@partshade Finally did the math just because not having it was driving me nuts. If I trust an old article (https://www.angrybirdsnest.com/forums/topic/economics-of-pig-cherry-juice/), the crystal cauldron yields, on average, 3.3 items. There is good reason to doubt these numbers, but until I’m by my phone again I’m not in a position to test true yields (even then making the assumption that all crafts have the same chance to receive 2 or 3 stars). I have also assumed a balanced wheel, which has been an underlying assumption of all calculations. Please let me know if anyone sees any errors.
FE = Friendship Essence, N = Energy, R = Resources.
With the above assumption, 1 FE = .66N. It has been demonstrated earlier that for the 3N wheel, 1N=.3333R, and for the 7N wheel, 1N=.2381R. Therefore, the average yield of a FE when used for stamina drinks is either .2222R (for the 3N wheel) or .1587R (for the 7N wheel). The first question is whether it is worthwhile to reroll at all, and if so, when.
For the 3N wheel, there are three reward possibilities: .3333R (twice), 1R (twice), or 1.3333R (four times). For the 7N wheel, there are five reward possibilities: .2381R (twice), 1 (once), 1.2381 (twice), 3 (once), and 3.2381 (twice). The decimal rewards correspond to the value of one stamina drink when playing that wheel (one or the other is strictly better, at least when always re-rolling, so it wouldn’t make sense to combine wheels in any way). Now to the first question.
For the 3E wheel, each spin of the wheel costs .2222R, or the value of 1FE, were it used for stamina drinks instead. Because you can re-roll an infinite number of times, and stop whenever you want, we are concerned only with the average payout of all better rolls. We are also concerned with the average number of FE that will be used. If you roll a .3333R reward, the average payout of better rewards is (1*2+1.3333*4)/6=1.2222R. This has a 75% chance to occur, so will cost, on average, 1.3333FE=.2963R. When you take the potential reward and subtract both the cost and the current reward, you get 1.2222R-.3333R-.2963R=.7037R. Therefore, on average, you end up .7037R ahead when rerolling after a .3333R reward, and should reroll. On the other hand, when you have rolled a 1R reward, you have a 50% chance of rolling a 1.3333R reward (the only better reward), which will cost an average of 2FE=.4444R. 1.3333R-1R-.4444R=-.1111R. Therefore, you end up an average of .1111R behind when rerolling, and should not do so.
For the 7N wheel, each spin of the wheel costs .1587R. I have skipped to the 1.2381R reward for sake of space and laziness. If you roll a 1.2381R reward, the average payout of better rewards is (3+3.2381*2)/3=3.1887R. This has a 37.5% chance to occur, so will cost, on average, 2.6667FE=.4232R. When you take the potential reward and subtract both the cost and the current reward, you get 3.1887R-1.2381R-.4232R=2.5274R. Therefore, on the 7E wheel, reroll until you have at least 3R. On the other hand, when you have rolled a 3R reward, you have a 25% chance of rolling a 3.2381R reward (the only better reward), which will cost an average of 4FE=.6348R. 3.2381R-3R-.6348R=-.3967R. Therefore, you end up an average of .3967R behind when rerolling, and should not do so.
So far, this has mostly just confirmed what people would already assume: reroll until you get the maximum resources possible, but don’t worry about getting energy drinks, so long as you have the crystal cauldron. Now for the next step.
For the 7N wheel, the optimal result (3.1887R) occurs in 37.5% of rolls, taking an average of 2.6667 rolls, and thus 1.6667FE. For the 3N wheel, the optimal result (1.2222R) occurs in 75% of rolls, taking an average of 1.3333 rolls, and thus .3333FE. For every 21N (an easy multiple; any amount of energy would work), the 7N wheel yields 9.5661R, but costs 5FE. On the other hand, the 3N wheel yields 8.5554R, but only costs 2.3331FE. The question, therefore, is which is worth more: 2.6669FE (on the 3N wheel) or 1.0107R? 2.6669FE is only worth .5926R, and .5926R+8.5554R=9.148R<9.5661R, so your intuition is correct, the 7N wheel better is better (by 4.57%). Since FE are worth fewer resources with worse cauldrons this will hold true for all cauldrons (assuming the initial assumptions are correct).
TLDR; if you are okay using Friendship Essences for resources *and always get 3 stars* the 7 energy level is best, but only slightly.
