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Go is known for having a vast space of possible board positions, which was one of the reasons why computers weren't able to compete for a long time.

I wonder how many of these board positions are actually used by humans. Go is absolutely deterministic, and it could seem reasonable for a person to always react to the same board position, in the same way, thus not exploring any new positions when the opponent does the same. Of course, one will try to explore new positions after losing a game to improve in the next round.

Still, humans often act the same in similar situations and are not good at doing something completely new. For example, humans usually cannot choose good passwords even when they try to avoid common mistakes that weaken passwords, so computer-generated passwords are way more random than what humans can come up with.

Are there any analyses about how many board positions are relevant, i.e., will be reached in many games, and what positions may never be reached because there is no way to reach them that would make any sense to the players? How many of the positions really matter when looking at the vast number of legal positions?

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    Awesome question!! Thanks for asking. Personally I'm not aware of any papers exploring this question. The total number of legal positions has been calculated exactly: Tromp, John. "The number of legal go positions." International Conference on Computers and Games. Springer, Cham, 2016.
    – Stef
    Mar 16, 2022 at 16:52
  • One problem is the subjectivity of what makes sense to players, which depends heavily on their level. I also wonder if you want the number of positions reached by humans in actual play or the number considered during reading, which is actually rather fuzzy – but those are certainly positions that “matter”!
    – PJTraill
    Mar 22, 2022 at 15:59
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    The number of positions reached in ❝many❞ games will be far smaller than that reached at least once. Even its logarithm will be much smaller.
    – PJTraill
    Mar 22, 2022 at 16:08

1 Answer 1

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I kind of disagree with the statement humans often act the same in similar situations. I know that my reaction to the same opening moves depends on my mood, what I ate earlier, the moon phase and the shoe size of my opponent modulo Pi. In other words, most players try to explore new moves and ideas at least some of the time. (I agree that humans suck at thinking of new passwords, but generating, say, 4 bits of entropy per digit when a true random source could generate 6, is still spanning a large space.)

The Tromp paper gives an upper bound. Naturally, this number will not (and cannot) ever be reached.

Practically speaking, the answer to how many positions are reached probably depends on the number of games played almost exclusively. I'm not aware of a proper scientific approximation, so I'm attempting an educated guess. Suppose there have been 100 million active players for 1000 years. Each player plays 30 games a day, and year has 333 days. Suppose each game lasts 200 moves, and half of them are uniquely new positions. That leaves us with 100,000,000 * 1,000 * 30 * 333 * 200/2 = 100,000,000,000,000,000 = 10^17 unique positions. Certainly more than one could ever hope to analyze.

Looking at it from another angle, in my experience there are usually only a handful (1 to 10 or so) options per move that seem worthy of consideration (*). The mode (most common value) is surely 1, i.e. there is only a single move as an option. The median (center of distribution) is probably around 2, and for the average I would guess around 3, but let's say just 2. So when playing a game, the number of reasonably reachable positions should be around 2 to the power of the move count. At 100 moves per game, that's around 10^36 positions.

The number of positions that is unlikely to be reached is the upper bound minus that, i.e. most of them.

(*) This is similar to what I (vaguely) recall about players in the game of chess and in other, more abstract situations (e.g. Newell & Simon, 1972: Human Problem Solving). It is also similar to the number of moves typically suggested by modern Go AI. A while ago I used this to losslessly compress the policy output of AI programs from 724 to just 8 bytes, which was required to keep large game trees in memory.

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  • With that many games, the number of positions reached must depend on the number of games played. But one interesting point is the probability of a position being reached. If the probability distribution is not uniform, an AI with an oracle that can tell if a position is "relevant" would be able to skip a lot of computations. But such an AI would still need to consider every position above a certain probability threshold, even when it is unlikely that a specific position will ever be played.
    – allo
    Mar 18, 2022 at 17:21
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    @allo That is true and has been, in fact, the key problem in Go AIs. The basic idea has been popularized a decade ago: Simply assign a weight to each possible next move and put more research effort into those moves it deems more likely to happen - this is "Monte Carlo Tree Search". The difficult question is, which moves are likely? That is more or less solved nowadays with modern "policy networks".
    – mafu
    Mar 19, 2022 at 6:14
  • One thing are these methods and their estimates relying on heuristics, tree search, and more advanced AI methods to find likely paths. But one can also ask what the size of the set of all games, which are reached with a probability greater than p, is. For most thresholds, the set will contain games that will never be played, and for some thresholds, it will only have games that will most likely never be played. And while we don't know the sets, I wonder if there are estimates about the size of these sets.
    – allo
    Mar 19, 2022 at 20:56
  • As far as I know, there are no such estimates yet. I suppose you could derive them using the (trivial) math in the answer, but it of course would be an extremely rough estimate. It also depends entirely on the assumptions you want to make.
    – mafu
    Mar 20, 2022 at 7:01
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    I keep it an open question here. Let's see what other answers the question gets. No need to wait for an definitive number, any approach and estimate or idea is interesting and your post already provided some interesting calculations as well.
    – allo
    Mar 21, 2022 at 9:38

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