# A Laboratory Experiment for Analyzing Electors’ Strategic Behavior in a First-Past-the-Post System

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The First-Past-the-Post System in the World

#### 2.1. The First-Past-the-Post System in the United Kingdom

#### 2.2. The First-Past-the-Post System in the USA

## 3. Macrolevel Implications of the First-Past-the-Post System

#### 3.1. Assumptions

_{i}is the i-th element of the string formed after using the equation; x

_{i}is the i-th element of the original string.

_{i}is the i-th of the string formed after using the equation, y

_{i}is the i-th element of the string formed after using the first equation and n is the total number of constituencies.

^{2}distributions do not yield realistic results and they were removed from further testing. Table 6 provides the results of testing the different distributions. The algorithm was run 10,000 times and an average result was considered.

#### 3.2. Parlamentary Mandates Simulation

#### 3.3. Discussions

## 4. Methodology for Analyzing the Microlevel Behavior of Voters

#### 4.1. Assumptions

#### 4.2. Conducting the Experiment and the Analysis

- Card 1: 4*Square + 2*Circle − 2*Hexagon − 4*Empty Circle
- Card 2: 2*Square + 4*Circle + 2*Triangle − 2*Empty Circle
- Card 3: 2*Circle + 4*Triangle + 2*Hexagon
- Card 4: −2*Square + 2*Triangle + 4*Hexagon + 2*Empty Circle
- Card 5: −4*Square − 2*Circle + 2*Hexagon + 4*Empty Circle

## 5. Results of the Case Study on the Strategic Behavior

## 6. Concluding Remarks and Discussions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

## Appendix B

## Appendix C

Square | Circle | Triangle | Hexagon | Empty Circle | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | |

Iteration 2 | 2 | 1 | 3 | 0 | 2 | 1 | 3 | 0 | 2 | 1 |

Iteration 3 | 3 | 0 | 3 | 0 | 2 | 1 | 3 | 0 | 1 | 2 |

Iteration 4 | 3 | 0 | 3 | 0 | 2 | 1 | 3 | 0 | 2 | 1 |

Iteration 5 | 3 | 0 | 2 | 1 | 2 | 1 | 3 | 0 | 3 | 0 |

Iteration 6 | 3 | 0 | 3 | 0 | 1 | 2 | 3 | 0 | 2 | 1 |

Iteration 7 | 3 | 0 | 3 | 0 | 1 | 2 | 3 | 0 | 2 | 1 |

Square | Circle | Triangle | Hexagon | Empty Circle | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | |

Iteration 2 | 3 | 0 | 2 | 1 | 3 | 0 | 3 | 0 | 3 | 0 |

Iteration 3 | 3 | 0 | 3 | 0 | 2 | 1 | 3 | 0 | 3 | 0 |

Iteration 4 | 3 | 0 | 3 | 0 | 2 | 1 | 3 | 0 | 2 | 1 |

Iteration 5 | 3 | 0 | 2 | 1 | 3 | 0 | 3 | 0 | 3 | 0 |

Iteration 6 | 3 | 0 | 3 | 0 | 0 | 3 | 3 | 0 | 3 | 0 |

Iteration 7 | 3 | 0 | 2 | 1 | 0 | 3 | 3 | 0 | 3 | 0 |

Square | Circle | Triangle | Hexagon | Empty Circle | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | |

Iteration 2 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |

Iteration 3 | 2 | 1 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |

Iteration 4 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |

Iteration 5 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |

Iteration 6 | 3 | 0 | 1 | 2 | 0 | 3 | 3 | 0 | 2 | 1 |

Iteration 7 | 2 | 1 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |

Square | Circle | Triangle | Hexagon | Empty Circle | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | Choices over Expected Value | Choices under Expected Value * | |

Iteration 2 | 1 | 2 | 0 | 3 | 3 | 0 | 3 | 0 | 2 | 1 |

Iteration 3 | 3 | 0 | 3 | 0 | 2 | 1 | 1 | 2 | 3 | 0 |

Iteration 4 | 2 | 1 | 3 | 0 | 1 | 2 | 2 | 1 | 1 | 2 |

Iteration 5 | 2 | 1 | 1 | 2 | 1 | 2 | 3 | 0 | 2 | 1 |

Iteration 6 | 2 | 1 | 3 | 0 | 2 | 1 | 3 | 0 | 3 | 0 |

Iteration 7 | 2 | 1 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |

Iteration 8 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 | 3 | 0 |

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**Figure 1.**The evolution of the number of seats (red line) based on the initial score (green line) for the largest party.

