# Quantifying Risk Perception: The Entropy Decision Risk Model Utility (EDRM-U)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### Practical Application Example (Notional)

- Part 1.
- Choice 1A: ($5, 0.5; −$1; −$1.11) (Selected) Choice 1B: (−$0.11).
- Part 2.
- Choice 2A: ($20M, 0.1; −$1M) Choice 2B: ($2M, 0.5; −$1M) (Selected).

## 2. Method

^{2}) and the nonparametric Spearman Rank Test (${r}_{s}$). Summary analyses are also performed using ANOVA at a standard 5% significance level to determine if the types of problems and the magnitude of the values have a significant effect on the results for both EDRM and EDRM-U. Note, in this case, that the assumptions of independence and constant variance can be presumed and normality will be confirmed by the use of the Shapiro–Wilk test. A flowchart illustrating the present research is provided in Figure 2.

## 3. Literature Review and Application

#### 3.1. Daniel Bernoulli’s Expected Utility

#### Comparison of Power Utility and Expected Utility

#### 3.2. Neutral Reference Point

#### 3.3. Entropy Decision Risk Model

- It accepts the use of the power utility ubiquitous to positive decision theory research to isolate the subjective-objective probability relationship, as shown in Equation (4);
- It assumes there is no difference in how the subjects valued gains or losses (i.e., that the value function exponent was constant for gains or losses and that no loss aversion was present).

#### 3.4. Mental Accounting and Transaction Utility

- Case 1
- Multiple gains (segregation): $v\left(x\right)+v\left(y\right)>v\left(x+y\right)$, which says that people prefer many smaller gains over a single larger gain (i.e., people like a greater number of smaller presents).
- Case 2
- Multiple losses (integration): $v\left(-x\right)+v\left(-y\right)<v\left(-x-y\right)$; affirms that subjects prefer grouped losses rather than separated ones (i.e., people prefer consolidated bills).
- Case 3
- Mixed gains (integration with cancellation): $v\left(x\right)+v\left(-y\right)<v\left(x-y\right),\text{}\mathrm{where}xy$; $v\left(x-y\right)$ is always positive, but the losses more quickly cancel out gains due to loss aversion.
- Case 4
- Mixed losses (segregation or integration with cancellation): $v\left(x\right)+v\left(-y\right)<v\left(x-y\right),\mathrm{where}\text{}xy$. Because $v\left(x-y\right)$ is always negative and gains are felt less than commensurate losses, Thaler states that there is a point at which there is a shift between segregation and integration; the introduction of neutral wealth permits defining this shift point.

#### 3.5. Gain-Loss Separability

#### 3.6. Risk Perception Measures: Risk Aversion and Risk Sensitivity

#### 3.6.1. Risk Aversion ($\beta $, Proximity Exponent)

#### 3.6.2. Risk Aversion Evaluation and Introduction of Risk Sensitivity

## 4. EDRM-U Model

## 5. Validation

#### 5.1. VNM Axiomatic Analysis

#### 5.2. Probability Evaluation Model (PEM)

#### 5.3. Cumulative Prospect Theory

#### 5.4. Thaler Mental Accounting (Riskless, with no Uncertainty)

#### 5.5. Prospect Theory

#### 5.6. Framing of Decisions and Psychology of Choice

#### 5.7. Rational Choice and the Framing of Decisions

- Survival Frame Choice A states: 68% survive to 1 year, 34% survive to 5 years, 90% survive surgery.
- Mortality Frame Choice A states: 32% die within 1 year, 66% die within 5 years, 10% die from surgery.

#### 5.8. Wu and Markle Gain-Loss Separability

#### 5.8.1. Gains Only

#### 5.8.2. Losses Only

#### 5.8.3. Mixed

#### 5.9. Birnbaum and Bahra Gain Loss Separability

#### 5.10. Birnbaum: Three New Tests of Independence That Differentiate Models of Risky Decision Making

- Safe: (100, 0.8; 44, 0.1; 40, 0.1) Risky: (100, 0.8; 96; 0.1; 4, 0.1).

#### 5.11. Wu and Gonzalez Weighting Function Curvature

#### 5.12. Prelec: A “Pseudo-Endowment” Effect, and Its Implications for Some Recent Nonexpected Utility Models

#### 5.13. Hershey et al. Sources of Bias in Assessment Procedures for Utility Functions

#### 5.13.1. Hershey et al. Loss Problems (Experiment 1)

#### 5.13.2. Hershey et al. Large Value Gain Problems (Experiment 2)

## 6. Summary of Analyses

#### 6.1. EDRM-U Comparison with EDRM and EU Using Averaged Test Results

#### 6.2. Correlation between Power Utility Exponent and Neutral Wealth

#### 6.3. Risk Perception (EDRM versus EDRM-U)

#### 6.4. ANOVA Analysis

## 7. Discussion

- Choices with large ranges of values;
- Choices involving mixtures of gains and losses;
- Treatment of risk aversion, which includes loss aversion.

