## Monday, January 29, 2018

### Roko's Basilisk and why people were afraid of it

I'm really late on this one, but I wanted to explain Roko's Basilisk, for all the people who heard about it a while ago and never really "got" it.

The idea first started going around the internet a few years ago, and apparently was seriously freaking out a number of people in the Less Wrong forums.  I think I first heard about it from this Slate article, maybe, then spent time trying to find somewhere to explain why this idea was considered so horrfying.  The RaionalWiki explanation likewise failed to shed any light about why anyone would actually be scared of the thing.

The concept builds on a number of premises that float around the Less Wrong community that relate to the technological singularity, namely "friendly god" AI, utilitarian ethical calculus, and simulated consciousness.

The Basilisk is a superhuman AI from the future.  The abilities of this AI are essentially infinite, even up to traveling backwards in time.  The idea of the Basilisk is that it wants you to contribute your money to helping it be built, and if you refuse to help it, it will create a simulation of you and torture the simulation forever.

And so I think a normal person quite understandably has trouble understanding why anyone would even think this is a good B-list villain for Star Trek, much less a cause for existential dread.

But it's actually not that silly.  And once you understand the background to it better, it all makes sense.  So let me explain to you what the Basilisk is in clearer terms, so that you too can experience the angst.   (That was your warning)

## Monday, January 15, 2018

### The Monty Hall Problem, Bayes Theorem, and a fault in Numberphile

I watch a lot of educational videos on YouTube, in particular the awesome channel Numberphile.  I recently saw their video on the Monty Hall Problem, and was kind of disappointed at what seemed to be a rather pointless calculation that didn't really show the result, and instead showed something that was already kind of obvious.

The video can be found here and explains everything, but let me explain it again for completeness.

The Monty Hall Problem is a classic apparent paradox in probability, named after gameshow host Monty Hall from Let's Make a Deal.  In the show, the contestants are shown three doors and told behind one of the doors is a brand new car.  Behind the other two doors are "worthless" prizes; anything works, but traditionally the problem says the other two doors hold goats.  The player gets to pick any of the three doors, and whatever is behind the door is what they win.  If they pick right they get a car, otherwise they get a goat.

To add tension, after the contestant picks, Monty Hall would walk to another door, a door that the player did not pick, and show them what was behind it.  And look!  It's a goat!  The car is still out there!

In the Monty Hall Problem (not necessarily the show), Monty then asks if the contestant would like to change their mind.

The question is, what is the probability of the player guessing correctly if they swap their pick?

The answer is quite obviously $\tfrac{1}{2}$, or 50%.  Except that isn't the correct answer, and the probability of winning if you switch is actually much higher.  It's actually $\tfrac{2}{3}$ probability, or 67%, if you switch.  But why?

## Monday, December 25, 2017

### Richard Feynman and the Message of Christmas

I often come across as a Grinch during Christmas.  It isn't that I don't like the holiday, it's that I find the actual celebration of the holiday so small compared to the actual ancient reason for celebrating.

The phenomenon of Christmas, as it exists today amongst moderns,  is largely a commercial platform to sell you movies, toys, electronics, and honey baked hams.  We sing about snow and various foodstuffs eaten and herd animals, we share some presents, spoil our children, and eat a lot of food.

To most modern people, this is what it's about.  Its about time with family and the magic of Santa and having fun singing Christmas songs.

I don't think modern people really understand Christmas.  I don't think they get it.

To explain Christmas, then, let me begin with a quote by Richard Feynman (from this video interview):
I can’t believe the special stories that have been made up about our relationship to the universe at large. They seem to be… too simple to conn- too local, too provincial! The Earth! He came to the Earth! One of the aspects of God came to the Earth mind, you. Look at what’s out there! It isn’t in proportion.
Richard Feynman gets Christmas.  In his own way, as a nonreligious Jew, Feynman understands the celebration of Christ's birth better than most people alive today.

The Christian religion properly centers itself, and distinguishes itself from other theistic religions, by two scandalous errors about God.  Two propositions about the divine that are rightfully offensive to the pious.  Two dogmas that are nothing short of pure blasphemy.  Statements which no human being should ever rightly dare to say about the Most High.

Christmas is the celebration of one of these blasphemies.  (The second is celebrated on Good Friday.)

