What a pass is worth to a theory

The rung below lists what its own model leaves out and ends on this:

The other thing not shown is the passing move. Real scoring games let a player pass — Go’s entire ending mechanism is two consecutive passes — and the coin row does not. Passing is what makes the incentive condition matter, because a player who would rather pass and cannot is a player in zugzwang, and a game that permits passing has no zugzwang at all. That is a substantive difference between the model here and Go, and it cuts in the direction of making the model harder than the real thing rather than easier.

That is a debt with a number attached to it, and paying it takes one line of the recursion. What comes back is more interesting than a repair.

The condition, and the rows that break it #

Milnor’s 1953 theory of scoring games — the ancestor of everything this site’s fourth field does with temperature — has a hypothesis attached: there must be a non-negative incentive to move. Formally, the score with Left to move is never worse for Left than the score with Right to move. Informally, nobody would rather pass.

The rung below finds the rows that break it. Give a coin row an odd length and a coin nobody wants — a negative one — and a player can be forced to take it. Fifteen of the twenty-seven three-coin rows over {−3, 1, 2} are like that: having the move is a disadvantage, and the bound Milnor proved breaks on three hundred of the pairs built from them.

A position where having the move is a disadvantage is a zugzwang, which is the same object a chess ending is about and the same object the incentives essay measures one field over. What a move is worth is the general quantity, and a negative one is exactly a position a player would rather not be in.

Allowing a pass #

The clause is one sentence:

Either player may pass instead of taking a coin. Two passes in a row end the game and the remaining coins are left on the table.

That is Go’s ending rule exactly, and it is the ending rule of essentially every scoring game anybody plays. The recursion needs one more bit of state — whether the previous move was a pass — and stays quadratic in the length of the row.

Run it and the fifteen violations become nought.

Not fifteen reduced to two, not fifteen on a wider range. Nought, on all twenty-seven, and the reason is a one-line argument rather than a computation: a player who may pass can always decline to be hurt by having the move, so the score with them to move is never worse than the score with the opponent to move, so the incentive is non-negative by construction.

The hypothesis has stopped being a hypothesis. It is no longer a condition on the position that a theorem has to check; it is a consequence of the rule set, true of every position of every passing game there will ever be.

Which is worth stating as a repair to the older theory #

That deserves a moment, because it says something about why Milnor’s condition looks like an awkward technical clause and is not.

Milnor and Hanner were writing about Go. Go permits passing. So the condition their theory needs is automatic for the game they were writing about, and the reason it appears as a hypothesis at all is that they stated the theory for a wider class than Go — positional games in general, some of which do not permit a pass.

Read that way, the condition is not a restriction chosen to make the arithmetic work. It is the fingerprint of the game the theory was built for, left in a definition that was written to cover more.

And the coin row, which is the rung below’s laboratory, is one of the games outside it. A row of coins with no pass is a scoring game that Milnor’s theory does not apply to, and the rung below’s demonstration that the bound breaks is a demonstration that the theory was right to exclude it.

The argument in one line, and why it is not a computation #

The count is nought, and a count of nought is the kind of number that should be argued rather than reported, because a sweep that finds no counterexample has not established that none exists.

Here the argument is available and it is one sentence. Let L be the score with Left to move and R the score with Right to move, in a game where either may pass. Left, to move, may pass; if Left passes, the position is the same one with Right to move, worth R. So Left has an option worth R, so the best Left can do is at least R, so L ≥ R — which is the condition.

Nothing in that mentions coins, or rows, or the values on them. It holds in every scoring game with a pass in it, at every position, for ever.

So the sweep is not evidence for the claim; it is a test of the code. A run that found a violation would mean the implementation had a bug in it, since the mathematics forbids one, and running the sweep anyway is the same habit every figure on this site follows: a proof about mathematics does not protect a program from an error in its own arithmetic.

That distinction is worth keeping because the other count on the page is of the opposite kind. Fifteen rows breaking the condition without a pass is an empirical statement about a stated family, and it would change if the family did.

