GoneBlank Posted September 13, 2016 Share Posted September 13, 2016 ...the long awaited sequel to Regarding ducats... [for the math inclined....] Let's assume that ducats have the following value schedule: common 15, uncommon 45, rare 100 Let's assume that the probabilities of these three states (assuming no traces are used) is: common 76%, uncommon 22% and rare 2%. If we have a preference to play solo then it is straightforward to work out the long-run average ducat value per mission: 15*0.76 + 45*0.22 + 100*0.02 = 23.3 If we play in a party of four - again assuming no uses of traces - I think the calculation gets a little trickier. After a little reading.... https://en.wikipedia.org/wiki/Multinomial_distribution I think the multinomial distribution is the right tool to help with the question. " ....the multinomial distribution is a generalization of the binomial distribution. For example, it models the probability of counts for rolling a k sided die n times. For n independent trials each of which leads to a success for exactly one of k categories, with each category having a given fixed success probability, the multinomial distribution gives the probability of any particular combination of numbers of successes for the various categories. " Or, to use our language, we have a k = 3 side die (common / uncommon / rare) and we roll that die n = 4 times if we play with a full squad of four people. Now let's assume that people will always select the most valuable (ducat wise) result that is returned. If I jump into my nearest stats package.... starting with probabilities of the three probabilities p = [0.76 0.22 0.02] n = 4 Let's generate reps = 10000 (i.e., 10000 relic missions). r = mnrnd(n, p, reps) ...it boils down to finding the highest ducat value per mission and averaging that result over the 10000 reps. >> mean(max( [(r > 0) .* repmat([15 45 100], reps, 1)], [], 2)) ans = 38.9330 So I've resorted to a numerical solution - which makes me feel dirty. Analytical ftw. I'm curious if other people consider this the right structure to analyze the problem. Summary: farming solo yields a long-run ducat value of 23.3 per mission. Farming in a full squad increases your long-run expected ducat per mission to 38.9. PS In many instances I've found myself taking a lot longer in PUG missions (than what I can do solo) - e.g., in both setting up a lobbies and searching for hard to find reactant - so I'd be the first to say that the above values are not the full story - just one aspect of it. PPS Why did I write this? I have an earache and I'm waiting for the paracetamol to work.... Link to comment Share on other sites More sharing options...
SeaUrchins Posted September 13, 2016 Share Posted September 13, 2016 (edited) I think you are missing a couple of an important thing. You don't take into consideration the time used to run missions and that the most valuable stuff (100 ducat items) is rarely traded for ducats in the first place, it is sold for plat in most cases. Edited September 13, 2016 by SeaUrchins Link to comment Share on other sites More sharing options...
GoneBlank Posted September 14, 2016 Author Share Posted September 14, 2016 A quick addendum... I got to wondering....imagine you have all prime parts and you have maxed out your capacity of void traces and want to do some ducat farming.... Might the value of your ducat farming be improved by upgrading your relics from intact to exceptional? If so - by how much? I raise that scenario because spending 25 void traces does not strike me as crazy (especially if you are maxed out on them) and it is probably not too far away from the rate at which you can replenish them. Assuming the probability of (common / uncommon / rare) is accurately represented as P_intact = [0.76 0.22 0.02] P_exceptional = [0.70 0.26 0.04] P_flawless = [0.6 0.34 0.06] P_radiant = [0.5 0.4 0.1] and the corresponding ducat payout rate is ducats = [15; 45; 100] Then we can calculate the solo long-run ducat payout per mission as P_intact*ducats = 23.3 P_exceptional*ducats = 26.2 P_flawless*ducats = 30.3 P_radiant*ducats = 35.5 So for solo play we move from an average ducat payout of 23.3 for a (non-upgraded) intact relic to 26.2 if we are are willing to spend 25 void traces a mission. It is probably worth noting that flawless technically represents the best extra ducat value per void trace spent; i.e. flawless yields an additional 0.14 ducats per trace compared to exceptional's 0.116 ducats per trace. However, practically speaking there is value in keeping your trace stockpile close to maximum capacity (for the next batch of prime farming). Using the framework outlined in the original post we can analyze the impact if every member of the four person squad utilizes the various relics to calculate the long-run ducats per mission average as follows: >> r = mnrnd(n,P_intact,reps); >> mean(max( [(r > 0) .* repmat([15 45 100], reps, 1)], [], 2)) = 39.2000 >> r = mnrnd(n,P_exceptional,reps); >> mean(max( [(r > 0) .* repmat([15 45 100], reps, 1)], [], 2)) = 46.2235 >> r = mnrnd(n,P_flawless,reps); >> mean(max( [(r > 0) .* repmat([15 45 100], reps, 1)], [], 2)) = 53.0840 >> r = mnrnd(n,P_radiant,reps); >> mean(max( [(r > 0) .* repmat([15 45 100], reps, 1)], [], 2)) = 62.1890 NB: the intact value slightly differs to the value calculated in the original post because I forgot to set a fixed random seed (dohh). Two interesting results: (1) in the group farming setting (where everyone is running the same key) we see that 25 void traces (exceptional) nets you an additional long-run average of 46.2235-39.2=7 ducats per mission (compared to the additional 3 ducats per mission in a solo setting) and (2) in the group farming setting the exceptional relic actually offers the best additional ducat per void trace. Summary: if you are sitting on a large pile of void traces and you need ducats then it is not crazy to consider upgrading relics to exceptional. For additional bang-for-the-buck do this in a group. PS @SeaUrchins I'll be the first to admit that I'm looking at this in a very narrow way - that's why I keep mentioning the phrase ducat farming.....not some ducat and a bit of plat farming..... I suspect that's why my wife tends to describe my thinking as 'linear'. However, the benefit to being quite specific is that it enables you to come up with some exact results.....which I like. Link to comment Share on other sites More sharing options...
