Over time, researchers have become all too aware of the limitations of expected utility theory, especially those raised by the St. Petersburg, Allais, and Ellsberg paradoxes. As a result, numerous alternative theories have been developed to overcome the limitations of expected utility theory without losing its explanatory power. Prospect theory, developed by Daniel Kahneman and Amos Tversky is perhaps the most well-known of these alternative theories. This section covers the following topics:

1. Motivation -- certainty, reflection, and isloation effects
2. The editing and evaluation phases
3. Properties of the value and weighting functions
4. Cumulative Prospect Theory

In 1979, Daniel Kahneman and Amos Tversky conducted a series of thought experiments testing the Allais Paradox in Israel, at the University of Stockholm, and at the University of Michigan. Everywhere, the results followed the same pattern. The problem was even framed in many different ways, with prizes involving money, vacations, and so on. In each case, the substitution axiom was violated in exactly the same pattern. Kahnemann and Tversky called this pattern the certainty effect - meaning, people overweight outcomes that are certain, relative to outcomes which are merely probable.

Using the term "prospect" to refer to what we have so far called lotteries or gambles, (i.e. a set of outcomes with a probability distribution over them), Kahnemann and Tversky also state that where winning is possible but not probable, i.e. when probabilities are low, most people choose the prospect that offers the larger gain. This is illustrated by the second decision stage in the Allais Paradox.

More generally, if x and y are outcomes; 0 < p,q,r < 1, where p, q, and r refer to probabilities, they state that:
(y, p*q) (x, p) (y, p*q*r) (x, p*r);
where the term (outcome, probability) refers to a prospect.

Kahnemann and Tversky also found strong evidence of what they referred to as the reflection effect. To illustrate:

Imagine an Allais Paradox-type problem, framed in the following way. You must choose between one of the two gambles, or prospects:

Gamble A: A 100% chance of losing $3000.
Gamble B: An 80% chance of losing $4000, and a 20% chance of losing nothing.

Next, you must choose between:
Gamble C: A 100% chance of receiving $3000.
Gamble D: An 80% chance of receiving $4000, and a 20% chance of receiving nothing.

Kahnemann and Tversky found that 20% of people chose D, while 92% chose B. A similar pattern held for varying positive and negative prizes, and probabilities. This led them to conclude that when decision problems involve not just possible gains, but also possible losses, people's preferences over negative prospects are more often than not a mirror image of their preferences over positive prospects. Simply put - while they are risk-averse over prospects involving gains, people become risk-loving over prospects involving losses.

As long as prospects are in the positive domain, the certainty effect leads to a risk-averse preference for a sure gain, rather than one which may be larger but be merely probable. However, once prospects are in the negative domain, people exhibit risk-loving preferences for larger losses which are probable, rather than smaller certain ones.

Note: One might imagine that if this finding held universally that one would never observe people buying insurance. As we will see in the section on probability transformations, what this really implies is that in the domain of losses with moderate or high probabilities, risk seeking is predicted. Prospect theory does, in fact, predict risk-aversion for small-probability losses, which is normally the case with insurance.

Imagine yet another lottery-choice problem. Given a choice between the following, which would you choose?

Gamble A: A 25% chance of receiving $3000.
Gamble B: A 20% chance of receiving $4000, and an 80% chance of receiving nothing.

Now imagine you are faced with a two-stage problem. The first stage involves a 0.75 probability of ending the game without winning or losing anything, and a 0.25 probability of moving to the second stage, where you are presented with the following choice:

Gamble C: A 100% chance of receiving $3000.
Gamble D: An 80% chance of receiving $4000, and a 20% chance of receiving nothing.

65% of people chose B, while 78% chose C. Why is this surprising? Well, the true probabilities involved in the second choice are:
0.25 * 1 = a 0.25 probability of receiving $3000, and
0.25 * 0.8 = a 0.2 probability of receiving $4000.

Kahnemann and Tversky interpreted this finding in the following manner - in order to simplify the choice between alternatives, people frequently disregard components that the alternatives share, and focus on those which distinguish them. Since different choice problems can be decomposed in different ways, this can lead to inconsistent preferences, as above. They call this phenomenon the isolation effect.

Given the effects observed above, Kahneman and Tversky designed a new theory of decision-making under risk, which they named prospect theory.

Prospect theory differs from expected utility theory in many fundamental ways. To begin with, it distinguishes two phases in the decision-making process: an editing phase, which is a preliminary analysis of the offered prospects, and an evaluation phase, which is when the prospect with the highest value is chosen from among the edited prospects.

Next: The editing and evaluation phases

1. Kahneman, Daniel, and Tversky, Amos, (1979), "Prospect Theory: An Analysis of Decision under Risk", Econometrica, vol. 47, no. 2
2. Wakker, Peter P., Timmermans, Danielle R.M., and Machielse, Irma A. (2003), "The Effects of Statistical Information on Insurance Decisions and Risk Attitudes," working paper, Department of Economics, University of Amsterdam