**Figure 2.**The evolution of the number of seats (red line) based on the initial score (green line) of the last 4 parties.

Party | Percentage of Votes | Seats | Percentage of Seats |
---|---|---|---|

Conservative Party | 42.4% | 318 | 48.92% |

Labor Party | 40.0% | 262 | 40.30% |

Scottish National Party | 3.0% | 35 | 5.38% |

Liberal Democrat Party | 7.4% | 12 | 1.84% |

Party | Percentage of Votes | Seats | Percentage of Seats |
---|---|---|---|

Scottish National Party | 36.9% | 35 | 59.32% |

Conservative Party | 28.6% | 13 | 22.03% |

Labor Party | 27.1% | 7 | 12.06% |

Liberal Democrat Party | 6.8% | 4 | 6.77% |

**Table 3.**The results of 8 elections of the House of Representatives of the Congress of the United States of America.

Year | Democratic Party | Republican Party | ||
---|---|---|---|---|

Percentage of Votes | Percentage of Seats | Percentage of Votes | Percentage of Seats | |

1974 | 58.2% | 66.9% | 40.1% | 33.1% |

1976 | 56.7% | 67.1% | 41.7% | 32.9% |

1978 | 53.9% | 63.7% | 44.8% | 36.3% |

1980 | 50.6% | 55.9% | 47.9% | 44.1% |

1982 | 55.5% | 61.8% | 43.2% | 38.2% |

1984 | 52.4% | 58.2% | 46.8% | 41.8% |

1986 | 54.9% | 59.3% | 44.3% | 40.7% |

1988 | 53.7% | 59.8% | 45.2% | 40.2% |

Party | 1923 | 1924 | 1929 |
---|---|---|---|

Conservative Party | 5,514,541 | 7,854,523 | 8,656,225 |

Liberal Party | 4,301,481 | 3,035,257 | 5,308,738 |

Labour Party | 4,439,780 | 5,489,087 | 8,370,417 |

**Table 5.**Number of seats won by first three parties during the 1923, 1924 and 1929 British elections.

Party | 1923 | 1924 | 1929 |
---|---|---|---|

Conservative Party | 258 | 415 | 260 |

Liberal Party | 158 | 44 | 59 |

Labor Party | 191 | 151 | 287 |

Total | 607 | 610 | 606 |

**Table 6.**The results of the algorithm for the 1923, 1924 and 1929 British elections using 3 random distributions.

Distribution | Party | 1923 | 1924 | 1929 |
---|---|---|---|---|

Normal Distribution | Conservative Party | 342.3406 | 470.5549 | 306.2346 |

Liberal Party | 122.2653 | 5.0417 | 28.3289 | |

Labor Party | 142.3943 | 134.4034 | 271.4365 | |

Uniform Distribution | Conservative Party | 285.9101 | 374.7783 | 273.5152 |

Liberal Party | 153.1286 | 43.742 | 78.7223 | |

Labor Party | 167.9613 | 191.4797 | 253.7625 | |

Exponential Distribution | Conservative Party | 242.4447 | 310.449 | 243.0705 |

Liberal Party | 178.5607 | 91.7721 | 129.2914 | |

Labor Party | 185.9964 | 207.7789 | 233.6381 |

Distribution | 1923 | 1924 | 1929 |
---|---|---|---|

Normal Distribution | 59.86 | 40.33 | 33.26 |

Uniform Distribution | 21.08 | 32.94 | 23.63 |

Exponential Distribution | 15.16 | 74.02 | 51.88 |

**Table 8.**Average scores, number of seats and error of representation for each of the five simulated parties.

P1 | P2 | P3 | P4 | P5 | |
---|---|---|---|---|---|

Average Score | 0.408 | 0.276 | 0.192 | 0.114 | 0.007 |

Average Seats/1000 | 0.567 | 0.278 | 0.120 | 0.033 | 0.0003 |

Error of representation | 0.159 | 0.002 | −0.072 | −0.081 | −0.0067 |

Sets of Experiments | Extreme 1 | Moderate 1 | Center | Moderate 2 | Extreme 2 |
---|---|---|---|---|---|

