Benford's law and naturally occurring prices in certain ebaY auctions

Abstract

We show that certain winning bids for certain ebaY auctions obey Benford's Law. One implication of this is that it is unlikely that these bids are subjected to collusion among bidders, or ‘shilling’ on the part of sellers. Parenthetically, we also show that numbers from the naturally occurring Fibonacci and Lucas sequences also obey Benford's Law.

Acknowledgement

I am very grateful to Sophie Hesling for her excellent research assistance.

Notes

¹ This number of observations is dictated by machine precision. The last number in the sequence is F(1476) = 0.1306989E + 309.

² Pietronero et al . (2001) provided an explanation for Benford's Law in terms of multiplicative dynamical processes and derived a generalization in terms of a power law. Gottwald and Nicol (2002) proved that systems satisfying Benford's Law do not have to be random or chaotic–they may be deterministic and are more related to the (near) multiplicative nature of the process than to its chaoticity.

³ For example, for n > 1, every Lucas number is the sum of two Fibonacci numbers and for all n, L(n) = [F(n +) − F(n − 2)] and [L(n − 1)L(n + 1) + F(n − 1)F(n + 1)] = 6[F(n)]². Moreover, and this relationship is approximated well even when ‘n’ is quite small. For example, for n = 8, 9, 10 we have L = 47, 76, 123; and , 75.999 and 123.046.

⁴ See McMinn (2003) and Scott (undated). Also, in a sequence of ‘n’ equal resistors, connected alternately in series and in parallel, then . If the wiring is alternately in parallel and in sequence, .

⁵ The last number in the sequence is L(1475) = 0.111630E + 309.

⁶ More specifically, the time-span is from 00:00:01 PST on 25 November to 24:00:00 PST on 3 December 2004.

⁷ Strictly, it is possible for other currencies to be used, but there were no such examples in our population. In any case, Benford's Law is scale and base invariant, so the choice of (a common) currency does not affect our results.

⁸ For a detailed technical discussion, see Durbin (1973), for example. The Glivenko–Cantelli Theorem implies that, under quite mild conditions, the empirical distribution function converges uniformly and with probability one to the population distribution function.

⁹ Note that if the data are circular, as is the case with the digits ‘1’ through ‘9’, the distribution of the test statistic should not depend on whether we begin counting at ‘1’ and end at ‘9’ (say), or alternatively begin counting at ‘6’ and end at ‘5’ (say).

¹⁰ The purpose of transforming from V_N to is to create a statistic whose null distribution is independent of the sample size. This is especially helpful for large ‘N’.