Two and four are not conventions

Every piece of arithmetic in the Dots and Boxes endgame has a two or a four in it. Declining a chain costs four boxes of margin because two are surrendered and two are not taken; a loop costs eight for the same reason doubled; the parity law’s threshold between short and long sits at three, which is the first length at which two boxes can be left behind.

Those numbers have never been argued for. They are worth arguing for, because they are not conventions and the endgame changes completely when they move.

Where the two comes from #

The fee is not a rule of the game. It is the smallest number of boxes a player can leave behind and still hand over the obligation to cut.

An opened chain runs from one end. The taker cuts along it, pocketing a box at each cut, and at each cut is obliged to cut again. When two boxes remain there is one string between them and a string at the far end, and the taker has a choice for the first time. Taking both means cutting that last pair, which completes two boxes and obliges another cut — into whatever is left. Leaving them means cutting the string between the two boxes, which completes nothing, ends the turn, and hands the opponent two free boxes and the obligation that follows them.

Two is the fewest that can be left, because the last cut of a chain always completes two boxes at once. There is no way to leave one: a single box at the end of a chain has one string on it, and cutting it takes the box.

And where the four comes from #

A loop is the same argument with the ends joined, and the join doubles it.

A loop has no end, so the taker cuts into it and then has a row of boxes to eat from both directions. When four remain they sit in a row of four with the two outer strings still holding them — and cutting to leave two would leave a row of two with a string at each end, which is a chain of two, which the opponent can take with one cut and continue. That does not hand anything back.

So the decliner has to leave four, cutting the middle of the remaining four and giving the opponent two boxes in each direction. Four is the fewest that can be left in a loop, and it is twice the chain’s fee for a reason of shape rather than of arithmetic.

The fee is the only place the shapes differ #

It is worth noticing what the two numbers do to the theory’s vocabulary, because a chain and a loop are otherwise almost the same object.

A chain of $n$ boxes and a loop of $n$ boxes have the same coins, each holding two strings, and differ by one string: the chain has two running to the ground and the loop has neither. Every coin in both holds exactly two, which is why both fall inside the endgame theory’s domain and both are worth nothing alone.

The fee is the entire difference between them. It is why a loop is worth avoiding more than a chain of the same size, why the parity law counts loops among the long components and never among the short — a loop of four cannot be declined for two and so is never cheap — and why the margin formula would need two coefficients rather than one.

So a sweep of the fee is not a sweep of an incidental constant. It is a sweep of the one quantity that distinguishes the two shapes the theory is built on, and that is the reason to expect it to matter before any of it is computed.

Sweeping the fee #

Both numbers come out of the geometry, so a game with different fees would be a different game. Solving it anyway is the only way to find out which of the endgame’s facts are about the shapes and which are about the fee.

Four hundred and eighteen endgames of chains and loops, solved five times over. Halving the chain’s fee from two to one changes the winner on 29 per cent of them and the margin on 88. Raising it to three changes the winner on 22 per cent. Changing only the loop’s fee, from four to three, changes the winner on 9 per cent and the margin on more than a third.

Those are large numbers for a parameter nobody thinks of as a parameter. The arithmetic of the endgame — which player is forced to open, and by how many boxes they lose — is a function of the fee at least as much as of the components.

It is worth putting the two columns of the sweep side by side, because they say different things. The winner column is about structure: which player ends up forced to open, which is the question the parity law answers and the question the Nimstring skeleton was built for. The margin column is about arithmetic: how many boxes separate them.

The margin moves far more than the winner, everywhere. At a loop fee of three, 91 per cent of winners survive and only 65 per cent of margins. At a chain fee of one, 71 per cent of winners survive and 12 per cent of margins.

That gap is itself a finding. It says the parity structure is the robust part of the endgame and the box count is the fragile part — which is the reverse of how a player experiences it, since the boxes are what they are counting and the parity is the abstraction somebody taught them.

What does not move #

Two things survive every fee in the sweep and it is worth saying which.

Every component is still worth nothing on its own. A chain alone is a second-player win whatever the fee, because the fee is about what a declining player leaves and a lone component has nothing to be forced into afterwards. So the fact that breaks the nim-sum is not a fact about the fee.

And the short-long threshold moves with the fee rather than away. A chain is short when it cannot be declined at all, which is when it is no longer than the fee — leaving the fee behind would mean leaving the whole chain, and an opener denies that by cutting the chain’s middle rather than its end. At a fee of two, chains of one and two are short; at a fee of three, chains of one, two and three are. So the law’s vocabulary rewrites itself consistently — which is why 91 per cent of winners survive a fee of three on loops and only 71 per cent survive a fee of one on chains.

That second point is the useful one. The fee does not appear in the law as a number; it appears as the definition of short. Change it and the law’s statement is unchanged and its meaning is not, which is exactly the kind of dependence a rule can carry without anybody noticing it is there.

Why a fee of one is the worst #

The most destructive change is the smallest-looking, and it has an explanation.

At a fee of one, every chain of two or more can be declined, and declining costs a single box. So control becomes almost free: the formula that prices it would charge two a chain rather than four, and a position of many short chains that currently cannot afford control could afford it easily. Twenty-nine per cent of winners change, and 88 per cent of margins.

At a fee of three the change runs the other way: control becomes expensive, more positions abandon it, and the margins compress. Twenty-two per cent of winners change.

So the real fee sits between two regimes and it is not in the middle of them. The endgame the game actually has is one where control is worth buying on most positions and not on all, and that balance is a consequence of two rather than a design.

