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

by abstractapplic2 min read12th Apr 202130 comments

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Exercises / Problem-Sets
Personal Blog

You were prepared for gratitude, a commendation from the Admiral, your own department, parades in your name. You were also prepared to hear that your ‘list of helpful suggestions for ensuring supply ships survive random encounters’ was an impudent insult to the collective intellect of High Command, and receive a public execution for your trouble. What you weren’t prepared for was what happened: being allocated a modest stipend, assigned to a vessel, and told that if you’re so clever you should implement your plans personally.

You have 100gp to spend, and your options are as follows:

InterventionCost
Coating the underside of the ship in shark repellent would ensure that no journey would feature shark attacks; however, Vaarsuvius’ Law (“every trip between plot-relevant locations will have exactly one random encounter”) means something else would attack instead.40gp
You’ve given up trying to understand what it is about woodwork that makes its practitioners so good at fighting Crabmonsters, but your findings are undeniable: arming the ship’s carpenters would halve the damage done by Crabmonster attacks.20gp
Offering tribute to the Merpeople would ensure they won’t attack the ship, similar to the effect of shark repellent.45gp
There’s enough space in the lower decks to add up to twenty more oars, so when fleeing is the best option, the entire crew can work together to escape. Each extra oar would decrease the damage done by Krakens and Demon Whales by 2%.1gp/oar
You wouldn’t think these ships could fit more artillery, but clever ergonomics allow you to add up to three more cannons. Your studies suggest each cannon would reduce the damage suffered in Nessie and Pirate attacks by 10%.10gp/cannon
Arming the Crow’s Nest with state-of-the-art rifles would give lookouts a 70% chance of ensuring a given Harpy attack does no damage.35gp
Giving the deck crew novelty foam swords to wield alongside their standard-issue cutlasses would improve their effectiveness when fighting Water Elementals, reducing the damage these creatures do by 60%.15gp

You’re completely confident in the effectiveness of your ideas, but much less confident that you know which combination would make the best use of your limited budget. To investigate this angle, you’ve procured a record of random encounters encountered by the ships travelling your assigned route; unfortunately, it’s missing some important information for the ships that sank, due to everyone who could fill in those details being dead.

As you board the Gray Swan (why do they give these ships such charmingly unique names when they’re all built and operated identically?), it occurs to you that this might have been intended as an execution after all. The dataset suggests that without any of your clever plans, the survival rate for a journey along your route is a little below 90%, and the Gray Swan is scheduled to make ten trips – five northbound voyages, five southbound – in quick succession. Hopefully this indicates nothing more than your superiors wanting to test your interventions very very thoroughly.

Your top priority is to save your skin. Secondary priorities are minimizing total damage taken and spending as little gold as possible, to impress High Command and return to their good graces.

What will you do?

(Notes:

  • As a passenger, you’ll be kept away from any fights, but the Gray Swan has no lifeboats; keeping the ship from sinking is necessary and sufficient to ensure your survival.
  • Ships are fully repaired every time they make port.
  • Interventions stack such that two 10% reductions are equivalent to one 20% reduction.
  • Interventions apply such that a 10% reduction to an attack that would do 80% damage does 72% damage instead.
  • Each journey takes a month; it is currently Month 5, Year 1406.)

I’ll be posting an interactive letting you test your decision, along with an explanation of how I generated the dataset, sometime next Monday. I’m giving you a week, but the task shouldn’t take more than a few hours; use Excel, R, Python, a priori knowledge, or whatever other tools you think are appropriate. Let me know in the comments if you have any questions about the scenario.

If you want to investigate collaboratively and/or call your decisions in advance, feel free to do so in the comments; however, please use spoiler tags or rot13 when sharing inferences/strategies/decisions, so people intending to fly solo can look for clarifications without being spoiled.

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I haven't looked much into dependence on time and direction yet, apart from

noticing that the pirates decline in relative frequency

but, I would like some clarification about whether the 100gp budget is for a single set of interventions we'll use on all trips, or is spent each time on a (potentially varying) arrangement. (edit: I see abstractapplic already responded to such a question from Measure: it's a single set obtained up front and not changed.)

My current thoughts:

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).

I just realized I may have misread things?

