### Archive

Archive for November 15th, 2008

## On Auctions, Part III and On Pricing, Part VI - (On Diversity)

Before going into details respecting the weighted function $\mu_*$ and the variance $\sigma^2_*$, I was thinking going a little bit into the mix of individuals at an auction or several auctions.  I've been loosely categorizing the types as clueless or "laymen," "in-betweens," and "experts".  The number of subdivisions is up to anybody, but three is a practical and manageable number to me.  Let's suppose I have access to the data as before for P1...Pn individuals at auction A1: $\{\mu_i, \sigma^2_i\}$.  Let's suppose then that you can be called an "expert" at any auction if you believe your quote $\mu_i$ is correct to within plus or minus (0-10]%, an "in-between" if you think it is correct to within between (10 and 50]% above or below, and a layman if you think your quote is correct within more than (50-100]% above or below.  These percentages can be translated back to appropriate bounds of variances and so we can place each individual's variance in one of the three categories.  If we count up the proportion of variances lying in each "box" ($p_k, k=1...3; \sum p_k=1$) we can then borrow from Information Theory the measure of surprise or entropy as an indicator of diversity!  This has already been done in Biological Information Theory to see how diverse in species an area is (link or reference forthcoming):
$H = -\sum_{k=1}^3 p_k log_2 p_k$
where conventionally $p_k log_2 p_k = 0$ whenever $p_k = 0$.  $H$ is maximal if the proportions across each box are equal: $p_1 = p_2 = p_3 = \frac{1}{3}$ and zero or close to zero whenever the proportion of one box is 1 or close to 1.