Elicited bid functions in (a)symmetric first-price auctions

Since the seminal contributions of Vickrey (1961) and Milgrom and Weber (1982) much of
the theoretical literature on auctions has focused on the allocative properties of mechanisms
that involve either weakly dominant bidding strategies, such as the ascending-price auction, or
the Nash equilibrium concept, such as the first-price sealed bid auction. From a practical
standpoint, however, while weakly dominant strategies are relatively straightforward to figure
out and thus very likely to be used, the rationality assumptions underlying the definition of
Nash equilibrium type of strategies may cast some doubt about their empirical observation.

For the simplest case of single-unit auctions with private independent values and symmetric
bidders, evidence from laboratory experiments shows that the revenue and bid predictions for
first-price sealed bid auctions are systematically violated, mostly because subjects tend to bid
above the risk neutral Nash equilibrium prediction. Cox, Roberson and Smith (1982) and Cox,
Smith and Walker (1988) explain such overbidding in terms of a Nash equilibrium model that
assumes constant relative risk averse bidders (the CRRA model) and numerous studies have
been conducted to further assess this model.¹ While these studies find a remarkable support
for the CRRA model of bidding, they do not assess the underlying assumption of a strategic
behavior. Chen and Plott (1998) observe in particular that if bidders’ valuations are uniformly
distributed, as it is usually the case in auction experiments, then the Nash equilibrium
strategies are linear and, hence, impossible to disentangle from linear ad hoc bidding rules
such as a percentage markdown strategy. They report on a series of experiments for which the
Nash equilibrium predictions are nonlinear and show that the CRRA model is outperformed
by a non-linear ad hoc model. However, their study also indicates that a belief-free version of
the CRRA model (i.e., which does not restrict the subjects’ beliefs about the distribution of

¹ See Kagel (1995) for an overview. See Goeree, Holt and Palfrey (2002) for an assessment of behavior in these
auctions in terms of a quantal response equilibrium.

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