Richard Feynman properly understands the blasphemy of Christmas.

This holiday commemorates the day when Christians claim that God came to Earth in the form of an infant baby -- what is called the Incarnation.

Those words may have lost their sting to you.  To many Christians it is a bromide that they've heard so many times they forgot what it means.  And many modern nonchristians don't understand what is meant by the word "God."  They confuse the ineffable creator of time and space with dudes in the clouds, or comic book characters, or weird tentacled things that live in space.

But Feynman was a major theoretical physicist, from a Jewish cultural background.  He was in a better position to understand what exactly the word "God" means.  Studying the universe and the mind-boggling complexity of the patterns obeyed at lower and lower levels, Feynman would not be capable of confusing what is signified by the proper noun "God" with something like a man in a cape with a big hammer — that is, with some regular ol' god.

When Christians say "God", they are not talking about a bearded dude in the clouds.  They are talking about the mind that first dictated and ordained the sort of phenomenon that Feynman struggled his entire life just to bring to a first-order approximate description.

Feynman also was not able to think provincially.  Humans have always had trouble understanding the stupefying scale of the Universe at large, but physicists are usually given a better feel of it.  The universe is so large that its size can only really be properly expressed with scientific notation — there are too many 0's otherwise.

When we talk about the Earth and humankind, we don't know anything else, so it is easy to have an inflated view of ourselves.  But in the cosmic sense, we are slightly above nonexistent.  We live on just one planet in our solar system.  Our sun is not particularly interesting, as far as solar properties go.  It's a pretty normal star.  In our same galaxy, there are some 250 billion (billion with a B) other stars, and we now believe most of those probably have their own solar systems like ours.  Our galaxy, the Milky Way, is just one galaxy amongst trillions suspected to exist in the observable universe (trillion, with a T).  Cosmologists often just model entire galaxies as point particles -- as insignificant atoms in a giant cosmic fluid.  And beyond the observable universe, who knows what else exists.  We are a speck in a speck in a speck in a speck compared to the Universe, and the universe itself is just a speck within the mind of God.

We are really nothing important, and Earth is nowhere special.

While this is a scientific truth, it is also a biblical truth.
Behold, the nations are like a drop from a bucket,
and are accounted as the dust on the scales;
behold, he takes up the coastlands like fine dust.
...
All the nations are as nothing before him,
they are accounted by him as less than nothing and emptiness.
-- Isaiah 40:15-17
All the inhabitants of the earth are accounted as nothing,
and he does according to his will among the host of heaven
and among the inhabitants of the earth;
and none can stay his hand
or say to him, “What have you done?”
-- Daniel 4:35
The moon and the stars, which You have ordained;
What is man that You take thought of him,
And the son of man that You care for him?
-- Psalm 8:3-4
In the Bible, all the nations of the Earth and all the inhabitants therein are accounted as nothing.  They are dust on a scale -- when you go to trade silver for grain, you zero out the trays, check the balance, remove the crumbs left over, but the dust is what you don't even bother brushing off.

Christians teach and celebrate that the infinite being to whom the Earth is but dust, this being who is immaterial, invisible, and who exists without limitations, this being took to himself a human nature and entered into the physical world, born as an infant child inside a tiny little town on this tiny little speck of dust we call a planet.

This is blasphemous, scandalous, offensive, absurd, and violates all notion of scale or proportionality.

And that scandal is precisely what makes the message of Christmas such a beautiful one.  It is why Christians have traditionally celebrated Christmas by great acts of charity.

God has no reason to care about us or to stoop so low to us.  Yet he did.

The infinite, finite.  The immortal, mortal.  The transcendent, physical.  The omnipotent, helpless.  The glorious, an infant child in diapers.  The self-sufficient, crying to be fed.

The Jewish priests of Jesus' time understood this blasphemy.  Jews and Muslims today still understand it.  Richard Feynman understood it.

To truly understand Christmas, you must hold both; that the Christian doctrine of the Incarnation is blasphemous and preposterous and entirely out of proportion; and that nonetheless, it is a blasphemy authored by God himself.

This Christmas, I hope you and your family will again be shocked and offended by the message of Christmas: that the Word became flesh, and dwelt amongst us.

