The Absorbing Barrier

Non-Ergodicity from the St. Petersburg Paradox to Long-Term Capital Management

On September 9, 1713, the Swiss mathematician Nicolaus Bernoulli mailed a letter from Basel to the French probabilist Pierre Rémond de Montmort in Paris containing a five-sentence gambling puzzle that paralyzed classical economics for three centuries. Suppose a casino offers a fair coin toss. A player wins 1 ducat if heads appears on the first throw, 2 ducats if heads first appears on the second throw, 4 ducats on the third, and $2^{k-1}$ ducats if the first heads appears on the $k$-th throw. Under the foundational rule of probability formulated by Christiaan Huygens and Blaise Pascal, the fair price of any gamble is its expected value—the sum of each possible payout multiplied by its probability:

Because the mathematical expectation is infinite, classical theory dictated that a rational agent should gladly trade their entire fortune for a single ticket to play. Yet as Nicolaus's cousin Daniel Bernoulli observed in Volume V of the Commentarii Academiae Scientiarum Imperialis Petropolitanae in St. Petersburg in 1738, no reasonable person would pay even 20 ducats to enter. Half of all games end on the first flip for 1 ducat; three-quarters end within two flips for at most 2 ducats. To explain why humans reject an infinite average return, Daniel Bernoulli proposed that people evaluate money through diminishing marginal utility, where satisfaction grows logarithmically with total wealth.

For 273 years, economists treated the St. Petersburg paradox as a psychological quirk. Then, in 2011, the physicist Ole Peters at the London Mathematical Laboratory demonstrated that Daniel Bernoulli's logarithmic rule requires no psychological explanation at all. The paradox arose because eighteenth-century probabilists had imported a false physical assumption known as ergodicity.

A stochastic process is ergodic only if the average outcome across an ensemble of identical systems at one instant equals the average outcome of a single system tracked over time. Wealth, biological survival, and institutional capital are strictly non-ergodic: they compound multiplicatively over time and are bounded below by an absorbing barrier at zero.

Consider a coin-toss investment where a player wagers their net worth each round: on heads, wealth rises 50 percent; on tails, wealth drops 40 percent. Across ten thousand players flipping once simultaneously, half gain 50 percent and half lose 40 percent, yielding a positive ensemble expected return of $+5$ percent per round. Yet for a single individual playing sequentially through time, winning 50 percent in round one and losing 40 percent in round two multiplies initial wealth $X_0$ by $1.5 \times 0.6 = 0.90$, a 10 percent loss every two rounds. As Peters formalized, a single compounding trajectory over $T$ steps is governed by the time-average growth rate $g$:

For the $+50$ percent and $-40$ percent game, $g = \frac{1}{2}\ln(1.5) + \frac{1}{2}\ln(0.6) = \frac{1}{2}\ln(0.90) \approx -0.0527$, or $-5.13$ percent per round. While the imaginary crowd grows richer at 5 percent per step, the individual playing over time is guaranteed to go broke.

This divergence between ensemble expectation and time-average survival destroyed Long-Term Capital Management in 1998. Founded in 1994 by former Salomon Brothers bond trader John Meriwether alongside economists Myron Scholes and Robert C. Merton, the Greenwich hedge fund exploited tiny pricing discrepancies between sovereign bonds that were statistically certain to converge over time. Because each spread yielded only a few basis points, the partners borrowed $125 billion against $4.72 billion in equity—a leverage ratio exceeding 26 to 1—before layering $1.25 trillion in notional derivatives on top.

In an ensemble of parallel markets without liquidity barriers, the spreads converged. In sequential time, however, on August 17, 1998, the Russian government defaulted on 281 billion rubles of domestic GKO treasury bonds. Global investors fled into liquid U.S. Treasury bills, causing Long-Term Capital Management's arbitrage spreads to widen violently. On Friday, August 21 alone, the fund lost $553 million in a single session. Margin calls forced the fund to liquidate at the bottom of the market, striking the absorbing barrier of insolvency before the long run arrived. After the Federal Reserve Bank of New York orchestrated a $3.625 billion rescue by fourteen banks on September 23, 1998, the exact bond convergence trades that bankrupted the partners turned profitable for the rescue consortium eighteen months later—inside a game from which the original equity holders had already been permanently erased.