Thieves' Cache vs Deep Blue Heaven Shrine

Fallen London
1

[ETA: SIGNIFICANT REVISIONS HERE]

So I’ve been seeing recurring comments lately that Thieves’ Cache expedition is better than Deep Blue Heaven. I ran an analysis on it and that’s not what I am getting but perhaps I am missing something.

First some assumptions. 1) I assume we are comparing EPA over a long (statistically relevant) run and am ignoring opportunity costs etc, 2) I am assuming 1 action to get all the C:R for Eyeless Skull conversion (as I suspect it will cost 1 favour eventually), 3) I assume Humbug! is the most profitable way of caching in Criminal Favours, 4) Approximately 10% Eyeless Skull drop, 40% Souls drop (I know it is generally viewed that the drop rate is lower but these are the rates that I get from my data). It may be that it is precisely due to these assumptions that I reach a different conclusions and if that’s the case then I am happy with that.

The two expeditions have different costs and different rewards so one would first need to establish the correct basis of comparison. Thieve’s Caches costs half the supplies of DBH but also requires criminal favour. To my mind the correct basis of comparison is as follows:

  1. 2 Thieves’ Caches; versus
  2. 1 Deep Blue Heaven + 2 Humbugs!

These two options both cost 20 Supplies and 2 Criminal favours, thus establishing parity to compare the value of rewards. Now you might say that Thieve’s Cache starts with 1 progress whereas DBH has a rare success starting at 3. Well, I analysed all the ways of getting through the expedition (also accounting for A Sign?) and my figures are that the expected run for the Thieves’ Cache requires 8.4 supplies (over 3 actions) and the expected run for DBH requires 16.74 supplies (over 6.46 actions). Thus we are comparing 16.8 supplies (for two Caches) and 16.74 supplies. These are sufficiently close that I think the parity of 2 Caches to 1 DBH remains.

Next I looked at the rewards from the unpredictable treasures, ignoring the warm amber (but not ignoring the venomous rubies). My figures give the following expected values:

  • Thieves’ Cache (UUT 1-80) - 16.19 echoes for 2 action
  • DBH (excluding A tightly wrapped package; UUT 1-90, 100-160) (94.38%) - 54.09 echoes for 2 action
  • DBH (A tightly wrapped package; UUT 91-99) (5.62%)- 65 echoes for 4 actions
    (One action to complete the expedition and one action to play the unpredictable treasure; 2 actions to get F:TC and cash it in)

Taking account of the additional action needed to start the expedition, an action to play each Humbug! and 2 actions to cash in the Skull (one for Revolutionary &quotFavour&quot and one for Trade in), I get the following breakdown of options:

2 x Thieve’s Cache
Glim 60%: 44.38 echoes for 16.8 supplies and 12 actions
Souls 40%: 56.38 echoes for 16.8 supplies and 12 actions

1 x DBH, 2 x Humbug!

Tightly wrapped package 5.06%: 73.8 echoes for 16.74 supplies and 13.46 actions
Other unpredictable treasures 84.94%: 62.09 echoes for 16.74 supplies and 11.46 actions
Eyeless Skull 10%: 74.8 echoes for 16.74 supplies and 12.46 actions

Assuming that the supply costs are sufficiently equal, this lets us compare the two expeditions in relative EPA terms:

Thieves’ Cache: 4.10
DBH: 5.51

ETA: This is a relative comparison that shows that (with the assumptions above) DBH is more efficient. It does not say that your profit is at 4.61 since it doesn’t include the cost of supplies

ETA2: Corrected DBH ETA from 4.61 to 5.51
edited by genesis on 4/9/2017
edited by genesis on 4/15/2017

2

I used to believe in Echoes per Action and parity comparisons until I realized that neither assessment method reliably accounts for the vagaries of luck and the opportunity costs of spending actions on something else.