## Wednesday, December 6, 2017

### Alternative to the d20 System

Early D&D used to be a mishmash of various different rules. Roll this die for this, that die for that, look on a table for this other thing. Thieves rolled a percentile die (two d10s with one treated as the "tens" place) and compared to a table on thieves skills; fighters rolled a d20 and looked up their result on a table organized by their level and the targets AC; later fighters rolled a d20 and compared THAC0 and AC to the result; some traps you rolled a d6 and found it on a 1, some doors you rolled a d8 and forced it open on a 1, some checks you rolled a d20 and tried to get under your ability score like STR.

It was kind of a crazy, scattered mess.

They started fixing this with the mentioned THAC0 system, which took the look-up tables for striking an enemy and condensed them down to a simple equation based on your roll on a d20. This simplifcation inspired centering the entirety of the rolls on a similar mechanic: roll a d20, add some modifiers, compare to a target number. Rather than tons of crazy rules and tables and different dice for everything, any time you wanted to check if you character succeeds at an event you had one basic mechanic you could turn do:
d20 + Modifiers >= DC,
where DC is "difficulty class", a target number meant to encapsulate how hard a task it. This is the d20 mechanic, and is usually listed in quick start and players guides as "the most important rule," or the "core mechanic."

It was such a powerful simplification that most D&D clones use it now. Even RPGs attempting to emulate the original 0E rules of D&D still rely on the d20 system, rather than duplicating the To-Hit tables and Thieves percentile tables.  So systems like DCC or OSRIC or Basic Fantasy all use the d20 core mechanic, despite being "old school" games trying to emulate an older version of the rules.

The d20 mechanic is still a table, though; just a very simple, linear table that can be written as an equation. It is a table where each entry differs from its nearest neighbors by 5%. A +1 sword means a sword that has an additional 5% chance to hit. An additional 2 to your (ascending) AC means the chance to hit you decreased by an amount of 10%. A lock with a DC of 15 and a lock with a DC of 10 differ by a step of 25% in the chance to pick them (that is New Chance = Old Chance + 25%).

The simple equation for your chances of succes is :
Chance success = (21 - DC + Mods)*5% = (21-DC+Mods)/20
In tabular form, the DC system would look like this:
Mod -3 -2 -1 0 +1 +2 +3
DC 9 45% 50% 55% 60% 65% 70% 75%
DC 10 40% 45% 50% 55% 60% 65% 70%
DC 11 35% 40% 45% 50% 55% 60% 65%
DC 12 30% 35% 40% 45% 50% 55% 60%
which is such a boring and simple table that you've probably never seen anyone bother to write it out. It's just down 5% for higher DC, up 5% for higher mod, constant steps of 5%. A DC of 10 is considered "baseline," and an average character has a 55% of success, higher or lower from there.

The d20 system is a really good, simple system. There are alternatives out there. Understandably, most alternative systems try to use six-sided dice, because those are the only dice most normal people know of. Apocalypse dice is one alternative, which uses 2d6. Tunnels and Trolls runs on d6s. The Hypereon systems use d6s.

I recently discovered an alternative system that sort of goes in the opposite direction. I saw it first proposed on the Goodman Games forum by the author of the blog PeopleThemWithMonsters (described in the linked post, somewhat expanded here).  It's a bit of a crazier system, though it produces probability tables very similar to those for the d20 system. It has some benefits to the standard d20 system and resolves some problems, and overall gives games a more wild "feel", even while sticking pretty close to the original d20 chances of success.

I'm going to call this the Target Dice Chain system. It is not prefaced on a single die, but on a long list of dice. Several games use this list of dice, calling it the "dice chain." It is as follows:
d0-d1-d2-d3-d4-d5-d6-d7-d8-d10-d12-d14-d16-d20-d24-d30-d48.
You may be thinking, "Hold on now, there's no such thing as a d3 or a d48," but you are wrong.  Most of these odd dice can still be simulated using "normal" polyhedra, as I detailed somewhat in a previous post.  You may be thinking "What even is a d0 or a d1?" You aren't likely to find them in your friendly local game shop, but you can simulate them by taking a blank d6 and marking every face with a "0" for d0 or "1" for d1.

In each check, there are two dice that get rolled: a target die and a check die.