ArcusVeles Posted September 14, 2016 Share Posted September 14, 2016 You can greatly lower the time to complete in a group if you bring a Nekros to your runs, and do your best to lead the group + kill with a slash weapon. I've been able to get ten reactants out of as low as twenty enemies just by turning each combatant in to chum with the Tigris Prime (first shot kills, second shot splits the corpse in to 2-4 pieces) and Desecrating the lot of it. Just get way ahead with some bullet jump mod in your exilus slot (I use Ice Spring because it looks awesome with the necromancer motif), wait for them to become corrupted, then make mince meat before your fellows can muck things up too badly. Link to comment Share on other sites More sharing options...
Xaranoth Posted September 14, 2016 Share Posted September 14, 2016 I can't tell for certain, but did you consider forma as a reward from relics? Because that adds in a chance of 0 ducats for a run. And it's a higher chance of that happening playing solo. Link to comment Share on other sites More sharing options...
GoneBlank Posted September 15, 2016 Author Share Posted September 15, 2016 (edited) @Xaranoth You raise an excellent point - the math above does not consider forma (which does not generate ducats). Therefore, the above math may paint an upwardly biased picture of ducat earning potential. That being said, I think the above math is good for relics that do not contain forma in their drop chances. But what is the impact if relics do contain forma? You encouraged me to have a look on the wiki. ... So it looks like three scenarios are possible: (1) there is no forma in the particular relic's drop table, (2) a forma can drop among one of three possible commons and (3) a forma can drop among one of two possible uncommons. Note: from what I could see on the wiki, forma is not a part of any rare drop tables. [Opinion: I have no issue with forma being a part of the common drop pool - but rolling an uncommon just to get forma... hmmm.] I won't repeat all the above results - but let's just consider the impact of this on solo play. In essence, when considering forma, we are now playing a game with four possible outcomes ducats = [0; 15; 45; 100] If I play solo with an intact relic with no forma in its drop tables P_intact = [0 0.76 0.22 0.02] then my long-run ducat payoff is expected to be P_intact * ducats = 23.3 That is, the same result as calculated above. If I play solo with an intact relic with a forma that is part of the common drop chance (and assuming that all names in the common pool have the same chance of dropping) then P_intact = [(1/3)*0.76 (2/3)*(0.76) 0.22 0.02] = [0.2533 0.5067 0.2200 0.0200] then my long-run ducat payoff is expected to be P_intact * ducats = 19.5 If I play solo with an intact relic with a forma that is part of the uncommon drop chance (and assuming that all names in the uncommon pool have the same chance of dropping) then P_intact = [(1/2)*0.22 0.76 (1/2)*0.22 0.02] = [0.1100 0.7600 0.1100 0.0200] then my long-run ducat payoff is expected to be P_intact * ducats = 18.35 For the sake of everyone's sanity I will leave out the various possibilities with relics (I think the above results give people a reasonable view of the impact). Summary If you want to maximize your ducat farming then avoid relics with forma (I don't think that point will win me a nobel prize) If you have to use a relic with forma in the drop tables there is a small advantage to choosing a relic with forma in the common drop pool (versus the uncommon drop pool). For a solo player, relics with forma in their common drop pool will on average earn 16.3% less ducats than a relic without forma in any drop pools. Edited September 15, 2016 by GoneBlank Link to comment Share on other sites More sharing options...
Xaranoth Posted September 15, 2016 Share Posted September 15, 2016 Interesting. I guess on the one hand it's nice to have a reliable way to get forma (because in my experience "rng" likes forma), but on the other it is kinda annoying to roll an uncommon only to have it be a forma. Link to comment Share on other sites More sharing options...
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