Arne Vanhoyweghen publsihed this blog post on the blog Ergodicity Economics. Corresponding paper.
Readers of this blog will probably be familiar with the Copenhagen experiment. We conducted a similar but much simpler experiment, and we did it in Brussels, so we’re now calling it the Brussels experiment. What did we find? Our results suggest two things: First, putting people under time pressure makes them act more intuitively and less analytically. Second, ergodicity economics tells us that optimizing long-term wealth is generally different from optimizing expected wealth. In such cases, people acting intuitively, under time pressure, tend to act in line with the principles set out by ergodicity economics and optimize for the long run, whereas people without time pressure can be led astray by standard notions of rationality.
I studied economics in the wake of the financial crisis, which was sometimes a surreal experience. At university, we enjoyed classes on mainstream economics, while in the outside world, many of the criticisms of mainstream economics enjoyed a resurgence. While we learned how to model the world using GDP, its shortcomings were laid bare again in academic journals and popular science media. My fellow students and I started questioning mainstream economics: were we truly learning the right things? Or were there flaws in our standard models of which we were unaware? In this context, “The ergodicity problem in economics” seemed to me as the spark to rethink many economic models and assumptions. Did ergodicity breaking offer a lens that could solve some of these problems and allow us to create more grounded models? The Copenhagen experiment seemed like a hopeful sign. With these questions simmering in my mind and inspired by the Ergodicity Economics framework, my team and I created an experiment to uncover some of the nuances of intuitive human decision-making.
Setting the Stage:
Our experimental structure has some basic similarities with the Copenhagen Experiment. Participants were presented with a pair of gambles and asked to pick the one they preferred to play, within two types of wealth dynamics:
Multiplicative wealth dynamic: In this dynamic, wealth changes by a random percentage with a stationary distribution. An example of this dynamic is the Peters coin toss. In the Peters coin toss, landing heads means that your wealth increases by 50%, and on tails you lose 40%. This principle remains consistent, regardless of your changing wealth. Formally, we write
\( x(t+1)=r_t x(t),\)
Where \(r_t\) is a realization of a random variable, \(r\). In the Peters coin toss, this variable takes the values \(r^{\text{tails}}=0.6\) and \(r^{\text{heads}}=1.5,\) each with 50% probability.
Additive wealth dynamic: Here, wealth changes by a random amount with stationary distribution.
Formally,
\(x(t+1)=x(t) + \Delta x_t,\)
where \(\Delta x_t\) is a realization of a random variable \(\Delta x\). For instance, wealth may increase by $500 or decrease by $400, each with 50% probability.
In line with the Copenhagen Experiment, we introduced respondents to both dynamics in no particular order, and they encountered gamble pairs in a randomized way. However, this is where our experiment starts to diverge. First, we presented respondents with explicit outcomes of each gamble (in euro or multiplicative factor terms) rather than showing an image tied to an outcome. Second, we designed gambles in a very particular way. Namely, we made respondents choose between two gambles with the same expected value but differing variances. To keep things simple, for any gamble, each outcome was realized with 50% probability. This way of designing gambles makes it possible to describe behavior in terms of an affinity for risk without assuming any particular family of utility functions.
As an illustration of the power of this way of designing gambles, consider the following two gamble pairs, where the outcomes are determined by a fair coin toss:
Imagine someone prefers ‘gamble 1’ over ‘gamble 2’. The nomenclature in classical decision theory defines risk neutrality as being indifferent between gambles with equal expected payout. This means that consistently choosing ‘gamble 1’ would be labeled as being ‘risk-seeking’ for both dynamics without having to fit any utility functions. Conversely, we label preferring ‘gamble 2’ over ‘gamble 1’ as being risk-averse.
In a simple situation like this, Ergodicity Economics suggests that people’s behavior will be influenced by what’s best in the long run (as opposed to what’s best in expectation). This can be determined by choosing the gamble with the greater time-average growth rate. We therefore first need to answer the question, “What is the growth rate?”.