## Author Contributions

## Funding

## Conflicts of Interest

## References

- Monroe, T.J.; Beruvides, M.G.; Tercero-Gómez, V.G. Derivation and application of the subjective-objective probability relationship from entropy: The Entropy Decision Risk Model (EDRM). Systems
**2020**, 8, 46. [Google Scholar] [CrossRef] - Sinn, H.-W. Weber’s law and the biological evolution of risk preferences: The selective dominance of the logarithmic utility function, 2002 geneva risk lecture. Geneva Pap. Risk Insur. Theory
**2003**, 28, 87. [Google Scholar] [CrossRef] - Allais, M. Le comportement de l’homme rationnel devant le risque: Critique des postulats et axiomes de l’ecole Americaine. Econometrica
**1953**, 21, 503–546. [Google Scholar] [CrossRef] - Bentham, J. An Introduction to the Principles of Morals and Legislation; T. Payne and son: London, UK, 1789. [Google Scholar]
- Ellsberg, D. Risk, Ambiguity, and the Savage Axioms. Q. J. Econ.
**1961**, 75, 643–669. [Google Scholar] [CrossRef] [Green Version] - Tapiero, C.S. Risk and Financial Management; John Wiley and Sons Ltd.: West Sussex, UK, 2004. [Google Scholar]
- Schumpeter, J.A. History of Economic Analysis; Oxford University Press: New York, NY, USA, 1954. [Google Scholar]
- Kahneman, D. Thinking, Fast and Slow, 1st ed.; Farrar, Straus and Giroux: New York, NY, USA, 2011. [Google Scholar]
- Bernoulli, D. Exposition of a new theory on the measurement of risk (1738). Econometrica
**1954**, 22, 23–36. [Google Scholar] [CrossRef] [Green Version] - Kahneman, D.; Tversky, A. Prospect theory—Analysis of decision under risk. Econometrica
**1979**, 47, 263–291. [Google Scholar] [CrossRef] [Green Version] - Markowitz, H. The utility of wealth. J. Polit. Econ.
**1952**, 60, 151–158. [Google Scholar] [CrossRef] - Thaler, R. Mental accounting and consumer choice. Mark. Sci.
**1985**, 4, 199–214. [Google Scholar] [CrossRef] - Kahneman, D.; Knetsch, J.L.; Thaler, R.H. Experimental tests of the endowment effect and the coase theorem. J. Polit. Econ.
**1990**, 98, 1325–1348. [Google Scholar] [CrossRef] [Green Version] - Hershey, J.C.; Kunreuther, H.C.; Schoemaker, P.J.H. Sources of bias in assessment procedures for utility functions. Manag. Sci.
**1982**, 28, 936. [Google Scholar] [CrossRef] - Shannon, C.E.; Weaver, W. The Mathematical Theory of Communication; The Univeristy of Illinois Press: Urbana, IL, USA, 1949. [Google Scholar]
- Pisano, R.; Sozzo, S. A unified theory of human judgements and decision-making under uncertainty. Entropy
**2020**, 22, 738. [Google Scholar] [CrossRef] - Ben-Naim, A. Entropy and information theory: Uses and misuses. Entropy
**2019**, 21, 1170. [Google Scholar] [CrossRef] [Green Version] - Jaynes, E.T. Information theory and statistical mechanics. Phys. Rev.
**1957**, 106, 620–630. [Google Scholar] [CrossRef] - Monroe, T.J.; Beruvides, M.G. Translating risk perception for standard risk analyses. In IISE Annual Conference; Cromarty, L., Shirwaiker, R., Wang, P., Eds.; Norcross, GA, USA, 2020; Available online: https://www.iise.org/Annual/details.aspx?id=6790 (accessed on 20 November 2020).
- Kullback, S. Information Theory and Statistics; Wiley: New York, NY, USA, 1959. [Google Scholar]
- Monroe, T.J. Entropy Decision Risk Model (Accepted Dissertation Proposal); Beruvides, M.G., Patterson, P.E., Smith, M.L., Tercero-Gómez, V.G., Eds.; Texas Tech University: Lubbock, TX, USA, 2019. [Google Scholar]
- Tversky, A.; Kahneman, D. Advances in prospect-theory—Cumulative representation of uncertainty. J. Risk Uncertain.
**1992**, 5, 297–323. [Google Scholar] [CrossRef] - Frigg, R. Probability in boltzmannian statistical mechanics. In Time, Chance and Reduction. Philosophical Aspects of Statistical Mechanics; Gerhard Ernst, G., Hüttemann, A., Eds.; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Keynes, J.M. A Treatise on Probability; Macmillan and co., Limited: London, UK, 1921. [Google Scholar]
- Birnbaum, M.H.; Bahra, J.P. Gain-loss separability and coalescing in risky decision making. Manag. Sci.
**2007**, 53, 1016–1028. [Google Scholar] [CrossRef] [Green Version] - Sjöberg, L. The methodology of risk perception research. Qual. Quant.
**2000**, 34, 407–418. [Google Scholar] [CrossRef] - Thompson, W. Electrical units of measure. Pop. Lect. Addresses
**1889**, I, 73–136. [Google Scholar] - Perrow, C. Normal accident at three Mile Island. Society
**1981**, 18, 17–26. [Google Scholar] [CrossRef] - Perrow, C. Normal Accidents; Princeton University Press: Princeton, NJ, USA, 1999. [Google Scholar]
- Rasmussen, N. The application of probabilistic risk assessment techniques to energy technologies. Annu. Rev. Energy
**1981**, 6, 123–138. [Google Scholar] [CrossRef] - Slovic, P. Perception of risk. Science (Am. Assoc. Adv. Sci.)
**1987**, 236, 280–285. [Google Scholar] [CrossRef] - Sharp, M.E.; Viswanathan, J.; Lanyon, L.J.; Barton, J.J.S. Sensitivity and bias in decision-making under risk: Evaluating the perception of reward, its probability and value. PLoS ONE
**2012**, 7, e33460. [Google Scholar] [CrossRef] [Green Version] - Weber, E.U.; Blais, A.-R.; Betz, N.E. A domain-specific risk-attitude scale: Measuring risk perceptions and risk behaviors. J. Behav. Decis. Mak.
**2002**, 15, 263–290. [Google Scholar] [CrossRef] - Man, S.S.; Chan, A.H.S.; Alabdulkarim, S. Quantification of risk perception: Development and validation of the construction worker risk perception (CoWoRP) scale. J. Saf. Res.
**2019**, 71, 25–39. [Google Scholar] [CrossRef] - Stanovich, K.E.; West, R.F. Individual differences in reasoning: Implications for the rationality debate? Behav. Brain Sci.
**2000**, 23, 645. [Google Scholar] [CrossRef] - Sjoberg, L. Factors in risk perception. Risk Anal.
**2000**, 20, 1–12. [Google Scholar] [CrossRef] [PubMed] - Wakker, P. Separating marginal utility and probabilistic risk aversion. Theory Decis.
**1994**, 36, 1–44. [Google Scholar] [CrossRef] [Green Version] - Wakker, P.P.; Zank, H. A simple preference foundation of cumulative prospect theory with power utility. Eur. Econ. Rev.
**2002**, 46, 1253–1271. [Google Scholar] [CrossRef] - Falmagne, J.-C. Elements of Psychophysical Theory; Oxford University Press: Oxford, UK, 2002. [Google Scholar]
- Tversky, A.; Kahneman, D. The framing of decisions and the psychology of choice. Science
**1981**, 211, 453. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Bertalanffy, L.V. General System Theory; George Braziller Inc.: New York, NY, USA, 1968. [Google Scholar]
- von Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior; Princeton University Press: Princeton, NJ, USA, 2007. [Google Scholar]
- Department of the Treasury. Treasury Reporting Rates of Exchange as of 31 March 1978; U.S. Government Printing Office: Washington, DC, USA, 1978.
- Tversky, A.; Kahneman, D. Rational choice and the framing of decisions. (proceedings from a conference held October 13–15, 1985, at the University of Chicago). J. Bus.
**1986**, 59, S251. [Google Scholar] [CrossRef] - Schelling, T. Choice and Consequence; Harvard University Press: Cambridge, MA, USA, 1984. [Google Scholar]
- Wu, G.; Markle, A.B. An empirical test of gain-loss separability in prospect theory. Manag. Sci.
**2008**, 54, 1322–1335. [Google Scholar] [CrossRef] [Green Version] - Tversky, A. Elimination by aspects: A theory of choice. Psychol. Rev.
**1972**, 79, 281–299. [Google Scholar] [CrossRef] - Wu, G.; Gonzalez, R. Curvature of the probability weighting function. Manag. Sci.
**1996**, 42, 1676–1690. [Google Scholar] [CrossRef] [Green Version] - Prelec, D. The probability weighting function. Econometrica
**1998**, 66, 497–527. [Google Scholar] [CrossRef] [Green Version] - Prelec, D. A “Pseudo-endowment” effect, and its implications for some recent nonexpected utility models. J. Risk Uncertain.
**1990**, 3, 247–259. [Google Scholar] [CrossRef] - Hershey, J.; Schoemaker, P. Probability versus certainty equivalence methods in utility measurement: Are they equivalent? Manag. Sci.
**1985**, 31, 1213. [Google Scholar] [CrossRef] - Tversky, A.; Kahneman, D. Loss aversion in riskless choice—A reference-dependent model. Q. J. Econ.
**1991**, 106, 1039–1061. [Google Scholar] [CrossRef] - ISO. ISO 31000:2018 Risk Management—Guidelines; International Organization for Standardization: Vernier, Geneva, Switzerland, 2018. [Google Scholar]

