A Review of Prospect Theory

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1. Introduction

Through a large number of experimental studies, behavioral psychologists have found that people’s decisions are not always rational, and their risk attitudes and behaviors often deviate from the assumption of optimal behavior patterns in traditional economic theories. It is concluded that in the decision-making process, people not only have intuitive biases, but also have frame dependence biases, representativeness biases, availability biases, anchoring effects, cognitive differences and affected groups. Therefore, people often make different and contradictory choices for the same problem at different times.

Kahneman and Tversky [1] found that when making decisions with uncertainty, the ultimate utility of an individual is not simply the expected value of the possible future utility. The individual first edits the ultimate possible prospect (Kahneman and Tversky uses the word “prospect” specifically, which is different from the “expect” used in traditional expected utility theory. It emphasizes that the “prospect” is different from the simple “expectation”) and then evaluates the edited prospect. They believe that individual psychological structure plays a key role in editing.

More than 30 years later, prospect theory is still widely viewed as the best available description of how people evaluate risk in experimental settings. Kahneman and Tversky’s papers on prospect theory have been cited tens of thousands of times.

Over the past decade, researchers in the field of behavioral economics have put a lot of thought into how prospect theory should be applied in various fields. This effort is bearing fruit. A growing body of empirical work tests the predictions of these new theories. In this essay, after first reviewing prospect theory and the difficulties inherent in applying it, I discuss some of this recent work. It is too early to declare this research effort an unqualified success, but the rapid progress of the last decade makes me optimistic that at least some of the insights of prospect theory will eventually find a permanent and significant place in various fields.

2. What Is Prospect Theory

Prospect theory believes that the individual doesn’t value the final gains and losses to make decisions but the gains and losses relative to the reference point. According to this discovery, Kahneman and Tversky (1979) obtained a value function of gains and losses relative to the reference point. Furthermore, they point out that the expected utility function theory is too simple to use probability as the weight directly. Under uncertainty, the individual, instead of weighing the probability, adds up the possible value function in the future by decision weighting function, thus obtaining the final decision value of the individual.

2.1. Decision Framework

According to prospect theory (Kahneman and Tversky, 1979), investors will go through two stages when making selection and decision: Editing phase and evaluation phase. Editing stage of main function is to collect and organize information, and the corresponding pretreatment. It consists of four parts, which is data coding, data combination, data separation and data Cancellation. The second stage is the evaluation stage. In this stage, investors value and choose each edited prospect, and then choose the best prospect. The framework of investor decision-making under prospect theory is shown in Figure 1.

Kahneman and Tversky shift the traditional approach to assessing aggregate effects. They measure the total value of a prospect (V), which is determined primarily by a combination of the value function (v) and the decision weight function

Figure 1. The framework of investor decision-making under prospect theory.

(π). The value function reflects the subjective value of the result, and the decision weight function represents the decision weight corresponding to the probability (P) of the result, which is essentially different from the objective probability and reflects the influence of probability on the whole prospect value.

2.2. Value Function

The value function in prospect theory replaces the utility function in expectation utility theory. An important feature of value function is the existence of “reference point”. The position of point O in Figure 2 is the reference point. When people evaluate a thing or make a choice, they will compare other certain reference objects intentionally or unintentionally. The comparison reference objects are different, even the same thing will get different results. Therefore, the value function values the variation value based on the reference point, namely “gain” and “loss”. In addition to the definition of wealth change, the function curve is s-shaped, which is convex for “gains” and concave for “losses”. As the development of both ends, the change of direction presents a decreasing trend of sensitivity. Moreover, it bends at the reference point. It is much steeper to the left of the reference point when the “loss” is small, compared to the case where the “gain” is small to the right of the reference point, as shown in Figure 2 below.