In the additive dynamic, the time-average growth rate is the rate of change of expected wealth or the expected payout per round.
\(g_a=\frac{\Delta \mathbb{E}(x)}{\Delta t}\)
Because both gambles have the same expected payout, growth-rate optimization predicts risk neutrality in additive dynamics. Systematic deviations from risk neutrality, therefore, reflect effects beyond long-term growth-rate optimization, such as idiosyncratic preferences or a reflection of people acting differently in a lab environment than they would in real life.
The multiplicative dynamic, however, generates exponential wealth trajectories, and the time-average growth rate can be computed as the rate of change of the expected logarithm of wealth,
\(g_m=\frac{\Delta \mathbb{E}(\ln x)}{\Delta t}.\)
Maximizing the time-average growth rate is now different from maximizing the expected payout. For each gamble, the expected payout is the same by design, but the time-average growth rate is
\(g_m=\frac{1}{2}(\ln r^{\text{tails}} +\ln r^{\text{heads}})\)
Where we’ve set [katex]\Delta t,[/katex] the duration of one round, to 1. Using the concavity of the logarithmic function, it is easy to show that this growth rate is always greater for the gamble with the smaller variance (‘gamble 2’ in our setup). The risk-averse option is the time-optimal choice.
Our respondents:
As an economics PhD student, the cohort of participants that was easiest to recruit for me was undergraduate economics students at my home university, Vrije Universiteit Brussels (VUB). Usually, it is problematic in behavioral experiments if the cohort is not representative of the general population, but in our case, this specific cohort was particularly interesting. Unlike in previous experiments in Ergodicity Economics, the participants had formal training in the type of problem we presented to them. With their background, one could expect them to be primed to use expected values when making decisions - after all, this is what is taught in economics courses. But what would happen if we forced them to answer more intuitively by putting them under time pressure? To explore this, we placed some participants under perceived time constraints while allowing others all the time in the world to deliberate.
Our Findings:
When we analyzed the students’ decisions, we observed that in the additive dynamic (on the left), both groups – those with unlimited time and those who were time-constrained – exhibited similar behavior.
However, the multiplicative dynamic (the middle graph) told a different story. Here, the time-pressured group consistently chose the safer gambles. Why did this happen?
Each participant played both dynamics in a randomized order. Each dynamic took 10 minutes to play, which leaves the change in wealth dynamic as the most plausible explanation. So, what is the relevant difference when you experience a multiplicative setting rather than an additive setting?
Ergodicity Economics has a strong suggestion. In the additive dynamic, the additive changes in wealth are ergodic, and the null-model of optimal behavior as proposed in Ergodicity Economics, namely optimizing over time, makes the same predictions as the model of optimizing expected wealth. Economics students with plenty of time to carry out explicit computations are likely to optimize expected wealth.
However, this equivalence does not hold in the multiplicative wealth dynamic. In a multiplicative dynamic, optimizing over time predicts behavior that differs from optimizing expected wealth. The answers of the students with plenty of time aligned closely with expected-wealth optimization, i.e., indifference between both gambles. Those who were under time pressure and couldn’t do the required computations acted more intuitively and were more risk averse than their non-timed counterparts, shifting their behavior towards growth-rate optimality as formulated within Ergodicity Economics.
This provides a fascinating counterpoint to traditional economic decision theory, where expected-value optimization (of wealth or utility) is the dominant definition of rational behavior. In the long run, decision makers who optimize expected wealth in an additive dynamic do well, but in a multiplicative setting, more risk-averse behavior performs objectively better – expected-wealth optimization underperforms here, and intuitive decision makers seem to pick up on this. Labelling them `irrational’ seems a misnomer. There seem to be good reasons for the misgivings my fellow students and I had in our undergraduate days.
Reference:
A. Vanhoyweghen, B. Verbeken, C. Macharis, and V. Ginis, The influence of ergodicity on risk affinity of timed and non-timed respondents. Sci Rep 12, 3744 (2022). https://doi.org/10.1038/s41598-022-07613-6