# An Experiment on Cooperation in a CPR Game with a Disapproval Option

## Abstract

**:**

## 1. Introduction

## 2. Experimental Design

#### 2.1. The Common Pool Resource (CPR) Game

#### 2.2. Treatments

## 3. Results

#### 3.1. Summary Statistics

#### 3.2. Detailed Results

#### 3.2.1. Cooperation under the Nash Threat

**Result 1**: The Nash threat reduces the over-extraction of the CPR and, therefore, increases cooperation among participants.

**Support for Result 1**: In order to support Result 1, we use the difference-in-differences (DiD) estimation method. The DiD procedure is used to measure the effectiveness of the treatment by comparing the difference between the control groups and the groups subjected to the Nash threat (the AM) before and after the introduction of the Nash threat. It allows us to control for group and temporal characteristics by isolating the real impact of the Nash threat in reducing the group over-extraction observed in the control groups. The first differences concern the differences in group extraction between sequence 2 and sequence 1 (S2–S1) of the $Control$ and $Introduction$ treatments (see columns (1) and (2) of Table 2). These values are $0.49$ and $-1.28$ for the $Control$ and $Introduction$, respectively. The second difference refers to the difference between S2–S1 of the treatments named Introduction and Control, i.e., $-1.28-0.49=-1.77$. The coefficient of interest in the DiD is highly significant, indicating that the Nash threat significantly reduces the over-extraction of the CPR, which is materialized by increasing the cooperation between subjects.

#### 3.2.2. The Driver of the Approval Decision

#### 3.2.3. The Order Effect of the Nash Threat

**Result 2**: Whatever the Nash threat is implemented in sequence 1 or in sequence 2, the group extraction is reduced, and therefore the cooperation increases among group members.

**Support for Result 2**: We performed the rank-sum test for the average group extractions by period in sequence 1 of the control treatment (CPR game) compared to those in sequence 1 of the withdrawal treatment (CPR game with AM). This rank-sum test (p-value = 0.000) shows that the average group extraction under the withdrawal treatment is lower than the average group extraction under the control treatment. Since there is no difference between sequences 1 and 2 under the control treatment (see the p-value of the signed-rank test in column 1 of Table 2), we also performed the rank-sum test for the average group extractions by period in sequence 2 of the control treatment (CPR game) versus those in sequence 2 of the introduction treatment (CPR game with AM). This second rank-sum test (p-value = 0.000) also shows that the average group extraction under the introduction treatment is lower than the average group extraction under the control treatment. According to the results of the above rank-sum tests and with respect to Result 1, we can conclude that regardless of whether the Nash threat is implemented in sequence 1 or in sequence 2, group extraction is reduced, and therefore the cooperation increases.

#### 3.2.4. Persistent Effect of the Nash Threat

**Result 3**: The Nash threat has a persistent effect on the reduction in the over-extraction of the CPR.

**Support for Result 3**: First, we conducted the rank-sum test with sequence 2 of the control treatment (CPR game) and sequence 2 of the withdrawal treatment (CPR game), because there is no difference between sequences 1 and 2 under the control treatment and under the withdrawal treatment, respectively (see p-values of the signed-rank test in columns 1 and 3 of Table 2). The rank-sum shows that the average group extraction is lower under sequence 2 of the withdrawal treatment than under sequence 2 of the control treatment. Second, we performed Equation (3), where $sequence$ is coded as 1 for sequence 2 of the withdrawal treatment and sequence 2 of the control treatment, and 0 for sequence 1 of the withdrawal treatment and sequence 1 of the control treatment. $period$ represents the periods under withdrawal and control treatments. $AM$ is coded as 1 for the withdrawal treatment and 0 for the control treatment. The variable $sequence\times AM$ is our variable of interest, capturing the effect of withdrawal of the Nash threat. The regression is reported in column 3 of Table 3. This coefficient is not significant (see column 3 of Table 3). Thus, withdrawing the Nash threat in sequence 2 does not significantly alter the average group extraction and cooperation level. For both reasons, we can conclude that the Nash threat has a persistent effect with regard to reducing CPR over-extraction.

