Pie Rule/ Discussion

DocHoliday -- Strictly speaking, one person cutting the pie and the second person choosing a portion is non-optimal. For there to be no chance that one person will feel cheated, the pie rule should be as follows:

• First person cuts pie into two portions;
• Second person makes adjustment (cutting a small piece from one portion to add to the other) so that she feels portions are equal;
• First person chooses one of the portions.

The first person gets to choose, so she shouldn’t feel cheated. The second person adjusted the portions, so she should feel that both portions were equal. To carry this over to Go, the first player places a black stone, the second player moves the stone if she feels that is required, then the first player chooses if she wants to play white or black.

ilan: I don't believe that this is at all necessary, nor do I understand what is meant by "strictly speaking." In particular, the extra steps in your protocol merely make the second person into the first: the second person adjusting the portions is equivalent to the second person making the first cut (similarily in your go example in which the second player gets to make the first move).

Moreover, you do not specify how this adjustment is made, since there are various models of adjustment, including the "moving knife." There are at least 2 books written on this subject of which I have read one. By the way, J.H. Conway who is responsible for "mathematical go" also discovered the "envy free" protocol for cutting a pie into 3 pieces (where no one feels someone else got a bigger piece).

blubb: If you replace "the second player moves the stone" by "the second player places a white stone", DocHoliday's idea makes much of sense to me. While it may be not so clear if one single move can provide fine enough an adjustment, two moves can for sure. Probably, the latter is already even better than is usual komi. I suppose, a number of initial moves bigger than two mainly would increase needless peculiarity only, rather than fairness. Hence two moves seems perfect to me - one stone of each color. (I have to admit, I particularly like this because of color balance reasons, too.)

->---(later)---<- Errr, sorry, I missed something here. The number of initial moves needs to be odd. Else black could play straightly (4,4) etc., and there would be no way to white to catch up.

Bart Massey, unrated rank amateur -- A potential problem with the pie rule in Go, where the range of fuseki is so wide, is that it may be an advantage to cut rather than to choose. A player with a bunch of experience with some opening move may be able to play it better than the opponent whether White or Black. In other words, the person who cuts still gets to make the opening move, and the person who chooses still has to respond. It's just two moves in a different game.

Notochord: Or how about a strong advantage to choose over cutting? Both tasks seem to require determining the value of the cutting position, but the cutter also must be able to actually find a position that is truly equal (either theoretically, or more practically). Picture a pie rule applied where both participants have jittery hands. Even in a continuum, the cutter would have to spend an eternity of computational time waiting for his hand to come exactly over the line where he plans to cut the pie. In practical time, he could produce only a pie that's approximately equal, but which gives an advantage to the chooser. The cutter would probably have to have memorized quite a lot of opening positions, but given the nontriviality of the problem of actually finding ones that nullify the opening move advantage (assuming that we use the rule as a replacement for komi) doesn't seem to mean that he will have a lot of variety to spring, unless he, as you said, produces a specific trademark position that's pretty clearly unequal, but which gives an advantage to himself in practical terms through memorized tactics. Personally, I have never thought that the pie rule produces particularly elegant positions, since it throws in intentionally nonoptimal play into what is supposed to be a mutual honest struggle between the two color. I prefer most of the time that the game begin from an empty board. Maybe better than a plain pie rule would be to play zero to a few moves into the game, then auction off black. This has the advantage that we can explore almost any board position, since komi is determined to fit, and that both players participate in making the initial board. Rules like these, however, generally shunt off a bit of skill from 'the actual game' to 'pregame stuff'. Maybe better to use fixed or auction komi (if you really must), since it seems like it would a lot simpler in the pregame.

SiouxDenim There is nothing special about making the choice after the first move. On 19x19, why not have Player A play the first 10 moves for both colours? The arguments about balance still apply. I suspect the latest the choice could be allowed is at the time of the first contact play. A komi auction can still happen at the choice. My personal view is that shedding all of the opening advantage fairly in one move is unlikely, but something close to it might be possible in 5 or 6.

uxs While letting one player make the first 10 moves may make the game more even, it would also make it a different game than normal Go. I don't think I would like to play that way.

TDerz UXS, it is and was still normal Go!
Check here http://go.yenching.edu.hk/korhis.htm on the history of Korean Baduk.
This diagram shows the ancient Korean Baduk starting position.
Can you imagine that it leads immedeatly to more fighting?

ancient Korean starting position  

Perhaps it lead to even more fighting than in the initial Chinese board set-up below:

ancient Chinese starting position  

ilan: From a theoretical standpoint, the pie analogy cannot be applied to Go. The reason is that pie division is a completely continuous problem which cannot be solved if there is a discrete element. In other words, if the pie dividers believe that there is an indivisible part, then they may not be able to agree, for example, if there is a cherry on top that both want but both cannot have. Since Go is completely discrete, and does not have any continuous element, one sees the limitation of the analogy.

sig?

in practice you can treat the game tree as a continuum

blubb: However, both aspects you mention (as I understand them), don't affect the practical usability of "pie komi" so much. The first (invisibility) one doesn't, because the obscure part remains unchanged from the moment of cutting to the moment of choosing. And the second one doesn't, because the "19x19 Go pie" is sufficiently fine-grained (at least, I think so). Moreover, there are discontinuous dividing problems the pie rule can be applied to. Take for an example, two identical coins in a basket. I guess 19x19 Go doesn't belong to this kind of "even" problems, though.

ilan: In fact, the pie problem is completely different in nature from what is being discussed here. In the pie problem, one is dealing with two people and their personal preferences, independently of an objective aspect. If this model is used in Go, then my comment about atomic pies easily applies, for example if both Go players believe that playing on the centre point is the winning move for black and all other moves lose for black. For this combination of player beliefs, there is no fair pie rule. Now, if this belief is actually The Truth (maybe in 5x5 Go?), then once again, there is no fair pie rule, this time in the formal sense.

sig?

