@Bopp

Let's say I have this fully heated Level 9 5-star weapon that's ready to reach Level 10, but I only have 68 Radiants -- just enough to go for a Medium chance with a supposed 70% chance of success, but not enough to get the Maximum chance. So I go for the Medium chance because the odds look favorable to me and I get to save some Radiants for later.
And then, worst possible scenario, it fails and I have no Radiants left. So now I'm going to have to get 68 more Radiants -- half of which I could have gotten beforehand in order to get the Maximum chance with a 100% chance of success -- and this can happen at any star rank and at almost any heat level, with differing success rates of course.

In a perfect world where the odds are always favorable, Minimum chance would be the best choice because you would be guaranteed to save about three times the amount of crystals you'd otherwise need on Maximum.
Unfortunately, we're in no such perfect world, and as such succeeding with Minimum or Medium chances is no different than winning on slot machines or roulettes -- if you win, congratulations, you won big; if you lose, too bad, try again.

Therefore, I would argue that the "long-term rational strategy" is actually going for the Maximum chance, as a +50% crystal price compared to Medium isn't that big in the long run, and it's the only chance of the three that offers an actual consistent result.

And admittedly I could just as easily say that players that use the always-Medium strategy are irrationally optimistic because they assume they will experience average or above-average luck.

Fact of the matter is, probabilities are still probabilities, not certainties, and any rational mind will take certainties over probabilities; something that only the Maximum chance offers.

I guess I forgot to add something important when I said that last paragraph; any rational mind will take certainties over probabilities *if* given the choice between one or the other in a given situation.

You are right; the world outside is so random that we usually can only rely on probabilities -- which is why, in the rare chance that we do get a choice between certainty and probability, the most rational choice would be to go for the certain one.

Needless to say, we're entering into quite the philosophical conundrum for a video game feature that arguably doesn't deserve the attention, but honestly I believe it's intriguing to look at things in such a way.

brownie, your logic is flawed. Bopp is correct. You can easily prove this using high school level math.

We're not saying you have to succeed every time on medium to be successful, we're saying that if you fail 25% of the time and succeed 75% of the time (which is the listed odds) you will have saved fire crystals over using the max option every time.

You are wrong brownie. Taking medium every time will on average save you crystals. (I still do max for the first 3 levels tho, for a chance at those prize boxes)

Here's the math:

Each level costs 2x crystals on medium, and 3x crystals on max.

Medium has a 80% chance for the first 3 forges, 75% for the next three, and 70% for the last three, so the average is 75%. Max always has a 100% chance.

you have to forge 9 times per item, so if you go max every time you use 9(3x) = 27x crystals.

Medium fails 25% of the time on average, but each time you fail you have to forge again, re-rolling, so using limit theory we can determine that it will cost you 133.33333...% of the listed crystal cost on average, taking into account the chance for failure (my equation)

This results in 9(2x*1.33333) = 24x crystals.

Therefore, using medium each time saves you (27-24)/27 = 11.111...%

You save 11% of your crystals on average.

This accounts for when you fail 25% of the time, brownie, which is the listed odds.

The only reason to take max is if you expect your luck to be extremely bad, which you should have no reason to believe, because your luck will be exactly average. That's how probability works.

Yes, all of the naive calculations, such as mine, ignore Forge bonuses entirely.

I did once try to integrate the rate of double level-ups, but I didn't have enough data. And I really didn't have enough data on Forge prize boxes. And I attach no value to the 25% heat bonus.

Thanks for reminding us of Cheshireccat's data and your data.

Yup, 25% heat bonus can be ignored. Forge prize box (for returned crystals) is too infrequent and small in quantity to influence the calculations. I'd even venture to guess that there are other factors for obtaining forge prize boxes... since they often appeared in pairs (one shortly after another).

It still kinda shocks me that I started that recording thread back in Nov, 2013.
And at least everyone is in clear agreement that 1x crystals = poorest decision.

But the thing is, that doesn't matter. You aren't supposed to succeed every time. You are supposed to fail 1 out of 4 times. I don't think you understand brownie, even if you fail 1 out of 4 times you will save crystals over going high every time.

Brownie, the 7.41% figure is completely irrelevant. I'm not saying I'll succeed every time. I'm saying I'll succeed 3 out of 4 times and fail one out of 4.

I will still save crystals, even failing 1 out of 4 times.

How can I make this any more clear? By providing that figure as 'food for thought' it is evident that you do not understand at all how probability works.

I'll say it again because you consistently miss the point:

forging on medium and failing 1 out of 4 times, which is expected, will still save you crystals over going high each time.

you are correct in that you on average save radiants with 2x, but you also have to think of how likely it actually is to do so.
if you heat a billion items, of course 2x is better, but what if you only heat a few?
so I did some math. or rather I let my computer do math, with dice rolls.
http://imgur.com/a/qFJRA
the graph is cut off at 750 for the sake of compactness, it does actually go even further.
this is what came out after simulating the heating of a million 5* items. in addition, the mean is 417.1685 and the standard deviation is 89.1814.
in case you dont know what those terms mean, the mean is basically the average, so your average 5* heating with 2x will take 417 radiants, 36 less than using 3x. the standard deviation on the other hand is more or less the "compactness" of the data points. a very low one would mean that all the results are very close together, while a very large one means that they are very far apart.
and here comes my problem with 2x, the standard deviation is massive. sure, you can end up saving a lot, but you are equally likely to lose a lot. that 453 easily falls within the standard deviation, so unless you feel particularly lucky or you are heating an obscene amount of gear, 3x is MUCH safer.
PS: if anyone wants to see my code, I can provide that, though it isnt very compact.

actually, going high is more gamble-y than going med! med will save you crystals on average, so going high is like paying extra crystals for a chance at the forge prize box!

My personal strategy is HHH MMM MMM for the 9 forges, although if you want max crystal efficiency, you'll go M the whole way.