I saw this video of a talk by Katherine Scott about calculating the values of loony endgames.

http://www.msri.org/publications/ln/msri/2000/gametheory/scott/1/banner/24.html

(video link on the top left)

Not much use on 5*5, but still interesting.

She wrote a paper with Berlekamp about this stuff. There is also her MSc thesis, which as far as I know is only available if you have access to the UC Berkeley library. I was in Berkeley over the summer and looked at the thesis—most of it was independently discovered by me (see my “technotes.pdf” at wccanard.wetpaint.com), although of course she did it much earlier; it's mostly just computing the values of endgames. But the stuff with Berlekamp (which is what the video is about) is pretty cool (and the results in that paper certainly weren't independently known to me—indeed I learnt them from reading the paper). Her analogy is that keeping control in a dots and boxes endgame is like a plane flight where you want to keep the plane from crashing. Although I've never really been able to use the precise results she proved with Berlekamp in practice on a 5x5 board, the analogy is still pertinent; the principle says (vaguely) that if you want to win a game in which you're in control, you need to make sure that *all* the chains (not just the total length of the chains) are long enough. Otherwise the defender can start opening the shortest chains early and hurt you more. As a concrete example, if you're two boxes down and have control and there are growing chains which have yet to grow to their full potential, and if each of them already have length 4, you're in good shape (by strategy-stealing) but if one has length 5 and another length 3 you might not be (opening the incomplete 3-chain might win for the defender).

thnx for the link.

Is the paper you are refering to, the one included at the More Games of No Chance volume? (21. Forcing your opponent to stay in control of a loony dot-and-boxes endgame, by Elwyn Berlekamp and Katherine Scott)

Yes. In that paper they don't discuss the algorithm Scott uses to colour the board but they explain in some depth how to fly the plane in the regions that ended up being coloured hazel in the algorithm she explained in the talk. This particular aspect of the theory never shows up on a 5x5 board. The kind of situations they're interested in are so-called “bracelets”, which are lots of loops all connected by long chains. The point is that you have to cut the chains in the right order to expose the loops at the right times. It's an interesting paper but the actual results about bracelets don't help on a 5x5 board.

As you know yper, there are two basic kinds of games that show up on a 5x5 board, and most (but not quite all) serious games played on this site fall into one of these two categories: first there are the games where 1 player wins the chain battle and where it's worth keeping control (i.e. there are more long chains than loops, or perhaps no loops at all). In those games the controller stays in control and hopes he hasn't sacrificed too much on the way (he might have lost a couple of boxes making sure no extra chain appeared). The second type are the games where the chain battle turns out to be worth very little (perhaps because there are lots of loops, even just two would be enough if they're small enough) and then play becomes messier; the winner of the chain battle might not win the game.

A 5x5 player can take the following away from the Berlekamp-Scott theory. It applies only to the “first type” of game above: they are interested only in games where the “controlled value” is >=0. The actual value of a position is always at least the controlled value and they give a criterion for equality. (equality will never hold if the controlled value is negative, that's why the ideas don't apply to the second kind of game). The criterion is in the form of an algorithm for the player not in control—what he needs to do is to make sure that it's never better for the controller to let go of control. So he wants the controlled value to go up (take-off) initially so should open the quads and 3-chains initially. Then he can move onto the longer stuff whilst he's arranging for the landing.

As a concrete example, imagine all that's left is a quad, a 3-chain and a 10-chain, and you are 7-1 up, and it's your move. The point is that if you open the 3-chain and the quad [ascent of the plane], your opponent will keep control to get the 10-chain but will find he's lost 13-12 at the end of it. On the other hand if you open the 10-chain first you crash the plane because the controlled value is only 5 and opening the 10-chain sends the plane plummetting down 8. And surprise surprise, if you open the 10-chain your opponent, who spent all that time winning the chain battle, will take all 10 boxes, open the quad, say “thanks” and win 14-11.