Prospect Theory - Model

Model

The theory was developed by Daniel Kahneman, a professor at Princeton University's Department of Psychology, and Amos Tversky in 1979 as a psychologically more accurate description of preferences compared to expected utility theory. It describes how people choose between probabilistic alternatives and evaluate potential losses and gains. In the original formulation the term prospect referred to a lottery.

The theory describes the decision processes in two stages, editing and evaluation. In the first, outcomes of the decision are ordered following some heuristic. In particular, people decide which outcomes they see as basically identical, set a reference point and then consider lesser outcomes as losses and greater ones as gains. In the following evaluation phase, people behave as if they would compute a value (utility), based on the potential outcomes and their respective probabilities, and then choose the alternative having a higher utility.

The formula that Kahneman and Tversky assume for the evaluation phase is (in its simplest form) given by

where is the overall or expected utility of the outcomes to the individual making the decision, are the potential outcomes and their respective probabilities. is a so-called value function that assigns a value to an outcome. The value function (sketched in the Figure) that passes through the reference point is s-shaped and asymmetrical. Losses hurt more than gains feel good (loss aversion). This differs greatly from expected utility theory, in which a rational agent is indifferent to the reference point. In expected utility theory, the individual only cares about absolute wealth, not relative wealth in any given situation. The function is a probability weighting function and expresses that people tend to overreact to small probability events, but underreact to medium and large probabilities.

To see how Prospect Theory (PT) can be applied in an example, consider the decision to buy insurance. Assuming the probability of the insured risk is 1%, the potential loss is $1,000 and the premium is $15. If we apply PT, we first need to set a reference point. This could be the current wealth or the worst case (losing $1,000). If we set the frame to the current wealth, the decision would be either to

1. Pay $15 for sure, which yields a PT-utility of ,

OR

2. Enter a lottery with possible outcomes of $0 (probability 99%) or -$1,000 (probability 1%), which yields a PT-utility of .

These expressions can be computed numerically. For typical value and weighting functions, the latter expression could be larger due to the convexity of in losses, and hence the insurance looks unattractive. If we set the frame to −$1,000, both alternatives are set in gains. The concavity of the value function in gains can then lead to a preference for buying the insurance.

In this example, a strong overweighting of small probabilities can also undo the effect of the convexity of in losses: the potential outcome of losing $1,000 is overweighted.

The interplay of overweighting of small probabilities and concavity-convexity of the value function leads to the so-called fourfold pattern of risk attitudes: risk-averse behavior in gains involving moderate probabilities and of small probability losses; risk-seeking behavior in losses involving moderate probabilities and of small probability gains.