A coin needs no tie-break

The auction never gets to the money leaves a recursion with nothing to be about. Every number it produces is still there, the order it respects is still respected, and the quantity the numbers were said to measure has gone.

There is a second derivation of the same recursion and it does not use money at all.

A fair coin decides who moves at each turn. The player it names moves, choosing among their own options; a player whose turn it is with no move has lost; and both play to win. That is a complete ruleset with no convention left over, and the probability it assigns is computed by the same three lines.

The two derivations #

They are written from different rules and they arrive at the same three lines.

The auction reading says: if Left wins the auction, Left moves to whichever option leaves Right needing the most; if Right wins it, Right moves to whichever leaves Right needing the least; nobody knows which will happen, so average.

The coin reading says: with probability a half the coin names Left, who takes whichever option leaves Left best off; with probability a half it names Right, who takes whichever leaves Left worst off; and the average is what a fair coin is.

The base cases match as well. A player named by the coin with no move has lost, so a position where Left cannot move contributes nought to Left’s chances and one where Right cannot move contributes one. Those are exactly the two ends the auction reading supplies, and they arrive for a different reason: there, they encode a penalty for winning an auction; here, they encode losing.

Two turns in a row #

The coin changes the game in one way that has nothing to do with money and everything to do with what a turn is. It can name the same player twice.

Under alternating play a free move for Left is a win for Left whoever starts: if Left starts, Left takes it and Right is stuck; if Right starts, Right is stuck immediately. Two lines, one answer, no hedging. Under the coin the same position is a three-quarter chance, and the missing quarter is the case where the coin names Left twice — Left takes the free move, and is then named again at the empty position with nothing to play.

A position won by moving last is lost by being made to move twice, and that is the whole difference between the two conventions in one sentence. Alternation is what guarantees a player their move is answered; a coin guarantees nothing of the kind, and a run of turns is a liability under a convention that punishes running out.

That is also why every probability on the pool sits strictly between nought and one rather than hitting either end. However good a position is for a player, a long enough run of turns exhausts it, and a fair coin gives every run some chance.

Whose probability it is #

The number belongs to Left, and it is worth pinning down because the auction reading attaches it to Right.

A single free move for Left comes out at three quarters. Under the coin, Left wins that position three times in four: the flip names Right half the time, and Right loses immediately with nothing to play; it names Left the other half, Left moves to the empty position, and the empty position is an even split. Three quarters, and the arithmetic is two lines.

Under the auction reading the same number was the share Right needs. Both readings put a large number on a position that suits Left, so nothing looks wrong from a distance, and they are saying opposite things about who the number is a number of. The coin settles it, because the coin’s number is a probability and a probability has an owner.

The position the coin decides entirely #

The empty position is where the two readings are furthest apart in what they mean by their shared number.

Under the coin its two branches are nought and one. Whichever player the flip names has no move and loses on the spot, so the flip is the game, and the half is a genuine half — one flip, two outcomes, nothing else happens.

Under the auction the same half was said to mean that the money decides. It does not, and the play-out shows it does not: both players bid nothing and the rule for equal bids settles it. So the auction reading’s half describes a quantity with no influence and the coin reading’s half describes a fair flip, and only one of those is a half of anything.

This is also where the difference between the two conventions is easiest to state. Alternating play calls the empty position a second-player win, which is a fact about whose turn it is. The coin removes whose turn it is and replaces it with a flip, so the same position becomes an even chance. Nothing about the position changed; the quantity that decided it was taken away and a fair one put in its place.

Where the flip decides nothing #

At the other extreme are the positions where the coin’s two branches are equal.

A star has one option and both players have it. Whoever the coin names moves to the empty position, so the two branches are the same number and the flip changes nothing about what happens next. The gap between the branches is nought.

Eight of the twenty-two values born by day two behave that way: star, up, down, up-star, down-star, star-two, and the two values a free move away from a star. Under alternating play they occupy three different outcome classes and are strictly ordered among themselves in a way no number expresses. Under the coin the flip is idle in every one of them.

Idle is not the same as balanced, and the two eights are different eights. Six of these sit at a half and two do not — a free move beside a star gives Left three quarters, and its mirror a quarter — while the set of values at a half contains the hottest thing born on day two, whose flip is worth a great deal. Six values are in both sets, and the overlap is what makes the two conditions easy to confuse.

That distinction is what the coin reading can say and the auction reading two essays below could not. There the collapse is read as the number failing to tell the infinitesimals apart. Here it is read off the branches: in these eight, the coin has nothing to tell apart, because both branches lead to the same chance, and that is a fact about the position rather than about the instrument.

The hot position, where the flip is worth a lot #

The hottest value on the day-two pool is also at a half, and for a completely different reason from the star. Its two branches are three quarters and a quarter — the coin genuinely decides a great deal — and they average to a half because the position is symmetric between the two players.

So a half under the coin means one of two things and the branches say which. Either the flip is irrelevant, as in the star, or the flip is enormously relevant and the position is balanced, as here. The single number cannot tell them apart and the pair of branches can, which is the same observation the auction reading makes about its bid — and it is better founded here, because a pair of probabilities is a pair of probabilities and a bid was a quantity nobody ever paid.

The half that four classes share #

That overlap deserves its own paragraph, because it is the strongest form of the collapse and it is easy to state.

On the day-two pool, eight values come out at exactly a half and they occupy all four outcome classes. The empty position is the second-player win; up is a Left win and down a Right win; star and star-two are first-player wins. The coin gives every one of them the same number.

Alternating play separates those four completely and has been separating them since the first thing this collection establishes — the four classes are the whole answer under that convention, and no two of them are ever confusable. One flip of a coin at each turn is enough to make them indistinguishable.

