Fire Crystals: A Study of Forge Amounts.
Fri, 01/27/2017 - 16:18
I wondered one day, "Is it even worth forging minimum or medium?"
So I made a chart of the average amount of fire crystals used in 4- and 5-star forges in each level, and of the different types: minimum, medium, and maximum(The chart can be found here).
To calculate the averages I used this formula: f = a*(100/c); where f is the forge, a is the amount of fire crystals and c is the chance percentage. (Unfortunately, I could not put into account the chance for level skips that medium and maximum forges can provide.)
From the data, I've gathered that
- The minimum forge is an unreliable wasteful forge, only implemented to drive up the crystal amounts for the other forges; and
- The medium forge is your best bet with the least average fire crystals, although not much lower than the maximum forge
If the minimum forge was removed, and the crystal costs for the medium and maximum forges were shifted down, it would bring the large crystal requirements down to a more tolerable level.
Sat, 01/28/2017 - 12:28
#2
"Erecting a Handgun Dispenser."
@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.
Sat, 01/28/2017 - 18:21
#3
objectively wrong
If you assume that your luck is average, and you analyze your expected forging costs under that assumption, then basic probability theory shows that Medium chances require an average of 420 Radiants to heat a 5-star item. Players who use the always-Maximum strategy are irrationally pessimistic. They are assuming that they will experience worse-than-average luck.
And then, worst possible scenario, it fails and I have no Radiants left.
This is it, right here. You are giving too much weight to the failure scenario. To do this rationally, you need to weight failure and success by their probabilities. On Medium chances, you succeed most of the time, and when you do, you save a bunch of fire crystals --- more than enough to make up for the failures.
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.
No one is arguing for that strategy. Again, basic probability theory shows it to be worse than using Medium chances.
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.
Yes. But, as in all games of chance, you need to consider the chances of the outcomes, not just the outcomes.
a +50% crystal price compared to Medium isn't that big in the long run
A lot of players would consider a 50% difference --- namely, 302 vs. 453 Radiants --- to be pretty big. However, the difference between Medium and Maximum is not that big, because of the failures. The difference is only 420 vs. 453, on average. That's only 33 Radiants per item. But, even if it's only 33 Radiants, why waste them?
Sun, 01/29/2017 - 06:01
#4
"Erecting a Handgun Dispenser"
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.
Sun, 01/29/2017 - 07:39
#5
okay
irrationally optimistic because they assume they will experience average or above-average luck
The terms "optimistic" and "pessimistic" don't have precise mathematical definitions. But I would say that expecting average luck is realistic, expecting better-than-average luck is optimistic, and expecting worse-than-average luck is pessimistic. This kind of thinking is built into all aspects of our society that deal with risk: investing, insurance, traffic laws, etc.
any rational mind will take certainties over probabilities
The world is uncertain. If you restrict yourself to acting only on certainties, then you will be almost totally paralyzed. For example, you will not be able to ride in a car, because you are far from certain of surviving the car ride. Humans make important decisions based on probabilities all the time.
But Spiral Knights is not worth fighting over. If you are happier with Maximum forging chances, then you are not alone, and you should do what makes you happy. Cheers.
Sun, 01/29/2017 - 21:42
#6
"Erecting a Handgun Dispenser"
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.
Mon, 01/30/2017 - 06:07
#7
one more try
the most rational choice would be to go for the certain one
In case I haven't explained myself well, here's a different example. Suppose I give you the following two options, and you have to choose one.
- I give you $2, no questions asked.
- We roll a (fair, six-sided) die. If the roll is a 1, then you pay me $1. If the roll is anything else, then I pay you $1,000,000.
Option 1 is certain, but Option 2 is rational. A player who chooses Option 1 is overly risk-averse.
Mon, 01/30/2017 - 07:20
#8
>_>
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.
Tue, 01/31/2017 - 05:52
#9
↑↑↓↓←→←→ba
One item that I've seen that all of these studies tend to ignore is the rate of double levelups (which completely skip a necessary quantity of radiants).
While the sample numbers are small (still), the data I've collected is still available. If we can trust that there is a measurable difference between 3x and 2x 25% heat bonus chance (for 5* items) and extend that to also mean there is a difference for double levelup...
Cheshireccat was most interested in this dynamic and actually went through the math (and gathered data to test it too!). His thread on the topic can be found here. He actually used the 4* estimates of double levelups to show that the odds are small enough to not influence the range much. It's funny because at the time the estimate of double levelups for 4* was 12.22% +/- 3.45%. Current data estimates 9.6% (still within the confidence range, but on the lower end of it).
