A Brief Introduction to Prospect Theory.

“For all we know, the handwriting might have been on the wall all along. The question is: was the ink invisible?”

Amos Tversky


Nicolaus Bernoulli was born in 1687 in Basel. Hailing from a family of great mathematical thinkers (a cousin of Daniel Bernoulli), it seemed he was destined to discover, or perhaps stumble upon, a seminal finding that would underpin an entire subject area for centuries to come. What Bernoulli is perhaps best known for today is his contribution to the field of economics in the form of the St. Petersburg Paradox, which set the cogs turning in works on the mathematical reasoning behind probabilistic thinking, thus laying the foundation for microeconomics and a great amount of academic squabbling.

The St. Petersburg Paradox is a simple yet arresting philosophical problem. It goes something like this:

Some individual, organisation, or other economic agent decides to offer willing participants an interesting opportunity. If they are willing to pay a fee, they will be given a coin to flip. Starting from some arbitrarily small prize, every flip that results in heads being landed on results in a doubling of the participant’s prize. However, when tails is landed on, it’s game over; the participant walks away, unscathed, with his earnings. Now, how much should an individual be willing to pay to participate in such an opportunity? Bernoulli reasons that they should be willing to stake an arbitrarily large number to compete, as their earnings are theoretically infinite; each flip represents a 50% chance to double their money. When considering the impacts of compound interest, this would appear a very attractive investment. Bernoulli suspected that the actual stake committed by any participant would in fact be much lower. The Stanford Encyclopedia of Philosophy notes that the maximum a modern-day participant would be willing to put on the line is around $25, vindicating Bernoulli even now, in a world supposedly more educated and rational; some things just don’t change.

Troubled by this hypothetical scenario, Bernoulli set about attempting to explain this irrational behaviour through logical reasoning. The end result is what helped to consolidate marginal utility as a fundamental concept in the then fetal field of microeconomics. Furthermore, by suggesting that the total utility of wealth diminished as additional units of wealth were added, he also offered the principle of diminishing marginal utility, which would also be used to explain how some individuals would be more likely to accept one proposition than another; the size of the risk the individual is willing to take is relative to their current stock of wealth. What this amounted to was a new way of looking at the interactions between agents in markets; consumers were assumed to be utility maximisers, and perfectly rational, or in Thaler’s words ‘Econs’. Whilst the far-reaching implications of Bernoulli’s findings would not be realised until many years after his death, it can not be understated the impact he had on the field, the curiosity he inspired in others, and the fastidious discussion that would result from his suggestions; a debate that still rages today in many faculties.

In the years following Bernoulli’s initial work, theorising in probability and individual decision-making proliferated. Work by brilliant minds such as Von Neumann, Savage, Ramsey, and others set to work giving the field a more rigorous mathematical framework and grounding, lending increasingly greater levels of credibility and prestige to what once was an operation only normatively described. Whilst differing in a technical sense, most of these theories alluded to a central, and now widely-accepted tenet of microeconomic study. Individuals have well-ordered and consistent preferences that can be revealed by studying an empirical record of choices made. Given that these preferences were assumed to be consistent, it was, therefore, a matter of assigning a technical relationship to what was going on. If an individual preferred option A to option B and option B to option C, then it must be that the individual prefers option A to option C. As Thaler notes, if preferences are not well-ordered and consistent then it would be near impossible to reach any level of optimisation or equilibrium, and economics textbooks would grind to a halt faster than you could say ‘Adam Smith’s pin factory’. And so, for the best part of a couple of hundred years, expected utility and hyper-rationality was the dominant way of looking at optimisation problems involving consumers. Soon enough, however, the ghost of Joseph Schumpeter wailed, and the gales of creative destruction came sweeping in.

There had, for a long time, been a running dispute between economists and psychologists regarding the latter’s utility (ironically) to the field of economics. Staunch defenders of perfect rationality, led by the likes of Friedman et al. had frequently expressed, often publicly, that the domain of economics was better off left to mathematicians than (in their view) borderline pseudoscientists. It was not until around 1979 that the cracks in economic thinking were beginning to be penetrated by individuals of alternative academic training, often in psychology, sociology, and other socially orientated sciences. The ignorance of irrationality borne out of conservatism in academic circles was becoming more costly, as economics was becoming increasingly forceful in swaying the nature of public and social policy. It was time to start paying attention to the real world, a world bubbling with irrationality, and seemingly little way to solve it. Enter Danny Kahneman and Amos Tversky.

