Inner Peace (Through Superior Firepower) – Episode 030

[MarbitChow]

The sheer fact that they popped a dollamancer anything like soon after losing the previous greatly favors models that skew the probabilities in a side-defined way.

Or shows that the caster generation method is not random at all. It may, in fact, actually be determined completely by the Titans (for example, to maximize the entertainment value of the conflict if they're malicious, or as a reward for piety if they're benevolent - which could explain peaceful Banhammer's abundance and warlike Jillian's complete lack). The truth is, we don't have anywhere near a large enough sample size to make an accurate guess.

[BLANDCorporatio]

The sheer fact that they popped a dollamancer anything like soon after losing the previous greatly favors models that skew the probabilities in a side-defined way.

Or shows that the caster generation method is not random at all. It may, in fact, actually be determined completely by the Titans (for example, to maximize the entertainment value of the conflict if they're malicious, or as a reward for piety if they're benevolent - which could explain peaceful Banhammer's abundance and warlike Jillian's complete lack). The truth is, we don't have anywhere near a large enough sample size to make an accurate guess.

Aye. Hypothesis testing/model fitting is one murky lake I will not tread for now. I'll stick with things like "assuming a 'nice' random process for casters, with possibly some bias to Thinkamancers, here are the chances of seeing various things".

So far, it's not unlikely that a side would have no Thinkamancer in all of its nine casters, under that hypothesis. It would, OTOH, be somewhat unlikely for a side that popped 9 casters to have no caster type repetition (~0.17 chance, if we assume all caster types are equally likely|caster pop; ~0.12 if thinkas are twice as likely compared to another one caster type). Immediately popping a given caster type after that particular caster type has died is also about as likely as popping a particular caster type (roughly 1/23).

[0beron]

Yes unless there is also a "clause" that prevents duplicates from naturally popping to a given side.
The fact that on a 8-caster side we didn't see any repetition suggests to me this is the case.

[drachefly]

You can assume even distribution for analysis, and we may not have a good handle on what exactly is the basis of caster selection (temperament? fate? titanic whim? feelings? Some combination of those?), but I think we have ample evidence to rule out the 'roll a d23' hypothesis as an accurate model of what's going on.

[MarbitChow]

You can assume even distribution for analysis, and we may not have a good handle on what exactly is the basis of caster selection (temperament? fate? titanic whim? feelings? Some combination of those?), but I think we have ample evidence to rule out the 'roll a d23' hypothesis as an accurate model of what's going on.

No, you really don't. You don't have a large enough sample size to rule out or prove ANYTHING. You've got 9 Casters in FAQ, 3 in Goodmitton, 2 in Transylvito, 5 in GK (several of whom overlap w/ FAQ), 5 or 6 in Jetstone. (Numbers are approximate based off a mental head-count.) You've got about 20-ish unique casters, with a few duplicate types (dollmancer, thinkamancer). Your sample size is smaller than all of your available options; you CANNOT conclude anything from that.

[BLANDCorporatio]

No, you really don't. You don't have a large enough sample size to rule out or prove ANYTHING.

Technically true, as for example I (and I don't think drachefly either) don't claim to prove, definitively, that a d23 model (or some such variant where Thinkas are more or less likely) IS what governs caster pops.

However, I've got a few calcs I showed you and I stand by the results (if I'm a bit waffly on the conclusion, that's a fault of mine):

"given BLAND's(*) model, how likely is it that a side which popped 9 casters didn't pop even one Thinkamancer?": even if Thinkamancers were twice as likely than any other one caster type, the probability is about 0.45. Within the realms of plausibility.

"given BLAND's model, how likely is it that a side which popped 9 casters has not popped two casters of the same type?": assuming all casters are equally likely, about 0.17; if Thinkas are twice as likely than any other caster type, that drops to about 0.12. I guess freaks are possible, and these are certainly better than lottery odds, but it suggests BLAND's model isn't what's going on.

(*): a model that attempts to describe caster type|caster pop probabilities that I've been using in the recent posts.

[0beron]

BLAND, how does your math change if we assume it's "impossible" (would require Titantic interference, but otherwise won't happen) for a side to naturally pop a duplicate caster (that is, a caster they presently have, not one which they had but croaked). What are the odds then for an 8-caster side lacking a Thinka?

[vintermann]

Yes unless there is also a "clause" that prevents duplicates from naturally popping to a given side.
The fact that on a 8-caster side we didn't see any repetition suggests to me this is the case.

