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Auctions with uncertain numbers of bidders

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Abstract

We investigate bidders’ and seller's responses to ambiguity about the number of bidders in the first price auction (FPA) and the second price auction (SPA) with independent private valuations. We model ambiguity aversion using the maxmin expected utility model. We find that bidders prefer the number of bidders to be revealed in the FPA, are indifferent between revealing and concealing in the SPA, and prefer the SPA to the FPA. If bidders are more pessimistic than the seller then the seller prefers to conceal the number of bidders in the FPA, and prefers the FPA to the SPA.

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      Bose et al. (2006) show that a full insurance auction is also the optimal standard auction when bidders are averse to ambiguity, in the form of maxmin expected utility preferences. Ambiguity-averse bidders also prefer the SPA to the FPA (Levin and Ozdenoren, 2004) but di Tillio et al. (2016) show that an ambiguous mechanism, in which certain rules may be hidden from bidders, may have higher revenue than any standard auction in such a setting. Lange and Ratan (2010) study behavior in auctions with loss-averse bidders, using the reference-dependent utility model of Kőszegi and Rabin (2007).

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      A natural benchmark of our analysis is the growing literature on auctions with ambiguity averse agents, which mostly focuses on static settings. For instance, in single-unit sealed-bid auctions, Salo and Weber (1995) show that over-bidding in experiments may be a consequence of ambiguity aversion, Lo (1998) examines maxmin bidders and uncertain distribution of values, Levin and Ozdenoren (2004) study the case with ambiguity about the number of bidders and Chen et al. (2007) experimentally study more general ambiguity preferences. A common insight of these papers and ours is that ambiguity aversion leads to pessimism and aggressive bidding.

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    Levin's work was supported by NSF Grant # 0136928. Much of the work was done while Dan Levin was visiting Harvard Business School whose hospitality is gratefully acknowledged.

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