1 | Prospect Theory was analyzed using Israeli Pounds (I£), which was replaced by the Shekel in 1980 and the New Shekel in 1985. |

2 | By convention, states with a value of zero are not explicitly mentioned. In this case, there is a 95% chance of no gain or loss. |

3 | First half of the Wu and Markle surveys and all of the Wu and Gonzalez surveys. |

4 | Problems: 2 Birnbaum (1–6 and 2–7), 2 Birnbaum–Bahra Gain-Loss (6 and 12). |

5 | Problems: 2 Birnbaum (1–6 and 2–7), 2 Birnbaum–Bahra Gain-Loss (6 and 12), 2 Prelec 3CD. |

6 | For EDRM, when the 18 Hershey et al. gains problems are removed from the data set, the remainder of the optimized EDRM analyses together are normal, with a Shapiro–Wilk p-value of 0.1531. This is consistent with the discussion of Section 5.13.2. which documents the significant differences between the calculated and actual results. |

**Figure 1.**EDRM-U risk perception plot for the example problem incorporating a small gamble and large investment decisions. The white region is where the executive will choose to take a gamble in a friendly penny-ante poker game, but will choose the safer option when it comes to a large investment. The line represents neutral risk aversion.

**Figure 2.**Flowchart for the present EDRM-U research showing established theories comprising the application of expected utility for EDRM and EDRM-U model development, validation, and analysis of risk perception at a system level.

**Figure 3.**Bernoulli’s Expected Utility 1738 (Bernoulli, 1954). AB represents the initial wealth and AC, AD, AE, and AF represent various states with corresponding probabilities of occurrence, which all sum to 1. AP is the final expected wealth calculated through adding the logarithmic expectations ($\mathrm{ln}\mathrm{AP}={p}_{1}\mathrm{ln}\mathrm{AC}+{p}_{2}\mathrm{ln}\mathrm{AD}\dots $), assuming ${\sum}^{\text{}}{p}_{i}=1$.

**Figure 4.**Power utility curve as a subset approximation of logarithmic utility. This Log-Log plot illustrates that, for a common reference point, power utility roughly follows logarithmic utility, and this is an explanation for why power utility works for smaller values but does not work for larger ones. This plot assumes ${w}_{n}=3755$ and $\alpha =0.88$.

**Figure 5.**Utility function curvature comparison. Since a utility function’s effect is based upon its curvature (2nd derivative), it is important to understand the behavior of the logarithmic (solid line) and power utilities (dashed line) about a neutral wealth, ${w}_{n}=3755$ in this example, since this is where they are defined as zero value ($\Delta v=\Delta u=0)$, but power utility converges much more quickly. Both curvatures converge to zero for extremely large amounts of wealth, where they approach an expected value function. At the neutral wealth point (i.e., neutral reference), the power utility is asymptotic. The logarithmic utility is smooth. Thaler’s four cases are annotated, as discussed in Section 3.3. Note that the Thaler case 2 curvature using negative ${w}_{n}$ is only defined for negative wealth differences (losses).