3

That’s an interesting and detailed analysis. I haven’t checked out all your assumptions and calculations, so I can’t tell if it’s accurate. However, I believe there are a couple of additional factors you haven’t taken into account:

  • It matters how you execute the expedition. For Thieve’s Cache you need to accumulate 9 progress, and for Deep Blue Heaven you need 19. If your Watchful is high enough for the Buccaneering approach, this means that TC needs 3 actions and DBH needs 4 (putting aside for a moment the chance to get &quotA Sign?&quot).[/li][li]While Humbug! can indeed be very profitable, it’s just one card, and there are many more options to collect Favours: Criminals. So, even assuming you’ll use Humbug any time it pops up, you will still probably find yourself with lots of Faovurs: Criminals (and when they reach 7 you can’t collect anymore). So now the question is - besides Humbug, what is the best use for these?
4

It’s a method of comparison. No method of comparison is going to account for all factors. Measuring opportunity costs is really difficult. Unless you are going to give up on the idea of comparison entirely or going to do so on the basis of subjective anecdotal approach, you need to have a benchmark. It’s completely fine to say that EPA isn’t a suitable benchmark for such-and-such purposes but then you’d need to provide an alternative benchmark.

As for luck, because I am using expected EPA this actually removes most of its vagaries. The main aspect of luck that remains is the concept of variance - effectively, how long you have to repeat a strategy until its average performance converges to the expected performance. So yes, it may be that you have to wait a while but in the long run the comparison holds.

[quote=dov]
It matters how you execute the expedition. For Thieve’s Cache you need to accumulate 9 progress, and for Deep Blue Heaven you need 19. If your Watchful is high enough for the Buccaneering approach, this means that TC needs 3 actions and DBH needs 4 (putting aside for a moment the chance to get &quotA Sign?&quot).[/quote]

Yes. So, I think I’ve accounted for that in the fourth paragraph. That’s why I say there that the expected run for the Cache is 3 actions and for DBH is 6.46. This in fact also accounts for the possibility of A Sign? The actual workings though are a bit too hairy to have included in the post.

That’s a fair point. But in a way I think it’s accounted for as well because I actually made a mistake when typing up the post (which I’ll edit in a moment). The relative ETA for Blue Deep Heaven is actually much higher:

0.056*(73.8/13.46) + 0.8494*(62.09/11.46) + 0.1*(74.8/12.46) = 5.51

This means that even with lower payout for Criminal Favours the comparison should hold. Suppose we say that a favour gives you only 2 echoes instead of 4.4. This is broadly in line with an Implausible Penance, for example. Then each possible outcome for DBH drops by (4.42-22)=4.8. So instead of payouts of 73.8, 62.09 and 74.8 we’ll have, instead: 59, 57.29 and 70. So we’d get:

0.056*(59/13.46) + 0.8494*(57.29/11.46) + 0.1*(70/12.46) = 5.05

It’s still higher than the Cache (as that’s not affected by your choice of Criminal Favour cash in).
edited by genesis on 4/9/2017

5

This isn’t a factor if you can 100% the watchful check, but for a lot of players the 3 action option isn’t a sure thing. And there’s a greater chance of having successive failures and raising rival’s progress to 10 over the longer expedition.

6

Few notes:

  • Beating the minister has best EPA of Criminal Favours.
  • Basic expedition cost is 9 and 19.
  • Getting and Eyeless Skull doesn’t require an extra AP as you don’t go through the treasure step. Save one AP, but use it to sell it
  • About once in 3 expeditions (Blue) you get the full benefit of a Sign
  • Avg of Thieves is around 37 E and Blue only reaches 53 E

Now, even with all these, Thieves has a way better EPA than Blue expeditions! Add the Glim and Souls, the CC from starting Thieves. My theory is that the higher values are harder to achieve so it’s probably not only RNG.
For those who finished Soul Trade it’s more than clear that they must go for Thieves only for that 40% chance to get Souls.

EDIT: Using Wiki info, excluding the Horsehead Amulet, assuming Glim has 60% chance, I got these values:

  • average Thieve 35.1625 for 9 crates which gives ~3.9069 E/crate
  • average Blue hits ~60.0949 for 19 crates which puts a crate’s value at ~3.2681. There’s almost 20% difference and the Eyeless skull is worth only ~4% more than the avg of Blue so it’s clear it won’t make a big difference.

