Finding Pricing Excellence on a Roulette Wheel

One of my recent posts, “You Are Not At the Mercy of the Market…”, attracted a rather thought-provoking response posted directly to the blog.  The crux of this response, and others sent directly to me, have all revolved around a similar theme:  With so much uncertainty surrounding consumer behavior, words such as “pinpoint” or “optimize” should not be uttered when it comes to the decisions that pricing and marketing managers must make.  This is indeed a compelling sentiment, and has stirred much discussion amongst my colleagues in industry and in academia (our research organization collaborates closely with professors within the University of Chicago and Carnegie Mellon University).  This discussion has taken on many twists and turns, which we hope to summarize in future posts.  But, there is one particular question that has resonated throughout our discussions:

What are the implications of the words “pinpoint” and “optimal” when market behavior is so uncertain?

In other words, is it possible to find a single decision that will maximize the odds of earning a handsome payoff when the outcome of any decision is uncertain?  In a rather extreme example, in the highly uncertain world of gambling, can I make some decisions that are clearly better than others in light of the uncertainty?

Let’s say that all of a sudden I have the urge to gamble, and head for Monaco.  I don my tuxedo, and enter the Monte Carlo casino, where I see 3 different tables offering 3 different games.  I have €1,000 to spend, but the house has imposed the constraint that I must pick a single table and commit myself to that table for the entire evening.  Now, let’s say that at each one of the 3 tables, the wager amount for a single bet is €10 (admittedly, a far-cry from what I should be prepared to spend at Monte Carlo), which allows me to play 100 games (€1000 total wallet size ÷ €10 per game) at each table.  So, how does the evening unfold?

Since I am not a gambler, I will have to fabricate some numbers to convey the point.  Let’s say that Table A offers a 49% chance of winning, and each win produces €18 (for a gain of €8 based on my €10 bet); Table B offers a 10% chance of winning, and each win produces €85; whilst Table C offers a 32% chance of winning and each win produces €25.  Needless to say, everything is all but certain inside Monte Carlo, and as a rational man I should stay out.  But if I do venture in and wish to risk my money, can I pinpoint the table that will optimize my returns?

To answer this question, I must travel through several centuries of mathematical thought, and invoke the laws of probability.  I know that if I flip a coin a sufficient number of times, I can say with great certainty that 50% of the outcome will be heads, and the other 50% of the time will be tails).  Notice, the key here is that I must flip the coin a “sufficient” number of times.  Due to the vagaries of random chance, a small number of throws may not reveal the true nature of the coin – with just 7 coin tosses it is possible that all 7 times I see heads.  But, with 700 throws, it is quite unlikely that I’ll see 700 heads.  Instead, I’ll probably see close to 350 heads and close to 350 tails.  This conclusion is driven by a theorem known as the Central Limit Theorem, which was originally put forward by the French-born mathematician Abraham de Moivre.  It states that if we know the probability of an outcome from some event (like a coin toss), we will see that outcome occur as often as the probability multiplied by the number of times the event occurs.

In our hypothetical Monte Carlo excursion, the gambling event can occur 100 times (given my €1,000 allowance and €10 per gamble – i.e., per event).  This means I can expect the following (as I hear de Moivre’s voice from centuries past):

Table A: 100 Events x €18 Won per Event x 0.49 Chance of Win per Event = €882 Expected Winnings

Table B: 100 Events x €85 Won per Event x 0.10 Chance of Win per Event = €850 Expected Winnings

Table C: 100 Events x €25 Won per Event x 0.32 Chance of Win per Event = €800 Expected Winnings

With this arithmetic, de Moivre guides us to Table A – indeed, once can say that he has pinpointed Table A for not in spite of the uncertainty, but in light of the uncertainty. Now, it’s important for us to recognize that the power of de Moivre’s insights rest on understanding the uncertainty – i.e., on “quantifying the chance” of an outcome.  Let’s say that Table A is a Roulette Wheel.  As the host spins the Wheel and we all place our bets, it is impossible to state exactly whether or not I will win or whether or where the ball will land.  There are too many factors to humanely consider:

1. The bounciness of the ball
2. The initial rate of spin with which the host turns the wheel
3. The amount of lubrication in the bearings that bind the wheel to the axle
4. The amount of humidity in the room which can slow down the wheel
5. The air currents in the room when the Air Conditioning comes on
6. The initial height from which the ball is dropped onto the wheel
7. Etc, etc, etc.

And even after knowing all of these factors, the equation used to predict the balls final resting position is nonlinear and the solution is going to be chaotic, as shown in the figure above.  We simply cannot predict where the ball will land.  But, there is hope!

We can predict the probability of winning, and we can do so using 2 different methods.  First, we can look at the fact that there are 37 pockets on a European Roulette wheel, and the odds that our ball will land in any pocket is therefore 1/37 = 0.027, which means we have almost a 3% chance of winning.  Alternatively, we can take a sample of all the previous people that have played in the prior 6 months and look at what fraction have won.  This gives us what is known as a “Frequentist Probability”, and can be used within the context of de Moivre’s principles to help guide us to the “optimal table” on which to place our bets.  The discovery of these probabilities is one of the fundamental pursuits of the entire discipline of Econometrics, and has become pivotal to achieving Pricing excellence (further exposition on this important topic can be found in the “Modeler’s Mechanics” of our blogs, and within our white papers).  As shown in the figure to the left, this probability discovery process leans heavily on historical market data (and remember, the past does not provide a definitive glimpse into the future, but does provide a very good glimpse into the probabilities of many different futures), with a heavy dose of computing power to produce predictions that have so far proven to dramatically improve the pricing manager’s decisions when compared to using their human instincts alone.

For instance, companies will often either adopt an Every Day Low Pricing (EDLP) or a High-Low Pricing (HLP) strategy.  The former seeks to keep prices low by achieving enormous economies of scale and tight supply-chain operations, whilst the latter seeks to lower prices to almost unprofitable levels a few times per month for select products in the hope that it will consumer attention and fuel the sales of additional products in the store.  A major problem with HLP strategies is that the momentary dip in price can wreak havoc on demand predictions for not only the low-price product, but for other products that are swept up within the consumer frenzy.  In one of Canada’s largest retailer, we found that their ability to predict demand to within +/-15% of actual demand occurred only 32% of the time.  But, with probabalistic methods and the recent advances of scientific pricing, this same retailer was able to predict demand to within +/- 15% of actual demand about 87% of the time!  Note, we are not pinpointing the demand value here, nor are we optimizing which specific products to invent and stock.  Rather, we are pinpointing a price that will maximize demand and revenue by understanding the probability of the market’s needs.

I believe that in order to respect the important points that several respondents to the “At the Mercy of the Market” blog have raised, it is important for us to be careful and augment words such as “pinpoint” and “optimal” with the phrase “expected returns.”  In a world of uncertainty, predicting the expected results using the Laws of Probability rather than the absolute results using a Crystal Ball is indeed the best we can do.  It is extremely important for us, and all other scientists / modelers, to inform the audience that we are in no way aspiring to peddle a crystal ball.

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