Set 1 (used in pre-experiment phase) | Red | Orange | Green | Blue | Purple |

Set 2 (used in experiment phase) | Square | Circle | Triangle | Hexagon | Empty Circle |

Set 1 | Red | Orange | Green | Blue | Purple |
---|---|---|---|---|---|

Card 1 | 1 | 0 | −1 | −2 | −3 |

Card 2 | 0 | 1 | 0 | −1 | −2 |

Card 3 | −1 | 0 | 1 | 0 | −1 |

Card 4 | −2 | −1 | 0 | 1 | 0 |

Card 5 | −3 | −2 | −1 | 0 | 1 |

Set 2 | Square | Circle | Triangle | Hexagon | Empty Circle |
---|---|---|---|---|---|

Card 1 | 4 | 2 | 0 | −2 | −4 |

Card 2 | 2 | 4 | 2 | 0 | −2 |

Card 3 | 0 | 2 | 4 | 2 | 0 |

Card 4 | −2 | 0 | 2 | 4 | 2 |

Card 5 | −4 | −2 | 0 | 2 | 4 |

Winning Shape | Utility |
---|---|

Square | 0 |

Circle | 6 |

Triangle | 8 |

Hexagon | 6 |

Empty Circle | 0 |

Trial | Rational Choices | Irrational Choices |
---|---|---|

3 | 74 | 16 |

4 | 78 | 12 |

5 | 82 | 8 |

6 | 81 | 24 |

Iteration | Square | Circle | Triangle | Hexagon | Empty Circle | Total | First Two Options |
---|---|---|---|---|---|---|---|

1 | 2 | 2 | 6 | 4 | 1 | 15 | 10 |

2 | 1 | 2 | 7 | 4 | 1 | 15 | 11 |

3 | 2 | 2 | 6 | 4 | 1 | 15 | 10 |

4 | 0 | 5 | 5 | 4 | 1 | 15 | 10 |

5 | 0 | 5 | 4 | 4 | 2 | 15 | 9 |

6 | 0 | 7 | 2 | 5 | 1 | 15 | 12 |

7 | 1 | 6 | 1 | 6 | 1 | 15 | 12 |

Iteration | Square | Circle | Triangle | Hexagon | Empty Circle | Total | First Two Options |
---|---|---|---|---|---|---|---|

1 | 2 | 2 | 8 | 1 | 2 | 15 | 10 |

2 | 2 | 2 | 5 | 3 | 3 | 15 | 8 |

3 | 2 | 3 | 4 | 3 | 3 | 15 | 7 |

4 | 0 | 7 | 2 | 3 | 3 | 15 | 10 |

5 | 0 | 7 | 2 | 5 | 1 | 15 | 12 |

6 | 0 | 7 | 0 | 8 | 0 | 15 | 15 |

7 | 0 | 4 | 1 | 10 | 0 | 15 | 14 |

Iteration | Square | Circle | Triangle | Hexagon | Empty Circle | Total | First Two Options |
---|---|---|---|---|---|---|---|

1 | 1 | 0 | 4 | 4 | 6 | 15 | 10 |

2 | 3 | 3 | 3 | 4 | 2 | 15 | 7 |

3 | 1 | 2 | 5 | 6 | 1 | 15 | 11 |

4 | 6 | 0 | 3 | 4 | 2 | 15 | 10 |

5 | 0 | 6 | 4 | 4 | 1 | 15 | 10 |

6 | 5 | 1 | 1 | 8 | 0 | 15 | 13 |

7 | 5 | 0 | 6 | 4 | 0 | 15 | 11 |

Card Number | Card Type | Iteration 1 | Iteration 2 | Iteration 3 | Iteration 4 | Iteration 5 | Iteration 6 | Iteration 7 |
---|---|---|---|---|---|---|---|---|

