D&D.Sci April 2021: Voyages of the Gray Swan

The most notable thing, to me, is that plausible extrapolation of the distribution of damage from the encounter types listed is far from being sufficient to explain how many ships are lost. Crabmonsters don't usually do a lot of damage and it looks as if maybe 1% of crabmonster encounters are terminal. Demon whales are the most unclear -- the PDF is still increasing at 100% damage -- but it looks as if maybe 100% is approximately the peak, in which case we should expect about as many sinkings as we've seen non-sinkings. Harpies are never known to do >24% damage. Krakens are never known to do >78% damage. Merpeople seem like they have a bit of a right tail, but there's no way to know how far it extends. It doesn't look as if it's likely to account for a huge numbre of sinkings. Nessie has what looks like a normal-ish distribution whose right tail extends a bit beyond 100% but I wouldn't expect more than a few percent of encounters to result in sinking. Pirates are never known to do more than 64% damage. Sharks are never known to do more than 56% damage. Water elementals are always between 74% and 85%. These seem like we might have maybe 10-20 sinkings from crabmonsters, 330 from demon whales, a couple of hundred from merpeople, 25ish from Nessie, and none from anything else. But actually there are 2367 sinkings.

This suggests that most of the sinkings come either from encounters that are way beyond the known distribution of damage, or from entirely different types of encounters we don't have a record of.

For the first option, maybe the demon whale distribution keeps growing rightward for a while longer. Not terribly implausible. Maybe the merpeople have a really long right tail. The water elemental distribution is pretty weird; maybe there are other types of water elemental with similarly narrow distributions living beyond 100%. For the second option, maybe sometimes the gods get angry and smite ships with lightning so that everyone dies, or maybe there's some horribly dangerous geographical feature that isn't on the maps because once you get close enough to see it you're already doomed. Or maybe sometimes the captain or crew decides that they can sail far off course, sell the cargo, keep the proceeds, and live happily ever after.

What can we do about all these? We can completely stop merpeople attacking. Maybe merpeople are a lot of the mystery sinkings. Might be worth it. We can bring krakens and demon whales down by 2% per oar, and weirdly this stacks additively, so buying 50 oars might be worth it since demon whales are our leading "in-sample" candidate to explain the sinkings. Except that we top out at 20 extra oars. Better than nothing, probably worth it. Shark repellent is a terrible idea because sharks are among the nicer encounter options. Rifles for harpy protection seem dumb since harpies are never known to do much, but if we're battling unknown unknowns something that offers to prevent all damage done by harpies is tempting. (Though it's hard to know how we could actually know that they will prevent all damage even in hypothetical cases where the harpies do >4x more damage than they have ever done to any ship that's returned to tell the tale.) Water elementals are pretty common and the foam swords will always save a lot of damage when they attack -- and if indeed there are currently-unknown super-water-elementals doing >100% damage, they might save us.

As for the completely unknown unknowns, maybe we can bring a priest or institute big rewards for foiling mutinies, or something.

(Perhaps I should take the name of the ship as a hint that we're facing the first kind of scenario rather than the second.)

Since my top priority is to survive, I buy off the merpeople (45gp) and fill up on extra oars (20gp). Now I could get rifles (35gp) just in case there are occasional super-harpies, but it's not clear that that's more likely than occasional super-elementals so I'll get the foam swords instead (15gp) since that does much more to reduce damage in the non-fatal cases. I have 20gp left, with which I can halve crabmonster damage (saving, by eyeball, maybe 20% damage on average in the 4% of cases when they attack) or knock 20% off Nessie-and-pirate damage (saving, by eyeball, maybe 16% of damage in 4% of cases when Nessie attacks, plus maybe 12% of damage in 20% of cases when pirates attack); the latter is clearly better damage-wise and might be better sinking-wise too. So: tribute to the merpeople, max extra oars, foam swords, two cannons.

But I am still painfully aware that I don't know where those sinkings are really coming from, and if there were some way to get more information about that (e.g., fit some ships with lifeboats) it would be a hell of a good idea to do it first.

I didn't see any obvious sign of large seasonal or directional effects. There are some long-term drifts; e.g., pirates seem to be becoming less common, harpies, merpeople, sharks, and maybe krakens seem to be becoming more common. I don't think these effects are large enough to change the decisions above. I haven't looked for correlations (e.g., maybe when a ship has been attacked by pirates it becomes more or less likely that the next ship going the same way will be). I do have more detailed notes on what the damage-distributions look like for the various kinds of encounter, but the level of detail above seems sufficient. (They don't appear to vary much with month, year, or direction.)

