[EV] Vanilla Variants

[mith]

Double Day: 3.9104538193 M^1.5 + 0.0815355751 M + 6.2675689608 (R² = 0.9999999996) for M in [2..1000] (there is some tweaking to my output that needs to be done for Double Day - for instance, it is catching 1:4 as the last <= 50% setup for M=1, but that is actually a no-lynch and 1:3 is what we want; the best fit will change once I fix that issue, but it's not going to be far off)

And we're back in business with my original hypothesis... or at least a modified version where the coefficient is close to but not exactly 4. (Interesting that the Double Day leading coefficient is too small by almost the same amount as Vanilla is too large.)

For Vengescum, the ratio is not fixed; there will be at most M-1 night cycles and at most (T+M-1)/2 day cycles. The fit is not linear, since N/D > 0, but it also cannot be any exponent > 1 since that would mean 2(M-1)/(T+M-1) -> 0 (since T grows faster than M). I have an idea why it would converge to O(M lnM) rather than some other sub-quadratic, but that's for another day.

[mith]

do you think it'd be easier for coming up with exact (rather than approximate) solutions for EV

to come up with models predicting EV for a fixed M or a fixed T

and then producing a three variable equation based on that?

and then of course setting EV to 0.5 and solving for T

i'm no expert in regression but it seems like this would produce a more holistic view of EV

I'm not entirely sure what you're getting at here.

As far as calculating exact EVs (for any setup other than Nightless)... I can't say for sure whether a closed form solution exists, but my suspicion is that the general answer is no.

If what you mean is coming up with a best fit like the ones provided and then using that fit to calculate EV at a given M:T, the issue is that these are typically least accurate when M is small (which are the setups we actually care about playing). The Double Day one in particular is terrible (if you plug in 1 you get a "balanced" setup of 1:9, which is silly), though I expect some of that is from the output issue. But even for Vengescum, my fit would spit out a 2:6 (43.21%) setup, rather than 2:7 (49.48%). It's pretty easy to calculate an exact EV for small games, recursively, these fits are only interesting asymptotically.

[mith]

So, this month's challenge prompted me to add the flag bearer mechanic to my script. Initially, I tested it on Flag Bearer Vengescum, finding that the growth is linear. I wasn't totally shocked that Flag Bearer Vanilla is

also

linear, but I was pretty surprised to find that the balance very quickly converges on Nightless balance.

Flag Bearer Vanilla

For M>=6, the closest setups to balanced are M:(3M-1) and M:(3M+1) - in fact, for larger M, the EVs are close to equidistant from 50%. (For example, 1000:2999 is 49.9875% and 1000:3001 is 50.00124%. Verified that this continues up to M=10000, and the extrapolated balance point seems to be converging on M:3M precisely.)

As far as practical setups go, 2:7 (48.25%) and 3:10 (49.10%) are both quite good - 4:13 and 5:16 are also just below 50%, before the trend above sets in. M:(3M+1) is always the closest to 50%.

Flag Bearer Nightless

Flag Bearer Nightless seems to be sub-linear in terms of excess townies (i.e. something like M:M+O(M^x), where 0 < x < 1). 1000:1028 has an EV of 49.8%, and the best fit up to M=1000 is somewhere around T=M+1.22*M^0.452.

2:4 (46.67%), 3:5 (44.05%), 4:6 (41.90%), and 5:7 (40.10%) would be reasonable to play; 3:6, 4:7, and 5:8 are closer to 50%, but on the town-sided side of things. 2:4 in particular is a better balanced version of Lovers Nightless (sort of a half-Lovers Nightless). After that 3 excess townies is below 50% (while 2 excess dips below 40%), up to 12:15.

Flag Bearer Vengescum

2:4 (42.78%) here is fine but Nightless is probably better for this count. 3:6 (47.57%) and 4:7 (42.50%) are good options.

Reasonable Micro and Mini Setups

2:4 Flag Bearer Nightless (46.67%)
2:5 Flag Bearer Vengescum (53.41%)
3:5 Flag Bearer Nightless (44.05%)
3:6 Flag Bearer Vengescum (47.57%), 2:7 Flag Bearer Vanilla (48.25%)
4:6 Flag Bearer Nightless (41.90%)
4:7 Flag Bearer Vengescum (42.50%)
5:7 Flag Bearer Nightless (40.10%)
3:10 Flag Bearer Vanilla (49.10%)

The Flag Bearer could also be combined with White Flag - which would effectively change the win condition to "lynch the Flag Bearer or lynch all the non-Flag Bearer Mafia" - but this would make very little difference to the asymptotic behavior. Which is a shame, because "White Flag Bearer" is such an obvious name.

[mith]

One thing I may look at in the future is the idea of multiple Flag Bearers - for example, town has to lynch both of the 2 Flag Bearers from however many Mafia total to win.

(In some sense, the Flag Bearer Mechanic is sort of an inverse Traitor - it effectively makes all the regular Mafia into Traitors except with full knowledge and communication between the Traitors and Mafia.)

[mith]

Double Flag Vanilla isn't that interesting; 3:16 up to 3:22 would all be fine EV-wise, but you might as well play with 2 Mafia and a Traitor (which would need even fewer Town), and probably everyone would hate it anyway. It does appear to be growing linearly for M>2 (something like M:5.5(M+1) isn't too far off; I didn't put this in the script, just eyeballing from my spreadsheet).

Double Flag Vengescum is fine at M:(2M+3) - 3:9 (48.83%), 4:11 (48.31%), 5:13 (47.86%), etc.; M:(2M+4) doesn't dip below 50% until 8:20.

Double Flag Nightless would be ok at 3:7 (47.78%) or 4:8 (45.86%)... even 8:12 is still above 40%, though 8:13 (46.19%) is better.