In 1713 Nicolaus Bernoulli posed a question: How much would you pay to play a game? In the game, a coin is flipped until it lands heads up, and you win $2n depending on the number of tosses n. If heads comes up on the first toss, you win $2. If heads doesn’t come up until the second toss, you win $4. A rational person should select their maximum buy-in based on their expected winnings. The probability of the game lasting for n flips is 1/2n, so the expected winnings are:
(1/2 × $2) + (1/4 × $4) + (1/8 × $8) … = $1 + $1 + $1 + … = +∞
With infinite expected winnings, the rational choice seems to be to pay any finite amount, no matter how large, to play. But with actual winnings of only a few dollars for the majority of games, people probably wouldn’t and shouldn’t spend millions. The unlikely 10100-flip game skews the expectation value with its massive payout. The problem is known as the St Petersburg paradox after the city of residence of Daniel Bernoulli, Nicolaus’s cousin and the first to propose a solution. The paradox highlights the contradiction between a typical outcome and an expected outcome that is sensitive to rare events.
Although casinos may not be enacting a version of the paradoxical game anytime soon, Jake Fontana of the US Naval Research Laboratory in Washington, DC, and Peter Palffy-Muhoray of Kent State University in Ohio have physically realized it through the failure probability of a fiber. The researchers found that the force at which the fiber fractured depended on its length with a relation derived from a reformulation of the St Petersburg paradox.
The problem is reframed as many games played over a finite number of total coin flips N. In the full sequence of coin tosses, which can be pictured as a string of 1s and 0s, one can calculate the average number of streaks with n tails or 0s in a row—that is, the number of games with n coin flips. The longest expected chain of 0s is proportional to the logarithm of N.
For a fiber, a coin toss coming up tails becomes the presence of a defect at one point in the fiber. A single, tiny defect won’t make a fiber easy to break, whereas a less likely large defect—or a long chain of defects—will. As a result, the force required to break the fiber should be linear with defect size and thus the logarithm of the fiber’s length.
Fontana and Palffy-Muhoray tested polyester and polyamide fibers with lengths from 1 mm to 1 km. (To test the 1 km fibers, they took over a long stretch of a bike trail.) They clamped one end of the fiber and to the other end applied a larger and larger load until the fiber fractured. Their experimental results (see the figure) of the normalized breaking force F/F0 as a function of the normalized fiber length L/L0 matched the St Petersburg model better than the Weibull model, which is widely used to describe breakdown. The paradox model’s success suggests it could be useful for other systems, such as weather forecasting and financial markets, that are sensitive to rare events. (J. Fontana, P. Palffy-Muhoray, Phys. Rev. Lett. 124, 245501, 2020.)
Highlight image credit: Jake Fontana, Peter Palffy-Muhoray, and Robert Gates
