# Derivation and Application of the Subjective–Objective Probability Relationship from Entropy: The Entropy Decision Risk Model (EDRM)

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. A New Approach

#### 1.2. Objectives

#### 1.3. Definitions

## 2. Literature Review

## 3. Method

**Hypothesis**

**1.**

**Hypothesis**

**2.**

## 4. Derivation of EDRM: Theoretical Framework

#### 4.1. Foundation of Utility Theory

#### 4.2. Entropy

#### 4.3. Two Types of Probabilities

#### 4.4. Entropy Decision Risk Model (EDRM) Framework

- Certainty of gains and the uncertainty of losses are more highly valued;
- Gains and losses are considered contiguously as two regions of the same scale;
- Relative certainty, or redundancy, is one minus the relative entropy;
- Proximity is represented by the subjective probability of reaching a state;
- Prospect can be stated as magnitude times proximity as a function of relative certainty;
- The choice with the greatest prospect, positive or negative, is preferred.

#### 4.5. Choices and States

#### 4.6. Prospect

#### 4.6.1. Derivation of Proximity from Information Theory Entropy (SMI) and Statistical Mechanics

#### 4.6.2. Very Small Probabilities

#### 4.6.3. Inflection and Preference Reversal Points

#### 4.6.4. Calculating Prospect of a Choice

#### 4.6.5. Applying a Proximity Exponent ($\beta $) to the Prospect of a Choice

## 5. EDRM Validation (Without Application of Any Factors or Corrections, $\mathbf{\beta}\mathbf{=}\mathbf{1}$)

#### 5.1. The Percentage Evaluation Model (PEM)

- Varies monotonically with the difference in prospect between choices;
- Scaled by the range, positive and negative, of values being evaluated in a given choice;
- Accounts for non-linearities of human perception;
- Equitably reports subject percentages for choices involving gains, losses, or mixtures of the two;
- Performs consistently across a range of studies (not tuned to a specific set of research).

#### 5.2. Allais Paradox

#### 5.3. Prospect Theory (Kahneman and Tversky)

#### 5.4. Cumulative Prospect Theory

#### 5.5. The Framing of Decisions and the Psychology of Choice (Tversky and Kahneman)

#### 5.6. Rational Choice and the Framing of Decisions (Tversky and Kahneman)

#### 5.7. Gain-Loss Separability (Wu and Markle)

## 6. Summary of Analyses

#### System-Level Analysis of Choices (Sensitivity)

## 7. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Derivation of Proximity from Entropy

**Figure A1.**Divergence, or relative entropy, is the distance between certainty and uncertainty for a given subjective probability. The arrow shows how the divergence curve is flipped when converted to Shannon’s redundancy, which is referred to herein as relative certainty and is an objective probability.

**Figure A2.**Plot of relative certainty versus proximity that relates objective and subjective probabilities, but is then flipped around the diagonal to place relative certainty on the horizontal. This plot is shown to graphically illustrate the steps of the mathematical derivation.

## Appendix B. Very Small Probabilities

## Appendix C. Statistical Analyses

#### Appendix C.1. Percentage Evaluation Model

**Table A1.**Statistical analysis of EDRM Percentage Evaluation Model using eight data sets from Birnbaum, Birnbaum and Bahra, Kahneman and Tversky, Tversky and Kahneman, Wu and Gonzalez, Wu and Markle, and Prelec with matching binary results and optimized values of $\alpha $ and $\beta $ to validate performance.

Regression analysis: Coefficient of determination (${R}^{2})$ | ||||||

Actual percentages compared with calculated for matching binary results only | 0.8026 | |||||

Spearman rank correlation coefficient (Rho) | 0.8899 | |||||

ANOVA ($\u2206$% actual vs. calculated) | Df | Sum Sq | Mean Sq | F-Value | Prob(>F) | Result |

Study source | 7 | 517.8 | 73.977 | 0.8601 | 0.5401 | Not Significant |

Type of choice (gain, loss, mix) | 2 | 96.0 | 48.005 | 0.8828 | 0.5737 | Not Significant |

Interaction between source and type | 5 | 379.7 | 75.932 | 0.8828 | 0.4947 | Not Significant |

Residuals | 128 | 11,009.5 | 86.012 | |||

Normality Assumption | ||||||

Shapiro–Wilk | W = 0.99522 | p-value = 0.9218 | Normal | |||

Conclusions | ||||||

1. Cannot reject and null hypothesis, which means that the EDRM evaluation model is likely effective at expressing relative differences in prospect as percentages. Criteria 4 and 5 are met. 2. T-statistic test confirms no survey source is significant. |

#### Appendix C.2 Prospect Theory

**Table A2.**Statistical analysis of EDRM performance with Prospect Theory showing 100% binary agreement and excellent alignment between reported percentages and those calculated using the PEM.

Binary matching (yes/no) (percentage) | 100% | |||||

Regression analysis: Coefficient of determination (${R}^{2}$) | ||||||

Actual percentages compared with calculated (all match) | 0.8581 | |||||

Spearman rank correlation coefficient (Rho) | 0.6966 | |||||

ANOVA ($\u2206$% actual vs. calculated) | Df | Sum Sq | Mean Sq | F-Value | Prob(>F) | Result |

Type of gamble (gain or loss) | 1 | 58.98 | 59.983 | 0.4648 | 0.5051 | Not Significant |

# of Non-zero States (1 or 2) | 1 | 36.66 | 36.658 | 0.2889 | 0.5983 | Not Significant |

Residuals | 15 | 2030.40 | 126.900 | |||

Normality Assumption | ||||||

Shapiro–Wilk | W = 0.94119 | p-value = 0.2771 | Normal | |||

Conclusions | ||||||

1. Cannot reject any of the null hypotheses, which means that EDRM reasonably predicts results of Prospect Theory. |

#### Appendix C.3. Cumulative Prospect Theory

**Table A3.**Statistical analysis of EDRM Performance with Cumulative Prospect Theory showing nearly perfect alignment between a priori EDRM and data reported by Tversky and Kahneman. These results suggest that there is some difference between gains and losses, but as a second-order effect. The number of states (1 or 2) has no effect.

