[Again, I got bored, and again, results are meaningless if you don’t publish them. At least I didn’t need WolframAlpha this time. The conclusion at the end of the Legendary-tier analyses even took me by surprise. OB]
A Mathematical Approach to Pet Levelling
19 Apr 2014
This thread attempts to determine the most effective routes to typical “final” pet levels (70/70 maxed Rare, 90/90/90 maxed Legendary, 100/100/100 maxed Divine). This is done through a mathematical analysis of the pet feeding system.
RotMG pet abilities are improved by feeding items to the pet; the feed power (FP) of the items are then applied towards the pet’s abilities, levelling them upwards. Once suitabily improved, two pets of the same tier can be fused together to create a pet of the next-higher tier (Common to Uncommon to Rare to Legendary to Divine).
However, the potential “max level” of these new pets directly depends on how much the original pets had been improved. To ensure that a post-fuse pet can be improved to its theoretical max level, it is critical that the pre-fuse pets first abilities have been completely improved. While it is not mandatory to improve the pre-fuse pets second/third abilities to their max level as well, it is often considered desirable to do so since the per-feed costs (in fame or gold) roughly triple from one tier to the next.
This leads to a number of interesting compromises. Intuitively, it makes sense that players with a lot of excess feed items but not a lot of excess fame will be best served by completely maxing their pets before fusing, and players with a lot of excess fame but not a lot of excess feed items will be best served by fusing as soon as possible (and thereby spending as little on the pre-fuse pets as possible).
The two most interesting questions here are cheapest route to a Maxed Rare pet (both in terms of fp and fame/gold) and cheapest route to a Maxed Legendary pet. (Maxed Divine is addressed here as well, although it is not as interesting since these players typically rely on realm gold and purchased pet food instead of fame and looted items).
The analysis and results discussed here are only valid if the following holds true:
- Only the routes that lead to fully maxed pets are discussed here. While pets can be fused before their first abilities are maxed, the end result is a post-fuse pet that cannot be improved to its theoretical max. There's a whole continuum of suboptimally maxed pets, but the routes capable of leading a maxed pet are more interesting.
- Fame is used to cover all per-feeding costs. Fame-per-feed costs consistently triple (roughly) from one tier to the next. On the other hand, gold-per-feed costs have different ratios (which vary from tier to tier) and are not addressed here.
The FP required to improve a pet's abilities from one level to the next can be calculated in advance. The best reference here is a Google Docs spreadsheet that was originally created by [url=http://www.realmeye.com/guild/THE%20OPPAI%20STARS]The Oppai Stars[/url] and privately shared with the author. There might occasionally be minor random variations in these values from pet to pet (which may depend on ability, but a solid link has not been established there). The key data here are that improving the pet's first ability from lvl1 to lvl2 requires 20fp, the second ability from lvl1 to lvl2 requires 31fp, the third ability from lvl1 to lvl2 requires 67fp, and everything from that point on increases by 8% from one level to the next. This has largely been verified to be accurate for most key values, but some anecdotal experiences may vary slightly from this.
(Alternately, it could be viewed that all abilities begin at 20fp, but FP directed towards the second ability is reduced to 70% and FP directed towards the third ability is reduced to 30%.)
- When fed, an item's FP value is applied in whole to all abilities. If the FP is at all divided (either evenly or unevenly) between abilities, then the analysis below would be incorrect. However, this has been verified to be true.
- Fame costs associated with Pet Fusion are not addressed here. Regardless of what path(s) you take, these remain constant.
- Fame costs associated with Pet Yard upgrades are not addressed here. Regardless of what paths(s) you take, these remain constant as well (77500 fame to fully upgrade to a Divine-level Pet Yard, or 27500 fame to stop at Legendary).
- All pets are hatched from Common eggs and fused upwards. Uncommon/Rare/Legendary eggs begin with their first abilities at a known starting level, but the abilities for the second/third abilities begin at a random level. As such, they won't sit on the graphs that will be presented later. (However, for pets that will be consumed in fusion, it is always better to hatch an Uncommon egg (if you have one) since it might be better (and will never be worse) than a freshly-fused 30/1 Uncommon pet. Still, that slight FP boost is not a compelling reason to buy Uncommon eggs; use them if you have them, but don't spend Realm Gold get them if you don't.)
Getting Started: Maxed Common and Maxed Uncommon Pets
The route to a maxed Common or Uncommon pet is very straightforward. However, it’s the first step along some paths that are considerably more interesting, so let’s go ahead and get these cases out of the way.
Lvl1 Common pets (1/0/0) begin with 0fp consumed. The first 20fp raise it to Lvl2 (2/0/0), 22fp more raise it to Lvl3, and so on. All in all, it takes 2080fp to raise a freshly-hatched Common pet to 30/0/0:
Once you have two maxed Common pets, they can be fused to create a 30/1/0 Uncommon pet. From that point, an additional 8527fp is enough to max the pet’s first stat; this raises the second stat to lvl41, for a 50/41/0 Uncommon pet. Finally, once you reach this point, an additional 7791fp is required to max the pet’s second stat:
On a mathematical note, one intermediate goal in this work was the development of a weighted directed acyclic graph depicting every possible route to a maxed pet of any tier. Note that there’s only one path to a maxed Uncommon pet (not very interesting so far), and the weights on these edges don’t account for the fact that a second Common pet has to be maxed for fusion. So, let’s go ahead and track total FP investment for the points on this graph; this will allow us to make reasonable FP-fame decisions when we have multiple options:
In this case, 2*2080 (for two maxed Common pets) + 8527 (for maxing the first ability) = 12687fp is the total FP required to build a 50/41 Uncommon pet.
