Estimation of Peaked Densities Over the Interval [0,1] Using Two-Sided Power Distribution: Application to Lottery Experiments

This paper deals with estimating peaked densities over the interval [0,1] using two-sided power distribution (Kotz, van Dorp, 2004). Such data were encountered in experiments determining certainty equivalents of lotteries (Kontek, 2010). This paper summarizes the basic properties of the two-sided power distribution (TP) and its generalized form (GTP). The GTP maximum likelihood estimator, a result not derived by Kotz and van Dorp, is presented. The TP and GTP are used to estimate certainty equivalent densities in two data sets from lottery experiments. The obtained results show that even a two-parametric TP distribution provides more accurate estimates than the smooth three-parametric generalized beta distribution GBT (Libby, Novick, 1982) in one of the considered data sets. The three-parametric GTP distribution outperforms GBT for these data. The results are, however, the very opposite for the second data set, in which the data are greatly scattered. The paper demonstrates that the TP and GTP distributions may be extremely useful in estimating peaked densities over the interval [0,1] and in studying the relative utility function.Density Distribution; Maximum Likelihood Estimation; Lottery experiments; Relative Utility Function.

Multi-Outcome Lotteries: Prospect Theory vs. Relative Utility

This paper discusses two approaches for the analysis of multi-outcome lotteries. The first uses Cumulative Prospect Theory. The second is the Relative Utility Function, which strongly resembles the utility function hypothesized by Markowitz (1952). It is shown that the relative utility model follows Expected Utility Theory with a transformed outcome domain. An illustrative example demonstrates that not only it is a simpler model, but it also provides more sound predictions regarding certainty equivalents of multi-outcome lotteries. The paper discusses estimation procedures for both models. It is noted that Cumulative Prospect Theory has been derived using two-outcome lotteries only, and it is hard to find any evidence in the literature of its parameters ever having been estimated by using lotteries with more than two outcomes. Least squares (mean) and quantile (including median) regression estimations are presented for the relative utility model. It turns out that the estimations for two- and three-outcome lotteries are essentially the same. This confirms the correctness of the model and vindicates the homogeneity of responses given by subjects. An additional advantage of the relative utility model is that it allows multi-outcome lotteries, together with the estimation results, to be presented on a single graph. This is not possible using Cumulative Prospect Theory.Multi-Prize Lotteries, Lottery / Prospect Valuation, Markowitz Hypothesis, Prospect / Cumulative Prospect Theory, Aspiration / Relative Utility Function.

Two Kinds of Adaptation, Two Kinds of Relativity

This paper presents a review of adaptation concepts at the evolutionary, environmental, neural, sensory, mental and mathematical levels, including Helson’s and Parducci’s theories of perception and category judgments. Two kinds of adaptation can be clearly distinguished. The first, known as level adaptation, refers to the shift of the neutral perception level to the average stimulus value. It results in a single reference point and stimuli changes represented in absolute terms. This concept is employed by Prospect Theory, which assumes that gains and losses are perceived as monetary amounts. The second kind of adaptation refers to the adjustment of perception sensitivity to stimuli range. It results in two reference points (minimum and maximum stimulus) and stimuli changes perceived in relative terms. Both range adaptation and range relativity are well documented phenomena and have even been confirmed by the creators of Prospect Theory. This makes room for another decision making theory based on the range relativity approach. As shown by Kontek (2009), such a theory would not require the concept of probability weighting to describe lottery experiments or behavioral paradoxes.Adaptation-Level Theory, Range-Frequency Theory, Prospect Theory

Linking Decision and Time Utilities

This paper presents the functional relationship between two areas of interest in contemporary behavioral economics: one concerning choices under conditions of risk, the other concerning choices in time. The paper first presents the general formula of the relationship between decision utility, the survival function, and the discounting function, where decision utility is an alternative to Cumulative Prospect Theory in describing choices under risk (Kontek, 2010). The stretched exponential function appears to be a simple functional form of the resulting discounting function. Solutions obtained using more complex forms of decision utility and survival functions are also considered. These likewise lead to the stretched exponential discounting function. The paper shows that the relationship may also have other forms, including the hyperbolic functions typically used to describe the intertemporal experimental results. This solution has however several descriptive disadvantages, which restricts its common use in the description of lottery and intertemporal choices, and in financial asset valuations.Discounted Utility, Hyperbolic Discounting, Decision Utility, Prospect Theory, Asset Valuation

What is the actual shape of perception utility?

