Random walks to colonoscopies: Modelling how memory of the past affects future behaviour – Fellows’ seminar by Rosemary Harris

2 November 2023

“As humans, how does memory of our past experiences affect our future choices? Are there psychological biases which might cause us to behave less rationally than we think? Can we make mathematical predictions about such situations? And what does all this have to do with theoretical physics research into ‘non-Markovian statistical mechanics’ – a branch of physics that combines statistics with the laws of mechanics to measure and explain large systems from knowledge of the properties and dynamics of their microscopic ingredients?” asked Rosemary Harris of the Department of Mathematics, University College London.

STIAS Fellow Rosemary Harris during her seminar on 26 October 2023

In her seminar Harris aimed to answer some of these questions starting from the statistical mechanics of random walkers and progressing (via discussion of colonoscopies!) to simple decision-making models of behavioural economics. “Along the way, I hope to convince you of the beauty of mathematical modelling as well as its limitations. There will be some equations too, but not that many!”

“I think we are all interested in memory,” she continued. “Memory and fear of forgetting permeates our everyday life.”

“Also in the real-world probability matters,” she said. “Most of my maths involves probabilities – the likelihood of random events. It tells us both about typical events and rare fluctuations (the realm of so-called large-deviation theory). I’m interested in both the abstract structure of probability theory and the applications, which range from biology to physics to finance, as well as the associated computational techniques involved.”

She explained that most probability theory deals with Markovian processes. A Markov process is a model describing a sequence of possible events in which the probability of each event may depend only on the state after the previous event but not on the previous history of events. So the probability of what happens in the future depends on the present, not on the past. However, we know that memory plays an important role in most real-life situations and that remembering past experiences affects future choices. A non-Markovian process is a process that depends on past history and is the basis of most of the work Harris undertakes.

She also described models as a “a simplification of reality. They are usually a bottom-up approach – which starts from the simplest level and adds layers of detail (like layers of an onion). You can make predictions about the role of particular parameters and about how a measured value might change. There is often beautiful maths along the way.”

Drunken walking and simple models

Harris explained the Drunkard’s Walk model, a statistical mechanical model memorably illustrated by polymath and theoretical physicist George Gamow. In its simplest form, the model describes a one-dimensional, random walk which is a simple model of decision making where an agent is repeatedly deciding between two options by tossing a coin. She pointed out that you can think about interesting mathematical questions by applying the model to a group or looking at it over a longer time scale.

Such simple decision making can be depicted visually by a Galton Board or Quincunx – a device invented in 1889 by English scientist Francis Galton. The board comprises marbles, evenly spaced pegs and slots. When it’s turned upside down, the marbles fall through the pegs and are randomly located in the slots. The marbles always settle in the familiar bell-shaped curve giving a very visual representation of a normal distribution.  “It’s about visualising order embedded in chaos,” explained Harris. “Of course, we can do all this now by computer simulation.”

But these models involve memoryless agents with two choices of equal probability – real life is generally much more complex which is where Non-Markovian statistical mechanics can assist.

Painful colonoscopies and complicating decision making

Harris explained the peak-end rule which is based on a series of experiments in the 1990s including those rating pain experienced during colonoscopies by psychologist and economist Daniel Kahneman (who won the 2002 Nobel Prize in Economic Sciences) and colleagues which suggested that pain ratings are based on a combination of peak pain and final pain.

“They found the experience was remembered by the worst and the last bits – so you remember the extremes, not everything. This has been tested in other scenarios and often appears to be a good predictor of memory,” explained Harris. It’s also supported by other psychological research showing that extreme and recent events are most dominant in our memories.

She also referred to the work of Jeremy Bentham, regarded as the father of modern utilitarianism, which suggests that the greatest happiness of the greatest number of people is the measure of right and wrong. “According to Bentham, humans should make decisions in a way that maximises happiness; utility is a measure of satisfaction, benefit and reward from a decision. Clearly memory plays a role in how we ascribe utility to decisions,” said Harris.

“We aim to investigate how peak-end memory affects future decisions. As a starting point, we make the random walk model more realistic by incorporating a utility score for each decision. The probability of a decision is now dependent on past experience – an agent remembers the best experience for choice A and the best experience for choice B. Neither choice is better but they remember the highest utility for each.” Significantly she explained how, even in this simple symmetric set-up, people can become trapped into thinking one choice is better and outlined three classes of behaviour depending on the distribution of utilities. The noise or fuzziness in decision making can play an important role here too.

“We also hope to get to a model that takes account of collective effects,” she continued. “We don’t make decisions in isolation. We interact with others. We have to account for our level of satisfaction being affected by the decisions of others, and shared memories as well. We are working to include such elements and correlated experiences.”

Harris believes that such modelling can be used for a variety of real-life decisions including those in the social and economics domains, however, she did point to the limitations. “Modelling is generally a consequence of assumptions not truths. Models depend on the questions asked and assumptions made. You have to be careful how you use them and how you interpret things. We need to interrogate models within the broader context. They can’t explain everything especially not when humans change their behaviours.”


Michelle Galloway: Part-time media officer at STIAS
Photograph: Noloyiso Mtembu

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