**Figure 6.**Proximity versus relative certainty. Relative certainty is objective probability and is the same as macro-probability observed at the thermodynamic level of physics. Proximity is subjective probability, which is otherwise unmeasurable. From a behavior perspective, subjective probabilities are degrees of belief, but in physics they are the micro-probability of a single atom being in a certain energy state [23,24]. This relationship is the foundation of this research into the difference between how people are supposed to make decisions (normative) and how they actually make decisions (positive). The preference reversal point and inflection point align well with prior descriptive studies [21].

**Figure 7.**Mixed gains plot taken directly from Thaler with “floating axis” overlay to illustrate the zero-wealth point in the center of the cross-hairs and neutral wealth at the “origin” [12]. The original plot assumes a significant loss aversion effect, as seen in the more pronounced effects due to losses, but this effect can be explained by the introduction of a neutral wealth reference point.

**Figure 8.**Mapping of Thaler’s four cases. If $x+y<0$, then the crossover wealth ($W$) is positive, but if $x+y>0$ then the crossover wealth is negative. $W=0$ along the diagonal.

**Figure 9.**Illustration of how neutral wealth affects each of Thaler’s four cases. For positive values of ${w}_{n}$, cases 1, 3, and 4 align with Thaler’s analysis; however, for the pure loss case 2 where integration is specified, ${w}_{n}$ must be negative—an observation consistent with application in PT and CPT.

**Figure 10.**Risk aversion using the proximity exponent ($\beta $). By first assuming that the neutral value of proximity exponent, or risk aversion, is $\beta =1$, then values less than 1 result in the inflation of proximity where subjects believe an outcome is more likely—i.e., risk seeking. For values greater than 1, the proximity is reduced from normal and the result is risk aversion. Nominal values of $\beta $ determined from this research are about 1.3, which supports the presence of mild risk aversion among subjects, as would be expected.

**Figure 11.**Decision types. The white region of each plot is where the safer option is preferred over a wide range of values $0\le \beta \le 3$ and $0.1\le {w}_{n}\le {10}^{6}$. In Type 0, the safer option is insensitive to either parameter over the range evaluated. Type 1 is very sensitive to risk changes. Type 2 is strongly dependent upon risk aversion. Type 3 is most common and varies with a positive slope. Type 4 is uncommon and varies with a negative slope to a point and begins to reverse. Type 5 is a compound type. Understanding the nature of these relationships enables assessing an individual’s or group’s nominal risk sensitivity (neutral wealth) and risk aversion, permitting the design of problems for a desired outcome.

**Figure 12.**(Left) 259 choice problems analyzed in this research corrected to 2020 U.S. Dollars assuming all subjects chose the safer option. When all the risk perception curves are superimposed, there is a strong dominance towards $\beta >1$ (white region) and low ${w}_{n}$, with 100% correlation of the 259 safer choices in the lower right-hand corner of the plot. This affirms that $\beta $ is a valid measure of risk aversion and ${w}_{n}$ is a measure of risk sensitivity. (Right) Similar plot illustrating aversion and sensitivity to cold as a physical representation of risk to aid in understanding the plot to the left.

**Figure 13.**VNM utility analysis example showing result of $B$ by comparing relative certainty via proximity using assumed values of $A=200$ and $C=100$ as examples in Equation (19). While none of the curves for any value $\beta $ meet VNM rationality for all values of relative certainty, 99.2% of the 518 individual choices align with the VNM axioms using optimized EDRM-U. The result is over 98.8% for optimized EDRM. Therefore, EDRM and EDRM-U are generally consistent with the normative definition of rationality for the practical examples considered in this research.

**Figure 14.**EDRM-U results for the problems presented by Prelec. The intersecting lines mark the optimal solution based upon subject choice percentages and the white region indicates the range of feasible solutions maximizing the number of matches (9 of 9). While neutral wealth may vary over a broad range, the risk aversion range is narrow, affirming that the subjects were generally all risk averse. By knowing the nominal values of neutral wealth and risk aversion for a group, sets of questions can then be engineered to achieve a desired outcome.

**Figure 15.**EDRM-U risk perception plot showing feasible solutions for Hershey et al., gain problems (white region) showing 100% matching between actual results and predications. This plot reveals that the subjects (Harvard MBA students) were most likely risk seeking ($\beta <1$). The lines indicate the optimized solution, but the results for risk neutral ($\beta =1$) are very similar.

**Figure 16.**Correlation between neutral wealth and power utility exponent ($\alpha $) showing results for each study (white circles), and two groupings of problems (black circles) from combined Wu–Markle and a compilation of the set of 259 problems. While there is a correlation between $\alpha $ and ${w}_{n}$, readers are reminded that the power utility applies to problems with a narrow range of values, whereas the expected utility is demonstrated to be valid over a wide range about neutral wealth. The two triangle points show the large value range studies (Prelec and Hershey et al.); neither these nor the compilations (circles or triangles) were used in the regression. The dashed line shows the optimal ${w}_{n}-\alpha $ correlation (see Figure 17). Note that at $\alpha =1$, ${w}_{n}\to \infty $.

**Figure 17.**Summary EDRM-U risk perception plot of the 259 uncertain economic problems showing the maximum of 209 matches at ${w}_{n}=8778.52$ (2020 USD) and $\beta =1.306$. Based upon the regression shown in Figure 16, this correlates to a power utility exponent value of $\alpha =0.918$, which is comparable to the $0.88$ value assumed by Kahneman and Tversky in Prospect Theory. For ${w}_{n}<1$, risk aversion approaches a constant $\beta \approx 0.225$. Note that nearly all subjects in these studies were undergraduate college students; a group of wealthy investment bankers will likely have adapted to a different level of risk sensitivity.

