# Conditional Cooperation and the Marginal per Capita Return in Public Good Games

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Linear Public Good Game

_{i}∈ [0,E] denote the contribution of member $i$ and let X = ∑

_{i}x

_{i}denote total contributions. Total contributions to the public good are multiplied by factor M > 0 and split evenly amongst group members. Let m = M/n enote the marginal per capita return (MPCR) on the public good. The final ‘monetary’ payoff of member i is given by

## 3. Theory

#### 3.1. Conditional Cooperation

**Hypothesis 1**: (a) The proportion of conditional cooperators is independent of the MPCR. (b) The cooperation factor of conditional cooperators is increasing in the MPCR.

**Hypothesis 2**: (a) The proportion of conditional cooperators is increasing in the MPCR. (b) The cooperation factor of conditional cooperators does not depend on the MPCR.

**Hypothesis 3**: The average cooperation factor is, ceteris paribus, lower in a leader-follower game than follower-average game.

#### 3.2. Leader Contribution

**Hypothesis 4**: There exists a critical value of the MPCR above which the leader has an incentive to contribute the full endowment towards the public good.

**Hypothesis 5**: The critical value above which a leader has an incentive to contribute is higher in a follower-average game than leader-follower game.

**Hypothesis 6**: The critical value above which a leader has an incentive to contribute is higher for a conditional cooperator than free-rider.

## 4. Experimental Design

Session | Type of Game | Part 1 | Part 2 | Part 3 | Subjects |
---|---|---|---|---|---|

1 | Follower-average | m = 0.4 | m = 0.8 | m = 0.2 | 31 |

2 | Leader-follower | m = 0.4 | m = 0.8 | m = 0.2 | 25 |

3 | Leader-follower | m = 0.2 | m = 0.4 | m = 0.8 | 19 |

4 | Leader-follower | m = 0.8 | m = 0.2 | m = 0.4 | 21 |

## 5. Experimental Results

**Figure 1.**Average total contributions in each of the six games distinguishing the role played by leaders and followers.

#### 5.1. Follower Behavior

#### 5.1.1. Proportion of Behavior Types

Leader-Follower Game | Follower-Average Game | |||||
---|---|---|---|---|---|---|

All Parts | m = 0.2 | m = 0.4 | m = 0.8 | m = 0.2 | m = 0.4 | m = 0.8 |

Conditional Cooperator | 47.69 | 63.08 | 53.85 | 41.94 | 70.97 | 70.97 |

Free-rider | 30.77 | 15.38 | 18.46 | 41.94 | 6.45 | 9.68 |

Hump-shaped | 3.08 | 3.08 | 6.15 | 0.00 | 3.23 | 3.23 |

Other | 18.46 | 18.46 | 21.54 | 16.13 | 19.35 | 16.13 |

Part 1 | ||||||

Conditional Cooperator | 47.37 | 72.00 | 66.67 | 70.97 | ||

Free-rider | 21.05 | 12.00 | 0.00 | 6.45 | ||

Hump-shaped | 5.26 | 4.00 | 4.76 | 3.23 | ||

Other | 26.32 | 12.00 | 28.57 | 19.35 |

**Table 3.**Results of two probit regressions with conditional cooperator as dependent variable, using the data from part 1 or the full sample. A panel regression is used for all parts.

Part 1 | All Parts | All Parts | |
---|---|---|---|

Constant | 0.552 ** (0.238) | 1.173 *** (0.416) | 1.068 ** (0.476) |

0.2 Return | −0.649 * (0.392) | −0.709 *** (0.272) | −0.839 (0.602) |

0.8 Return | −0.152 (0.389) | −0.187 (0.271) | 0.507 (0.474) |

Leader-follower Game | 0.030 (0.359) | −0.326 (0.419) | −0.095 (0.560) |

0.2 Return * Leader-follower | - | - | 0.139 (0.638) |

0.8 Return * Leader-follower | - | - | −0.905 * (0.519) |

Part 2 | - | −0.311 (0.277) | −0.507 * (0.303) |

Part 3 | - | −0.647 ** (0.301) | −0.613 * (0.338) |

Number of Observations | 96 | 288 | 288 |

Wald χ^{2} | 3.48 | 17.45 *** | 19.33 *** |

Pseudo R^{2} | 0.03 | - | - |

#### 5.1.2. Cooperation Factor

Leader-Follower Game | Follower-Average Game | |||||
---|---|---|---|---|---|---|

m = 0.2 | m = 0.4 | m = 0.8 | m = 0.2 | m = 0.4 | m = 0.8 | |

All Parts | 0.619 | 0.633 | 0.695 | 0.509 | 0.798 | 0.834 |

Part 1 | 0.773 | 0.642 | 0.731 | - | 0.798 | - |

Part 1 | Entire Sample | |||
---|---|---|---|---|

Cooperation Factor | All Subjects | Conditional Cooperators | All Subjects | Conditional Cooperators |

Constant | 0.643 *** (0.149) | 0.904 *** (0.110) | 0.606 *** (0.155) | 0.938 *** (0.104) |

