# Learning to Set the Reserve Price Optimally in Laboratory First Price Auctions

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Benchmark

## 3. Experimental Design

## 4. Results

#### 4.1. Expected Revenues

#### 4.2. Reserve Prices Chosen

#### 4.3. Univariate Treatment Comparisons

#### 4.4. Average Revenues and Reserve Prices Chosen

#### 4.5. Obtaining Information through Practice

#### 4.6. Multivariate Treatment Comparisons

## 5. Behavior of Less Experienced Sellers

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. Instructions

#### Appendix A.1. Instructions for 2L and 2H

#### Appendix A.2. Instructions for 4L and 4H

## References

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1. | Our study therefore focuses on predictable bidders, with sellers aware of the bidding rule. Automated bidding and informed sellers are standard design elements in the literature. These features simplify the problem of reserve price determination, which in practice may depend on a seller’s belief about the rule, by removing the strategic element. For a related experiment with non-automated bidding, see Chen, Katuščák and Ozdenoren (2010) [10]. |

2. | The purchasing power parity exchange rate between the Indian Rupee and the U.S. dollar for 2010 was 16.8 rupees to a dollar according to the Penn World Tables (Heston, Summers and Aten 2012) [11]. |

3. | Since subjects knew the formula being used to generate bids, they could be viewed as simultaneously sampling from conditional lotteries and choosing the value of the conditioning variable (the reserve price). |

4. | Forty subjects had registered for each session, but there were last minute no-shows in all sessions except 405, yielding 151 subjects in total. |

5. | Averages in 205 were significantly lower than in 415, while those in 405 and 215 were indistinguishable. |

6. | Kolmogorov–Smirnov and Kruskal–Wallis tests also indicated that the samples generated by these two treatments came from the same distribution and population, respectively. Kolmogorov–Smirnov gave an exact p-value of 0.121, and the Kruskal–Wallis chi-squared test with ties gave a p-value of 0.2334. |

7. | Evidence of an increasing relationship between reserve price chosen and the number of bidders was also found for second-price auctions by Davis, Katok and Kwasnica (2011) [3] using the standard protocol with a fixed number of rounds. |

8. | Variances were not insignificantly different at the 10% level for any other treatment pair. |

9. | The mean across Treatments 205 and 405 was 83.6, higher compared to 60, the maximum number of rounds permitted in prior studies. |

10. | If an increase in the number of bidders is interpreted as an increase in the complexity of the problem, then a possible explanation for the observed relationship between the number of bidders and the number of choices could be that increased complexity prompts more information gathering (more practice), which is infeasible when the subject is resource-constrained (the experience phase is short). |

11. | $w(.)$ can also be interpreted as a function representing errors in calculations of probability. |

12. | See Ingersoll (2008) [19] on problems of non-monotonicity associated with the TK function for low values of $\alpha $. |

13. | The differences in results between the current study and prior ones should be interpreted with caution however, as prior studies differed from the present one on more than one dimension. |

**Figure 1.**Screenshot. The horizontal axis is reserve price, and the vertical axis is revenue. The black and blue dots respectively represent actual and average revenue for a given reserve price. The black dots on the horizontal axis represent outcomes with zero revenue.

Treatment | 205 | 405 | 215 | 415 |

Mean | 31.73 | 40.4 | 45.89 | 49.23 |

t-Test | <0.0001 *** | 0.0028 ** | 0.1466 | 0.8142 |

Median | 35 | 42 | 47 | 53 |

Wilcoxon Test | <0.0001 *** | 0.0082 ** | 0.1294 | 0.9055 |

Comparison | 205/405 | 205/215 | 405/415 | 215/415 |

t-Test | 0.0379 * | 0.0005 *** | 0.0497 * | 0.4412 |

Mann–Whitney Test | 0.0321 * | 0.0005 *** | 0.0442 * | 0.2334 |

Treatment | 205 | 405 | 215 | 415 |

Mean | 90.5 | 77.2 | 126.3 | 277.7 |

Median | 85 | 63 | 131 | 223 |

Comparison | 205/405 | 205/215 | 405/415 | 215/415 |

t-Test | 0.1285 | <0.0001 *** | <0.0001 *** | <0.0001 *** |

Mann-Whitney Test | 0.1094 | 0.0005 *** | <0.0001 *** | <0.0001 *** |

Comparison | 205/405 | 205/215 | 405/415 | 215/415 |

Treatment Dummy | 8.3982 * (4.1914) | 14.7005 ** (4.5823) | 5.2955 (6.1263) | 0.3087 (5.3058) |

Number of Practice Choices | −0.0204 (0.0548) | −0.0152 (0.0657) | 0.0176 (0.0211) | 0.02 (0.0203) |

Constant | 33.5725 *** (5.7846) | 33.1075 *** (6.5463) | 39.0403 *** (3.5157) | 43.354 *** (4.0536) |

Dependent Variable | Final Reserve | No. of Practice Choices |
---|---|---|

dumN (0 if $N=2$, 1 if $N=4$) | 5.713 (3.1892) | 67.3519 *** (14.6553) |

dumphase (0 if 5 min, 1 if 15 min) | 10.736 ** (3.6132) | 122.0284 *** (14.6424) |

number of practice choices | 0.0052 (0.0167) | - (-) |

constant | 32.8324 *** (2.7234) | 48.5834 *** (12.7681) |

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## Share and Cite

**MDPI and ACS Style**

Banerjee, P.; Khare, S.; Srikant, P. Learning to Set the Reserve Price Optimally in Laboratory First Price Auctions. *Games* **2018**, *9*, 79.
https://doi.org/10.3390/g9040079

**AMA Style**

Banerjee P, Khare S, Srikant P. Learning to Set the Reserve Price Optimally in Laboratory First Price Auctions. *Games*. 2018; 9(4):79.
https://doi.org/10.3390/g9040079

**Chicago/Turabian Style**

Banerjee, Priyodorshi, Shashwat Khare, and P. Srikant. 2018. "Learning to Set the Reserve Price Optimally in Laboratory First Price Auctions" *Games* 9, no. 4: 79.
https://doi.org/10.3390/g9040079