Initial observations and results (warning, for all I know right now it could be substantially complete solution):
It looks like the non-Carver bidders for newer carcasses are consistent, and drop linearly with age in days:
snow serpents: 60 − 20*age
winter wolf: 50-12*age (technically we don’t have a third point to see that this is linear, but see below)
yeti: 55-6*age
Presumably, this is due to a single bidder. The trend continues until it drops below the distribution for older carcasses, so I guess that this bidder continues to bid this pattern down to 0 sp.
For older carcasses, there is a range of variation that doesn’t seem to depend much on age. I therefore assume for now that it really doesn’t depend on age, though for all I know it does.
For winter wolf, Carver won all the auctions for 2 and 3 day carcasses. However, Carver bid high enough for this data to be consistent with 2 day old winter wolves following the linear trend and 3 day old winter wolves following the old-carcass distribution, so I am assuming that they do follow these trends.
One potential pitfall in assessing this distribution for older carcasses is that where Carver also bids within the range, the lower non-Carver bids are more likely to be overbid by Carver. Ignoring this for now. For 4 day old yetis, the new-Carver bidder would fall within the expected older-carcass non-Carver bid distribution, though all actual bids were higher. Ignoring this as well.
The old-carcass distribution appears to be:
snow serpent: 10-17, but heavily skewed to higher values
winter wolf: 20-23, possibly some skew to higher values
yeti: uniformish 30-35, though with few data points, could also have some skew to higher.
In terms of revenue obtained, Yetis and Winter Wolves appear to show a linearish decline with age, while Snow Serpents appear roughly constant. Assuming such linearity/constantness:
snow serpent expected revenue: 27.15
winter wolf expected revenue: about 39 −2*age
yeti expected revenue: about 75-3.4*age
Solution based on the above:
Lot 1, Yeti, 0 days: bid 56 to beat expected new-carcass bid of 55. Expected profit about 19sp, we have 344 sp remaining.
Lot 2, Snow Serpent, 2 days: bid 21 to beat expected new-carcass bid of 20, 7 sp expected profit. 323 sp remaining.
Lot 3, Snow Serpent, 1 day: no profitable bid expected, skip.
Lot 4, Winter Wolf, 1 day: no profitable bid expected, skip.
Lot 5, Yeti, 5 days: bid 36 as there is enough likelihood of a bid up to 35 to avoid losing the ~22sp expected profit. 287 sp remaining.
Lot 6, Winter Wolf, 1 day: skip as with previous 1 day winter wolf
Lot 7, Snow Serpent, 1 day: skip as with previous 1 day snow serpent
Lot 8, Snow Serpent, 5 days: bid 18 as there is enough likelihood of a bid of 17 to make it not worth risking losing the ~9sp expected profit. 269 sp remaining.
Lot 9, Winter Wolf, 3 days: bid 24 as there is enough likelihood of a bid of 23 to make it not worth risking the ~9sp expected profit. 245 sp remaining.
Lot 10, Winter Wolf, 7 days: at an expected revenue of ~25, it is worth going down to 23sp to get ~2sp profit though we might lose. Going down to 22sp looks like it will reduce the chances of getting it by more than a third for less than 50% gain, not worth it. So, bid 23sp, ~2sp profit, 222 sp remaining.
Lot 11, Winter Wolf, 8 days: at an expected revenue of a bit over 23, it is worth going down to 22sp, but not to 21sp (which halves or slightly more than halves success chance for less than a doubling of revenue. So, bid 22sp, ~1sp profit, 200 sp remaining.
Lot 12, Snow Serpent, 8 days: bid 18 for same reason as any other old snow serpent. ~9sp profit, 192 sp remaining.
Lot 13, Winter Wolf, 2 days: bid 27 to beat expected new-carcass bid of 26 (even though we have no 2-day winter wolf non-Carver bids to confirm this); expected profit ~8sp, 165 sp remaining.