You're missing the point of the "pie rule": if there's a cherry in the pie, it's about splitting the pie so that it's obscure where the cherry is. So it's not about The Truth but about practical evenness. The Truth about komi isn't even either. In your center-wins example, cutter would NOT choose the center point because it would immediately hand over the cherry to the chooser. Even though all other squares would be "losing", pie rule would suggest cutter take the least obviously losing one. This is the most even you can do, keeping the rest of the game intact. In practice of course the situation isn't symmetric since players have different knowledge of the game tree. Cutter would then choose a "losing" opening move that she's studied very thoroughly.

blubb: Right, the bigger the Go board, the better the pie rule usually applies. 5x5 Go with the "cut" consisting of one initial move seems to be neither exactly "pie-solvable" nor sufficiently "pie-approximable". I suppose, also 3, 5, 7 or 9 initial moves couldn't make it fine-grained enough to be fair.However, on big boards the pie rule can work very well - even better than normal komi. (How big boards have to be for that? And how many initial moves are needed on a 19x19 for that?)

ilan: I believe that my above argument shows that the pie rule has no formal applicability to Go. This is also true of the usual komi for unsolved board sizes, however, those values are based on empirical data so are significant for practical play. Since there are no such statistics compiled for the pie rule, it does not have similar significance. Also, the number of moves played is irrelevant: If one chooses an initial move with theoretical value closest to zero, it doesn't matter how deeply one goes into the line leading to this value.

blubb: If we assume that the pie rule doesn't balance 19x19 Go in general exactly, three initial moves can approximate equality at least some closer than can a single one. On a 5x5 board something around 23 moves even allows exactly balanced positions: neutral sekis (move 23 is a pass). Chances are, that there is also a pie-rule consistent path leading to such a seki.

ilan: OK, you are correct, but this shows that application of the pie rule is not about each side playing the move which maximises his score, but both sides cooperating to get an even score. A totally different conception of game playing.

blubb: Huh? By "there is ... a pie-rule consistent path" I mean that each side plays in a way which does maximize it's score, given the pie rule. Do you disagree about that concept, or do you doubt there is at least one such path at any unsolved board size leading to a sufficiently neutral position? (I am not sure about the ladder, either.)

ilan: Yes, I understood you. Just call me ilanpie.

blubb: Ok, I'll do. :o) . . . Anyway, further thinking about the pie-path question, I encountered a problem with setup moves' numbers bigger than one: The choosing party gets influence on the way of cutting. This tends to undermine the balancing effect of all setup moves past the first one, particularly if it's easier to find smallest or biggest than most neutralizing moves.

Let's see what's supposed to happen to deviations from equilibrium (white is the chooser):

• If the setup favors white at any time (e. g. right after S1 because it's too small), white can continue as strongly as possible and black then has to answer as strongly as possible. White will choose to stay white in the main game.

• If the setup favors black at any time, white can continue playing as small moves as possible, forcing black to do the same. White will choose to exchange colors for the main game.

In both cases, the given imbalance is (the average of what is) expected to remain as profit to white.

So, white is interested in making the setup most unbalanced and will try to do so with every (setup) move, while black has a rather hard job to keep it balanced all the time. A single move setup seems to be fair to the cutting as well as to the choosing party, but every further pair of moves seems to increase the advantage to the chooser.

To avoid this, one could introduce a modificated pie rule, saying: If the cut is conducted by both players, then colors have to be chosen by none of them, but random. Similarly (though not equivalently), black could get the power to secretly decide in advance if or not to inverse white's choice. Also, the chooser could be random determined.

Using these variants, strong players can create very accurately neutralized setups (even better than with auction komi). Moreover, they eliminate the need to restrict the setup to odd move numbers. However, their balancing effect is less reliable, because it's based on players' (questionable) aiming for security rather than on score maximation. So, after all, the one-move pie-rule may be the best for practical purposes.

hk: There's no question that the pie rule works well on discrete games. It's been used in Hex since the fifties, as far as I know. The question is whether you need it in Go. You have komi. A pie rule turns a game into a subtly different game. If you used a pie rule instead of komi, I bet the typical first move would be very different from what it is today! A danger with multiple-move pie rules, is that the "splitter" can set up a position he's studied extensively, so that he gets an advantage whatever side he plays. This is relevant in Hex, since a bad second move can easily lose you the game. But I do not think it is the case in Go. If you should for some reason choose to abandon the fully functional tradition (komi) in favour of a pie rule, I think more moves before the choosing will only make a situation more balanced.

MF: I am new to the game of go. But what about this idea: First player chooses first move and komi (positive or negative) and second player have the option swap colours. A conservative player would have the option of playing a standard game at least half of the time (when he is the one cutting) by playing a optimal move but adjusting komi. A non conservative player could impose a strange opening half the time (when he is the one cutting) by playing a suboptimal and offering a small or even negative komi. But all the player would need to be confortable with standard and strange opennings because they only cut the pie half the times. The person who cuts have only the advantage of forcing the opening, but only one move so the advantage shouldn't be huge. Moreover I think this could make things interesting even on a small board like 7x7 or even 5x5.