What it changes about the scores #

The condition is not the only thing a pass touches. Fifteen of the twenty-seven rows have a different score with the pass available, and that count is worth reading carefully because it is not the same fifteen twice over.

A pass changes the score exactly where somebody would have been forced into a coin they did not want. On the rows made only of positive coins, both players take greedily to the end and a pass is never worth playing, so the score is unchanged. On rows with a negative coin in them, the pass is what lets a player leave it.

There is one shape worth naming: a row of three negative coins is worth nothing at all with a pass in force. Both players pass immediately, the game ends, and nobody has taken anything. Without the pass the same row forces one side to take two coins and the other to take one, and the score is whatever that division gives.

That is the clearest possible statement of what a pass is. It is not a move that does nothing; it is a move that refuses. A game where refusal is available is a game where an unwanted position simply does not have to be entered, and a whole class of forced loss disappears.

The pass and the parity strategy #

There is a second thing a pass takes away, and the rung below is where it comes from.

That essay finds the coin row’s normal-play answer useless as a value and load-bearing as a condition: the skeleton computes exactly one thing about a row, the parity of its length, and the parity of its length is exactly the hypothesis the best cheap strategy anybody has for the game needs. In an even row the two ends carry different parities, so the first player may choose a parity class and take the whole of it, guaranteeing at least half the coins — and on 761 of 819 rows that guarantee is the exact answer.

The pass destroys the argument at its first step. The strategy works by an induction on whose turn it is at each length, and a pass breaks the correspondence between the number of coins taken and the number of moves made. After one player passes, the remaining row is the same row with the other player to move — which is precisely the situation the parity argument is built to prevent.

So the row loses the guarantee along with the convention. That is the second half of the cost and it is the more practical half: without a pass a player has two additions and a comparison that never lose; with a pass they have a search.

What it costs, and the cost is the whole of the rest of this site #

Now the other side, and it is not a footnote.

A player who may pass is never without a move.

That sentence removes the normal-play convention. Every field on this site but this one rests on the player who cannot move loses, and in a passing game nobody can ever be in that position — there is always a pass available, so no position has an empty option list, so the base case the whole value recursion bottoms out at does not exist.

A pass is not a move makes the general form of that argument, and the coin row supplies the sharpest instance of it. The same clause is what makes a ko fight terminate two anchors over, from the other direction: there a player is forced to spend something to continue, and here a player is allowed to spend nothing. Without the pass, the row’s normal-play skeleton is a correct and completely useless statement: the game lasts exactly as many moves as there are coins, so the last move goes by the parity of the length. With the pass, there is no normal-play skeleton at all, because there is no last move to allocate — the game ends when both players decline, which is a scoring rule and not a last-move rule.

So the trade is stark. A pass makes the scoring theory’s hypothesis automatic and makes the normal-play theory inapplicable. The two theories want opposite things from the same clause, and a game either has a pass or does not.

Why a scoring game needs an ending rule at all #

There is a structural point underneath all of this and it is easy to miss.

A normal-play game ends because it runs out. Nobody has to decide when; the moves stop, and the player facing the stop has lost. That is why the convention is a convention rather than a rule — it is a way of reading a position that has already ended.

A scoring game does not end that way. Its score is settled long before its moves are, and once the score is settled there is no reason for anybody to move at all. So a scoring game needs a stopping rule, and every scoring game anybody plays has one: two passes in Go, the board being full in Dots and Boxes, the row being empty in this model.

That is a genuine difference in kind rather than a detail of the rules. It is the reason the rung below can say that a scoring game is the more primitive object — a normal-play game can always be turned into a scoring one by counting moves, and a scoring game cannot generally be turned into a normal-play one — and it is why the two theories have different shapes rather than one being a special case of the other. It is also why the mean and the error term arrived first in the scoring theory and were rebuilt afterwards in the other one: a game with a score has quantities to be a mean of, and a game with a last move has to invent them.