It doesn't to me. We don't know how many of old Faq's casters are home-popped. It may seem unlikely that they were grabbed in conflict, but since we know nothing about how Faq came to be or how long it has existed, we can't really say - if Faq is 200 times as old as any other side existing at its destruction, which is quite possible, it should probably have had plenty of chances for grabbing of casters during emergency annihilation of small neighbours that found them.

Another possibility is that they've simply adopted compatible barbarian casters from the Magic Kingdom. Being a barbarian in the Magic Kingdom isn't without worries, odds are you have to take on risky and/or unsavory work from time to time. If a promising caster expressed displeasure at that, and Marie's predictamancy was positive, he might be approached with a job offer from a highly veiled Jack.

There's one hint that Faq may assemble their casters selectively: We know they will pick up Wanda. Although she has a special that's an anathema to Faq just as much as shockamancy is, she ended up there, which suggests to me they care about diversity/completeness.

[0beron]

That's possible, but I'm not buying it, given Banhammer's nature. FAQ only engages in Mercenary work because they would run the treasury dry otherwise, and even so Banhammer seems to hate it. I don't think he's take the risk of taking on any MK caster, especially given the cost (Barbarian casters have higher upkeep than those native to a side).

As for snatching up casters from small conflicts, that's also possible but the same logic makes it more plausible that they have so many casters. If the kingdom is phenomenally old, then it means there is that much more time for them to have been popping warlords and hoping for casters.

I'll agree though, there isn't much proof either way as of yet, my position is motivated in part by opinion.

[MarbitChow]

Technically true, as for example I (and I don't think drachefly either) don't claim to prove, definitively, that a d23 model (or some such variant where Thinkas are more or less likely) IS what governs caster pops. ... I guess freaks are possible, and these are certainly better than lottery odds, but it suggests BLAND's model isn't what's going on.

The problem with that logic is that you're looking at a predefined set of results, and working the probability backwards from it. Using that same logic, let's say we have a series of coin flips that come up Heads three times in a row. That's the only sample we have. We calculate the probability of this to be 12.5% likely, and thus conclude that, because we got this result, a 50%/50% distribution isn't probable. Assuming that it isn't a 2-headed coin, however, we KNOW that the 50/50 model is the 'correct' one, but your logic is 'disproving' it. Those are about the same odds that your model came up with, correct?

[vintermann]

A barbarian, caster or otherwise, can freely turn to any side that wants it as member. Otherwise, Wanda couldn't have been with Haffaton. I'm suggesting Faq offers selected barbarian casters the opportunity to turn. That explains the diversity far better than my admittely poor snatching theory...

For you age theory, keep in mind they would have had to have been willing to sacrifice/disband warlords wantonly moments after they popped. If they practiced such ruthless euthanasia, why did they keep Jillian?

I do not think sending them out to die on mercenary work is an alternative. They could get captured and reveal Faq's location. Since they would probably not be flowing over with loyalty after being sent out to die, turning would also be a risk.

Actually, I think Faq could get by without mercenary activity if they tightened their belts. I do not think Faq did mercenary work until Jillian came along. She probably convinced her father, being fed up with them (and them being very distressed by her prescence, not least because she is a reminder to them that their side will fall).

Now that I think about it, that offers another explanation for the diversity: From Jillian's mercenary business, Faq got extra funds with which they could support casters beyond the 4 core ones (math, money, predict, fool). Some of those schmuckers they probably used to pop warlords as reinforcements for Jillian's inevitable losses - and on rare occasion they would get a caster. But Jillian may have orders to grab casters missing in Faq's lineup - maybe that's how they get Wanda.

But they wouldn't get thinkamancers that way: too risky, as a disloyal thinkamancer may communicate with outsiders. Unfortunately for them, thinkamancy was one of Wanda's side talents...

[DevilDan]

Barbarian casters end up in the Magic Kingdom.

Which is not a surprise, as they can live there freely, not answering to an overlord and not risking being croaked unless they choose to take up mercenary work.

It doesn't seem likely to me that Banhammer would willingly jeopardize Faq's secrecy (even given the precautions outlined by Jillian) just to get his hands on more casters. First and foremost, he wants Faq left alone.

[BLANDCorporatio]

The problem with that logic is that you're looking at a predefined set of results, and working the probability backwards from it. Using that same logic, let's say we have a series of coin flips that come up Heads three times in a row. That's the only sample we have. We calculate the probability of this to be 12.5% likely, and thus conclude that, because we got this result, a 50%/50% distribution isn't probable. Assuming that it isn't a 2-headed coin, however, we KNOW that the 50/50 model is the 'correct' one, but your logic is 'disproving' it. Those are about the same odds that your model came up with, correct?