0.2 Return | −0.221 (0.278) | 0.210 (0.194) | −0.483 ** (0.192) | −0.180 (0.116) |

0.8 Return | 0.062 (0.237) | 0.097 (0.163) | 0.175 (0.186) | 0.160 (0.107) |

Leader-follower Game | −0.235 (0.219) | −0.197 (0.148) | −0.140 (0.198) | −0.149 (0.130) |

0.2 Return * Leader-follower | - | - | 0.241 (0.217) | 0.167 (0.134) |

0.8 Return * Leader-follower | - | - | −0.250 (0.212) | −0.074 (0.121) |

Part 2 | - | - | −0.241 ** (0.109) | −0.132 * (0.067) |

Part 3 | - | - | −0.229 ** (0.109) | −0.090 (0.069) |

Number of observations | 96 | 63 | 288 | 164 |

F-test | 1.00 | 0.73 | 35.95 *** | 20.30 *** |

Pseudo R^{2} | 0.017 | 0.024 | - | - |

#### 5.2. Leader Behavior

Leader-Follower Game | Follower-Average Game | |||||
---|---|---|---|---|---|---|

All parts | m = 0.2 | m = 0.4 | m = 0.8 | m = 0.2 | m = 0.4 | m = 0.8 |

Overall average | 3.85 | 6.34 | 8.49 | 2.84 | 7.23 | 9.97 |

Conditional cooperator | 5.13 | 6.90 | 9.31 | 3.54 | 6.68 | 11.18 |

Free-rider | 0.35 | 2.60 | 3.25 | 2.15 | 2.50 | 1.00 |

Hump-shaped | 7.50 | 7.50 | 8.75 | n/a | 10.00 | 8.00 |

Other | 5.75 | 7.33 | 10.86 | 2.80 | 10.33 | 10.40 |

Part 1 | ||||||

Overall average | 4.58 | 4.64 | 9.05 | 7.23 | ||

Conditional cooperator | 5.22 | 5.22 | 9.00 | 6.68 | ||

Free-rider | 1.50 | 0.00 | n/a | 2.50 | ||

Hump-shaped | 5.00 | 3.00 | 5.00 | 10.00 | ||

Other | 5.80 | 6.33 | 9.83 | 10.33 |

**Table 7.**The results of Tobit regressions with unconditional contribution as a proportion of total endowment as the dependent variable. A panel regression is used for all parts.

