Thursday, 23 February 2012

Icon Update

My Icon experiments have not stopped - nor are they likely to in the near future - and I have some
additional results to share.  Some of this might not be entirely coherent if you haven't read my previous post on Icons.

Larger Reveres
For one thing, average Reveres are higher than I had said.  After around 100 reveres I settled in on low 700s and then stayed there for for another few hundred data points.  The average climbed very slightly from revere 100 to revere 400, but only about 5 points, so I assumed I was just slowly approaching the true average.

At that point I stopped recording my reveres in one big list and started recording them by icon in the order they appeared.  I did this because I was starting to have suspicions that the first revere after tithing was larger than the others (it is, as discussed earlier this week) and I wanted to be able to compare first reveres to later ones.  I also thought I might be making mistakes with when I was tithing, because that one time I got seven reveres out of an icon before tithing again seemed more likely to be an error on my part than something that really happened.  It's hard to say for sure, but I haven't had another tithe that lasted over five reveres yet.

I now have well over 300 reveres in my new sheet, and I started working with them.  I was stunned to find out that they averaged to 837.

This seems really, really weird to me, but the average of my last 300 reveres is over 100 points higher than the average of my first 400.  I'm not sure whether to average them all together or assume something changed and use the new values.  For the moment I'm going with the latter for three reasons:

One, the variability that the distribution would have to have any likelihood of showing this kind of change is staggering.  Now I know it is a fallacy to say that if X is very unlikely given Y, and we know X, then Y is unlikely, but it's really only a logical or mathematical fallacy, in a lot of real life scenarios it works pretty well.1

Two, maybe what changed is just that my better record keeping eliminated some kind of error I was making in the first place.  I don't actually think this is likely, but I do trust my newer data more than my older data.

Three, there is conclusive evidence that Tiny Speck took notice of my (somewhat erroneous) icon calculations and it is far from impossible that based on them (and, more likely, based on a corrected version of them that they could easily make since they have access to the code) they said to themselves, "Maybe icons aren't quite as good as we meant them to be."  This is hardly a change they would have to advertise.

I'm not trying to say that  I think I changed the world, just that it is not inconceivable that I did.  After all, if I didn't think that staff read forum posts or listened to player feedback then... who am I kidding, I would be doing all of these calculations anyway.

Level, Not Energy
Also, the size of your Reveres is not tied to your energy pool, it is tied to the size of your tithe.  Your tithe is tied to your level, and your level, of course, determines your energy pool, so you may ask what this means for you and your icons.

Tithe is tied linearly to levels above 20 while maximum energy is non-linear, so this means that my estimate of 25% of your total energy is off.  Depending on what level you are, the average Revere could range from 13% to 30% of your total energy (it actually goes up to 70% if you count levels lower than 10, but you aren't even allowed to tithe at these levels and you probably only have icons if you are an alt anyway).  The average currants per energy is unaffected, but the amount of energy you can extract from each icon each day changes, so whether or not to carry icons with you ends up depending very heavily on what level you are.  Factoring in stew stacks as the alternative, the worth per inventory slot of icons is as follows.

Since it's hard to tell from that graph, the value at levels 10 through 20 is very close to 32, or about 60% better than an emblem.  I could see the argument that you wouldn't carry icons at this level, but the value at level 23 is already 60 per slot, and it reaches 100 per slot at level 26.  Because the size of a revere climbs at a fixed rate per level, the value of the icon climbs much faster than it would if it were based on energy, and it is hard to make the case that Icons aren't worth carrying at any level that you are likely to have them.  Also, compared to the analysis that I did before, the higher reveres combined with improved calculations based on level means that icons are around 55% better than I estimated at level 31 (the 1000 energy mark) and about 250% better than I estimated at level 60; as if they weren't good enough at maximum level already.

The Near Future
I think more interestingly, Icon Reveres not being tied to maximum energy makes it a very open question what will happen to them after the big change to imagination.  Will the size of our tithes and the size of our reveres still be tied to level?  If so, it would be possible to raise your average revere to your maximum energy.  While this would be wasteful because it would mean that most reveres would have wasted energy, it would mean that teleporting would become a trivial expense.  I think I'd want my maximum energy pool to be slightly above my maximum revere, but I could certainly see making do with only 1500 to 1800 maximum energy and increasing my revere amount to teleport cost ratio substantially.  Then again, if quoin value stays tied to maximum energy then as far as I'm concerned more energy is good energy and I'm going to the moon.

To the moooon!!!

1. This fallacy is the basis of the famous Ontological Argument argument from design.  Classically, the argument says that because the universe is so complicated there must have been someone who created it - much like a watch has to have a creator.  That classical phrasing has been reinterpreted by some contemporary philosophers theologians people to be a sort of Bayesian argument.  Basically, they say, there are so many little factors that have to be just so for the universe to be as it is that it is super unlikely that life would have ever evolved at all, so there must have been some intelligence at the helm to make sure that things worked out this way.