Regression analysis: Coefficient of determination (${R}^{2}$) | ||||||

Actual values (not percentages) compared with calculated values | 0.9971 | |||||

Actual values (not percentages) compared with calculated values (Positive only) | 0.9885 | |||||

Actual values (not percentages) compared with calculated values (Negative only) | 0.9980 | |||||

Spearman rank correlation coefficient (Rho) | 0.9982 | |||||

ANOVA ($\u2206\mathrm{CE}$ actual vs. calculated) | Df | Sum Sq | Mean Sq | F-Value | Prob(>F) | Result |

Type of gamble (gain or loss) | 1 | 172.62 | 172.62 | 3.9040 | 0.05339 | Marginal |

# Non-zero states (1 or 2) | 1 | 48.40 | 48.40 | 1.0946 | 0.30020 | Not significant |

Residuals | 53 | 2343.49 | 44.217 | |||

Normality Assumption | ||||||

Shapiro–Wilk | W = 0.97213 | p-value = 0.2196 | Normal | |||

Conclusions | ||||||

1. The coefficient of determination values for the comparison of actual and calculated values indicates near-perfect alignment and affirms Hypothesis 1. The ANOVA results for type of gamble do not reject the null hypothesis of no significant effect; however, the probability is very close to the 5% significance value indicating there is some difference between gains and losses, but that they can be considered as a secondary effect in this research given there is nearly no difference in the ${R}^{2}$ for positive (0.9885) and negative (0.9980) problems. Using a value of $\beta =0.947$ rather than 1 increases the type of gamble Prob(>F) to nearly 0.35 from 0.053. |

#### Appendix C.4. Wu and Markle Gain-Loss Separability

**Table A4.**Statistical analysis of EDRM Performance on Wu and Markle Gain-Loss Separability Study. Because there were non-matching binary results, binomial and nonparametric tests are shown to confirm general alignment between the EDRM and the reported results.

Binary matching (yes/no) (percentage) | 82.3% | |||||

Binomal test (Probability > 50%) | # Y:28, # Trials: 34 | p-value 1.95 × 10^{−4} | ||||

Nonparametric analysis using Wilcoxon test | V = 206 | p-value 0.1207 | Agreement likely | |||

Spearman rank correlation coefficient (Rho) | 0.6946 | |||||

ANOVA($\u2206$% actual vs. calc, matching only) | Df | Sum Sq | Mean Sq | F-Value | Prob(>F) | Result |

Survey (6 surveys total) | 5 | 1602.40 | 320.48 | 8.7410 | 1.60 × 10^{−4} | Significant |

Prospect signs (both pos, both neg, mix) | 2 | 66.56 | 33.28 | 0.9077 | 0.4194 | Not significant |

Residuals | 20 | 733.28 | 36.66 | |||

Normality Assumption | ||||||

Shapiro–Wilk (All-including non-matching) | W = 0.81802 | p-value = 5.832 × 10^{−5} | Not normal | |||

Shapiro–Wilk (matching binary result only) | W = 0.96881 | p-value = 0.5492 | Normal | |||

Conclusions | ||||||

1. Wilcoxon null hypothesis cannot be rejected, so bias between calculated and actual values is unlikely. Additionally, this result further strengthens the PEM validation. 2. The sign of the resulting choice prospects has no significant effect. 3. The survey number is significant. All of the non-matching problems come from the surveys 1 through 3, which were conducted differently than surveys 4, 5, and 6; Survey 1 has a significantly higher difference mean than the other surveys. |

