Go is better than chess because

General Go Discussion
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Man, I has forgotten how technical Uni professors required proofs to be in a test. Well, for a Maths degree anyway.

I just wanted to add, in @ArsenLapin1 's favor, that @Vsotvep 's proof is existential. However, there is also a constructive proof (and, therefore, an algorithm).

Notice there are, at most P = 3361 × 2 positions that you can create in a board, that is 3 for every state (black, white, empty) times 2 for which player is to move next. I’m including the last bit so that it works for both positional and situational superko.

So basically, you create a tree of board positions, stemming from the empty board, and branching into all possible moves from every given node. The tree is finite because there’s always a finite number of branches off of every node, and it cannot have a depth larger than P, or it would repeat a board position (I suppose you can include passing as a valid move, and it would make the tree a bit larger, but still finite).

You’d then take all end positions and score them (without komi, just the value on the board). This part may go into trouble; simplest solution would be “everything on the board is alive, territory is empty space adjacent to a single color, everything else is dame”.

Then you work up the tree using Minimax which gives you a specific number for every node. That number is the best score that the player whose turn it is to move can hope for (has a strategy for).

Now, there is a number k for the root node, the empty board. That means, optimal strategy ends up with k points for Black. k, of course, could be positive, or negative, or even zero, but it is a unique integer (heuristically, k is probably around 7, I guess). So k would be called the Theoretically Fair Komi, because it is what you need to subtract from the final score in order to get a tie, if both players played optimally.

Now, this may be a bit convoluted, but the difference with Vsotvep’s proof is that it is technically possible for a computer (or a human, for that matter) to just do this and solve Go. It is just unfeasible, both in time and resources; you’d need several universes worth of everything.

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Good point, so you’ve proved the existence of a construction, without demonstrating that the construction actually works (in practice). A real hardcore constructive mathematician would object to both proofs, moderate constructive mathematicians would accept yours but not mine, and probably most mathematicians would accept both and not care about the difference all too much. :stuck_out_tongue:

To be honest, I tried to close all holes as carefully as possible as a matter of exhibition of mathematical rigour. Professionally, it seems most mathematician would be more than happy with @benjito’s “proof” and leave the details for the reader.

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6o31sa

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Memes are so much better when you need to have extensive knowledge to understand them

(I guess that’s why the Go memes thread can be so much better than 9gag ever was)

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I have some doubts about this part. It’s possible to tie by having the same number of points, but also by making a triple ko. Making a triple ko might be possible for several values of komi.

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A triple ko does not result in infinite play under any of the superko rules.

Note that superko is necessary for the proof to work, since under the Japanese ko rule Go is not a finite game (even using a rule like “no more than 3 repetitions of the same sequence of moves” does not solve this issue, since one can use the Thue-Morse sequence to infinitely keep the game going without ever repeating the same sequence of moves more than twice; see also this nice video about this sequence in chess, as discovered by Max Euwe).

If Go is not a finite game, the proof breaks down at the start, where we use Zermelo’s theorem to claim that the game is determined (one of the players wins or both can draw). In fact, Zermelo’s theorem cannot be generalised to infinite games without making some very strong assumptions about mathematical reality. In particular, one needs a form of the Axiom of Determinacy. It is not at all consensus that this Axiom is “obviously true”, and many areas of mathematics prefer to use a different axiomatic system that includes the Axiom of Choice. Unfortunately, Choice and Determinacy are incompatible with each other, since one can construct an infinite game that is not determined using the Axiom of Choice.

(if anybody is interested, I could go on for hours about these axioms or infinite games, feel free to ask :slight_smile: )

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But under Japanese rules, your proof that there exists a smallest integer X for which White has a winning strategy still works, I think, so that with komi X-1, both players can force a tie. So this proves existence of fair komi under Japanese rules, but uniqueness is not guaranteed.

Or did I miss something?

(And back to the topic: go is better than chess because you can have endless discussions about rules and komi.)