Since we have lots of sp remaining, I go back and put in bids for the 1 day winter wolves and snow serpents I skipped. Specifically, I’ll put in 18 for the one day snow serpents and 24 for the one day winter wolves to beat the old-carcass bid distribution in case the new-carcass bidder doesn’t show up.
A potential model of the full problem (involves questionable numerology):
There are 13 lots currently, and the number of carcasses in the record is divisible by 13 (129*13). If we include the current auctions, that’s 13*130, or 13*13*10.
So, I’ll assume that all auctions have 13 lots.
The individual monster types aren’t divisible by 13 (except Snow Serpents), nor are they if we include the current auctions (except Winter Wolves). However, the Wolf:Yeti ratio seems very close to 3, and if the overall ratios were 2:5:6 that would fit in with the 13 theme and seems close enough to the Yeti:Serpent:Wolf ratio.
The age distributions look fairly triangularish, with a maximum age of 11. One possible way to express that would be there is on average 1 carcass of age 11 for every 2 aged 10, up to every 12 aged 0. And what’s 1+2+...12? Of course—a multiple of 13. Specifically 13*6.
Now, it would be nice if looking at the data in blocks of 13*6 showed some pattern, but I don’t see one, nor is the data a multiple of 6. No matter, we will press on without such empirical validation.
I also note that, in the current auction, the early lots look newer than the late auctions. Coincidence? Probably.
So, model: Base Lot Generation (low confidence): Auction of 13 lots each day Each lot assigned an age from 0 to 11 by a random distribution weighted by {12-age} Each lot assigned an animal type by a random distribution weighted 2:5:6 for Yeti:Serpent:Wolf
Bids: There are two non-Carver bidders, one of whom bids: 60-20*age for Snow Serpents 50-12*age for Winter Wolves 55-6*age for Yeti
and the other one bids: 9+d8 for Snow Serpents 19+d4 for Winter Wolves 29+d6 for Yeti
whereas Carver bids (not high confidence): 7+2d10 for Snow Serpents 31+2d8-3*age for Winter Wolves 32+2d20-2*age for Yeti
Revenue: (credit to GuySrinivasan) 20+2d6 for Snow Serpents 25+4d6-2*age for Winter Wolves 72+1d6-{age}d6 for Yeti (assuming that the prior for d6′s over d5′s is stronger than the 45 times better fit for a d5 in the base part of this formula)
Still gonna guess that somehow this is from a distribution more easily generated than the triangular. But I may be overfitting on the last time I said something was triangular and it turned out to be exponential. Still, it looks to me like “start at age=0; roll a d6; if you rolled 1-5, add 1 to age and repeat, otherwise on 6 stop; if you reach age 12, discard and start again from age=0” is a decent match for this distribution, and it involves a d6, so that’s my guess I think.
Looks like the likelihood for triangular is over a million times better (to log-nearest order of magnitude ~10^-1672 v. ~10^-1679) than the 1⁄6 drop per turn exponential.
Anyway, in the spirit of tumbling platonic solids:
One possible distribution for the age numbers would be the distribution generated by min(d12,d12)-1. This is not the same as the 1,2,3,4...12 triangular distribution, but rather a 1,3,5,7,...23 triangular distribution. (The 1,2,3...12 distribution would be generated by min (d12, d13)-1).
And checking the likelihood—this one is actually better.
LOG10 likelihood
-1672.05 for 1,2,3,...12
-1671.43 for 1,3,5,...23
P.S. I was terse in the previous comment because of time constraints. About the difficulty of the triangular distribution, I was thinking it wasn’t that unlikely anyway because in the previous problem abstractapplic generated a weighted average by taking a random entry from a list that contained duplicates, and a suitable list could be generated easily enough using a for loop.
To save anyone time if they want to read my decision, it’s the same as GuySrinivasan’s better formatted decision except on lots 10 and 11, where I took seriously the statement that we’re indifferent to work and bid 1 sp higher than GuySrinivasan did.
Above I explicitly noticed that Carver’s bids would distort the distribution, set that aside, then, when I encountered a skewed distribution, failed to think to attribute it to Carver’s bids. But of course that would be the obvious explanation.