**Table 1.**Comparison of EDRM, EU, and EDRM-U. Expected utility is the accepted normative standard for most economic small and large economic decisions. EDRM uses power utility, which is generally used in economic behavior models, but only performs well over small ranges of values. EDRM-U advances EDRM by using logarithmic expected utility instead of exponential power utility. As a result, EDRM-U is a general model that predicts subject choice for small and large economic values.

Positive Theory | Normative Theory | Combined |
---|---|---|

EDRM | EU | EDRM-U |

Proximity (subj prob) | Objective Probability | Proximity (subj prob) |

Power Utility | Logarithmic Utility | Logarithmic Utility |

Pure Gain or Loss (No Cancellation) | Mixed (Cancellation) | |
---|---|---|

Segregation (e.g., prefer many separate gifts) | w_{n} > W | w_{n} < W |

Integration (e.g., prefer a single billing statement) | w_{n} < W | w_{n} > W |

**Table 3.**Comparison of risk perception factors showing that those used in this research are consistent with the two factors identified in prior studies and are aligned with established theory.

Economic Decision Theories | Dual Process Theory | Risk Perception | Risk Perception | Risk Perception |
---|---|---|---|---|

Stanovich, et al. 2011 | Slovic & Peters 2006 | Sjöberg 2000 | This Research | |

Positive, behavioral | Type 1 (intuition) | Risk as feelings | Risk attitude | Risk aversion (proximity exponent) |

Normative, expectation | Type 2 (deliberate) | Risk as analysis | Risk sensitivity | Risk sensitivity (neutral wealth) |

**Table 4.**Decision type examples (see Figure 11).

Type | Risky Choice (Value, Probability) | Safer Choice (Value, Probability) | Problem Source |
---|---|---|---|

0 | (96, 0.45; 4, 0.45; 2, 0.1) | (58, 0.45; 56, 0.45; 2, 0.1) | Birnbaum 3–12 |

1 | (96, 0.45; 4, 0.45; 2, 0.1) | (44, 45; 40, 0.45; 2, 0.1) | Birnbaum 1–12 |

2 | (98, 0.85; 96; 0.05; 11, 0.1) | (99, 0.9; 14, 0.05; 12, 0.05) | Birnbaum 4–5 |

3 | (5000, 0.001) or (−5) | (5) or (−5000, 0.001) | Prospect Theory 14 or 14′ |

4 | (10,000, 0.01; 2000, 0.04) | (7000, 0.05)2 | Prelec 3AB |

5 | (1200, 0.3; −200, 0.7) | (400, 0.7; −800 0.3) | Wu-Markle 23 Mixed |

Number matching | Number of binary matches and number of problems (used for optimization) |

Binomial Prob (>50%) | Binomial probability of result being >50% (same as ratio) |

Binomial p-Value | Resulting binomial p-value, values $\le 0.05$ are acceptable |

Matching $\Delta \%$ Std dev ($\sigma $) | Standard deviation of % difference of matches only (used for optimization) |

All % Coeff of Det (${R}^{2})$ | Coefficient of determination between all actual and calculated percentages |

All % Spearman rank (${r}_{s}$) | Nonparametric rank test between all actual and calculated percentages |

Unit: 1991 USD | EDRM | EDRM Optimal | EDRM-U | EDRM-U Optimal |
---|---|---|---|---|

Power utility exp ($\alpha $) | 1 | 1 | ||

Risk aversion ($\beta $) | 1 | 0.9473 | 1 | 0.9473 |

Neutral Wealth (${w}_{n}$) $ | $\to \infty $ | $\to \infty $ | ||

All Coeff of Det (${R}^{2})$ | 0.9971 | 0.9973 | 0.9971 | 0.9973 |

All Spearman rank (${r}_{s}$) | 0.9983 | 0.9985 | 0.9983 | 0.9985 |

Unit: 1982 USD | ${\mathit{w}}_{\mathit{n}}=50.61,\mathit{\beta}=\mathbf{n}/\mathbf{a}$ | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

Problem (Values Only) | EDRM-U | Calc% | Actual% | Diff | Match | |||||

Outcome A | Outcome B | $\mathit{T}{\mathit{U}}_{\mathit{A}}$ | $\mathit{T}{\mathit{U}}_{\mathit{B}}$ | A | B | A | B | $\Delta \mathit{\%}$ | Y/N | |

1 | (50, 25) | (75) | 55.08 | 46.00 | 82 | 18 | 78 | 22 | −4.2 | Yes |

2 | (−100, −50) | (−150) | −89.96 | −69.70 | 10 | 90 | 18 | 83 | 7.8 | Yes |

3 | (−20, 100) | (80) | 29.75 | 47.98 | 30 | 70 | 27 | 73 | −3.0 | Yes |

4 | (25, −200) | (−175) | −501.83 | −512.88 | 66 | 34 | 77 | 23 | 10.7 | Yes |

Unit: 1983 USD | EDRM Uncorrected | EDRM Optimal | EDRM-U Optimal |
---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.688 | |

Neutral Wealth (${w}_{n}$) $ | 50.61 | ||

Number matching | 4/4 | 4/4 | 4/4 |

Binomial Prob (>50%) | 1 | 1 | 1 |

Binomial p-Value | 0.125 | 0.125 | 0.125 |

Matching $\Delta \%$ Std dev ($\sigma $) | 13.297 | 6.476 | 7.539 |

All % Coeff of Det (${R}^{2})$ | 0.962 | 0.960 | 0.948 |

All % Spearman rank (${r}_{s}$) | 1 | 1 | 1 |

Unit: 1978 Israeli Pound | EDRM Uncorrected | $\mathbf{EDRM}\text{}\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U at $\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.87 | 0.853 | |||

