The plate before this asserted that the worth of a gamble is its expected utility and not its expected value. This plate tells the paradox that first forced that correction upon the world — a puzzle posed by one Bernoulli and resolved by another, in the city that gave it its name. The puzzle is a game. A fair coin is tossed until it first falls heads; if heads first appears on the second toss you are paid four ducats, if on the third, eight, if on the tenth, more than a thousand — the prize doubling with every toss the coin survives. How much should one pay to play? The arithmetic returns an answer that cannot be believed: the game's average payout is infinite, and by the rule then universally accepted, a man should stake his entire fortune, and more, for a seat at it. Yet no one of sound mind will offer more than a handful of coins. This plate will set out the game and the scandal of its infinite price; the rule it broke, that a gamble is worth its mathematical average; Bernoulli's resolution, which founded the whole theory of risk by ruling that the worth of a prize is not its size but what its size is worth to the winner; and the two lessons it leaves the volume — that the worth of risk lives in the one who bears it, and that an average ruled by rare and dazzling extremes is no guide at all to what bearing the risk will ordinarily bring.
Set the game out plainly. The prize is two ducats if the coin falls heads on the first toss, and this happens half the time; four ducats if it falls tails then heads, which happens a quarter of the time; eight if tails, tails, heads, an eighth of the time; and so on without end, the prize doubling as its probability halves. To find the average payout one multiplies each prize by its chance and sums: a half times two is one; a quarter times four is one; an eighth times eight is one; and so for every term, each contributing exactly one ducat to the sum. But the terms are infinite in number, and a sum of infinitely many ones is infinite. The game's expected value — its mathematical average payout — is therefore not large but unbounded, greater than any sum that could be named. By the rule that a gamble is worth its expected value, one ought to pay a thousand ducats to play, or a million, or everything one owns and everything one could borrow, for whatever the price, the average winnings exceed it. And yet set the game before any sensible person and he will pay two ducats, or four, and laugh at ten. The rule and the man are in flat contradiction, and the man is plainly right.
Consider what the man's good sense convicts. Before the paradox, the worth of a gamble was reckoned to be its expected value — the probability-weighted average of its outcomes, the "fair price" at which one might as well take either side. For ordinary bets this serves well enough: a fair coin for a hundred ducats is worth fifty, and few would quarrel. The rule had the authority of the founders of probability behind it, and the elegance of plain arithmetic. But the St. Petersburg game holds it up to the light and shows it hollow, for it prescribes a price — infinite — that no living person would pay, and so cannot be the measure of what the game is worth to anyone. The flaw, hidden until the paradox exposed it, was an assumption buried in the averaging: that a ducat is a ducat is a ducat, each worth exactly as much as any other, so that two thousand of them are worth precisely a thousand times two. The rule counted the prizes by their size and never asked what a prize of that size would be worth to the one who held it.
By the rule then accepted, a sane man should have staked his whole fortune on the game; that none would was the proof the rule was wrong.
It was Daniel Bernoulli, writing in St. Petersburg in 1738, who supplied the want — though his cousin Nicolas had posed the puzzle a quarter-century before, and Gabriel Cramer had glimpsed the answer in between. Bernoulli's stroke was to deny that the worth of the game is the average of its prizes, and to assert instead that it is the average of their worth — that one must weigh not the ducats but the good the ducats do, and that the two diverge because a ducat does less good the more ducats one already has. The hundredth ducat is welcome; the millionth, won on a long run of tails, is very nearly a matter of indifference, heaped upon a fortune already vast. Bernoulli proposed that worth rises as the logarithm of wealth, a curve that climbs ever more gently, and with that curve the sum is transformed. The dazzling far prizes, their worth now growing far slower than their size, contribute less and less; the infinite sum of ducats becomes a modest, finite sum of worth; and the game proves to be valued at a few coins after all — exactly what the sensible man would pay. The paradox dissolves not by correcting the arithmetic, which was right, but by replacing the quantity to which it is applied. And in that replacement the whole edifice of the previous plate is born: the worth of the uncertain is expected utility, the curve that yields it is concave, and the concavity is risk aversion.
