Maybe you need to clarify what exactly do you mean by depth. Otherwise I interpret this as follows: at one end of the scale you have complete beginners. At the other end, you have extremely talented people who have dedicated their lives to the game. Since the extremely talented people are still far from perfect play, the game itself is not the bottleneck as far as depth goes. Therefore, all such games (this certainly includes go and chess, but also almost every other board game) have the exact same depth, which is simply the general human analytical capacity to comprehend board games. The extra depth which may be inherent in the game is irrelevant, as it's not accessible to humans.

What's depth?

http://senseis.xmp.net/?EGFRatingSystem is pretty comprehensive.

Thanks! Embarrassingly, all those values of a are also on the EGD page, as well. It seems they use the formula a=205-(GoR/20), but it's not clear whether they think this formula extrapolates below 100 or above 2700.

Recalculating, we get ~48 levels of depth at 35%, or 40 levels if we assume that a should be as high as 255 and as low as 60.



In general, the depth of something is how many measuring-units deep something is, from the surface down. A pool might be 10' deep, the Marianas Trench is 11km deep, and so on. Now, obviously we aren't talking about physical depth - if we were, go would be deeper on a floor board than on a table board, and Herman's 3-go might be as many as 18 sun deep! I assume we are talking about depth of strategy or of game play.

A shallow or superficial game is one where a line of play that appears reasonable to someone who knows nothing about the game other than the rules actually is reasonable. (And likewise, the lines that appear unreasonable actually are unreasonable.) A game is less shallow if, lurking below the line that appears reasonable from the surface, there is another line that appears reasonable to another player who has thought more about how to win, and the more such lines that are hiding under one another, the deeper the game is. As I've probably made clear, I think the proper measuring-unit is bands of mastery such that there is a constant winning percentage between people at the top of one band and the top of the next band.

As you say, the maximum amount of effort someone can devote to any game is (a good chunk of) a human life. I don't think this has too much to do with depth. This would be sort of like measuring the height of a mountain by the weight of a hiker's backpack, or the depth of swimming pool by the number of rungs leading up to its diving board. You can see a connection between these things, but just as you can overpack for hiking, you can lavish effort on a comparatively shallow game. I can see the point that learning to play a ladder-breaker is a comparatively dull accomplishment compared to some of the exquisite tesuji that pros find, but I think it's equally important to realize that a high-dan player knows dozens of things about the game that collectively contribute less to his finding winning sequences than a 15k's shaky grasp of ladders and nets contribute to his.

Sorry, but this tells me very little. It also doesn't seem to contradict my point above, which is that depth is limited by the human ability to "see more lines".

This, on the other hand, is awfully specific. You don't show how or why this is related to what you say above. You also ignore a number of arguments people in this thread have brought up against using ranks (which is another word for "bands of constant winning percentage") for anything more than ranks. By this point I'm not sure that you really mean to say what you are actually saying. Let me restate the arguments:

* So you think there's the same depth between 30k and 20k as there is between 1k and 9d?

* Consider a new game, Go 2.0. In Go 2.0, we first play a game of go, then half the time the winner of the go game actually wins, and the other half we just flip a coin to determine the overall winner. This game has much fewer "constant winning probability bands" than Go does. So you think it is much less deep than Go?

* Consider a game where everyone in the world is randomly pre-assigned a number from a vast range. Whenever two people meet, they just ask a computer and it picks a winner with the probability based on the ratio of their numbers. That's all there is to this game. Because the numbers chosen can be very different, there's a large number of "constant winning probability bands". Do you think this game has a large depth?

With your definition being as specific as it is, you can't just brush these arguments aside. I suggest you think more about what you think depth is and why it should or shouldn't be related to winning probability. Personally I see no reason why winning probability is at all related to something like what you are describing above about lines lurking beneath the surface.

This really says it all to me. It's meaningless to try and compare the depth of things that are far deeper than we're capable of comprehending.



It would be more like measuring the height of a mountain by how high up it hikers can get. Saying a game is deeper because it has a wider ELO range is like saying a mountain is taller because hikers get higher before giving up/dying.

I mean what I say, and say what I mean!

First: You may be slightly mislead by how we were reckoning things with the Elo scores above. The depth bands aren't linear in points (under the EGF system). 30k to 20k would cover 8 EGF bands of 35% probability of winning, assuming a=200 for every rank; it would cover even fewer bands if we calculated it for an average a= ~225 (the a you would extrapolate for 25k). 1k to 9d would cover 18 bands of 35% probability, at an average a=82.5. So the the latter range you chose covers about 2.25 times as much distance, whether we call that distance depth or something different.

I may be responsible for this confusion, since I'm the one who started talking loosely about the first 1000 Elo points and the last 1000 Elo points earlier in the thread. But this is, imho, an advantage of using quantitative models: even if they are not perfect, they force me to be consistent, and indicate how I should change my understanding when I get more information. In the case of the EGF, I should compare the first 2200 points to the last 1000 points (but in the FIDE system, the points are linear and a direct comparison is appropriate).

("a", for people following along casually, is a variable the EGF uses to make ranks tighter at the top and looser at the bottom.)

But anyway, I understand that your point is broader than this. "Is there the same depth between (some equivalent range at the beginning of a go player's progression) as there is between 1k and 9d?" And I would say, if we are talking about the contribution each range makes to the total depth of the game, both ranges contribute the amount amount. If we are asking whether one range is deeper than the other or if they are at the same depth, then I would say that every single point in the second range is deeper than any point in the first range, so of course the second range is deeper than the first. Is there any other coherent way of rephrasing your question about "the depth between point A and point B" that doesn't boil down to one of these?

Second: yes, I agree it's a less deep game then Go. If we treat getting more points on the go board as the moral equivalent of winning, then of course we're just playing Go (fun! deep!) and then flipping a coin at the end for no apparent reason. But if we actually care about winning go-2, and winning by a flip is as good as getting more points on the board, which is hard to imagine but that's the thought experiment you set up, then go-2 would be less deep. There would be a lot of information to assimilate, but it would be relatively less interesting than go to the tune of it not mattering in 25% of games.

(Side question: what is n such that n-go-2 is equivalent in depth to 1-go-1?)

Considering the following two games. Game one is the board game Olympics: You play one game each of go, chess, and then three other games of strategy which aren't particularly important to the example; best of three wins. In game two, you play go and chess and then you play three luck-games that are equivalent to coin-tosses. If you really think the goal is to win, which game do you consider deeper?

Third: I don't really consider that to be a game. No strategic interaction, and person-specific starting positions. Sort of like roulette salted in with feudalism. I think there should be descriptions of hypothetical games that challenge my position in various ways, though.

To answer your very first point, about seeing more lines: again, I don't think the raw number of lines or the number of lines humans can see matters if they don't translate into any ability to beat other people.

I'm fine with this distinction, but you've got it entirely backwards. I'm not too concerned about some extra-human conception of depth. Suppose we have two games that exhibit the same depth in the sense I'm talking about, but computers have much more depth of play in one than the other. How important is that?

What I care about is the fact that for humans, go and chess are very deep games, and tic-tac-toe is not, and draughts is rather deep, but probably not deep enough (humans can play close enough to perfectly that there's a game killing frequency of draws at the expert level--far more than chess).

Q: Why do I care about this?
A: Because I'm a human.

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Actually that's exactly how I feel too. It's well worth making a distinction between games we can master and games we can't, such as tic-tac-toe and go, but attempts to compare the depth in two games we can't master just seems pointless.

Any game that I've played for 20 years, but some 6 year olds can still beat me, is not very deep.