Risk aversion ($\beta $) | 1 | 1 | 0.9775 | 1 | 0.894 | |

Neutral Wealth (${w}_{n}$) I£ | ${w}_{o}=483$ | 6750 | 3755 | |||

Number matching | 19/19 | 19/19 | 19/19 | 12/19 | 19/19 | 19/19 |

Binomial Prob (>50%) | 1 | 1 | 1 | 0.632 | 1 | 1 |

Binomial p-Value | 3.8 × 10^{−6} | 3.8 × 10^{−6} | 3.8 × 10^{−6} | 0.359 | 3.8 × 10^{−6} | 3.8 × 10^{−6} |

Matching $\Delta \%$ Std dev ($\sigma $) | 10.868 | 10.825 | 10.821 | 10.844 | 9.903 | 9.642 |

All % Coeff of Det (${R}^{2})$ | 0.859 | 0.859 | 0.859 | 0.519 | 0.882 | 0.888 |

All % Spearman rank (${r}_{s}$) | 0.697 | 0.701 | 0.701 | 0.645 | 0.705 | 0.699 |

Unit: 1980 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.65 | 0.65 | |||

Risk aversion ($\beta $) | 1 | 1 | 1 | 1 | 0.668 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=0.23$ | 17.8 | 0.315 | |||

Number matching | 8/8 | 8/8 | 8/8 | 7/8 | 8/8 | 8/8 |

Binomial Prob (>50%) | 1 | 1 | 1 | 0.875 | 1 | 1 |

Binomial p-Value | 0.008 | 0.008 | 0.008 | 0.070 | 0.008 | 0.008 |

Matching $\Delta \%$ Std dev ($\sigma $) | 8.757 | 6.023 | 6.023 | 8.613 | 7.810 | 6.312 |

All % Coeff of Det (${R}^{2})$ | 0.945 | 0.968 | 0.968 | 0.960 | 0.958 | 0.965 |

All % Spearman rank (${r}_{s}$) | 0.850 | 0.850 | 0.850 | 0.850 | 0.850 | 0.850 |

Unit: Years | EDRM | EDRM-U | EDRM-U | ||
---|---|---|---|---|---|

Problem (Years, Probability) | $\mathit{\alpha}=0.479$ $\mathit{\beta}=0.997$ | ${\mathit{w}}_{\mathit{n}}=0.130$ $\mathit{\beta}=0.892$ | ${\mathit{w}}_{\mathit{n}}\to \mathit{\infty}$ $\mathit{\beta}=1$ | ||

A-Surgery | B-Radiation | Act%-Calc% | Act%-Calc% | Act%-Calc% | |

1a (1 year) | (1, 0.68; 5, 0.34; 1, 0.9) | (1, 0.77; 5, 0.22) | 6% | 11% | 20% |

1b | (−1, 0.32; −5, 0.66) | (−1, 0.23; −5, 0.78) | 1% | 1% | 1% |

1a (5 years) | (1, 0.68; 5, 0.34; 5, 0.9) | (1, 0.77; 5, 0.22) | −13% | −17% | 0% |

1b | (−1, 0.32; −5, 0.66) | (−1, 0.23; −5, 0.78) | 1% | 1% | 1% |

Unit: 1985 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.649 | 0.328 | |||

Risk aversion ($\beta $) | 1 | 1 | 0.503 | 1 | 1.420 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=0.48$ | 11.885 | $\to \infty $ | |||

Number matching | 12/14 | 14/14 | 14/14 | 11/14 | 14/14 | 14/14 |

Binomial Prob (>50%) | 0.857 | 1 | 1 | 0.786 | 1 | 1 |

Binomial p-Value | 0.013 | 1.2 × 10^{−4} | 1.2 × 10^{−4} | 0.057 | 1.2 × 10^{−4} | 1.2 × 10^{−4} |

Matching $\Delta \%$ Std dev ($\sigma $) | 7.911 | 8.410 | 8.334 | 11.124 | 10.110 | 8.720 |

All % Coeff of Det (${R}^{2})$ | 0.793 | 0.910 | 0.911 | 0.883 | 0.899 | 0.898 |

All % Spearman rank (${r}_{s}$) | 0.864 | 0.917 | 0.917 | 0.900 | 0.917 | 0.917 |

Unit: 2006 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.557 | 0.557 | |||

Risk aversion ($\beta $) | 1 | 1 | 1 | 1 | 1.530 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=3776$ | 625 | 7599 | |||

Number matching | 23/34 | 31/34 | 31/34 | 31/34 | 29/34 | 31/34 |

Binomial Prob (>50%) | 0.676 | 0.912 | 0.912 | 0.912 | 0.853 | 0.912 |

Binomial p-Value | 0.058 | 7.7 × 10^{−7} | 7.7 × 10^{−7} | 7.7 × 10^{−7} | 3.9 × 10^{−5} | 7.7 × 10^{−7} |

Matching $\Delta \%$ Std dev ($\sigma $) | 7.884 | 11.297 | 11.297 | 12.236 | 12.126 | 11.202 |

All % Coeff of Det (${R}^{2})$ | 0.514 | 0.565 | 0.565 | 0.671 | 0.467 | 0.618 |

All % Spearman rank (${r}_{s}$) | 0.725 | 0.776 | 0.776 | 0.766 | 0.688 | 0.780 |

Unit: 2006 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.581 | 1 | |||

Risk aversion ($\beta $) | 1 | 1 | 1.721 | 1 | 1.721 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=20.2$ | 51.5 | $\to \infty $ | |||

Number matching | 19/34 | 28/34 | 28/34 | 20/34 | 21/34 | 28/34 |

Binomial Prob (>50%) | 0.559 | 0.824 | 0.824 | 0.588 | 0.618 | 0.824 |

Binomial p-Value | 0.608 | 2.0 × 10^{−4} | 2.0 × 10^{−4} | 0.392 | 0.230 | 2.0 × 10^{−4} |