In 1979, Danny Kahneman and Amos Tversky were awaiting approval on their new paper, Prospect Theory: An analysis of decisions under risk. Both hailing from the Hebrew University, it was almost a Robert Redford-esque adventure that took them precariously close to offending anyone who had any allegiance whatsoever to classical economic theories. They would, however, be paving the way for a cross-germination of subjects; a cross-germination unparalleled in terms of economic impact on wider society. Well-known individuals and academics such as Richard Thaler and Cass Sunstein sit high on the shoulders of the great adventurers; they walked so others could run. What they challenged was the supposed revealed preferences of so-called ‘rational’ agents, and, through a series of experiments and questionnaires, they found something quite puzzling.

In their paper Prospect Theory: An analysis of decisions under risk, they highlight a subtle, but critical, difference in how expected utility theory is not always entirely consistent with real-life human behaviour. The classical view is that consumers will utility maximise in a given situation, and using Bayesian reasoning compute the expected outcome of such a decision relating to the situation. This is done fairly simply by multiplying the potential earning by the probability of those earnings being realised; an expected value can be calculated to weigh up which outcome should be chosen based on which yields the greatest expected value. This computation and the consequent decision should be consistent across loss and gain situations if a consumer is perfectly rational, as they amount to effectively the same thing. This assertion is, however, either misleading or false, or perhaps both. In reality, what occurs is a ‘certainty effect’ when in a ‘gain’ scenario, and a desperate aversion to deduction in a ‘loss’ scenario. In the ‘gain scenario’, this means that when offered a choice of either a sure win or a probable win of a higher expected value, the typical individual would rather the sure win. The ‘certainty effect’ coined by K+T (as they shall be known from now on) relates to the way the participant prefers the sure gain, with the majority of respondents in their studies (often around 80%) selecting this option. You might assume, therefore, that individuals might repeat this choice when a loss is involved rather than a gain, right? Wrong. When it comes to a ‘loss’ scenario, we suddenly become incredibly loss averse, and would rather gamble at the risk of a higher loss than take a definite hit. These two scenarios are effectively identical, yet display two very different behaviours. What this shows is, heretically, that expected values do not always correspond with the actions of consumers; they are inconsistent in their preferences depending on the nature of a scenario, suggesting some loss of utility when considering each transaction. This would suggest that consumers are not perfectly rational when making decisions, and optimisation may be difficult. As previously mentioned, Thaler notes that this is not a good thing for economics; ‘preference reversals’ undermine the authorities on economic behaviour and decision-making. It would turn out, alas, that this would paradoxically bring economics greater leverage in the decisions made over public policy, given the subject a greater realised impact on the lives of many of us.

The UK Behaviour Insights Team is a legacy of the work that Kahneman and Tversky started. Richard Thaler and Cass Sunstein were as close to founding fathers of the BIT as you get, and they were both greatly influenced by that initial work by K+T. Instead of denying irrationality, it is now studied in much greater detail and alleviated somewhat by what Thaler calls ‘libertarian paternalism’, experimenting with different policies to maximise consumer rationality. It is quite possible that this project would never have come about, should Thaler have not picked up the Prospect Theory paper some years ago, or had K+T settled on working strictly within their domain. Reading Prospect Theory: An analysis of decisions under risk is the best way to truly understand the magnitude of their findings, and appreciate how their involvement in the field of economics altered the subject’s trajectory for the better. Furthermore, Thaler’s book Misbehaving gives a fascinating account of his ventures into behavioural economics alongside K+T. I would recommend both works to anyone with even the slightest interest in the field; it hooked me and is yet to release its grip. Kahneman’s Thinking, Fast and Slow is another great work and serves as a kind of symbolic obituary to his friend Amos, who sadly passed away in 1996, aged just 59. Our physical presence may be ephemeral, but the manifestation of our ideas, and the creative destruction they catalyse, is perpetual.

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