If figuring out how to maximize your revenues by charging the right price is hard when people actually need your product, imagine how much harder it is when they don’t need your product or don’t necessarily even need to pay to enjoy your product. The lessons learned from how to maximize revenue in this regard, which is a much more formidable challenge, can profoundly impact your ability to maximize earnings in the less difficult situation where people have no alternate choice but to pay for your product. In a stroll down a busy street, we will once in a great while receive a good that can stir our soul yet require no payment. We receive this good from the ubiquitous street musician who earns his income as a mendicant who lets you set the price (which is often nil), rather than setting his own price for “services tendered.”
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. Continue reading →
“Burn the mathematics” wrote economist Alfred Marshall in a letter to a friend, musing about the proper role of mathematics and scientific inquiry in the field of economics. That 19th century cogitation would seem to be a prêt-a-porter soundbite for these latter days of the 21st century’s first decade – a time in which the mathematical infrastructure that underpins longstanding economic and financial theories stands accused of all manner of malfeasance, particularly given its presumed role in the decade’s signature economic event – the financial market meltdown of 2008. The logic behind the accusation goes roughly thus: More complex (but not necessarily more “accurate”) models allow for more complex instruments to be created. Increased complexity means it takes more time to process and then fully comprehend what the numbers may be telling you. At the same time, though, technology allows buy and sell orders to be executed almost instantaneously through electronic trading systems. Time is of the essence, and ponderously complex computations simply won’t do. A seemingly elegant (and fast, and commercially viable) shortcut is discovered and becomes the currency of the day. The models’ outputs come to be trusted blindly simply because there is no time to question them (and too much money to be made by using them). The impenetrable Greek letters obfuscate the sensitivity of the models to changes in important assumptions – which is fine for a few years because those assumptions (e.g. rising housing prices) don’t change – but then all of a sudden they do. The models start losing more money than they make. Then the chasm widens further as the high levels of leverage in the system make themselves felt. The losses accelerate dramatically, wiping out years of profits in just a few months. Burn the mathematics, indeed.
But let’s take a different look at this apparent tight coupling of mathematics and dire outcomes. Our recent correspondence with an author who has been widely published on the subject of Wall Street’s use of mathematical models recently offered to us an interesting opinion. His point was that the problem with the models was not so much their complexity, but rather that they were models in the first place. His argument was that you can’t ever perfectly hedge model risk. Now, I agree with that observation: a model by definition selects some aspects of reality to represent and omits others, and the choice of what to include and what to omit is subject to human error, therefore fallible and not perfectly hedgable. But I take issue with the idea that the fault lies in the existence of the models themselves. Models can be misused – I think that much is clear. But the notion that models are all doomed to failure obscures a deeper truth about the goals of predictive modeling; namely that you can seek either to reduce the world or truly explain it. By trying to elegantly reduce the world to as few predictor variables as possible, you are more likely to be sowing the seeds of future failure, because complexity and actual drivers of outcomes are taken out of the equations to make them more solvable (or perhaps sellable, as in the case of the Gaussian copula function that was behind Wall Street’s demise, as we discussed in a previous posting “You Can’t Punt Away the Dimensionality Curse”). Predictive modelers don’t have to go down that road, however: they can also set out with the goal not of reducing an entire system to a single neat, tractable equation, but to quantify and explain all of the relationships that dictate outcomes to the absolute fullest extent possible. Tractability and computability are things to address later in the process, through technological means, but they should not dictate the fundamental mathematical approach at the outset. Continue reading →