What a value leaves out

A game value is a remarkably complete object — computed from the position by recursion, reduced to a canonical form, and answerable for everything the position does. It decides who wins the position. It decides who wins any sum the position appears in, alongside any other games. It supports arithmetic, comparison, and substitution.

There is one thing it does not decide, and the omission is total: how long the game takes.

Two moves, or eighteen #

Two heaps of one counter is worth zero. So is a position of heaps 8, 9 and 1.

The first is over in two moves. The second can run for eighteen. They are equal as games in the strongest sense the theory has: compare returns =, either may be substituted for the other anywhere, and no outcome of any sum containing one differs from the outcome of the corresponding sum containing the other.

The value is not being coarse or approximate. It is being complete about a different question.

Why “optimal play” does not fix a length #

It is tempting to define the length of a position as how long it lasts under optimal play, and that definition does not work.

Optimal play means: win where winning is available. It says nothing about how. A player who has already won a position may take one counter at a time or clear an entire heap, and both are optimal, because both win. So a position has a range of lengths compatible with optimal play, not a length.

That is why the figure draws bars rather than points. The shortest legal play from a Nim position takes each heap in one move — as many moves as there are non-empty heaps. The longest takes one counter at a time — as many moves as there are counters. Everything between is available.

And length is not a function of the shape either, which is worth separating from the claim about values. Take the positions that are two equal heaps — the plainest zero there is, and one anybody would call the same position at different sizes.

So the omission is not a coarseness in how positions were described. Two heaps of n is as complete a description of a position as anybody could ask for, and it fixes the value, the outcome and the shortest play while leaving the longest free to grow without limit.

What the value does determine, for contrast #

The omission is easier to size once the rest of the list is in view, and the rest of the list is long.

A value determines who wins the position. It determines who wins any sum the position appears in, whatever the other components are. It determines whether the position is greater, less, equal or confused with any other. It determines the position’s temperature and mean, what it is worth in a large number of copies, and whether it is a number.

That is a great deal of information, and every item on it is a statement about outcomes of sums. That is not a coincidence — it is the definition of equality at work, and it is what the next section is about.

The equality on which all of that rests is the one thing on this page it would be careless to inherit rather than measure, so it is measured.

Twenty-one comparisons find nothing, and the same seven positions differ in length by up to sixteen moves. That gap between what the comparison can see and what a reader can count is the whole subject of this essay.

Where it matters: the same value, a different game #

If length were merely uninteresting the omission would not be worth an essay. It matters in three places.

In a sum, when somebody is counting tempo. The number of moves available in a component is a resource. A player who needs to pass — who would rather the opponent moved — cannot, under these rules, and a component with many cheap moves in it is a place to spend a turn. The value does not record how many.

When the rules attach anything to move count. Change the scoring so that the game ends after twenty moves, or so that each move costs something, and two positions of equal value stop being interchangeable immediately.

And in the partizan case, where the shapes differ. The integer 2 is $\{1 \mid\ \}$: Left has a move, Right has none. The position $\{1 \mid 3\}$ is also worth 2, by the simplicity rule — the simplest number strictly between 1 and 3. They are equal. Only the second gives Right a legal move at all.

Three other things it discards #

Length is the cleanest omission and it is not the only one. The same argument — anything invisible to sums is discarded — has three more consequences worth naming.

Which move achieves the value. A position worth ↑ tells a reader nothing about what to play. That is why this site’s figure standard requires the move to be stated separately wherever it matters, and it is why a value is not a strategy.

How the position looked. The canonical form is reached by removing dominated options and replacing reversible ones, and the original shape is gone. Two positions reducing to the same form are indistinguishable afterwards, and one of them may have had twenty options and the other two.

How expensive it was. The value of a 4×4 Domineering board has fourteen nodes in it and took six million recursion steps to find. Nothing in the fourteen nodes records the six million.

The surprise: temperature is the closest thing to a clock, and it is not one #

Temperature measures what is at stake in a position, and it falls as a game progresses, so it looks like it might serve as a clock.

It does not. A position may have temperature zero — nothing at stake, everything settled — and still have many moves left in it, because a number is worth what it is worth however many moves it takes to collect. And two positions of the same temperature may be one move from the end or twenty.

What temperature governs is which component to move in, not how long anything lasts. The two quantities are independent, and treating one as a proxy for the other is a real mistake that the arithmetic catches immediately.

