Bayes Bayes Revolution, Aumania

Writing the Aumann game posts got me thinking about games designed to promote rationality. Nick Bostrom and commenters discussed the topic on Overcoming Bias. Tom McCabe had a post about rational debating. In this post I’ll brainstorm about two ideas I’ve had for computer/video games.

First, Bayes Bayes Revolution.

The basic tool here is a probability slider that you move left/right with two keys or a mouse. It runs from 0 to 1, but probably not in a linear way. There should be more distance between 0.01 and 0.02 than 0.50 and 0.51, just for convenience. And maybe the results of your setting the slider should show up on a view-only linear scale so your intuitions don’t get warped the same way. Maybe a two-color pie diagram.

The game starts out posing you a question, maybe whether a hidden animal is red or green. You have some initial guess for the probability, either from having played the game before (50-50 if you haven’t), or because the game told you.

Then, one by one, at a rate depending on difficulty level, the game gives you pieces of evidence. You might see a blue ball flying across the screen, and you might know (again, from experience or because you’re told) that red animals throw blue balls three times as often as green animals. If your initial probability for a red animal was, say, 0.2, then if you crank the Reverend you find:

p(red animal | blue ball) = p(red animal) * p(blue ball | red animal) / p(blue ball) = 0.2 * 3q / (0.2 * 3q + 0.8 * q) = 3/7.

The game scores you based on how close you put the slider to this correct answer. Then when the next piece of evidence comes it scores you based on how close you put the slider to the correct answer after taking into account both pieces of evidence. And so on. From what I understand about existing games of this genre, they rate you “OK” or “excellent” or whatever each time according to some formula. (You’re being scored here on following the right procedure rather than placing a high probability on the right outcome; that would be cleaner in a way, but would introduce a luck factor.)

Starting from there, you can make the game as simple or complicated as you want. It can be spartan and abstract or full of bells and whistles, using simple independent drawings from an urn or bombarding the player with baroque causal diagrams.

In the end it would be like following a sort of dance routine. But the Bayes Dance is a very special kind of dance, because if you do it right, you never have any idea where you’re going. In fact, to those who can’t hear the Music of the Evidence, the Bayes Dance looks exactly like a random walk. Just like “if we knew what we were doing, it wouldn’t be research”, one can say that “if we knew where we were going, it wouldn’t be Bayes”. Take a Bayesian to a regular dance lesson, and he will say, “if I already know I’m going to have to step to the left, then that must be a better place for my foot, so why not put it there to start with?” Or he will say, “my leg has to stay here on average, so if you’re that sure it’s going to the left, then in the unlikely case that it does go to the right, it will have to fly off, all the way out of the window”.

So in Bayes Bayes Revolution, as indeed in life, there is no predictable routine you can practice. All you can do is align your gut feelings with the math so they can work with any input.

not photoshopped!

Second, Aumania.

I seem to have turned BBR into a rationality lecture more than an actual game idea. I guess if you really hated rationality lectures, you wouldn’t come here. Aumania is a different game idea, one that greatly resembles the Aumann game we’ve played on this blog before, and unlike BBR I could see it being quite enjoyable.

To stay with the metaphor from earlier on a little longer: Aumann’s agreement theorem says that if two Bayes dancers have gone out of synch because they heard different music, then when they watch each other after the music stops (or perhaps between notes if they’re quick), they will do a sequence of steps, each reacting to the last, that ends in them gradually merging back and standing still.

Unlike BBR, Aumania is necessarily multi-player. The point of the game is to react smartly to the opinions of others.

Here’s how it works. Each player has an individual probability slider, like in BBR. Maybe they’re arranged in a circle on the screen. The game presents the players with randomly generated claims, perhaps like the ones we used for the Aumann game. Players have maybe ten seconds to move their slider. The key is that they should take special note of how all the other sliders are being set. When time is up, the answer is revealed and everyone gets scored.

Your score is the logarithm of the probability you put on the correct answer. That way, entering your true subjective probability maximizes your expected score. Score, defined this way, is always a negative number; the game could either accept this, or transform back to percentages, or give players a fixed amount of free points for every question, or whatever.

Score could also map to hit point damage. Learning to be rational so you can help humankind find and act on true beliefs is nice and all, but learning to be rational because else Mario gets eaten just before killing that level 6 boss, that’s motivation.

For every setting on the probability slider there is a corresponding score size if the claim is true and if the claim is false. These could be displayed as two bars to show the trade-offs you’re making.

Aumania is partly about knowing your trivia, but not as much as you might think. Players will do better in groups with much knowledge dispersed in them than in groups with little. But the world’s most ignorant person can score the same as the world’s most knowledgeable person, just by always copying the probability. If he’s also the world’s most stubborn person, that’s when he has a problem.

I’m not sure whether Aumania would work best as a cooperative game (players maximize the group’s total score), an independent game (players maximize their own score), or a competitive game (players minimize the rank order of their score compared to the other players). In a cooperative or competitive game, there might be incentives for deception, which means Aumann’s theorem won’t work. Unpredictable deadlines might help.

Again, you can add as many complications as you want. Random pieces of evidence, as in BBR, would be one possibility. Other means of communicating evidence, such as chat, would be another.

If we can teach the next generation to update beliefs as smoothly as this, our species will be utterly unstoppable.

