A.D&D.Sci May 2021: Interdimensional Monster Carcass Auction

Mild Boar is the best name I've seen in a while.

We're bidding against other people, not against essentially-us, as the problem initially states. :D 

I wonder how good the NPCs are? If e.g. there are 6 PCs, it should be two groups of 3 apiece with an extra NPC thrown into each group.

How concerned should we be that this man's best available method of making money off of his carcasses is to create a portal to 4 alternate dimensions???? Also, from a distant timeline? I am disturbed, not at the existence of an interdimensional traveler, but at the implications that this is his efficient path.

My first thought was that if I'm bidding against different versions of me then we should all co-operate and bid at most 1sp in order to maximise our profits. However we would most likely be up against NPC who won't do this which wrecks this strategy. In the absence of any indicateion of how the NPC and other bidders would bid I generated a random price between 0.9* average price and 0.9* minimum price, and then manually adjusted a couple that didn't  look right which resulted in the following bids:

Lot 1 Red Dragon 1 days since killed       73sp
Lot 2 Jungle Mammoth 1 days since killed     35sp
Lot 3 Mild Boar 5 days since killed        14sp
Lot 4 Jungle Mammoth 5 days since killed     22sp
Lot 5 Mild Boar 1 days since killed        14sp   
Lot 6 Green Dragon 2 days since killed      63sp
Lot 7 Mild Boar 2 days since killed        16sp
Lot 8 Mild Boar 5 days since killed        7sp
Lot 9 Mild Boar 8 days since killed        5sp
Lot 10 Mild Boar 6 days since killed       10sp
Lot 11 Mild Boar 8 days since killed       2sp
Lot 12 Blue Dragon 8 days since killed      18sp
Lot 13 Jewel Beetle 1 days since killed      4sp
Lot 14 Mild Boar 1 days since killed       14sp 
Lot 15 Jungle Mammoth 4 days since killed     29sp
Lot 16 Jungle Mammoth 2 days since killed     29sp
Lot 17 Mild Boar 5 days since killed       10sp
Lot 18 Red Dragon 6 days since killed       44sp
Lot 19 Mild Boar 5 days since killed       14sp
Lot 20 Jungle Mammoth 1 days since killed     34sp

Decision on how aggressively to bid

With some exceptions for the jewel beetle and mild boars, discussed below, I generally estimated the EV and bid lower by a scaling factor. The scaling factor was pretty ad hoc and not based on some sophisticated game theory, as I don't really know how aggressively people are going to bid. I did not adjust the scaling factor based on the lot number.

One Schelling point is to bid a total of 300, so I figure I should probably bid higher than that on average (given the revenue up for grabs is more than twice that). Another would be to bid at the minimum end of the observed range for each lot, so I could have tried to beat that if the minimums were reasonable, but didn't get around to actually checking this, except that I did note that my bids were above my expectations for what the true minimums were in the cases where I got around to estimating that. 

I assume other people are also bidding above these points. If that is not the case, I will win a lot of bids, but likely lose in profit to someone making higher per-lot profit on fewer lots.

Analysis of revenue from different carcass types:

Jungle Mammoths:

The Jungle Mammoths (=elephants?) looked consistent with a formula of 31+4d6-3dsd so I assumed that their EV was 45-3dsd.

Dragons:

The dragons look like they all have similar characteristics in their drops over time, with in particular a big drop of around 30 value between 4 and 5 dsd (except gray dragon which has too little data to tell). One possibility would be that each has their own non-time variant distribution which is added with a "dragon curve". If I had more time, I would have tried to figure out the dragon curve and the separate distributions based on comparing the different dragon types (or rule it out and look for another hypothesis). As it is, I estimated the dragons in a pretty ad-hoc manner (eyeballing graphs mostly).

I do note that red dragon has some interesting even/odd behaviour, as it is always odd from 1 dsd to 6 dsd, and always even from 7 dsd to 10 dsd. If the "dragon curve" hypothesis is true, then this could be explained by an always-even or always odd "red distribution" (e.g. 2*2d12?) combined with a "dragon curve" that switches from odd to even at that point.

Mild boars:

For the mild boars (=pigs?), I tried to figure some model out that would match the observed qualitative behaviour and came up with rolling two d20s and setting each individually to 0 if less than or equal to the dsd. However, this did not match the quantitative characteristics, as it was consistently too pessimistic at low dsd and too optimistic at high dsd.

So, instead of taking the hint that I was wrong, I doubled down and added some epicycles. Namely, rolling 3 dice, setting each to zero if below dsd, then taking the top two, except that if you rolled a zero, you had to include the zero. (That's a pretty crazy hypothesis as stated, but maybe slightly less crazy in the equivalent formulation of adding the dsd to each die, taking the top two dice, and then setting any die over 20 to zero).

This seemed to predict the low-dsd mild boars a lot better, but was still optimistic on the high-dsd mild boars. Due to low numbers, a close fit on the high-dsd boars might be less necessary though. It also predicts a bimodal distribution with a trough at around 22 and while you can sort of see something like a hint of that in the data, it is not very convincing. Going to 4 dice adversely affected the early mild boar fit and seemed worse overall.

Anyway, I decided to roll with it (the 2 out of 3 d20s model), but since I am not super convinced, I limited my bids on the 8 dsd mild boars (lots 9 and 11) to 9sp, equal to the ceiling of the average of observed value for 8 dsd mild boars. Due to the "winner's curse", in the very likely event that I am wrong on their distribution I will probably take a loss on these.

Jewel Beetle

As previously remarked on by other commenters, the jewel beetle (or "lottery ticket beetle" as I think of it) has a high variance distribution. It looks more or less like a power law. In fact, it looks like it's such an extreme power law that it won't even have a finite expected value, as the extreme low frequency outliers will have value disproportionate to the low frequency.

So, if I were in the position of the hypothetical scenario provided, I would probably bid a lot for the lottery ticket beetle.

However, I'm not in that situation. I am instead competing for the glory of being Numbah One. And while the jewel beetle might have an extreme value, it probably doesn't. So, I reduced my jewel beetle bid to the median jewel beetle value of 12 instead of gambling on an outlier here.

I also note that new jewel beetles seem to tend to be lower in value than old ones. Not sure if this is random and my prior is generally against this.

Actual Bid

71,33,16,24,20,51,19,16,9,15,9,18,12,20,26,31,16,42,16,33

Beetle hands! NPCs will all bid 300 on the beetle, I assume. HODL!