Combinatorial Game Theory
who wins, and by how much
Theme

The thread: Equal, better, or neither — page 3

Two positions can be equal, one can be better, or the pair can be genuinely incomparable — a fourth relation with its own symbol. Deciding which is a search rather than a look, and equality quantifies over every game there is.
Self-negative values, day by day. How many values each day of the construction supplies and how many of them are their own negatives. Day four cannot be counted; a corner of it supplies 571. Sums and comparison

At least five hundred and seventy-one

The rung below dated the self-negative values — the earliest of temperature t is born the day after t — and left the count to a day-four census nobody can run. The construction settles it instead: a value is its own negative exactly when its form is a mirror, so the subgroup can be built from subsets of the day below rather than sifted out of the day above. Day four supplies at least 571 against day three's 26, and the share of a day that is self-negative keeps falling.

 Add, then reduce again. The arithmetic the homomorphism promises, measured: summing two reduced forms gives a reduced form on 88 per cent of pairs and needs a second reduction on the rest. Sums and comparison

Add, then reduce again

The homomorphism promises that a sum's reduced form can be computed from its parts', and says nothing about what the operation is. It is addition followed by a second reduction — needed on 431 of 3,600 pairs of day-three values, and on not one of the 1,751 pairs with a cold part. What the second pass removes is an option that only becomes dominated once the two fights are side by side.

 Every failure is on one board. The eight Domineering boards of the mobility census with the number of failing pairs on each. Seven of them contribute none; every failure at a margin of two is on the largest board, at two depths, and sixteen positions up to symmetry. Values

The threshold was a fact about the census

Two rungs failed to account for the seventy-two pairs where a mobility count gets the direction of a comparison wrong, and the third looks at them one at a time. They are not a class of shapes. All seventy-two are on the largest board in the census, at two depths, and sixteen positions up to symmetry — and one board larger the count fails at a margin of three, which the ladder has been quoting as the point at which it never does.

 The excess is not a flat fee. The excess fitted against the birthday inside each ruleset with enough values to fit a line. A fee would have a slope of nought; every slope here but one is negative, so the excess is largest on the values born earliest. Values

The entry fee was the cap

Two rungs measured how much bigger a position has to be than the value it exhibits, and attributed what was left to the ruleset — Toads and Frogs paying 2.25 squares on everything, green Hackenbush paying nothing. Neither number is a property of the rules. Inside every ruleset the excess falls as the birthday rises, because the sweep's size cap censors exactly the values that would pay most — and three squares past the cap, Toads and Frogs exhibits values born later than the strip is long.

 The list saturates at three. Rules added greedily, each chosen to answer the most decisions given the ones already on the list. Three rules answer 94.5 per cent, and the fourth and fifth answer not one more. Values

Three rules and a tie-break

An exhaustive table of what a Domineering strategy has to remember is 3,308 lines. Three rules applied in order answer 94.5 per cent of it — leave the opponent fewest replies, then keep the region whole, then take whichever placement comes first — and the fourth and fifth rules answer not one more. The residue is 181 decisions in which every rule scores the candidates the same and one of them is worse.

 Eight ways to name it, and none of them works. Candidate rules for which option the second reduction deletes, scored on every pair where it deletes exactly one. The best reaches four in five and none is exact. Sums and comparison

The option nothing names

The rung below found the arithmetic on reduced forms to be add and reduce again, needing the second pass on 431 of its sums, and asked whether the option that pass deletes can be named from the parts. Eight rules were scored and the best reaches four in five — and on a pool closed under negation it falls to under half, which says the near-miss is a property of the population. What the second pass does have is a shape and a cheap test that rules it out.

 The collapse happens twice. How many subsets, antichains and values there are. Domination takes 1,793 subsets to 96 antichains and the rest of the reduction takes those to 30 values. Sums and comparison

What identifies two subsets

Every subset of a day gives a self-negative value by mirroring it, and 1,793 subsets of day two give thirty values. The collapse happens in two stages with different characters: domination takes the 1,793 to 96 antichains and is a theorem, and the rest of the reduction takes 96 to 30 and is concentrated almost entirely on two values — nought, which has an exact description, and star, which has none.

 Star's fibre, described. The fourteen antichains whose mirror value is star, with the two conditions that pick them out of the ninety-six. Sums and comparison

A mex with no impartial game in it

The rung below described the zero fibre of the mirror map and left star's fourteen undescribed. Star's fibre is 'some element is at least nought, and none is at least star' — and the two rules are one rule: the mirror value is the least nimber no element of the set reaches. That is a mex, in a construction built entirely from partizan values.

