# Avoiding the Worst Decisions: A Simulation and Experiment

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Methods of Simulation

#### 3.2. Method of Experiment

#### 3.2.1. Participants in the Experiment

#### 3.2.2. Experimental Apparatus

#### 3.2.3. Tasks and Strategies Used in the Experiment

#### 3.2.4. Procedure of the Experiment

## 4. Results

#### 4.1. Simulation

#### 4.2. Experiment

## 5. Discussion

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Simulation

Strategy | Explanation |
---|---|

Weighted Additive (WAD) | Examine the weighted value of all attributes for each option and choose the option with the highest value. |

Equal-weighted Additive (EQW) | Examine the value of all attributes for each option and choose the option with the highest value. However, no weighting is assigned to the attribute values. |

Additive Difference (ADF) | The evaluation values of any pair of alternatives are compared for each attribute. When there are three or more alternatives, the winning alternatives are compared one after another in a tournament-like fashion, and the final alternative is adopted. |

Conjunctive (CON) | Each attribute has a set of necessary conditions, and if any one of the attributes does not satisfy the necessary conditions, the information processing of the alternative is terminated and the alternative is rejected, regardless of the values of the other attributes. In the case of choosing only one option with this decision strategy, the first option that exceeds the necessary conditions across all attributes will be chosen. |

Disjunctive (DIS) | Sufficient conditions are set for each attribute, and if any one attribute satisfies the sufficient condition, the alternative is adopted regardless of the values of the other attributes. |

Lexicographic (LEX) | The option with the highest rating is chosen for the most important attribute. If there is a tie for the most important attribute, the next most important attribute is chosen. |

Lexicographic Semi-order (LEX-S) | The option with the highest rating is chosen as the most important attribute. If there is a tie for the most important attribute, the next most important attribute is chosen. However, if a small difference within a certain range is also considered at the same rank, the next most important attribute is used for judgment. |

Elimination by Aspects (EBA) | The option with the highest rating is chosen as the most important attribute. If there is a tie for the most important attribute, the next most important attribute is chosen. However, if a small difference within a certain range is also considered at the same rank, the next most important attribute is used for judgment. |

Majority of Confirming Dimensions (MCD) | The evaluation values of any pair of alternatives are compared for each attribute in a brute-force fashion. The evaluation method differs from the additive difference method in that it compares the number of dominant attributes and adopts the option that has the highest winning rate in the round-robin comparison. |

The First Stage | The Second Stage | The First Stage | The Second Stage |
---|---|---|---|

CON | CON | LEX | CON |

CON | DIS | LEX | DIS |

CON | EBA | LEX | EBA |

CON | LEX | LEX | LEX |

CON | LEX-S | LEX | LEX-S |

CON | WAD | LEX | WAD |

CON | EQW | LEX | EQW |

CON | ADF | LEX | ADF |

CON | MCD | LEX | MCD |

DIS | CON | LEX-S | CON |

DIS | DIS | LEX-S | DIS |

DIS | EBA | LEX-S | EBA |

DIS | LEX | LEX-S | LEX |

DIS | LEX-S | LEX-S | LEX-S |

DIS | WAD | LEX-S | WAD |

DIS | EQW | LEX-S | EQW |

DIS | ADF | LEX-S | ADF |

DIS | MCD | LEX-S | MCD |

EBA | CON | ||

EBA | DIS | ||

EBA | EBA | ||

EBA | LEX | ||

EBA | LEX-S | ||

EBA | WAD | ||

EBA | EQW | ||

EBA | ADF | ||

EBA | MCD |

#### Appendix A.1. RA and MRA in Two-Stage Decision-Making

**Figure A1.**RA in two-stage decision-making when only two alternatives are left in the second stage. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A2.**RA in two-stage decision-making when the number of alternatives left in the second stage is three. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A3.**RA in two-stage decision-making when only two alternatives are left in the second stage for the different numbers of attributes. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A4.**RA in two-stage decision-making when three alternatives are left in the second stage for the different numbers of attributes. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A5.**MRA in two-stage decision-making when only two alternatives are left in the second stage for the different numbers of alternatives. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A6.**MRA in two-stage decision-making when three alternatives are left in the second stage. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A7.**MRA in two-stage decision-making when only two alternatives are left in the second stage for the different numbers of attributes. ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A8.**MRA in two-stage decision-making when three alternatives are left in the second stage for the different numbers of attributes. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

#### Appendix A.2. Worst-Choice Rate in Two-Stage Decision-Making

**Figure A9.**Worst-choice rate in two-stage decision-making when only two alternatives are left in the second stage. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A10.**Worst-choice rate in two-stage decision-making when three alternatives are left in the second stage. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A11.**Worst-choice rate in two-stage decision-making when only two alternatives are left in the second stage for the different numbers of attributes. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A12.**Worst choice rate in two-stage decision-making when three alternatives are left in the second stage for the different numbers of attributes. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

#### Appendix A.3. Best-Choice Rate in Two-Stage Decision-Making

**Figure A13.**Best-choice rate in two-stage decision-making when only two alternatives are left in the second stage. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A14.**Best choice rate in two-stage decision-making when three alternatives are left in the second stage. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A15.**Best-choice rate in two-stage decision-making when only two alternatives are left in the second stage for the different numbers of attributes. ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A16.**Best-choice rate in two-stage decision-making when three alternatives are left in the second stage for the different numbers of attributes. ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