1 | Empty Circle | Triangle | Empty Circle | Hexagon | Empty Circle | Hexagon | Hexagon | Hexagon |

2 | Empty Circle | Triangle | Empty Circle | Hexagon | Circle | Empty Circle | Hexagon | Hexagon |

3 | Empty Circle | Empty Circle | Empty Circle | Empty Circle | Empty Circle | Hexagon | Hexagon | Hexagon |

Card Number | Card Type | Iteration 1 | Iteration 2 | Iteration 3 | Iteration 4 | Iteration 5 | Iteration 6 | Iteration 7 |
---|---|---|---|---|---|---|---|---|

4 | Hexagon | Triangle | Triangle | Hexagon | Hexagon | Hexagon | Hexagon | Hexagon |

5 | Hexagon | Empty Circle | Hexagon | Triangle | Empty Circle | Hexagon | Hexagon | Hexagon |

6 | Hexagon | Triangle | Hexagon | Empty Circle | Hexagon | Triangle | Hexagon | Hexagon |

Card Number | Card Type | Iteration 1 | Iteration 2 | Iteration 3 | Iteration 4 | Iteration 5 | Iteration 6 | Iteration 7 |
---|---|---|---|---|---|---|---|---|

7 | Triangle | Triangle | Triangle | Empty Circle | Triangle | Triangle | Hexagon | Hexagon |

8 | Triangle | Hexagon | Triangle | Triangle | Hexagon | Circle | Hexagon | Hexagon |

9 | Triangle | Triangle | Triangle | Triangle | Triangle | Circle | Circle | Hexagon |

Card Number | Card Type | Iteration 1 | Iteration 2 | Iteration 3 | Iterations 4 | Iterations 5 | Iteration 6 | Iteration 7 |
---|---|---|---|---|---|---|---|---|

10 | Circle | Triangle | Circle | Circle | Circle | Circle | Circle | Circle |

11 | Circle | Square | Hexagon | Circle | Circle | Circle | Circle | Circle |

12 | Circle | Circle | Triangle | Square | Circle | Hexagon | Circle | Hexagon |

Card Number | Card Type | Iteration 1 | Iteration 2 | Iteration 3 | Iterations 4 | Iterations 5 | Iteration 6 | Iteration 7 |
---|---|---|---|---|---|---|---|---|

13 | Square | Triangle | Square | Triangle | Circle | Circle | Circle | Triangle |

14 | Square | Square | Circle | Circle | Circle | Circle | Circle | Circle |

15 | Square | Circle | Square | Square | Circle | Circle | Circle | Circle |

Card Number | Card Type | Iteration 1 | Iteration 2 | Iteration 3 | Iteration 4 | Iteration 5 | Iteration 6 | Iteration 7 | Expected Total |
---|---|---|---|---|---|---|---|---|---|

1 | Empty Circle | 0 | 4 | 2 | 4 | 2 | 2 | 2 | 16 |

2 | Empty Circle | 0 | 4 | 2 | −2 | 4 | 2 | 2 | 12 |

3 | Empty Circle | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 22 |

4 | Hexagon | 2 | 2 | 4 | 4 | 4 | 4 | 4 | 24 |

5 | Hexagon | 2 | 4 | 2 | 2 | 4 | 4 | 4 | 22 |

6 | Hexagon | 2 | 4 | 2 | 4 | 2 | 4 | 4 | 22 |

7 | Triangle | 4 | 4 | 0 | 4 | 4 | 2 | 2 | 20 |

8 | Triangle | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 18 |

9 | Triangle | 4 | 4 | 4 | 4 | 2 | 2 | 2 | 22 |

10 | Circle | 2 | 4 | 4 | 4 | 4 | 4 | 4 | 26 |

11 | Circle | 2 | 0 | 4 | 4 | 4 | 4 | 4 | 22 |

12 | Circle | 4 | 2 | 2 | 4 | 0 | 4 | 0 | 16 |

13 | Square | 0 | 4 | 0 | 2 | 2 | 2 | 0 | 10 |

14 | Square | 4 | 2 | 2 | 2 | 2 | 2 | 2 | 16 |

15 | Square | 2 | 4 | 4 | 2 | 2 | 2 | 2 | 18 |

Card Number | Card Type | Iteration 1 | Iteration 2 | Iteration 3 | Iteration 4 | Iteration 5 | Iteration 6 | Iteration 7 | Expected Total |
---|---|---|---|---|---|---|---|---|---|