Suppose a ship was about to take damage such that our records would log it as "80%".

If we "halve" that damage, I think our records would read "40%".

If we "reduce by 10%" that damage, would our records read "70%" or would they read "72%"?

72%

(Good question, by the way; I added a note to the main post to clarify this.)

I thought it was unambiguously 72% except for the weirdness of "2%, ten times, is 20% rather than ~18%".

Thoughts after poking at the data for a short while:

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.

Looks like you've put in more work than I did below and come up with very similar answers, the one diff I see is that you tried 

foam swords as a speculative guess that water elementals are non-linear and explain the extra sinkings/a way to reduce non-sinking-related damage

whereas I just went for

a third cannon for a bit of marginal equity against Nessie

General:

I didn't notice any variation over years or months except that in the last five years Pirate attacks were less common and did slightly less damage. Considering we're making voyages in both directions over ten consecutive months, such variation shouldn't change the analysis much anyway. The most important factors are the distribution of possible damage amounts for each type of encounter and the relative frequency of each type.

Encounters:

  • Crabmonsters have a wide variation in damage with a long tail into the lethal range.
  • Harpies consistently do minimal damage.
  • Krakens do moderate damage with moderate variation but never lethal.
  • Merfolk do moderate damage with a wide variation into the lethal range.
  • Nessie does high damage with moderate variation into the lethal range.
  • Pirates do low-to-moderate damage but never lethal.
  • Sharks do low-to-moderate damage but never lethal.
  • Water Elementals do consistently high damage in a tight spread from 70-90. Never lethal.

We can ignore everything except Demon Whales, Crabmonsters, Merfolk, and Nessie since those are the only lethal threats.

Interventions:

  • Shark Repellent - Worse than useless since it increases the chance of a lethal encounter of another type.
  • Armed Carpenters - Good since it's cheap and should effectively reduce the lethality of Crabmonsters.
  • Merfolk Tribute - Possibly worthwhile even though it's expensive.
  • Additional Oars - Over half of all lethal encounters are Demon Whales, so as many oars as we can get.
  • Additional Cannons - We don't care about Pirates, but Nessie has a moderate chance of lethal damage and is one of the more common lethal encounters, so reducing that is worthwhile. Given the apparent damage distribution for Nessie, two is somewhat better than one and three might be slightly better than two.
  • Rifles - Useless since Harpies aren't lethal.
  • Foam Swords - Water Elementals aren't lethal, but if we have leftover budget we can considerably reduce repair costs.

Allocations:

  • Spend 20gp to arm the carpenters.
  • Spend 45gp for Tribute to the Merfolk.
  • Spend 20gp for +20 additional oars.
  • Spend 10gp for +1 additional cannon.

There was a choice at the end between [Merfolk tribute + 1 Cannon] or [Foam swords + 3 cannons]. I'm eyeballing that without intervention Merfolk and Nessie have about equal share of lethal encounters and 1 Cannon should significantly reduce lethal Nessie encounters. Eliminating Merfolk encounters does slightly raise the chance of encountering a Demon Whale, but overall I think it's still an improvement.

After reading gjm's point about unexplained excess sinkings, I'm less confident in my choices. I had briefly looked at them at the beginning of my analysis and concluded that most of the unknowns were Demon Whales, but Super-Water-Elementals seem like a real possibility as well considering their unusual damage distribution (and thematically that makes more sense than Super-Harpies). If I could have a do-over, I'd give up arming the carpenters to get the foam swords and a second cannon.

As long as we're considering things thematically, Pirates could probably board and commandeer a vessel without doing enough damage to risk sinking it, and they're the most common encounter, so maybe some of the excess unknowns are successful Pirate attacks?

I love your novel theory at the end there. Full marks.

Evidence against that theory is that

pirate attacks have changed somewhat in frequency, but ship-sinkings don't seem to have.

Possible counter-evidence:

Pirates have a bimodal distribution (around 20% and 40% damage) and only the 40% part of the distribution seems to have declined. So, this looks like two different populations and theoretically, the 20% pirates could be the strong, smart pirates who win a lot and back off early if they won't get an easy win, while the 40% pirates could be weak, stupid pirates who go all out every time. 

