Hi all :)

I’ve a proposal on how to improve the swap rule: One player draws one position over the board with how many red and blue pieces as he wants. Let’s suppose he uses N total pieces to draw the position. The other player can choose between playing as red or playing as blue or to draw another position with at most to N-1 total pieces and hand over the 3 choices to the other player. At every step N decreases by some arbitrary amount. Obviously, if and when N will be 0, there will be no longer the option to draw a new position.

This seems perfect swap rule because it’s almost completely symmetric and It’s guaranteed to terminate. Moreover, this does not create any problems for hex players that are using the current swap rule, because those knowing how to play N=1 can always go back to that case without waste of clock time or any effort. It would also make easy to set up games with handicaps. Am i missing something?

Endorsements? Criticisms? :)

I've forgot to say, each position drawn on the board comes also with a fixed color with the right to move. I'm used to playing hex 11x11 on iggamecenter.com and i was just assuming that blue was the color with the right to move. It also goes without saying, this proposal brings tangible advantages only for “small” boards. But even 11x11, as far as i know, is not so as “small” as it seems.

Introducing a size-cap for N should be worth considering. I see people playing 11x11 on 19x19, because they like smaller boards more or something. Not sure if theres a problem with that but it seems kind of weird.

purgency, do you've understood my modified swap rule? in my proposal N is not the board size but the number of “stones”/“pieces”. On 11x11 board, for example, my N is capped by 121 (11 * 11)  at the first round, but then it decreases at each round. Maybe i should have user another letter instead of capital N. Let's use K if you prefer.

Let me give a sample imaginary game so you get the idea. There are two players, Alice and Bob. Alice puts K0 stones, some red and some blue, with the understanding that blue has the right to put the next stone on that position, and then asks Bob, do you want to play hex on this position with a color of your choice or you prefer to draw another? Bob says, no, i think you've studied this position too much at home, and i want to draw another and pose you the same question. Alice says to Bob, ok, but you must draw a position with K1 stones, with K1 K2 > K3 … until one player accepts the position and picks a color for himself. Is this all clear now? :)

Ignoring the issues involved in actually implementing this on a server, I could draw a position familiar to me, which I have studied. If my opponent plays this position, the “game” that results may be less of a challenge to my ability than a memory test.

yeah, i understood it perfectly. what's your point anyways. 361 stones are not enough to turn a 19x19 board into 11x11? because that possibly happening is all i said.

David, the point is your opponent will not allow you to draw a position where home preparation plays an important role. But perhaps your objection is that it's hard for one to estimate if his opponent has special preparations for a given position…

purgency, ah you said they may on purpose “downgrade” the game into a smaller board? i wouldn't say that this a defect?!

and drawing a smaller board is not advisable because your opponent will likely know how to play it and it'll have the theoretical win at hand…

yeah right, it's not a defect. just as i said “Not sure if theres a problem with that but it seems kind of weird.”

i find it weird, thats all. not criticizing at all just providing something to think about.

Hex is weird anyway :)

“those knowing how to play N=1 can always go back to that case without waste of clock time or any effort.”

Without any effort? If my opponent begins this process with N>1, then if I just want the plain swap rule, I must reject the position.

The only way I could be certain to use the swap rule would be when I place the first stone. From my perspective, this makes the

game less symmetric, not more. At least, with the plain swap rule, my opponent would place the first stone about half the time.

David, if opponent starts with many stones (N much larger than 1), you reject his proposed position and draw a position with a single stone. Then either he accepts it, and you're playing normal hex with normal swap rule or he must draw a position with zero stones, which is the same as saying playing hex without any swap rule at all, and with you going first…

Isn't that what I said?

David, well, in my swap rule there is trade-off between theoretical win and having more home preparation. You can't have both theoretical win without giving more home preparation, and you can't get home preparation without giving theoretical win. I think this is the fairest.

This change doesn't alter the theoretical win for Player 2.  Regardless of whatever position Player 1 draws, it is either a winning position for white or a winning position for black.  If player 2 chooses the right color and plays perfectly, it's a win for them.

Interesting proposal, but here's a potential criticism. Suppose that Alice and Bob are both good enough that with N=1, the second player has a big advantage. Then when N>=2, the first player (say, Alice) has an advantage: she can study a difficult 2-stone position really well at home, and then put Bob in the uncomfortable situation of having to choose between a 2-stone position Alice has studied, and having to place an N=1 stone (giving Alice an advantage).

Shalev, well, if Bob can play perfect N=1, then he'll will be capable of at least approximating perfect play on N=2. And he'll start that game with a theoretically winning position. Overall, this is the best that a swap rule can provide, i suppose…

Knyx, yes but that is a problem only if player 2 can solve over the board ANY position that can show up in a board of the given size. This is harder to do than being able to solve all 1 stone positions. And this task is known to be hard in theoretical computer science. To sum up, the goal of the proposal is to squeeze the max out of the game of hex, not to invent a new game.

I spent 10 min trying to come up with some 2 stones examples on 11x11. Let's see what people think about this http://www.trmph.com/hex/board#11,f10f2 with black to play freely. On first sight, I've absolutely no idea about on who has the theoretical win here…

Lazyplayer, do you propose this variant of swap rule, because regulary swap rule has been exhausted for you?

