Oct 24, 2008

42 comments

Imagine that I'm playing chess against a smarter opponent. If I could predict *exactly* where my opponent would move on each turn, I would automatically be at least as good a chess player as my opponent. I could just ask myself where my opponent would move, if they were in my shoes; and then make the same move myself. (In fact, to predict my opponent's *exact* moves, I would need to be superhuman - I would need to predict my opponent's exact mental processes, including their limitations and their errors. It would become a problem of psychology, rather than chess.)

So predicting an exact move is not possible, but neither is it true that I have *no* information about my opponent's moves.

Personally, I am a very weak chess player - I play an average of maybe two games per year. But even if I'm playing against former world champion Garry Kasparov, there are certain things I can predict about his next move. When the game starts, I can *guess* that the move P-K4 is more likely than P-KN4. I can *guess* that if Kasparov has a move which would allow me to checkmate him on my next move, that Kasparov will not make that move.

Much less reliably, I can guess that Kasparov will not make a move that exposes his queen to my capture - but here, I could be greatly surprised; there could be a rationale for a queen sacrifice which I have not seen.

And finally, of course, I can guess that Kasparov will win the game...

Supposing that Kasparov is playing black, I can guess that the final position of the chess board will occupy the *class*
of positions that are wins for black. I cannot predict specific
features of the board in detail; but I can narrow things down relative
to the class of *all possible* ending positions.

If I play chess against a superior opponent, and I don't know for
certain where my opponent will move, I can still endeavor to produce a
probability distribution that is *well-calibrated* - in the sense
that, over the course of many games, legal moves that I label with a
probability of "ten percent" are made by the opponent around 1 time in
10.

You might ask: Is producing a *well-calibrated* distribution over Kasparov, beyond my abilities as an inferior chess player?

But there is a trivial way to produce a well-calibrated probability distribution - just use the maximum-entropy distribution representing a state of total ignorance. If my opponent has 37 legal moves, I can assign a probability of 1/37 to each move. This makes me perfectly calibrated: I assigned 37 different moves a probability of 1 in 37, and exactly one of those moves will happen; so I applied the label "1 in 37" to 37 different events, and exactly 1 of those events occurred.

Total ignorance is not very useful, even if you confess it honestly. So the question then becomes whether I can do better than maximum entropy.
Let's say
that you and I both answer a quiz with ten yes-or-no questions. You assign
probabilities of 90% to your answers, and get one answer wrong. I
assign probabilities of 80% to my answers, and get two answers wrong.
We are both perfectly *calibrated* but you exhibited better *discrimination* - your answers more strongly distinguished truth from falsehood.

Suppose that someone shows me an arbitrary chess position, and asks me: "What move would Kasparov make if he played black, starting from this position?" Since I'm not nearly as good a chess player as Kasparov, I can only weakly guess Kasparov's move, and I'll assign a non-extreme probability distribution to Kasparov's possible moves. In principle I can do this for any legal chess position, though my guesses might approach maximum entropy - still, I would at least assign a *lower* probability to what I *guessed* were obviously wasteful or suicidal moves.

If you put me in a box and feed me chess positions and get probability distributions back out, then we would have - theoretically speaking - a system that produces Yudkowsky's guess for Kasparov's move in any chess position. We shall suppose (though it may be unlikely) that my prediction is well-calibrated, if not overwhelmingly discriminating.

Now suppose we turn "Yudkowsky's prediction of Kasparov's move" into an *actual chess opponent*, by having a computer *randomly* make moves at the exact probabilities I assigned. We'll call this system RYK, which stands for "Randomized Yudkowsky-Kasparov", though it should really be "Random Selection from Yudkowsky's Probability Distribution over Kasparov's Move."

Will RYK be as good a player as Kasparov? Of course not. Sometimes the RYK system will *randomly* make dreadful moves which the real-life Kasparov would never make - start the game with P-KN4. I assign such moves a low probability, but sometimes the computer makes them anyway, by sheer random chance. The real Kasparov also sometimes makes moves that I assigned a low probability, but only when the move has a better rationale than I realized - the astonishing, unanticipated queen sacrifice.

Randomized Yudkowsky-Kasparov is definitely no smarter than Yudkowsky, because RYK draws on no more chess skill than I myself possess - I build all the probability distributions myself, using only my own abilities. Actually, RYK is a far worse player than Yudkowsky. I myself would make the best move I saw with my knowledge. RYK only occasionally makes the best move I saw - I won't be very confident that Kasparov would make exactly the same move I would.

Now suppose that I myself play a game of chess against the RYK system.

RYK has the odd property that, on each and every turn, my probabilistic prediction for RYK's move is exactly the same prediction I would make if I were playing against world champion Garry Kasparov.

Nonetheless, I can easily beat RYK, where the real Kasparov would crush me like a bug.

The *creative* unpredictability of intelligence is not like the *noisy* unpredictability of a random number generator. When I play against a smarter player, I can't predict *exactly* where my opponent will move against me. But I can predict the end result of my smarter opponent's moves, which is a win for the other player. When I see the randomized opponent make a move that I assigned a tiny probability, I chuckle and rub my hands, because I think the opponent has randomly made a dreadful move and now I can win. When a superior opponent surprises me by making a move to which I assigned a tiny probability, I groan because I think the other player saw something I didn't, and now *I'm* about to be swept off the board. Even though it's exactly the same probability distribution! I can be *exactly* as uncertain about the actions, and yet draw very different conclusions about the eventual outcome.

(This situation is possible because I am not logically omniscient; I do not explicitly represent a joint probability distribution over all entire games.)

When I play against a superior player, I can't predict *exactly* where my opponent will move against me. If I could predict that, I would necessarily be at least that good at chess myself. But I can predict the *consequence* of the unknown move, which is a win for the other player; *and the more the player's actual action surprises me, the more confident I become of this final outcome.*

The unpredictability of intelligence is a very special and unusual kind of surprise, which is not at all like noise or randomness. There is a weird balance between the unpredictability of actions and the predictability of outcomes.