Still all totally speculative of course.

Nice idea.

Let me just say I love the concept on this one.

There is a chance that the extra missing ships come from unknown unknowns. I will spend gold as if they do not.

I give the merpeople 45gp tribute, reducing my absolute risk by 5.6% per trip.

I outfit the lower decks with 20 more oars for 20gp, reducing my absolute risk by 4.9% per trip. All from demon whales.

I purchase one cannon for 10gp, reducing my absolute risk by 0.1% per trip; Nessie can't sink me.

I arm the carpenters for 20gp, reducing my absolute risk by 0.2% per trip; crabs can't sink me.

Altogether I spend 95gp to reduce my risk from 10.6% per trip to 0% per trip, and report that costs may be saved if desired by not really telling anyone about cannons or carpenters, if the value of a successful voyage plus the cost of replacing a boat and crew is greater than about 10000 gp.

My method was to inspect graphs, noting that most encounters do not sink ships, noting that only the kraken seems to have any time-based concerns but that krakens do not sink ships and that I'll likely take the full insurance against them anyway due to demon whales, and noting that crabs appear to have some sort of simple linear decline, Nessie is likely normalish (I'm guessing something like 4d12+60), lots of things are normal and some look bimodal, demon whales are probably normal - fit - 109+-13ish, and merpeople are VERY SCARY because if they're bimodal then the second hump kills a lot of sailors. So then I estimated how many ships sink due to crabs (42), Nessie (23), and whales (1070), leaving 1135 from unknown causes. PROBABLY THE VERY SCARY MERPEOPLE.

I've put my attempt in my Github but I'll try to summarise it here.

Initial thoughts:

Edit: Changed to Rot13 as I failed on spoiler tags 

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Ships are fully repaired every time they make port.

Ah, I missed this. Now things look much simpler.

Fixing this mistake, I see that now the best result for my model is from buying

Woodworker weapons

Merpeople tribute

20 extra oars

1 extra cannon

with results

Survival rate for all 10 trips was estimated to 95 %

Average damage 41 % per trip were you survive

95 gold spent

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

Spoilered:

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.

Is the quip about Vaarsuvius' Law intended to be taken seriously here? I.e., are we to assume that without exception every voyage involves exactly one encounter? (And that that's why there are no entries in our records saying "no encounters"?)

Similarly, are we to assume that every voyage does have an entry in our records?

All your assumptions are correct, with the debatable exception of the last one: you have a record for every voyage

  • made along the same route you're taking
  • by a supply ship
  • working for the same Navy as you
  • in the span 1396-1405.

 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. 

Is it assumed that we spend some amount on interventions up front and then undergo all ten voyages with that setup? Are repair costs unaffected by interventions?

Yes to the first question.

To the second: you can aim to reduce repair costs with your interventions, but since they won't come out of your budget that's more of an optional extra for further impressing the brass.

Do I understand correctly that if a ship sank we simply have no record for it at all?

... Ah, no, those are the "100%+" lines, right? So for those we know that the ship sustained enough damage to sink it, we don't know what amount of damage it "would" have suffered if it had been tough enough to take more than 100%, and (after looking briefly, haven't looked more carefully yet) in all these cases we don't know what the cause was, only that something caused the ship to sink.

That's exactly right.

Only four creatures have been known to do significant damage, implying that they've probably sunk some ships:

  • demon whales
  • nessie
  • merpeople
  • crabs

All attacks by these creature appear to be roughly the same relative frequency on a yearly basis.

Demon whales look the scariest, let's max out at 20x oars: -20% for 20 gp.

Nessie has probably taken out a few ships but she rarely does more than 90%. Definitely worth a cannon: -10% for 10 gp.

Merpeople and crabs both have weird long tails. Let's skip the merpeople entirely for 45 gp.

What's sinking all these ships? Is it...murdercrabs? Let's arm our carpenters, -50% for 20 gp.

We've never seen any of the other creatures come close to sinking a ship, so we won't worry about sharks with lasers on their heads or mega-harpies. I have 5 gp remaining, which I keep as a fee for my services. Alternately, I'd consider trimming off 5 oars for a second cannon, but I'm not sure it's worth it.

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.