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

November 15th, 2008 No comments

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.

Therefore, we can compare several different auctions' diversity or population mix and determine whether it's attended by mostly experts, in-betweens, or laymen (by proportion) or whether there is a happy jumble of all (how "ordered" the mix is).

In fact, why the method of measuring diversity (by measuring information-theoretical entropy) is not more greatly exploited by Mankinde is really a bugging question in my mind: it can be applied everywhere!  For example, I was just at the mall and thought "Hmm, this winter season seems to be only purples and blues.  I wonder what the most diverse season in terms of color in men's shirts is... probably summer?" I also thought at the time of my visiting the mall "This measure of diversity could really be applied to scale countries in terms of mix of ethnicity or nationality - is the US most diverse because it is (purportedly) a melting pot?  Has it gotten less diverse post-9/11 with all the added restrictions on foreign nationals?" or something similar for a social-networking site or a school/university (I do wonder about my alma mater - e.g.?), or if I'm a company producing a number of different SKUs, "did I produce many different SKUs or more of a specific type?" or for a chain of restaurants one could determine whether the population is within an age range at a location compared to another (others) more uniform, or does the population at Chain 1 request more of a particular kind of menu item or it's more or less evenly distributed amongst all menu items, or does it peak by days or months of the year, etc.  These questions answered numerically can then help decide whether "I should buy more ingredients of this specific type, during such-and-such period."

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