# A Note on Buyers’ Behavior in Auctions with an Outside Option

## Abstract

**:**

## 1. Introduction

## 2. The Model

## 3. Comparing Cutoff Strategies

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A. The Derivative of the Cutoff Function

## References

- Etzion, H.; Pinker, E.; Seidmann, A. Analyzing the simultaneous use of auctions and posted prices for online selling. Manuf. Serv. Oper. Manag.
**2006**, 8, 68–91. [Google Scholar] [CrossRef] - Caldentey, R.; Vulcano, G. Online auction and list price revenue management. Manag. Sci.
**2007**, 53, 795–813. [Google Scholar] [CrossRef] [Green Version] - Sun, D. Dual mechanism for an online retailer. Eur. J. Oper. Res.
**2008**, 187, 903–921. [Google Scholar] [CrossRef] - Etzion, H.; Moore, S. Managing online sales with posted price and open-bid auctions. Decis. Support Syst.
**2013**, 54, 1327–1339. [Google Scholar] [CrossRef] - Hummel, P. Simultaneous use of auctions and posted prices. Eur. Econ. Rev.
**2015**, 78, 269–284. [Google Scholar] [CrossRef] - Reynolds, S.S.; Wooders, J. Auctions with a buy price. Econ. Theory
**2009**, 38, 9–39. [Google Scholar] [CrossRef] - Mathews, T.; Katzman, B. The role of varying risk attitudes in an auction with a buyout option. Econ. Theory
**2006**, 27, 597–613. [Google Scholar] [CrossRef] - Anderson, S.; Friedman, D.; Milam, G.; Singh, N. Buy It Now: A Hybrid Internet Market Institution. J. Electron. Commerce Res.
**2008**, 9, 137–153. [Google Scholar] [CrossRef] [Green Version] - Chen, J.R.; Chen, K.P.; Chou, C.F.; Huang, C.I. A Dynamic Model of Auctions with Buy-It-Now: Theory and Evidence. J. Ind. Econ.
**2013**, 61, 393–429. [Google Scholar] [CrossRef] [Green Version] - Anwar, S.; Zheng, M. Posted price selling and online auctions. Games Econ. Behav.
**2015**, 90, 81–92. [Google Scholar] [CrossRef] - Maslov, A. Competition in Online Markets with Auctions and Posted Prices; KSU Coles College of Business Working Paper Series. Available online: https://coles.kennesaw.edu/research/docs/fall-2019/FALL19-02.pdf (accessed on 19 February 2019).
- Vickrey, W. Counterspeculation, auctions, and competitive sealed tenders. J. Financ.
**1961**, 16, 8–37. [Google Scholar] [CrossRef]

1 | In 2000s Yahoo discontinued its services in most of the countries except Japan and Taiwan. |

2 | This has been done in another paper by [11]. |

3 | The model could be generalized to n risk-averse buyers and include a reserve price, but it does not affect the intuition based on a technically simpler framework used in this paper. |

4 | The following allocation rule may potentially create discontinuity (also well documented in auctions with a BIN price) in the cutoff function, because some buyers can do better by exiting at the start of the auction rather than bidding in the auction where everyone follows symmetric strategy. |

5 | If $x<c$, then x drops out from the auction, and it ends before the clock reaches c. If $x>v$ then the buyout option is used before c, and the auction again ends before the clock reaches c. |

6 | Note that c(i), c(ii) and c(iii) can be summed up to: ${\int}_{p}^{{z}^{-1}\left(c\right)}{\int}_{\underset{\xaf}{v}}^{x}(v-p)2f\left(x\right)f\left(y\right)dydx$. |

7 | Note that ${z}^{-1}\left(c\right)$ may be to the left or right of the “jump-down.” The same pertains to x and y when $x,y>{z}^{-1}\left(c\right)$. Also, it is easy to show that when $p<v<{z}^{-1}\left(c\right)$ the domains of integration remain the same, i.e., ${\int}_{p}^{v}{\int}_{\underset{\xaf}{v}}^{x}(v-p)2f\left(x\right)f\left(y\right)+{\int}_{v}^{{z}^{-1}\left(c\right)}{\int}_{\underset{\xaf}{v}}^{x}(v-p)2f\left(x\right)f\left(y\right)dydx={\int}_{p}^{{z}^{-1}\left(c\right)}{\int}_{\underset{\xaf}{v}}^{x}(v-p)2f\left(x\right)f\left(y\right)dydx$. However, when ${z}^{-1}\left(c\right)<v<\overline{v}$ domains of integration change conditional on whether $v>x$ or $x>v$ (the figure shows the case when $v>x$). |

8 | Notice that this condition is identical to the one in auctions with a BIN price, except that now it is based on the second order statistic rather than the first order statistic. |

9 | Note that $F\left(\underset{\xaf}{v}\right)=0$, $F\left(\overline{v}\right)=1$ and $\frac{d}{dc}{z}^{-1}\left(c\right)=\frac{1}{{z}^{\prime}\left({z}^{-1}\left(c\right)\right)}$. |

**Figure 1.**Equilibrium bidding functions7.

© 2020 by the author. 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/).

## Share and Cite

**MDPI and ACS Style**

Maslov, A.
A Note on Buyers’ Behavior in Auctions with an Outside Option. *Games* **2020**, *11*, 26.
https://doi.org/10.3390/g11030026

**AMA Style**

Maslov A.
A Note on Buyers’ Behavior in Auctions with an Outside Option. *Games*. 2020; 11(3):26.
https://doi.org/10.3390/g11030026

**Chicago/Turabian Style**

Maslov, Alexander.
2020. "A Note on Buyers’ Behavior in Auctions with an Outside Option" *Games* 11, no. 3: 26.
https://doi.org/10.3390/g11030026