You can add more &quotspice&quot to these numbers even if it lowers EPA. When talking about Dock Favours the point is to get the most juice out of one!
edited by Skinnyman on 4/9/2017

7

[quote=Skinnyman]Few notes:

  • Beating the minister has best EPA of Criminal Favours.
    [/quote]

Humbug! has EPA of 4.4. That’s vastly better than Beating the minister. Dov is right that one would have to account for the cases where you fail to draw Humbug! (which I partially have started doing in my last reply) but on pure EPA there is no doubt that Humbug is better.

This is already taken account of. What I hadn’t taken account of is that if you have a Sign as your last action then you could overshoot your target progress. My initial calculations had assumed you would always stop on 9/17/19 exactly. I am revising my workings and will post an update once done. It’s possible that the outcome would be affected.

That’s precisely what I said… The extra AP comes from having to procure revolutionary connections to pay for the trade in.

It’s not clear to me how you are getting these numbers. Once I update my workings for the action points I’ll also share my calculations for unpredictable treasures. But the current summary is in the original post and as you can see it does not agree with your estimates.

If you look at my post you’ll see that all of that is accounted for. So if you are getting a different EPA there must be a different reason (which could be my mistake of course!).

That’s possible, I suppose. As I am not a spirifer I don’t have a sense how that affects the economy.

[quote] Using Wiki info, excluding the Horsehead Amulet, assuming Glim has 60% chance, I got these values:

  • average Thieve 35.1625 for 9 crates which gives ~3.9069 E/crate
  • average Blue hits ~60.0949 for 19 crates which puts a crate’s value at ~3.2681. There’s almost 20% difference and the Eyeless skull is worth only ~4% more than the avg of Blue so it’s clear it won’t make a big difference.
    [/quote]

Well for one we are clearly disagreeing on these averages. These are not at all the figures I am getting, also from the wiki info. But I’ll wait till I can make my detailed calculations public.

However, my much bigger objection is to how you seem to be defining EPA! EPA is echoes per action, not per crate. If there were a consistent and definitive relationship between actions and crates that would be one thing but there isn’t. The number of crates you actually end up using varies due to A Sign and the rare success on the Heaven. To account for that you need to provide the appropriate parity comparison by calculating the expected crate use per run (as I have done). If you wanted to, you could of course argue that &quotechoes per crate&quot is a more meaningful benchmark but even in that case you would need to account for the variability of crate use and not just use 9/19. Secondly, the number of actions varies beyond just the varying amount of supplies. Different reward outcomes also come with different action expenditures. And unlike echoes where you can approximate or assume away small errors in echo values a difference of one action can change the calculations significantly.

8

Humbug actually has an EPA of 2.2. The total profit is 4.4 E, yes, but it’s spread across two actions - one for getting the Criminal Favours and one for spending it.[li]

9

Humbug has the additional cost of not earning another favor. That’s opportunity cost, which is hard to calculate, but considering the small difference between Humbug and Catch a Glimpse of the Notorious Smuggler, it’s quite certainly better to spend singular favors on the latter (barring certain combinations of card draws where you need to drop a favor quick but don’t have Implausible Penance available).

Dunno if I’ll bother to pull out my old math, but from what I remember, Crime or Punishment is the best way to spend criminal favors. I didn’t compare it to thief’s cache expeditions though, mostly because I knew it’d be impossible to gain enough expedition supplies to fund a thief’s cache for every criminal favor I got.

10

[quote=Optimatum]
Humbug actually has an EPA of 2.2. The total profit is 4.4 E, yes, but it’s spread across two actions - one for getting the Criminal Favours and one for spending it.[/quote]

A very fair point! I should have qualified it as &quotconditional EPA&quot. In the sense that the point of the exercise is to compare the efficacy of the two expeditions once you have the necessary supplies and favours. In other words, controlling for the input, to see how the output differs.

