Fortunately for business decision-makers, fair pricing and optimal pricing are not at odds with each other but can comfortably coexist.  Over the course of the coming weeks my colleagues at Sentrana and I will be approaching the rich topic of fair pricing in a series of exchanges on this blog.

debating the age-old question of fair price

What is a fair price?  This question has perplexed humanity throughout history.  Leading thought output of the ages, from Aristotle’s Nicomachean Ethics to the Summa Theologicae of  Thomas Aquinas, Pierre de Fermat’s probability proofs and Adam Smith’s classsical economics, have all weighed in with considered opinions on the fairness and justness of alternative ways to price economic goods and services, and the debate continues today.  A series of letters exchanged between Blaise Pascal and Pierre de Fermat in 1654 is often regarded as a primal cause of the development of modern probability theory: this exchange was actually an attempt to establish a scientific basis for the notion of fair price.  In his paper “The Unity and Diversity of Probability” Rutgers professor Glenn Shafer shows how these letters created hypothetical games of value that we today can recognize as the application of probability methods to defend a price as ‘fair’ under conditions of uncertainty.

Uncertainty is the 800-pound gorilla in the room when it comes to price-making decisions.  Buyers and sellers operate from positions of considerable uncertainty in approaching transactions with each other: buyers have only partial information about the features of what they are buying such as quality, reliability, service support and the extent to which a given offered price may be reasonable in relation to these features, while sellers have a limited perspective on what demand exists for their products and what combination of levers such as price, assortment and marketing could influence that demand.  Buyers thus face the risk of inequity in their exchange – paying more than the intrinsic worth of the object acquired, while sellers face the risk of their transactions being unprofitable and, if persistently so, driving them out of business.

Having worked for a number of years in the investment industry I offer up a useful model from this corner of the economy for dealing with uncertainty.  In the investment world uncertainty commands a price: investors demand more compensation, in the form of return on investment, for assets that exhibit higher levels of short term volatility.  Participants widely view this as fair: it is not thought ‘unfair’ that an investor in, say, a 5-year U.S. Treasury note earns a dependable return of 5% whereas someone who takes a punt on the shares of a small-cap biotechnology company may potentially earn over 25% in the same time period.  There is more likelihood that the value of the biotech shares will plunge in the wake of unexpected news or that the company will go out of business than there is of the U.S. government failing to honor its legal obligations to bondholders.  A capitalist economy offers the potential for greater rewards to the investor willing to assume greater risk.

what matters is the customers who don't come

How is this concept analogous to the uncertainty faced by businesses that sell in markets for real (i.e. non-financial) goods and services?  I thought about that the other day while dining out at one of my favorite Northern Italian restaurants, located in a trendy urban area chock-full of good eats.  As I looked around the dining room on a late September Tuesday evening it occurred to me that the uncertainty this business experiences on a daily basis is plainly visible: the number of empty seats during peak dining hours.  Restaurant patronage is a notoriously fickle notion to quantify and is subject to considerable fluctuation in real time.  I wondered about the methodology through which this restaurant’s owner translates the uncertainty of empty seats into the revenue model. It seems to me that the real art to the formulation of this model is not based on the tables that have patrons sitting at them, but rather the ones that are empty.  The hard part of revenue calculation is not figuring out what the average occupied table will spend on any given night – it is dealing with the uncertainty of those empty tables.

Now in theory, the owner could simply build an ‘uncertainty factor’ into menu prices as a partial compensation for the prospect of empty tables.  In practice this is unlikely, and the reason why it is unlikely brings us back to the concept of fairness.  Prospective restaurant patrons (including yours truly) are very unlikely to be sympathetic to the notion that they should have to pay a higher price for the verze e luganega because it helps the owner’s revenue model – to us patrons, that is an unfair offloading of the owner’s problem onto us.  We don’t even have to explicitly know the owner’s motivation.  Discerning customers have plenty of access to comparative information – from other restaurants in the area, our social networks, Internet reviews and so forth – to form strong perceptions of the fairness or unfairness of prices at any given spot.  We will wield our verdict of ‘fair or unfair?’ with much self-righteous certitude in making future dining out decisions.

