Cumulative prospect theory
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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).
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[edit] Outline of the model
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.
CPT encorporates 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:
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.
[edit] 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).
[edit] 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.
[edit] 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.