#### Appendix A.4. Average EIPs in Two-Stage Decision-Making

**Figure A17.**Average EIPs in two-stage decision-making when only two alternatives are left in the second stage. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A18.**Average EIPs in two-stage decision-making when three alternatives are left in the second stage. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A19.**Average EIPs in two-stage decision-making when only two alternatives are left in the second stage for the different numbers of attributes. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure A20.**Average EIPs in two-stage decision-making when alternatives left in the second stage is three for the different numbers of attributes. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

## Appendix B. Experiment

Attribute | Level | ||
---|---|---|---|

Price | 60,000 yen | 80,000 yen | 100,000 yen |

Weight | 0.8 kg | 1.2 kg | 1.6 kg |

Battery | 5 h | 10 h | 15 h |

Warranty period | 1 year | 2 years | 3 years |

CPU | 1.5 GHz | 2.0 GHz | 2.5 GHz |

Memory | 4 GB | 8 GB | 16 GB |

**Figure A23.**Best-choice rate of the conjoint analysis and the weight estimation task for single- and two-stage decision-making. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

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**Figure 1.**Worst-choice rate in two-stage decision-making when the number of alternatives left in the second stage is two for the different number of alternatives. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure 2.**Average elementary information processes in two-stage decision-making when the number of alternatives left in the second stage is two for the different number of alternatives. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure 3.**Efficiency rates (relative accuracy ($RA$)/elementary information processes (EIP)) in two-stage decision-making when the number of alternatives left in the second stage is two for the different number of alternatives. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

**Figure 4.**Worst-choice rate of the conjoint analysis and weight estimation for the single- and two-stage decision strategies. Note: ADF: additive difference; CON: conjunctive; DIS: disjunctive; EBA: elimination by aspects; EQW: equal-weighted strategy; LEX: lexicographic; LEX-S: lexicographic semi-order; MCD: majority of confirming dimensions; STR: strategy; WAD: weighted additive.

${\mathit{p}}_{1}$ | ${\mathit{p}}_{2}$ | Type | Method | Mean | SD | 2.5% | 97.5% | ${\mathit{n}}_{\mathit{eff}}$ | $\widehat{\mathit{R}}$ |
---|---|---|---|---|---|---|---|---|---|

DIS | CON | Best | Conjoint | −0.216 | 0.107 | −0.423 | −0.005 | 6562 | 1.00 |

DIS | ADF | Best | Conjoint | −0.322 | 0.106 | −0.521 | −0.111 | 7962 | 1.00 |

DIS | LEX-WAD | Best | Conjoint | −0.218 | 0.105 | −0.419 | −0.011 | 5416 | 1.00 |

CON | 1/6 | Best | Conjoint | 0.318 | 0.085 | 0.152 | 0.487 | 4421 | 1.00 |

CON | 1/6 | Best | Weight | 0.178 | 0.077 | 0.034 | 0.334 | 4349 | 1.00 |

EBA | 1/6 | Best | Conjoint | 0.293 | 0.085 | 0.131 | 0.456 | 3464 | 1.00 |

LEX | 1/6 | Best | Conjoint | 0.266 | 0.081 | 0.113 | 0.430 | 4572 | 1.00 |

LEX | 1/6 | Best | Weight | 0.291 | 0.079 | 0.137 | 0.443 | 4085 | 1.00 |

ADF | 1/6 | Best | Conjoint | 0.387 | 0.079 | 0.230 | 0.539 | 4241 | 1.00 |

ADF | 1/6 | Best | Weight | 0.255 | 0.078 | 0.105 | 0.413 | 4527 | 1.00 |

EQW | 1/6 | Best | Conjoint | 0.254 | 0.080 | 0.103 | 0.420 | 4488 | 1.00 |

EQW | 1/6 | Best | Weight | 0.149 | 0.075 | 0.014 | 0.302 | 4099 | 1.00 |

EQW | 1/6 | Worst | Conjoint | −0.114 | 0.037 | −0.160 | −0.0024 | 4498 | 1.00 |

EQW | 1/6 | Worst | Weight | −0.113 | 0.037 | −0.160 | −0.022 | 5260 | 1.00 |

WAD | 1/6 | Best | Conjoint | 0.265 | 0.080 | 0.114 | 0.426 | 3725 | 1.00 |

WAD | 1/6 | Best | Weight | 0.213 | 0.079 | 0.065 | 0.372 | 4094 | 1.00 |

WAD | 1/6 | Worst | Weight | −0.112 | 0.037 | −0.159 | −0.018 | 5203 | 1.00 |

WAD | 1 | Best | Conjoint | −0.569 | 0.080 | −0.721 | −0.410 | 4202 | 1.00 |

WAD | 1 | Best | Weight | −0.621 | 0.079 | −0.768 | −0.462 | 3797 | 1.00 |

WAD | 1 | Worst | Conjoint | −0.919 | 0.044 | −0.983 | −0.816 | 4486 | 1.00 |

WAD | 1 | Worst | Weight | −0.946 | 0.037 | −0.994 | −0.855 | 4809 | 1.00 |

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## Share and Cite

**MDPI and ACS Style**

Takemura, K.; Tamari, Y.; Ideno, T.
Avoiding the Worst Decisions: A Simulation and Experiment. *Mathematics* **2023**, *11*, 1165.
https://doi.org/10.3390/math11051165

**AMA Style**

Takemura K, Tamari Y, Ideno T.
Avoiding the Worst Decisions: A Simulation and Experiment. *Mathematics*. 2023; 11(5):1165.
https://doi.org/10.3390/math11051165

**Chicago/Turabian Style**

Takemura, Kazuhisa, Yuki Tamari, and Takashi Ideno.
2023. "Avoiding the Worst Decisions: A Simulation and Experiment" *Mathematics* 11, no. 5: 1165.
https://doi.org/10.3390/math11051165