A position with no value, and the rule that gives it one

The rung below gives each king a stated, finite supply of waiting moves, and defends the decision in one line: without it the kings could shuffle indefinitely and the position graph would have a cycle in it.

That line is the whole of this rung. A king shuffling indefinitely is not a modelling nuisance to be legislated away — it is a fortress, the one class of pawn ending every endgame manual has a chapter about, and the one class this site’s entire apparatus has nothing to say about.

What a fortress is, in the vocabulary of this site #

An endgame manual describes a fortress as a position the stronger side cannot break down: the extra material is there, the plan is not, and the defender shuffles a king between two squares while nothing happens.

Restated as a graph, that is: play can return to a position it has already been at, for ever, and neither side can force it not to.

Every value on this site is defined by a recursion on options, and the recursion needs the options to run out. Allow a position to recur and the definition has nothing to stand on: the value of the position is defined in terms of itself, and no amount of unrolling reaches a base case. There is no number to compute and none to quote.

So the correct statement about a fortress is not that it is hard to evaluate. It is that it has no value, and the reason has nothing to do with how much material is on the board.

The model, stated exactly #

Three clauses, and each is a real feature of a pawn ending.

A shuffle. The defender has two squares its king may move between, and may do so as often as it likes. Nothing else happens when it does, and nothing else is available to it. That is what makes the graph cyclic.

A march. The attacker needs a stated number of moves of preparation — a king walking round to a square from which the breakthrough works — before the winning advance is available at all. In a real ending that is where the moves go.

A counter. After a stated number of moves with no pawn move, the game is a draw. That is chess’s fifty-move rule, modelled exactly: the counter counts king moves, and both the march and the shuffle are king moves, so nothing in the model resets it except the breakthrough that ends the game.

Setting the counter to no rule removes the third clause, and setting the march to none removes the second. Those two settings are what make the table a measurement rather than a demonstration.

Row one: no way through, and no label #

With no breakthrough available, the defender shuffles and the attacker has nothing to do but shuffle back. Nobody is ever stuck, so nobody ever loses, and the game does not end.

The labelling agrees, and the way it agrees is worth watching. Backward induction starts at the positions where somebody has no move at all and works outward. There are no such positions here, so round zero is empty, so the propagation never starts — and every position-and-mover pair in the graph is left with no label at all.

That is the same statement a draw is not a value makes, on a position from an endgame manual rather than on a two-node example. A drawn position is not a fifth outcome class the procedure computes. It is the residue the procedure never reaches, and a fortress is a residue of exactly that kind.

Two independent computations say so here, and they say it in different vocabularies. The three-valued solver reports a draw as a verdict; the retrograde labelling reports four labels never settled out of four. Neither is checking the other’s arithmetic, and the figure refuses to draw if they disagree.

Row two: a cycle, and a value anyway #

The second row is the one a reader should be suspicious of, because it contradicts the obvious reading of the first.

Give the attacker a breakthrough four moves away and remove the counter. The defender still shuffles for ever; the graph still has a cycle in it; and the position is a win. The labelling settles every pair, in nine rounds, with nothing left over.

The reason is that the defender’s shuffle goes nowhere. Each of the attacker’s moves advances the march by one, the march is finite, and after four of them the breakthrough is available. A cycle that one player traverses while the other makes progress is not an obstacle to anything.

That is the same finding off makes in the loopy field — a position where only one side may move, for ever, and which is settled in two rounds despite being a cycle by any definition. The condition backward induction needs is not no cycles. It is that round zero is reachable from everywhere: every position must have some route to a place where somebody is stuck.

A fortress cuts that route and a shuffle beside a march does not, and the two look identical on a chessboard. Whether a king walking back and forth is a fortress or a nuisance is a question about whether the other side is making progress, which is exactly the question a player is trying to answer and exactly the question the drawing cannot show.

Row three: the rule, and what it takes away #

Now switch the counter on.

The graph becomes finite immediately, and it becomes finite for a reason worth stating precisely: the state now carries a number that increases on every move and is bounded, so no line of play can revisit a state. That is a decreasing quantity in the sense the whole subject rests on, added to a game that did not have one.

