I Introduction
In Chapter 5 of his work "Prospect Theory", Wakker (2010) substantially aims to demonstrate the superiority of a rank-dependent utility (meaning a hierarchical sorted utility) over the "old" expected utility theory, which pre-sumes a linearity of utility. The relevant parameters by which the expected utility of its "founders" (von Neumann and Morgenstern (1944) is com-posed, are the utility, which is linked to a decision option, and the probability of occurrence, the likelihood that the decision option will lead to the desired outcome. A decision problem for person 1 between the decision options A and B, therefore, is demonstrated as the choice of probabilities of occurrences, which by Neumann and Morgenstern (1944) were shaped linearly as follows:
D = pA + (1-p)B
Thus the probability after von Neumann and Morgenstern becomes an output-oriented choice between the different option's probabilities. Figure 1 shows the relationship between two options A and B for two randomly chosen distributions as a probability distribution function.
As can be learnt from figure 1, the probability of the choice of option A increases linearly, whereas the corresponding probability for option B can be seen as a directly opposed probability, which decreases to the same degree to which the probability of option A increases. The two options are essentially inversely correlated with one another.
This graphic illustration implies the conviction, that the choice between two options is a choice between equally weighted risks, whereas the one option is chosen, for which the product of risk (probability) and utility is most beneficial. An outcome of 10 (see figure 1) therefore is more likely for option B, while an outcome of 25 is more likely for option A.
Contrary to this assumption of linearity a number of objections were pro-duced, which were also addressed by Tversky and Kahnemann in their work published in 1986.
In his contributing article, Wakker (2010) aims to identify an intuitive heu-ristic, pointing out that the improvement of identifying risk as a non-linear parameter determined by a rank dependent utility should be made. There-fore he divides Chapter 5 into three different parts:
1) The first part argues that risk aversion is not linear
2) Part two shows, that former suggested psychological solutions in order to do justice to the non-linearity of risk aversion are not sufficient
3) Finally, part three presents the concept of rank dependent utility
Table of Contents
I Introduction
II Non-Linearity of Risk Aversion
III Proposal For a Solution to the Problem of Non-Linearity of Risk Aversion
IV Rank-Dependent Utility
V Summary
Research Objectives & Topics
The primary objective of this seminar paper is to analyze the limitations of traditional expected utility theory, specifically its assumption of linearity, and to demonstrate how rank-dependent utility provides a more descriptive and empirically accurate model for individual decision-making under risk.
- Limitations of linear probability assumptions in expected utility theory.
- Analysis of risk aversion through concave utility functions.
- Identification of theoretical anomalies in simple probability weighting models.
- Practical application and calculation of rank-dependent utility.
Excerpt from the Book
I Introduction
In Chapter 5 of his work "Prospect Theory", Wakker (2010) substantially aims to demonstrate the superiority of a rank-dependent utility (meaning a hierarchical sorted utility) over the "old" expected utility theory, which presumes a linearity of utility. The relevant parameters by which the expected utility of its "founders" (von Neumann and Morgenstern (1944) is composed, are the utility, which is linked to a decision option, and the probability of occurrence, the likelihood that the decision option will lead to the desired outcome. A decision problem for person 1 between the decision options A and B, therefore, is demonstrated as the choice of probabilities of occurrences, which by Neumann and Morgenstern (1944) were shaped linearly as follows:
(1) D = pA + (1-p)B
Thus the probability after von Neumann and Morgenstern becomes an output-oriented choice between the different option's probabilities. Figure 1 shows the relationship between two options A and B for two randomly chosen distributions as a probability distribution function:
Summary of Chapters
I Introduction: This chapter introduces the critique of the linear expected utility theory and establishes the context for rank-dependent utility as a more robust framework.
II Non-Linearity of Risk Aversion: The chapter explores why risk aversion is empirically non-linear and discusses how marginal utility leads to a concave utility function.
III Proposal For a Solution to the Problem of Non-Linearity of Risk Aversion: This section examines simple weighting parameters to address non-linearity and highlights the theoretical failures of these initial attempts, such as the violation of stochastic dominance.
IV Rank-Dependent Utility: This chapter defines the concept of "rank" and provides a step-by-step methodology for calculating rank-dependent utility to better evaluate prospects.
V Summary: The concluding chapter synthesizes the arguments and emphasizes that rank-dependent utility functions as a descriptive, empirical theory distinct from normative game theory.
Keywords
Rank-Dependent Utility, Expected Utility Theory, Risk Aversion, Prospect Theory, Probability Weighting, Marginal Utility, Decision Theory, Non-Linearity, Stochastic Dominance, Utility Function, Decision Weights, Rational Choice, Behavioral Economics, Economic Psychology, Strategic Decision-Making.
Frequently Asked Questions
What is the core focus of this paper?
The paper examines the transition from traditional, linear expected utility theory to rank-dependent utility, aiming to better model human decision-making under risk.
What are the primary thematic areas covered?
The main themes include the limitations of von Neumann and Morgenstern's theory, the psychological necessity of concave utility functions, and the practical application of rank-based weighting systems.
What is the research goal?
The goal is to demonstrate that rank-dependent utility offers a more realistic and descriptive framework for evaluating individual decisions compared to normative models.
Which scientific methodology is applied?
The paper employs a theoretical analysis based on existing literature (primarily Peter P. Wakker's work) and illustrates the mathematical shifts from linear weighting to rank-dependent weighting through examples and data models.
What topics are discussed in the main body?
The main body covers the objections to utility linearity, the identification of risk as a non-linear parameter, the technical failure of simple probability transformations, and the specific calculation steps for rank-dependent utility.
Which keywords define the work?
Key terms include Rank-Dependent Utility, Risk Aversion, Prospect Theory, and Probability Weighting.
Why did the author argue that the initial weighted models were unsound?
The author highlights that simple weighting models violate the assumption of stochastic dominance and continuity, rendering them theoretically inconsistent for decision modeling.
How is the "rank" defined within this utility model?
A rank is defined as the probability that an expected outcome generates a higher value than a specific benchmark outcome, allowing for a hierarchical evaluation of prospects.
How does this theory differ from traditional normative theory?
While traditional theories focus on how decisions "should" be made based on linear rationality, the presented rank-dependent model serves as a descriptive theory that accounts for actual, often biased, human decision-making behavior.
- Quote paper
- Julia Plagemann (Author), 2011, Heuristic arguments for probabilistic sensitivity and rank dependence , Munich, GRIN Verlag, https://www.hausarbeiten.de/document/183122