Cumulative Prospect Theory (Kahneman, Tversky, 1979, 1992) holds that the value function is described using a power function, and is concave for gains and convex for losses. These postulates are questioned on the basis of recently reported experiments, paradoxes (gain-loss separability violation), and brain activity research. This paper puts forward the hypothesis that perception utility is generally logarithmic in shape for both gains and losses, and only happens to be convex for losses when gains are not present in the problem context. This leads to a different evaluation of mixed prospects than is the case with Prospect Theory: losses are evaluated using a concave, rather than a convex, utility function. In this context, loss aversion appears to be nothing more than the result of applying a logarithmic utility function over the entire outcome domain. Importantly, the hypothesis enables a link to be established between perception utility and Portfo-lio Theory (Markowitz, 1952A). This is not possible in the case of the Prospect Theory value function due its shape at the origin.Prospect Theory, value function, perception utility, loss aversion, gain-loss separability violation, neuroscience, Portfolio Theory, Decision Utility Theory.

The Illusion of Irrationality

This short paper shows that the Allais Paradox and the Common Ratio Effect regarded as classic examples of the violation of the Expected Utility Theory Axioms – may be easily explained by assuming that changes in wealth (i.e. gains and losses) are perceived in relative terms. The preference reversal observed in experiments is therefore predictable and the choices shall consequently be assumed to be rational. By contrast, the assumption that wealth changes are perceived in absolute terms leads to the conclusion that the choices violate the axioms underlying Expected Utility Theory, and are therefore irrational. This state of affairs is called the illusion of irrationality.Expected Utility Theory, Relative Utility Function, Allais Paradox, Common Ratio Effect, Prospect Theory

Are People Really Risk Seeking for Losses?

This short paper demonstrates that the claim of Cumulative Prospect Theory (CPT) that people are risk seeking for loss prospects, which confirmed a hypothetical assumption of the earlier Prospect Theory (PT), appears to be merely a result of using a specific form of the probability weighting function to estimate the power factor of the value function. Using experimental data and the form of the probability weighting function presented by CPT gives a power factor for losses of less than 1. This would mean that people are risk seeking for loss prospects. However, once more flexible, two-parameter forms are used, the power factor takes on values between 1.04 and 1.10. This, however, makes the value function convex, which indicates risk aversion. It follows that people are generally risk averse both for gains and for losses. This contradicts one of the main theses of Prospect Theory.Prospect Theory; Value Function; Probability Weighting Function; Risk Attitude

Mean, Median or Mode? A Striking Conclusion From Lottery Experiments

This paper deals with estimating data from experiments determining lottery certainty equivalents. The paper presents the parametric and nonparametric results of the least squares (mean), quantile (including median) and mode estimations. The examined data are found to be positively skewed for low probabilities and negatively skewed for high probabilities. This observation leads to the striking conclusion that lottery valuations are only nonlinearly related to probability when means are considered. Such nonlinearity is not confirmed by the mode estimator in which case the most likely lottery valuations are close to their expected values. This means that the most likely behavior of a group is fully rational. This conclusion is a significant departure from one of the fundamental results concerning lottery experiments presented so far.Lottery experiments; Least Squares, Quantile, Median, and Mode Estimators; Nonparametric and Parametric Estimators; Relative Utility Function; Prospect Theory.

Decision Utility Theory: Back to von Neumann, Morgenstern, and Markowitz

Prospect Theory (1979) and its Cumulative version (1992) argue for probability weighting to explain lottery choices. Decision Utility Theory presents an alternative solution, which makes no use of this concept. The new theory distinguishes decision and perception utility, postulates a double S-shaped decision utility curve similar to one hypothesized by Markowitz (1952), and applies the expected decision utility value similarly to the theory by von Neumann and Morgenstern (1944). Decision Utility Theory proposes straightforward risk measures, presents a simple explanation of risk attitudes by using the aspiration level concept, and predicts that people might not consider probabilities and outcomes jointly, on the contrary to the expected utility paradigm

Mean, Median or Mode? A Striking Conclusion From Lottery Experiments

This paper deals with estimating data from experiments determining lottery certainty equivalents. The paper presents the parametric and nonparametric results of the least squares (mean), quantile (including median) and mode estimations. The examined data are found to be positively skewed for low probabilities and negatively skewed for high probabilities. This observation leads to the striking conclusion that lottery valuations are only nonlinearly related to probability when means are considered. Such nonlinearity is not confirmed by the mode estimator in which case the most likely lottery valuations are close to their expected values. This means that the most likely behavior of a group is fully rational. This conclusion is a significant departure from one of the fundamental results concerning lottery experiments presented so far