Every position now gets a label, and the fortress row gets its draw as a computed verdict rather than as a residue. That is a real improvement in kind: a drawn position under the rule is a position the procedure has examined and filed, and a drawn position without the rule is a position the procedure never looked at.

And then the cost.

Eleven cells of the sweep are wins the rule takes away. A march of four with a counter of three is drawn; the same position with no counter is won. The attacker’s plan is sound, the plan takes longer than the rule allows, and the rule does not care.

This is not a modelling artefact #

It is tempting to read the last row as a consequence of building a toy, and it is not. Chess’s own fifty-move rule has been amended, more than once, for particular endings — because positions were found in which the win is real, forced, and longer than fifty moves.

The mechanism there is exactly the one modelled here. The rule counts moves since the last capture or pawn move; the winning plan in those endings consists of manoeuvres that are neither; the plan takes more moves than the counter allows; and the game is declared drawn while a forced win is on the board.

So the third row is not an approximation of a chess fact. It is the chess fact, on a model small enough to sweep, and the sweep says which cells it happens in and why.

What the amendments do, incidentally, is worth reading as a piece of rule design. They do not remove the counter — a rule set with no repetition rule at all has the first row’s problem everywhere, and chess would stop being a finite game. They raise it, for named position classes, which is a rule that buys termination and tries not to buy any particular answer with it.

What the counter costs in states #

The rule’s price is visible in the size of the object before any of its answers are read.

Without the counter the model’s state is where the march has got to and which square the defender’s king is on: a handful of positions, and the same handful however long anybody plays. With the counter the state is that pair and the number of moves since the last pawn move, so the position count is multiplied by the counter’s own size.

That is not a rounding cost. A rule that adds a coordinate makes the graph as large as the coordinate’s range, and chess’s coordinate ranges to a hundred half-moves. Nothing in the model is troubled by it because the model is tiny; a solver working on real endgame tablebases is troubled by it a great deal, which is why tablebases are usually built without the fifty-move rule and then queried with it afterwards — the two produce different answers, and the difference is a documented list of positions rather than a theoretical worry.

So the rule is not free in either currency. It costs states, and it costs wins, and it buys the one thing without which none of the rest is defined.

A rule that deletes edges, and a rule that adds a coordinate #

There are two ways to make a cyclic game finite and chess uses both, and the difference is visible in what happens to the position graph.

Threefold repetition deletes edges. A move into a position that has occurred twice already is not available, or rather is available and ends the game. The graph loses arrows, and once enough arrows are gone there is no cycle left. That is what Go’s ko rule does, and what superko does more thoroughly.

The fifty-move rule adds a coordinate. The position is no longer a board; it is a board and a number, and the number never decreases within a stretch of play. The graph is not pruned — it is replaced by a larger one that cannot cycle by construction.

The two are not interchangeable and the difference shows in the state count. Deleting edges leaves the number of positions alone; adding a coordinate multiplies it, here by the size of the counter. That is the same price superko pays in Go, where a position stops being a board and starts being a board plus a history, and players find it counterintuitive for exactly that reason.

Both prices buy the same thing, which is the finiteness hypothesis every theorem on this site needs.

The values are still not there #

One thing the counter does not do, and it is the thing a reader might hope for.

With the rule in force every position has an outcome — win, loss or draw — and none of them has a value. A drawn position is still not a number; three outcome classes is not four; and a position that the rule declares drawn cannot be added to anything.

So the counter restores the theorem — every position is determined — and restores none of the arithmetic. That is the honest summary and it is the reason the loopy field exists as a separate subject: the games with cycles in them needed a theory built on graphs and backward analysis rather than on a recursion over options, and what that theory produces is outcomes rather than values.

The distinction is easiest to feel by asking what a player could do with each. An outcome answers one question about one position: this ending is drawn. A value answers every question about it in every context: this ending is worth two, so it can be set beside a fight elsewhere and the total computed. A pawn ending that is a fortress on one wing and a race on the other cannot be added up at all — not because the arithmetic is hard, but because one of the two summands is not the kind of thing that adds.

That is the sharpest available statement of what a cycle costs, and it is a cost the counter does not refund. The loopy theory does have an arithmetic of its own, built by hand for a small vocabulary of named primitives, and it is a much smaller object than the theory it replaces.