If your goal is a maxed Uncommon Pet, you really have no options: max two Common pets, fuse them, and keep on feeding it until it’s done. 50/41/0, meanwhile, is where the first ability is first maxed; this will be important when dealing with “feed or fuse?” decisions in later tiers.
Fuse First or Feed First? Maxed Rare Pets
Now we’re getting to our first interesting decision. Should you fully max your Uncommon pets to 50/50 first, or should you fuse your Uncommon pets as soon as you possibly can (to hit 70/70, this would be at 50/41)?
Assuming that those are the only two fusing points of interest, this presents two different routes to 70/70:
Path #1: If you fully max your Uncommon pets to 50/50, it will take 39746fp to max the Rare pet’s first ability to lvl70. This is enough fp to bring the pet up to 70/65. From there, it takes an additional 21403fp to max the pet’s second ability to 70/70.
Path #2: If you fuse your Uncommon pets at 50/41…well, the first abilities were maxed, at least, so it will still take 39746fp to max the Rare pet’s first ability to lvl70, but this only brings the pet up to 70/63 instead. From there, it takes another 7791fp to “catch up” to the pet on Path #1…and from there, the same 21403fp is required to bring the pet up to 70/70.
Comparing the two paths, both paths require the 39746fp to reach 70/*. Both paths require the 21403fp to reach 70/70. The difference is the two 7791fp edges. By maxing first, you have to feed that 7791fp twice (since you’re maxing two Uncommon pets before fusing) at 30 fame/feed. By fusing first, you have to feed that 7791fp only once, but at the higher Rare-tier cost of 100 fame/feed. So from a FP-only context, fusing first is cheaper, but by feeding first, you require more FP but at a lower fame/feed cost:
(You’ll note the two “total FP investment” numbers differ by…yup, you guessed it, 7791fp.)
As a result, this decision is a compromise. There is no right answer, and the best route varies from person to person. However, to better address this decision in more depth, let’s make some additional assumptions.
Assume that the player has no fame to spend, and has no items to feed. As a result, they can only feed the items they farm, and they can only spend the fame that their characters earn (and die with, of course).
For Path #1 (maxing first), every 30 fame earned allows them to feed the best item they’ve earned in that time. Let’s say that all these items have an average fp value of X fp/feed. Since you’ll have to do this twice (for two Uncommon pets), we can calculate the amount of fame required:
2*7791 fp / X fp/feed = 15582/X feedings
15582/X feedings * 30 fame/feed = 467460/X total fame required to raise two 50/41 Uncommon pets to 50/50.
(Example: this would require 104 para rings, costing 3,120 fame)
For Path #2, every 100 fame earned allows them to feed the best item they’ve earned in that time. Let’s say that all these items have an average fp value of Y fp/feed. Thus, the total amount of fame required for this edge would be:
7791 fp / Y fp/feed = 7791/Y feedings
7791/Y feedings * 100 fame/feed = 779100/Y total fame required to raise one 70/63 Rare pet to 70/65.
(Example: this would require 26 exa rings, costing 2,600 fame)
Assuming that item-farming characters get about +50% in achievement stacking, that 30 fame comes from 20 basefame, and 100 fame comes from 67 basefame. So now the player must ask himself: what is the best feed item I can reliably earn in 20 basefame? In comparison, what is the best feed item I can reliably earn in 67 basefame? This leads to this comparison: which approach would cost less fame?
467460/X fame < 779100/Y fame
If the term on the left is less, than maxing first will cost less fame. If the term on the right is less, then fusing first will cost less fame. So, let’s simplify:
467460/X < 779100/Y
467460 < 779100 X/Y
467460 Y < 779100 X
Y < 1.66666… X
What exactly does this mean? This can be interpreted as follows:
Path #1: If the best feed item you can farm in ~65 basefame is not at least 66% better than the best feed item you can farm in ~20 basefame, then you will be best served by maxing your Uncommon pets before fusing.
Path #2: If the best feed item you can farm in ~65 basefame is at least 66% better than the best feed item you can farm in ~20 basefame, then you will be best served by fusing as soon as possible and feeding your Rare pet.
(Looking at the “para rings vs. exa rings” examples above, X=150 while Y=300. In this case, the (Y < 1.666…X) test does not hold true, so one should conclude that fusing first will cost less fame. The above examples show this to be true: feeding the para rings to Uncommon pets would cost 3,120 fame while feeding the exa rings to a Rare pet would cost 2,600 fame.)