A number that comes out of the drawing #

There is a general habit worth extracting, because this is one of the clearer cases of it in the whole collection.

Most constants in this subject are chosen. The convention that the player who cannot move loses is a choice with a whole other convention beside it; the decision to count a game’s value as a number rather than as a class is a choice; the threshold of three boxes is a definition. Each of them could have been otherwise and the theory would have been different.

The two and the four could not have been otherwise. They are read off a picture of a chain and a picture of a loop: a chain’s last cut completes two boxes because the last two boxes share one string, and a loop’s four because its four share two. Nobody decided them; they were counted.

That is why the sweep is worth running rather than merely noting. A chosen constant invites the question what if it were different, and the answer is usually another theory. A counted constant does not, and the only reason to vary it is to find out how much of the theory is resting on it — which is a question about the theory rather than about the rules.

The answer here is: a great deal. Nearly a third of the winners and nearly all of the margins.

The threshold moves with it, and that is the surprise #

The most interesting row of the sweep is the one that shows the least change, and the reason is worth pulling out.

Raising the loop fee from four to three leaves 91 per cent of winners alone. Halving the chain fee from two to one leaves 71 per cent. The loop fee is the larger number and changing it matters less, which is the reverse of what a sensitivity analysis usually reports.

The explanation is the short-long threshold. A chain is short when it is no longer than the fee, so the chain fee defines which components are short, and the parity law counts short chains. Move it and the law’s own vocabulary changes underneath it: a chain of two stops being short at a fee of one, and the parity is being taken over a different set.

The loop fee does the same job on a much smaller population. Only two loop sizes exist in the sweep, four and six, and the loop fee crosses one of them once: at a fee of six a loop of four can no longer be declined and becomes short, and at three or four both loops stay long. So the loop fee reclassifies one size in one of the four alternatives, while the chain fee reclassifies a size in every one of them — the sweep holds chains of one to six and the fee moves between one and three, straight through the middle of them.

The chain fee is load-bearing twice over and the loop fee once, and that is why the two rows differ by three to one despite the loop fee being the bigger number.

What is varied, and what is held fixed #

Dots and Boxes with the score kept, and the fee as the one thing varied. Everything else is unchanged: a cut that frees a coin obliges another, the game ends when the strings run out, and both players maximise their own boxes.

Three conventions of the sweep.

A chain of exactly the fee cannot be declined at all. Leaving the fee behind would mean leaving the whole component, which is not a decline but a refusal to move. The solver denies that branch, and a sweep that allowed it would be pricing a move the game does not have.

Loops of four and six only. A loop of two is not a shape a board produces and a loop of eight runs past the budget, so the loop half of the sweep is two sizes.

And the comparison is of winners rather than margins. A margin moves whenever a fee does, almost by definition, so counting margin changes would report every fee as catastrophic. The winner changing is the structural question and it is the one the shares above are about.

A fee is not really free to vary #

A fee is not really free to vary. Every alternative in the sweep describes a game nobody plays, so the shares are a sensitivity analysis rather than a comparison of rulesets. What they measure is how much of the endgame’s structure rests on a number that comes from the shape of a chain.

Nothing here varies the capture rule. The fee is downstream of a cut that frees a box obliges another cut, which is the rule that makes Dots and Boxes what it is and the rule that breaks the disjunctive sum. Changing that would change everything rather than a share.

The alternatives are not games with rules attached. Changing the fee in the solver is changing what a declining player is allowed to leave behind, which on a real board is not a choice anybody has — the geometry decides it. So the sweep varies a quantity by fiat and reports what depends on it, which is a different thing from comparing two rulesets.

And the budget is eighteen boxes over four components. The positions where control is marginal are the interesting ones and they cluster at many small chains, which the budget reaches; the positions where it is obviously worth keeping are long chains, which it reaches less well.

What a player is really choosing #

The fee has a reading that is not about the numbers at all, and it is the reading that makes the endgame make sense.

A player handed an opened chain is not choosing between more boxes and fewer boxes. They are choosing between boxes and the turn, and the fee is the exchange rate. Two boxes buys the obligation to open, handed back; four boxes buys the same thing at a loop.

Seen that way the endgame is a market with one commodity and a fixed price, and the whole of the analysis is working out when the commodity is worth buying. It is worth buying when the chains average more than the fee, and the three positions where the margin formula fails are exactly the ones where it is not.

That also says why the exchange rate being fixed matters more than what it is. A game in which the price of handing back the turn varied with the position would have no formula at all — every component would need its own accounting, and the controlled value would be a search rather than a subtraction. The endgame is tractable because the same two boxes buy the same thing every time.

Which is a property of the geometry and not of the game’s design. Every chain ends the same way, so every chain charges the same, so the arithmetic is a subtraction.

Still open: the fee a board could have had #

The sweep varies the fee as a number and the deeper question is whether a board could produce a different one.

The two comes from a chain’s last cut completing exactly two boxes, which follows from every coin holding two strings. A component in which some coin held three would have a different endgame at its end — and a fork is worth something rather than nothing, which is the same observation from the other side.

So the question with an answer is what the fee is for a fork: how few boxes can be left behind at the end of a branching component while still handing over the obligation. If it is two, the fork’s endgame is a chain’s and the value differences are about something else; if it is three or four, then the fee is a function of the component’s shape and the endgame theory has a parameter per kind rather than one per game. That is a measurement the same apparatus could make, and it is the one that would say whether two is a fact about chains or a fact about Dots and Boxes.