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All-pay auctions with private signals about opponents’ values

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Abstract

We study all-pay auctions where each player observes her private value as well as a noisy private signal about the opponent’s value, following Fang and Morris’s (J Econ Theory 126(1):1–30, 2006) analysis of winner-pay auctions with multidimensional private signals. A unique symmetric monotonic equilibrium exists if the signal is not informative enough. When the signal is sufficiently informative, there exists a symmetric non-monotonic equilibrium in which all types of players randomize in overlapping supports. The revenue is lower than that in the standard independent private value setting. Fixing the signal’s informativeness, the all-pay auction raises lower revenue than the second-price auction, whereas the revenue ranking between the all-pay and the first-price auction is ambiguous.

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Notes

  1. Such as rent-seeking and lobbying (Ellingsen 1991; Baye et al. 1993; Che and Gale 1998), research and development (Che and Gale 2003), job promotion (Clark and Riis 1998), R&D procurement contests (Dasgupta 1986; Kaplan 2012; Fu et al. 2014; Kovenock et al. 2015), and crowdsourcing (Liu et al. 2014; Chawla et al. 2019). As an auction format, all-pay auctions also play a critical role in fund-raising and charity auctions (Goeree et al. 2005; Engers and McManus 2007; Schram and Onderstal 2009).

  2. A similar pattern of behavior is observed in the study by Chen (2019), who focuses on incentives to spy on other players before a contest.

  3. See also discussions in Fang and Morris (2006) for a comparison of the private signals environment and the APV setting.

  4. We show that players with the low value randomize in an interval strictly below the low value in the APA. Fang and Morris (2006) show that low-value players bid their value in the FPA.

  5. When it comes to fund-raising or charity auctions, it is generally true that a carefully designed APA raises greater revenue than the SPA and the FPA (Goeree et al. 2005; Engers and McManus 2007; Schram and Onderstal 2009).

  6. In proving Proposition 2 we show that type \((v_h,h)\) is indifferent across all bids in \([\underline{b}, \widehat{b}]\) and \((v_l,h)\) is indifferent across all bids in \([0,\overline{b}]\), for all \(q \in [q^{*}, 1]\).

  7. According to Fang and Morris (2006), there exists a unique SME in the SPA and the FPA in the independent private binary value setting. Furthermore, the revenue in the two auction formats is given by \(p^2_h v_h + (1-p^2_h) v_l\). It is straightforward to show that the same is true for the APA when \(q=\frac{1}{2}\) based on our proof of Proposition 3.

  8. Low-value players bid their value \(v_l\) in the FPA (Fang and Morris 2006) but randomize in an interval whose upper bound lies strictly below \(v_l\) in the SPA, as indicated by Proposition 1 and 2.

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  1. School of Economics and Management, Harbin Institute of Technology, Shenzhen, HITSZ campus of University Town, Nanshan District, Shenzhen, 518000, China

    Zhuoqiong Chen

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This work was supported by the National Natural Science Foundation of China No. 71903046 and the “Shenzhen Peacock Plan” start-up research grant.

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Chen, Z. All-pay auctions with private signals about opponents’ values. Rev Econ Design 25, 33–64 (2021). https://doi.org/10.1007/s10058-020-00242-3

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