The cure was not quite complete, and the manner of its incompleteness is instructive. For if the worth of money rises without limit, however gently, one can always build a crueller game — let the prizes balloon fast enough, doubling the worth rather than the sum — whose expected utility is itself infinite, and the paradox returns in a higher key. Karl Menger showed, two centuries after Bernoulli, that no unbounded measure of worth can escape every such game; only a worth that is bounded — that approaches some ceiling beyond which no fortune, however vast, can lift one higher — defeats them all. The full resolution, then, requires that the good of wealth not merely diminish but cease to grow at all past some summit. And the world lends its own help besides: no real bank could honour the unbounded prize, for a finite purse caps the winnings and restores a finite average; and the great prizes hang on runs of tails so long that one might play for ten thousand lifetimes and never see them. Diminishing worth, bounded worth, and the plain finitude of things together tame what the bare arithmetic could not.
Beneath the arithmetic lies the lesson that matters most for this volume, and it is the first of two the paradox bequeaths. It is that the worth of a gamble is not a property of the gamble. The very same game is worth more to a pauper than to a prince, for the early ducats that the pauper would win mean far more to him; worth more to the bold than to the timid; worth one thing to a man who can bear a loss and another to a man who cannot. There is no single number that is the worth of the St. Petersburg game, no objective price stamped upon it; its worth lives in the one who would play, in the shape of his wants and the depth of his purse and the strength of his nerve. And this carries straight into the price of risk that the volume pursues, for it means that the premium a man demands to bear a risk is not read off the risk alone but off himself — off how steeply his own worth curve bends. Risk has no one price, because its weight is felt differently by every shoulder that takes it up.
There is no single worth to the game; its worth lives not in the dice but in the one who must throw them.
The second lesson is subtler, and it is the one the volume will return to at its end. The whole of the game's infinite average came from its tail — from the vanishingly rare, prodigious prizes, the thousand-ducat win that waits on ten straight tails, the million that waits on twenty. Strike those off, keep only the outcomes one is at all likely to see, and the average collapses to a few ducats. The mathematics of the game is the mathematics of an average ruled by extremes that almost never happen — and such an average, however large, tells one almost nothing about what playing the game will ordinarily bring. This is the shape of what a later plate will call the fat tail, and it carries a warning the volume must keep. For if the reward of bearing some risk is likewise governed by its tail — by the rare, immense success that lifts the average while almost no one attains it — then the average reward is a deceiving figure, swollen by a handful of prodigious winners and never received by the many who bear the same risk and meet the ordinary, modest, or ruinous result. The dazzling average belongs to the few who drew the long run of luck; the typical bearer gets the typical outcome. What looks, in the aggregate, like a handsome wage for courage may be, on inspection, a lottery whose mean is the property of its winners. The paradox that opens the theory of risk thus also plants the doubt that closes the volume.
So the St. Petersburg game was the first reckoning that risk is more than its average — the paradox that broke the rule of expected value and, in breaking it, founded the theory the volume rests upon: that the worth of the uncertain is the worth of its outcomes to the one who bears them, weighed on a curve that bends and is bounded, which is to say weighed by risk aversion. It leaves the volume two gifts and a warning. The gifts are that the price of risk is real, rooted in the diminishing worth of wealth, and that it is the same diminishing worth a richer posterity invokes when we discount the future on its behalf. The warning is double: that risk carries no single price, its weight differing on every shoulder; and that an average ruled by rare extremes is a treacherous measure of an ordinary fate — a warning the plates on the equity puzzle, the black swan, the survivor, and the fortune will spend themselves making good. Risk is more than its average; and whose average it is, and whether the many who bear a risk ever see it, the volume has only begun to ask. As the table says of every price, first to last: the argument remains open.