I looked at this, and found that the actual distribution does match pretty well what I’d expect if the true distribution of non-Carver bids were a uniform distribution for old carcasses, though the skew effect was only really dramatic for Snow Serpents anyway.
(edit): Assuming this is correct, we can now better calculate the probability of losing if we bid lower.
For Lot 10: 23 v. 22 sp on 7 day wolf, we cut win rate by exactly 1⁄3 going down to 22, the revenue is a bit over 25 by my linear fit or exactly 25 by GuySrinivasan’s formula; meaning, if GuySrinivasan is right then the average profit is exactly the same either way.
For Lot 11: 22 v. 21 sp on 8 day wolf, we exactly halve win rate going down to 21, the revenue is a bit over 23 by my linear fit or exactly 23 by GuySrinivasan’s formula; meaning, if GuySrinivasan is right then the average profit is exactly the same either way.
(edit2): A further note—Even though accounting for Carver’s bids when calculating the old-carcass bid distribution dramatically reduces the loss probability from bidding low on Snow Serpents, it’s still not worth it to bid low on them (since 1⁄8 probability of losing more than makes up for profit going from 9 to 10). However, if the parameters of the problem had been changed a bit, this could have been a nice trap.
(edit3): Or wait—do we actually know that the non-Carver bid distribution for Snow Serpents only goes down to 10? what if it goes down to 8 or lower, but we miss those because carver bids over those? Carver’s bids go down to 9, so we could theoretically see a 9 non-Carver bid, but the data might not include any by coincidence. We can’t possibly see a non-Carver bid below 9. Looking at this—no, if it goes down to 8, the number of non-Carver bids we’d expect to see seems too low—I’m reasonably confident it doesn’t go down that low.
Initial observations and results (warning, for all I know right now it could be substantially complete solution):
It looks like the non-Carver bidders for newer carcasses are consistent, and drop linearly with age in days:
snow serpents: 60 − 20*age
winter wolf: 50-12*age (technically we don’t have a third point to see that this is linear, but see below)
yeti: 55-6*age
Presumably, this is due to a single bidder. The trend continues until it drops below the distribution for older carcasses, so I guess that this bidder continues to bid this pattern down to 0 sp.
For older carcasses, there is a range of variation that doesn’t seem to depend much on age. I therefore assume for now that it really doesn’t depend on age, though for all I know it does.
For winter wolf, Carver won all the auctions for 2 and 3 day carcasses. However, Carver bid high enough for this data to be consistent with 2 day old winter wolves following the linear trend and 3 day old winter wolves following the old-carcass distribution, so I am assuming that they do follow these trends.
One potential pitfall in assessing this distribution for older carcasses is that where Carver also bids within the range, the lower non-Carver bids are more likely to be overbid by Carver. Ignoring this for now. For 4 day old yetis, the new-Carver bidder would fall within the expected older-carcass non-Carver bid distribution, though all actual bids were higher. Ignoring this as well.
The old-carcass distribution appears to be:
snow serpent: 10-17, but heavily skewed to higher values
winter wolf: 20-23, possibly some skew to higher values
yeti: uniformish 30-35, though with few data points, could also have some skew to higher.
In terms of revenue obtained, Yetis and Winter Wolves appear to show a linearish decline with age, while Snow Serpents appear roughly constant. Assuming such linearity/constantness:
snow serpent expected revenue: 27.15
winter wolf expected revenue: about 39 −2*age
yeti expected revenue: about 75-3.4*age
Solution based on the above:
Lot 1, Yeti, 0 days: bid 56 to beat expected new-carcass bid of 55. Expected profit about 19sp, we have 344 sp remaining.
Lot 2, Snow Serpent, 2 days: bid 21 to beat expected new-carcass bid of 20, 7 sp expected profit. 323 sp remaining.
Lot 3, Snow Serpent, 1 day: no profitable bid expected, skip.
Lot 4, Winter Wolf, 1 day: no profitable bid expected, skip.
