Cumulative Prospect Theory is a model for descriptive decisions under risk which has been introduced by Amos Tversky and Daniel Kahneman in 1992 (Tversky, Kahneman, 1992). It is a further development and variant of prospect theory. The difference from the original version of prospect theory is that weighting is applied to the cumulative probability distribution function, as in rank-dependent expected utility theory, rather than to the probabilities of individual outcomes. In 2002, Daniel Kahneman received the Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel for his contributions to behavioral economics, in particular the development of Cumulative Prospect Theory (CPT). Amos Tversky (March 16, 1937 - June 2, 1996) was a pioneer of cognitive science, a longtime collaborator of Daniel Kahneman, and a key figure in the discovery of systematic human cognitive bias and handling of risk. ... Daniel Kahneman Daniel Kahneman (born March 5, 1934 in Tel Aviv, in the then British Mandate of Palestine, now in Israel), is a key pioneer and theorist of behavioral finance, which integrates economics and cognitive science to explain seemingly irrational risk management behavior in human beings. ... 1992 (MCMXCII) was a leap year starting on Wednesday. ... The prospect theory was developed by Daniel Kahneman and Amos Tversky in 1979. ... The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel (in Swedish Sveriges Riksbanks pris i ekonomisk vetenskap till Alfred Nobels minne), is a prize awarded each year for outstanding intellectual contributions in the field of economics. ...

Outline of the model

A typical value function in Prospect Theory and Cumulative Prospect Theory. It assigns values to possible outcomes of a lottery.

A typical weighting function in Cumulative Prospect Theory. It transforms objective cumulative probabilities into subjective cumulative probabilities.

The main observation of CPT (and its predecessor Prospect Theory) is that people tend to think of possible outcomes usually relative to a certain reference point (often the status quo) rather than to the final status, a phenomenon which is called framing effect. Moreover, they have different risk attitudes towards gains (i.e. outcomes above the reference point) and losses (i.e. outcomes below the reference point) and care generally more about potential losses than potential gains (loss aversion). Finally, people tend to overweight extreme, but unlikely events, but underweight "average“ events. The last point is a difference to Prospect Theory which assumes that people overweight unlikely events, independently of their relative outcomes. In economics, framing means the manner in which a rational choice problem has been presented. ... In prospect theory, loss aversion. ...

CPT incorporates these observations in a modification of Expected Utility Theory by replacing final wealth with payoffs relative to the reference point, by replacing the utility function with a value function, depending on this relative payoff, and by replacing cumulative probabilites with weighted cumulative probabilities. In the general case, this leads to the following formula for the subjective utility of a risky outcome described by the probability measure p: The expected utility hypothesis is the hypothesis in economics that the utility of an agent facing uncertainty is calculated by considering utility in each possible state and constructing a weighted average. ... This article is about utility in economics and in game theory. ...

where v is the value function (typical form shown in Figure 1), w is the weighting function (as sketched in Figure 2) and , i.e. the integral of the probability measure over all values up to x, is the cumulative probability.

This formula is a generalization of the original formulation by Tversky and Kahneman which allows for arbitrary (continuous) outcomes, and not only for finitely many distinct outcomes.

Differences to Prospect Theory

The main modification to Prospect Theory is that cumulative probabilites are transformed, rather than the probabilities itself. This leads to the aforementioned overweighting of extreme events which occur with small probability, rather than to an overweighting of all small probability events. The modification helps to avoid a violation of first order stochastic dominance and enables the above generalization to arbitrary outcome distributions. Prospect Theory can instead only be applied to finitely many outcomes. CPT is therefore on theoretical grounds an improvement over Prospect Theory. The predictions of CPT and Prospect Theory are often very similar, however, there are subtle differences, and it is in particular remarkable that violations of stochastic dominance have been frequently observed in experiments (Birnbaum & Navarrete, 1998). The term Stochastic dominance is used in decision theory to refer to situations where one lottery (a probability distribution over outcomes) can be ranked as superior to another, with only limited knowledge of preferences. ...

Applications

Cumulative prospect theory has been applied to a diverse range of situations which appear inconsistent with standard economic rationality, in particular the equity premium puzzle, the asset allocation puzzle, the status quo bias, various gambling and betting puzzles, intertemporal consumption and the endowment effect. The equity premium puzzle refers to the phenomenon that observed average annual returns on stocks over the past century are higher, by approximately 6 percentage points, than returns on government bonds. ... Status quo bias is cognitive bias for the status quo; in other words, people like things to stay relatively the same. ... The endowment effect is a hypothesis that people value a good (object) more once their property right to it has been established. ...

References

• Amos Tversky and Daniel Kahneman. Advances in prospect theory: Cumulative

representation of uncertainty. Journal of Risk and Uncertainty, 5:297–323, 1992.

• Birnbaum, M. H. and Navarrete, J. B. Testing descriptive utility theories: Violations of stochastic dominance and cumulative independence. Journal of Risk and Uncertainty, 17, 49-78, 1998.