# Prospect Theory: An Analysis of Decision Under Risk

Updated: Oct 24, 2020

Kashmira Sahoo and Naehal Talwar

The analysis of decision making under risk has been mainly dominated by expected utility theory. Expected utility theory states that the decision-maker chooses between risky or uncertain prospects by comparing the respective utility values. This is calculated by summing the utility values of outcomes multiplied by their respective probabilities.

However, in practice, a wide range of choice problems violate the axioms of expected utility theory. Since this model was not an adequate descriptive model, Kahneman and Tversky came up with an alternative model of choice under risk which commonly is known as the prospect theory.

**Prospect theory** addressed the challenges presented by the expected utility theory.

There are two major probable reasons why the prospect theory is inconsistent with the expected utility theory: While in expected utility theory, utility is dependent on final wealth, under the prospect theory values assigned to gains and losses.

To understand decision under risk as a choice between a prospect or gamble, the discussion is restricted to prospects with standard probabilities.

The application of expected utility theory to choices between prospects is based on three tenets:

a) **Expectation-**overall utility of a prospect** **

b) **Asset Integration**-the domain of utility function is final states(including asset position) rather than gains or losses.

c) **Risk Aversion**- Risk aversion is the concavity of the utility function.

To have a fair idea of decision making under prospect theory, the theory highlights two phases under the choice process and reveals the three probable effects.

As individuals come across choices necessarily in most decisions they make under risk their probabilities

This theory distinguishes two phases in the choice process an early phase of editing and a subsequent phase of evaluation.

The editing phase consists of an undertaking preliminary analysis of the offered prospects for a simpler understanding of the prospects available. In the second phase, the edited prospects are then evaluated on basis of their values with the prospect of the highest value being chosen.

1**.Editing phase **

The editing phase refers to how people involved in decision-making characterize the options for choice. The effects explain how a person’s choice is influenced by the wording, order, or method in which the choices are presented. This can also be seen why several marketing strategies present options to people in different manners to influence decision making.

2. **Evaluation phase **

In the evaluation phase, people tend to make a decision based on the potential outcomes and choose the option with a higher utility. The evaluation phase comprises two indices, i.e., the value function and the weighting function, which are used to compare the prospects.

To present a fair idea, the overall value of an edited prospect denoted **v**, is expressed in terms of two scales,** 𝜋**,** **and **v**.

The first scale, **v**, associates with each probability **p **a decision weight **𝜋(p)**, which reflects the impact of **p **on the overall value of the prospect. The second scale, **v**, assigns to each outcome **x** a number **v**(**x**), which reflects the subjective value of that outcome. Hence, **v** measures the value of deviations from that reference point, i.e., gains and losses.

The basic equation of the theory describes how **𝜋** and **v** are combined to determine the overall value of regular prospects.

If **(x,p;y,q) **is a regular prospect, (i.e.,either** p+q<1,or x ≥ 0 ≥ y or x ≤ 0 ≤ y**), then –

(1) **V(x, p; y, q)=𝜋(p)v(x)+r(q)v(y);** where **v(0) = 0, 𝜋(0)=0, and 𝜋(1)= 1**.

As in utility theory, V is defined on prospects, while v is defined on outcomes. The two scales coincide for sure prospects, where **V(x, 1.0) = V(x) = v(x)**.

This Equation generalizes expected utility theory by relaxing the expectation principle.

The evaluation of strictly positive and strictly negative prospects follows a different rule. In the editing phase, such prospects are segregated into two components:

(i) the riskless component, i.e., the minimum gain or loss which is certain to be obtained or paid;

(ii) the risky component, i.e., the additional gain or loss which is actually at stake. The evaluation of such prospects is described in the next equation.

If **p+q =1** and either **x > y > 0 or x<y<0**, then

(2) **V(x, p; y, q) = v (y) + 𝜋(p)[v(x) - v (y)].**

That is, the value of a strictly positive or strictly negative prospect equals the value of the riskless component plus the value-difference between the outcomes, multiplied by the weight associated with the more extreme outcome.

**The Certainty Effect **

After conducting experiments and understanding people’s decision-making behavior, a series of effects were observed.

Outcomes that are considered certain are overweighed by people relative to merely probable outcomes. This is called the **certainty effect**. When people were given the choice:

**A**: 2500 with probability 0.33,

2400 with probability 0.66,

0 with probability 0.01;

**B**: 2400 with certainty,

**A significant percentage of people chose B **

When the probabilities of winning are substantial, more people choose the prospect where winning is more probable. When probabilities of winning are minuscule where winning is possible but not probable, people choose the prospect which offers more gain.

**The reflection effect**

The reflection effect explores preferences between negative outcomes where instead of gaining, you lose. Preferences between negative outcomes are the mirror image of preferences between the positive outcomes. The reflection effect implies that risk aversion in the positive realm is accompanied by risk-seeking in the negative realm. For example, the majority of people will be willing to accept a risk of .90 to lose 4000, in preference to a sure loss of 2000, although the gamble has a lower expected value. As preferences between the positive prospects are inconsistent with the expected utility theory, preferences between the corresponding negative prospects also violate the theory in the same manner. In the positive realm, the certainty effects lead to a risk-averse preference for a sure gain over a larger gain which is merely probable. The same effect leads to a risk-seeking preference in the negative realms, for a loss that is merely probable over a smaller loss that is certain. This is due to the overweighting of certainty. The reflection effect eliminates aversion for uncertainty as an explanation of the certainty effect.

**Probabilistic Insurance **

The prevalence of the purchase of insurance is regarded as strong evidence for the concavity of the utility function. An examination of the relative attractiveness of various forms of insurance does not support the hypothesis that the utility function is concave everywhere. Probabilistic insurance is the type of insurance problem in which people’s responses are inconsistent with the concavity hypothesis. The superiority of probabilistic insurance over regular insurance is implied by the expected utility theory. 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 Isolation Effect **

The isolation effect occurs when people are presented with two prospects with the same outcome, but different routes to the outcome. In this case, people are likely to cancel out information that is alike to lighten the cognitive load, and the choices will depend on how the options are framed.

Going deeper into the theory, A change of reference point alters the preference order for prospects. In particular, the present theory implies that a negative translation of a choice

the problem, such as arises from incomplete adaptation to recent losses, increases risk-seeking in some situations.

As in this theory, 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 one's current assets.

Although this is probably true for most choice problems, however, there are situations in which gains and losses are referred relative to an expectation level that differs from the asset status. For example, an entrepreneur who is managing a slump with greater success than his competitors may interpret a small loss as a gain, relative to the larger loss he had reason to expect.

The reference point in the preceding example highlights an asset position that one had expected to attain. A discrepancy between the reference point and the current asset position may also arise because of recent changes in wealth to which one has not yet adapted or is not anticipated.

The main observation from the analysis suggests that a person who has not made peace with his losses is likely to accept gambles that would be unacceptable to him otherwise.

To give general limitations of the theory will include the inability to take decisions involving more than 2 prospective outcomes or not being able to explain the psychological rationale behind decision even though it has been able to outline and validate certain challenges of expected utility theory

Prospect Theory with its certain limitations has application not only in consumer decision making but can be applied to decide various gambling and betting puzzles, political science, and also the much-known endowment effect. To make it more interesting it can be even used during war times by the political leaders of a country or to make certain policy decisions like a radical policy to ensure ensuring 90% employment rather than 10% unemployment.

References

https://www.uzh.ch/cmsssl/suz/dam/jcr:00000000-64a0-5b1c-0000-00003b7ec704/10.05-kahneman-tversky-79.pdf