## References

- von Neumann, J. Mathematical Foundations of Quantum Mechanics; Princeton University Press: Princeton, NJ, USA, 1955. [Google Scholar]
- DoD Risk, Issue, and Opportunity Management Guide for Defense Acquisition Programs; Department of Defense (Ed.) Office of the Deputy Assistant Secretary of Defense for Systems Engineering: Washington, DC, USA, 2017.
- DoD System Safety (MIL-STD-882E); Department of Defense (Ed.) Air Force Material Command, Wright-Patterson Air Force Base: Dayton, OH, USA, 2012.
- Monroe, T.J.; Beruvides, M.G. Risk, Entropy, and Decision-Making Under Uncertainty. In Proceedings of the 2018 IISE Annual Conference, Orlando, FL, USA, 19–22 May 2018. [Google Scholar]
- Kahneman, D. Thinking, Fast and Slow, 1st ed.; Farrar, Straus and Giroux: New York, NY, USA, 2011. [Google Scholar]
- Taleb, N.N. The Black Swan, 2nd ed.; Random House Trade Paperbacks: New York, NY, USA, 2010. [Google Scholar]
- Stanovich, K.E.; West, R.F. Individual differences in reasoning: Implications for the rationality debate? Behav. Brain Sci.
**2000**, 23, 645. [Google Scholar] [CrossRef] [PubMed] - Schnieder, M. Dual Process Utility Theory: A Model of Decisions Under Risk and Over Time; Economic Science Institute, Chapman University, One University Drive: Orange, CA, USA, 2018. [Google Scholar]
- ISO. ISO 31000:2018 Risk Management—Guidelines; International Organization for Standardization: Geneva, Switzerland, 2018. [Google Scholar]
- Rasmussen, N. The Application of Probabilistic Risk Assessment Techniques to Energy Technologies. Annu. Rev. Energy
**1981**, 6, 123–138. [Google Scholar] [CrossRef] - Tapiero, C.S. Risk and Financial Management; John Wiley and Sons Ltd.: West Sussex, UK, 2004. [Google Scholar]
- Ariely, D. Predictable Irrational: The Hidden Forces That Shape Our Decisions; HarperCollins: New York, NY, USA, 2009. [Google Scholar]
- Cohen, D. Homo Economicus, the (Lost) Prophet of Modern Times; Polity Press: Malden, MA, USA, 2014. [Google Scholar]
- 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] - Bernoulli, D. Exposition of a New Theory on the Measurement of Risk (1738). Econometrica
**1954**, 22, 23–36. [Google Scholar] [CrossRef] [Green Version] - Tversky, A.; Kahneman, D. Advances in Prospect-Theory—Cumulative Representation of Uncertainty. J. Risk Uncertain.
**1992**, 5, 297–323. [Google Scholar] [CrossRef] - Karmarkar, U.S. Subjectively weighted utility: A descriptive extension of the expected utility model. Organ. Behav. Hum. Perform.
**1978**, 21, 61–72. [Google Scholar] [CrossRef] - Gonzalez, R.; Wu, G. On the Shape of the Probability Weighting Function. Cogn. Psychol.
**1999**, 38, 129–166. [Google Scholar] [CrossRef] [Green Version] - Quiggin, J. A theory of anticipated utility. J. Econ. Behav. Organ.
**1982**, 3, 323–343. [Google Scholar] [CrossRef] - Luce, R.; Ng, C.; Marley, A.; Aczél, J. Utility of gambling II: Risk, paradoxes, and data. Econ. Theory
**2008**, 36, 165–187. [Google Scholar] [CrossRef] - Buchanan, A. Toward a Theory of the Ethics of Bureaucratic Organizations. Bus. Ethics Q.
**1996**, 6, 419–440. [Google Scholar] [CrossRef] - 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] - Bentham, J. An Introduction to the Principles of Morals and Legislation; Kitchener, Ont.: Batoche, SK, Canada, 2000. [Google Scholar]
- Mill, J.S. Utilitarianism; Heydt, C., Ed.; Broadview Editions: Buffalo, NY, USA, 2011. [Google Scholar]
- Introduction to Aristotle; The Modern Library: New York, NY, USA, 1947.
- Ellsberg, D. Risk, Ambiguity, and the Savage Axioms. Q. J. Econ.
**1961**, 75, 643–669. [Google Scholar] [CrossRef] [Green Version] - Wakker, P. Separating marginal utility and probabilistic risk aversion. Theory Decis.
**1994**, 36, 1–44. [Google Scholar] [CrossRef] [Green Version] - Kahneman, D.; Wakker, P.P.W.; Sarin, R.S. Back to Bentham? Explorations of Experienced Utility. Q. J. Econ.
**1997**, 375. [Google Scholar] [CrossRef] [Green Version] - Kahneman, D.; Thaler, R.H. Anomalies: Utility Maximization and Experienced Utility. J. Econ. Perspect.
**2006**, 20, 221–234. [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] - Boltzmann, L. Über die Mechanische Bedeutung des Zweiten Hauptsatzes der Wärmetheorie [On the Mechanical Importance of the Second Principles of Heat-Theory]. Wien. Ber.
**1866**, 53, 195–220. [Google Scholar] - von Neumann, J.; Morgenstern, O. Theory of Games and Economic Behavior; Princeton University Press: Princeton, NJ, USA, 2007. [Google Scholar]
- Shannon, C.E.; Weaver, W. The Mathematical Theory of Communication; The Univeristy of Illinois Press: Urbana, IL, USA, 1949. [Google Scholar]
- Nawrocki, D.N.; Harding, W.H. State-Value Weighted Entropy as a Measure of Investment Risk. Appl. Econ.
**1986**, 18, 411–419. [Google Scholar] [CrossRef] - Yang, J.; Qiu, W. Normalized Expected Utility-Entropy Measure of Risk. Entropy
**2014**, 16, 3590–3604. [Google Scholar] [CrossRef] - Belavkin, R.V. Asymmetry of Risk and Value of Information; Middlesex University: London, UK, 2014. [Google Scholar] [CrossRef] [Green Version]
- Belavkin, R.; Ritter, F.E. The Use of Entropy for Analysis and Control of Cognitive Models. In Proceedings of the Fifth International Conference on Cognitive Modeling, Bamberg, Germany, 9–12 April 2003; pp. 21–26. [Google Scholar]
- Tversky, A. Preference, Belief, and Similarity; The MIT Press: Cambridge, MA, USA, 2004. [Google Scholar]
- Hellman, Z.; Peretz, R. A Survey on Entropy and Economic Behaviour. Entropy
**2020**, 22, 157. [Google Scholar] [CrossRef] [Green Version] - Zingg, C.; Casiraghi, G.; Vaccario, G.; Schweitzer, F. What Is the Entropy of a Social Organization? Entropy
**2019**, 21, 901. [Google Scholar] [CrossRef] [Green Version] - Pisano, R.; Sozzo, S. A Unified Theory of Human Judgements and Decision-Making under Uncertainty. Entropy
**2020**, 22, 738. [Google Scholar] [CrossRef] - Keynes, J.M. A Treatise on Probability; Macmillan and Co., Limited: London, UK, 1921. [Google Scholar]
- Hume, D. A Treatise of Human Nature: Being an Attempt to Introduce the Experimental Method of Reasoning into Moral Subjects; Batoche Books Limited: Kitchener, ON, Canada, 1998. [Google Scholar]
- Jaynes, E.T. Probability Theory: The Logic of Science; Bretthorst, G.L., Ed.; Cambridge University Press: Cambridge, UK; New York, NY, USA, 2003. [Google Scholar]
- Waismann, F. Logische Analyse des Wahrscheinlichkeitsbegriffs. Erkenntnis
**1930**, 1, 228. [Google Scholar] [CrossRef] - Carnap, R. The Two Concepts of Probability: The Problem of Probability. Philos. Phenomenol. Res.
**1945**, 5, 513–532. [Google Scholar] [CrossRef] - Abdellaoui, M. Uncertainty and Risk: Mental, Formal, Experimental Representations; Springer: Berlin/Heidelberg, Germany; London, UK, 2007. [Google Scholar]
- Tversky, A.; Kahneman, D. Judgment under Uncertainty: Heuristics and Biases. Science
**1974**, 185, 1124–1131. [Google Scholar] [CrossRef] - Bayes, T.; Price, R. An Essay towards Solving a Problem in the Doctrine of Chances. By the Late Rev. Mr. Bayes, F.R.S. Communicated by Mr. Price, in a Letter to John Canton, A.M.F.R.S. Philos. Trans. (1683–1775)
**1763**, 53, 370–418. [Google Scholar] - Frigg, R. Probability in Boltzmannian Statistical Mechanics. In Time, Chance and Reduction. Philosophical Aspects of Statistical Mechanics; Gerhard Ernst, G., Huttemann, A., Eds.; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Waismann, F. Philosophical Papers; D. Reidel Pub. Co.: Dordrecht, The Netherlands, 1977. [Google Scholar]
- Popper, K. The Logic of Scientific Discovery; Routledge: London, UK; New York, NY, USA, 1992. [Google Scholar]
- Shackle, G.L.S. Expectation in Economics, 2nd ed.; Cambridge University Press: Cambridge, UK, 1952. [Google Scholar]
- ANSI/ASSE/ISO 31000-2009. Risk Management Principles and Guidelines; American Society of Safety Engineers: Des Plaines, IL, USA, 2011. [Google Scholar]
- Prelec, D. The Probability Weighting Function. Econometrica
**1998**, 66, 497–527. [Google Scholar] [CrossRef] [Green Version] - Wu, G.; Gonzalez, R. Curvature of the Probability Weighting Function. Manag. Sci.
**1996**, 42, 1676–1690. [Google Scholar] [CrossRef] [Green Version] - Lichtenstein, S.; Slovic, P. Reversals of preference between bids and choices in gambling decisions. J. Exp. Psychol.
**1971**, 89, 46–55. [Google Scholar] [CrossRef] [Green Version] - Tversky, A.; Sattath, S.; Slovic, P. Contingent Weighting in Judgment and Choice. Psychol. Rev.
**1988**, 95, 371–384. [Google Scholar] [CrossRef] - Schumpeter, J.A. History of Economic Analysis; Oxford University Press: New York, NY, USA, 1954. [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] - Tversky, A.; Kahneman, D. Rational choice and the framing of decisions. J. Bus.
**1986**, 59, S251. [Google Scholar] [CrossRef] - Birnbaum, M.H. Three New Tests of Independence That Differentiate Models of Risky Decision Making. Manag. Sci.
**2005**, 51, 1346–1358. [Google Scholar] [CrossRef] [Green Version] - 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] - 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] - 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] - Allais, M. An Outline of My Main Contributions to Economic Science. Am. Econ. Rev.
**1997**, 87, 3–12. [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] - Machina, M.J. Choice Under Uncertainty: Problems Solved and Unsolved. J. Econ. Perspect.
**1987**, 1, 121–154. [Google Scholar] [CrossRef] [Green Version] - Conlisk, J. The Utility of Gambling. J. Risk Uncertain.
**1993**, 6, 255–275. [Google Scholar] [CrossRef] - Thaler, R.H. Transaction Utility Theory. Adv. Consum. Res.
**1983**, 10, 229–232. [Google Scholar] - Hoseinzadeh, A.; Mohtashami Borzadaran, G.; Yari, G. Aspects concerning entropy and utility. Theory Decis.
**2012**, 72, 273–285. [Google Scholar] [CrossRef]