 One-sided, all three. The three option tests with their disagreements split by direction. None ever refuses a comparison that holds. Where it stops

Wrong in one direction only

The rung below asked for the simplified comparison test the dead-ending hypothesis is supposed to license, and predicted it would agree with the quantifier on the dead-ending rulesets and not on Toads and Frogs. Written three ways and scored on 492 pairs, it agrees best on the ruleset that is not dead-ending — and never once refuses a comparison that holds, which makes it a sound filter and not a test.

 Not a domination, in the order the rung below meant. The second pass's deletions scored as dominations in two orders: the partial order on games, and the order on stops. Sums and comparison

Not a domination, in that order

The rung below asked which pair the second reduction acts on, taking for granted that the operation is a domination. It is not: on none of the 525 deletions is a surviving option greater than or equal to the deleted one. In the order the reduced form actually works in — both stops at least as good — every deletion with a survivor is a domination, the dominator is unique on all but twelve, and it always comes from the other part.

 One board, all the way down. The mobility rule's failure rate on a three by five board at every depth, with the threshold each depth gives. Values

A heuristic that becomes a theorem

The mobility rule's failure rate had been measured at two depths on each of five boards and found to fall. Swept at every depth it does not merely fall — it accelerates, and it reaches exactly nought before the endgame. From four to eight empty squares onwards the rule has no exceptions at all, which turns a rule of thumb into a guarantee for the last few moves.

 Four moves, three arguments. Every first move in the sum, with what answers it and how many cases of each the census holds. Sums and comparison

The case that was supposed to be hard

The mex rule for the mirror construction was to be proved by induction, and the step flagged as needing care was the one where an option is incomparable with the nimber. There is no induction: the argument is four lines, and incomparability is what makes two thirds of the cases go through — because a fuzzy sum is a first-player win and the first player is the opponent.

 The test, scored. The recognition test run on every deletion the second reduction makes, against what actually happens. Sums and comparison

A side about to lose its move

A fifth of the second reduction's work removes the last option a player had on a side, and no rule on the ladder had looked at one — because a deletion with no survivor has no pair in it. The recognisable object is not which option goes but whether the side is one an option can go from, and two comparisons on the parts decide it on all 525.

 The theorem a proof would have needed. The mobility rule's failures on decomposed positions against connected ones, across every board in the depth sweep. Values

The easy case was not the reason

The rung below found the mobility rule reaching a failure rate of exactly nought near the endgame and named what a proof would need: that a decomposed board's comparable options are ordered by reply count. That statement is false on all five boards, at margins up to two — and split positions go exact two squares of depth before whole ones, so decomposition is the easy case rather than the cause.

 The proposed case, scored. The gift-horse theorem and the domination argument proposed for it, each scored over every gift horse added. Sums and comparison

The proof needs both reductions

The gift-horse theorem was to be proved by showing the added option dominated. It is, on 97.8 per cent — and the other 232 are reversible instead, with nothing left over. The case the proposal missed is almost entirely one follower: none under a positive number, 190 under a negative one.

 The fall stops. Comparability on days two, three and four, the last built and corrected. Sums and comparison

A floor, and not a decline

Comparability fell eighteen points from day two to day three and the next day cannot be enumerated. It can be built — and the construction's bias measured one day lower, where the truth is known. Corrected, day four comes to 60.6 per cent against day three's 59.7: the fall was a one-day event.

 Same group, three different yields. The rulesets with a trivial symmetry group, which the conjecture predicts must all have a yield of one. Values

Three groups, and three yields

The conjecture was that each ruleset's yield tends to the reciprocal of its symmetry group's order. Three rulesets have a trivial group and predicted yields of one; they measure 1.000, 0.531 and 0.204. And every colliding value in Push and Shove — all 175 of them — has two rows no symmetry relates.

 Two fibres and a tail. The values reached by the most antichains. Nought takes half of them and star fourteen more; twenty-three of the thirty values are reached by exactly one. Sums and comparison

Twenty-six other values

The mex rule accounts for sixty-six of the ninety-six antichains and is silent on the other thirty. Every one of those thirty is worth a self-negative value born by day three — and the same mex, run over that family instead of over the nimbers, is exact on all ninety-six. The nimber rule is this one cut short after its fourth member.