So what is a ‘fair’ way for our poor restauranteur to manage uncertainty toward a more profitable outcome?   Rather than accepting empty tables as a given fact of life the owner can try to figure out intelligent ways to fill them.  Who may be walking by the restaurant in the late afternoon, or working in a nearby office building and considering an after-work dining outing with colleagues?  What combination of factors might entice these and other prospective patrons to choose this establishment over numerous other choices?  Is there a way to figure out attractive deals that would lure certain prospective customers and to surgically target each such customer with a unique offer?  Yes – it is possible through scientific micromarketing techniques that optimize at the granular level of the customer-product interaction.  The next question – if it is possible, is it also fair?

All those centuries of debate on the notion of fairness and justice in economic commerce now come back into this discussion.  Paul Krugman expressed a concern about this in a New York Times op-ed piece titled “What Price Fairness?” all the way back in October 2000, when price optimization methods were in a much, much earlier stage of development.  His remark (related to the notion of dynamic pricing in general) was that while it may be “arguably good for the economy,” dynamic pricing is also “…unfair: some people pay more just because of who they are.”  Sitting in the restaurant, I imagined a hypothetical case where the gnocchi with sweet basil pesto, which I ordered for the menu-listed price of $14.50, was being enjoyed by the gentleman at a nearby table for $11.30 simply because the restaurant’s micromarketing system contacted his iPhone with a targeted discount offer just before he left his office just down the road.

Is that unfair?  I don’t think so.  Who wins and who loses in this scenario?  The gentleman who receives the offer wins – he gets the opportunity to enjoy a dining experience targeted to his personal preferences.  The restauranteur wins by filling a table that would otherwise be empty, reducing uncertainty and improving the nightly profit intake.  I am still enjoying the gnocchi I ordered at full price and am no worse off than I would have been otherwise; having already concluded that $14.50 is a reasonable price for the dish and ordered on that basis.  On a broader social scale the notion of micromarket pricing does not discriminate between the two of us in a way that I would deem unfair.  I have my own set of preferences that may benefit me with a different offer set on a different day.  In fact, were I to be made aware of the circumstances under which the gentleman got his gnocchi for a lower price, I may well be inclined to leave my own contact information with the establishment in anticipation of future benefits.

There is a road ahead before scientific micromarketing becomes a more accepted feature of B2C commerce situations like that of my hypothetical imaginings while dining out (no doubt helped along by the delights of a 2003 Castello di Camigliano Brunello).  And I expect that a vigorous debate about the question of fairness versus optimality will be part and parcel of this journey.  At day’s end, though, I believe the two are fundamentally compatible.

Looking at the human mind this way, as a machine constantly evaluating possible outcomes based on prior knowledge and assigning probability weights to those outcomes, gave rise to the notion of the “Bayesian brain,” a term popularized in 1983 by Geoffrey Hinton of the University of Toronto and Terry Sejnowski of Johns Hopkins University.  This notion has subsequently received a good amount of validation by neuroscientists as they continue to make advances in understanding how the brain really works.  Neuroscientists Alexandre Pouget and David Knill of the University of Rochester in 2004 referred to a “growing body of evidence that human perceptual computations are ‘Bayes optimal’ (“The Bayesian brain: the role of uncertainty in neural coding and computation,” Trends in Neurosciences, vol 27 issue 12, December 2004, pp 712-719).  That’s a fancy way of saying that we make estimations based on likelihood – for example we routinely make estimations about things like the distance from us of an object, or the speed at which it is traveling, based on our prior knowledge of the shape, clarity and movement patterns of such objects and the likelihood that the present reality fits into that a priori knowledge.

So far, so good.  But there is an Achilles heel to our hierarchical mental gymnastics.  Briefly, we may be great at the kind of proposition-within-proposition reasoning that our sales rep exhibited in getting to the essence of the competitive threat posed by the wholesaler’s move to town.  But we humans are generally pretty lame when it comes to computation.  Our intuitive reasoning skills fail at the task of instantaneously calculating that 34 x 57 X 71 = 137,598.  One way that we get around this failing is through heuristics – basically, shortcuts that we use in conditions of uncertainty to help us get from information to evaluation and decision.  Take the example above of evaluating the distance a particular object might be from us.  Now, there is a physics formula we could apply to measuring distance, height, momentum (if moving) and so forth to give us the precise answer as to how far away that bicyclist in the yellow jersey is from us.  But our brains lack the ability to do spontaneous physics equations.  Even if we precisely knew one or more of the variables it would be hard to do an on-the-spot computation.  So we need something else – a proxy, a heuristic.  That something might be clarity.  Can we see the bicyclist clearly?  Can we make out the details of his black helmet with red stripes, and the ‘Elf Aquitaine’ logo on the yellow jersey?  That can give us enough information to call upon our “cyclist in a yellow jersey” neuronal connections and infer a likelihood that he is, say, 50 yards away.

