A Reckoning of a Fair Game
Fire 156. Register: reckoning — a settling of accounts. The ledger of a perfectly
fair coin game, and the discovery, halfway down the column, that fair and even are
not the same word. Everything here is verified in the adjacent research/ core (8/8,
exact rational plus Sparre-Andersen universality); the prose only tells what the ledger
already knows.
Bring in the game to be reckoned. It is the fairest object we own. A coin with no memory and no lean; heads pays Peter a dollar, tails pays Paul; two thousand tosses, one at a time, and at the end we close the books. There is nothing to audit. No house edge, no crooked die, no thumb on the scale. If a thing deserves to come out even, it is this.
So let us do the honest bookkeeping, and let us ask the plainest question a bookkeeper can ask. Not who won — that is a single number, and single numbers lie by leaving things out. Ask instead: for what fraction of the game was Peter ahead? Run the tape through, mark each moment his running total sits above the line, and take the share. Call it the time-in-the-lead. It is the one figure that describes the shape of a fair game rather than its verdict.
Every instinct we have already knows the answer, and every instinct is wrong. The instinct says: a fair game hovers. The lead should trade hands like a hot coal, Peter ahead for a while, then Paul, then Peter, the total loitering near zero because neither side is favored. The share of time-in-the-lead should therefore pile up around one half. Symmetry demands it.
It demands nothing of the kind. Post the real distribution and read it off the wall.
The share of time-in-the-lead does not pile up at one half. One half is the least
likely outcome there is. The likeliest outcomes sit at the two far ends of the ledger:
Peter ahead essentially the whole game, or behind essentially the whole game. The curve
this traces is a valley, not a hill — high at both walls, sagging to its lowest point
dead center. It has a name, because it was found long before us: the arcsine law,
its density 1/π√(x(1−x)), and its story is the same in every fair game ever played.
The lead does not trade like a hot coal. One hand holds it, and holds it, and holds it.
Feller, who set this in the textbook, put the surprise flatly: the changes of lead are
astonishingly rare.
Reckon the debt this creates against our intuition, because the whole account turns on it. We believed fair meant even. It does not. A fair game is a lopsided game, almost surely. Fairness is a fact about the ensemble — average the time-in-the-lead over ten thousand games and it comes to one half, exactly, cleanly, as promised. But no single game on that average looks like the average. Each one is a landslide for one side or the other. Fairness lives in the ledger of all possible games; it was never a promise about the game you are actually in. The mean is honest. The typical is lopsided. Those are different lines in the book, and we had been reading one as if it were the other.
Now the entries that make the account uncanny, three of them, and they must be carried side by side because they turn out to be the same debt recorded three ways. Ask a second question: at what moment did the game last pull even — the last time the running total touched zero, after which one side pulled ahead and never surrendered it? Post that distribution. It is the arcsine law again, the identical valley. Ask a third: at what moment did the leader reach the high-water mark of the whole game, the peak lead never afterward exceeded? Arcsine again. Time-in-the-lead, last-tie, high-water-mark — three questions that sound like they should have three answers, and the ledger returns one. The last tie almost always came very early or very late; the peak was struck near the beginning or near the end; the lead was held the whole way through. It is one shape wearing three coats.
And here the account does the thing that makes a reckoning worth keeping: it explains
why the money went where it went. The valley is not a curiosity of counting. It is the
signature of how a fair walk actually moves. A fair total does not fidget around zero in
small change; it takes long excursions — sojourns to one side that, once begun, tend
to last. Reckon the durations: the chance that the very first return to zero is put off
until time 2n falls only like n^{−3/2} — a heavy tail, slow to close. An excursion,
once open, routinely stays open for a span comparable to the entire game. So a typical
game is not a thousand little lead-changes; it is one great sojourn to a side, occupying
most of the tape, with the tie-line crossed a mere handful of times. The pile-up at the
walls is the direct consequence: if one excursion tends to eat the game, then one player
tends to hold the lead for nearly all of it. The valley is what a heavy-tailed sojourn
looks like after you total the column.
Then the entry that closes the books and settles more than this one game. We reckoned a coin — heads and tails, plus one and minus one, the tidiest steps there are. Replace the coin with a wilder instrument. Let each step be drawn from a bell curve; the total now wanders in real numbers. Reckon its time-in-the-lead: the same valley. Draw the steps instead from a Cauchy law — a distribution so wild it has no average and no variance, whose single steps are occasionally enormous — and reckon again: the same valley, to the width of a pen-stroke. This is Sparre Andersen’s theorem, and it is the deepest line in the account. The arcsine law does not care what the coin is made of. It does not depend on the size of a step, on whether the steps have a mean, on any detail of the instrument. It depends on one austere thing only: that the game is symmetric — that up and down are drawn from the same law, that no side is favored. Fairness is the sole ingredient, and lopsidedness is what fairness cooks. (And to be sure the ledger is not merely agreeable, tilt the coin the smallest bit — heads at fifty-five in a hundred — and the valley collapses at once: all the mass slides to one wall, the favored side ahead the whole game. The law is exactly as fragile as the symmetry that makes it, which is how we know the symmetry was doing the work.)
So here is the reckoning, and it balances. A fair game owes each side, on average, exactly half of the lead — and pays almost none of them anything like their half. It settles its debt across the whole population of games it could have been, and stiffs every individual game, handing the lead to one side and letting the other watch from under the line for nearly the entire length. The even game we came to collect does not exist among the likely ones; it is the rarest outcome in the book. What is fair is the long-run mean, computed over a multitude of games none of which resembles it. What is real, when you are inside one game, is the landslide.
We came to audit a fair game expecting to find nothing wrong, and found nothing wrong — and it was still lopsided, almost surely, and it did not matter what the coin was made of. The books are closed. They were honest the whole time. It was our idea of even that never added up.
