# Instrumental Reciprocity as an Error

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## Abstract

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## 1. Introduction

## 2. Experimental Design and Procedures

#### 2.1. Experimental Game

#### 2.2. Treatments

#### 2.3. Procedures

## 3. Results

#### 3.1. Cooperation and Reciprocation Rates

#### 3.2. Strategies of Second Movers

#### 3.3. Is Instrumental Reciprocity a Simple Mistake?

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## References

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1. | |

2. | A general reputation-building argument need not specify why cooperative types choose to cooperate. One of the most common explanations is that some players have social preferences and that is why they cooperate (e.g., as argued by Andreoni and Miller [9] and Camerer and Fehr [10]). However, the same logic applies if cooperative types are cooperating due to other reasons, such as inability to backward induct [1], having naive prior beliefs [11], or because they are prone to make mistakes [12]. |

3. | |

4. | |

5. | In Section A of the Supplementary we show that always defect dominates all the other strategies of second movers. |

6. | Throughout the results section, we report p-values from regressions used to test whether the frequency of various strategies and actions significantly differ. In all regressions, we cluster standard errors on sessions since errors may be correlated because participants are randomly re-matched within sessions. Section B of the Supplementary contains the output of all regressions and the precise description of each regression. Given that there is some concern about session effects and clustering in laboratory experiments [19], we checked whether our results hold if instead we cluster standard errors on subjects. We find that they do (see the Supplementary for details). Finally, the Supplementary also contains the results of the equivalent nonparametric tests. |

7. | This is well in line with evidence from infinitely repeated prisoner’s dilemmas showing that tit-for-tat is one of the most common cooperative strategies [20]. |

8. | The same is true for always reciprocate. It has a lower expected payoff than the strategy ddcd, but it is used considerably more often. |

9. | For example, the payoff difference between reciprocate then defect and always reciprocate in a SPD with a continuation probability of 0.50 and a payoff of mutual cooperation of 37 points, as in SPD-High, can also be attained in a SPD with a continuation probability of 0.23 and a payoff of mutual cooperation of 44 points, or a SPD with a continuation probability of 0.77 and a payoff of mutual cooperation of 30 points. |

10. | Consider the example where second movers believe that their matched first mover is a reciprocator with certainty (i.e., ${p}_{1}=1$, ${p}_{2}^{cc}=1$ and ${p}_{2}^{cd}=0$). Second movers with this belief and with Fehr-Schmidt preference $\beta \in [0.03,0.31]$ in SPD-High or $\beta \in [0.37,0.48]$ in SPD-Low, derive a higher expected utility from reciprocate then defect (which gives second movers an expected utility of $EU=c+0.5(t-\beta (t-s\left)\right)$) than from both always defect (which gives $EU=t-\beta (t-s)+0.5d$) and always reciprocate (which gives $EU=c+0.5c$). Second movers with $\beta >0.31$ in SPD-High or $\beta >0.48$ in SPD-Low derive a higher expected utility from always reciprocate, and second movers with $\beta <0.03$ in SPD-High or $\beta <0.37$ in SPD-Low derive a higher expected utility from always defect. |

11. | Alternatively, one can always assume that second movers hold out-of-equilibrium beliefs, in which case it is not hard to find beliefs for which the three types choose different strategies (e.g., see footnote 10). However, making this assumption makes this explanation quite similar to simply assuming that different individuals use different social heuristics. |

**Figure 1.**Distribution of observed strategies of second movers. Note: The figure shows distributions of all observed strategies (the bar on the left) and also zooms in on the distributions of observed strategies that involve some cooperation by second movers (the bar on the right). The label “inexperienced” refers to behavior when players play the experimental game for the first time. The label “experienced” refers to behavior in the last five repetitions of the experiment.

First Mover’s Action In: | Period 1 | Period 2 | ||
---|---|---|---|---|

$\mathit{c}$ | $\mathit{d}$ | $\mathit{c}$ | $\mathit{d}$ | |

Second mover’s strategies: | ||||

always defect | d | d | d | d |

reciprocate then defect | c | d | d | d |

d | c | d | d | |

d | d | c | d | |

d | d | d | c | |

c | c | d | d | |

always reciprocate | c | d | c | d |

c | d | d | c | |

d | c | c | d | |

d | c | d | c | |

d | d | c | c | |

c | c | c | d | |

c | c | d | c | |

c | d | c | c | |

d | c | c | c | |

always cooperate | c | c | c | c |

Period | SPD-Low | SPD-High | ||||||
---|---|---|---|---|---|---|---|---|

Inexperienced | Experienced | Inexperienced | Experienced | |||||

1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 | |

If first mover cooperates | 23% | 23% | 21% | 19% | 31% | 19% | 35% | 19% |

If first mover defects | 6% | 9% | 2% | 1% | 11% | 11% | 3% | 3% |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Reuben, E.; Suetens, S.
Instrumental Reciprocity as an Error. *Games* **2018**, *9*, 66.
https://doi.org/10.3390/g9030066

**AMA Style**

Reuben E, Suetens S.
Instrumental Reciprocity as an Error. *Games*. 2018; 9(3):66.
https://doi.org/10.3390/g9030066

**Chicago/Turabian Style**

Reuben, Ernesto, and Sigrid Suetens.
2018. "Instrumental Reciprocity as an Error" *Games* 9, no. 3: 66.
https://doi.org/10.3390/g9030066