Combinatorial Game Theory
who wins, and by how much
The collection

Every essay — page 17

One idea per essay, ordered so that the earlier ones set up the later ones — but nothing here depends on being read in sequence.

Where it stops

Misère play and loopy games: the two places the theory itself gives way — values that stop composing, and a recursion with no bottom.

on + off: what the backward analysis settles. A position graph in which the moves can lead back to where they started. The labels are the order in which a backward analysis settles each position, starting from the ones where a player has already run out of moves. Positions the analysis never reaches are drawn — and there is no test for that; being unreachable is what a draw is.

One part that never ends

The game called `on` has one move and it is back to itself. Add anything to it — a star, a point, its own mirror image — and the whole board is drawn. So `off` is exactly the negative of `on` and their sum is not zero, which is the group law failing for a reason that has nothing to do with who is winning.

9 figures · Loopy The mirror strategy, and the ending that punishes it. A position beside its negative and the sum of the two, with the outcome under both endings. Under normal play the sum is worth zero every time, because the second player answers every move with its mirror image. Under misère the same answers are available and the same player runs out last, so every one of these sums is a first-player win — there is no zero, and no subtraction.

Misère play has no negatives

Put a position beside its own mirror image and answer every move with the mirror move. Under normal play the answerer wins and the sum is worth zero. Under misère the answerer still has every reply and loses because of it — so there is no zero, no subtraction, and no comparison, which is why the misère theory had to be rebuilt rather than adjusted.

7 figures · Misère play The genus of Kayles ·77, heap by heap. One row per heap: the genus symbol, the misère outcome it implies, and whether the symbol is one a Nim heap has. A game all of whose positions are tame is played in a misère sum exactly as Nim is; a single wild heap ends that, and the normal-play Grundy value gives no warning of which heaps those will be.

Tame and wild

The genus is a Grundy value with a tail — the misère values of the position with 0, 1, 2, … heaps of ∗2 added — and a game is tame when its symbols are the ones Nim heaps have. Computed here for seven games over heaps 1 to 14: Kayles goes wild at heap 5, Dawson's chess at heap 9, the octal game ·6 at heap 7, and heaps 3 and 11 of Dawson's chess are both worth ∗2 under normal play with only one of them tame.

8 figures · Genus What the two outcome classes of the parts settle. For each pair of outcome classes, the set of outcomes the sums actually took. A cell with one letter is a pair of classes that decided the answer; a shaded cell with several is a pair that did not. Both conventions have ambiguous cells — the difference is that normal play repairs them with values and misère play has nothing to repair them with.

Two misère outcomes are not enough

Knowing who wins each part does not say who wins the sum. Over 676 sums built from a pool of twenty-six positions, nine of the sixteen pairs of outcome classes settle the answer under normal play and not one of the sixteen settles it under misère — and the nine that work are theorems about a value being zero, which is exactly the thing misère play does not have.

8 figures · Misère play Top Entails, one heap at a time. Each heap with the outcome of playing it alone, the Grundy value an ordinary solver would give it, and the moves that win from it. Taking the top coin of a heap forces the opponent to answer in that heap, which is a kind of move an ordinary sum of games has no room for.

A move that must be answered

Every argument about sums assumes the parts are independent: a move in one leaves the others alone, and the reply may go anywhere. Top Entails denies it — take the top coin of a heap and the opponent must answer in that heap. The nim-sum then misreads 9 of 36 two-heap positions, and two heaps of two coins are a first-player win, which no impartial game the theory covers can be.

8 figures · Entailing a loop with a way out under three rules for never ending. One graph, one labelling, and three ways of reading the residue the labelling never reaches. A draw is not a computed outcome here — it is what is left over — so declaring infinite play a win for one side is a legal alternative that costs no extra search and changes who wins.

When never ending is a win

Retrograde analysis labels a position a win when somebody can force the opponent to be stuck, and leaves everything else blank. Calling the blanks draws is a rule from outside the game — and two other rules are available. The labelling does not change under any of them; only the residue does, and on a three-cycle that residue is every position on the board.

6 figures · Loopy Kayles ·77: what each heap may be replaced by. Each heap with its genus, the Nim position carrying that genus, and the Nim heap a reader would substitute from the normal-play value alone. The two columns agree except where the genus belongs to no single heap — and there the second one is wrong, in sums, by exactly the amount the census counts.

What a tame heap may be replaced by

Calling a heap tame is only worth anything because a tame heap can be swapped for a Nim position with the same genus in any misère sum. The swap is not always a single heap: Kayles' heap of eight is worth ∗ under normal play and carries the genus of 2 + 3, and substituting ∗ instead gets three of the twenty-eight Kayles pairs wrong.

