On prospect theory

Imagine that you are proposed the following gamble on a coin toss:

If the coin shows tails, you lose $100.
If the coin shows heads, you win $150.
Do you accept the gamble? Yes/No

According to Daniel Kahneman's experiments, most people will not accept the gamble. Despite the positive expected value, the discomfort of losing $100 outweighs the possible gain of $150. Behavioural economist Richard Thaler coined the term Econs to represent the fictitiously rational agents that exist in economic theory contrasted with the oft-irrational Humans.

While basic economic theories are useful, there are many examples like in the gamble above that act as counterexamples against the assumption of rationality. As an aspiring Econ, I wanted to better understand how Humans make decisions by exploring prospect theory¹. Published by Kahneman and Tversky in 1979, prospect theory made significant contributions to the field of behavioural economics; which led to Kahneman winning the 2002 Nobel prize in economics.

Simple gambles (such as “40% chance to win $300”) are to students of decision making what the fruit fly is to geneticists.
-from Thinking, Fast and Slow²

According to prospect theory, there are three principles that affect how we evaluate situations. First, evaluation is relative to a neutral reference point. This is also called adaptation level. You evaluate opportunities based on a reference point which is usually the status quo but can sometimes be the outcome you expect or the outcome you feel entitled to. The unit of measurement is often monetary in these scenarios for ease of comparison. Consider the following proposals.

Problem A: In addition to whatever you own, you have been given $1000. You are now asked to choose one of these options:
50% chance to win $1000 OR get $500 for sure

Problem B: In addition to whatever you own, you have been given $2000. You are now asked to choose one of these options:
50% chance to lose $1000 OR lose $500 for sure

In problems A and B, the reference points are ignored; people barely pay attention to the initial $1k/2k that are given.

Humans also tend to compare things on a relative scale instead of in absolutes. The subjective difference between $900 and $1000 is much smaller than the difference between $100 and $200. I am guilty of this sin, having spent more effort deliberating which menu items has better cost per calorie than the McDouble compared to the hundreds or thousands of dollars I spend on new tech or my racing addiction. This concept of diminishing marginal gains is also seen in Bernoulli's Utility theory.

The third principle is loss aversion. Faced with a gain or loss of the same amount, people would weigh the loss more heavily. This would explain why most people are unwilling to accept gambles such as the one introduced in the beginning. However, people become risk seeking when faced with loss such as in Problem B. The pain of losing $500 for sure is greater than 50% of the pain of losing $1000.

Not all theories are perfect, however. Kahneman is wary of developing the psychological theory equivalent of rose-coloured glasses and offers some criticisms of his own theory. For instance, prospect theory cannot deal with disappointment. If you had a 90% chance of winning $1M and 10% chance to win nothing only to end up winning nothing, you will be quite disappointed. Even though the status quo ought to be the reference point objectively, remaining there is somehow immensely disappointing.

This might be explained by the brain’s reward signals reacting to something unexpected even if it's not a sure thing. Once you are aware of the proposal, your brain sends out the reward signals. This is similar to telling people about your new year's resolutions. The act of telling others about it gets you a partial reward signal just by talking about it and gaining social acceptance.

Another criticism of prospect theory is that it does not deal with regret, as the following example shows.

Problem C: Choose between 90% chance to win $1 million OR $50 with certainty

Problem D: Choose between 90% chance to win $1 million OR $150,000 with certainty

People will feel bigger regret if they fail to win in Problem D. Perhaps the emotion of regret is the difference between the option not chosen ($150K) and the end state ($0). In the case of uncertain gambles (e.g. 90% chance to win $1 million), the value of the potential outcome is multiplied by the probability.

When examining these probabilistic gambles, I was reminded of a story in Kahneman’s Thinking, Fast and Slow that demonstrated that even statistics professors are terrible intuitive statisticians. If that is the case, can we expect Humans to fairly weigh probabilities in evaluating highly likely or unlikely events?

People tend to buy lottery tickets even though the expected value is negative (otherwise the lottery operator would lose money), and are disappointed when they do not win despite having astronomically low odds of winning. Burns et. al (2010)³ gives two explanations for this behaviour: availability heuristic and possibility effect.

Availability heuristic is a mental shortcut that relies on examples that immediately come to mind. When examining your own happiness, if all your can recall are social media stories of people’s best experiences, your life will feel miserable in comparison. Lottery players can only think of the few publicized winners, not the millions of losers that do not get reported on. This leads them to believe that the odds of winning seem more likely than it actually is.

Possibility effect describes the phenomenon where people overweight small probabilities compared to moderate ones. Although nuclear fission reactors are quite safe (scientists haven't quite gotten fusion working yet), people worry about the negative implications more than the probabilities demand. Perhaps availability heuristic is at work here with images of Fukushima or Chernobyl.

So, if people tend to overweight highly unlikely events (and also underweight highly likely events compared to certainty), can we figure out how the average person actually weighs different probabilities? We can then build a probability to weighting mapping function such as the following. \[f(p) = w, \textrm{for probability } p \textrm{ and weight } w \textrm{, with } 0 \leq p,w \leq 1\]

Luckily, some smart people already thought about this and describe a weighted cumulative probability function as part of cumulative prospect theory (CPT), a successor to prospect theory. It is interesting to note that I came about this idea before knowing about CPT. This provides some evidence that the underlying concepts of CPT are already circulating within the general public.

One criticism I would offer for CPT’s probability weighting function is that the overweighting can be done at infinitesimally small scales. I posit that at extreme scales (small or large), people lose the concept of scale. What things are measured in picometer (10-12 m) versus a femtometer (10-15 m)? Most people would not have a decent reference point to compare against. The Andromeda galaxy is a stone’s throw⁴ of 2.5 M light years away, on a collision course with the Milky Way in 4 billion years whereas the Eye of Sauron is watching from 52 M light years away (or ago?). Instead of an exponential function at the extremes, it may be a step function until there is a reasonable reference point that makes sense.

In order to empirically estimate this mapping function, I propose simple gambles between certainty to win such as:

Problem E: Choose between 1% chance to win $1 million OR $2,500 with certainty

Problem F: Choose between 99% chance to win $1 million OR $900,000 with certainty

With the right combination of chance and certain monetary values, the gambles that have close to 50-50 split will be the right trade-off. In the second example, 99% chance gets mapped to $900,000/$1 million = 90% provided that the answers are split 50-50.

Prospect theory, introduced in 1979, helped to more accurately model human behaviour, including the irrational ones. While not without its limitations, prospect theory introduced concepts such as loss aversion and evaluation from a reference point which greatly improved upon the prevailing theories of the time.

Studying decision making is useful because you can often catch yourself making irrational choices. Recall the very first problem that I introduced: you lose $100 if it's tails and win $150 if it's heads. If this offer was proposed to me, my first question would be "how many times can I play?". And my second question would be "how much time can you give me to liquidate all my assets?".

However, I want to avoid condemning all irrational behaviour. The lottery is simply a negative expected value event for Econs. But the thrill of anticipation and perhaps even the subsequent disappointment makes it a worthwhile experience for many Humans. If life was only about maximizing expected value, it would be a pretty boring one.

¹Kahneman, D., & Tversky, A. (1979). Prospect Theory: An Analysis of Decision under Risk. Econometrica, 47(2), 263-291. doi:10.2307/1914185
²Kahneman, D. (2011). Thinking, fast and slow. New York: Farrar, Straus and Giroux.
³Burns et al. (2010). Overweighting of small probabilities. https://doi.org/10.1002/9780470400531.eorms0634
⁴Technically, it will get there eventually if one throws it faster than escape velocity and the stone is freed from Earth’s gravity.