## 4. Discussion and Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Instructions

#### Appendix A.1. Welcome

#### Appendix A.2. General Procedure

#### Appendix A.3. Types of Investments

#### Appendix A.3.1. Earnings Activity A

**This part is only for the set of treatments of Separated Payoff:**We present the different possibilities of earnings in activity A. They are described in the earnings Table (see Table A1). The first column corresponds to your investment in activity A (between 0 and 10). The other columns correspond to the other player’s investment in activity A (between 0 and 10). Your earnings in activity A and the other player’s earnings n activity A are measured in points. There are two values in each cell of the Table: Your total earnings in points (in blue) and the other player’s total earnings in points (in black). For example, you decide to invest 8 tokens in activity A and therefore 2 tokens in your activity B. The other player decides to invest 6 tokens in activity A and therefore 4 tokens in his activity B. Your earnings in activity A for the period is 300 points. The total earnings of the other player of your group are 225 points.

#### Appendix A.3.2. Earnings from the Investment in Activity B

#### Appendix A.3.3. Total Earnings

**This part is only for the set of treatments of Aggregated Payoff:**We present the different possibilities of total earnings. They are described in the earnings Table (see Table A2). The first column corresponds to your investment in activity A (between 0 and 10). The other columns correspond to the other player’s investment in activity A (between 0 and 10). Your total earnings and the other player’s earnings are measured in points. There are two values in each cell of the Table: Your total earnings in points (in blue) and the other player’s total earnings in points (in black). For example, you decide to invest 8 tokens in activity A and therefore 2 tokens in your activity B. The other player decides to invest 6 tokens in activity A and therefore 4 tokens in his activity B. Your total earnings for the period are 330 points. The total earnings of the other player of your group are 285 points.

#### Appendix A.4. Sequence Corresponding to the CPR Game

#### Appendix A.5. Sequences Corresponding to the CPR with AM (The Threat of Nash Equilibrium)

- total earning as well as the total earning of the other player.
**[under the treatments of aggregated payoff]** - earning of activity A and your earning of activity B as well as earnings of activity A and activity B of the other player.
**[under the treatments of separated payoff]**

Mate | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||

self | 0 | 0 ; 0 | 0 ; 66.75 | 0 ; 129 | 0 ; 186.75 | 0 ; 240 | 0 ; 288.75 | 0 ; 333 | 0 ; 372.75 | 0 ; 408 | 0 ; 438.75 | 0 ; 465 |

1 | 66.75 ; 0 | 64.5 ; 64.5 | 62.25 ; 124.5 | 60 ; 180 | 57.75 ; 231 | 55.5 ; 277.5 | 53.25 ; 319.5 | 51 ; 357 | 48.75 ; 390 | 46.5 ; 418.5 | 44.25 ; 442.5 | |

2 | 129 ; 0 | 124.5 ; 62.25 | 120 ; 120 | 115.5 ; 173.25 | 111 ; 222 | 106.5 ; 266.25 | 102 ; 306 | 97.5 ; 341.25 | 93 ; 372 | 88.5 ; 398.25 | 84 ; 420 | |

3 | 186.75 ; 0 | 180 ; 60 | 173.25 ; 115.5 | 166.5 ; 166.5 | 159.75 ; 213 | 153 ; 255 | 146.25 ; 292.5 | 139.5 ; 325.5 | 132.75 ; 354 | 126 ; 378 | 119.25 ; 397.5 | |

4 | 240 ; 0 | 231 ; 57.75 | 222 ; 111 | 213 ; 159.75 | 204 ; 204 | 195 ; 243.75 | 186 ; 279 | 177 ; 309.75 | 168 ; 336 | 159 ; 357.75 | 150 ; 375 | |

