### Archive

Posts Tagged ‘uncertainty’

## 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.

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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## Some Thoughts On Pricing, Part IV

On the other hand, if instead of making the variance or standard deviation tight we allow it to be relaxed, the same Gaussian distribution becomes more and more like a uniform distribution over the entire real domain.  If there is one other company competing against me, and it's a real coin toss regarding how F1 will price, it pays for me to price above the mean given my belief of the mean and standard deviation.  Recall that for a uniform distribution expected profit looks like an inverted parabola, and great uncertainty around ten pesos will occasion my expected profit to look just similarly so.

Something akin happens when there are 2, 3 companies competing against me, except the maximum of this "inverted parabola" (it actually isn't parabolic, but sort of) is closer to 10 (the mean) and my maximum expected profit is lesser.  When there is a lot of competition pricing all over the place it really becomes a coin toss as to where I should price, as maximum expected profit will be more or less the same regardless (in fact, close to zero).

The graphs (soon-coming) perhaps will make this more obvious.

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## Some Thoughts On Pricing, Part III

What happens when competition decides they will become organized and the price of a product is exactly 10 pesos?  If I am bound by the price too due to politics (perhaps the government itself sets the price because it has such powers) or some other factor, and I have to price myself at 10 pesos, then the rational consumer is faced with identical products at identical prices to choose from.  Perhaps he will then choose at random.  If F1 is my only competitor, he will choose me at the shops half the time.  If instead there are Fn competing companies, I will be chosen perhaps $\frac{1}{n+1}$ of the time.  My expected profit in such a situation is easily calculated as:

$E_p(10) = \frac{1}{n+1} \cdot (9) + \frac{n}{n+1} \cdot (-1) = constant$

Having established such, let us assume that I am not bound by any politics.  Then it is only obvious that I would want to price at 9.99, since this virtually guarantees that the rational consumer choosing as by the C1 axiom will pick me over any other product: I am guaranteed in effect selling 100 percent of my product, and furthermore at maximal expected profit, since selling at anything less than 9.99 would mean obtaining less for product I am sure to give away.  In terms of expected profit, we can make

$E_p(x) = x - 1$

if I price at less than ten $x < 10$, or

$E_p(x) = -1$

if I price at more than ten $x > 10$, for any amount of competition against me (does not depend on n, since everyone is pricing at 10).  Maximal expected profit is at a price of 9.99 in these cases.

I like to consider this particular example the limiting case in which the Gaussian distribution is tightly wound around 10 pesos. The tighter the certainty around ten pesos, the closer I am to the above distribution (except the definition at 10 pesos, which we reasoned in a different way).  This is because the probability of selling my product at less than 10 pesos is essentially 1, where selling at a price above 10 pesos is essentially zero.  Perhaps this can be more easily seen upon inspection of the following graphs.

First, I have graphed what happens as the certainty of F1's pricing becomes tighter and tighter (one other competing firm).

In this next graph, I have shown what happens as the certainty of F1...F5's pricing becomes tighter.

For 40 competing firms, this is what happens.

All these graphs show that indeed the limiting distribution will not depend on the number of companies competing against me as they converge or stack upon a single price quote.  The more in agreement companies are about what the price should be, the less their ability to sell (oppositely for me) and the more of the pie I can take, and my expected profit per product will balloon to an absolute maximum of 8.99 pesos (9.99 revenue - 1 cost).

For a company thinking this way we have differences of pennies across comparable products, much perhaps as Elisa remarks in her comments about pricing in Switzerland.

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## Some Thoughts On Pricing, Part II

October 16th, 2008 1 comment

Let's say instead of a uniform distribution, as described in "Some Thoughts On Pricing, Part I" and its simulation, comparable products' (say apple juice) prices hover around 10.00 pesos, with standard deviation 3.00 pesos, and the distribution is normal (by omniscience).  If there is just one other competing company (F1) against me I could have ascertained the mean and variance (or standard deviation) by sampling its pricing across 1000 shops, and obtained them by maximum likelihood.  If there are more than one companies (several tens) in the market competing against me perhaps I could have derived the distribution parameters by maximum likelihood across a shelf of a single shop (by ergodicity), always assuming prices across companies are realizations of the same distribution.

An advantage of the Gaussian distribution is that the domain is the entire real line, and so probabilities are assigned for every conceivable price (even negative prices, meaning I as a supplier choose to pay you the consumer to take the product!).  Expected profit is likewise therefore defined everywhere of practical interest.

Following as before, let's say again that the cost of producing a single unit of product is 1 for me (*).  The probability of my selling by  being the lowest price at a particular shop is one minus the cumulative density of a Gaussian:

$1 - \frac{1}{2}\big(1+erf(\frac{x - \mu}{\sigma \sqrt{2}})\big)$,

or in this case:

$1 - \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)$,

and the probability of failing at a sale is the cumulative density:

$\frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)$

As before, for one competing company, my expected profit is:

$E_p(x) = \Big(1 - \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)\Big) \cdot (x-1) + \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big) \cdot (-1)$

and if there are n competing companies,

$E_p(x) =\Big(1 - \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)\Big)^n (x-1) +$

$+ \Big(1 - \Big(1 - \frac{1}{2}\big(1+erf(\frac{x - 10}{3 \sqrt{2}})\big)\Big)^n\Big)(-1)$

Here I have graphed price to expected profit for 1...5, 10, 20, and 40 competing companies.  I also used a computer to obtain approximations of zeroes and extrema.

One of the remarkable things about this family of functions is that maximum expected profit can be achieved for any competition in the market, no matter if there are few companies competing against me or thousands, although if there are several hundreds or thousands at some point my maximum expected profit is negative (but never less than -1, of course, since I cannot lose more than what I cannot sell).  Here's a graph of this phenomenon:

Another remarkable thing is that maxima seem to lie on a line defined for $E_p(x) > -1$, although I am not entirely sure of the behavior at near $E_p(x) = -1$ (maxima may in fact be asymptotic to such).  Break-even prices become lower and lower the more competition there is, until finally there is a point at which, even though I may sell some product of apple juice, I can never make up for my expense (graph above).  Lastly, here again maximal expected profit is achieved at lower pricing the more competition there is in the market.

At any rate, the lesson would seem to be: the more competition there is, the more the piece of the pie is divided up, the lesser the maximal profit for me.

In my next post, I will describe what happens to expected profit when the Gaussian distribution of prices across competing firms is more certain: by this I mean the standard deviation or variance is smaller, which translates to there being a general agreement or consensus across suppliers about what the price of a product should be.

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