Also a fair point. I am willing to concede that the opportunity costs renders Humbug not uncontestably superior. Personally, I still think it’s vastly superior to beating the minister. But it’s a moot point because, as I said in response to dov, even allowing for a much lower payout than Humbug my conclusion stand. (assuming my revisions I mentioned in the last post don’t change the picture).

11

For Spirifers, it’s actually better to spend favours on One Stone in a Thousand instead. It takes 6.25 favours for the necessary 1000 Souls for the efficient trade-in option, making 13.5 actions. With 30.30 echoes profit, it’s 2.24 EPA compared to Try to Glimpse a Notorious Smuggler’s 2.14 EPA. As the Spirifer trade-in also gives 30 CP of C: Hell, two more actions to cash it in for another 4.80 echoes in the FQ brings us up to 15.5 actions and 35.10 echoes, for a total of 2.26 EPA.

That doesn’t make any sense. The input for all situations is actions, and we control for the input by calculating the profits relative to actions. Ignoring some of the actions entirely because of where they fall along the grinding process doesn’t make those actions unnecessary, so it can throw off the math entirely. For example, using your math on the example earlier in my post:

[ul][li]Spirifers can spend 6.25 favours (in as many actions) on An Implausible Penance for 1000 Souls. In one action those 1000 Souls become 30.30 echoes and 30 CP C: Hell. That 30 CP becomes 4.80 echoes in two actions. Overall that’s 35.10 echoes in 9.25 actions, for 3.79 EPA.[/li][li]Non-Spirifers would spend 1 favour on An Implausible Penance for 4.28 echoes and a pittance of C: Constables. That is 4.28 EPA.[/li][/ul]Using this math the Spirifage method has a lower EPA than the other method, even though my earlier math shows the opposite.
edited by Optimatum on 4/9/2017

12

Hm. Ok. I wasn’t ignoring it but rather trying to condition. Having thought a bit more about it, I think you are right that in general I can’t do that but I don’t think it’s for the reasons that you give and I also think that in this specific case it doesn’t matter.

Say you are comparing two strategies. Strategy 1 pays out x in a actions but also costs some resource that takes n actions to accumulate. Strategy 2 pays out y in b actions and also costs the same resource. The first pay out rate is x/(a+n) and the second is y/(b+n). Suppose you happen to know that the former is more efficient so x/(a+n) > y/(b+n) but you want to avoid calculating n. So you really want to conclude that x/a > y/b but as you say there are circumstances where you can’t conclude that.

In fact, a sufficient condition for that conclusion is that a > b. This is basically the case in my original post and so while you are right and in general I can’t just assume that conclusion, it should be ok there.

Having said that, I’ll try to incorporate the action count for favours more explicitly when I finish with the revisions.

13

If the relevant items are things you just tend to accumulate, such as whispered hints, then it is fair to exclude gathering them from calculations.

But for favours/supplies you spend actions getting them and no other benefit. You need to go out of your way to get them, so you need to include gathering them in your action count.

14

The only ‘benchmark’ which I use myself these days is the amount of money I get from selling surplus items every Sunday - in other words, my income per week.

I know that this is not a direct measurement of any specific endeavour (expeditions in this case), but it does include the effect of consequences - including opportunity costs - of any changes which I make to my money-making schemes.

And I will tell you this: where previously my characters had barely been able to reach past the ~1500 Echoes per week mark with Deep Blue Shrine expeditions, they are now breaching well past ~2000 Echoes per week after I switch over to Thieves’ Cache.

I used to think that too, after making all those Echoes per Action calculation. After hundreds of expeditions and other endeavours, I don’t anymore.
EPA calculations are poor for long-term revenue gauging.
edited by Rostygold on 4/10/2017

15

Oh, sorry for my previous post, but I was very busy this weekend and I opened this thread hours before posting. I had the stupid thought that I refreshed the page and even before the edit I somehow missed those extra posts.
As Rostygold mentioned, experience and hundreds of expeditions got me to this conclusion and that’s why I think bigger values of treasure are more rare than they should be.
Finishing Soul Trade offers a boost of Thieves expedition no matter of your side.

