Combinatorial Game Theory
who wins, and by how much
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The thread: It has to end — page 2

Every value here is defined by a recursion that needs play to stop. Sometimes that is obvious, sometimes it is a theorem, and sometimes the game ends with nothing bounding when.
A game every play of which ends, and no round settles. A game whose first move chooses how long the game will be, cut off at several sizes. Every play of it is finite and no position is drawn, so the fourth outcome class has nothing to do with what goes wrong. What goes wrong is the round counter: the opening is a loss, a loss settles only when the last of its options is known, and there is no last option. Cut the game off larger and the round grows, so no number in the column is the answer for the untruncated game — and the induction that labels it has to run past every finite stage. How it was found

Every play ends and no round settles

Take the finiteness hypothesis away carefully — not by adding a cycle, which has already been priced twice, but by adding infinitely many positions to a game every play of which still ends. Nothing is drawn, every line finishes, and the round the opening settles in grows with every cut: two, four, six, eight, twelve, sixteen, and no number in the column is the answer.

 A shuttle and a loop, judged by what the play returns to. A three-node loopy game drawn as a graph, with Left's moves in blue, Right's in red and position a marked. Beside it, each position-and-mover pair under the backward labelling and under the rule that a never-ending play goes to Left when it returns to a infinitely often. Four pairs are drawn by the labelling; the new rule gives two to Left and two to Right and leaves the decided pairs as they were. How it was found

What the play keeps coming back to

A draw is what the backward labelling never reaches, and handing every never-ending play to one player turns the draws into wins wholesale. Judge an infinite play instead by what it keeps returning to, and every draw gets a winner of its own: over the 262,144 three-node games, 15,432 send some of their draws to one player and some to the other, which no wholesale rule can do. Finding those winners takes a fixed point inside a fixed point.

 A hub with two spokes, and the bit of memory it needs. A three-node loopy game in which Left, at a hub, chooses between two spokes and Right must return from either. Left wins a never-ending play that passes through both spokes infinitely often. With one bit of memory recording which spoke is owed, Left wins from the hub; with a strategy that depends only on the position, Left always takes the same spoke and loses. Three position-and-mover pairs change hands. How it was found

One bit of memory

Judge an infinite play by whether one position keeps recurring and every winner can play from a table of one move per position, with nothing remembered. Ask for two positions to keep recurring and that stops being true. At a hub with two spokes a player has to alternate, and a table cannot alternate: over every three-node game, 49,487 position-and-mover pairs are won with one bit of memory and lost without it.

 Right, wrong, and right again. A 4 × 5 Domineering position with Right to move, which Right loses, beside what a search cut at each depth from 0 to 9 says about it when it guesses that the player with more placements wins. The guess alone is right, a search one move deeper is wrong, and every deeper search is right. What it costs

A verdict that changes with the depth

A who-wins search of 4 × 5 Domineering cut at a fixed depth, guessing that the player with more placements wins where it stops, is right about 72.5 per cent of positions at depth 0 and about every one of them by depth 7. On the way, 4,697 positions are right at one depth and wrong at a deeper one. With a guess that knows nothing, going one move deeper makes the search worse — and its errors alternate in kind with the parity of the depth, so that half its verdicts are proofs.

 Two depths that agree. A 4 × 5 Domineering position with Right to move, which Right wins, beside what a search to each depth from 0 to 6 says under the guess that any mover wins. The verdicts alternate until depths 2 and 3 agree, which certifies the answer 3 moves before the longest line. What it costs

Where a search may stop

A search deepened until two consecutive depths agree carries a proof of its verdict, and on 4 × 5 Domineering it stops before the longest line on 17,589 of 48,670 positions. It also costs three times what the search that simply finishes costs. The rule that pays is the other one. Search on wherever the two players' counts of placements are within one, and at depth 2 the wrong verdicts fall from 2,140 to 86 for about a quarter more work per search.

 Two ways to search on, one position. A 4 × 5 Domineering position with Right to move, which Right wins, beside what deepening says at each depth when it declines to guess where the counts of placements are close. Searching on one move at a time, depths 0 and 1 agree on the wrong verdict; searching on two moves at a time, the search stops at depth 3 with the right one. What it costs

Search on in pairs of moves

Deepening until two depths agree gives a proved verdict, and searching on where the counts are close gives a better one; put together the obvious way, they stop on a wrong verdict at 3,231 positions of 4 × 5 Domineering. A guess one move past the cut has the other player to move and flatters the wrong side. Searching on two moves at a time keeps the proof, and the window that suits it is one-sided — but however it is widened, the certificate gets cheaper only by turning into the search that finishes, and on four boards it never gets below it.

 Every two-position loopy region, as two names. The 256 loopy regions of two positions, placed by the names of their onside and offside as identified against the 1,474 values born by day three. Ten names cover every side: 0, 1, −1, ∗, on, off, over, under, upon + ∗ and −upon + ∗. The largest groups are on & off with 94 regions and off & off and on & on with 53 each; 25 regions need only finite names. How it was found

A loop is written with two names

A region with a cycle in it has no brace expression, and every one of the 256 regions of two positions can be written anyway — as two names, the game it is when a play that never ends goes to Left and the game it is when it goes to Right. Checked against all 1,474 values born by day three, ten names cover every side, 25 regions need only finite ones, and the pair predicts every sum with a finite game, draws included: a draw arrives exactly where the two names disagree.

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