Unconditional Contribution | Part 1 | All Parts | All Parts |
---|---|---|---|

Constant | 0.318 *** (0.097) | 0.142 (0.099) | 0.176 * (0.104) |

Conditional Cooperator | 0.031 (0.091) | 0.232 *** (0.065) | 0.230 *** (0.066) |

0.2 Return | 0.041 (0.118) | −0.223 *** (0.059) | −0.348 *** (0.117) |

0.8 Return | 0.340 ** (0.153) | 0.163 *** (0.057) | 0.134 (0.111) |

Leader-follower Game | −0.220 * (0.121) | −0.033 (0.096) | −0.096 (0.119) |

0.2 Return * Leader-follower | - | - | 0.167 (0.133) |

0.8 Return * Leader-follower | - | - | 0.039 (0.126) |

Part 2 | - | 0.034 (0.057) | 0.044 (0.065) |

Part 3 | - | −0.024 (0.058) | 0.016 (0.066) |

Number of Observations | 96 | 288 | 288 |

LR χ^{2} test/Wald χ^{2} | 8.79* | 71.24 *** | 73.07 *** |

Pseudo R^{2} | 0.061 |

## 6. Conclusions

## Acknowledgements

## Appendix

## Author Contributions

## Conflicts of Interest

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^{1}The idea of a ‘representative’ other player is not new. See, for example, chapter 7 of Camerer [17].^{2}Everyone contributing 0 is the unique Nash equilibrium of both the leader-follower and follower-average game.^{3}If ${\rho}_{n}=0$ then set ${c}_{n}\left(L\right)=0$.^{4}This claim is only true if ${c}_{n}\left(E\right)>0$. But, this will be the case except for relatively extreme parameter values. For instance, the claim is valid for the games used in our experiments.^{5}The only exception is if a follower believes other followers will give more than the leader. One could, for instance, obtain an equilibrium where the leader contributes 0 but followers contribute a positive amount because they believe other followers will contribute a positive amount. Hypothesis 3 is based on the assumption that a follower believes other followers will contribute (weakly) less than the leader.^{6}In the Falk and Fischbacher [12] model of sequential reciprocity the leader can anticipate intentional unkindness from contributing a positive amount. This lessens the incentive to contribute the stronger are her reciprocal preferences.^{7}See Fischbacher, Schudy and Teyssier [7] for a similar approach. Subjects were given feedback on all three parts of the experiment.^{8}To explain how this is calculated consider the follower-average game. If we randomly select three subjects and take the average of their unconditional contribution we have the average leader contribution. If we randomly select a fourth subject and look at their contribution table we have the follower contribution. Combining these gives the total contribution. Repeating this exercise for all possible permutations of subjects gives the expected total contribution. Data from all parts of a session is used in deriving Figure 1.^{9}This criterion is different from that introduced in Section 3 but clearly in the same spirit.^{10}A similar order effect was observed by Fischbacher, Schudy and Teyssier [7].^{11}An alternative explanation is that choices in parts 2 and 3 were influenced by the game played in part 1. We shall see, however, that there was a basic shift towards lower contributions in parts 2 and 3 irrespective of the game played in part 1. This does not suggest a priming effect. Moreover, ‘learning’ effects have been observed from playing the same game many times with no feedback (e.g., [27]).^{12}An efficiency seeking follower will contribute a positive amount irrespective of the average leader contribution when the MPCR is 0.4 or 0.8. None of our subjects did that. Neither did any of those in Fischbacher et al. [10]. Note that prior evidence of efficiency seeking preferences (e.g., [28]) is in a context where efficiency can be enhanced without sacrifice in own material payoff.^{13}The MPCR reflects the ‘price’ of making a mistake, reducing inequality, reciprocating etc. So, a decrease in the MPCR may influence, say, the decision to reciprocate. But, there is no ‘discontinuity’ around the MPCR at which contributing becomes socially efficient, m = 0.25 in our case. This is captured in the analysis of Section 3.^{14}It is also reasonable that she would have preferred the leaders to contribute zero in order to avoid a costly round of ‘gift giving’!^{15}Recall that in part 1 we only have data for the leader-follower game with an MPCR of 0.4.^{16}We cannot reject the null that the part 2 and part 3 coefficients are the same (p = 0.70, F-test).^{17}If we look at the net-effect then there is no significant difference between the follower-average game with an MPCR of 0.8 and leader-follower game with an MPCR of 0.4 (p = 0.40, F-test) but there is a difference the follower-average game with an MPCR of 0.8 and leader-follower game with an MPCR of 0.8 (p = 0.09).^{18}Of the remaining subjects: 24 were a conditional cooperator for at least one game while 17 were not a conditional cooperator in any game. Only 6 subjects were classified as a free-rider in all three games.^{19}For each subject and each game we have 21 data points from his or her completion of the contribution table (because the average leader contribution can range from 0 to E). When own contribution is regressed against leader contribution (and a constant term) the cooperation factor ${\theta}_{i}(\text{\Gamma})$ is given by the coefficient on leader contribution.^{20}The difference between a marginal return of 0.2 and 0.8 is highly significant (p < 0.01) for both the leader-follower and follower-average game.^{21}That is, we look at expected payoff for all possible values of unconditional contribution given the actual conditional contributions of our subjects.^{22}There was one free-rider who contributed 20 as leader when the MPCR was 0.8, but this was clearly the exception. The overall proportion of subjects making an unconditional contribution of 20 was around 6%, 12% and 19% for an MPCR of 0.2, 0.4 and 0.8 respectively.^{23}We focus here on the treatments of most direct relevance to us, i.e. those with exogenous leadership, symmetric returns from the public good, no exclusionary power and complete information.^{24}As an intermediate step we can write the standard formulation$${u}_{i}^{s}\left({x}_{1},\dots ,{x}_{n}\right)=E-{x}_{i}+mX-{\alpha}_{i}\frac{1}{n-1}{\displaystyle \sum}_{j\ne i}max\left\{0,{u}_{j}-{u}_{i}\right\}-{\beta}_{i}\frac{1}{n-1}{\displaystyle \sum}_{j\ne i}max\left\{0,{u}_{i}-{u}_{j}\right\}$$^{25}This would be inaccurate as she can influence inequality relative to the other followers through her contribution. Nevertheless it may approximate how a follower would behave.^{26}This comes from$${u}_{i}^{s}=E-{x}_{i}-{\alpha}_{i}\left(1-p\right){x}_{i}-{\beta}_{i}p\left(\frac{n-2}{n-1}\right)\left(E-{x}_{i}\right)-{\beta}_{i}p\left(\frac{1}{n-1}\right)\left(\frac{{x}_{i}+pE(n-2)}{n-1}-{x}_{i}\right)+m\left({x}_{i}+p\left(n-2\right)E+p\left(\frac{{x}_{i}+pE(n-2)}{n-1}\right)\right).$$

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**MDPI and ACS Style**

Cartwright, E.J.; Lovett, D. Conditional Cooperation and the Marginal per Capita Return in Public Good Games. *Games* **2014**, *5*, 234-256.
https://doi.org/10.3390/g5040234

**AMA Style**

Cartwright EJ, Lovett D. Conditional Cooperation and the Marginal per Capita Return in Public Good Games. *Games*. 2014; 5(4):234-256.
https://doi.org/10.3390/g5040234

**Chicago/Turabian Style**

Cartwright, Edward J., and Denise Lovett. 2014. "Conditional Cooperation and the Marginal per Capita Return in Public Good Games" *Games* 5, no. 4: 234-256.
https://doi.org/10.3390/g5040234