Why the first row is the one to hold onto #

Of the three rows, the second and third are about a rule and the first is about a game, and it is the first that says what this rung is for.

A fortress is not a rare shape. It is one of the standard defensive resources, it appears in composed studies and in real games, and a player who reaches one has reached a position about which the whole apparatus of this site is silent — not approximate, not expensive, silent. There is no number, there is no comparison, and there is nothing to add it to.

That is worth setting against the rung at the bottom of this ladder, which finds the theory saying something exact about a real position class and concludes that the theory earns its place exactly where the material is level. Both are true and they are two halves of one boundary. The theory reaches the positions that end, and a pawn ending is a position class that sometimes does not. Which of the two a diagram belongs to is decided by whether either side can make progress, and that is the question a player is trying to answer rather than a hypothesis they can check first.

What the picture cannot show #

The march is a number and a real breakthrough is a plan. In this model the attacker needs some stated count of preparatory moves and then wins; in a real ending the preparation is a king walking to a square, the defender may have counterplay, and whether the plan works at all is the question. The model assumes the plan works and measures only how long it takes.

And the shuffle is drawn as two squares. A real fortress is a defensive setup — a king and pawns arranged so that no entry square exists — and what makes it a fortress is a geometric fact about the whole board. Nothing here is geometric. What survives the simplification is the only question this rung is about, which is whether the position graph has a reachable cycle, because that question does not depend on why.

Nor does anything here model the fifty-move rule’s actual clause. Chess resets the counter on a capture or a pawn move, and this model has no captures and exactly one pawn move, which is the one that ends the game. A real ending resets the counter repeatedly and the interesting cases are the ones where it does not.

The convention, named #

Two conventions are in play and it is worth keeping them apart, because they disagree about what a draw is.

Normal play says a player with no move loses. Under it there are four outcome classes and a fortress is in none of them; the fourth possibility — neither side can force a win — is not a class the procedure produces, it is what is left when the procedure stops.

Chess says a game may be drawn by rule, and a draw is a result: half a point, recorded, final. Under that convention a fortress is a perfectly ordinary outcome and the fifty-move rule is what makes it official.

The substitution this essay performs is to use the first convention’s machinery on the second convention’s rule, and it is legitimate for a narrow reason: whether a graph has a reachable cycle does not depend on what anybody scores. The labelling reports the same set of positions either way. What changes is whether that set is a defect or a result, and the answer to that is not mathematics.

The surprise: the rule that makes the theory possible is the rule that gets the answer wrong #

The natural way to read the three rows is as a progression from worse to better. No rule, no answers; a rule, answers; therefore the rule is an improvement.

The sweep says something less comfortable. The counter is what makes every position labelled, and it is what labels eleven of them wrongly. There is no setting of it that does one without the other, because the two are the same mechanism: the rule bounds the length of play, bounding the length of play is what makes the graph finite, and a win longer than the bound is a win the bound denies.

That trade has no good side. A shorter counter settles the theory faster and steals more wins; a longer one steals fewer and leaves the graph larger; and removing it entirely gives back every win and takes away the theorem. Chess’s answer — fifty, then amendments for named endings — is a series of patches on a trade that does not have a clean solution, and it looks like a series of patches because it is one.

The general shape, and it is the one to carry: a termination rule is not neutral. Go’s ko rule buys finiteness by deleting an edge and the three candidate rules give three different answers to the same fight. Chess’s counter buys finiteness by adding a coordinate and gives different answers at different settings. In both games the rule was adopted to stop something annoying and turned out to be the hypothesis under every theorem about the game — and in both, somebody had to notice afterwards that the cheap rule and the correct rule are not the same rule.

Where the ladder goes next #

chess now has four rungs and they are the four things the theory can say about a pawn ending: it is a sum, its values depend on which clause of the move rule holds, the sum fails when a piece is shared, and the whole apparatus stops where the position stops ending.

The rung above is the one Elkies’s paper spends its length on and none of these have taken: which real pawn structures reduce to which of these models. Every value here is exact for a stated rule set and every rule set is a model, and the step from a diagram in an endgame manual to one of these four models is a claim about chess that no computation on this site can make. It is a rung about the dictionary running the other way — from a board to a clause — and it needs a reader who plays chess rather than a solver.