The GM rolls the target die. Increased difficulty corresponds to moving to a die higher up the dice chain, starting at d6.

The player rolls the check die. Modifiers correspond to one step up (or down) the dice chain, starting at d6.

In checks, the smaller die is always rolled first. This helps maintain the tension and makes sure both parties roll.

Your baseline check for an average character corresponds to rolling one d6 for the GM, one for the player, and player wins if his die meets or beats the GM's. This corresponds to a check with a DC 10.

If the character has a +2 to his check, then roll a d6 for the GM and a d8 for the player (two steps up the chain), and player wins if his die meets of beats the GM's. This corresponds to a  DC of 10 and a +2 bonus.

If the check is actually fairly difficult, and the player has a -2 to this check, then you roll a d12 for the GM and a d4 for the player, and the player wins if his die meets of beats the GM's. This corresponds to a DC of 15 with a -2 bogus.

The underlying mechanic works like this:

The GM picks a target die (TD) according to the difficulty of the task. The player rolls a check die corresponding to his ability, starting at 6 and doing up or down one step in the dice chain for each point of modifier. You then make an opposed roll with the check die against the TD, and tie goes to player.

You can express the die roll symbolically as:
d6 + (MOD)d >= TD
where +d is a way of saying "up one step in the dice chain", and +3d means "up 3 steps in the dice chain", and +(MOD)d means "up (MOD) number of steps in the dice chain."

The TD can be chosen sort of arbitrarily, but should correspond to the difficulty. A d6 is the baseline TD (TD 6), and corresponds to a DC 10. A TD 12 (that is, a d12 as target die) is a fairly difficult TD, and corresponds to a DC 15. For even more difficult checks, doubling seems to work, with a TD 24 being close to a DC 20 and a TD 48 being close to a DC 25 (though the correspondence weakens at higher DC). A TD 3 is pretty close to a DC 5. Use dice in between for DCs in between. Maybe a TD 8 for a DC 12, or a TD 20 for a DC 18.

But why would anyone use this system, anyway?

This system is a bit "swingier" than the usual d20 system. In d20, certain DCs are simply impossible unless you have a bonus for that check: so, even with a +3 bonus, a DC of 25 is simply impossible. You will never succeed. In the TD system, there is always a chance for success. Even for a TD 48 check, with a -4d, there is still a chance for success. Contrariwise, in the d20 system, you can reach a point where failure is impossible. If you have +10 to a skill, then you always succeed on a DC 10 check. In the TD system, failure is also always possible. Even the best mess up and fail, even the most difficult task can be bested by a stroke of luck. The whole system sort of spreads probability around, leading to more unpredicted outcomes.

Here is a chart for comparison, showing the probabilities of success for different mods to a check for different DCs and the corresponding DC.

Normal dice chain, d6+(MOD)d to beat target
MOD -3 -2 -1 0 +1 +2 +3
DIE d3 d4 d5 d6 d7 d8 d10
DC5 65% 70% 75% 80% 85% 90% 95%
TD3 67% 75% 80% 83% 86% 88% 90%
DC10 40% 45% 50% 55% 60% 65% 70%
TD6 33% 42% 50% 58% 64% 69% 75%
DC15 15% 20% 25% 30% 35% 40% 45%
TD12 17% 21% 25% 29% 33% 38% 46%
DC20 0% 0% 0% 5% 10% 15% 20%
TD24 8% 10% 12% 15% 17% 19% 23%
DC25 0% 0% 0% 0% 0% 0% 0%
TD48 4% 5% 6% 7% 8% 9% 11%

As you can see, it actually sticks pretty close to the usual d20 results.

One major issue this resolves is the opposed check mechanic. Sometimes in D&D, you may find yourself grappling with an orc. In the simplest resolution of this, you'd just make an opposed STR check; you roll a d20 and the orc rolls a d20, add the modifiers to the respective rolls, meet or beat the orc's roll. While this maybe has the feel of a battle of strength, it's actually identical to a base DC of 10 (proven mathematically here), plus your mods and minus the orc's mods, in terms of probability. If you are at +2 on your STR and the orc is at +3, then it's just
d20 + 2 - 3 >= 10.
It turns all opposed rolls into static DCs. You could replace opposed rolling with a constant DC and get the same statistical results.