1 | The factors used in equations by Gonzalez and Wu ($\left(\gamma ,\delta \tau \right)$) are not those used in EDRM but are quoted in their original form for accuracy. Additionally, this relationship is nearly identical to that stated by Karmarkar. |

2 | As this paper is focused upon the application of an entropy model for positive decision theories, the apparent isomorphology between Boltzmann’s Principle and Daniel Bernoulli’s expected utility theory will be more deeply addressed in subsequent research. |

3 | This case is identical to that of the classical or frequency definition of probability, where each state is assumed to have to same probability due to a lack of knowledge about the states. |

4 | The Authors have chosen to use T out of respect for Amos Tversky who passed before being awarded the Nobel Prize alongside Daniel Kahneman. |

5 | Prelec’s relationship is provided as written; however, the constant $\alpha $ is not the same as that used for power utility. |

6 | To separate decision weights in the two-value CPT actual data, the following was assumed: $\frac{w\left(p\right)}{w\left(1-p\right)}\propto \frac{\tau \left(p\right)}{\tau \left(1-p\right)}$. |

**Figure 1.**There are two groups of decision theories: positive and normative. Normative theories are those applied in standard economic decisions and are tied with deliberate choices (i.e., system 2). In contrast, positive theories counter the normative to address how subjects make choices, often involving intuition (system 1). In other words, normative theories are viewed as how people should make decisions, whereas positive theories address how people actually make decisions. The Entropy Decision Risk Model (EDRM) provides a translation between the two domains. Subsequent research will report on the use of EDRM to apply Expected Utility Theory in the positive domain.

**Figure 2.**PT decision weight. Contrary to expected utility theory, Prospect Theory (1979) empirically determined that subjects make decisions based upon a weighting factor rather than objective probability. Kahneman and Tversky provided this plot as a notional relationship [14].

**Figure 3.**CPT weighting factor. To address concerns identified with Prospect Theory (PT), Tversky and Kahneman later developed cumulative prospect theory (1992) [17]. These curves are formed from a linear regression model based on their data using Equations (1) and (2) with $\gamma =61$ and $\delta =0.69$

**Figure 4.**Flowchart for the present EDRM research showing established theories comprising the EDRM framework, model development, and validation. Section numbers are noted in parentheses.

**Figure 5.**EDRM mathematically and philosophically relates subjective and objective probabilities, which are referred to as proximity and relative certainty, respectively. Proximity encompasses the group of unmeasurable subjective probabilities and relative certainty relates to those probabilities which are directly measurable, as listed in the respective boxes.

**Figure 6.**This figure illustrates basic two- and three-state choices, where ${x}_{i}$ is the magnitude of a state. Consistent with the application of statistical mechanics, as discussed, proximity ($\tau $) is used rather than objective probability ($p$) because this is a depiction of a set of choice states. Although only two and three-state choices are shown, a choice can be made up of any number of states.

**Figure 7.**Uncorrected EDRM plot of proximity ($\tau $) versus relative certainty ($p$) provides a monotonic relationship between subjective and objective probabilities, in general. Of particular importance for comparison with prior research are the preference reversal point and the inflection point, which closely match previous empirical results.

**Figure 8.**Overlay of four plots to show alignment of EDRM with Cumulative Prospect Theory (CPT). The base layer showing the CPT weighting factor curves, which includes the axes, is taken directly from the original CPT paper [17]. The second layer of blue dots represent the actual positive and negative data points from the original report 6. The next (orange) layer is a 5th order linear regression trendline calculated from the original results. The final layer shows the uncorrected EDRM, which more closely trends with the original data than the reported weighting factor curves.

**Figure 9.**Indifference plots of cumulative prospect theory (

**a**,

**b**) and EDRM (

**c**,

**d**) for nonnegative prospects (${x}_{1}=0,\text{}{x}_{2}=100,\text{}{x}_{3}=200)$ and nonpositive prospects (${x}_{1}=-200,\text{}{x}_{2}=-100,\text{}{x}_{3}=0)$. Figures (

**a**,

**b**) are taken directly from the original text [17]. Figures (

**c**,

**d**) are calculated using EDRM. The dashed lines represent probability ${p}_{2}$. It is noteworthy that EDRM generally matches the original CPT indifference plots, except along the edges. This may be explained by the fact that proximities calculated from relative certainties ($p$) are not required to sum to 1 (Section 4.6)

**Figure 10.**To illustrate the effect of $\beta $ on proximity, this plot graphically shows the shape of the proximity curve for various values to show change in preference reversal, as annotated. For $\beta \ge 2$, there is no preference reversal. While this paper will only apply $\beta =1$ for comparison to prior studies to validate the a priori model, in Section 6 the proximity exponent is varied along with the value exponent to further validate the use of $\alpha =0.88$ and $\beta =1$ across all prior studies as a system.

**Figure 11.**This illustration shows a new model for converting the prospect (T) of two choices into the relative percentages of subject responses for direct comparison with prior studies, which universally report these percentages. No prior works reviewed attempt to compare results in this manner, making this the first to do so, to the authors’ knowledge. This model is based upon the Weber-Fechner law of human perception, which is logarithmic, and scaled by the minimum and maximum values. Asinh was chosen because it is likewise logarithmic and permits comparison of positive and negative prospects contiguously along a single scale.

**Figure 12.**Comparison of the results of PT versus uncorrected EDRM are shown above for all problems provided in the original PT paper. EDRM predictions match all results reported by Kahneman and Tversky.

**Figure 13.**This plot shows a high degree of alignment of EDRM compared with actual CPT data for one and two-state choices. The plot scales are different on the horizontal and vertical axes to amplify the results. The dashed line represents a linear trendline using all the data, which shows excellent alignment with the positive and negative extremes. There is a slightly tighter correlation of the model for negative values. The negative slope of the trendline shows there is a very small difference between gains and losses (loss aversion), but is considered a minor effect in this research.

**Figure 14.**Wu and Markle Gain/Loss versus EDRM. This study was chosen because it provides a challenging test of EDRM’s ability to handle mixtures of gains and losses. The plot shows the actual and calculated percentages (H) first by survey number and then by problem number. Note that all the non-matching binary results occur in surveys 1, 2, and 3, which were conducted differently in the original study.

