Left always wins, and loses more often than not

Alternating play sorts every position into one of four classes, and two of them are unconditional: Left wins whoever moves, or Right does. Those are the strongest statements the convention makes. The coin replaces the turn order with a flip and answers with a probability instead, and there is no reason in either definition why the two should line up.

They very nearly do, and where they fail to is worth the whole essay.

A certainty becomes a rate #

The first thing the coin does to an unconditional answer is take the un off it.

A lead of three free moves is as decided as a position gets. Left has three moves and Right has none, so under alternating play Left wins from it whoever starts and there is nothing to discuss. Under the coin, Left wins it fifteen times in sixteen.

The missing sixteenth has a shape: it is the coin naming Left four times in a row. Left spends the three free moves and is named once more with nothing to play. Four flips, one in sixteen, and the lead is gone.

That generalises exactly. A lead of $n$ is lost with probability $2^{-(n+1)}$ — a lead of ten one time in two thousand and forty-eight, a lead of twenty one time in two million — and never with probability nothing. A coin turns every certainty into a rate, and the rate is a power of two set by how many free moves there are to burn.

It also explains why no probability on either pool is ever nought or one. However decided a position is, there is a run of flips that exhausts the winner, and a fair coin gives every run some chance.

The four bands on the small pool #

Over the day-two values the classes sort themselves almost perfectly.

The six values Left wins whoever moves run from a half to seven eighths; the six Right wins run from an eighth to a half; the nine first-player wins run from three eighths to five eighths and average exactly a half; and the single second-player win sits at a half. The bands overlap only at the endpoints, and they overlap there for a reason rather than by accident — a half is where the positions the flip cannot touch all end up.

So on this pool the two conventions never contradict each other in the strong sense. A position Left wins whoever moves is never a position the coin gives Left less than an even chance. That reads like a theorem, and the margin by which it holds is exactly nothing: the lowest such value is at a half, not above it.

A claim that holds with no margin on the pool where it is checked is a claim to check on a bigger one.

Seven positions that go the other way #

On day three the claim fails, and it fails on fourteen values.

Seven of the 217 values Left wins whoever moves are given a chance of seven sixteenths, which is to say that Left, holding a position that cannot be lost under alternating play, loses it more often than not when the turns are random. Seven more do the same for Right, by the mirror symmetry the whole subject has.

The smallest of them has one Left option, the empty position, and one Right option, a first-player win. Alternating play reads it in two lines. Left moving first reaches the empty position, where Right has to move and cannot, so Left wins. Right moving first reaches a first-player win with Left to move next, so Left wins that too. Both branches are Left wins and the class is unconditional.

The coin reads the same two options and gets a half and three eighths. The empty position is a Left win under alternating play only because the turn passes to Right; under the coin it is a fair flip, worth a half. The first-player win Right can move to is a Left win under alternating play only because Left moves next; under the coin nobody is guaranteed to move next, and it is worth three eighths to Left. Average a half and three eighths and the answer is seven sixteenths.

Both of alternating play’s wins were wins about whose turn it was. That is the whole mechanism, and it is why the failures cluster where they do: every one of the seven has a Left option worth exactly a half and a Right option worth three eighths or less, which is to say every one of them is built out of positions whose alternating verdict is entirely a matter of the turn passing.

Alternation guarantees that a player’s move is answered. A coin does not, and a position whose safety is assembled out of that guarantee is a position a coin can take apart piece by piece.

Why the first-player class sits exactly on a half #

One column in both censuses is exact rather than approximate, and it is worth a paragraph because it is the one thing here that is a theorem rather than a count.

The nine first-player wins on day two average exactly a half, and so do the 1,039 on day three. That is not a tendency: the mean is a half to the last digit on both pools.

The reason is the mirror. Negating a game exchanges the two players, and it sends the coin’s number to one minus itself — a position giving Left a chance of five eighths becomes one giving Left three eighths — which follows from the recursion by induction, since negation swaps the maximum and the minimum and swaps the two base cases. The first-player class is closed under negation, because a position is a first-player win exactly when its negative is. So the class is a union of pairs each summing to one, together with any self-negative members sitting at a half, and the mean of such a set is a half whatever else is true of it.

The same argument applies to the second-player class, which is why the single value there is at a half exactly.

It also says that the two unconditional classes are each other’s mirror image, so the seven positions pointing the wrong way on one side had to be matched by seven on the other. The fourteen are seven facts, drawn twice.

What a second-player win becomes #

The single second-player win on either pool is the empty position, and what happens to it is the clearest small case of the same thing.

Under alternating play it is the most decided position there is: whoever must move has lost, full stop, and every recursion in this collection stops there. Under the coin it is the most undecided position there is — the flip is the entire game, and the answer is a half for a reason that has nothing to do with the position and everything to do with the coin.

That is the trade in one line. Alternating play’s certainty about the empty position is certainty about a turn, and there is no turn under the coin to be certain about.

It is also why the second-player class cannot grow. A position is a second-player win when both players moving first lose, and both of those are statements about who moves next; a coin that may name the same player twice makes both of them probabilistic at once. So the class alternating play treats as the spine of the subject — the one every value is measured against — is the class the coin flattens hardest.

What the classes are not #

The census is easy to over-read in a particular direction, and it is worth blocking.

An up is a Left win whoever moves, and the coin gives it exactly a half. The two answers are not in conflict and they are not in agreement either: one of them is saying that Left cannot lose and the other that the flip decides, and both are correct about their own convention.