The value function converts surface values such as dollars into decision values. The specific form of value function proposed by Kahneman and Tversky is:

$v\left(x\right)=\{\begin{array}{l}{x}^{\alpha}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\left(x\ge 0\right)\\ -\lambda {\left(-x\right)}^{\beta}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.05em}}\text{\hspace{0.05em}}\text{\hspace{0.17em}}\left(x<0\right)\end{array}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}v(\cdot )\pi \left(p\right)<p$

Figure 2. Value function.

where x is the gain and loss of the surface value, and the gain is positive and the loss is negative; α and β are risk attitude coefficients, 0 < α < 1, 0 < β < 1, the larger the α and β are, the more risk prone the decision maker is; λ is the loss aversion coefficient, or if it is λ > 1, the decision maker is more sensitive to losses. Υ is the decision value, and obviously, υ(0) = 0.

2.3. Weighting Function

The decision weight function in prospect theory replaces the probability in expected utility theory. Decision weight is a kind of subjective judgment made by decision makers according to the probability of the occurrence of event results. It is not probability, nor does it follow the axioms of probability theory, but gives probability a weight, which can be regarded as the psychological probability of decision makers. Decision weight function has the following properties: π(ρ) is the monotone function of ρ. For small probability, it always gives great weight, π(ρ)>ρ; And for the large probability is always given a small weight, that is, π(ρ)<ρ. Its general shape is shown in Figure 3.

The weight function converts probability into decision weight. As mentioned above, the profit and loss decision weights defined by Kahneman and Tversky are:

$\begin{array}{l}{\pi}_{t}^{+}={w}^{+}\left({\displaystyle \underset{j=i}{\overset{n}{\sum}}{P}_{j}}\right)-{w}^{+}\left({\displaystyle \underset{j=i+1}{\overset{n}{\sum}}{P}_{j}}\right)\\ {\pi}_{t}^{-}={w}^{-}\left({\displaystyle \underset{j=1}{\overset{i}{\sum}}{P}_{j}}\right)-{w}^{-}\left({\displaystyle \underset{j=1}{\overset{i-1}{\sum}}{P}_{j}}\right)\end{array}$

where, w^{+} and w^{−} are non-linear weight functions of gains and losses respectively.

Figure 3. Weighting function.

For situations where the risk outlook is two or more outcomes, the w^{+} and w^{−} functions given by Prelec D (2005) are:

$\begin{array}{l}{w}^{+}\left({\displaystyle \underset{j=i}{\overset{n}{\sum}}{P}_{j}}\right)=\mathrm{exp}\left(-{\gamma}^{+}\left(-\mathrm{ln}\left(\underset{j=i}{\overset{n}{{\displaystyle \sum}}}\right)\phi \right)\right)\\ {w}^{-}\left({\displaystyle \underset{j=1}{\overset{i}{\sum}}{P}_{j}}\right)=\mathrm{exp}\left(-{\gamma}^{-}\left(-\mathrm{ln}\left(\underset{j=1}{\overset{i}{{\displaystyle \sum}}}\right)\phi \right)\right)\end{array}$

where, γ^{+}, γ^{−} and φ are parameters of the model, γ^{+} > 0, γ^{−} > 0, φ > 0.

2.4. Contributions

To sum up, there are four main contributions of prospect theory:

1) People not only value the absolute amount of wealth, but also the change in wealth. Compared with the total amount of investment, investors are more concerned about the profit or loss of investment.

2) People are more inclined to take risks and gamble when faced with the prospect of loss with similar conditions (risk preference), while they are more inclined to achieve certain profits when faced with the prospect of profit with similar conditions (risk aversion).

3) The pain of a decrease in wealth is not equal to the pleasure of an increase in wealth by the same amount, and the former is greater than the latter.

4) The actual results of the early decision affect the later risk attitude and decision. The early profits can enhance people’s risk preference and smooth the later losses. Early losses exacerbated the pain of later losses and increased risk aversion.

And there are two defects of prospect theory:

1) Prospect theory lacks the strict theory and mathematical deduction, only on people’s behavior, so the prospect theory research can only make its descriptive is getting better and better, in other words, it just shows how people will do, and don’t tell people how to do.

2) Prospect theory, as a descriptive model of decision-making under risk, has great application value and wide application scope. However, the current application research mainly focuses on the financial market, so the application scope needs to be expanded.

3. The Development of Prospect Theory

3.1. Overview of Theoretical Development

The development time line of prospect theory was roughly put forward in 1979, and developed by leaps and bounds in 1982. After the 1990s, hundreds of flowers blossomed. The following Table 1 clearly shows the development of prospect theory.

