# Prospect theory: an analysis of decision under risk

## Summary (4 min read)

### 50% chance to win nothing;

- To appreciate the significance of the amounts involved, note that the median net monthly income for a family is about 3,000 Israeli pounds.
- Several forms of each questionnaire were constructed so that subjects were exposed to the problems in different orders.
- The problems described in this paper are selected illustrations of a series of effects.
- The pattern of results was essentially identical to the results obtained from Israeli subjects.
- The reliance on hypothetical choices raises obvious questions regarding the validity of the method and the generalizability of the results.

### Certainty, Probability, and Possibility

- In expected utility theory, the utilities of outcomes are weighted by their probabilities.
- The best known counter-example to expected utility theory which e*ploits the certainty effect was introduced by the French economist Maurice Allais in 1953 [2] .
- The following pair of choice problems is a variation of Allais' example, which differs from the original in that it refers to moderate rather than to extremely large gains.
- The number of respondents who answered each problem is denoted by N, and the percentage who choose each option is given in brackets.
- The certainty effect is not the only type of violation of the substitution axiom.

### The Reflection Effect

- The previous section discussed preferences between positive prospects, i.e., prospects that involve no losses.
- Second, recall that the preferences between the positive prospects in Table I are inconsistent with expected utility theory.
- In the positive domain, the certainty effect contributes to a risk averse preference for a sure gain over a larger gain that is merely probable.
- Third, the reflection effect eliminates aversion for uncertainty or variability as an explanation of the certainty effect.

### Probabilistic Insurance

- The prevalence of the purchase of insurance against both large and small losses has been regarded by many as strong evidence for the concavity of the utility function for money.
- To illustrate this concept, consider the following problem, which was presented to 95 Stanford University students.
- In this program you pay half of the regular premium.
- In case of damage, there is a 50 per cent chance that you pay the other half of the premium and the insurance company covers all the losses; and there is a 50 per cent chance that you get back your insurance payment and suffer all the losses.
- Under these circumstances, would you purchase probabilistic insurance: Yes, No.

### The Isolation Effect

- In order to simplify the choice between alternatives, people often disregard components that the alternatives share, and focus on the components that distinguish them (Tversky [44] ).
- This approach to choice problems may produce inconsistent preferences, because a pair of prospects can be decomposed into common and distinctive components in more than one way, and different decompositions sometimes lead to different preferences.
- The essential difference between the two representations is in the location of the decision node.
- Thus, the outcome of winning 3,000 has a certainty advantage in the sequential formulation, which it does not have in the standard formulation.
- The isolation effect implies that the contingent certainty of the fixed return enhances the attractiveness of this option, relative to a risky venture with the same probabilities and outcomes.

### The preceding problem illustrated how preferences may be altered by different representations of probabilities. We now show how choices may be altered by varying the representation of outcomes.

- Consider the following problems, which were presented to two different groups of subjects.
- Evidently, the subjects did not integrate the bonus with the prospects.
- The choice between a total wealth of $100,000 and even chances to own $95,000 or $105,000 should be independent of whether one currently owns the smaller or the larger of these two amounts.
- The responses to Problem 12 and to several of the previous questions suggest that this pattern will be obtained if the individual owns the smaller amount, but not if he owns the larger amount.

### The Value Function

- An essential feature of the present theory is that the carriers of value are changes in wealth or welfare, rather than final states.
- An individual's attitude to money, say, could be described by a book, where each page presents the value function for changes at a particular asset position.
- Many sensory and perceptual dimensions share the property that the psychological response is a concave function of the magnitude of physical change.
- These preferences are in accord with the hypothesis that the value function is concave for gains and convex for losses.
- Hence, the derived value function of an individual does not always reflect "pure" attitudes to money, since it could be affected by additional consequences associated with specific amounts.

### The Weighting Function

- Decision weights are inferred from choices between prospects much as subjective probabilities are inferred from preferences in the Ramsey-Savage approach.
- For any reasonable person, the probability of winning is .50 in this situation.
- The two scales coincide (i.e., 77(p) = p) if the expectation principle holds, but not otherwise.
- The authors turn now to discuss the salient properties of the weighting function 7r, which relates decision weights to stated probabilities.
- The pattern of preferences in Problems 7 and 7', however, suggests that subadditivity need not hold for large values of p.

### 4. DISCUSSION

- In the final section the authors show how prospect theory accounts for observed attitudes toward risk, discuss alternative representations of choice problems induced by shifts of reference point, and sketch several extensions of the present treatment.
- Expected utility theory is violated in the above manner, therefore, whenever the v-ratio of the two outcomes is bounded by the respective 7r-ratios.
- The same analysis applies to other violations of the substitution axiom, both in the positive and in the negative domain.

### Risk

- The authors next prove that the preference for regular insurance over probabilistic insurance, observed in Problem 9, follows from prospect theory-provided the probability of loss is overweighted.
- This analysis restricts risk seeking in the domain of gains and risk aversion in the domain of losses to small probabilities, where overweighting is expected to hold.
- In prospect theory, the overweighting of small probabilities favors both gambling and insurance, while the S-shaped value function tends to inhibit both behaviors.
- A comprehensive theory of insurance behavior should consider, in addition to pure attitudes toward uncertainty and money, such factors as the value of security, social norms of prudence, the aversiveness of a large number of small payments spread over time, information and misinformation regarding probabilities and outcomes, and many others.
- Some effects of these variables could be described within the present framework, e.g., as changes of reference point, transformations of the value function, or manipulations of probabilities or decision weights.

### Shifts of Reference

- So far in this paper, gains and losses were defined by the amounts of money that are obtained or paid when a prospect is played, and the reference point was taken to be the status quo, or one's current assets.
- The well known observation [31] that the tendency to bet on long shots increases in the course of the betting day provides some support for the hypothesis that a failure to adapt to losses or to attain an expected gain induces risk seeking.
- The preceding argument entails that insurance is likely to be more attractive in the former representation than in the latter.
- Another important case of a shift of reference point arises when a person formulates his decision problem in terms of final assets, as advocated in decision analysis, rather than in terms of gains and losses, as people usually do.

### Extensions

- Some generalizations are immediate; others require further development.
- When the number of outcomes is large, however, additional editing operations may be invoked to simplify evaluation.
- The manner in which complex options, e.g., compound prospects, are reduced to simpler ones is yet to be investigated.
- The theory is readily applicable to choices involving other attributes, e.g., quality of life or the number of lives that could be lost or saved as a consequence of a policy decision.
- In such situations, decision weights must be attached to particular events rather than to stated probabilities, but they are expected to exhibit the essential properties that were ascribed to the weighting function.

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