What size of hexboard makes the swap rule unuseful ?

Hey guys,

The title says it all : the swap rule is important in 11*11, 13*13, but what about 19*19 ? what about bigger sizes ?

THanks by advance for your help.

Neither size makes the swap rule unuseful.

The swap rule is to compensate first player advantage (there is a theorem that 1st player has a winning move).
Do you say on bigger boards he has less of an advantage ? Maybe on infinite boards it is the case but how to quantify it for 19x19 ? It must still be there.

It depends how you define "not useful," but by any reasonable definition you need extremely large board sizes to make it not useful. For example, if you play 19x19 without swap, KataHex thinks it has a 95% win rate for the first player in self-play, which is a massive first-player advantage. Top humans are much weaker than KataHex, but I'd guess that it still translates to a 75-85% win rate on 19x19 for strong humans (against an equal opponent). KataHex thinks the first-player advantage is 85-15 on 27x27, and 70-30 on 37x37 (although I don't think it was specifically trained on 37x37). If your definition of useful is "without it, the game is still practically fair in that the first player doesn't tend to win more than 55% of the time against an equal opponent," I would say the cutoff point is roughly size 50 for a superhuman bot like KataHex, size 37 for top humans, and size 25 for intermediate players.

Thank you guys for your explanations. I clearly overestimated the importance of the board size.

This is a little bit of a stretch, but Hex is in some aspects very similar to Go. And in Go, the komi (compensation for having 2nd move) is the same, irrespective of the board size.

Well, the "komi" for Hex is "0.5 of a stone" on any board size, because in Hex without swap, going first is fully equivalent to going second where your opponent passes for a turn (so 0-komi is exactly as advantageous for Red/Black as 1-komi is for Blue/White) :)

So I think the relevant question is at what board size does "half of a stone" stop becoming an overwhelming first-player advantage.

Agreed, there is exactly same reasoning in Go, using AI it shows that value of a move there in the beginning is around 13 points which makes half of that fair value of komi. Apparently in Go, the value is still similar irrespective of board size (of course without taking into account some ridiculous examples of board of very small sizes, where first move would basically win the game).

What others said already, but to provide another level of intuition, I would probably lose 100% of games vs KataHex on 19x19 without the swap rule and lose "often" on 13x13 despite the first move advantage.

Hex is a richer game than I had thought before Kata existed, the resources it finds are out of reach for the current human level.

Sorry to be pedantic Arek, but are you saying that before Kata, you thought humans could plumb the depths of even 19x19? Or did you think computers could not exceed humans?

I was completely aware of human deficiencies and I had no doubt computers would far exceed human strength, but I imagined they would kick our asses by applying similar patterns that we were already familiar with. I was wrong in that, when analysing positions with Kata I have seen minutiae that I considered irrelevant, or areas that I thought were dead, being sometimes decisive for the end result.

I don't know if my answer is satisfactory, but I find it hard to explain. I will try to find an example.

It has been similar in Go few years ago, AI started playing moves that totally rejected parts of theory established by human and broaden perspective of what is available to play. It was funny, because it has shown, that there are more good moves that we can choose from, not less. Funny, that there were also old patterns used by humans that AI totally agreed with.

Ok, let me make a quite a vague example, but I don't have anything else at the moment.

If you consider the joseki played in the picture, I used to think that the last move (g12) is never ever an option. In an analysis with Kata I have seen this move to be a very important resource, because if black chooses to play "to the left" g11 then it can no longer threaten with L8

And if black chooses to play h11 then white has theoretical possibility to utilize that tiny g12 advantage.

I am not able to create a "working" example, but Kata really squeezes water from a stone with a variety of little seemingly meaningless moves.

@hexanna This is why 53x53 is so awesome. You can play a random first move every time and never run out of new openings.