For reference, my old data thread is located here: Link. It has a big spreadsheet that has all the raw data, different levels of filtering, and as much transparency as possible on the calculations. It's still updated very slowly.
Tue, 01/31/2017 - 09:06
#10
yep, good point
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.
Tue, 01/31/2017 - 09:37
#11
honestly
the chance of getting a forge prize box is so low I don't factor it in at all. Out of the countless things I have forged, I've only received a single prize box (and it was a 4*). However, the double level up can be a factor, but I doubt the increase of chance from med to high would make a significant difference in the outcome, if there even is a difference.
Tue, 01/31/2017 - 17:07
#12
↑↑↓↓←→←→ba
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.
Tue, 01/31/2017 - 18:14
#13
"Erecting a Handgun Dispenser"
Amidst all of your supposed calculations, I only want to say this:
While it is true that the average probability of success for a single 5-star level up on Medium chance is 75%, the average probability of success for all level ups combined is actually 7,41%.
Consider that as some food for thought for whenever you're calculating the average amount of Radiants used for all 5-star level ups when always done on Medium chance.
Tue, 01/31/2017 - 18:18
#14
huh?
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.
Tue, 01/31/2017 - 18:19
#15
seriously
I'm honestly not sure if you are trolling at this point, or just haven't taken highschool math
it's not a hard concept to grasp.
Tue, 01/31/2017 - 21:55
#16
lets look at this from another angle
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.
Wed, 02/01/2017 - 07:56
#17
that's the point
Yes, 420 is just the average, and any single player will experience variation about that average, and the variation is bigger than 33. A non-trivial fraction of players will end up spending more than 453 Radiants using the 2x strategy.
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.
But this is where you go wrong. You are not equally likely to lose a lot. Your chances of losing a lot are much less than 50% (even though they are substantially bigger than 0%).
To put it another way: For every 2x player who spends more than 453 Radiants and is sad, there is a 2x player who spends less than 390 Radiants and is joyous. The 420 is just an average. But the fact that this average is less than 453 means that the 2x strategy is better, for most players, than the 3x strategy.
Wed, 02/01/2017 - 08:02
#18
___
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.
Wed, 02/01/2017 - 08:25
#19
update
I just wrote and ran my own simulation of 1,000,000 5-star forgings at 2x chances. The summary is:
* The average cost was 419.22..., in agreement with theoretical calculations.
* 20.9% of forgings required more than 489.
* 30.2% of forgings required more than 453.
* 43.1% of forgings required fewer than 385.
* 24.9% of forgings required fewer than 351.
Sun, 02/05/2017 - 12:21
#20
Everyone loves numbers
Using a double-levelup rate of 0%, my numbers are the same as Bopp's:
- Average: 419.3
- Percentiles: 354 / 400 / 468
- Odds of needing more than 453: 30.2%
If you factor in a double-levelup rate of 5% for any given forging:
- Average: 399.7
- Percentiles: 330 / 382 / 454
- Odds of needing more than 453: 25.4%
The numbers are very clear that in the long run, using the Medium setting saves radiants. It actually looks even better if you flip the logic - what if you assume that over your SK career you will forge 100 5* items? (Parentheses if factoring in 5% double-levelup chance.)
- Average: 41,934.8 (39,969.1)
- Percentiles: 41,292 / 41,916 / 42,556 (39,302 / 39,952 / 40,622)
- Odds of needing more than 45,300 crystals: 0.04% (effectively 0%)
Even if you go for a modest 20 items, your odds are:
- Average: 8,387.5 (7,993.7)
- Percentiles: 8,096 / 8,368 / 8,660 (7,690 / 7,978 / 8,280)
- Odds of needing more than 9,060 crystals: 6.1% (1.2%)
All this really goes to show is that humans are risk-averse.
Your link is broken. I think that you linked to a file on your own computer, not on a public web server.
We've been talking about this since the Forge came out. You can find many threads where I advise people to use Medium chances, because it costs about 420 Radiants on average. And then players respond that you should use Maximum chances, even though it costs 453 Radiants, because the disappointment of failing a forging is not worth the 33 Radiants that you save. And then I respond that the real disappointment is spending 50% more Radiants per forging. And we go on like that for a while. And in the end most players seem to go with the "emotional" strategy (Maximum), not the long-term rational strategy (Medium).