5 | 288.75 ; 0 | 277.5 ; 55.50 | 266.25 ; 106.5 | 255 ; 153 | 243.75 ; 195 | 232.5 ; 232.5 | 221.25 ; 265.5 | 210 ; 294 | 198.75 ; 318 | 187.5 ; 337.5 | 176.25 ; 352.5 | |

6 | 333 ; 0 | 319.5 ; 53.25 | 306 ; 102 | 292.5 ; 146.25 | 279 ; 186 | 265.5 ; 221.25 | 252 ; 252 | 238.5 ; 278.25 | 225 ; 300 | 211.5 ; 317.25 | 198 ; 330 | |

7 | 372.75 ; 0 | 357 ; 51 | 341.25 ; 97.5 | 325.5 ; 139.5 | 309.75 ; 177 | 294 ; 210 | 278.25 ; 238.5 | 262.5 ; 262.5 | 246.75 ; 282 | 231 ; 297 | 215.25 ; 307.5 | |

8 | 408 ; 0 | 390 ; 48.75 | 372 ; 93 | 354 ; 132.75 | 336 ; 168 | 318 ; 198.75 | 300 ; 225 | 282 ; 246.75 | 264 ; 264 | 246 ; 276.75 | 228 ; 285 | |

9 | 438.75 ; 0 | 418.5 ; 46.5 | 398.25 ; 88.5 | 378 ; 126 | 357.75 ; 159 | 337.5 ; 187.5 | 317.25 ; 211.5 | 297 ; 231 | 276.75 ; 246 | 256.5 ; 256.5 | 236.25 ; 262.5 | |

10 | 465 ; 0 | 442.5 ; 44.25 | 420 ; 84 | 397.5 ; 119.25 | 375 ; 150 | 352.5 ; 176.25 | 330 ; 198 | 307.5 ; 215.25 | 285 ; 228 | 262.5 ; 236.25 | 240 ; 240 |

Mate | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | ||

self | 0 | 150 ; 150 | 150 ; 201.75 | 150 ; 249 | 150 ; 291.75 | 150 ; 330 | 150 ; 363.75 | 150 ; 393 | 150 ; 417.75 | 150 ; 438 | 150 ; 453.75 | 150 ; 465 |

1 | 201.75 ; 150 | 199.5 ; 199.5 | 197.25 ; 244.5 | 195 ; 285 | 192.75 ; 321 | 190.5 ; 352.5 | 188.25 ; 379.5 | 186 ; 402 | 183.75 ; 420 | 181.5 ; 433.5 | 179.25 ; 442.5 | |

2 | 249 ; 150 | 244.5 ; 197.25 | 240 ; 240 | 235.5 ; 278.25 | 231 ; 312 | 226.5 ; 341.25 | 222 ; 366 | 217.5 ; 386.25 | 213 ; 402 | 208.5 ; 413.25 | 204 ; 420 | |

3 | 291.75 ; 150 | 285 ; 195 | 278.25 ; 235.5 | 271.5 ; 271.5 | 264.75 ; 303 | 258 ; 330 | 251.25 ; 352.5 | 244.5 ; 370.5 | 237.75 ; 384 | 231 ; 393 | 224.25 ; 397.5 | |

4 | 330 ; 150 | 321 ; 192.75 | 312 ; 231 | 303 ; 264.75 | 294 ; 294 | 285 ; 318.75 | 276 ; 339 | 267 ; 354.75 | 258 ; 366 | 249 ; 372.75 | 240 ; 375 | |

5 | 363.75 ; 150 | 352.5 ; 190.50 | 341.25 ; 226.5 | 330 ; 258 | 318.75 ; 285 | 307.5 ; 307.5 | 296.25 ; 325.5 | 285 ; 339 | 273.75 ; 348 | 262.5 ; 352.5 | 251.25 ; 352.5 | |

6 | 393 ; 150 | 379.5 ; 188.25 | 366 ; 222 | 352.5 ; 251.25 | 339 ; 276 | 325.5 ; 296.25 | 312 ; 312 | 298.5 ; 323.25 | 285 ; 330 | 271.5 ; 332.25 | 258 ; 330 | |