In the TD system, roll a die corresponding to where you are in the dice chain. In the example of a player at +2 and orc at +3, you roll a d8 and the orc a d10, meet or beat the orc. Here an opposed roll is central to the mechanic, and you get a more immediate feel for the power mismatch between the player and the orc in terms of the physical dice being rolled. Imagine then a dragon rolling a d30 vs an average human's d6, or an advanced player rolling a d12 against a d4 mook.

Another major advantage is that it tracks so well with the d20 system. In any game using the d20 system, you can just slide the target die system right in the d20 system's place without a lot of difficulty. If you use the TD system and want to have some particular check be a DC check instead, just slide that in there, keeping the same bonus. It converts almost instantly.

But there are a few major drawbacks to the target dice system.

One, you have to get all those dice. And they aren't very cheap. If you use a computer dice roller like Roll20 then this isn't even an issue: typing 1d48 will work just fine. But if you like the feeling of physically rolling, you'll have to buy more weird dice. I don't know if a GM should be opposed to owning more dice (who doesn't like newer, weirder dice?), but the availability is an issue, as is the money. Luckily, you can find the "off" dice in the dice chain (d3, d5, d7, 14, d16, d24, d30) from several sets offered by Koplow or Impact! miniatures intended for games like DCC RPG which use the dice chain already, usually for around $8 or$12 -- which is still expensive compared to the standards, but about the same as an adventure module.

Two, is that there is no way to make something impossible. Even if you had the ridiculous TD 120, with a d1 check die, there's a chance of success 1 in 120 times. I reported this earlier as a benefit, but maybe in some instances it's a drawback. Maybe you want that lock to be impossible to pick. Maybe you want that armor to be impossible to hit. There's also no way to make something guaranteed. Even if your PC rolls the ridiculous d120 against a TD2, you can fail 1 in 120 times. The swinginess of the system can be a cool and exciting way to mix things up, or it could topple tough challenges through sheer stupid luck of the draw.

## Tuesday, December 5, 2017

### Advanced Berenst#in Bears Theory - Complex Euclidean Space

I wrote a blog post several years ago about the Berenst#in Bears that, somehow, against all odds, managed to get picked up on the internet.  It's been posted on Reddit and other outlets several times, and accounts for about 95% of all traffic to my blog.

The post is of course intended to be silly, but part of the silliness is that hidden inside this ridiculous proposition of universes merging over children book names is a very mathematically complicated thesis about the nature of space and time.

I don't think I did a very good job explaining what that thesis is, because I often see my blog being referenced as claiming either time travel, the many-worlds hypothesis, or something else about quantum mechanics are responsible for us finding ourselves in this piteous state of being in the wrong universe with the wrong cartoon bears.

In this post, my goal is to outline what exactly it is that I was "claiming," so that if the story gets picked up again (unlikely now that more major outlets have picked it up), maybe people can actually at least summarize what I'm saying correctly.

TL;DR the theory is about changing the geometry of spacetime by allowing all four dimensions to be general complex numbers.

## Saturday, November 4, 2017

### A Rule for Shields from Descending Armor Class, Applied to Ascending Armor Class

I've been playing 'D&D 'ever since I was a young boy and found my dad's Dungeon Master's Guide on his bookshelf (the original, by Gygax).  I was still too young to understand a lot of it (Gygax opens with a discussion of uniform vs binomial probability distributions...), but I spent hours of days flipping through the pages, looking at the pictures and reading the descriptions of a fantastic world.  It inspired me to make my own games about exploring through planned dungeons with friends and having them use ability scores to get through.

By the time I was old enough to try to really figure out the rules, version 3.0 was out, which made a lot of changes.  One of the biggest was replacing the original Descending AC and THAC0 system with the slightly more intuitive Ascending AC system.  This system always just made sense to me, whereas DAC and THAC0 always seemed weird and complicated.  I never understood how DAC worked, and never really saw a good explanation of it, and so just never used it.  It was only recently, when getting into old school tabletop RPGs, that I decided to look up an explanation of how DAC and THAC0 used to work.

## Wednesday, October 25, 2017

### test

test for commenting

Update: I think I figured it out.  Comments should work better now.

From my phone I can comment.  From my computer, I'm only able to  comment if I use a separate page for  comments.