The reason is the one above: in every one of those eight, the flip decides nothing, because both players reach the same kind of place whoever is named. What alternating play is measuring in them is who runs out first, and running out first is a fact about parity — which a coin destroys by handing out turns at random.

So the collapse is not the number being coarse. It is that the quantity alternating play is reading in those positions has been removed from the game.

Why this derivation survives being played #

The play-out that destroyed the auction reading has nothing to bite on here.

Refusing was available because a bid of nothing was available. A coin does not ask for a bid, so there is nothing to refuse: the flip names a player and that player moves. A player who would rather not move does not get the choice, which is exactly the obligation normal play is built on and exactly the thing bidding removed.

So the coin restores the one clause the auction took out, and restores it without restoring whose-turn-it-is. Play still does not alternate — the coin can name the same player twice in a row, and that is what makes the numbers differ from the outcome classes — but somebody has to move at every turn, and that is enough for the game to end and the recursion to mean something.

The two derivations agree on every number and disagree about whether there is a game underneath. That is a sharper distinction than a disagreement about a number would have been, and it is why the coin reading is the one the essays that follow are built on.

What the coin does to the four classes #

Lining the two conventions up class by class is the clearest way to see what has been traded.

Alternating play answers with one of four outcome classes and the answer is exact: a position is a Left win, a Right win, a first-player win or a second-player win, and the classification is complete. What it does not say is how much — two first-player wins are the same class and can be nothing alike.

The coin answers with a number between nought and one, which orders the positions finely and loses the classes. The six values Left wins whoever moves are spread across four different probabilities — a half, five eighths, three quarters and seven eighths — and the nine first-player wins across three. Worse, the two lists overlap: a half is a probability held by a Left win, a Right win, a first-player win and the second-player win at once. Neither reading can be recovered from the other.

The honest summary is that they are not competing answers to one question. Alternating play answers who wins, which is a question with four answers and no size. The coin answers how often, which is a question with a size and no certainty. The bid the auction reading quotes was reaching for the second of those and could not get there, because it had no game underneath it to count outcomes over.

Why the two derivations were ever confused #

It is worth saying how one recursion came to carry two readings, because the confusion is instructive rather than careless.

Both settings produce an average of two branches, and for the same formal reason: neither player knows in advance which of the two things will happen. In the auction that ignorance is about who will be willing to pay; under the coin it is about how the coin lands. The arithmetic cannot tell those apart, and a recursion is arithmetic.

What distinguishes them is whether the ignorance is warranted. A coin is genuinely unpredictable and the half is a fact about it. An auction is not: both players choose their bids, and if one of them can guarantee the outcome — which under normal play they can, by bidding nothing for ever — then the ignorance was an assumption rather than a feature, and the half was standing in for a choice one player controls.

An average is only an average when nobody can fix the coin. That is the whole of the difference, and it is invisible from inside the three lines.

The coin, stated exactly #

Random turns throughout: a fair coin, independent at every turn, naming the player who must move; a player named with no move has lost; and both players play to maximise their own chance rather than to reach any particular position.

Three things about that are choices rather than facts.

The coin is fair and independent. A biased coin gives a different recursion — the average becomes a weighted one — and every number here moves. Nothing on this page measures that, and the fair case is the one the auction reading claims to match.

Both players optimise a probability, which is a different objective from winning under alternating play and is the subject of its own essay. It is also why the maximum and minimum in the recursion are over probabilities rather than over outcome classes.

And the number is a probability of winning, not of anything else. There are no draws here: every game ends, somebody is named with no move, and that player has lost. A loopy game would break that and no loopy game is in the pool.

Where the identity is checked and not proved #

Nothing here is a theorem about the equality of the two derivations. They are computed separately and compared on twelve positions, which is a check rather than a proof — though the two sets of three lines can be read against each other and the base cases lined up, which is most of one.

The probability is not a value. Two positions with the same probability are not interchangeable in company, for the same reason an earlier essay here gives: there is no sum theory here, and adding the parts of a board does not add their chances.

And a probability is not a strategy. Knowing that Left wins three times in four says nothing about what Left plays. The move comes from the maximum inside the recursion, and whether that move is the move an alternating player would want is a question with a surprising answer, and it belongs to a essay of its own.

The same census a day later #

The day-three pool sharpens all of it and changes nothing.

There, 1,474 values produce fifteen probabilities. The 217 values Left wins whoever moves occupy nine of them and the 1,039 first-player wins occupy seven, so the classes stay thoroughly mixed at forty times the size. And the highest chance any value gives Left is fifteen sixteenths — a lead of three free moves, which alternating play calls an outright win and the coin calls a one-in-sixteen risk.

That last number is the one worth carrying. It is not a rounding: a lead of three really is lost one time in sixteen, and the losing line is the coin naming Left four times running. A lead of ten would be lost one time in two thousand and forty-eight, and never with probability nothing at all.

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.

Still open: what a biased coin does #

The fair coin is the one that matches the auction’s arithmetic, and it is not the only one worth asking about.

A coin that names Left with probability $p$ gives a recursion with $p$ and $1-p$ in place of the two halves, and the numbers it produces are no longer dyadic — they are polynomials in $p$. Two questions follow and neither is settled here. The first is where the outcome classes reappear: as $p$ runs from nought to one, a position Left wins under alternating play should become a certainty at some point, and the value of $p$ at which each position crosses a half is a quantity the fair coin cannot see. The second is whether the crossing points respect the game order, which the fair coin’s number does on every comparable pair.

That would make the coin a family rather than a convention, with the alternating theory at one end of it and nothing at all at the other.