Wrong in one direction only

What the class does not buy measured misère comparison inside four rulesets’ universes and found the dead-ending class explaining none of the counts. It closed by naming the measurement that would put the hypothesis to work:

Implementing the simplified test and checking that it agrees with the quantifier — on the dead-ending rulesets, where it should, and on Toads and Frogs, where it should not — is the census that would put the hypothesis to work. That is the rung above and it needs the test written.

The test is written here in three forms. It does not agree with the quantifier anywhere, it agrees best on the ruleset that is not dead-ending, and the way it fails is worth more than the way it was supposed to succeed.

What the tests are #

Misère comparison over a universe is a quantifier: $A \ge B$ when for every $X$ in the universe, the misère outcome of $A + X$ is at least that of $B + X$. That is expensive — a sum and an outcome for every member — and a simplified test would replace it with a condition on the two games’ options.

Three candidates are scored:

The recursion. $A \ge B$ unless some Right option of $A$ is at most $B$, or some Left option of $B$ is at least $A$ — checked recursively by the same test. Under normal play that is a theorem; under misère it is a candidate.

The recursion with an end condition. The same, with the clause dead-endedness is about: under misère play an end is a win for the player who has run out, so a position that is an end for Left is one Left is happy in, and $A \ge B$ ought to require that a Left end is only ever at most another Left end.

The two outcomes compared, which is the control: a test with no content, included so that a recursion beating it means the recursion is reading something.

These are this site’s constructions and not the literature’s theorem. The rung below refers to a simplified test stated under the dead-ending hypothesis; what a negative result about these three establishes is a negative result about these three.

They agree with nothing #

The recursion agrees with the quantifier on 308 of 492 ordered pairs across the four rulesets — 64 per cent on Domineering, 51 on Shove, 68 on Toppling Dominoes.

That is not a characterisation by any reading. A test that is right two times in three on a relation that holds on 30 of the 492 pairs is a test that is mostly saying yes.

The base rate is the number to read it against. Only 30 of the 492 pairs compare at all — misère comparison over a universe is a very strong condition, which is the whole reason equal in this company had to make it computable before anything could be said about it. A test that simply refused every pair would agree on 462 of the 492, which is 94 per cent. So the recursion’s 64 per cent is not merely below a characterisation; it is far below the trivial rule that says no to everything.

That comparison is worth making because it inverts the natural reading of the agreement column. The recursion is not a nearly-right test with a residue. It is a test that is wrong in a very particular way — it accepts far too much — and the rest of this page is about what that particular way is good for.

And the prediction inverts. The recursion agrees on 83 per cent of Toads and Frogs’ pairs — the one ruleset here that is not dead-ending — against Domineering’s 64 and Shove’s 51.

That is the third measurement on this anchor to find the class explaining nothing, after the comparison counts and the stability under enlargement. Three negative results with three different instruments is a different kind of statement from one, and it is worth saying plainly: on this site’s evidence the dead-ending hypothesis does no measurable work.

The end condition makes it worse #

The end condition is the piece of the hypothesis that ought to be load-bearing, so its effect is the sharpest thing here.

Adding it makes the test agree on 288 pairs against the plain recursion’s 308. On Domineering it falls from 117 of 182 to 102; on Toppling Dominoes from 38 of 56 to 25. It helps on Shove — 93 to 103 — and on nothing else.

A condition that a hypothesis licenses should not make a test worse on the rulesets the hypothesis holds for. That it does is not proof the hypothesis is empty; it is evidence that the version of the condition available to this site is not the one the hypothesis is about, and that the difference matters. Getting a proviso slightly wrong and losing fifteen agreements on one ruleset is the ordinary fate of a condition guessed rather than derived, which is why the limits below say exactly what these three tests are and are not.

The control behaves as a control should. Comparing the two misère outcomes and nothing else agrees on 265 pairs, below both recursions — so the recursions are reading something more than the outcome, which is the minimum a test has to clear.

The way they fail #

The interesting column is the one nobody asked for. Split each test’s disagreements by direction and the false negatives are nought — across all three tests, all four rulesets, all 492 pairs.