3.2. Cumulative Prospect Theory

On the basis of the prospect theory, Tersky and Kahneman (1992) [12] further

Table 1. The development of prospect theory.

proposed the cumulative prospect theory. Different from previous theories, Cumulative prospect theory is based on the rank-dependent function, or cumulative function. In the cumulative prospect theory, they summarized the phenomena violating the classical rational man hypothesis in the experiment into five categories:

1) Framing effects: In the traditional rational man hypothesis, the description of choice set does not affect the order of individual utility. However, experiments have shown that different organizational frameworks of choice sets have a systematic effect on individual preferences.

2) Nonlinear preferences: The traditional unascertainable expected utility function is a linear function with probability as the weight and different individual choices, but the experiment shows that this linear relationship is often broken.

3) Source dependence: They found that when dealing with uncertainty, individuals should consider not only the size of uncertainty, but also the source of uncertainty. That is, in the case of the same size of uncertainty, individual decisions may be different simply because of different sources of uncertainty.

4) Risk seeking: In contrast to the classical hypothesis about individuals, in some cases individuals may be risk-averse rather than risk-averse.

5) Loss aversion. They found that the fundamental uncertainty is that individuals are more sensitive to loss than to gain under the same conditions.

Based on these experimental results, Tersky and Kahneman (1992) revised the value function and weight function of the original prospect theory. In the end, they get a more specific s-shaped value function, and an anti-s-shaped weight function (a big change from the previous outlook theory), moreover, the value function and weight function are no longer independent. According to them, after comprehensive consideration of s-shaped value function and anti-s-shaped weight function, the individual finally presents four risk attitudes:

a) risk avoidance for benefits under high probability; b) seek for the risk of loss with high probability; c) seek for the risk of return under small probability; d) risk avoidance of loss with small probability.

4. Application Fields

Prospect theory is, first and foremost, a model of decision making under risk. As such, the most obvious places to look for applications are areas such as finance and insurance where attitudes to risk play a central role. We therefore start by discussing efforts to integrate prospect theory into these two fields and then turn to other areas. According to the number of published year, recently ten years it is rising until 2014, nearly four years slowed, may be a mature prospect theory, not the iconic new viewpoint is put forward. The application of the theory of value according to the number of related literature on is very high in Table 2 below brief introduction some reference of the related applications.

5. Conclusions

Reviewing the emergence and development of prospect theory, we can find that the framework of prospect theory has been formed, and it has better explanatory power to those phenomena that can’t be explained by the theory of expected efficiency. From the point of view of the existing research, the focus is mainly on the value function, the specific form of the weight function and the determination of parameters. Relative “gains” and “losses” based on reference points are also applied in many fields. However, as can be seen from the literature review, the prospect theory may have some limitations in the following two aspects:

First of all, the value function and weight function need to be further studied. The existing theoretical model of prospect still has some phenomena inconsistent with the empirical evidence. The main reason is that the value function and the weight function have some subjectivity in the form selection, and there is no uniform method and standard in the function parameter fitting.

Secondly, prospect theory is also one of the main foundations of behavioral finance, but the current research is basically in the category of risk, and most of the research methods are using experimental data fitting or constructing discrete model simulation. There is not much research on the behavioral decision-making of enterprises in the market environment. If the relevant research

Table 2. Application fields of prospect theory.

model for the uncertainty of the natural environment can be adapted to the internal and external environment of enterprises, the prospect theory may also be developed.

Although this paper has made a systematic introduction to the prospect theory, including its theoretical content, development and application fields, there are still some deficiencies in the research, for example: 1) the model introduction of the prospect theory is not detailed enough; 2) there is a lack of reference materials about the recent progress of the prospect theory; 3) the application field of the prospect theory is not detailed enough.

At the end of the paper, the application scope of prospect theory is still expanding, but its disciplinary applicability is not enough. I plan to apply prospect theory to the field of product innovation for research. Then it can help enterprises reasonably judge and control the risk of product innovation, so as to improve the success rate and benefits of product innovation.

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