A thermograph makes that visible by what it does not have. It plots what a position is worth against the cooling parameter, and the parameter is a price rather than a clock: there is no axis in it for the number of moves, and nowhere a length could be read off even in principle.

What the theory does record, and what it deliberately discards #

The definition of equality is the place to look. Two games are equal when $G - H$ is a second-player win — when the difference is worth zero.

That definition quantifies over sums: it says the two behave identically in every context. So anything a value fails to record is something that makes no difference in any sum, and length is exactly that: two positions of equal value and different lengths cannot be told apart by any game placed beside them.

The omission is therefore not an oversight but a theorem. Length is discarded because it is invisible to the equivalence the theory is built on, and any theory that recorded it would be making distinctions no sum can detect.

The same argument disposes of shape as well as of length. The forms $\{0 \mid 1\}$, $\{0, \ast \mid 1\}$, $\{-1, 0 \mid 1\}$ and $\{-1, 0, \ast \mid 1\}$ all reduce to one canonical form, and they differ in how many options each player has and in how deep the position runs. What survives the reduction is precisely what a sum could have detected, so the canonical form is a poor guide to how the original looked, exactly as it is a poor guide to how long it takes.

The one place inside the theory where it shows #

There is a place in the standard machinery where length leaks back in, and it is worth pointing at because it is the exception that fixes the rule.

Tempo in the endgame account. Once the temperature has fallen to zero and every component is a number, the game is a sequence of alternating collections and the total depends on who has the move. That is a parity fact about the number of moves remaining, and parity is a fact about length.

The theory handles it by making the move itself an item in the account rather than by making values record length. Whose turn it is at the moment the temperature reaches zero is worth something, and the something is computed.

So the theory is not blind to tempo. It declines to put it in the value, because the value quantifies over all sums and tempo is a fact about a particular sequence of play, and it recovers it where it matters as a separate term.

Where the model stops #

This is a normal-play statement. Under conventions where the number of moves is scored — Go, in effect — length is not discardable, and the theory that applies is the one about temperature and accounting rather than the one about values alone.

And “length” here means the number of moves to the end of the game, not the depth of the search needed to evaluate it. The second is a computational quantity and is enormous where the first is small.

The measurement, restated #

The numbers behind the hero figure are worth having in the text as well as in the drawing, because they are the essay’s evidence.

Nine Nim positions were taken, seven of them worth zero. Their shortest legal plays run from 2 moves to 4; their longest from 2 to 18. Every one of the seven is equal to every other as a game — compare returns = for each pair — and the spread in longest play across them is 16 moves.

Sixteen moves of difference, between objects the theory declares identical. That is not a small discrepancy being tolerated; it is a quantity the theory has no slot for.

What the picture cannot show #

The bars in the hero figure are a range, and a bar is a claim: it says that every intermediate length is achievable and not merely the two ends. For Nim it is true, and it is worth checking rather than asserting, since a bar drawn over a set with holes in it would be describing plays that do not exist.

That is not true of games in general. A partizan position may have lengths that skip values — no legal play of exactly seven moves, but plays of six and eight — and the bar would be lying about it. The figure draws Nim positions for that reason, and the honest general statement is a set of lengths rather than an interval.

When the omission becomes a problem #

For the games on this site it never does, and it is worth being clear about why, because “the theory ignores time” sounds like a limitation waiting to bite.

Normal play has exactly one thing at stake: who moves last. Length is not scored, not bounded, and not otherwise constrained, so a difference in length is a difference in nothing that the rules attend to. Discarding it loses no information about the games the theory is a theory of.

The omission becomes a problem the moment the rules mention moves, and three variants do.

Scoring play, where points are accumulated rather than the last move being decisive — Go, most obviously, whose endgame theory needs the tempo correction precisely because of this.

Bounded games, where play stops after a fixed number of moves and whoever is ahead wins.

And loopy games, where a play may be infinite and the outcome depends on that rather than on a final position. There the length is not merely relevant, it is the whole classification: a draw is an infinite play.

So the discipline is not that length is uninteresting. It is that under one convention length is invisible to every question, and the theory is built exactly to that convention and says so.

A worked case: two positions, one value, different games to be in #

It helps to have one instance where the difference would matter to somebody, even though under these rules it does not.

Take a sum containing a component worth zero. If that component is two heaps of one, it offers two moves and then nothing. If it is heaps of 8, 9 and 1, it offers up to eighteen.