The New Adventures of Aumann

I should still say something about lessons from the last Aumann Game, but in the mean time here are some new claims. Remember, the object is to maximize the sum of the logarithms of the probabilities you assign to the right answers. Putting a .1 probability on a true claim means you get one negative cookie, putting a .01 probability on a true claim means you get two negative cookies, and so on. In the best case you get no cookies. To play, post your honest estimates and update them based on those posted by others. Initial scores will reflect your calibration and general knowledge/judgment. Later scores will reflect your calibration, your ability to correctly weigh the opinions of others, and possibly still your general knowledge/judgment.

Where possible I usually added the question before knowing the answer. I’d be curious to know if there are some kinds of question you think are particularly (un)suitable for this. It probably wouldn’t be too hard to generate these automatically and write a human brain calibrator program.

First, some more statistics claims, using NationMaster and StateMaster.

1. Belgium has more inhabitants than Sweden.

2. Bolivia has a greater total GDP (PPP) than Kenya.

3. Life expectancy at birth is greater in Israel than in Uruguay.

4. Hungary has more land area than Utah.

5. South Dakota has more murder and nonnegligent manslaughter per capita than Vermont.

6. Louisiana consumes more oil than Washington (the state).

7. California has a greater fraction of male inhabitants than Kentucky.

8. Austrians have won more total gold medals in the summer Olympics (all time) than New Zealanders.

9. Argentina is more urbanized than Latvia.

10. There are more tractors in use in Belarus than Zimbabwe (2000).

Some pure probability theory questions.

11. I will shuffle a deck of cards, take the first ten cards, and multiply the numbers on them. Face cards count as 1. Claim: The result will be over one hundred thousand.

12. I will shuffle a deck of cards and flip them over one by one. Claim: I will reach the third 8 or the third 9 before I reach the second 3.

13. I will add the results of ten 6-sided dice. Claim: The result will be at least 43.

14. I will flip a coin until I get heads four times in a row or tails five times in a row. Claim: This will happen before or on the 30th flip.

15. I will place two black rooks, two black bishops, a black queen, five white pawns, and a white king on a chessboard at random (no pawns on the first or last row). Claim: the white king will be in check.

Times and distances.

16. Cairo and Bombay are farther apart than Vancouver and Mexico City.

17. London and Dublin are farther apart than Milwaukee and Philadelphia.

18. Bucharest and Prague are farther apart than Sao Paulo and Rio de Janeiro.

19. Jerusalem and San Francisco are farther apart than Sydney and Helsinki.

20. Nietzsche died before Wagner.

21. Eisenhower died before Truman.

22. Leonidas died before Socrates.

23. Caesar was born before Virgil.

24. Aristotle was born before Ptolemy.

25. Shakespeare died before Newton was born.

26. Mao Zedong died before Britney Spears was born.

27. The Chinese invented paper before the Romans invaded Britain.

28. Buddhism was introduced in Japan before Norwegians started colonizing Iceland.

29. Ankara was founded before Kiev.

30. The Eighth Crusade happened before the War of the Roses.

Some internet questions. Google searches are including quote marks.

31. I will choose a random message from the Extropians mailing list from January 1998. Claim: The message body (including any quoted text) will contain the word “technology”.

32. Same as previous claim, but for SL4, January 2008, and “maybe”.

33. Googling “cow” gives more search results than “horse”.

34. Googling “Paris Hilton” gives more search results than “Eiffel Tower”.

35. Googling “Oort cloud” gives more search results than “Kuiper belt”.

36. Googling “Lance Armstrong” gives more search results than “Neil Armstrong”.

37. Googling “John McCain” gives more search results than “Michael Jackson”.

38. This blog gets more visitors from India than Ireland.

39. Gandalf has a longer Wikipedia page than James Randi.

40. Robert Aumann has a longer Wikipedia page than Spiderman.

Chess for the Warcraft Generation

Just for fun, from the Chess Variants Page, here’s Fantasy Grand Chess, with six possible armies.

Some other variants that look intriguing:

And people say immortals will get bored! Really, Chess is a genre, not a game. I wonder whether serious players of standard chess haven’t long reached the point of diminishing returns in fun. Probably one of those lock-in things.

Speedrunning through Life

An amusing thing you can find on the internet these days is the speedrun, where people try to find the quickest possible way of completing some particular video game, possibly using tools like save states, but always with legal input sequences. There are examples at various different sites.

I suspect that if you went back and asked people who played these games without knowing about speedruns, their estimate of the minimum time needed to complete a game would almost always be far too high. They would think of some simple improvements they themselves could imagine making, then maybe adjust slightly further downward — failing to account for all the tricks they couldn’t imagine.

In these games, despite some glitches, the basic physics corresponds quite closely to the surface rules we intuitively use to make predictions of what can and what can’t be done. In real life, the gap between basic physics and intuitive surface rules is much wider. We probably once thought of fire as an uncontrollable force with a will of its own. We once thought of atoms as, well, atomic — unbreakable things that you could rearrange, but that you otherwise had to take as given. But reality gives us much greater scope for putting in information than any video game, and now we’ve found ways to make many of the rules obsolete.

A superintelligent posthuman being could speedrun through life like some people speedrun through video games. The only difference is that reality has more loopholes to exploit — perhaps no longer in fundamental physics, but in things like engineering, computer security, and human interaction. That is why the consequences of a technological singularity are predicted to be so quick and so extreme.
I could watch this thing for hours.

(image by this fellow)