All-pay auction

In economics and game theory, an all-pay auction is an auction in which every bidder must pay regardless of whether they win the prize, which is awarded to the highest bidder as in a conventional auction. As shown by Riley and Samuelson (1981),^[1] equilibrium bidding in an all pay auction with private information is revenue equivalent to bidding in a sealed high bid or open ascending price auction.

In the simplest version, there is complete information. The Nash equilibrium is such that each bidder plays a mixed strategy and expected pay-offs are zero.^[2] The seller's expected revenue is equal to the value of the prize. However, some economic experiments and studies have shown that over-bidding is common. That is, the seller's revenue frequently exceeds that of the value of the prize, in hopes of securing the winning bid. In repeated games even bidders that win the prize frequently will most likely take a loss in the long run.^[3]

The all-pay auction with complete information does not have a Nash equilibrium in pure strategies, but does have a Nash equilibrium in mixed-strategies.^[4]

^ Riley, John; Samuelson, William (1981). "Optimal Auctions". American Economic Review (3). doi:10.1257/aer.100.2.597.
^ Jehiel P, Moldovanu B (2006) Allocative and informational externalities in auctions and related mechanisms. In: Blundell R, Newey WK, Persson T (eds) Advances in Economics and Econometrics: Volume 1: Theory and Applications, Ninth World Congress, vol 1, Cambridge University Press, chap 3
^ Gneezy, Uri; Smorodinsky, Rann (2006). "All-pay auctions—an experimental study". Journal of Economic Behavior & Organization. 61 (2): 255–275. doi:10.1016/j.jebo.2004.09.013.
^ Hillman, Arye L.; Riley, John G. (March 1989). "Politically Contestable Rents and Transfers". Economics and Politics. 1 (1): 17–39. doi:10.1111/j.1468-0343.1989.tb00003.x. ISSN 0954-1985.

[1] Riley, John; Samuelson, William (1981). "Optimal Auctions". American Economic Review (3). doi:10.1257/aer.100.2.597.

[Moldovanu-2] Jehiel P, Moldovanu B (2006) Allocative and informational externalities in auctions and related mechanisms. In: Blundell R, Newey WK, Persson T (eds) Advances in Economics and Econometrics: Volume 1: Theory and Applications, Ninth World Congress, vol 1, Cambridge University Press, chap 3

[Gneezy-3] Gneezy, Uri; Smorodinsky, Rann (2006). "All-pay auctions—an experimental study". Journal of Economic Behavior & Organization. 61 (2): 255–275. doi:10.1016/j.jebo.2004.09.013.

[4] Hillman, Arye L.; Riley, John G. (March 1989). "Politically Contestable Rents and Transfers". Economics and Politics. 1 (1): 17–39. doi:10.1111/j.1468-0343.1989.tb00003.x. ISSN 0954-1985.

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