[EDITED to add:] Hmm, but buying off the merpeople might be risky? It would mean more attacks by other things that might turn out to be potentially ship-sinking. In particular, more demon whales, and our best demon whale countermeasure isn't very good. I'm not terribly sure that paying the merpeople is actually wise.

noticing that the pirates decline in relative frequency

From looking at the probability distributions, I mostly agree with gjm; my current recommendations are the same as GuySrinivasan's and Measure's.

Demon Whale distribution scares the crap out of me, and I probably panic buy all 20 oars allowed. I mostly agree that the distribution looks like it could be peaking near 100% damage, but I am not at all confident of this and think it could be consistent with something growing a lot bigger beyond the cutoff. I expect 250-1000 of the destroyed vessels to have been destroyed by demon whales. While it looks like it will be in control before needing 20 oars, I am uncertain enough to value oars pretty high. Budget spent = 20gp.

Merpeople have a very wierd looking distribution. It doesn't seem to tailing off at the end, and so (like gjm) I am very uncertain about what happens after the 100% cutoff. I think (like GuySrinivasan) there's a possibility that it has a multimodal distribution with another peak beyond 100% (not saying bimodal since there even looks like there might be a small peak around 25% distorting the main peak of  about 50%, though this could very easily be random). I figure merpeople are responsible for around 200 (assuming no extra peak) or potentially vastly more (assuming an extra peak) of the losses. Merpeople are potentially another solid choice for mitigation imo, unless removing them from the encounter table puts in demon whales in as the substitute. Budget spent = 45+20=65gp

I do not agree with gjm that only about 1% of crabmonster encounters are terminal. The distribution seems to be tailing off very slowly, visually more or less consistent with a triangle-shaped distribution. A simple linear extrapolation would suggest a few percent which I would take as a lower bound. But it might not be linear, but slowing in how it tapers off, so it might be much much more than this. For all I know (apart from the finite number of sinkings) it might not even sum to a finite value. On the other hand, we only really care about crabmonster attacks that do less than 200% damage, since the only relevant intervention reduces damage by 50%. I estimate that between about 50 and about 250 of the destroyed vessels to have been destroyed by the relevant part of the crabmonster damage distribution, with potentially unlimited numbers destroyed by crabmonsters outside that range. Despite the expected max of about 250 mitigatable losses I consider arming carpenters a pretty solid choice for 20gp. Budget spent = 20+65=85gp.

Nessie looks like a pretty straightforward distribution where I assume about 10-20 losses are from the bit of the distribution we can't see. A single cannon (which is all that we can afford after the other purchases) should suffice. Budget spent = 10+85=95gp.

Conclusion: 20 oars (20gp) + pay off merpeople (45gp) + arm carpenters (20gp) + one cannon (10gp), total 95 gp spent. This is the same plan previously recommended by GuySrinivasan and Measure.

Nothing else looks like it can kill us, unless e.g. some bimodal distribution has one of its humps located entirely within the >=100% zone. 

However, stepping out of the pure data analysis and into reasoning about the fantasy world, it seems strange that pirates would bother to attack us if they only ever do 64% damage. They are intelligent, after all. Maybe they run away if in a hard fight, not sticking around to do more damage than that, and take over the ship entirely if they win? I might consider a second cannon (also provides extra insurance in case nessie's distribution extends further than expected). Dropping 3 oars would provide the funds, and will probably still be enough for the demon whale. 

(and...also anticipated by Measure on the pirate theory).

 By looking at the data, squinting at some sumcurves, fitting some very speculative lines and sacrificing a pidgeon to the diviners I guess that of the 2300 ships that were lost were caused by:

1000 were due to Crab people (whose damage distribution doesn't look like any of the others', so I am guessing 50% of their attacks do more than 100%)

700 were due to Demon Whales (Looks like at least 2/3 of their attacks do more than 100% damage)

300 were due to merpeople (their damage might have 10% above 100)

200 were due to Nessie (seems around 15 - 20 % of damage above 100)

The last two are the most imprecise. None of the remaining encounters look like they could have any serious chance of sinking a ship, so let us leave the last 100 or thereabouts to them.

Judging by this I will suggest getting Armed Carpenters, 20 Oars and either 3 cannons or bribe the merpeople. I wil swing with the cannons, since Varsuvius law tells me that fewer merpeople means more other monsters, potentially Demon Whales. At last, if less damage is more important than saving money we could get foam swords, they won't save our lives but might save on the ship repair costs in the long run.