Lot 5, Yeti, 5 days: bid 36 as there is enough likelihood of a bid up to 35 to avoid losing the ~22sp expected profit. 287 sp remaining.
Lot 6, Winter Wolf, 1 day: skip as with previous 1 day winter wolf
Lot 7, Snow Serpent, 1 day: skip as with previous 1 day snow serpent
Lot 8, Snow Serpent, 5 days: bid 18 as there is enough likelihood of a bid of 17 to make it not worth risking losing the ~9sp expected profit. 269 sp remaining.
Lot 9, Winter Wolf, 3 days: bid 24 as there is enough likelihood of a bid of 23 to make it not worth risking the ~9sp expected profit. 245 sp remaining.
Lot 10, Winter Wolf, 7 days: at an expected revenue of ~25, it is worth going down to 23sp to get ~2sp profit though we might lose. Going down to 22sp looks like it will reduce the chances of getting it by more than a third for less than 50% gain, not worth it. So, bid 23sp, ~2sp profit, 222 sp remaining.
Lot 11, Winter Wolf, 8 days: at an expected revenue of a bit over 23, it is worth going down to 22sp, but not to 21sp (which halves or slightly more than halves success chance for less than a doubling of revenue. So, bid 22sp, ~1sp profit, 200 sp remaining.
Lot 12, Snow Serpent, 8 days: bid 18 for same reason as any other old snow serpent. ~9sp profit, 192 sp remaining.
Lot 13, Winter Wolf, 2 days: bid 27 to beat expected new-carcass bid of 26 (even though we have no 2-day winter wolf non-Carver bids to confirm this); expected profit ~8sp, 165 sp remaining.
Since we have lots of sp remaining, I go back and put in bids for the 1 day winter wolves and snow serpents I skipped. Specifically, I’ll put in 18 for the one day snow serpents and 24 for the one day winter wolves to beat the old-carcass bid distribution in case the new-carcass bidder doesn’t show up.
A potential model of the full problem (involves questionable numerology):
There are 13 lots currently, and the number of carcasses in the record is divisible by 13 (129*13). If we include the current auctions, that’s 13*130, or 13*13*10.
So, I’ll assume that all auctions have 13 lots.
The individual monster types aren’t divisible by 13 (except Snow Serpents), nor are they if we include the current auctions (except Winter Wolves). However, the Wolf:Yeti ratio seems very close to 3, and if the overall ratios were 2:5:6 that would fit in with the 13 theme and seems close enough to the Yeti:Serpent:Wolf ratio.
The age distributions look fairly triangularish, with a maximum age of 11. One possible way to express that would be there is on average 1 carcass of age 11 for every 2 aged 10, up to every 12 aged 0. And what’s 1+2+...12? Of course—a multiple of 13. Specifically 13*6.
Now, it would be nice if looking at the data in blocks of 13*6 showed some pattern, but I don’t see one, nor is the data a multiple of 6. No matter, we will press on without such empirical validation.
I also note that, in the current auction, the early lots look newer than the late auctions. Coincidence? Probably.
So, model:
Base Lot Generation (low confidence):
Auction of 13 lots each day
Each lot assigned an age from 0 to 11 by a random distribution weighted by {12-age}
Each lot assigned an animal type by a random distribution weighted 2:5:6 for Yeti:Serpent:Wolf
Bids: There are two non-Carver bidders, one of whom bids:
60-20*age for Snow Serpents
50-12*age for Winter Wolves
55-6*age for Yeti
and the other one bids:
9+d8 for Snow Serpents
19+d4 for Winter Wolves
29+d6 for Yeti
whereas Carver bids (not high confidence):
7+2d10 for Snow Serpents
31+2d8-3*age for Winter Wolves
32+2d20-2*age for Yeti
Revenue: (credit to GuySrinivasan)
20+2d6 for Snow Serpents
25+4d6-2*age for Winter Wolves
72+1d6-{age}d6 for Yeti (assuming that the prior for d6′s over d5′s is stronger than the 45 times better fit for a d5 in the base part of this formula)
Love it!
