The best chance is the wrong move

A coin needs no tie-break leaves a number with a sound derivation and says nothing about play. The recursion has a maximum in it — at Left’s turn, Left takes whichever option leaves Left best off — so it does name a move, and whether that move is the move an alternating player would want is a question with two answers.

Two objectives #

The two players are optimising different things and the difference is one word.

Under alternating play, Left wants an option from which Right, moving next, loses. That is a statement about the opponent’s position and it has two answers: the option is winning or it is not. There is no better and worse among the winning ones.

Under the coin, Left wants the option with the highest chance. That is a real number and it orders every option strictly. A position that is a certain win under alternating play may have a lower chance than a position that is not, because the chance is about surviving runs of flips and the class is about who moves next.

Nothing forces those two orders to agree, and on the small pool they do.

Three positions and no disagreement #

The day-two pool contributes three positions with a choice and a win available, and on all three the coin’s move is a winning move.

All three are the same shape. A star-two, an up-star and the day-two switch with a star in its left option each offer Left the empty position or a star, and both of those have a chance of exactly a half. The coin is indifferent at every one of them.

That is a sweep with nothing in it. Three positions, no disagreement, and no possibility of one, because the coin never expressed a preference to be wrong about. A reading of the small pool alone would have concluded that the two conventions share a strategy, on the evidence of three positions where one of them declined to have an opinion.

It is also a warning about how thin the day-two pool is for questions about choices rather than about values. Twenty-two values, and only three of them put Left in front of a genuine decision with a win available. Most of the pool has one option or none, which is the same thinness the census of deciding turns finds on small boards in a different subject.

The smallest disagreement #

The larger pool has 904 such positions and the coin’s move loses on 189 of them. The smallest is two options wide.

Left’s two options are the empty position and a switch. The empty position gives Left a chance of a half; the switch gives five eighths. The coin takes the switch.

Under alternating play the arithmetic runs the other way. Moving to the empty position leaves Right to move with nothing to play, so Right loses and Left wins. Moving to the switch leaves Right to move at a first-player win, so Right takes it and Left loses. One of the two options wins and the coin chose the other.

Why a first-player win looks good to a coin #

The mechanism is the same one running through this whole anchor, and here it is at its shortest.

A first-player win is a position whose whole value is being to move in it. Under alternating play, moving into one hands it to the opponent, which is why it is never a good move when a win is available elsewhere. Under the coin, nobody is handed anything: the flip decides who moves next, so a first-player win is worth an even chance of a good thing rather than a certainty of a bad one.

So the coin systematically over-values exactly the positions alternating play treats as poisoned. The switch here is worth five eighths precisely because Left has a fair chance of being the one to move in it, and that fair chance is what alternating play does not give.

The empty position is the mirror image of the same fact. Alternating play calls it decisive, because the opponent has to move in it; the coin calls it an even flip, because nobody has to.

The two conventions disagree most about the positions whose content is whose turn it is, and those are the positions a normal-play theory is mostly made of.

What a move to a first-player win is for #

The over-valuation has a second reading worth setting out, because it says something about the alternating theory rather than about the coin.

Under normal play a first-player win is the one class that cannot be handed over safely. A Left win can be handed to Right and stays a Left win; a second-player win handed to Right is a Left win; a first-player win handed to Right is a Right win. That asymmetry is the whole content of the class and it is a fact about whose turn it will be.

So the first-player class is the class that exists only because turns alternate. Remove the alternation and there is nothing for it to be: a coin gives both players a fair shot at moving in it, so it becomes an ordinary position with an ordinary chance attached, and a rather good one.

That explains the direction of every disagreement here. The coin’s errors are all the same error — moving into a first-player win when something else was available — and they are errors only relative to a convention in which handing over the move is a real transaction. The 189 are not 189 different mistakes; they are one mistake, made 189 times, by a player using a number that has no concept of handing anything over.

Two options and eighty-one of them #

Of the 189 disagreements, 81 are positions where Left has exactly two options — the smallest a disagreement can be. The rest are wider.

That matters because a two-option disagreement admits no excuses. There is no question of the coin having ranked several near-equal moves and picked the wrong one of them; there are two options, one wins and one loses, and the number prefers the loser. Whatever the coin is measuring, it is not a refinement of the alternating verdict.

The share is worth stating plainly as well. 189 of 904 is better than one in five, on positions where a win was there to be taken.

What this does not say #

Two readings are available and only one is right.

It does not say the coin plays badly under its own convention. The option it takes is the option maximising Left’s chance under random turns, and that is exactly what a random-turn player should do. Measured against random turns, every one of the 189 moves is correct.

It says the two conventions do not share a strategy. A player who has learned to play well under one is not playing well under the other, and the failure is not at the margins — it is on a fifth of the positions where a choice matters, at the very smallest sizes, with two options to choose between.

That is worth holding against the temptation to treat random turns as a slightly noisier version of alternating play. A coin turns every certainty into a rate, which sounds like noise; here it turns a move into its opposite, which is not.

Where it does not go wrong #

The 715 agreements are worth a paragraph too, because a fifth going wrong means four fifths going right and that is not nothing.

Most of the time the option with the highest chance is also a winning option, and the reason is that the two orders are correlated even though they disagree. A position good for Left under alternating play tends to be good for Left under the coin as well — the coin’s number respects the game order on every comparable pair, so a strictly better position never gets a worse chance.

