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.

And then there are those rare occasions where we encounter a street musician whose music soars so high that we are forced to refer to him simply as a “musician,” for using the adjective “street” would be nothing short of a criticism.  About 2 years ago, this is what I encountered at one of Washington D.C.’s busiest Metro (subway) stations during the morning rush hour.  It wasn’t until much later in the day that I discovered the musician in whose masterly hands the violin “sobbed and laughed and sang” was the great virtuoso Josh Bell.  In the middle of the morning rush hour, 1,097 commuters passed by and all heard soul-stirring music at a price of their own choosing that just a few days earlier fetched more than $100 a seat at Boston’s Symphony Hall.  Josh Bell played to a rush hour herd, and demanded no price for priceless music.

His income depended not on the value he provided to those 1,097 passersby, but the overwhelming value he provided – for, if he failed to stir, we listless commuters would feel no compunction to pause and forfeit even a meager fraction of our purse.  And stir he did, with a masterly performance of Bach’s Chaconne from Partita No.2 in D Minor.  Of the almost 2,000 pedestrians that filed by, only 27 gave money for a total of $32.  In other words, for a performance that was described by the Washington Post as “pearls before breakfast,” less than 3% of us offered any payment (for “a man whose talents can command $1,000 a minute”).  Did the service deserve such scant payment, or was there more to the revenue than just the greatness of the service itself.  This is a question that goes right to the root of just how complex the endeavor of pricing can be.

Beyond Josh Bell’s performance and the payment it merited, there is a litany of other factors that affected his earnings power (or pricing power if he were to charge a price).  First and foremost, there were 3 possible locations at which Bell could have positioned himself (see figure below):

1. At a location of high transient pedestrian traffic (between the entrance door to the subway station and the escalator bank that leads to the underground train platform).
2. At a location of stationary traffic waiting for a subway train to arrive (i.e., on the underground platform).
3. At a location where the acoustics of the Metro arcade would create the most perfect sound possible within the subway station.

Of the 3 possible locations, Bell perhaps chose the worst location to generate earnings.  No matter how good his music, rush hour traffic has no time to stop.  It wasn’t their purse that the commuters failed to contribute, but their precious time.

The attributes that govern your power to generate revenue transcend the product itself.  Bell’s performance at the L’Enfant Plaza Metro stop highlights several fundamental truths of pricing science:

1. Could being situated at one of the locations where stationary traffic was high have yielded more revenue?
2. Or, perhaps could being situated at a location where the sound would be even more magnificent have yielded more revenue?
3. What if the perfect acoustic spot had only scant stationary traffic?  Then, sadly, we can surmise his income would be lower even though the quality of the product would be it’s highest.
4. To complicate matters even further, what if Bell chose a different day for his performance?
5. What if he played on a sunny, spring day where spirits are higher instead of a dreary winter morning?
6. What if he played on payday (in the Federal government, payday typically lands on the 2nd and 4th Friday of the month) instead of an arbitrary day?
7. What if Bell advertised with a sign around his neck that he was indeed Josh Bell?
8. What if underneath the sign, Bell posted a requested donation of $5 for his performance?

By not focusing on these questions, and concentrating solely on his music (i.e., the tangible product itself), Bell failed to create a high market price for his product, and earned only $32 dollars from a 43 minute performance that in a proper venue would have earned 6 figures.   All of these factors would have dramatically affected the revenues Bell earned that morning.  But, identifying the 12 square inch box out of the 15,000 square feet of space in which Bell would have obtained maximum revenue (i.e., gotten an “optimal price” for his public performance) is an optimization problem of mind-boggling complexity, and simply cannot be solved without data and quantitative analysis.  After all, the analysis is not just about where Bell should stand to optimally balance sound quality with foot traffic, but also about how this optimal location varies with changes in any of the above 8 questions.  The lessons here are deep, and profoundly shape our responsibilities as pricing managers.

Before you jump to match your competitor’s price, you should recognize the market’s willingness to offer you payment that goes beyond the value of the product itself.  It is important for you to optimize the marketing attributes of the product in order to optimize the price of the product.  You are not at the mercy of the market. Through your actions you can greatly influence the market price of your product. As a pricing manager, you should not just view the setting of price as your only responsibility.  You have access to much data about your market and your previous sales and your customers, which you can leverage to determine the interplay between all of the marketing attributes of the product and the price of the product.  The price you are able to make is inextricably linked to the actions of your marketing managers, your category managers, and your product and sales managers.  And once those actions are executed, you can digest all of your data to pinpoint the optimal price you can charge in the market.  Last but not least, as a pricing manager, you are the final decider of the trade-offs your enterprise should make to maximize immediate earnings and establish a long-term competitive advantage:  Should you play your violin where it sounds the best but generates the least income or do you play it where it sounds the worst but stands to generate the most income because of high foot traffic?  If you do the latter, will those that pay you today have the loyalty to pay you again tomorrow?

So, before you give in to the seduction of lowering your prices to beat your competitors, remember to identify and carefully influence all of the attributes (i.e., don’t just make great music, stand in a great location) that will compel the market to pay you your just reward for the goods and services you provide.  The crux of price optimization is not about touching price at all, but touching all the other things that make a price.