Economists thinking about anything from housing policy to climate change need to model how people make decisions, especially when facing uncertainty and risk. The standard theoretical approach is known as Expected Utility Theory (EUT), and asserts that people generally act so as to maximize their expected “utility,” this being a subjective measure of how much value they get from a certain outcome. A $1,000 reward gives more utility to a poor person than to a billionaire. In simplest form, EUT asserts that people assess the likelihood of different outcomes, the utility they would gain from each, and then act to maximise the expected gain – which would also be the average gain, if they played the gamble many times.

EUT is simple, easy to apply, and so familiar as to seem the obvious right approach. Yet this recipe carries a subtle flaw, buried in its procedure of averaging over a set of alternative outcomes. Anyone who faces risky situations over time — and that’s essentially everyone — needs to handle those risks well, with one thing happening after the next. The seductive genius of the concept of probability is that it removes this history aspect by imagining the world splitting with specific probabilities into parallel universes, one thing happening in each. The expected value doesn’t reflect an average over time, but over possible outcomes considered outside of time.

Unfortunately, in many cases of practical importance, averages through time and over probable outcomes aren’t the same, and the latter calculation offers a dangerously misleading guide to risky choices. Especially when downside risks get large, real outcomes averaged through time are much worse than the expected value would predict. There are often sound mathematical reasons for being unwilling to take on gambles (or projects), despite wildly positive expected pay-offs. This fact has important practical implications, the focus of Ergodicity Economics, currently being explored at the London Mathematical Laboratory.

So – how do real people behave? Do they consistently act to optimize some kind of utility, as EUT asserts, or do they instead act in a way that optimizes their actual time average rate of gaining wealth? A team of neuroscientists led by Oliver Hulme of Copenhagen University has recently tested this question in experiments, finding strong evidence against the classical view of EUT. The experiments also conflict with so-called Prospect Theory, another set of ideas proposed to provide more psychological realism by fixing some shortcomings in EUT. Importantly, however, the experimental results give strong support to the foundations of Ergodicity Economics, and the view that people actually act to try to maximise the time-average growth rate of their winnings.

As the researches point out, behavioural scientists generally explore decision making using simple gambles with “additive dynamics.” In each gamble, a person wins or loses fixed amounts of money – say, gaining £1 for a win, and losing £0.50 for a loss. Gambles of this kind turn out to be “ergodic” in the sense that averages over the possible outcomes really are equal to averages over time if the gamble is played many times.

Because of this, using Expected Utility Theory with a linear utility function – utility simply being proportional to the amount won or lost – turns out to be optimal for maximising the growth of wealth over time.

But this additive paradigm isn’t a useful model for many real-world decisions where the amount gained or lost in a gamble depends on how much wealth an individual already has. An investment in the stock market, for example, gives uncertain gains or losses, and the magnitude is proportional to the amount already invested. Each day effectively multiplies the investment value by a number slightly larger or smaller than one. Mathematically, this situation corresponds to gambles having multiplicative dynamics, for which the rate of change of wealth is non-ergodic. This means that the expectation value of changes in wealth, calculated by averaging over possible outcomes, is a poor guide to the actual time averaged growth of wealth. Many gambles with positive expectation values – seemingly worth playing from the EUT perspective – actually yield consistent losses when played out in time. In situations with multiplicative dynamics, it turns out that one can still use the EUT framework, but only if the utility is considered to depend logarithmically on the outcome.

This implies that the actually optimal behaviour depends very sensitively on the nature of the gamble dynamics. Or, in the setting of EUT, that the utility a person assigns to an outcome – winning £1, for example – should change if the gamble flips between additive and multiplicative dynamics. If people really act this way, it’s a big problem for both expected utility theory and prospect theory which assert that utility functions should be fixed and stable, indifferent to dynamics. Hulme and colleagues experiments were designed to find out how people really do behave.

In the experiments, they gave 18 subjects an initial $155 to play with, and had them engage for two days in a gambling paradigm with either additive or multiplicative wealth dynamics. At the start of each day they first took part in a passive session during which they could observe and learn the gamble dynamics, watching as random stimuli caused gains or losses to their wealth. On the first, additive day, the stimuli caused additive changes in wealth, whereas on the second, multiplicative day, the stimuli caused multiplicative changes. On each day, the subjects then took part in an active session during which they could choose between gambles, and try to choose wisely to build their wealth.

Overall, the results showed strong evidence that the subjects choice behaviour shifted significantly when moving from the additive to the multiplicative dynamic. Interpreted from within the utility maximisation framework, the experiments show that there was not a single unique utility value that subjects assigned to a particular outcome, but that the utility value shifted when the gamble dynamics changed. In other words, it seems that individuals don’t have a utility function reflecting their own psychology, but that utility depends strongly on the dynamic. The subjects behaviour could be more accurately and naturally explained as the result of their adaptive effort to maximise their average rate of wealth gain through time, rather then by averaging over alternative outcomes.

The concept of expected utility maximisation plays a fundamental organising role in economics, and seems to provide a way to make useful predictions, if one takes past behaviour as a reliable guide to future behaviour. Researchers generally assume that people have stable or unchanging preferences. Watch their behaviour in one setting, and estimate the utility function for which this behaviour would be optimal, and one can then use the same utility function to predict their behaviour in the future. This entire procedure falls apart, however, if, as these experiments demonstrate, the way people assign utility to outcomes is more fluid and changes when the environment changes.

These experiments lend powerful empirical support to the perspective of Ergodicity Economics, which does not rely on arbitrary utility functions. As the authors note, their observations simply seem to show that people manifest a stable preference for growing wealth over time, and adjusting their behaviour to do so across different circumstances. This makes it imperative that economic and psychological models of decision-making take the influence of environmental dynamics much more seriously.