# Deciding on a Continuum of Equivalent Alternatives Engaging Uncertainty through Behavior Patterning

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## Abstract

**:**

## 1. Introduction

## 2. Can Humans Be Random?

## 3. A Triple Experiment on Choosing on a Continuum of Equivalent Alternatives

- Experiment 1:

- Experiment 2:

- Experiment 3:

## 4. Methods: Machine Learning Algorithms for Data Clustering

#### 4.1. Data Embedding and Related Riemann Metric

#### 4.2. Mapping the Data by Stochastic Neighbor Embedding

#### 4.3. Soft Clustering by Gaussian Mixture Model

## 5. Results

#### 5.1. Experiment 1: Engaging Uncertainty through Behavior Patterning

- Group 1, Experiment 1:

- Group 2, Experiment 1:

- Group 3, Experiment 1:

- Group 4, Experiment 1:

#### 5.2. Experiment 2: Breaking Behavior Patterns through Imposed Restrictions

#### 5.3. Experiment 3: No Common Choice Preferences when More Restrictions Imposed

## 6. Conclusions and Discussion

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AI | Artificial intelligence |

SNE | Stochastic Neighbor Embedding |

GMM | Gaussian Mixture Modeling |

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**Figure 1.**The scheme of the triple quasi-experiment on a continuum of equivalent alternatives. (

**a**) Experiment 1: subjects were allowed to draw lines in any direction. (

**b**) Experiment 2: subjects could draw lines at any angle, excepting those in the first quadrant. (

**c**) Experiment 3: the first and second quadrants were barred, but subjects could draw lines at any angle within the left side of circle.

**Figure 3.**(

**a**) The fine structure of the distance matrix ${D}_{ij}^{\left(1\right)}$ is visualized by a phylogenetic tree constructed using the neighbor-joining method. The ellipses mark the four groups of subjects following different random choice strategies. (

**b**) The results of GMM clustering in Experiment 1 shows four groups of subjects. Features 1 and 2, the coordinates obtained for the subjects when the GMM iteration process converged, are measured in procedure-defined units.

**Figure 4.**(

**a**) A scatter plot representing male and female participants in the feature space of experimental data collected in Experiment 1. (

**b**) The kernel density plots of the line-drawing angles in Experiment 1 for male and female subjects. The Mann–Whitney U-test of the distributions of line-drawing angles shows that there is no statistically significant sex-specific difference in line angel preferences between men and women.

**Figure 5.**(

**a**) The autocorrelation function (10) of the angle lags between the sequential lines observed in Group 1 shows peaks at the lag multiple of ${45}^{\circ}$. The dashed horizontal lines mark the level of statistically significant correlations. (

**b**) The radial histogram (not normalized) represents the approximation for a kernel density estimation of the line angles drawn in particular directions in Group 1.

**Figure 6.**(

**a**) The autocorrelation function (10) represents a single statistically significant lag (of ${180}^{\circ}$) between the sequentially drawn lines in Group 2. The dashed horizontal line indicates the level of statistically significant correlations. (

**b**) The radial histogram (not normalized) of the line-drawing angles observed in Group 2.

**Figure 7.**(

**a**) The autocorrelation function (10) represents the statistical significance of lags between the sequential line angles in Group 3. The dashed horizontal lines indicate the level of statistically significant correlations. (

**b**) The radial histogram (not normalized) of the line-drawing angles observed in Group 3.

**Figure 8.**(

**a**) The autocorrelation function (10) represents the statistical significance of lags between the sequential line angles in Group 4. The dashed horizontal line marks the level of statistically significant correlations. (

**b**) The radial histogram (not normalized) of the line-drawing angles observed in Group 4.

**Figure 9.**(

**a**) The structure of the distance matrix ${D}_{ij}^{\left(2\right)}$ visualized by a phylogenetic tree with the use of the neighbor-joining method. (

**b**) The results of GMM clustering aggregates all participants of Experiment 2 into a single cluster. Features 1 and 2 (the coordinates obtained for subjects when the iteration process converged) are measured in procedure-defined units.

**Figure 10.**(

**a**) The autocorrelation function (10) of the angle lags between the sequential lines observed for the participants of Experiment 2. The dashed horizontal line marks the level of statistically significant correlations. (

**b**) The radial histogram (not normalized) of the line-drawing angles observed in Experiment 2.

**Figure 11.**(

**a**) The autocorrelation function (10) of the angle lags between the sequential lines observed in the group of participants in Experiment 3. The dashed horizontal line marks the level of statistically significant correlations. (

**b**) The radial histogram (not normalized) of the line-drawing angles observed in the third experiment.

**Figure 12.**(

**a**) The probability density plot of the line-drawing angles in Experiment 3, with a reference line representing the normal distribution $\mathcal{N}(180.{29}^{\circ},8.18)$ fitted the data best. (

**b**) The quantile–quantile normal probability plot shows quantiles of the empirical drawing angle distribution against quantiles of the normal distribution $\mathcal{N}(180.{29}^{\circ},8.18)$. The similarity of both distributions are justified by the fact that points are almost perfectly aligned along the diagonal line $y=x$. The shaded area defines the 95% confidence bounds for the normal distribution quantiles, with mean and standard deviation calculated from data (angles).

**Figure 13.**The main compass axes featured by the line-drawing strategies on a continuum of equivalent alternatives.

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**MDPI and ACS Style**

Rathnayake, K.; Lebedev, A.; Volchenkov, D.
Deciding on a Continuum of Equivalent Alternatives Engaging Uncertainty through Behavior Patterning. *Foundations* **2022**, *2*, 1080-1100.
https://doi.org/10.3390/foundations2040071

**AMA Style**

Rathnayake K, Lebedev A, Volchenkov D.
Deciding on a Continuum of Equivalent Alternatives Engaging Uncertainty through Behavior Patterning. *Foundations*. 2022; 2(4):1080-1100.
https://doi.org/10.3390/foundations2040071

**Chicago/Turabian Style**

Rathnayake, Kusal, Alexander Lebedev, and Dimitri Volchenkov.
2022. "Deciding on a Continuum of Equivalent Alternatives Engaging Uncertainty through Behavior Patterning" *Foundations* 2, no. 4: 1080-1100.
https://doi.org/10.3390/foundations2040071