Multiball setups gets difficult in a hurry even for Vanilla games, never mind with abilities. See [EV] Vanilla Multiball if you want to break your brain again.
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Post Post #12 (isolation #1) » Wed Feb 21, 2018 4:18 pm

Post by mith »

Also keep in mind that the Doctors don’t only affect EV by preventing kills (which actually matters less in Multiball, and already doesn’t matter much in single faction). The bigger effect is that they are a named role that can be claimed. In this particular setup, both Doctors claiming and protecting each other is very powerful, and scum counterclaiming puts them at a significant disadvantage relative to the other group. I would guess best strategy for both groups is to try to take out the other and save the strongman to break the Docs when the other group is gone. I expect this is very townsided with optimal play by all sides (though bad play by one scum group could bump up the other significantly).
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Post Post #13 (isolation #2) » Wed Feb 21, 2018 4:21 pm

Post by mith »

FWIW, back in the early days we played a 3:3:4 setup (including at the first scummeet in 2005) where all town were 1-shot Docs. Multiball is self-balancing to some degree.
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Post Post #15 (isolation #3) » Wed Feb 21, 2018 5:10 pm

Post by mith »

You misunderstand what I'm saying - it is not optimal for any scum to claim in that situation (at least, that's my intuition). It's a Prisoner's Dilemma situation (as is multiball in general); just as scum want to shoot other scum first (if they have a choice they want to eliminate the other group before killing town at all - in a perfect knowledge situation, this leads to a town win), they also don't want to be the group that counters Doc (it only helps the other group).

In particular, if *all* scum claimed Doc, there are three confirmed VT and town should win easily (they'd have to be very unlucky for scum to even get close to winning).

The town's EV is likely worst if there is one counter from each group; but from the scum perspective, if I think there will be one counter from the other group, it's better for ours not to counter at all. If both groups reason that way, neither will counter and there are two confirmed Docs protecting each other. (Again, intuition. I could be totally wrong and one counter is best in all cases.)
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Post Post #17 (isolation #4) » Wed Feb 21, 2018 5:49 pm

Post by mith »

I think you’re still not following, but I want to think about this some more anyway. I’m coming around to the one claim each strategy.
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Post Post #19 (isolation #5) » Thu Feb 22, 2018 11:00 am

Post by mith »

To simplify things, let's just talk about the basic Prisoner's Dilemma situation. Consider that we are in some game state, and there is some action that each scum group can either take or not take (independently of one another). This could be something like "fake claiming Doctor" or "shooting confirmed Townies" or "using strongman on a confirmed Doctor" or whatever else.

Consider the following (example) outcomes:

Neither group takes action: Really good for town, really bad for both scum groups (say 80-10-10 in terms of EV percentages)
Both groups take action: Less good for town, less bad for both scum groups (say 40-30-30)
Only group A takes action: Moderately good for town, really bad for group A, good for group B (say 60-5-35)
Only group B takes action: Moderately good for town, good for group A, really bad for group B (say 60-35-5)

If both groups were able/willing to cooperate, the 40-30-30 is best combined result! However, regardless of what the other group does, each group is better off if it does nothing. For example, if B takes action, then A gets a 30 if it also takes action, but a 35 if it doesn't. If B doesn't take action, A gets a 5 if it does but a 10 if it doesn't.

The argument I was making yesterday is basically this: that whether or not group B offers one fake claim, A may be better off not fake claiming; the reverse is also true, that B is better off not fake claiming. If that's true, then neither should fake claim, which leaves town in the amazing position of having two confirmed Doctors. (There are further PD situations in what the scum groups should do afterward, of course.)

(I am no longer convinced that the claiming strategy is actually a Prisoner's Dilemma, though. It may be best for each group to claim once.)
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Post Post #21 (isolation #6) » Thu Feb 22, 2018 12:33 pm

Post by mith »

I'm detailing why it's bad for scum to do those actions
if
the outcomes are such that it is a Prisoner's Dilemma.

As of post 17, I am not convinced that the particular case we were discussing (whether scum should fake claim) actually has a Prisoner's Dilemma payoff.

Imma stop trying to explain this now. :)
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Post Post #23 (isolation #7) » Thu Feb 22, 2018 12:52 pm

Post by mith »

That's because I haven't arrived at that conclusion yet. I'm not convinced that it *is* a Prisoner's Dilemma, but I'm not sure it isn't either. I need to crunch numbers.
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Post Post #25 (isolation #8) » Thu Feb 22, 2018 1:23 pm

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Will do, but no promises that it will be anything useful. As I said, EV for Multiball (and finding the correct strategies along the way) is already tricky, and the Doctors (in particular) may make it intractable even for a 2:2:5. Even just having them as powerless Named Townies introduces a slew of potentially nasty game theoretic equilibria to sort out. They may all turn out to be not at all nasty, but determining that will take some effort.

That said, I think it's a worthwhile extension of Multiball - basically making it "Neighborhood" Multiball, where the distributions of each group of claims is known. Say 2:2:5 splits into 1:1:3 and 1:1:2. And then you have to figure out which group town should lynch from (it's not obvious to me yet that it's the 1:1:2 group, even though that has a higher chance of success), and which group the scum groups try to nightkill into. The Doctors and Strongmen add something extra, but first you have to figure out the "Vanilla" part of the problem.
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Post Post #27 (isolation #9) » Fri Feb 23, 2018 11:02 am

Post by mith »

I should really just write a guide to calculating EV and finding optimal strategies. There's a lot of good stuff to be said there.

As for your specific question, the main thing is that it is a lot of tedious work (if the setup is at all complicated), mostly in establishing what the best strategy is in LyLo or near-LyLo situations. Towns (and sometimes Mafia) often have non-obvious ways to increase their EV, and significant changes in small setup EVs carry over to the larger setups.

Other than that, it's all logic and math. If you want to generalize to an arbitrary size, the idea is to set up a recursive relationship to smaller setups (simple example: 2:5 Vanilla can either reduce to 2:3 or 1:4 after a lynch and nightkill, depending on whether you lynch Mafia or Town; to calculate the EV of 2:5, you need to know the EV of 2:3 and 1:4, along with the probabilities of reach both states, and similarly you need the EV of 1:2 to calculate those). In some setups, you can find a pattern and go back and prove the pattern holds via induction (see the Vanilla Variants thread for examples - that complicated mess in the most recent post is just the result of me putting things into a spreadsheet until I found the pattern). Some setups don't have nice non-recursive EV formulas (Vanilla doesn't), but some do (Nightless is (T-M)/(T+M)), and once you know what the answer is it's pretty easy to prove it.

The other thing to remember is that small changes can lead to huge complications. You want to approach EV in an incremental way, understanding the simplest setups first before you add even a minor role ability.
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Post Post #29 (isolation #10) » Fri Feb 23, 2018 11:14 am

Post by mith »

Was going to add this to the above post, but got quote sniped:

Investigative roles are a nightmare even at small counts, because of the addition of claiming strategies and fake claiming and so on. Killing roles are slightly better (counter claims aren't as effective if the real role can just kill the claimer). You can simplify these roles by making them non-counterable, though - that is, a Cop that can choose to claim at any time, and is then confirmed to be the Cop by the Mod.

And yes, having a baseline for EV is useful, that's why I started the Vanilla Variants thread and try to expand on it whenever I get in the mood. :)
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