Spoiler protection

Known harpy, kraken, merpeople, and shark attacks seem to be on a gentle rising trend.

Known pirate attacks fall substantially from 1401 onwards, but are still the most common. Despite this the year and month doesn't appear to have much impact on losses, and there is no obvious correlation between the month and the frequency of a given attack.

The direction doesn't appear to have much impact either.

Looking at how the damage is distributed across known encounters

- Harpy peaks at around 10-15 then falls away, none have done more than 23

- Kraken peaks at around 55 then falls away, none have done more than 78

- Pirates peaks around 15-20, and falls away, none have done more than 64

- Sharks peaks at around 20 -25 then falls away, none have done more than 56

- water elemental is strongly peaked around 80, but falls away rapidly, none have done more than 85

Suggests the only serious threats are:

- crabmonsters

- demon whale

- merpeople

- nessie

All of these peaked below 100 percent damage with 1 exception the demon whale, which suggests this is the biggest threat. With this in mind I should definitely invest in all 20 oars for 20 gp.

Nessie is responsible the causing the second highest number of incidents with very heavy damage, so investing another 30gp in cannons looks sensible.

Merpeople are the next biggest threat, but bribing them could make matters worse if it leads to more dangerous encounters. There were 2678 known merpeople encounters. If we assume they are spread evenly across the other 8 encounter types that means at least 334 demon whale attacks. Looking at the distribution of damage caused by the demon whale attacks it looks like the peak is likely to be well over 100 percent damage, and they represent a very real threat even with all the extra oars. It is not at all clear if this would represent a net reduction in risk, and when I consider that the high cost would prevent me investing in crabmonster defenses the benefits of this look dubious.

The crabmonsters are the only other encounter that do a large amount of damage, so arming the carpenters for 20gp looks sensible.

For the remaining 35 gp investing money to reduce a minute risk to my life and impressing the admiralty with less damage done to the ship seems a lot more attractive than minutely increasing the risk to my life and impressing them with saving money. Of the lesser threats the water elementals do the most damage, so if any of them are going to get me it is most likely to be them, so I will invest the last of the gold in the foam swords.

Final decision:

- 20 oars @ 20gp

- 3 cannons @ 30gp

- Arm the carpenters @ 20gp

- Foam swords for the deck crew @ 15gp

- 15 gp under budget

Looking at the 99% and 98% damaged ships:

• We see 46 99%-damaged ships that were attacked: 31 by demon whales, 6 by Nessie, 1 by crabmonsters, 8 by merpeople.
• We see 44 98%-damaged ships that were attacked: 22 by demon whales, 6 by Nessie, 6 by crabmonsters, 10 by merpeople.

These look fairly similar, and suggest that those are the threats that are likely to be most dangerous (until of course some more serious threat is reliably unsurvivable so we don't find out about it, or some threat deals non-linear damage...but let's ignore that for now.  I was assured in Linear Algebra class that the real world never works like that.)

Given this: 

• Oars, Cannons and Armed Carpenters (in descending order) counter the most common threats.
• Tribute to the merpeople reduces the chance that they attack, though it increases the chance that something else does.
• Rifles and Foam Cutlasses counter attacks that don't seem to be serious enough to matter anyway.
• Shark repellent is actively harmful, since it prevents sharks from attacking (no shark attack has ever done more than 9% damage (EDIT: wrong, I sorted these as strings because I am a dullard, but they still never seem to do near-lethal damage), which makes sense because...well...how are sharks going to get into your boat?) and makes something more dangerous attack instead.

Adding up Armed Carpenters (20) + Tribute to Merpeople (45) + 20 Oars (20) + 3 Cannons (30) makes 115gp, which we can't quite afford.  My best guesses:

• If trying purely to survive, buy Tribute to Merpeople + 20 Oars + 3 Cannons for 95gp.
• If trying to survive while also impressing the Admiralty with your budget-consciousness (and your refusal to pay Danegeld to those damned merfolk!), buy Armed Carpenters + 20 Oars + 3 Cannons for 70gp.

 If I'm using all the gold, I would spend 20 on arming carpenters, 45 on mermaid tribute, 15 oars and 2 cannons. If I want to save gold, I would avoid the cannons. I think the big risks are evenly spread between demon whales, merpeople and crabmonsters. With some risk from nessie. 

 So, Oars, Bribes, one cannon, and then either Axes or another 2 cannons. I'm guessing the cannons will do more, based on speculating the extrapolations of Crab vs Nessie.