“All of these questions have something to do with randomness, and the way to deal with them in the 17th century was to imagine parallel worlds representing everything that could happen,” Gell-Mann said. “To assess the value of some uncertain venture, an average is taken across those parallel worlds.”
This concept was only challenged in the mid-19th century when randomness was used formally in a different context—physics. “Here, the following perspective arose: to assess some uncertain venture, ask yourself how it will affect you in one world only—namely the one in which you live—across time,” Gell-Mann continued.
“The first perspective—considering all parallel worlds—is the one adopted by mainstream economics,” explained Gell-Mann. “The second perspective—what happens in our world across time—is the one we explore and that hasn’t been fully appreciated in economics so far.”
The real impact of this second perspective comes from acknowledging the omission of the key concept of time from previous treatments. “We have some 350 years of economic theory involving randomness in one way only—by considering parallel worlds,” said Peters. “What happens when we switch perspectives is astonishing. Many of the open key problems in economic theory have an elegant solution within our framework.” …
This concept reaches far beyond this realm and into all major branches of economics. “It turns out that the difference between how individual wealth behaves across parallel worlds and how it behaves over time quantifies how wealth inequality changes,” explained Peters. “It also enables refining the notion of efficient markets and solving the equity premium puzzle.”
One historically important application is the solution of the 303-year-old St. Petersburg paradox, which involves a gamble played by flipping a coin until it comes up tails and the total number of flips, n, determines the prize, which equals $2 to the nth power. “The expected prize diverges—it doesn’t exist,” Peters elaborated. “This gamble, suggested by Nicholas Bernoulli, can be viewed as the first rebellion against the dominance of the expectation value—that average across parallel worlds—that was established in the second half of the 17th century.”
Exploring gambles reveals foundational difficulty behind economic theory (and a solution)
The dynamic approach to the gamble problem makes sense of risk aversion as optimal behavior for a given dynamic and and level of wealth, implying a different concept of rationality. Maximizing expectation values of observables that do not have the ergodic property of Section I cannot be considered rational for an individual. Instead, it is more useful to consider rational the optimization of time-average performance, or of expectation values of appropriate ergodic observables. We note that where optimization is used in science, the deep insight is finding the right object to optimize (e.g. the action in Hamiltonian mechanics, or the entropy in the microcanonical ensemble). The same is true in the present case – deep insight is gained by finding the right object to optimize – we suggest time-average growth. Laplace’s Criterion interpreted as an ergodic growth rate under multiplicative dynamics avoids the fundamental circularity of the behavioral interpretation. In the latter, preferences, i.e. choices an individual would make, have to be encoded in a utility function, the utility function is passed through the formalism, and the output is the same as the input: the choices an individual would make.
Evaluating gambles using dynamics