[quote=genesis][quote=Skinnyman]
Using Wiki info, excluding the Horsehead Amulet, assuming Glim has 60% chance, I got these values:

  • average Thieve 35.1625 for 9 crates which gives ~3.9069 E/crate
  • average Blue hits ~60.0949 for 19 crates which puts a crate’s value at ~3.2681. There’s almost 20% difference and the Eyeless skull is worth only ~4% more than the avg of Blue so it’s clear it won’t make a big difference.
    [/quote]

Well for one we are clearly disagreeing on these averages. These are not at all the figures I am getting, also from the wiki info. But I’ll wait till I can make my detailed calculations public.
[/quote]

We’ll exclude the grinds that give you Favours so stick to cards. Because of this I consider that it’s best to squeeze as much as you can from one if the final EPA is decent.

As a Shepherd, before resetting my expedition sheet (doing both) I had an EPA of ~2.7; doing only Thieves and I’m sitting on a value of 3+ EPA. I only excluded some values because of the black swans I had, but I’ll include those as well after some time.
If I’ll cut down everything I will end up with a higher EPA, I’ll do less expeditions, use less AP and the Echo output will be way lower. No point to stop improving and extending a grind as long as the EPA is bigger than your possible average!

16

Just wanted to say thank you for the detailed analysis.

Personally I’m a huge fan of DBH. One of its advantages not mentioned here is that you get resources that are otherwise difficult to obtain (Airag, Night Whispers, Enigmas, Skulls). Those are in my opinion worth more than the price tag suggests simply because there are no straightforward grinds.

I’m doing a lot of DBH runs and in my experience Skulls are rarer than your numbers suggest. But I’m not really keeping notes and maybe I’m just unlucky (or am I actually being lucky if I don’t get skulls? ;)). But even if you’re are slightly off, I doubt the differences are large enough to make DBH less profitable than TC.

17

I agree with the other posters who are saying that you are likely underestimating the value of the Thieves’ Cache Unpredictable Treasure. I’ll be interested in seeing your numbers on that, but just with a quick look on the wiki, I see:

23.75% chance of An Instrument? for 14.5 echoes
22.5% chance of A Trophy? for 22.5 echoes
2.5% chance of An Amulet for 2 echoes
25% chance of A Wooden Mask? for 30.55 echoes (plus Warm Amber)
25% chance of The Seal of the Hooded Hawk for 40 echoes
1.25% chance of A fragile flake of vellum for 62.5 echoes

When you factor in 60/40 glim/souls and the clues from the start, those numbers give me ~35.23, which is in the same neighborhood as Skinnyman’s. How do your calculations differ such that you’re getting a value that’s less than half that?

Rahv7 does raise a good point in that if you are trying to achieve other goals that pure echoes (Master’s Blood, Noman Tattoo, etc.) then Deep Blue Heaven has its benefits. Personally, one of the fringe benefits of Thieves’ Caches for me is that Warm Amber a quarter of the time, which helps me get Rubbery Favours without having to worry about Scandal cures.
edited by Kaigen on 4/10/2017

18

While the Warm Amber can’t be sold directly, I find it is worth a decent amount, thanks to all the actions saved by getting Rubbery Favours without scandal.
edited by Optimatum on 4/10/2017

19

If you use your warm amber cleverly, you can avoid drawing the Afternoon of Good Deeds? card as much as possible. I only use the warm amber option if I’m < 2 scandal already, since The Devil and the Child conflict card significantly reduces scandal for some reason (I guess manipulating children out of their souls is commendable in london?).

20

Ugh. This is an embarrassing mistake. I had mistyped an 80 as a 60 in my spreadsheet and that’s why I was getting the incorrect (and lower) figures. The trade-off between automating complicated calculations is it’s more accurate in principle but more prone to errors!

It’s definitely helpful to have all the feedback and it’s clear that I need to revise a few things. I am working on a new version but also travelling this week so might be a bit slower in coming!

Interesting. I tried pricing WA in terms of the Trembling Amber tradein but not in terms of Scandal avoidance. Is it really more action efficient to use WA for favour rather than just spend a similar amount of actions on Scandal cure?