1 | Empty Circle | 0 | 0 | 0 | −2 | −2 | 2 | 2 | 0 |

2 | Empty Circle | 0 | 0 | 0 | −2 | −2 | 2 | 2 | 0 |

3 | Empty Circle | 0 | 0 | 0 | −2 | −2 | 2 | 2 | 0 |

4 | Hexagon | 2 | 2 | 2 | 0 | 0 | 4 | 4 | 14 |

5 | Hexagon | 2 | 2 | 2 | 0 | 0 | 4 | 4 | 14 |

6 | Hexagon | 2 | 2 | 2 | 0 | 0 | 4 | 4 | 14 |

7 | Triangle | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 20 |

8 | Triangle | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 20 |

9 | Triangle | 4 | 4 | 4 | 2 | 2 | 2 | 2 | 20 |

10 | Circle | 2 | 2 | 2 | 4 | 4 | 0 | 0 | 14 |

11 | Circle | 2 | 2 | 2 | 4 | 4 | 0 | 0 | 14 |

12 | Circle | 2 | 2 | 2 | 4 | 4 | 0 | 0 | 14 |

13 | Square | 0 | 0 | 0 | 2 | 2 | −2 | −2 | 0 |

14 | Square | 0 | 0 | 0 | 2 | 2 | −2 | −2 | 0 |

15 | Square | 0 | 0 | 0 | 2 | 2 | −2 | −2 | 0 |

Number | Card Type | Expected Total | Obtained Total | Difference |
---|---|---|---|---|

1 | Empty Circle | 16 | 0 | −16 |

2 | Empty Circle | 12 | 0 | −12 |

3 | Empty Circle | 22 | 0 | −22 |

4 | Hexagon | 24 | 14 | −10 |

5 | Hexagon | 22 | 14 | −8 |

6 | Hexagon | 22 | 14 | −8 |

7 | Triangle | 20 | 20 | 0 |

8 | Triangle | 18 | 20 | 2 |

9 | Triangle | 22 | 20 | −2 |

10 | Circle | 26 | 14 | −12 |

11 | Circle | 22 | 14 | −8 |

12 | Circle | 16 | 14 | −2 |

13 | Square | 10 | 0 | −10 |

14 | Square | 16 | 0 | −16 |

15 | Square | 18 | 0 | −18 |

**Table 25.**The number of cases in which the participants changed their option and the number of cases in which they did not change their options based on the utility received at the previous iteration.

Utility Received at the Previous Iteration | Number of Participants Who Changed Their Vote | Number of Participants Who Did Not Change Their Vote | Total |
---|---|---|---|

−2 | 5 | 4 | 9 |

0 | 16 | 11 | 27 |

2 | 18 | 18 | 36 |

4 | 6 | 12 | 18 |

Total | 45 | 45 | 90 |

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**MDPI and ACS Style**

Chiriță, A.; Delcea, C.
A Laboratory Experiment for Analyzing Electors’ Strategic Behavior in a First-Past-the-Post System. *Symmetry* **2020**, *12*, 1081.
https://doi.org/10.3390/sym12071081

**AMA Style**

Chiriță A, Delcea C.
A Laboratory Experiment for Analyzing Electors’ Strategic Behavior in a First-Past-the-Post System. *Symmetry*. 2020; 12(7):1081.
https://doi.org/10.3390/sym12071081

**Chicago/Turabian Style**

Chiriță, Andrei, and Camelia Delcea.
2020. "A Laboratory Experiment for Analyzing Electors’ Strategic Behavior in a First-Past-the-Post System" *Symmetry* 12, no. 7: 1081.
https://doi.org/10.3390/sym12071081