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Well, technically a triple ko is a “no result”, so it is neither a draw nor a win / loss. I’d guess it would be “undefined” mathematically. You could of course call such a result a draw, and then I see no problem with your generalisation. Indeed one cannot prove uniqueness (with my proof at least).

Empirically and intuitively, I feel that probably there will still be only a unique komi for which both players have a drawing strategy, since it seems hard to ‘aim’ for getting a triple ko through strategy. But mathematically this has litlte value, of course.

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Theorem: There is at least one fair komi for every ruleset of Go.

Proof:

This is left as an easy exercise to the reader.

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Many things have been said in this thread with which I agree, like the greater “conceptual depth” and rule-simplicity (“elegance”) of go. In the parts I have read however, I did not see the following idea.

Arguably, Go is a “complete war”, while chess is a skirmish.

This can be exemplified with a kind of do or die attack strategy being more a chess phenomenon. Sure, in Go you can go all in, and in chess a do or die attack quite often is a horrible idea, but it is far more common in (19x19) Go that you get a set of more or less isolated battles as the game goes on. You can attempt a do or die approach in each isolated area, but typically one such attempt that fails will cost you so much that it doesn’t much matter that the other skirmishes are going well.

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Under a suitable formalization of “Japanese-like” rules (for instance “when a position is repeated for the third time, it’s a no-result”, making the game finite again), I believe you can prove something similar to Zermelo’s theorem, you just get some additional cases.

It is still the case that there is a least integer komi for which White can force a win (from White’s perspective, just treat no-result as a Black win and apply Zermelo’s theorem), and similarly for Black.

For similar reasons as before, these two integer komis must differ by at least 2. (in this case, they may also differ by more, since there could be a whole range of komi where either side can force no-result to avoid losing)

For concreteness, let’s assume that with komi 5 black can force a win and with komi 7 white can force a win. Then there are these different possibilities for komi 6:

  • The result will be either draw or no-result, black can force either.
  • The result will be either draw or no-result, white can force either.
  • Either player can force a draw: it will only be no-result if both players want that outcome.
  • Either player can force a no-result: it will only be draw if both players want that outcome.
  • It will always be draw. Neither player can force no-result.
  • It will always be no-result. Neither player can force a draw.

(I think that covers all the possibilities, let me know if I missed any)

And if you have a whole range of komi where neither player can force a win, you could conceivably have multiple of these possibilities come up at different komi.

What remains true is that there is at least one komi where neither player can force a win.

This was not a rigorous proof, I’m just saying that I believe that a proof can be made along these lines :slightly_smiling_face:

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A 25x25 GoBan is a good thing… I mean of course, if you can’t get over to the Little Golem Game Server and play 37x37 WeiQi! :wink:

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go players aren’t paranoid about hidden cheat devices

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Our scandals just arent interesting enough to hit the mainstream media…

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The better handicap system in Go is why Go is better.

On the other hand, you have this in Chess:

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original-f12cc34645177175ed1369fb5c73b42d1080×1349 145 KB

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Go is better than Chess because chess has a chequered past.

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Go has rules. While chess has list of different vehicles instead.

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Go is better than cheese because I can play Go with vegan materials.

Oh wait, this is about CHESS, not CHEESE?

Mh… :thinking: … sry, I’m out then … I like both but prefer Go but that’s just a personal preference (just like with vegan Go gear :wink:).

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If I may be allowed to say, in my opinion, the difference between Chess and Go is the same as between boxing and jujutsu. The goal of a boxer is to knock out an opponent, and this can be compared to checkmate in Chess. The purpose of the jitser is to gain a dominant position in the fight, and this does not require punching the opponent in the face. This can be compared to Go, where the task is not to destroy the opponent, but to surpass him, even if only slightly.

I hope it’s clear what I mean. I hope we don’t start figuring out which is better, boxing or jujutsu. :wink:

Peace and goodness to all.

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Yes. But i think this is 1 difference only. There are more.

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