**Figure 15.**(Sample) Wu and Markle Problem 25 (Actual Choice: A). This contour plot illustrates predicted subject choice for varying values of exponents $\alpha $ and $\beta $. For the standard values of $\alpha =0.88,\beta =1$, and $\lambda =1$, choice A will be preferred. Plots such as this were generated for all the choices considered in this research for evaluation as a system.

**Figure 16.**EDRM Multiple Study $\alpha $-vs-$\beta $ Sensitivity Analysis ($\lambda =1$). This plot represents a compilation of all 63 choices evaluated in this research for which EDRM correctly predicted the binary result for 57 (90.5%). The legend shows the z-axis representing the percentage of the problems with a correct binary result as $\alpha $ and $\beta $ are varied, up to a maximum of 57, which correlates to 100% on the plot. The results clearly demonstrate that the standard values of $\alpha =0.88$ and $\beta =1$ are valid, affirming the original work by Kahneman and Tversky and EDRM.

**Figure 17.**EDRM Multiple Study $\alpha $-vs-$\lambda $ Sensitivity Analysis ($\beta =1$). Formatted similarly to Figure 16, this plot shows that as $\lambda $ (loss aversion factor) increases, slightly wider ranges of $\alpha $ will correctly predict the binary result for a maximum of 57 of 63 choices analyzed. This shows that loss aversion is present for negative state values, but validates its consideration as a secondary effect, since the standard values of $\alpha =0.88$ and $\beta =1$ are valid assuming loss aversion is not present (i.e., $\lambda =1$). Plotting of $\beta $-vs-$\lambda $ has nearly identical results.

**Table 1.**Allais Paradox performance using EDRM. ${T}_{j}$ is the prospect of the choice $\alpha =0.88$ is the standard power utility used by Kahneman and Tversky and others. Calc % provides percentage comparison from the PEM. Underlines indicate the greater values.

$\mathit{\alpha}=0.88$ | |||||||
---|---|---|---|---|---|---|---|

Problem (Value, Probability) | EDRM | Calc % | Match | ||||

Choice A (1 and 3) | Choice B (2 and 4) | ${T}_{A}$ | ${T}_{B}$ | A | B | Y/N | |

1 and 2 | (1M) | (5M, 0.10; 1M, 0.89) | 190,456 | 131,265 | 90 | 10 | Yes |

3 and 4 | (5M,.10) | (1M,.11) | 112,312 | 28,925 | 90 | 10 | Yes |

**Table 2.**EDRM performance with Prospect Theory. ${T}_{j}$ is the prospect of the choice. Calc % and Actual % provide comparison of model results to those reported by Kahneman and Tversky. Underlines indicate the greater values.

Problem | EDRM | Calc % | Actual % | Diff | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Choice A | Choice B | ${\mathit{T}}_{\mathit{A}}$ | ${\mathit{T}}_{\mathit{B}}$ | A | B | A | B | Δ% | ||

1 | (2500, 0.33; 2400, 0.66) | (2400) | 824.66 | 943.16 | 16 | 84 | 18 | 82 | 2 | |

2 | (2500, 0.33) | (2400, 0.34) | 308.94 | 304.55 | 64 | 36 | 83 | 17 | 19 | |

3 | (4000, 0.8) | (3000) | 978.90 | 1147.80 | 15 | 85 | 20 | 80 | 5 | |

4 | (4000, 0.2) | (3000, 0.25) | 330.73 | 298.54 | 74 | 26 | 65 | 35 | −9 | |

5 | (10000, 0.5) | (4320)1 ^{1} | 1430.49 | 1582.07 | 19 | 81 | 22 | 78 | 3 | |

6 | (10000, 0.05) | (4320, 0.1) ^{1} | 308.95 | 226.24 | 77 | 23 | 67 | 33 | −10 | |

7 | (6000, 0.45) | (3000, 0.9) | 840.31 | 879.79 | 25 | 75 | 14 | 86 | −11 | |

8 | (6000, 0.001) | (3000, 0.002) | 20.70 | 16.64 | 60 | 40 | 73 | 27 | 13 | |

3′ | (−4000, 0.8) | (−3000) | −978.90 | −1147.80 | 85 | 15 | 92 | 8 | 7 | |

4′ | (−4000, 0.2) | (−3000, 0.25) | −330.73 | −298.54 | 26 | 74 | 42 | 58 | 16 | |

7′ | (−3000, 0.9) | (−6000, 0.45) | −879.79 | −840.31 | 25 | 75 | 8 | 92 | −17 | |

8′ | (−3000, 0.002) | (−6000, 0.001) | −16.64 | −20.70 | 60 | 40 | 70 | 30 | 10 | |

10 ^{2} | (4000, 0.8) | (3000) | 978.90 | 1147.80 | 15 | 85 | 22 | 78 | 7 | |

11 | (1000, 0.5) | (500) | 188.57 | 237.19 | 19 | 81 | 16 | 84 | −3 | |

12 | (−1000, 0.5) | (−500) | −188.57 | −237.19 | 81 | 19 | 69 | 31 | −12 | |

13 | (6000, 0.25) | (4000, 0.25; 2000, 0.25) | 549.43 | 593.50 | 25 | 75 | 18 | 82 | −7 | |

13′ | (−6000, 0.25) | (−4000, 0.25;−2000, 0.25) | −549.43 | −593.50 | 75 | 25 | 70 | 30 | −5 | |

14 | (5000, 0.001) | (5) | 17.63 | 4.12 | 67 | 33 | 72 | 28 | 5 | |

14′ | (−5000, 0.001) | (−5) | −17.63 | −4.12 | 33 | 67 | 17 | 83 | 2 |

^{1}. Estimated trip values using certainty equivalent from CPT: CE(10000, 0.5) = 4320;

^{2}. Problem 10 is the second stage of a two-stage problem where there is only a 25% chance of proceeding past the first stage; however, as stated by Kahneman and Tversky in problem 10 of Prospect Theory, people tend to disregard the first stage [14]. Therefore, the first stage is not applied in this model.

**Table 3.**EDRM Performance with Cumulative Prospect Theory through comparison of calculated and actual certainty equivalents (CE), which is equivalent to prospect, T. Proximities (${\tau}_{i}$) calculated for each state are also shown. Data are as reported by Tversky and Kahneman.