Matching $\Delta \%$ Std dev ($\sigma $) | 10.205 | 12.237 | 12.237 | 8.976 | 10.086 | 12.237 |

All % Coeff of Det (${R}^{2})$ | 0.475 | 0.505 | 0.505 | 0.084 | 0.042 | 0.505 |

All % Spearman rank (${r}_{s}$) | 0.816 | 0.678 | 0.678 | 0.241 | 0.354 | 0.678 |

Unit: 2006 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.8205 | 1 | |||

Risk aversion ($\beta $) | 1 | 1 | 1.208 | 1 | 1.39 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=420$ | 12975 | 4375 | |||

Number matching | 28/34 | 28/34 | 28/34 | 20/34 | 28/34 | 30/34 |

Binomial Prob (>50%) | 0.824 | 0.824 | 0.824 | 0.588 | 0.824 | 0.882 |

Binomial p-Value | 2.0 × 10^{−4} | 2.0 × 10^{−4} | 2.0 × 10^{−4} | 0.392 | 2.0 × 10^{−4} | 2.0 × 10^{−6} |

Matching $\Delta \%$ Std dev ($\sigma $) | 8.818 | 8.603 | 8.361 | 12.919 | 9.309 | 8.557 |

All % Coeff of Det (${R}^{2})$ | 0.352 | 0.380 | 0.384 | 0.135 | 0.305 | 0.456 |

All % Spearman rank (${r}_{s}$) | 0.703 | 0.687 | 0.697 | 3.4 × 10^{−11} | 0.725 | 0.624 |

Unit: 2004 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.898 | 0.848 | |||

Risk aversion ($\beta $) | 1 | 1 | 0.846 | 1 | 0.968 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=5.01$ | 993.27 | 995.75 | |||

Number matching | 14/18 | 14/18 | 14/18 | 13/18 | 17/18 | 17/18 |

Binomial Prob (>50%) | 0.778 | 0.778 | 0.778 | 0.722 | 0.944 | 0.944 |

Binomial p-Value | 0.031 | 0.031 | 0.031 | 0.096 | 1.5 × ^{−4} | 1.5 × 10^{−4} |

Matching $\Delta \%$ Std dev ($\sigma $) | 8.800 | 8.751 | 7.187 | 12.981 | 8.855 | 8.773 |

All % Coeff of Det (${R}^{2})$ | 0.615 | 0.619 | 0.685 | 0.313 | 0.678 | 0.685 |

All % Spearman rank (${r}_{s}$) | 0.800 | 0.800 | 0.860 | 0.317 | 0.800 | 0.804 |

Unit: 2003 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.5 | 0.5 | |||

Risk aversion ($\beta $) | 1 | 1 | 0.93 | 1 | 1.001 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=23.0$ | 22.75 | 22.75 | |||

Number matching | 11/21 | 19/21 | 19/21 | 16/21 | 18/21 | 18/21 |

Binomial Prob (>50%) | 0.524 | 0.905 | 0.905 | 0.762 | 0.857 | 0.857 |

Binomial P-Value | 1 | 2.2 × 10^{−4} | 2.2 × 10^{−4} | 0.027 | 1.5 × 10^{−}^{3} | 1.5 × 10^{−}^{3} |

Matching $\Delta \%$ Std dev ($\sigma $) | 8.257 | 7.290 | 6.975 | 7.376 | 5.841 | 5.850 |

All % Coeff of Det (${R}^{2})$ | 0.244 | 0.746 | 0.731 | 0.571 | 0.712 | 0.713 |

All % Spearman rank (${r}_{s}$) | 0.250 | 0.880 | 0.885 | 0.719 | 0.878 | 0.878 |

Unit: 1995 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.5885 | 0.387 | |||

Risk aversion ($\beta $) | 1 | 1 | 1.220 | 1 | 1.221 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=1300$ | 99 | 23.4 | |||

Number matching | 26/40 | 30/40 | 33/40 | 18/40 | 29/40 | 34/40 |

Binomial Prob (>50%) | 0.650 | 0.750 | 0.825 | 0.450 | 0.725 | 0.850 |

Binomial P-Value | 0.081 | 2.2 × 10^{−3} | 4.2 × 10^{−5} | 0.636 | 6.4 × 10^{−3} | 8.4 × 10^{−6} |

Matching $\Delta \%$ Std dev ($\sigma $) | 5.998 | 7.191 | 9.031 | 8.290 | 7.110 | 7.990 |

All % Coeff of Det (${R}^{2})$ | 0.473 | 0.482 | 0.393 | 1.0 × 10^{−5} | 0.499 | 0.474 |

All % Spearman rank (${r}_{s}$) | 0.449 | 0.484 | 0.645 | 0.114 | 0.522 | 0.712 |

Unit: 1989 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.543 | 0.467 | |||

Risk aversion ($\beta $) | 1 | 1 | 1.268 | 1 | 1.12 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=35096$ | 4765 | 22170 | |||

Number matching | 5/9 | 7/9 | 9/9 | 7/9 | 8/9 | 9/9 |

Binomial Prob (>50%) | 0.556 | 0.778 | 1 | 0.778 | 0.889 | 1 |

Binomial P-Value | 1 | 0.180 | 3.9 × 10^{−3} | 0.180 | 0.039 | 3.9 × 10^{−3} |

Matching $\Delta \%$ Std dev ($\sigma $) | 6.940 | 7.575 | 16.201 | 12.617 | 9.160 | 7.245 |

All % Coeff of Det (${R}^{2})$ | 0.320 | 0.720 | 0.558 | 0.124 | 0.803 | 0.924 |

All % Spearman rank (${r}_{s}$) | 0.433 | 0.850 | 0.700 | 0.333 | 0.900 | 0.983 |

