Theory and individual behavior of first-price auctions (original) (raw)

1988, Journal of Risk and Uncertainty

First-price auction theory is extended to the case of heterogeneous bidders characterized by Mparameter log-concave utility functions. This model, and its specific two-parameter constant relative risk averse special case, is generally supported by the results of 47 experiments. The one-parameter special case that comprises most of the theoretical literature is not supported by the experiments. One anomaly for the two-parameter model is that too m~ny of the subjects exhibit positive (or negative) intercepts in their linear estimated bid functions. Accordingly, we develop a specific three-parameter model, which introduces a utility of winning, and a threshold utility of surplus. The new model, tested directly by introducing lump-sum payments or charges for winning, is not falsified by the new experiments. 62 JAMES c. COx, VERNON L. SMITH, AND JAMES M. WALKER constant relative risk averse model, all of which have been analyzed in the literature! There are two reasons why our extension of bidding th~ory to agents with heterogeneous risk preferences is important: (1) some such extension is demanded by the bidding data, since these data are highly inconsistent with all identical bidders models; (2) a recent comprehensive survey of over 50 bidding theory papers (McMee and McMillan, 1987) lists only one contribution that is not based on the assumption that bidders have identical risk preferences (Cox, Smith, and Walker, 1982). We also present the results of 47 F auction experiments, which are designed to discriminate among the various noncooperative equilibrium models of bidding~.:f that are contained in the generalM-parameter model. These results do not support either the linear or the one-parameter concave model. The results do support im-' portant features of the constant relative risk averse model (CRRAM), such as ..; linearity of bid functions and invariance of bidding behavior to a multiplicative transformation of payoffs. However, the homogeneity property of CRRAM bid functions is violated by 22% of the subjects. In addition, subject bids are generally inconsistent with the predictions of the conjunction of CRRAM and square or square root transformations of payoffs. We develop a specific three-parameter model, CRRAM*, based on a utility of winning and a threshold utility of surplus, to account for the observed nonzero bid function intercepts. Five new experiments (30 subjects) yield results that are consistent with the testable implications of CRRAM*. 1. The log-concave equilibrium model of bidding in the first-price auction Consider anF auction where the seller's reservation price is zero and, therefore, no nonpositive bid can be a winning bid. Let there be n > 2 bidders. Each bidder's monetary value, Vi' i = 1, ..., n, for the auctioned object is independently drawn from the probability distribution with cdf H(') on [O,v]. H(.) has a continuous density function that is positive on (O,v). Bidders are assumed to know their own Vi but to know only the distribution from which their rivals' values are drawn. The utility to any bidder i of a winning bid in the amount hi is the von Neumann-Morgenstern utility U(Vi-hi, 9;), where 9i is anM-1 vector of para meters that is independently drawn from the probability distribution with integrable cdf(.) on the convex set 0. Each bidder knows his or her own 9i but knows only that his or her rivals' 9's are drawn from the distribution <1>(.). Thus, a bidder is' represented by (vi,9;), where Vi is his or her (scalar) auctioned object value and 9i is his or her M-1 vector of other individual characteristics that affect bidding behavior in thisM-parameterlog-concave model. Assume thatu(y,9) is twice continuously differentiable and strictly increasing in monetary payoff y and normalized such that u(0,9) = 0, for all 9 E 0. Finally, assume that u(y,9) is strictly log-concave in y, for each 9 E 0; that is, assume that ul(y,9)/u(y,9) is strictly de-'c.::