Under normal play the two are interchangeable, and the reason is that neither player wants to move in a zero component: moving there hands the opponent the reply that restores it to zero, and the exchange nets nothing. So the eighteen available moves are never taken and their availability costs nothing.

Change the rules so that a player may be obliged to move — remove the option of moving elsewhere — and the two components stop being interchangeable immediately, because one of them provides sixteen more spare turns than the other.

Two heaps of one is the two-move version of zero and heaps of 8, 9 and 1 the eighteen-move one, and under normal play each is exactly as good as the other, because nobody will move in either.

That last clause is doing the work, and it is what the disjunctive sum guarantees: there is always somewhere else, until the game is over. A convention that removes the guarantee makes length matter, which is the general form of the three cases above.

Why the definition is the right one anyway #

It would be possible to build a theory whose values recorded length. It would be a worse theory, and the reason is worth stating because it is not obvious that more information is ever a cost.

Values with length in them would not add — or so the argument usually runs. It is the weaker of the two reasons and the section below takes it apart: on the impartial positions this page measures, length composes perfectly well.

And the equivalence would shatter. This is the reason that survives.

The canonical form exists because the equivalence classes are large; make the classes finer by distinguishing lengths and there are vastly more of them, with no corresponding gain in what can be decided.

So the omission buys the two properties the theory is for. That is the usual shape of a good definition — it discards exactly what would break the operation it is built around — and it is worth recognising as a design rather than as a gap.

Length adds, for the positions on this page #

The argument two sections above gives two reasons a value could not carry length, and the first of them is too quick. It says a value with length in it would not add. On the positions this page measures, length adds exactly.

Take two Nim positions side by side. Every move of the sum is a move in one component or the other, and the sum ends when both components are empty, so the number of moves in a play of the sum is the number made in the first plus the number made in the second — and each of those is a complete clearing of its own component. A Nim position of $h$ heaps and $c$ counters can be cleared in any number of moves from $h$ to $c$, and two components can be interleaved freely, so

$$\operatorname{lengths}(G + H) = \operatorname{lengths}(G) + \operatorname{lengths}(H)$$

as sets. That is an equation and it is cheap to evaluate, since a sum of Nim positions is just the two heap lists written side by side.

So the composition objection does not bite here, and the honest reason is the second one. Length is not a function of the value. Seven positions on this page share a value and have seven different lengths, so a theory whose objects were value-and-length pairs would not be a theory of the equivalence classes at all — it would separate positions that no sum can separate, and canonical form and substitution would both go with it. What breaks is not the arithmetic; it is the equivalence the arithmetic is defined on.

The partizan case needs more care than either reason admits, and this page does not settle it. The moves made inside one component during a play of a sum need not alternate between the players — Left can move twice in a row in $G$ while Right answers in $H$ — so the trace inside a component is a line of play and not a game of that component, and the clean interval arithmetic above is an impartial fact rather than a general one.

The name #

This site calls the omitted quantity tempo, which is borrowed from chess and is not quite the chess meaning.

In chess, tempo is a move gained or lost relative to the opponent — a piece developed with a threat gains one. Here it is the coarser thing: how many moves a component has left in it, and who ends up holding the move when the components run out.

The borrowing is deliberate and the mismatch is worth flagging, because a chess player reading tempo will import an idea about initiative that the theory does not carry. What is meant is closer to the number of spare moves available, which chess players also count and call something else.

Where the ladder goes next #

This is the first rung of an anchor about the quantities the value does not carry, and the rungs above it count the others.

How many moves are worth making takes the first: not how long a position lasts but how many of its moves achieve the value. Over 1,034 Domineering regions carrying 125 values between them, 63 values have two regions that disagree about how many placements are good enough and 52 disagree about whether there is any choice at all — so the value does not settle the size of the choice, and the count of good moves stays near one and a half however large the region gets.

What a strategy has to remember sizes the whole gap. A player who wants to win rather than to predict has to store 3,308 choices across 4,269 positions carrying 128 values — twenty-six entries for every number the theory supplies, which is the price of the omission stated as a table.

Three rules and a tie-break then compresses that table: three geometric rules applied in order answer 94.5 per cent of it, and the fourth and fifth answer not one line more. And seventy-two of them were not silence goes back over the residue and finds that two fifths of it is the rules speaking and being wrong rather than falling silent, which is a different failure and a repairable one.

Read in order they make one measurement: the value is complete about outcomes and empty about play, and what a player needs beyond it is large, structured, and mostly reducible to a handful of rules about shape.