Point well made, but I'd argue not for the right reasons. What I'm hinting at, but won't do now, is something akin to checking whether a coin is fair. It's perfectly valid to ask, having tossed a coin n times (or, having run the caster roulette n times) whether the coin is fair (or BLAND's distribution holds).

The real problem is that as you pointed the first time, the data is not large enough to offer a significant result.

BLAND, how does your math change if we assume it's "impossible" (would require Titantic interference, but otherwise won't happen) for a side to naturally pop a duplicate caster (that is, a caster they presently have, not one which they had but croaked). What are the odds then for an 8-caster side lacking a Thinka?

Hmm. I'll assume equally likely caster types for this quick sketch.

Spoiler: show

Let arr(n, m) be the number of ways to arrange m elements selected from n. So the total number to normalize against is the number of all possible sequences of 8 casters selected from 23 types. This happens to be arr(23, 8).

Which itself is 23!/(23-8)! and is rather huge.

Of these sequences, arr(22, 7) is the number of sequences starting with a thinkamancer, and continuing with other caster types. If the second caster was a thinka, the number of sequences is again arr(22, 7) (just remove the thinka, and now once again you have a sequence of 7 casters selected from the remaining 22 caster types). Etc etc, so you have 8*arr(22, 7) sequences in which a thinka appears at some position in the sequence of 8.

These sequences are all equally likely and mutually exclusive, so the probability of there being a thinka somewhere in the sequence of 8 is 8*arr(22, 7)/arr(23, 8) or

8*22!*(23-8)!/((22-7!)*23!) = 8*22!/23! = 8/23.

The probability of no thinka is 1-that, so 15/23 IF all caster types are equally likely.

[drachefly]

You can assume even distribution for analysis, and we may not have a good handle on what exactly is the basis of caster selection (temperament? fate? titanic whim? feelings? Some combination of those?), but I think we have ample evidence to rule out the 'roll a d23' hypothesis as an accurate model of what's going on.

No, you really don't. You don't have a large enough sample size to rule out or prove ANYTHING. You've got 9 Casters in FAQ, 3 in Goodmitton, 2 in Transylvito, 5 in GK (several of whom overlap w/ FAQ), 5 or 6 in Jetstone. (Numbers are approximate based off a mental head-count.) You've got about 20-ish unique casters, with a few duplicate types (dollmancer, thinkamancer). Your sample size is smaller than all of your available options; you CANNOT conclude anything from that.

That criterion is silly. What if there were 20 thinkamancers? We could exclude the d23 hypothesis with extreme strength. The real evidence isn't that strong, but you're going to have to come up with a much stronger justification for rejecting my rejection.

But you're really ignoring the real evidence we have of nonuniform distribution:

1) Characters have said that it's based on something nonrandom, and Rob has no reason to mess around with us like that;

2) odd specific things that get lost if you just dump them in a bucket:

Gobwin Knob has a ton of gems under it. They get a dirtamancer out of three popped casters. (1 - (22/23)^3)
Jetstone gets two dollamancers, one shortly after the other dies. (1/23)
Wanda... what other caster types would be able to help there, to a similar extent? Very few would be comparably useful. (Let's call it 4/23)
FAQ1 gets no thinkamancer, when they really wouldn't want one. (1-(22/23)^8 or so)
FAQ2 gets no casters at all. (??? but low enough to be noted by the queen)

That's, what, 5 out of 8 sides we know a significant amount about, that pop warlords (i.e. not Charlie), that have some notable coincidences around them?

[MarbitChow]

That criterion is silly. What if there were 20 thinkamancers? We could exclude the d23 hypothesis with extreme strength.

That's because 20 Thinkamancers in a sample size of 20 is DRAMATICALLY different odds.
A 12% (the likelihood based on BLAND's calculations) gives a 1 in 8 chance, which is the likelihood of heads coming up three times in a row.
20 Thinkamancers out of 20 casters, with 23 possible casters for each pop gives a 1 in 1,716,155,831,334,586,342,923,895,201 (or 1 / 23^20) chance.
The first is not sufficient to exclude the purely random hypothesis. The second is.

However, if you're going to introduce extra (and mostly irrelevant, like having gems + a dirtamancer) data into the equation, then we can go a step further and conclude that it is COMPLETELY deterministic, because Rob wrote it that way. There is nothing random about it at all.

[drachefly]

That criterion is silly. What if there were 20 thinkamancers? We could exclude the d23 hypothesis with extreme strength.

