simon

D&D.Sci(-Fi) June 2021: The Duel with Earwax

Initial impressions:

Even though Janelle probably always uses Beta and Maria probably always uses Delta, we can get an idea of the characteristics of each resonance type by comparing their hypothetical results against heteropneums weak enough for them to overwhelm.

From eyeballing graphs of strength v. heteropneum amplitude for each resonance type and both pilots:

The qualitative behaviour of each resonance looks similar between Janelle and Maria, but quantitatively different (likely a simple multiplicative factor, but I should check!). The multiplier per pilot is different for the different resonance types (so, e.g, Janelle is about as strong as Maria with Beta resonance, but weaker with other resonance types).

And for the different resonance types the graphs are as follows:

Alpha does not depend on enemy strength and maybe has two clumps.

Beta does not depend on enemy strength.

Gamma's points seem to line up on straight, mostly slanted lines from a more-or-less common origin at zero enemy strength. Suggesting a strong dependence on enemy strength but one of the lines is flat and too low, so need to find a way to find out which line you'll end up on.

Delta has a gentle upward trend for Maria (too noisy to detect for Janelle). This does not appear to be a selection effect as Maria is always handily beating the heteropneums.

Epsilon has a curve that looks like a parabola at first, but then slows down, so maybe a sine curve? It is very consistent looking (not noisy) so should be possible to have an accurate fit for it. The curve looks a bit distorted in places for Janelle but this is likely just rounding due to her very low values at this resonance.

Zeta and Eta have points lined up on flat lines. For Zeta one of those lines is at zero.

Based on this, some candidate responses:

- If we can figure out which line we'll end up on, possibly Gamma as used by Janelle. We need to be confident however that we'll end up on a good line.
- Ditto for Zeta as used by Corazon, but with the additional caveat that we need to know that her past hypothetical results of 1.98 were low tier results (and she'll get high tier this time). If the 1.98's were high tier, she'll lose.
- If we can't figure out the information needed for either of the previous, the safe choice appears to be...Epsilon as used by Will. This may seem surprising at first glance since Will's only hypothetical result with Epsilon resonance was a measly 0.21. However, this result was at Heteropneum amplitude 0.57, near the bottom of the Epsilon power curve as seen for Maria and Janelle. If Will has the same Epsilon power curve but with a multiplier, he is around twice as strong as Maria with Epsilon resonance (but check rounding error bars!), and should confidently beat Earwax as long as the Epsilon power curve doesn't take a surprisingly sharp turn to decline between 3.12, where Maria last overwhelms heteropneums with Delta, and Earwax's amplitude of 3.2. However, Will will not overwhelm Earwax, and either option 1 or 2 could do so if successful, so if we can figure out the necessary information for either of those options, they would be preferable.

A.D&D.Sci May 2021 Evaluation and Ruleset

It* requires less effort because 'cooperation' reduces effort, while 'competition' increases it**.

In general, one would define cooperation in games as strategies that lead to better overall gains, and ignore effort involved in thinking up the strategy. In this case, there was an easy cooperative strategy, but it's not in general true, for example, in the Darwin Game designing a cooperative strategy was more complicated than a simple 3-bot defect strategy. 3-bot didn't do well but possibly could have if there were a lot of non-punishing simulators submitted (there weren't).

Also, even in this particular case, you could have had better results if you had taken the effort to get more to follow the same strategy. The rules did not explicitly forbid coordination, even by non-Lesswrongers, so you could have recruited a horde of acquaintances to spam 1-bids. (that might have been against the spirit of the rules, but you could have asked abstractapplic about it first I, I guess).

A.D&D.Sci May 2021 Evaluation and Ruleset

Good point. I should have anticipated strategies that require less effort to be more popular.

A.D&D.Sci May 2021 Evaluation and Ruleset

are the 2-bidders stable against 'defection'?

Of course not, they lose to 3-bidders. I wouldn't consider that "defection" in the same way though, since the 1-bidding is presumably an attempt at coordination and the 2-bidding would be exploiting that coordination and not directly a coordination attempt.

There weren't any 2-bidders.

Sure, but if 1-bidding were to become popular in similar problems, there would start to be 2-bidders.

A.D&D.Sci May 2021 Evaluation and Ruleset

Yeah, and actually 1-bidding can be a good strategy even from a selfish perspective if you can get enough people to coordinate on it, since a small enough number of high bidders will run out of money and the 1-bidders make a large profit on what they do win, though it's not stable against defection (2-bidders win in the 1-bidder-filled environment).

A.D&D.Sci May 2021 Evaluation and Ruleset

Bidder G reporting in...

Looks like my incorrect speculations on the exact models were likely not helpful, I also did not expect the 1 bidders (fine strategy against real duplicates like in the scenario given, but we're trying to have a competition here!).

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

I'm assuming that BST is British Summer Time and the deadline has passed. Remarks about the problem and my bid before abstractapplic posts the results:

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

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

Is our profit evaluated based on actual results, or based on expected value?

Does butterfly affect?

Sure, the butterfly is really minor compared to everything else going on, and so only "causes" the hurricane if you unnaturally consider the butterfly as a variable while many more important factors are held fixed.

But, I don't believe the assassination of Franz Ferdinand is in the same category. While there's certainly a danger that hindsight could make certain events look more pivotal than they really were, the very fact that we have a natural seeming chain of apparent causation from the assassination to the war is evidence against it being a "butterfly effect".

Update on Epsilon resonance:

It's cubic not sine; I can fit Maria's Epsilon data so that the curve rounds to the exactly correct value for every data point, and also for Janelle's data (separately) to round to the exactly correct value; I still need to check if I can make a single curve and multiplier between Maria and Janelle to round exactly for both, but it does look like the curves are at least fairly close to exact multiples of each other.

Interestingly, no x-value rounding needs to be assumed, at least to get the correctly rounding values for Maria and Janelle separately. So, perhaps the x (heteropneum amplitude) values are exact?

The cubic curve

doestake a big dive at high heteropneum amplitudes, but fortunately not until after Earwax's ~3.2 amplitude. Also, the fit for Maria's 0.57 amplitude result of 0.1 is actually around 0.096. Will getting 0.21 suggests he is at least around 2.13 times stronger than Maria using Epsilon and is projected to get at least about 3.85 against a 3.2 amplitude heteropneum. So, Will using Epsilon still looks like a safe pick to survive if we can't find guaranteed survival another way.