$\mathit{\alpha}=1,\mathit{\beta}=1$ | ||||||
---|---|---|---|---|---|---|

Problem | EDRM | Results | ||||

Outcomes | Gamble | ${\mathit{\tau}}_{1}$ | ${\mathit{\tau}}_{2}$ | $\mathbf{Calc}\text{}\mathbf{CE}\text{}\mathit{T}$ | Actual CE | Diff Δ CE |

(0, 50) | (50, 0.1) | 0.1430 | 7.15 | 9 | 1.85 | |

(50, 0.5) | 0.4320 | 21.60 | 21 | 0.6 | ||

(50, 0.9) | 0.7665 | 38.32 | 37 | 1.325 | ||

(0, −50) | (−50, 0.1) | 0.1430 | −7.15 | −8 | 0.85 | |

(−50, 0.5) | 0.4320 | −21.60 | −21 | −0.6 | ||

(−50, 0.9) | 0.7665 | −38.32 | −39 | 0.675 | ||

(0, 100) | (100, 0.005) | 0.0933 | 9.33 | 14 | −4.67 | |

(100, 0.25) | 0.2601 | 26.01 | 25 | 1.01 | ||

(100, 0.5) | 0.4320 | 43.20 | 36 | 7.2 | ||

(100, 0.75) | 0.6183 | 61.83 | 52 | 9.83 | ||

(100, 0.95) | 0.8372 | 83.72 | 78 | 5.72 | ||

(0, −100) | (−100, 0.005) | 0.0933 | −9.33 | −8 | −1.33 | |

(−100, 0.25) | 0.2601 | −26.01 | −23.5 | −2.51 | ||

(−100, 0.5) | 0.4320 | −43.20 | −42 | −1.2 | ||

(−100, 0.75) | 0.6183 | −61.83 | −63 | 1.17 | ||

(−100, 0.95) | 0.8372 | −83.72 | −84 | 0.28 | ||

(0, 200) | (200, 0.01) | 0.0361 | 7.22 | 10 | −2.78 | |

(200, 0.1) | 0.1430 | 28.60 | 20 | 8.6 | ||

(200, 0.5) | 0.4320 | 86.40 | 76 | 10.4 | ||

(200, 0.9) | 0.7665 | 153.30 | 131 | 22.3 | ||

(200, 0.99) | 0.9284 | 185.68 | 188 | −2.32 | ||

(0, −200) | (−200, 0.01) | 0.0361 | −7.22 | −3 | −4.22 | |

(−200, 0.1) | 0.1430 | −28.60 | −23 | −5.6 | ||

(−200, 0.5) | 0.4320 | −86.40 | −89 | 2.6 | ||

(−200, 0.9) | 0.7665 | −153.30 | −155 | 1.7 | ||

(−200, 0.99) | 0.9284 | −185.68 | −190 | 4.32 | ||

(0, 400) | (400, 0.01) | 0.0361 | 14.44 | 12 | 2.44 | |

(400, 0.99) | 0.9284 | 371.36 | 377 | −5.64 | ||

(0, −400) | (−400, 0.01) | 0.0361 | −14.44 | −14 | −0.44 | |

(−400, 0.99) | 0.9284 | −371.36 | −380 | 8.64 | ||

(50, 100) | (50, 0.9; 100, 0.1) | 0.1430 | 0.7665 | 52.62 | 59 | −6.375 |

(50, 0.5; 100, 0.5) | 0.4320 | 0.4320 | 64.80 | 71 | −6.2 | |

(50, 0.1; 100, 0.9) | 0.7665 | 0.1430 | 83.80 | 83 | 0.8 | |

(−50, −100) | (−50, 0.9; −100, 0.1) | 0.1430 | 0.7665 | −52.62 | −59 | 6.375 |

(−50, 0.5; −100, 0.5) | 0.4320 | 0.4320 | −64.80 | −71 | 6.2 | |

(−50, 0.1; −100, 0.9) | 0.7665 | 0.1430 | −83.80 | −85 | 1.2 | |

(50, 150) | (50, 0.95; 150, 0.05) | 0.0933 | 0.8372 | 55.85 | 64 | −8.145 |

(50, 0.75; 150, 0.25) | 0.2601 | 0.6183 | 69.93 | 72.5 | −2.57 | |

(50, 0.5; 150, 0.5) | 0.4320 | 0.4320 | 86.40 | 86 | 0.4 | |

(50, 0.25; 150, 0.75) | 0.6183 | 0.2601 | 105.75 | 102 | 3.75 | |

(50, 0.05;150, 0.95) | 0.8372 | 0.0933 | 130.24 | 128 | 2.245 | |

(−50, −150) | (−50, 0.95; −150, 0.05) | 0.0933 | 0.8372 | −55.85 | −60 | 4.145 |

(−50, 0.75; −150, 0.25) | 0.2601 | 0.6183 | −69.93 | −71 | 1.07 | |

(−50, 0.5; −150, 0.5) | 0.4320 | 0.4320 | −86.40 | −92 | 5.6 | |

(−50, 0.25; −150, 0.75) | 0.6183 | 0.2601 | −105.75 | −113 | 7.25 | |

(−50, 0.05; −150, 0.95) | 0.8372 | 0.0933 | −130.24 | −132 | 1.755 | |

(100, 200) | (100, 0.95; 200, 0.05) | 0.0933 | 0.8372 | 102.38 | 118 | −15.62 |

(100, 0.75; 200, 0.25) | 0.2601 | 0.6183 | 113.85 | 130 | −16.15 | |

(100, 0.5; 200, 0.5) | 0.4320 | 0.4320 | 129.60 | 141 | −11.4 | |

(100, 0.25; 200, 0.75) | 0.6183 | 0.2601 | 149.67 | 162 | −12.33 | |

(100, 0.05; 200, 0.95) | 0.8372 | 0.0933 | 176.77 | 178 | −1.23 | |

(−100, −200) | (−100, 0.95; −200, 0.05) | 0.0933 | 0.8372 | −102.38 | −112 | 9.62 |

(−100, 0.75; −200, 0.25) | 0.2601 | 0.6183 | −113.85 | −121 | 7.15 | |

(−100, 0.5; −200, 0.5) | 0.4320 | 0.4320 | −129.60 | −142 | 12.4 | |

(−100, 0.25; −200, 0.75) | 0.6183 | 0.2601 | −149.67 | −158 | 8.33 | |

(−100, 0.05; −200, 0.95) | 0.8372 | 0.0933 | −176.77 | −179 | 2.23 |

**Table 4.**EDRM compared with results of Framing of Decisions and the Psychology of Choice. ${T}_{j}$ is the prospect of the choice. Calc % and Actual % provide comparison of model results to those reported by Tversky and Kahneman. Note that problem 4 makes use of the dominance effect. Underlines indicate the greater values.