The age distribution is very clearly off, though. Age 0 has about 2.5x the count of age 1, for each species. I didn’t see anything come up clearly...
The age 0 amount is higher than expected from the model distribution, but it’s nowhere near 2.5x the age 1 amount. I have:
Overall 289 age 0, 233 age 1 (expected 258, 236.5)
Snow Serpents 103 age 0, 92 age 1
Winter Wolves 141 age 0, 108 age 1
Yeti 45 age 0, 33 age 1
Thank you, my bin size was bad. :(
Still gonna guess that somehow this is from a distribution more easily generated than the triangular. But I may be overfitting on the last time I said something was triangular and it turned out to be exponential. Still, it looks to me like “start at age=0; roll a d6; if you rolled 1-5, add 1 to age and repeat, otherwise on 6 stop; if you reach age 12, discard and start again from age=0” is a decent match for this distribution, and it involves a d6, so that’s my guess I think.
Looks like the likelihood for triangular is over a million times better (to log-nearest order of magnitude ~10^-1672 v. ~10^-1679) than the 1⁄6 drop per turn exponential.
To acquire karma: respond to my posts with actual data.
Anyway, in the spirit of tumbling platonic solids:
One possible distribution for the age numbers would be the distribution generated by min(d12,d12)-1. This is not the same as the 1,2,3,4...12 triangular distribution, but rather a 1,3,5,7,...23 triangular distribution. (The 1,2,3...12 distribution would be generated by min (d12, d13)-1).
And checking the likelihood—this one is actually better.
LOG10 likelihood
-1672.05 for 1,2,3,...12
-1671.43 for 1,3,5,...23
P.S. I was terse in the previous comment because of time constraints. About the difficulty of the triangular distribution, I was thinking it wasn’t that unlikely anyway because in the previous problem abstractapplic generated a weighted average by taking a random entry from a list that contained duplicates, and a suitable list could be generated easily enough using a for loop.
To save anyone time if they want to read my decision, it’s the same as GuySrinivasan’s better formatted decision except on lots 10 and 11, where I took seriously the statement that we’re indifferent to work and bid 1 sp higher than GuySrinivasan did.
Above I explicitly noticed that Carver’s bids would distort the distribution, set that aside, then, when I encountered a skewed distribution, failed to think to attribute it to Carver’s bids. But of course that would be the obvious explanation.
I looked at this, and found that the actual distribution does match pretty well what I’d expect if the true distribution of non-Carver bids were a uniform distribution for old carcasses, though the skew effect was only really dramatic for Snow Serpents anyway.
(edit): Assuming this is correct, we can now better calculate the probability of losing if we bid lower.
For Lot 10: 23 v. 22 sp on 7 day wolf, we cut win rate by exactly 1⁄3 going down to 22, the revenue is a bit over 25 by my linear fit or exactly 25 by GuySrinivasan’s formula; meaning, if GuySrinivasan is right then the average profit is exactly the same either way.
For Lot 11: 22 v. 21 sp on 8 day wolf, we exactly halve win rate going down to 21, the revenue is a bit over 23 by my linear fit or exactly 23 by GuySrinivasan’s formula; meaning, if GuySrinivasan is right then the average profit is exactly the same either way.
(edit2): A further note—Even though accounting for Carver’s bids when calculating the old-carcass bid distribution dramatically reduces the loss probability from bidding low on Snow Serpents, it’s still not worth it to bid low on them (since 1⁄8 probability of losing more than makes up for profit going from 9 to 10). However, if the parameters of the problem had been changed a bit, this could have been a nice trap.
(edit3): Or wait—do we actually know that the non-Carver bid distribution for Snow Serpents only goes down to 10? what if it goes down to 8 or lower, but we miss those because carver bids over those? Carver’s bids go down to 9, so we could theoretically see a 9 non-Carver bid, but the data might not include any by coincidence. We can’t possibly see a non-Carver bid below 9. Looking at this—no, if it goes down to 8, the number of non-Carver bids we’d expect to see seems too low—I’m reasonably confident it doesn’t go down that low.