What the order does not constrain is the comparison between incomparable positions, and a winning option and a losing option need not be comparable. The empty position and a switch are incomparable in the game order: neither is at least as good as the other for Left. So the coin is free to rank them however its arithmetic falls, and the game order has nothing to say about it.

Every one of the 189 disagreements is a choice between options the game order does not rank — checked, and it could have failed. A blunder involving a comparable pair would be the coin preferring a strictly worse position to a strictly better one, which would contradict the monotonicity the number has on every comparable pair of both pools, and would take that result down with it. There are none.

So the coin’s total order is exactly as much more than the game order as it can be and no more. A total order on a partially ordered set has to invent comparisons somewhere, and this is where the inventions cost something.

The wider census behind it #

The census behind the disagreement is the overlap between the classes. Day three’s first-player wins spread from five sixteenths to eleven sixteenths, and its Left-wins-whoever-moves class spreads from seven sixteenths to fifteen sixteenths. The two ranges overlap over most of their length, so a first-player win with a chance above some Left win is not a rarity — it is the normal case.

Every disagreement is one instance of that overlap being read as an ordering. The coin has a total order on positions and alternating play has four classes, and a total order that crosses the classes will disagree with them wherever it crosses.

What would make the two agree #

It is worth asking what a random-turn player would have to know to play the alternating game correctly, because the answer is short and unflattering to the number.

They would have to know the outcome class, which the chance does not determine: a half is held by a Left win, a Right win, a first-player win and the second-player win at once. No function of the chance can recover the class, so no amount of care with the number gets a player to the right move.

What they would need instead is the class itself, which is to say the alternating theory. And having it, they would have no use for the chance, since the class already names every winning move.

That is the sharpest statement of what the two conventions are to each other. Neither number is a refinement of the other and neither is recoverable from the other, and a player has to choose which game they are playing before they can be told anything. The auction reading two essays below reached the same wall from a different side — its number was fine and it had no game underneath it — and this one has a game and a number that does not play it.

One more way of putting it #

The shortest statement of the finding is about what a number can be asked to do.

A total order on positions is a strong thing to have. It ranks everything, it never hedges, and it is exactly what a player wants when choosing a move. The value theory declines to provide one: its order is partial, two positions are often incomparable, and the four outcome classes are the coarse answer left when the fine one refuses.

The coin provides a total order and the price is that it is wrong about a fifth of the choices that matter. That is not a defect of the coin — it is answering its own question correctly — and it is a fair summary of what a total order on a partially ordered set costs.

A number that always answers is a number that sometimes answers the wrong question, and here the wrong answers are countable.

Which positions count, and which are excluded #

Alternating play for the verdicts and random turns for the chances, with normal play throughout.

Three conventions of the count.

A position is counted only when Left has more than one option and at least one of them wins. With one option there is nothing to choose; with no winning option there is nothing to get wrong. Both exclusions make the count smaller and the share larger, and both are necessary for the question to be about choosing.

A tie among the coin’s preferred options counts as agreement if any of them wins. A coin-flipping player indifferent between two options might take either, so counting such a position as a blunder would be counting a position where the coin has no opinion. On the small pool that is every position, which is why it reports nothing.

And only Left’s choices are counted. Right’s are the mirror image by the negation symmetry, so counting both would double every number without adding a fact.

One move is not a game #

Constructed values, not a game’s positions. The census is over the values born by day three, and a real game’s positions are distributed quite differently — with far more of them close and far fewer of them extreme, which would probably raise the share rather than lower it.

One move, not a game. The count is of single choices made from a position, not of games lost. A player making the coin’s move at every turn would lose more often than 189 of 904 suggests, or less, depending on how the errors compound, and nothing here follows either.

And the coin’s move is not the only random-turn move. Where several options share the best chance, the coin has no reason to prefer any of them, and this counts such a position as agreement whenever one of them wins. A stricter count — requiring every best option to win — would be larger, and it would be measuring a player with no tie-breaking judgement rather than a player with one.

The same shape one subject over #

The finding has a close relative in a different corner of this collection, and setting them beside each other says what kind of fact it is.

Replacing an opponent by a fixed rule recovers more than four fifths of Nim’s lost positions and almost none of Domineering’s, because a Nim loss is held by an invariant somebody has to keep restoring and a Domineering loss is held by a shortage that needs nobody. The measurement there is what happens when the opponent stops choosing.

Here the opponent is still choosing and what has been removed is the guarantee that they choose next. The 189 are the price of that, and the positions where the price is paid are the ones whose verdict is held by whose turn it is — which is the same distinction, between a verdict somebody has to maintain and a verdict already written into the position.

Both experiments find that a normal-play verdict is a maintained thing rather than a possessed one. One takes away the maintainer and one takes away the schedule, and each finds the same set of positions collapsing.

Still open: whether the errors compound #

The obvious next measurement is the one a player would ask for: what does it cost to play the coin’s move every turn under alternating play?

The count above is of single positions. A game is a sequence of them, and the two ways it could go are opposite. A blunder at one turn might be recoverable at the next, in which case the per-game loss is far below the per-move share. Or a blunder might put the position where every later move is also a blunder, in which case one mistake decides the game and the per-game loss is far above it.

The measurement is a tournament rather than a census: the coin’s player against the alternating player, from every position of a real game with each side moving first, and the winner recorded. Nothing here needs new theory and the answer is not guessable from anything on this page — which is the shape of a question worth asking.