Prospect Theory: A Framework for Understanding Cognitive Biases

A prospect theory agent tasked with a decision first sets it within a frame with a convenient zero point, allowing em to classify the results of the decision as either losses or gains. Ey then computes a subjective expected utility, where the subjective expected utility equals the subjective value times the subjective probability.

If a person objects to singular they, I'm having a hard time seeing them not objecting to this. So why not just use singular they? It'd make this a lot more readable.

I agree that known biases can be explained by curves like those, plus the choice of a "frame". But how do we know we're not overfitting?

In other words: does prospect theory pay rent?

I'd want to at least see that we're identifying some real differences between people when we fit their curves from a bunch of measurements of their behavior - I'd expect their personally fit model to describe their (held-out from fitting) future actions better than one fit over the whole population, etc.

It seems like the additional degree of freedom "well, it depends on how they chose their frame in this instance" needs to be nailed down as part of testing the model's fit on future actions.

I have a paper in press at the Journal of Applied Psychology that used both hypothetical scenarios and real money in prospect theory experiments. We looked at whether people shifted their reference points post hoc-- after they had learned the outcomes of their decisions. Our results showed that people shifted their reference points to either maintain positive moods or repair negative moods.

If you are interested, you can see the paper here:

http://faculty.washington.edu/mdj3/Johnson,%20Ilies,%20&%20Boles%20%28in%20press%29.pdf

It can be interesting to look at prospect theory curves that are based on experimental data. Here are the best fit curves for 10 subjects in one study, Gonzalez & Wu (1999), for the value function for gains (v) and the probability weighting function (w). Each subject in the study made 165 (hypothetical) decisions about gambles with various possible outcomes and probabilities, in the domain of gains only (no losses).

The images are not working anymore.

The wayback machine has a stored copy here.

This post mis-uses the term "utility". Expected utility theory does not treat utility as linear in money, as you suggest.

... trying to decide whether or not to buy an hurricane insurance policy costing $5000/year. Prospero owns assets worth $10,000, and estimates a 50%/year chance of a hurricane destroying his assets; to make things simple, he will be moving in one year and so need not consider the future. Under expected utility theory, he should feel neutral about the policy.

See http://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem, or perhaps also

http://lesswrong.com/lw/244/vnm_expected_utility_theory_uses_abuses_and/

The main descriptive difference between prospect theory and EU theory is that for monetary decisions, EU theory uses one curve (utility function), whereas prospect theory uses two curves (a value function and weight function) as well as a framing variable... it's about three times as suspect for overfitting, so I think I'll wait until it pays a little more rent :)

It may not be an official bias, but the "but there's still a chance, right" outlook looks a lot like the sharply rising curve of the subjective probability function near zero.

(RETRACTED) This is an official bias, known as the certainty effect. (/RETRACTED)

EDIT (thanks, Vaniver): This is closely related to the certainty effect, which describes the sharp change in weighting near p=1 when an outcome switches from a sure thing to merely a likely possibility. The sharp change in weighting near p=0 is similar, as an outcome switches from an impossibility to merely an unlikely possibility, but I don't think it has a handy name.

I recommend Kahneman & Tversky's 1984 paper Choices, Values, and Frames (republished as chp 1 in their book of the same name) as a more readable (and shorter) introduction to prospect theory than their 1979 paper which Yvain has linked. It contains several examples demonstrating consequences of the shape of the functions and exploring the possibilities for framing effects.

I'm learning about utitlity theory just now, but I hadn't heard about prospect theory before. Thanks for posting it.

I know the main point of the post was to introduce prospect theory, but I wanted to add a comment about standard utility theory. In the text you write that standard utility theory predicts Prospero should be indifferent between a certain $5,000 and a 50-50 chance of either $0 or $10,000. This isn't quite right, maximising expected utitlity isn't the same as maximising expected wealth.

In standard utility theory you have a utility function U(W), so Prospero has the choice between U(5,000) and a 50-50 chance of U(0) or U(10,000). The expected utility need not be the same for both cases. In fact, most investors are assumed to have a utility function such that each addttional dollar adds less utility than the previous one (diminishing marginal utility of wealth). E.g $10 adds less utility to a millionaire than it would to the same person if he were broke and homeless. An investor with diminishing marginal utility of wealth would always take the insurance since, taking the certain $5,000 as the base case, the 50% chance of losing that $5,000 would cost more utility than the 50% chance of the gain of an extra $5,000 would add.