$\mathit{\alpha}=0.88,\mathit{\beta}=1$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Problem (Value, Probability) | EDRM | Calc % | Actual % | Diff | Match | |||||

Choice A | Choice B | ${T}_{A}$ | ${T}_{B}$ | A | B | A | B | Δ% | Y/N | |

1 | (200) | (600, 1/3) | 106 | 89 | 75 | 25 | 72 | 28 | −3 | Yes |

2 | (−400) | (0, 1/3; −600, 2/3) | −195 | −154 | 18 | 82 | 22 | 78 | 4 | Yes |

3i | (240) | (1000, 0.25) | 124 | 114 | 71 | 29 | 84 | 16 | 13 | Yes |

3ii | (−750) | (−1000, 0.75) | −339 | −270 | 16 | 84 | 13 | 87 | −3 | Yes |

4 | (240, 0.25; −760, 0.75) | (250, 0.25; −750, 0.75) | −180 | −176 | 0 | 100 ^{2} | 0 | 100 | 0 | Yes |

5 | (30) | (45,.8) | 20 | 19 | 59 | 41 | 78 | 22 | 19 | Yes |

6 ^{1} | (30) | (45,.8) | 20 | 19 | 59 | 41 | 74 | 26 | 15 | Yes |

7 | (30,.25) | (45,.2) | 5.2 | 6.4 | 41 | 59 | 42 | 58 | 1 | Yes |

^{1}. Problem 6 is the second stage of a two-stage version of problem 5 where there is only a 25% chance of proceeding past the first stage; however, as stated by Kahneman and Tversky in problem 10 of Prospect Theory, people tend to disregard the first stage [14]. Therefore, the first stage is not applied in this model;

^{2}. Dominance is present, so the evaluation model returns 100% for the choice with the greater prospect.

**Table 5.**EDRM compared with results of select problems from Tversky and Kahneman’s Rational Choice and the Framing of Decisions having more than three states and mixtures of gains and losses.${T}_{j}$ is the prospect of the choice. Calc % and Actual % provide comparison of model results to those reported. Similar results were achieved with a wide range of power utility function exponents. Underlines indicate the greater values.

$\mathit{\alpha}=0.88,\mathit{\beta}=1$ | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Problem (Value, Probability) | EDRM | Calc % | Actual % | Diff | Match | |||||

Choice A | Choice B | ${\mathit{T}}_{\mathit{A}}$ | ${\mathit{T}}_{\mathit{B}}$ | A | B | A | B | Δ% | Y/N | |

7 | (0, 0.9; 45, 0.06; 30, 0.01;−15, 0.01; −15, 0.02) | (0, 0.9; 45, 0.06; 45, 0.01;−10, 0.01, −15, 0.02) | 2.71 | 3.14 | 0 | 100 ^{1} | 0 | 100 | 0 | Yes |

8 | (0, 0.9; 45, 0.06, 30, 0.01; −15, 0.03) | (0, 0.9; 45, 0.07; −10, 0.01,− 15, 0.02) | 2.95 | 2.40 | 52 | 48 | 58 | 42 | 6 | Yes |

^{1}. Dominance is present, so the evaluation model returns 100% for the choice with the greater prospect.

**Table 6.**EDRM Performance with Wu and Markle Gain–Loss Separability Study (mixed gambles). ${T}_{j}$ is the prospect of the choice, ${v}_{ij}$ is the state value, ${p}_{ij}$ is the state relative certainty, and ${\tau}_{ij}$ is the state proximity. Note that in this analysis, 6 of 34 problems have non-matching binary results, which are italicized. Underlines indicate the greater values.

Choice H | Choice L | Proximity | Prospect | Results (%) | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Calc | Actual | Eval | ||||||||||||||||||

${\mathit{v}}_{1,\mathit{H}}$ | ${\mathit{p}}_{1,\mathit{H}}$ | ${\mathit{v}}_{2,\mathit{H}}$ | ${\mathit{p}}_{2,\mathit{H}}$ | ${\mathit{v}}_{1,\mathit{L}}$ | ${\mathit{p}}_{1,\mathit{L}}$ | ${\mathit{v}}_{2,\mathit{L}}$ | ${\mathit{p}}_{2,\mathit{L}}$ | ${\mathit{\tau}}_{1,\mathit{H}}$ | ${\mathit{\tau}}_{2,\mathit{H}}$ | ${\mathit{\tau}}_{1,\mathit{L}}$ | ${\mathit{\tau}}_{2,\mathit{L}}$ | ${\mathit{T}}_{\mathit{H}}$ | ${\mathit{T}}_{\mathit{L}}$ | H | L | H | L | Δ% | Y/N | |

1 | 150 | 0.3 | −25 | 0.7 | 75 | 0.8 | −60 | 0.2 | 0.30 | 0.58 | 0.66 | 0.22 | 14 | 21 | 38 | 62 | 22 | 78 | −16 | Y |

2 | 1800 | 0.05 | −200 | 0.95 | 600 | 0.3 | −250 | 0.7 | 0.09 | 0.84 | 0.30 | 0.58 | −20 | 8 | 37 | 63 | 21 | 79 | −16 | Y |

3 | 1000 | 0.25 | −500 | 0.75 | 600 | 0.5 | −700 | 0.5 | 0.26 | 0.62 | 0.43 | 0.43 | −33 | −17 | 39 | 61 | 28 | 72 | −11 | Y |

4 | 200 | 0.3 | −25 | 0.7 | 75 | 0.8 | −100 | 0.2 | 0.30 | 0.58 | 0.66 | 0.22 | 21 | 17 | 59 | 41 | 33 | 67 | −26 | N |

5 | 1200 | 0.25 | −500 | 0.75 | 600 | 0.5 | −800 | 0.5 | 0.26 | 0.62 | 0.43 | 0.43 | −13 | −35 | 62 | 38 | 43 | 57 | −19 | N |

6 | 750 | 0.4 | −1000 | 0.6 | 500 | 0.6 | −1500 | 0.4 | 0.36 | 0.50 | 0.50 | 0.36 | −96 | −108 | 60 | 40 | 51 | 49 | −9 | Y |

7 | 4200 | 0.5 | −3000 | 0.5 | 3000 | 0.75 | −6000 | 0.25 | 0.43 | 0.43 | 0.62 | 0.26 | 171 | 160 | 59 | 41 | 52 | 48 | −7 | Y |

8 | 4500 | 0.5 | −1500 | 0.5 | 3000 | 0.75 | −3000 | 0.25 | 0.43 | 0.43 | 0.62 | 0.26 | 439 | 411 | 62 | 38 | 48 | 52 | −14 | N |

9 | 4500 | 0.5 | −3000 | 0.5 | 3000 | 0.75 | −6000 | 0.25 | 0.43 | 0.43 | 0.62 | 0.26 | 213 | 160 | 63 | 37 | 58 | 42 | −5 | Y |

10 | 1000 | 0.3 | −200 | 0.7 | 400 | 0.7 | −500 | 0.3 | 0.30 | 0.58 | 0.58 | 0.30 | 68 | 43 | 63 | 37 | 51 | 49 | −12 | Y |