The coin row without a pass is a hybrid: a scoring game whose stopping rule happens to coincide with running out of moves. That coincidence is what makes it a laboratory, and putting a pass into it is what removes the coincidence and makes it an ordinary scoring game.

The model, stated #

Nothing above is a claim about Go, and the model is worth stating so it is clear what is.

A row of coins. Each player in turn takes the coin at one end and keeps it, or passes. Two passes in a row end the game. When it ends, the higher total wins.

Every number is that game solved exactly by an interval recursion with one extra bit of state, over every row of three coins from a stated set. The comparison run is the same recursion with the pass clause removed, which is the rung below’s model exactly.

What the model gets right about Go is the shape of the ending rule. What it does not model is anything else about Go, and in particular it has no notion of a position where passing costs something — which under area scoring it does, since a stone on a neutral point is worth a point and declining to place it gives that point away.

What the picture cannot show #

The pass here is free and in a real game it is not always. Passing in Go under territory scoring costs nothing; under area scoring it costs a point, because the opponent will fill the neutral point instead. So the clause modelled here is one of the two passes Go has, and the other one is a move with a price.

And nothing here measures how often a pass is played. The counts are of rows where the pass changes something, not of positions where a player would use it; a pass that is available and never worth playing is invisible in a score and central to what a game feels like.

The rows are three coins long. The condition’s failure needs an odd length and a coin nobody wants, and three is the shortest row that supplies both. Whether the counts scale is a question about longer rows, and the answer over five coins from {−2, 1, 2} is that a hundred and fifty-six of two hundred and forty-three rows break the condition without a pass and none breaks it with one — the same shape, further out.

The convention, named #

Three conventions are in play and the essay is about the collision between two of them.

Scoring: the game ends by agreement or exhaustion and the higher total wins. That is the convention the coin row is played under and it is the one that needs a stopping rule.

Normal play: the player who cannot move loses. That is the convention every other field on this site uses, and a passing game satisfies it vacuously — nobody can ever fail to move — so it has nothing to say.

And the incentive condition, which is not a convention but is treated like one. It is a hypothesis about positions, it is what the older theory needs, and the finding here is that a rule which makes the second convention useless makes this hypothesis automatic. A clause cannot serve both.

The surprise: the older theory’s awkward hypothesis was a description of Go #

The natural way to read Milnor’s non-negative-incentive condition is as a technical restriction: the theory works on some scoring games and not others, and the ones it works on are picked out by a condition somebody had to notice.

The measurement says the condition is not about which games are tractable. It is the fingerprint of a pass, and a pass is what Go has and what the coin row was built without. Every scoring game with a pass in it satisfies the condition automatically; every counterexample the rung below exhibits is a game with no pass; and the theory’s hypothesis, read that way, is not a restriction on scoring games at all but a description of the ending rule the games it was written about happen to use.

That inverts what a reader takes from the older theory. It is not a general theory of scoring games with an awkward clause. It is a theory of passing scoring games, stated for a class wide enough that the passing has to be assumed rather than assumed away.

And the general shape is worth carrying. A hypothesis that looks like a restriction on positions is often a consequence of a rule the author was not thinking about, because the games in front of them all had it. The same reading applies to the normal-play convention itself, which reads as a fact about games to anybody who has never been shown an alternative. The way to find out which is to build a game without the rule and watch the hypothesis fail — which is what the coin row is for, and is the whole reason a laboratory is worth having.

Where the ladder goes next #

scoring now has three rungs: what the last-move convention costs on a game that keeps score, and what a pass does to the hypothesis that costs it.

The rung above is about the hypothesis itself, and it is sharper than it looks. Milnor’s condition is checked here on the row, and the bound it supports is proved by induction over the play — so the condition a proof can use has to hold at every position the play reaches, not at the position it starts from. Those are different conditions, a row can satisfy one and not the other, and the gap between them is where the bound actually breaks.