# Prospect theory: an analysis of decision under risk

## SummaryÂ (6 min read)

### 1. INTRODUCTION

- The present paper describes several classes of choice problems in which preferences systematically violate the axioms of expected utility theory.

### 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.

### N = 66 [73]* [27]

- Note that in Problem 7 the probabilities of winning are substantial (.90 and .45), and most people choose the prospect where winning is more probable.
- Most people choose the prospect that offers the larger gain.
- Similar results have been reported by MacCrimmon and Larsson [28] .
- The above problems illustrate common attitudes toward risk or chance that cannot be captured by the expected utility model.
- The results suggest the following empirical generalization concerning the manner in which the substitution axiom is violated.

### The Reflection Effect

- The previous section discussed preferences between positive prospects, i.e., prospects that involve no losses.
- His subjects were indifferent between (100, .65; -100, .35) and (0), indicating risk aversion.
- 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.

### N=95 [20] [80]*

- It is worth noting that probabilistic insurance represents many forms of protective action where one pays a certain cost to reduce the probability of an undesirable event-without eliminating it altogether.
- The installation of a burglar alarm, the replacement of old tires, and the decision to stop smoking can all be viewed as probabilistic insurance.
- The responses to Problem 9 and to several other variants of the same question indicate that probabilistic insurance is generally unattractive.
- In contrast to these data, expected utility theory (with a concave u) implies that probabilistic insurance is superior to regular insurance.

### pu (w-x) + (1-p) u (w) = u (w-y) implies

- (1r)pu(w -x) + rpu(wy) + (-p)u(w ry)> u(wy).

### Without loss of generality, we can set u(w -x) = 0 and u(w) = 1. Hence, u(wy) = 1-p, and we wish to show that rp(1-p)+(1-p)u(w-ry)> 1-p or u(w-ry)> 1-rp which holds if and only if u is concave. This is a rather puzzling consequence of the risk aversion hypothesis of utility theory, because probabilistic insurance appears intuitively riskier than regular insurance, which entirely eliminates the element of risk. Evidently, the intuitive notion of risk is not adequately captured by the assumed concavity of the utility function for wealth.

- The aversion for probabilistic insurance is particularly intriguing because all insurance is, in a sense, probabilistic.
- The most avid buyer of insurance remains vulnerable to many financial and other risks which his policies do not cover.
- There appears to be a significant difference between probabilistic insurance and what may be called contingent insurance, which provides the certainty of coverage for a specified type of risk.
- Thus, two prospects that are equivalent in probabilities and outcomes could have different values depending on their formulation.
- Several demonstrations of this general phenomenon are described in the next section.

### 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.
- In fact, Problem 12 is obtained from Problem 11 by adding 1,000 to the initial bonus, and subtracting 1,000 from all outcomes.
- 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.

### 3. THEORY

- The two scales coincide for sure prospects, where V(x, 1.0) = V(x) = v(x).

### ) equals r(p)v(x) +[1 -r(p)]v(y). Hence, equation (2) reduces to equation (1) if wr(p) + r(lp) = 1. As will be shown later, this condition is not generally satisfied. Many elements of the evaluation model have appeared in previous attempts to modify expected utility theory. Markowitz

- The replacement of probabilities by more general weights was proposed by Edwards [9] , and this model was investigated in several empirical studies (e.g.,.

### [3, 42]). Similar models were developed by Fellner [12], who introduced the concept of decision weight to explain aversion for ambiguity, and by van Dam [46] who attempted to scale decision weights. For other critical analyses of expected utility theory and alternative choice models, see Allais [2], Coombs [6], Fishburn

- The equations of prospect theory retain the general bilinear form that underlies expected utility theory.
- In order to accomodate the effects described in the first part of the paper, the authors are compelled to assume that values are attached to changes rather than to final states, and that decision weights do not coincide with stated probabilities.
- These departures from expected utility theory must lead to normatively unacceptable consequences, such as inconsistencies, intransitivities, and violations of dominance.
- Such anomalies of preference are normally corrected by the decision maker when he realizes that his preferences are inconsistent, intransitive, or inadmissible.
- In these circumstances the anomalies implied by prospect theory are expected to occur.

### 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.

### v(y)+v(-y)>v(x)+v(-x) and v(-y)-v(-x)>v(x)-v(y).

- Thus, the value function for losses is steeper than the value function for gains.
- Decision weights could produce risk aversion and risk seeking even with a linear value function.
- Nevertheless, it is of interest that the main properties ascribed to the value function have been observed in a detailed analysis of von Neumann-Morgenstern utility functions for changes wealth (Fishburn and Kochenberger [14] ).
- The functions had been obtained from thirty decision makers in various fields of business, in five independent studies [5, 18, 19, 21, 40] .
- With a single exception, utility functions were considerably steeper for losses than for gains.

### 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.

### N=72 [17]

- Note that in Problem 14, people prefer what is in effect a lottery ticket over the expected value of that ticket.
- A similar quantum of doubt could impose an upper limit on any decision weight that is less than unity.
- The following example, due to Zeckhauser, illustrates the hypothesized nonlinearity of ir.
- Most people feel that they would be willing to pay much more for a reduction of the probability of death from 1/6 to zero than for a reduction from 4/6 to 3/6.
- Because prospect theory has been proposed as a model of choice, the inconsistency of bids and choices implies that the measurement of values and decision weights should be based on choices between specified prospects rather than on bids or other production tasks.

### 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.
- If this hypothesis is correct, the decision to pay 10 for (1,000, .0 1), for example, is no longer equivalent to the decision to accept the gamble (990, .01; -10, .99).

### 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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