In this case, what is the difference between standard theory and prospect theory? Taking the first graph, you could regard this as a plot of a standard utility function with wealth on the x axis and utility on the y axis. The differences seem to be:

• in the second plot, it is shown that a prospect theory agent seems to behave as if small probabilities are larger than they actually are, and as if large probabilities are smaller than the actually are;
  • the fact that Prospero's utility function is different depending on how the question is framed;
  • the shape of the utility function has the form shown in the first graph, wheras in standard utility theory it can take a wider variety of possible shapes.

An excellent introduction, and I love how you've tied it in with LW discussion on cognitive biases.

Also check out temporal motivation theory (2006), which tries to integrate (cumulative) prospect theory with other theories of human behavior.

One thing I'm a bit confused about: How would weighted probabilities work when there are more than two possible outcomes? "sum Probability(x) = 1" does not imply "sum Weighted Probability(x) = 1", and furthermore, you can get a different weighted probability distribution by grouping similar outcomes and applying the weighting stepwise, first to groups of similar outcomes, and then to specific outcomes within the groups.

Prospero's change from rejecting the insurance when framed as gains, to buying it when framed as losses, directly mirrors the change in preferred survival strategies mentioned in Circular Altruism.

Did you mean?

Prospero's change from rejecting the insurance when framed as losses, to buying it when framed as gains, is directly opposite to the change in preferred survival strategies mentioned in Circular Altruism. Apparently we use radically different decision theories when thinking about our own well-being than we do when thinking about the well-being of abstract others.

I'd upvote this twice if I could.

The weighted probability curve reminded me of some other research I first heard of a couple of years ago, to do with human choices being made by comparing them to their neighbouring choices, rather than on an absolute scale of utility. The result of this being that people find it hard to appraise things on more than five levels of gradation ("worse", "this", "better", and intervals between them). This provides a plausible explanation for why we rank so many things out of five.

I looked for the research in question, and found Decision by Sampling. Having now had a look at the actual paper, it actually references prospect theory twice. I really should follow these things up more.

Fascinating. I'm amazed that nobody has brought this up here before - this is something I should have read about years ago.

The observation that gains saturate has a fairly simple explanation from evolutionary theory: the increased evolutionary fitness advantage from large material gains saturates (especially in a hunter-gatherer environment). Successfully hunting a rabbit will keep me from starving for a day; but if I successfully hunt a mammoth, I can't keep the meat for long enough for it to feed me for years. The best I can do is feed everyone in the village for a few days, hoping they remember this later when my hunting is less successful, and do a bunch of extra work to make some jerky with the rest. The evolutionary advantage is sub-linear in the kilograms of raw meat. In more recent agricultural societies, rich and powerful men like Ramses II who had O(100) children needed a lot more than 50 times the average resources of men in their society to achieve that outcome (and of course that evolutionary strategy isn't possible for women). Even today, if I'm unlucky enough to get pancreatic cancer it doesn't matter how rich I am: all that money isn't going to save me, even if I'm as rich as Steve Jobs.

Similarly, on the downside, from a personal evolutionary fitness point of view, saturation also makes sense, since there is a limit to how bad things can get: once I, my family, and everyone else in the tribe related to me are all dead, it's game over, and my personal evolutionary fitness doesn't really care whether everyone else in the region also died, or not.

So it seems to me that at least the first diagram above of prospect theory may be an example of humans being aligned with evolution's utility function.

I don't have a good evolutionary explanation for the second diagram, unless it's a mechanism to compensate for some psychological or statistical bias in how hunter-gatherers obtain information about and estimate risks, and/or how that compares to modern mathematical risk measures like probabilities and percentages.

What happens if you graph subjective expected value to probability and outcome? It looks like a lot of it could cancel, giving something close to true expected value.

I think the distinction between decisions (as an end result) and other brain processes can be useful in fields of behavioral economics and the like on the short term, as it reahes results quite fast. But the complexity of decisions makes me visit the examples of unifications in physics. Perhaps if all decisions (not only final output) are treated as the same phenomena, aspects like framing can be understood as altering sub decisions by using their constant value functions, leading to a different decision later in time (which just happens to be the output decision). The idea is perhaps understanding the building blocks of decisions (on a level smaller than final outputs and bigger than single nueron firings) can provide a better model for decision making