No option test ever refuses a comparison the quantifier accepts.

That makes the recursion an upper bound on misère comparison rather than a characterisation of it. It accepts everything true and a good deal besides; what it never does is throw away a real comparison.

Whether that holds in general is not settled here. It is a measurement over 492 pairs and four rulesets, and its being true of the outcome-only control as well suggests it is a property of the direction the tests are written in rather than a deep fact — a test that looks for obstructions and finds none says yes, and misère play makes obstructions harder to find than they are in normal play. What would settle it is a single counterexample — one pair a recursion refuses and the quantifier accepts — and 492 pairs did not produce one.

Why misère breaks the normal-play test #

The recursion is a theorem under normal play and a bad guess under misère, and the reason is worth spelling out because it is the whole difficulty the anchor exists inside.

Under normal play, $A \ge B$ fails exactly when one of two things happens: Right has a move in $A$ that already reaches something at most $B$, or Left has a move in $B$ that reaches something at least $A$. The argument is a strategy argument — whichever of those exists, the player who benefits plays it, and the comparison collapses. It needs one fact: a player with no move has lost, so the recursion has a base case that agrees with the order.

Under misère a player with no move has won, and the base case inverts. A position with no Left options is one Left is delighted by, so the recursion’s bottom is upside-down relative to its top, and the strategy argument that works at every internal node fails at the leaves. That is not a small correction: two misère outcomes are not enough is the anchor’s record of how much it changes.

What the measurements here add is the shape of the failure. The recursion still finds every genuine obstruction — an obstruction is a move, and a move is a move under either convention — but it no longer finds enough of them, because under misère a comparison can fail for a reason that is not a single move. That is exactly one-sidedness, and it says the recursion is looking in the right place and not looking far enough.

Nobody comes back is where the class was supposed to help with precisely this: in a dead-ending universe a finished component stays finished, so the inverted base case is at least stable. The measurements say it is not enough stability to repair the recursion.

What a one-sided test is good for #

A test that never refuses a true comparison is a filter. A program can run it first, and put the expensive quantifier only to the pairs it passes, with no risk of a missed comparison.

It removes between 49 and 64 per cent of the quantifier’s work: 107 of Domineering’s 182 pairs never reach the quantifier, 89 of Shove’s 182, 36 of Toppling Dominoes’ 56. Across the four rulesets that is 60,396 sums and misère outcomes never computed, because a quantifier skipped is a whole universe skipped.

That is a larger saving than it looks, because the quantifier’s cost grows with the square of the ruleset — a universe is the ruleset’s positions plus their sums, so doubling the games quadruples the universe and quadruples every quantifier call. A filter removing half the calls therefore removes half of a quadratic, and it is the difference between a sweep of fourteen positions and a sweep of twenty.

And the saving is the same whether the ruleset is dead-ending or not — Toads and Frogs saves 64 per cent, the best of the four. So the one useful thing this page produces is available without the hypothesis, which is a fourth way of saying the same thing.

Three instruments, one answer #

It is worth collecting what the anchor has now asked of the dead-ending hypothesis, because three different questions have returned the same answer and that is a stronger statement than any of them alone.

Does the class make comparison commoner? What the class does not buy counted comparisons inside each ruleset’s own universe and found the dead-ending rulesets comparing on 2 to 5 per cent of their pairs against Toads and Frogs’ 19.

Does it make comparison stable under enlargement? The same page enlarged each universe by another ruleset’s and found ten comparisons lost — all of them to Toppling Dominoes, which is dead-ending.

Does it license a simplified test? This page, and the answer is no in the direction asked and inverted in the measurement.

Three instruments, four rulesets, and nothing separates. What that licenses saying is narrow and worth saying exactly: on the games this site can evaluate, membership of the dead-ending class predicts nothing measurable about misère comparison. What it does not license is the class is empty, because the class is a hypothesis about how a proof goes rather than about how many comparisons there are, and a proof is not something a census can see.

