International Game Theory Review, Vol. 12, No. 1 (2010) 75–81 c World Scientific Publishing Company
DOI: 10.1142/S0219198910002520
A NOTE ON DISCRETE BID FIRST-PRICE AUCTION
WITH GENERAL VALUE DISTRIBUTION
GANGSHU (GEORGE) CAI
Department of Management, Kansas State University
Manhattan, KS 66506, USA gcai@ksu.edu PETER R. WURMAN
Kiva Systems, Inc., 225 Wildwood Avenue
Woburn, MA 01801, USA pwurman@kivasystems.com XITING GONG
Guanghua School of Management
Peking University, Beijing 100871, China gongxiting@gsm.pku.edu.cn This paper evaluates the discrete bid first-price sealed-bid (FPSB) auction in a model with a general value distribution. We show that a symmetric Bayesian Nash equilibrium exists for the discrete bid FPSB auction. We further prove that the discrete bid FPSB equilibrium conditionally converges to that of a continuous bid FPSB auction.
Keywords: First-price sealed-bid auction; discrete bid; Bayesian Nash equilibrium.
1. Introduction
In a first-price sealed-bid (FPSB ) auction, each bidder submits a single bid without observing others’ bids, and the bidder with the highest bid pays the value of her bid. FPSB auctions are popular in practice, e.g., the FPSB auctions utilized by the
Chicago Wine Company (TCWC.com). In terms of bidding space, FPSB auctions can be categorized into continuous bid and discrete bid FPSB auctions. Discrete bid auctions can be seen in the real world, since a minimum bid increment is usually default to one cent. In most online auctions, bidders are required to submit a minimum increment.
A voluminous theoretical literature has been developed on FPSB auctions. Many models have assumed that bidders are symmetric and risk neutral (McAfee and
McMillan, 1987; Milgrom and Weber, 1982; Myerson, 1981; Riley and Samuelson,
1981; Vickrey, 1961). More recently, researchers have studied models with asymmetric information (Engelbrecht-Wiggans, 1993; Lebrun, 1999; Maskin and Riley,
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2000a, b), affiliated values (Campo et al., 2003; McAdams, 2003; Milgrom and
Weber, 1982), or other variations (Krishna, 2002; Lebrun, 2002; Levin and Smith,
1994; Lizzeri, 2000). Most of the literature assumes incomplete information and continuous bids. With bidders having complete information and continuous bids, equilibrium does not exist due to the discontinuity of the payoff function (Lebrun,
1996). To enable the existence of equilibrium in this situation, Lebrun (1996) suggests an “augmented” first-price auction to break the tie so that the bidder having higher valuation wins the item by submitting a “message” marking her higher valuation. Maskin and Riley (2000b) propose using a second-round Vickrey auction to break the tie if any. In a model of incomplete information, Athey (2001) concludes the existence of pure strategy equilibrium in FPSB auctions with finite strategies, when the single crossing condition holds. A few papers have discussed on auctions with discrete bids, such as Grimm et al. (2003); Lengwiler and Haller
(1998); Menezes (1996). However, the properties of discrete bid FPSB auctions, especially the relationship between the equilibrium and the size of bid increment, are not well studied.
When the number of discrete bids is finite or bounded, the existence of equilibrium in the FPSB auctions with complete information can be easily proven using the
Nash theorem (Nash, 1950, 1951). In a discrete bid FPSB auction with incomplete information, if the bidders’ valuations follow a uniform distribution, Chwe (1989) shows that the discrete bid strategy converges to the continuous bid strategy. We extend his work by examining the equilibrium under a general value distribution.
The remainder of this paper is organized as follows. Section 2 presents the FPSB
auction