I spent some time in #rotmg asking players about the best feed item they felt they could reliably farm in 15 basefame, and the results were generally along the lines of para rings and other 150-200fp items. In comparison, when asked what they felt they could reliably farm in 50 basefame, the answers were more along the lines of minor UTs and 400+fp items. This anecdotal data would seem to imply that diehard item farmers using a “zero-sum game” to feed pets would have the best success on Path #2.
Extending This Model
Of course, very few people use this “zero-sum” model to feed their pets. Lots of players stockpiled inventories of 500+fp items to apply towards their Rare pets. Players similarly had tens of thousands of fame saved up when pets were first introduced.
The above terms (467460/X and 779100/Y) can be applied to existing inventories as well. Think about the items you’re comfortable feeding to an Uncommon pet—their FP value would be X. Think about the items you’re comfortable feeding to a Rare pet—their FP value would be Y. Which approach will cost less fame, and how do those calculations compare to the amount of fame you have to spend?
It should be noted that there are in fact an entire continuum of options between these two paths. A player could choose to fuse at 50/42, or 50/43, etc., all the way up to 50/49:
As the previous sections discussed, fusing later will require more items at a lower average fame-per-feed cost, while fusing earlier will require fewer items but at a higher average fame-per-feed cost. For players with existing stockpiles of feed items and fame, calculating the ideal point for fusing two 50/* Uncommon pets to take the optimal route to a 70/65 Rare pet is not discussed here, but will be addressed in a future work.
If your goal is a maxed Rare pet, then there are two primary options (with a continuum of potential choices between them). In general terms, players who want to minimize fame costs should max their Uncommon pets and fuse at 50/50, while players who want to minimize item farming and number of feedings should fuse their Uncommon pets as soon as possible (50/41) and feed the single Rare pet from that point. Readers can choose which path would be best by evaluating their efficiency at farming high-FP feed items, with respect to basefame earned in that time.
And Now Things Get Interesting: Maxed Legendary Pets
As we saw from the previous post, there are three types of semi-maxed Rare pets that are interesting from a fusing standpoint: the 70/63 pet created by fusing as quickly as possible, the 70/70 fully maxed pet, and an intermediate point (70/65). Let’s evaluate all of these in parallel.
To begin with, they all require just over 185k FP to max their first ability. This will raise them all up to 90/*/71, where the second ability depends on the Legendary pet’s starting point.
From here, things become very…nonintuitive. If you begin at 70/70/0 (“feed first”), the first ability will be maxed at 90/85/71. Continued feeding will max the second ability at 90/90/76:
On the other hand, if you begin at 70/63/0 (“fuse first”), the first ability will be maxed at 90/84/71 (the second ability lags behind by just over 28k FP). However, continued feeding here will max the second ability at 90/90/78:
What is happening here is that the additional fp required to max the second ability here was also applied towards the third ability, so the end result is that the 90/90/* pet here ends up ahead of the 90/90/* pet created by maxing before fusing. By maxing the Rare pets’ second abilities before fusing, the third ability ends up lagging behind.
(Meanwhile, the 70/65 starting point ends up midway between the two, and is otherwise unremarkable.)
If you look at the FP requirements for these two paths, it is surprising to find that they both require the exact same amount of FP:
It doesn’t matter where you start…by the time the paths all converge, you’re still investing the same amount of FP to get there. Here’s the full map for the Legendary tier if you’d like to verify it for yourself (only the lowest “total FP investment” values are shown here):
Note that all three paths require 314204fp to reach 90/90/78:
- 185257+99753+21403+7791 = 314204
- 185257+121156+7791 = 314204
- 185257+128947 = 314204
As with the 70/* Rare pets in the previous post, there are in fact a continuum of potential fusing points into the Legendary tier (70/64, 70/65, 70/66…). However, unlike the previous compromise, the newly-fused Legendary pets here all still need:
- 185k FP to reach 90/*/71,
- 314k FP to reach 90/90/78, and
- 785k FP to reach 90/90/90.
If your goal is a maxed Legendary Pet, then you should fuse your Uncommon and Rare pets as soon as you possibly can (50/41 and 70/63). This will get you to the Legendary tier as soon as possible, where the feed items will be applied towards the third ability. Any FP you spend towards maxing your Uncommon or Rare pets’ second abilities is wasted.
If your goal is a 90/90/* Legendary Pet, then…this is actually is a bit of a compromise. If you really don’t care about the final level of the third ability, then fully maxing your Rare pets to 70/70 will minimize the amount of Legendary-tier feeding that is required. This leads to a compromise similar to what we saw in the Rare tier: maxing first means the player must feed 29,194fp twice at 100 fame/feed while fusing early means the player must feed 29,194fp once at 350 fame/feed. This particular compromise has also been set aside as an area for future work.
If your goal is a 90// Legendary Pet, then you should fuse you Uncommon and Rare pets as soon as you possibly can (50/41 and 70/63). Regardless of starting point, it will always take 185k FP to improve the pet’s first ability to 90. Maxing your Rare pets first doesn’t save you anything here, either. Granted, your pet’s second ability will be better if you max your Rare pets first, but only by one level (lvl85 instead of lvl84) and your third ability will still be the same (lvl71).