Technical Appendix of Sinha and Greenleaf "The Impact of Discrete Bidding and Bidder Aggressiveness on Sellers' Strategies in

All variables and functions are real valued. Assumption 1: f(v)/(1-F(v)) is strictly monotonically increasing in v (see Footnote 4 for discussion). Fact 1: Suppose v(n-1), the second highest order statistic of n random variables drawn from F(v), is known. Then the distribution of the highest order statistic v(n) is the distribution F(v) truncated to the left at v(n-1) and normalized (Arnold, Balakrishnan, and Nagaraja, 1992, Theorem 2.4.1, p. 23). Definition 1: In discrete bid auction, the set of permissible bids is Θ = {sa,sa+I,...,sa+KI}, where sa is the starting auction bid, and K is an integer ≥2. Here sa+KI ≤ v 1. Definition and Properties of ℘(⋅) Here, we define ℘(⋅) formally, based upon the discussion of bidding aggressiveness in the text (pgs. 8-9). Let ℘(j)Aggressive denote the bid probability for a relatively aggressive Bn, recalling that the "bid probability" is the probability that Bn obtains the current bid when j other bidders also remain. In a parallel manner, ℘(j)Neutral and ℘(j)Unaggressive describe the bid probabilities for a neutral and an unaggressive Bn. Further, note that ℘(⋅) is a function of j and some other parameter, the latter directly representing aggressiveness in any functional form appropriate for ℘(⋅). We require that ℘(⋅) satisfy the following three properties: (a) For any given j, ℘(j)Aggressive > ℘(j)Neutral > ℘(j)Unaggressive, with the inequalities being strict; and that ℘(j) is strictly increasing as aggressiveness increases, regardless of the type of aggressiveness. Note ℘(j)Neutral=1/(j+1). (b) (j+1)℘(j) be nondecreasing in j for aggressiveness, nonincreasing in j for unaggressiveness, and constant in j for neutrality, for all j corresponding to all n. This sufficient condition ensures that for each bidder, ℘(j) 1/(j+1) is always either > 0, or < 0, or = 0, for all j. Also, note that different bidders are allowed to have different types of aggressiveness. (c) ℘(j) is not a constant in j for aggressiveness, neutrality, or unaggressiveness, for all j. It follows from properties (a) and (c) that ℘(j) is a monotonic function of the parameter directly representing aggressiveness, for a given j. For example, γ is that parameter in the numerical examples in the text §5. Since the proofs do not require that parameter explicitly, it does not appear as an argument in ℘(⋅).