It seems to me that if both players has their own home preparations, they probably will understand it and won't let the opponent play it. So it will soon be decrased to usual swap rule. Or may be to some interesting 2 stone position, like you have mentioned.

Anyway it's worth trying, and probably possible to do something like that on IGgamecneter.

JKB, on 11x11, it would seem almost all interesting moves win as first move with relative ease. But i believe if the other side would play perfectly, then it would be able to win some of those position that seem “obviously” lost to my (or our) eyes.  As it is now, i don't “need” this on 11x11. But i think it extends the scope of the game no matter what board size. Because every time you refuse to play on the position that hs been brought up by adversary, you're giving up the game theoretical win. So ambitious people would not do this in entirely automatic way…

I would say in theory it depends on the clock used for the game. The more time in the clock, the more we should be for willing to understand the position that has been brought up by opponent and beat him on it…

In theory, this sounds like a reasonable swap rule to me.  (In practice, it might be too cumbersome for some players.)

The following is obvious, but I'll say it anyway: The issues here are not really specific to Hex.

 I've thought about this in the context of Hex-like games where the usual 1-move swap rule doesn't work.  (Example: when you play on a torus all first moves are equivalent.)

A simpler alternative is just to fix N (and perhaps even specify k red stones and N-k blue stones).  A good value of N is one which is as small as possible but which still allows for many N-stone initial positions which are difficult to analyze.

I find this a painful sounding rule, one that tends to favor the player who memorizes the most positions–not what I want out of a game. Do you all think that the current rule is inadequate for Hex?

Ignatius J. Reilly makes a good point about the value of this for other games (e.g., toroidal board).  And maybe it would help with a game whose openings have been analyzed to death, but that isn't a problem with, say, 13x13 Hex; and if it ever got to be a problem, I'd rather increase the board size than add this rule.  Just personal preference–I like how Hex starts with an empty board.

hyperpape, i think the current rule is adequate, but it's not beautiful because it's somewhat arbitrary. With my rule the reward for opening preparation is at most equal to the reward for going as 2nd player in a game with normal swap rule. The only source of troubles is when one player is over-optimistic on his ability to play perfectly in a position that has been prepared at home as a trap by his opponent.

Ignatus, i agree. My knowledge of geometry is terrible so i don't know what a torus is or how to visualize it. How do you visualize the torus?

Tom, i agree that currently there is no need for this on 13x13. In fact i think not even on 11x11 this is needed. Maybe it's not needed anyway indeed! The real goal of this is mathematical elegance and beauty! :)

On a rectangle or a rhombus grid, when you glue together left and right borders and top and bottom, you get a torus.

You must usually also change the winning condition, for example complete a loop…

I like your swap rule.

I guess that as for hex on the projective plane (where you glue the edges reversed), the loop must cross the border an odd time to make a big loop, see http://sagme.blogspot.fr/2007/03/bill-taylor-this-article-is-chiefly-for.html

Caroll, thanks, i also like it, but i think Caroll's objection has some merit. In essence, the problematic situations are those where the normal swap rule is unfair for first player but it would still lead to games that are worth playing. This is similar to chess, which is unfair for 2nd player, but is still an interesting game to play. In this case, with my proposal, players would try to avoid reducing to N=1 and therefore they would be encouraged to have a large list of opening traps. And this could be considered worse than the problem it was supposed to solve…

I like the proposal and I would prefer to play this swap rule given a choice. According to this exact proposal I have 2 remarks:

1) If the rule was considered to be introduced I would suggest, for simplification, to restrict the N to a “good value” (quoting Ignatius).

2) I suggest a change:

Round 1 of negotiations: player A picks a position with N1 stones and hands over the choice, then player B can also pick a position of N1 stones. In the next rounds N(k) < N(k-1).

Certainly there are 2 sides of the problem. First is that we want to make a game perfectly fair to all players. The other is we don't want to pay a cost of complication to rules. In the face of fairness there are even better options than posed in this thread. For example this:

There is no limit on numer of stones in a pattern. Player A selects an opening pattern and both players are forced to secretly pick their favourite side. If they pick the same side the negotiation proceeds so that player B selects a pattern etc. The actual game starts only when the players pick opposing sides so that they both play their preffered color.

I don't know if this approach is practical, what do you make of it?

Arek, i think your approach of both players agreeing is worse than mine, because there is a larger reward for memorization. But your proposal of decrementing N only after two turns seems an improvement. I think fairness is welcomed but is not essential. Chess is playable even if unfair. What's essential is that 1) the game is not trivial to play perfect and 2) that in some sense it is interesting and beautiful to play. For hex, we can apparently always ensure the game is not trivial by increasing the board size, but at the expense of point 2. My proposal was trying to improve this.

And i would say it's only moderately successful at this, because anyway it rewards boring memorization…

Another proposal, let's call it Lazy's 2nd proposal, is this: One player at random is asked to propose a number N. Then N stones of random color are placed at random in the board. Then players are asked to secretly pick a color. If they secretly pick different colors, the normal game can begin. Otherwise this procedure is repeated. This ensures perfect fairness and minimizes value of brutal memorization. The cost is that we added randomness and secret moves. Many players may dislike these, especially randomization…