That's because 20 Thinkamancers in a sample size of 20 is DRAMATICALLY different odds.

You didn't qualify your complaint that way, did you?

What you said was treating it as a distribution, ignoring the information that made each draw different.

... The first is not sufficient to exclude the purely random hypothesis. The second is.

Exclusion comes to degrees. We can exclude the random hypothesis to a fair degree, <1%, using just the hard data we have. Forget about the dirtamancer bit.
And throw out Wanda since she was subject to strong post-selection.

If we consider our pre-most-evidence hypothesis-space, it looks something like:
Totally random 1/23
Depends on existing caster types only (correlation, anti-correlation, other specific pattern)
Depends on ruler's (unconscious?) wishes/preferences/personality/appearance
Depends on a side's 'theme'
Depends on location of popping
Depends on circumstances - 'need'.
Depends on Titanic whim (i.e. complicated, arbitrary-seeming rules, nonetheless distinguishable from chance)
Depends on new factors not considered up to this point.

Now, the first of these has the highest prior, by a fairly substantial margin, due to its minimal complexity.
However, we do know that sides and cities have different overall capabilities and potentials in other ways. This reduces the new information brought in by such hypotheses. So does the notion of Signamancy - which creates new links to and from appearance, in causal graphs. This relieves much of the information cost of the appearance of a side or ruler affecting their caster production.
So too with the Fate axis. There's a whole FATE axis out there.

And so on. We are surrounded by effects that may be able to perturb this symmetry.

However, if you're going to introduce extra (and mostly irrelevant, like having gems + a dirtamancer) data into the equation, then we can go a step further and conclude that it is COMPLETELY deterministic, because Rob wrote it that way. There is nothing random about it at all.

No.

Even if it's completely deterministic which characters Rob talks about, he could describe a world which is largely ruled by randomness and he just happened to pick an interesting spot in it.

But he is not describing such a world.

[BLANDCorporatio]

*sniff* this is becoming the best thread ever.

EDIT: because of FAIL on my part.

[vintermann]

It doesn't seem likely to me that Banhammer would willingly jeopardize Faq's secrecy (even given the precautions outlined by Jillian) just to get his hands on more casters. First and foremost, he wants Faq left alone.

* Banhammer has a predictamancer, so they have a pretty good idea of whether a prospective candidate will say yes or not (provided Marie tells the truth, but they're already on her mercy).

* A barbarian caster has nothing to gain from revealing the existence of Faq.

* A barbarian caster can't reveal anything more than the existence of a bubble side, not where it is, its name or even which casters/portal are associated with it. Remember, they have Jack. The offer can be made anonymously.

* Barbarian casters' existence in the MK isn't as blissful as all that. They need schmuckers for upkeep, they have to get them by providing something the sides of Erfworld want. Florists may supplement their upkeep with boosted farming, and moneymancers with whatever tricks moneymancers use, and probably they can cooperate to boost each other (florist + dirtamancer, moneymancer + dittomancer, etc.), but even so, I don't think the MK is a perpetuum mobile. Otherwise why work for sides at all?

* Banhammer wants enlightening philosophical discussions, not just safety at all costs.

[BLANDCorporatio]

Okay, after the gushing is done, I've got a few things piquing my curiosity.

If I get your argument right, drachefly, you're saying that the various alternative hypotheses about caster popping are less information costly because of Signamancy and similar effects (possibly) dictating or indicating something deeper that controls not just caster selection, but various other aspects of a side. So, while the alternatives to "just uniformly independently distributed" are more complex, they may explain more.

So if that bit of understanding is correct, and correct me if I'm wrong, you reject the random u.i.d. hypothesis not based on some posterior argument based on a wealth of statistical data about what was actually popped (since we lack such data) but on a, hmm, prior argument about how prior probability should be assigned to the various hypothesis (ie, while it looks like random u.i.d. deserves the bigger slice of the pie, it's actually not the case).

(I'm referring here to your latest post, in which you make no mention of characters claiming caster popping is not random)

My question is how can one go about actually assigning numbers to such things, be it in this or some simpler example that may illustrate the process better.

[MarbitChow]

Exclusion comes to degrees. We can exclude the random hypothesis to a fair degree, <1%, using just the hard data we have.

This is the part I'm not understanding. BLAND's calculations showed that, in a purely random distribution, FAQ's lack of duplicates would occur about 1 time in 8 (0.12 => 12%), which is well within plausibility (as the 3-heads-in-a-row example shows). How do you go from that to excluding the random hypothesis completely?