I am Andrew Hyer, currently living in New Jersey and working in New York (in the finance industry).
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Shouldn't that be counting the number squared rather than the number?
If you enjoyed the concept there have been sequels to this: if you want one that's currently being actively played I have one running now, or there's a D&D.Sci tag with a list of all past scenarios.
I'm likely not to actually quantify 'relative to' - there might be an ordered list of players if it seems reasonable to me (for example, if one submission uses 10 soldiers to get a 50% winrate and one uses 2 soldiers to get a 49% winrate, I would feel comfortable ranking the second ahead of the first - or if all players decide to submit the same number of soldiers, the rankings will be directly comparable), but more likely I'll just have a chart as in your Boojumologist scenario:
with one line added for 'optimal play' (above or equal to all players) and one for 'random play' (hopefully below all players).
Overall, I don't think there's much optimization of the leaderboard/plot available to you - if you find yourself faced with a tough choice between an X% winrate with 9 soldiers or a Y% winrate with 8 soldiers, I don't anticipate the leaderboard taking a position on which of those is 'better'.
Will that extra credit be worth...uh...at least 1.98 gp?
1I can't help but read this post as something like this:
- Current government mandates around children are very harmful to children.
- Enforcement of current cultural norms around children is very harmful to children.
- ???
- We need to add on and enforce these three new government mandates around children and these two new cultural norms around children.
There is one section arguing that schools are prisons that children hate and are miserable in. And then there is another section advocating for the schools to crack down harshly on children using their phones in school. I find this somewhat depressing.
Haven't found anything particularly good, but I've probably gone as far as I'll go. I've done some analysis trying to predict how much variance we expect from each turtle so that I know how much to overestimate, and for the non-special turtles I'm predicting:
Abigail: 23.0lb
Bertrand: 19.0lb
Chartreuse: 26.2lb
Donatello Dontanien: 21.1lb
Espera: 17.3lb
(Flint is already estimated as a gray turtle as 7.3lb)
Gunther: 30.0lb
(Harold is already estimated as a six-segmented clone as 20.4lb)
Irene: 23.7lb
Jacqueline: 20.0lb
I'm rounding these to 0.1lb even though I'm allowed to go more granular, because if the Tyrant weighs to the same precision we do he will also be rounding to 0.1lb, which means we don't gain anything from more precision (estimating 7.25lb gives a payoff exactly halfway between estimating 7.3 and 7.2).
I'll put these estimates in the parent comment for ease of GM extraction.
The one interesting thing I've turned up is that Abnormalities appear to convey a very large amount of variance: each abnormality adds ~1lb of average weight, but actually slightly over 1lb of stdev-weight. I suspect that abnormalities are adding weight in a highly-random way: my weight estimates for Espera, Irene and Jacqueline (0-abnormality turtles) are relatively low as a result because my confidence was higher, while my estimate for Gunther (6 abnormalities?) has a lot more safety margin built in.
A simple linear regression analysis on the remaining turtles (everything that isn't a Fanged Gray Turtle or a Six-Segmented Harold Clone) gives the following formula:
- 10.56lb base weight if green...
- +2.02lb if grayish-green,
- +5.47lb if greenish-gray,
- +0.359lb/Wrinkle
- +0.142lb/Scar
- +0.598lb/Segment
- +1.000lb/Abnormality
This does a reasonable job of prediction, but has a residual with a fairly-large ~2lb standard deviation. Our standard-deviation math suggests that this means we should give the Tyrant answers overestimating each turtle by 2.4-2.5lb, and should expect to lose on average ~35gp/turtle to error.
That seems like we might be able to improve on it, but I'm not sure how. I haven't been able to find any useful interactions yet. There does seem to be an obvious explanation of all the traits except Abnormalities being driven by some hidden Age variable: old turtles start getting grayish, are wrinklier, have grown more shell segments and accumulated more scars, and are larger. However, I'm not sure how actionable this is for us.
The one thing it does look like I can do is adjust the amount of overestimation I do: it does seem that our estimate is less accurate as turtles get older and larger, and so rather than overestimating by 2.44lb for every turtle I should overestimate the larger ones by more and the smaller by less. That's not going to be a very large improvement, though. I feel like there ought to be something else to do, but haven't found anything yet.
What is QC?
The Fanged Gray Turtle seems relatively simple, so we look at that first.
The weight of a Fanged Gray Turtle seems well-approximated by (0.425 + 0.4568*#segments) lb.
This leaves behind a residual that looks roughly like a normal distribution with stdev ~0.357lb. I'm not able to find any interaction of this residual with any other properties of the turtles - scars, mutations, etc. all seem unpredictive for the Fanged Gray Turtle.
Some quick math reveals that the Tyrant's asymmetric payoff distribution encourages us to overestimate a turtle's weight by ~1.22 standard deviations. Therefore, we're going to bump up all our weight estimates by 0.435lb in order to flatter His Tyranny.
(We could bump them up a bit further if we thought that reducing the odds of him having an unflattering portrait of us was worth trading off money for. However, I actually think we can plausibly use that to extract more money: whatever itinerant artist he kidnaps to do that portrait, we can demand that they give us part of their commission in exchange for us being helpful and sitting for the portrait! Kaching!)
There's only one Fanged Gray Turtle among the Tyrant's pets: Flint, with 14 Shell Segments. Our best guess of Flint's true weight is 6.8lb, but we're going to overestimate this to 7.3lb in order to optimize our payoff.
And two(low-priority) questions for the GM:
- Are we unusually careful and competent at weighing turtles in a way that the Tyrant is not likely to be? If he is careless about weighing his turtles, and introduces additional error, that increased variance makes us want to slightly increase how far we overestimate by.
- What level of granularity are we able to give the Tyrant in our weight estimates? I think that an estimate of 7.25lb for Flint is slightly higher-payoff than 7.3lb in expectation, but don't know if that's something I'm allowed to give.
Understood, I'll refrain from posting the solution tomorrow until I've heard from you - if you want more time, let me know and I can push that further back.