7 | 417.75 ; 150 | 402 ; 186 | 386.25 ; 217.5 | 370.5 ; 244.5 | 354.75 ; 267 | 339 ; 285 | 323.25 ; 298.5 | 307.5 ; 307.5 | 291.75 ; 312 | 276 ; 312 | 260.25 ; 307.5 | |

8 | 438 ; 150 | 420 ; 183.75 | 402 ; 213 | 384 ; 237.75 | 366 ; 258 | 348 ; 273.75 | 330 ; 285 | 312 ; 291.75 | 294 ; 294 | 276 ; 291.75 | 258 ; 285 | |

9 | 453.75 ; 150 | 433.5 ; 181.5 | 413.25 ; 208.5 | 393 ; 231 | 372.75 ; 249 | 352.5 ; 262.5 | 332.25 ; 271.5 | 312 ; 276 | 291.75 ; 276 | 271.5 ; 271.5 | 251.25 ; 262.5 | |

10 | 465 ; 150 | 442.5 ; 179.25 | 420 ; 204 | 397.5 ; 224.25 | 375 ; 240 | 352.5 ; 251.25 | 330 ; 258 | 307.5 ; 260.25 | 285 ; 258 | 262.5 ; 251.25 | 240 ; 240 |

**This value corresponds to the predicted symmetric Nash equilibrium of the CPR game**) and the rest of the 10 tokens is invested in activity B. Then the computer will display the investments (tokens in activities A and B respectively) and the total earnings.

#### Appendix A.6. Sequence 2 of Control Treatment

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Treatments | Control | Introduction | Withdrawal |
---|---|---|---|

Sequence 1 (S1) | CPR | CPR | CPR + Nash threat |

Sequence 2 (S2) | CPR | CPR + Nash threat | CPR |

Number of group | 17 | 30 | 19 |

Treatments | Control | Introduction | Withdrawal |
---|---|---|---|

(1) | (2) | (3) | |

sequence 1 (S1) | 15.74 | 15.35 | 14.34 |

sequence 2 (S2) | 16.23 | 14.07 | 14.81 |

S2–S1 | 0.49 | −1.28 | 0.47 |

p-value of signed-rank test | 0.168 | 0.012 ** | 0.153 |

p-value of K-S test | 0.994 | 0.002 *** | 0.418 |

Introduction | Withdrawal | ||
---|---|---|---|

(1) | (2) | (3) | |

$sequence\times AM$ | $-1.77$ *** | $-0.02$ | |

$\left(0.307\right)$ | $\left(0.307\right)$ | ||

$Mutual$$approval$ | $-3.06$ *** | ||

$\left(0.281\right)$ | |||

$sequence$ | $-0.14$ | 0.17 | |

(0.355) | (0.347) | ||

$period$ | $0.06$ | $-0.01$ | $0.03$ |

$\left(0.025\right)$ | $\left(0.019\right)$ | $0.026$ | |

constant | $15.14$ *** | $16.08$ *** | $14.82$ *** |

$\left(0.175\right)$ | $\left(0.346\right)$ | $\left(0.182\right)$ | |

$GroupFE$ | yes | yes | yes |

$Prob>F$ | $0.000$ | $0.000$ | $0.011$ |

$Observation$ | 940 | 300 | 720 |

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Yao, K.S.W.
An Experiment on Cooperation in a CPR Game with a Disapproval Option. *Games* **2021**, *12*, 83.
https://doi.org/10.3390/g12040083

**AMA Style**

Yao KSW.
An Experiment on Cooperation in a CPR Game with a Disapproval Option. *Games*. 2021; 12(4):83.
https://doi.org/10.3390/g12040083

**Chicago/Turabian Style**

Yao, Koffi Serge William.
2021. "An Experiment on Cooperation in a CPR Game with a Disapproval Option" *Games* 12, no. 4: 83.
https://doi.org/10.3390/g12040083