What makes it possible is that the flip is idle here — both branches are a half, so the coin changes nothing about the position and everything about who runs out. Left’s guarantee under alternating play is the guarantee of getting the last move, and an up is precisely a position where Left has one more move than Right in a sense too small for a number to express. A coin hands out moves at random, so one more move stops being a reliable advantage and becomes a coin flip.

The infinitesimals are exactly the positions whose whole content is the parity of the moves, which is the quantity a coin destroys. That is why all eight of them land on a half and why their classes are scattered across all four.

It is worth putting that beside the auction reading’s account of the same eight. There the collapse was read as the auction being unable to tell them apart — an advantage smaller than every number being worth nothing when a move can be bought. The coin says something different and better founded: the flip is idle in all eight, because both branches of the average are equal, so there is nothing to tell apart in the first place. One reading blames the instrument and the other locates the sameness in the positions, and only the second is computable from the branches.

And the eight are not all the infinitesimals there are — they are the ones born by day two. On day three the same behaviour spreads to eighty-two values whose flip decides nothing, every one of them at temperature nought, spread across seven different chances rather than all sitting at a half.

Where the ordering survives #

It would be easy to leave this reading with the impression that the two conventions are simply different, and they are more closely related than that.

The coin’s number respects the game order. If Left is at least as well off in one position as in another — the comparison the whole value theory is built on — then the coin gives Left at least as good a chance in the first. That holds on every comparable pair of both pools, and it is the structural property the auction reading also claimed and kept.

So the failure above is not a failure of order. The fourteen positions pointing the wrong way are positions where the class and the chance disagree, and a class is not a point in the order: it is a statement about two particular comparisons, with nought and with the positions reachable in one move. The coin agrees with the order everywhere and disagrees with the classes in fourteen places, which is only possible because the classes are not determined by the order either.

The classes and the order are two different readings of alternating play, and the coin preserves one of them. That is a much sharper description of what survives the change of convention than mostly agrees, and it is the reason the disagreements are rare rather than random: they need a position assembled from parts whose classes come from the turn passing, and most positions are not.

The chance a number gets #

Numbers deserve their own paragraph, because they are the positions where alternating play is most confident and the coin is least impressed.

A lead of $n$ free moves comes out at $1 - 2^{-(n+1)}$: three quarters at one, seven eighths at two, fifteen sixteenths at three. Every one of those is an unconditional win under alternating play and none of them is a certainty under the coin, and the shortfall halves with each extra move.

What makes the progression readable is that it is the only thing a number carries here. A number is a position neither player wants to move in, so its whole content under alternating play is a count of spare moves and its whole content under the coin is the chance of a run long enough to spend them. Those are the same quantity read twice, which is why the formula is so clean — and it is the one family where the two conventions agree about what is being measured while disagreeing about whether the answer admits doubt.

Two conventions running side by side #

Alternating play for the classes, random turns for the chances, and normal play throughout: the player who cannot move loses, and under the coin that means the player the flip names.

Two conventions of the census itself.

The values are counted, not the forms. Two ways of writing the same value have one canonical form and therefore one class and one chance, so a pool of 1,474 is a pool of 1,474 distinct values rather than of the many more expressions that name them.

The comparison is with alternating play and not with the auction reading, which has no play-out to compare against. Every chance here comes from the coin.

Day three is complete rather than sampled. Every value born by day three is in the census, which is what makes fourteen a count rather than an estimate — a sample could have missed all fourteen, and on the evidence of day two it very nearly would.

What two pools cannot settle #

Two pools is two pools. Whether the disagreement grows, and how fast, is a question about day four, which has too many values to enumerate the way these are enumerated.

And the fourteen are not shown to be extreme. Seven sixteenths is the lowest chance found on a Left-always-wins value in this range; nothing here says a later day does not go lower, and nothing says it does. The obvious guess is that the floor falls with each day as the recursion gains depth, and it is a guess.

Nor does any of this say a class is wrong. Alternating play’s classes are exact statements about alternating play, and the coin’s numbers are exact statements about the coin. What is measured is the relationship between two conventions, and both of them are still right about themselves.

Why the mirror makes the count even #

The seven and the seven are not two findings and it is worth being exact about why, because the argument is short and it is the same one that fixes the first-player class at a half.

Negating a position exchanges the two players: Left’s options become the negatives of Right’s and the other way round. The values that are their own negatives are the fixed points of that operation, and everything else comes in pairs.

The coin’s number sends a negation to one minus itself, and the check is exhaustive: over all 1,496 values across both pools, the chance the coin gives Left in a position and in its negative always sum to exactly one. So a position Left wins whoever moves with a chance below a half corresponds to a position Right wins whoever moves with a chance above a half, and the seven on each side are the same seven read backwards.

That also fixes what a larger sweep would find. Any count of such positions on any pool closed under negation is even, and half of it is the real number. Fourteen is seven.

Still open: whether the floor keeps falling #

The measurement the next essay of this would make is the floor, day by day: the lowest chance the coin gives on a position its owner wins whoever moves.

Day two puts it at a half exactly, with no position below and none strictly above it either — the floor and the boundary coincide. Day three puts it at seven sixteenths. If the pattern is that the floor falls by one grid step a day it will reach a quarter around day six and approach nothing eventually, which would mean there is no bound at all on how badly the two conventions can disagree. If instead it settles, the settling point is a real constant of the subject and nothing here has a reason to expect one.

Enumerating day four is out of reach by the route used here. What is not out of reach is building the deep examples directly: the seven positions found on day three have a shape, and a family built to that shape, driven one essay deeper at a time, would say which of the two answers the floor gives.