 A game beside its own mirror, and what is left over. Every coin row added to its own negative, played out exactly, with the resulting scores counted. Under the last-move convention every such sum is worth nothing, because the mirroring strategy guarantees the second player the last move. Here the same strategy is available and the score it produces is not nothing: the mirror of a coin conceded is another coin conceded. Gold is the sums that do come to nothing, which are a minority. Out in the world

Nothing to subtract with

Comparison is defined by contexts and computed by subtraction, and the equivalence between the two is a theorem about groups. A scoring game is not one — sixty-six of eighty-one coin rows do not cancel against their own negatives — and the difference test then fails on a row compared with itself, which every context accepts and nothing certifies.

 What a short key gets wrong. A memoised who-wins search of 4 × 5 Domineering using a Zobrist key of 10 to 32 bits, run under 60 random keys at each length and checked against the exact answer: how many stored positions share a key, how many runs store a wrong verdict or name the wrong winner, and what checking the whole position would cost instead. At 16 bits 59 runs store a wrong verdict and 19 name the wrong winner. What it costs

A key shorter than the position

A who-wins table for 4 × 5 Domineering addressed by a 16-bit Zobrist key stores a wrong verdict in 59 runs of 60 and names the wrong winner of the empty board in 19. The pairs of positions sharing a key follow the birthday count exactly while addresses are scarce, and fall away to nothing once the key has more bits than the board has squares, because a Zobrist key is linear. Symmetry and value identify positions that really are the same; a short key identifies positions that differ, at a rate set by arithmetic.

 One family of confused positions. Two 4 × 5 Domineering positions with Left to move that share every bit of an 18-bit Zobrist key, differing only on 4 marked squares whose words cancel. Under that key 100 of 136 confused pairs differ on exactly those squares; its other families hold 25 and 11. What it costs

A check bit halves the average and not the key

Real transposition tables keep a few of a key's bits beside each verdict and trust an entry only when they match. On 4 × 5 Domineering each such check bit halves the average number of wrong verdicts, exactly as the birthday count says. It does not halve any one key's. A Zobrist key confuses positions in families — every pair that differs on one set of squares whose words cancel — and a bit removes a family whole or not at all, so from eighteen bits to nineteen thirty of fifty-eight keys lose every confusion and fourteen keep every one.

 Four restrictions, and what each one buys. Four candidate classes of scoring game — every row, the incentive condition at the top, the same condition at every subposition, and the rows that cancel against their own negatives — scored on two families of coin rows for the mean-value bound, for comparison by subtraction, and for cancellation. Out in the world

The restriction that buys the most

Four candidate classes of scoring game, scored on the same two families and the same three questions. The class everyone expects to be tiny — the rows that cancel against their own negatives — is empty on rows of three and the widest restriction on rows of four, where it holds fifteen rows against the hereditary class's twelve and gets all 225 of its comparisons right against 108 of 144. The trade everyone expected does not exist.

 Two squares a key never needs. A 4 × 5 Domineering board shaded like a chessboard, with two squares of the same shade in rows 1 and 2 marked. Because a domino covers one square of each shade, a vertical domino covers one square in an odd row and one in an even row, and turns alternate, the other eighteen squares determine both marked squares: a key that leaves them out confuses none of the 48,670 reachable positions. What it costs

A key is a code, and two squares come free

The families of positions a Zobrist key confuses are the words of a binary linear code — the sets of squares whose words cancel — so choosing a key is choosing a code. On 4 × 5 Domineering the textbook choice, a code with the largest minimum distance, confuses more stored positions than a random key at fourteen, sixteen and nineteen bits. The choice that reads the board confuses none at eighteen: two squares of one shade, in rows of different parity, are decided by the other eighteen, and no seventeen-bit key is exact.

 A position Left always wins, and not always. Values grouped by the outcome class alternating play assigns them, with the range of probabilities the coin gives Left inside each class. A class that alternating play calls a win for Left every time holds no position the coin makes certain. Where it stops

Left always wins, and loses more often than not

Alternating play answers with one of four classes and the coin answers with a chance, and the two do not have to agree. Over the twenty-two values born by day two they never disagree and the margin is exactly nothing — the lowest chance on a position Left wins whoever moves is a half. Over the 1,474 born by day three, seven of them sit at seven sixteenths, and seven mirror them on the other side.

All themes