7 figures · Genus What the outcome of a loopy sum can be. One row and one column per loopy outcome class, and each cell lists every outcome a sum of two such positions was found to have. Most cells hold several. The cell where both parts are drawn holds one.

The one outcome that adds

Finite outcomes do not add: two first-player wins can sum to anything. Loopy play has seven outcome classes instead of four and adds even less — of the 28 cells in the table, eleven hold several answers. Two do not, and they are the two worth having: a second-player win added to anything leaves the outcome alone, and a draw added to a draw is a draw. A draw added to anything else is not.

6 figures · Loopy The same position, two conventions, two winners. Three-player Nim with the last counter winning. The two columns differ only in what a player does when they cannot win themselves, which is a question the rules do not answer — and the answer decides who wins.

Three players and no answer

Every theorem here is about two players, and the reason is not convenience. With two players the game is zero-sum, so 'play well' needs no further explanation. Add a third and the winner of a Nim position becomes a fact about the convention: two reasonable ones disagree on 56 of the 71 positions swept. The one question no convention touches — can a player force a win against the other two together — is answered 'nobody' in 65 of the 71.

8 figures · Multiplayer A pass that may not end the game is not a component at all. The same grouping with the pass forbidden as the final move. Each group now holds several values, and a group with several values is a proof that the parts do not determine the whole.

A pass is not a move

Put a single pass token on a Nim board and one clause decides everything. If it may be taken at any time — including as the move that ends the game — the value of the whole is the nim-sum with a one added, in all 120 positions swept: the pass is a heap of one. Forbid it as the final move and the value stops being a function of the nim-sum at all, and 3 and 1 + 2 come apart.

8 figures · Pass What the auction can and cannot see. Values under both conventions. The Richman value is the share of the money the second player needs; a half means the position itself decides nothing and whoever has more money wins. Every infinitesimal on the list, and zero with them, comes out at a half.

Nobody has to move

Every convention here rests on one sentence nobody examines — the players move alternately. Replace it with an auction and a position stops having an outcome class and starts having a number: the share of the money the second player needs. The 22 values born by day two collapse to seven of those numbers, eight of them landing on exactly a half; the new number respects the game order on all 179 comparable pairs, and is not determined by the parts under addition on 14 of 49.

8 figures · Bidding Hydras, and how long each takes to kill. Six small hydras with the ordinal the termination proof assigns to each and the exact number of chops it takes to finish it. Two of them are not finished here: the fight is guaranteed to end and the machine runs out of memory long before it does, which is the gap between a termination proof and a bound.

It ends, and nothing says when

The recursion this site runs needs every line of play to reach a position with no moves, and the condition is usually met by an obvious decreasing quantity. The hydra meets it with no such quantity anywhere: the tree grows at nearly every step and the fight ends regardless, because the only thing that decreases is an ordinal. A four-node hydra dies in twenty chops; one level deeper and 279 chops reach forty thousand nodes with no end in sight.

8 figures · Termination How much company equality needs. Each row restricts the quantifier in the definition of equality to the games named, and counts how many of the 22 values born by day two survive as distinct. The bar is the same number drawn; the jump from nine numbers to four games is the whole argument.

Equal in this company

Equality quantifies over every game there is, and the quantifier can be made smaller. Restricted to a company of nine numbers, the twenty-two values born by day two collapse to seventeen; restricted to four games — nought, one, minus one and star — they stay twenty-two, and no three of the four will do. The company that decides equality is tiny, and it has to contain a star.

8 figures · Universes Two ways to be certain and ignorant at once. Ten positions from two games that both terminate for reasons no bound comes out of. Sylver Coinage's proof counts something that goes down and can be counted; the hydra's counts an ordinal, which cannot, and the last column shows what that difference is worth.

Two ways to end with no bound

Sylver Coinage and the hydra are both guaranteed to finish and neither will say when. The difference is that one of them carries its own bound: every move in Sylver removes at least one gap, the gaps can be counted in a moment, and over ten openings the longest play uses every single one. The hydra has no decreasing quantity a solver can hold — three hydras of five nodes each take seven chops, twenty-one, and a number past two hundred and seventy-nine that this machine never reaches.

8 figures · Termination The genus of a sum. Every pair of heaps up to 9 counters, from nine impartial games, filed by the genus symbols of its two parts. The claim under test is that the file determines the answer; it does, and neither half of the symbol determines it alone.

The genus of a sum

A genus symbol is meant to be carried one per heap, so that a solver never has to look at the heap again. That is a claim that the pair of symbols determines the sum's, and across nine games and 405 pairs it holds without exception — while the bases alone determine it in only 38 of 50 cases and the superscripts alone in 70 of 74. Both halves of the symbol are load-bearing, and two wild heaps can add to a tame sum.

8 figures · Genus

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