11 | 4800 | 0.5 | −1500 | 0.5 | 3000 | 0.75 | −3000 | 0.25 | 0.43 | 0.43 | 0.62 | 0.26 | 480 | 411 | 65 | 35 | 54 | 46 | −10 | Y |

12 | 3000 | 0.01 | −490 | 0.99 | 2000 | 0.02 | −500 | 0.98 | 0.04 | 0.93 | 0.05 | 0.90 | −175 | −170 | 42 | 58 | 59 | 41 | 17 | N |

13 | 2200 | 0.4 | −600 | 0.6 | 850 | 0.75 | −1700 | 0.25 | 0.36 | 0.50 | 0.62 | 0.26 | 178 | 53 | 67 | 33 | 52 | 48 | −15 | Y |

14 | 2200 | 0.2 | −1000 | 0.8 | 1700 | 0.25 | −1100 | 0.75 | 0.22 | 0.66 | 0.26 | 0.62 | −94 | −112 | 61 | 39 | 58 | 42 | −4 | Y |

15 | 1500 | 0.25 | −500 | 0.75 | 600 | 0.5 | −900 | 0.5 | 0.26 | 0.62 | 0.43 | 0.43 | 16 | −52 | 65 | 35 | 51 | 49 | −14 | Y |

16 | 5000 | 0.5 | −3000 | 0.5 | 3000 | 0.75 | −6000 | 0.25 | 0.43 | 0.43 | 0.62 | 0.26 | 281 | 160 | 65 | 35 | 65 | 35 | 0 | Y |

17 | 1500 | 0.4 | −1000 | 0.6 | 600 | 0.8 | −3500 | 0.2 | 0.36 | 0.50 | 0.66 | 0.22 | 8 | −110 | 66 | 34 | 59 | 41 | −7 | Y |

18 | 2025 | 0.5 | −875 | 0.5 | 1800 | 0.6 | −1000 | 0.4 | 0.43 | 0.43 | 0.50 | 0.36 | 183 | 209 | 37 | 63 | 72 | 28 | 35 | N |

19 | 600 | 0.25 | −100 | 0.75 | 125 | 0.75 | −500 | 0.25 | 0.26 | 0.62 | 0.62 | 0.26 | 37 | −18 | 66 | 34 | 58 | 43 | −8 | Y |

20 | 5000 | 0.1 | −900 | 0.9 | 1400 | 0.3 | −1700 | 0.7 | 0.14 | 0.77 | 0.30 | 0.58 | −48 | −229 | 67 | 33 | 40 | 60 | −27 | N |

21 | 700 | 0.25 | −100 | 0.75 | 125 | 0.75 | −600 | 0.25 | 0.26 | 0.62 | 0.62 | 0.26 | 47 | −29 | 67 | 33 | 71 | 29 | 4 | Y |

22 | 700 | 0.5 | −150 | 0.5 | 350 | 0.75 | −400 | 0.25 | 0.43 | 0.43 | 0.62 | 0.26 | 102 | 56 | 66 | 34 | 63 | 37 | −3 | Y |

23 | 1200 | 0.3 | −200 | 0.7 | 400 | 0.7 | −800 | 0.3 | 0.30 | 0.58 | 0.58 | 0.30 | 90 | 7 | 67 | 33 | 70 | 30 | 3 | Y |

24 | 5000 | 0.5 | −2500 | 0.5 | 2500 | 0.75 | −6000 | 0.25 | 0.43 | 0.43 | 0.62 | 0.26 | 355 | 55 | 68 | 32 | 79 | 21 | 11 | Y |

25 | 800 | 0.4 | −1000 | 0.6 | 500 | 0.6 | −1600 | 0.4 | 0.36 | 0.50 | 0.50 | 0.36 | −89 | −121 | 64 | 36 | 58 | 43 | −6 | Y |

26 | 5000 | 0.5 | −3000 | 0.5 | 2500 | 0.75 | −6500 | 0.25 | 0.43 | 0.43 | 0.62 | 0.26 | 281 | 15 | 67 | 33 | 71 | 29 | 4 | Y |

27 | 700 | 0.25 | −100 | 0.75 | 100 | 0.75 | −800 | 0.25 | 0.26 | 0.62 | 0.62 | 0.26 | 47 | −58 | 68 | 32 | 73 | 28 | 5 | Y |

28 | 1500 | 0.3 | −200 | 0.7 | 400 | 0.7 | −1000 | 0.3 | 0.30 | 0.58 | 0.58 | 0.30 | 123 | −16 | 68 | 32 | 75 | 25 | 7 | Y |

29 | 1600 | 0.25 | −500 | 0.75 | 600 | 0.5 | −1100 | 0.5 | 0.26 | 0.62 | 0.43 | 0.43 | 25 | −85 | 67 | 33 | 73 | 28 | 6 | Y |

30 | 2000 | 0.4 | −800 | 0.6 | 600 | 0.8 | −3500 | 0.2 | 0.36 | 0.50 | 0.66 | 0.22 | 112 | −110 | 68 | 32 | 65 | 35 | −3 | Y |

31 | 2000 | 0.25 | −400 | 0.75 | 600 | 0.5 | −1100 | 0.5 | 0.26 | 0.62 | 0.43 | 0.43 | 88 | −85 | 68 | 32 | 80 | 20 | 12 | Y |

32 | 1500 | 0.4 | −700 | 0.6 | 300 | 0.8 | −3500 | 0.2 | 0.36 | 0.50 | 0.66 | 0.22 | 67 | −194 | 69 | 31 | 78 | 23 | 9 | Y |

33 | 900 | 0.4 | −1000 | 0.6 | 500 | 0.6 | −1800 | 0.4 | 0.36 | 0.50 | 0.50 | 0.36 | −75 | −147 | 66 | 34 | 70 | 30 | 4 | Y |

34 | 1000 | 0.4 | −1000 | 0.6 | 500 | 0.6 | −2000 | 0.4 | 0.36 | 0.50 | 0.50 | 0.36 | −61 | −173 | 67 | 33 | 78 | 23 | 10 | Y |

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.
Derivation and Application of the Subjective–Objective Probability Relationship from Entropy: The Entropy Decision Risk Model (EDRM). *Systems* **2020**, *8*, 46.
https://doi.org/10.3390/systems8040046

**AMA Style**

Monroe T, Beruvides M, Tercero-Gómez V.
Derivation and Application of the Subjective–Objective Probability Relationship from Entropy: The Entropy Decision Risk Model (EDRM). *Systems*. 2020; 8(4):46.
https://doi.org/10.3390/systems8040046

**Chicago/Turabian Style**

Monroe, Thomas, Mario Beruvides, and Víctor Tercero-Gómez.
2020. "Derivation and Application of the Subjective–Objective Probability Relationship from Entropy: The Entropy Decision Risk Model (EDRM)" *Systems* 8, no. 4: 46.
https://doi.org/10.3390/systems8040046