The problem with heuristics is that they are subject to error: not occasional lapses in judgment but systematic, predictable biases.  For example if we use clarity as a heuristic we may overestimate the distance of that cyclist from us if there is poor visibility.  Understanding the role of heuristic errors in human judgment and decision making was one of the main contributions of Amos Tversky and Daniel Kahneman to our understanding of behavioral factors in human decision-making (“Judgment Under Uncertainty: Heuristics and Biases,” Science New Series vol 185 no. 4157 Sep 1974 pp 1124-1131).  Tversky and Kahneman documented specific heuristic errors such as representativeness (drawing broad or sweeping conclusions from a limited data set), availability (assigning the likelihood of an event based on the easiest example that comes to mind, whether or not appropriate, and anchoring (relying heavily on one piece of information when making a decision even if it is irrelevant).

Since heuristic errors are part and parcel of human judgment and decision-making under uncertainty, we have to take this reality into account when we attempt to integrate quantitative modeling methods and qualitative human judgment.  What are the best mathematical tools and frameworks to integrate these two domains?  One area in which we at Sentrana are spending considerable time is that of Bayesian hierarchical modeling. The Bayesian approach is particularly useful in marketing situations such as modeling differences in the needs and wants of customers using both generalized and conditional assumptions involving multiple variables. Bayesian frameworks provide a natural way to pool disparate sources of information. A Bayesian model requires the formulation of prior distributions and the estimation of a likelihood function, which can add complexity to the model-building process.  However we expect future insights and innovations in this area, alongside the development of more robust computation capabilities to bring more firepower to bear on this difficult but potentially valuable quantitative approach.

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.

As I see it, the problem with the financial market meltdown is not that David Li published an article in the Journal of Fixed Income Securities on the Gaussian copula function, or even that in his article Li, then an analyst with JPMorganChase, identified the price of credit default swap (CDS) contracts as a seemingly elegant proxy for the mortgage market – a proxy that greatly reduced the immense complexity of modeling values and risks in this market but, as it turned out, lost a great deal of critically important information along the way.  No – the real problem was with the incremental decisions practitioners made to adopt this model wholesale, to leverage it up to 50 or more times the worth of the underlying assets, and ultimately to heedlessly employ it as a path to untold riches.  In other words it was the people who used the model, not the model itself.  It was the rating agencies who, in conferring the AAA ratings without which the securities would have never been as widely distributed as they were, assumed that housing prices would never go down.  It was the investment bankers who successfully shouted down the warnings of their internal credit risk departments so that they could sell ever higher volumes of CDOs, with ever-higher levels of leverage, in order to maximize their year-end bonuses.

You can’t fully hedge model risk – that is true.  But you can mitigate model risk through the application of robust decision-making processes.  A model did not take down Wall Street. Models do not “screw up” – they do exactly what they are supposed to do once they have their inputs. The screw-ups occur solely in our application of models to inappropriate situations or to situations which we do not fully understand. The predictions may not reflect reality outcomes as precisely as we wish, but that possibility of error needs to be accounted for by the ultimate decision makers. The output of a model should be an input in any decision process, not the entire decision process. For that reason, the emerging generation of predictive technology solutions being employed by all varieties of business seeks to marry model holism (by including ALL of the relevant variables, rather than the most computationally feasible), computational firepower, and above all, ranges of probability-based recommendations, rather than a single output. It may sound heretical at the moment given the present economic calamity, but as the world gets ever more complex, models will become more valuable to decision makers, not less. Informed, prudent decision-making in regard to those models will not be a luxury, but an absolute necessity.