That distinction is the honest boundary of the whole anchor. A hypothesis that makes a theorem provable and changes no count is doing real work in a place a measurement cannot reach, and three null results are compatible with that. What they rule out is the reading a reader most easily forms — that dead-ending games are somehow better behaved — and ruling that out is what a null result is for.

A one-sided error is a usable error #

An error with a known sign is worth much more than a smaller error with an unknown one, and it is worth stating why, because the distinction decides whether a reading can be used at all.

A two-sided error has to be treated as a range. If a reading can be wrong in either direction, the only safe use is to bracket it, and the bracket is twice the error wide. Anything read from the estimate itself is a guess.

A one-sided error is a bound. If the reading is never too small, it is an upper bound — exactly, with no widening — and a bound is a statement that can be used in an argument rather than a number that has to be hedged. The same reading, with the same magnitude of error, delivers a theorem instead of a guess.

That is why the direction is worth establishing separately from the size, and why a census reporting only the reading is right 80 per cent of the time has left out the more useful half. Eighty per cent right and always in one direction is a tool; eighty per cent right in both directions is a hint.

It also says which repairs are worth attempting. A one-sided reading can be tightened by looking for the cases that make it loosest, and every improvement keeps the guarantee. A two-sided one cannot be tightened safely at all without re-establishing the direction, because a correction that fixes the common case can easily break the sign on the rare one — which is the failure the weighted count records on a different ladder.

What this does not say #

Three candidate tests, not the literature’s. The rung below refers to a simplified test stated in the misère literature under the dead-ending hypothesis. What is implemented here is a recursion in the shape of the normal-play test with and without an end condition, and a null result about it does not refute a theorem it may not be.

Four rulesets and 492 pairs. Fourteen positions each at most, with universes of 45 to 210, which is what a quadratic-in-a-quadratic sweep affords. The rung below’s own limits apply unchanged.

Three dead-ending against one that is not. The population available is what nobody comes back found, and a comparison of three against one is weak however it comes out. The right instrument is a game family with a clause that can be switched, and this site’s nearest is the same strip without the jump.

And the filter is measured, not proved. Nought false negatives on 492 pairs licenses the filter on these rulesets. A program using it elsewhere would be relying on a measurement, and the honest form of the claim is that no counterexample was found rather than that none exists.

The convention, named #

Misère play throughout: the player who cannot move wins.

An end for a player is a position from which that player has no move. A ruleset is dead-ending when every end is dead — once a player has no move, no continuation ever gives them one again.

A universe here is a ruleset’s positions, everything reachable from them, and the sums of two of those, which is the company a solver stays inside. $A \ge B$ over a universe means the misère outcome of $A + X$ is at least that of $B + X$ for every $X$ in it, where the outcome order puts a Left win above a first-player or second-player win and those above a Right win.

An option test decides $A \ge B$ from the two games’ options rather than from the universe. It accepts a pair when it says the comparison holds. A false positive is a pair it accepts and the quantifier refuses; a false negative is one it refuses and the quantifier accepts.

The population is every ordered pair of distinct games from each ruleset’s list, which is 492 pairs across the four.

Where the ladder goes next #

The dead-ending anchor has three rungs: nobody comes back, what the class does not buy, and now the test it was supposed to license.

The rung above is the switchable clause. Three negative results on four rulesets is as far as this population can be pushed, and the fault is the population rather than the instruments: three dead-ending rulesets against one that is not cannot separate a hypothesis from a coincidence. What is wanted is one game with a clause that turns dead-endedness on and off, so that everything else about the ruleset is held fixed — and Toads and Frogs with and without the jump is very nearly that pair, since the jump is what lets a stuck player move again. Running this census on the two of them, with everything else identical, is the experiment the anchor has been unable to do for three rungs.

Two neighbours are worth the trip. Equal in this company is where restricted equality was made computable, and it is the quantifier this page is trying to replace. And misère quotients is what the subject does instead of a general theory, and it is worth reading beside three pages of a hypothesis failing to do measurable work. Two misère outcomes are not enough is why a quotient is needed in the first place, and it is the page that names this class from the other side.