Unit: 1980 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 0.250 | 1 | |||

Risk aversion ($\beta $) | 1 | 1 | 4.003 | 1 | 4.003 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}=0.81$ | $\to $0 | $\to \infty $ | |||

Number matching | 6/10 | 10/10 | 10/10 | 10/10 | 10/10 | 10/10 |

Binomial Prob (>50%) | 0.6 | 1 | 1 | 1 | 1 | 1 |

Binomial P-Value | 0.705 | 2.0 × 10^{−3} | 2.0 × 10^{−3} | 2.0 × 10^{−3} | 2.0 × 10^{−3} | 2.0 × 10^{−3} |

Matching $\Delta \%$ Std dev ($\sigma $) | 15.484 | 9.586 | 9.586 | 10.231 | 10.704 | 9.586 |

All % Coeff of Det (${R}^{2})$ | 0.206 | 0.272 | 0.272 | 0.181 | 0.191 | 0.272 |

All % Spearman rank (${r}_{s}$) | 0.345 | 0.564 | 0.564 | 0.394 | 0.418 | 0.564 |

Unit: 1980 USD | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Power utility exp ($\alpha $) | 0.88 | 1 | 0.891 | |||

Risk aversion ($\beta $) | 1 | 1 | 0.748 | 1 | 0.325 | |

Neutral Wealth (${w}_{n}$) $ | ${w}_{o}\to 0$ | 274.25 | 1.494 | |||

Number matching | 12/18 | 13/18 | 13/18 | 9/18 | 18/18 | 18/18 |

Binomial Prob (>50%) | 0.667 | 0.722 | 0.722 | 0.500 | 1 | 1 |

Binomial P-Value | 0.238 | 0.096 | 0.096 | 1 | 7.6 × 10^{−}^{6} | 7.6 × 10^{−6} |

Matching $\Delta \%$ Std dev ($\sigma $) | 14.877 | 14.304 | 13.849 | 7.254 | 7.829 | 6.332 |

All % Coeff of Det (${R}^{2})$ | 0.075 | 0.116 | 0.151 | 0.584 | 0.869 | 0.913 |

All % Spearman rank (${r}_{s}$) | −0.044 | −0.055 | −0.064 | 0.717 | 0.890 | 0.956 |

Weighted Averages | EDRM Uncorrected | EDRM $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM Optimal | EU Optimal | EDRM-U $\mathbf{at}\text{}\mathit{\beta}=1$ | EDRM-U Optimal |
---|---|---|---|---|---|---|

Number matching | 183/259 | 221/259 | 226/259 | 174/259 | 219/259 | 236/259 |

Binomial Prob (>50%) | 0.707 | 0.853 | 0.873 | 0.672 | 0.846 | 0.911 |

Matching $\Delta \%$ Std dev ($\sigma $) | 9.108 | 9.591 | 9.973 | 10.261 | 9.115 | 8.954 |

All % Coeff of Det (${R}^{2})$ | 0.471 | 0.554 | 0.541 | 0.352 | 0.527 | 0.636 |

All % Spearman rank (${r}_{s}$) | 0.589 | 0.658 | 0.683 | 0.497 | 0.676 | 0.760 |

Optimal EDRM | All | Gain Only | Loss Only | Mix Only |
---|---|---|---|---|

Power utility exp (α) | 0.7095 | |||

Risk aversion ($\beta $) | 1.012 | 1.012 | 0.988 | 0.969 |

Number matching | 202/259 | 120/154 | 52/61 | 33/44 |

Optimal EU | ||||

Initial wealth (${w}_{o}$) $ | 24.90 | |||

Number matching | 144/259 | 88/154 | 43/61 | 12/44 |

Optimal EDRM-U | ||||

$\mathrm{Neutral}\text{}\mathrm{Wealth}\text{}({w}_{n}$) $ | 8778.52 | |||

Risk aversion ($\beta $) | 1.306 | 1.245 | 1.604 | 1.306 |

Number matching | 209/259 | 120/154 | 52/61 | 39/44 |

EDRM | EDRM-U | |||
---|---|---|---|---|

Actual-Calc % Diff (prob > F) | ||||

Type (Gain, Loss, Mix) | 0.5160 | Not Significant | 0.9206 | Not Significant |

Match (yes, no) | 0.1992 | Not Significant | 0.2364 | Not Significant |

Interaction (Type and Match) | 0.0329 | Significant | 0.2785 | Not Significant |

Magnitude (prob > F) (Corrected to 2020 USD) | ||||

Act-Calc % Diff | 0.7101 | Not Significant | 0.4448 | Not Significant |

Match (yes, no) | 0.9461 | Not Significant | 0.5930 | Not Significant |

Interaction (% Diff and Match) | 0.0672 | Marginal | 0.6293 | Not Significant |

Normality (p-value) | ||||

Shapiro–Wilk | 8.21 × 10^{−4} | Not Normal6 | 0.6774 | Normal |

Normality Plots | Normal, except at the lower extreme due to Hershey et al. gains (See Section 5.13.2). | Normal |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Monroe, T.; Beruvides, M.; Tercero-Gómez, V.
Quantifying Risk Perception: The Entropy Decision Risk Model Utility (EDRM-U). *Systems* **2020**, *8*, 51.
https://doi.org/10.3390/systems8040051

**AMA Style**

Monroe T, Beruvides M, Tercero-Gómez V.
Quantifying Risk Perception: The Entropy Decision Risk Model Utility (EDRM-U). *Systems*. 2020; 8(4):51.
https://doi.org/10.3390/systems8040051

**Chicago/Turabian Style**

Monroe, Thomas, Mario Beruvides, and Víctor Tercero-Gómez.
2020. "Quantifying Risk Perception: The Entropy Decision Risk Model Utility (EDRM-U)" *Systems* 8, no. 4: 51.
https://doi.org/10.3390/systems8040051