Let's consider a less loaded and metaphysical analogy.  Suppose you roll an ordinary die 25 times and it comes up one every time.  The odds of that happening with a fair die are less than 1 in 1,000,000,000,000,000,000 so we feel pretty comfortable concluding that it is probably not a fair die.

Both arguments have the same form: Event X (life exists or the die came up 25 ones) was very unlikely given Y (the universe was created randomly or the die is fair).  We know event X happened, so Y probably isn't the case.

Let's scale back the die example, though.  Suppose you rolled three ones in a row instead of twenty-five.  Now how sure are you that the die is not fair?  If the die was fair the chance of the die coming up that way would be 1 in 256.  Even though that is unlikely, you probably feel intuitively against declaring the die to be loaded.  In fact, we've probably all seen dice come up the same way three times in a row, are we to suspect that all dice are loaded?

Somewhere out there there is a dice factory.  It produces some good dice and some bad dice.  Depending on which factory, the rate of defective dice may be higher or lower.  Also there are people who intentionally make loaded dice, given any die and the amount of its history you know, the odds that it is one of these intentionally loaded dice will vary as well.  We might guess that dice factories have a defect rate of less than 1 in 1000 and that intentionally loaded dice are a rarity in the general population of dice.

What's more, a die that is defective is unlikely to be loaded to always roll ones.  It may be loaded to be more likely to roll ones than sixes, but always rolling ones is very unlikely.  Defective dice are also just as likely to be defective in a way that makes them roll any number more, not just ones (this may be factually untrue because of a certain way that popular die-making machines are made, but even then there is no reason to suspect that if there is a systematic bias to produce dice loaded in one direction that that direction would be to roll ones).  Finally, intentionally loaded dice are probably much more likely to be loaded to roll high numbers than low ones.

All of this together gives us an intuitive guess of the odds that the three-in-a-row was caused by the die being unfair, and it seems awfully unlikely.  Let's just go with one in 100,000 for the sake of argument.  Now we can actually estimate the chance the die is not fair (about 0.3%) and our estimate is as reliable as our estimate of the probability of an arbitrarily chosen die being loaded in a way that makes it significantly more likely to roll a one.

So as an everyday argument, the logical fallacy often works because we are able to quickly, and often subconsciously, estimate the likelihood of certain events that the argument depends on.  In the particular case of icons, I know that the way reveres are generated is done by a computer, so it is based on an algorithm, and that algorithm was designed by a human, and that human had some idea of what he or she would like the outcome to be.  All random algorithms designed by humans who have some idea what they would like the outcome to be have certain features in common.  Based on this I can say that I am highly skeptical that an algorithm that produces numbers in that range would return an average so varied over two data sets with a few hundred many points.

If the fallacious argument worked in a vacuum, without the background assumptions about reality, then you could reason as follows: it is very unlikely that you will win the lottery (event X) given that the sun will come up tomorrow (event Y) so if you win the lottery (event X) then you are in big, big trouble (probably not event Y).  That clearly doesn't work, and the reason it doesn't work is because while it is unlikely that you win the lottery, it is far more unlikely that the sun doesn't come up tomorrow (often people will hear this example and say it doesn't count because there is no causal relationship between the two things, but neither probabilities nor logic actually care one bit about causal relationships).  The argument, therefore, only works for the existence of God if we can estimate the odds of an arbitrary universe containing a designing intelligence (or being contained by a designing intelligence?).  Good luck with that.

None of this is an argument against the existence of a designing intelligence.  Rather, it is an argument for two things: 1) I am well within my rights to say that the observed data on reveres is suspicious; and 2) the Ontological Argument, or argument from design, for the existence of God essentially assumes its conclusion - something that is intuitively understood by nearly everyone (as evidenced by the fact that people who believe in God seem to think it works great and people who don't believe in God seem to think it doesn't work at all) but is rarely formally stated.  No matter how complex, wonderful, or seemingly unlikely the universe is, that should not give us cause to think that is was designed intentionally unless we can estimate the chance that it was designed intentionally to begin with.


  1. The argument you refer to as the Ontological Argument is actually the Teleological Argument. The Ontological argument is something different.

    The Ontological Argument says that a perfect being must exist, because if it didn't exist, it wouldn't be perfect.

    There are two traditional versions of the Teleological Argument: one from complexity, as you mention, and the other is from beauty (ie., the world would not be beautiful unless it was designed).

    -- Carthago Creases

  2. Again, how embarrassing! How could I get such things confused in my head, when the argument above is clearly not Ontological in any way. To correct this, I will have to figure out how to make the Ontological Argument extremely tangentially related to another post so I can write a lengthy footnote about it.

  3. No big deal! As a non-math person it was really the only constructive thing I had to offer. Your main point is interesting but I have nothing interesting or of value to say!