Gaming the Numb3rs

Last night at the Old Globe here in San Diego I got to see one of my favorite plays, Rosencrantz and Guildenstern are Dead, presented as part of the Globe’s 2013 Shakespeare Festival. Doing so brought the following post to mind in that it uses the play as a springboard for discussing probability and investing. I hope you will enjoy it — or enjoy it again.

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Tom Stoppard’s Rosencrantz and Guildenstern are Dead  presents Shakespeare’s Hamlet from the bewildered point of view of two of the Bard’s bit players, the comically indistinguishable nobodies who become headliners in Stoppard’s play.  The play opens before our heroes have even joined the action in Shakespeare’s epic. They have been “sent for” and are marking time by flipping coins and getting heads each time (the opening clip from the movie version is shown above).  Guildenstern keeps tossing coins and Rosencrantz keeps pocketing them. Significantly, Guildenstern is less concerned with his losses than in puzzling out what the defiance of the odds says about chance and fate. “A weaker man might be moved to re-examine his faith, if in nothing else at least in the law of probability.”

The coin tossing streak depicted provides us with a chance to consider these probabilities.  Guildenstern offers among other explanations the one mathematicians and investors should favor —“a spectacular vindication of the principle that each individual coin spin individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does.”  In other words, past performance is not indicative of future results.

Even so, how unlikely is a streak of this length? Continue reading

Gaming the Numb3rs

 

Tom Stoppard’s Rosencrantz and Guildenstern are Dead  presents Shakespeare’s Hamlet from the bewildered point of view of two of the Bard’s bit players, the comically indistinguishable nobodies who become headliners in Stoppard’s play.  The play opens before our heroes have even joined the action in Shakespeare’s epic. They have been “sent for” and are marking time by flipping coins and getting heads each time (the opening clip from the movie version is shown above).  Guildenstern keeps tossing coins and Rosencrantz keeps pocketing them. Significantly, Guildenstern is less concerned with his losses than in puzzling out what the defiance of the odds says about chance and fate. “A weaker man might be moved to re-examine his faith, if in nothing else at least in the law of probability.”

The coin tossing streak depicted provides us with a chance to consider these probabilities.  Guildenstern offers among other explanations the one mathematicians and investors should favor —“a spectacular vindication of the principle that each individual coin spin individually is as likely to come down heads as tails and therefore should cause no surprise each individual time it does.”  In other words, past performance is not indicative of future results.

Even so, how unlikely is a streak of this length?

The probability that a fair coin, when flipped, will turn up heads is 50 percent (the probability of any two independent sequential events both happening is the product of the probability of both). Thus the odds of it turning up twice in a row is 25 percent (½ x ½), the odds of it turning up three times in a row is 12.5 percent (½ x ½ x ½) and so on.  Accordingly, if we flip a coin 10 times (one “set” of ten), we would only expect to have a set end up with 10 heads in a row once every 1024 sets {(½)10 = 1/1024}.

Rosencrantz and Guildenstern got heads more than 100 consecutive times. The chances of that happening are: (½)100 = 1/7.9 x 1031. In words, we could expect it to happen once in 79 million million million million million (that’s 79 with 30 zeros after it) sets. By comparison, the universe is about 13.9 billion years old, in which time only about 1017 seconds (1 with 17 zeros after it) have elapsed.  Looked at another way, if every person who ever lived (around 110 billion) had flipped a 100-coin set simultaneously every second since the beginning of the universe (again, about 13.9 billion years ago), we could expect all of the 100 coins to have come up heads two times.    

If anything like that had happened to you (especially in a bet), you’d agree with Nassim Taleb that the probabilities favor a loaded coin.  But then again, while 100 straights heads is less probable than 99, which is less probable than 98, and so on, any exact order of tosses is as likely (actually, unlikely) as 100 heads in a row:  (½)100.  We notice the unlikelihood of 100 in a row because of the pattern and we are pattern-seeking creatures.  More “normal” combinations look random and thus expected.  We don’t see them as noteworthy.  Looked at another way, if there will be one “winner” selected from a stadium of 100,000 people, each person has a 1 in 100,000 chance of winning.  But we aren’t surprised when someone does win, even though the individual winner is shocked.

The point here is that the highly improbable happens all the time.  In fact, much of what happens is highly improbable.  This math explains why we shouldn’t be surprised when the market remains “irrational” far longer than seems possible.  But we are.

Much of that difficulty arises because we neglect the limits of induction.  Science never fully proves anything.  It analyzes the available data and, when the force of the data is strong enough, it makes tentative conclusions.  But these conclusions are always subject to modification or even outright rejection based upon further evidence gathering.  Instead, we crave and claim certainty, even when we have no basis for it. 

In his brilliant book, On Being Certain, neurologist Robert Burton systematically and convincingly shows that certainty is a mental state, a feeling like anger or pride that can help guide us, but that doesn’t dependably reflect anything like objective truth. One disconcerting finding he describes is that, from a neurocognitive point of view, our feelings of certainty about things we’re right about is largely indistinguishable from our feelings of certainty about things we’re wrong about (think “narrative fallacy” and “confirmation bias”).

As Columbia’s Rama Cont points out, “[w]hen I first became interested in economics, I was surprised by the deductive, rather than inductive, approach of many economists.” In the hard sciences, researchers tend to observe empirical data and then build a theory to explain their observations, while “many economic studies typically start with a theory and eventually attempt to fit the data to their model.”  As noted by Emanuel Derman:

In physics it’s fairly easy to tell the crackpots from the experts by the content of their writings, without having to know their academic pedigrees. In finance it’s not easy at all. Sometimes it looks as though anything goes.

I suspect that these leaps of ideological fancy are a natural result of our constant search for meaning in an environment where noise is everywhere and signal vanishingly difficult to detect.  Randomness is difficult for us to deal with.  We are meaning-makers at every level and in nearly every situation.  Yet, as I have noted before, information is cheap and meaning is expensive.  Therefore, we tend to short-circuit good process to get to the end result – typically and not so coincidentally the result we wanted all along.

As noted above, science progresses not via verification (which can only be inferred) but by falsification (which, if established and itself verified, provides relative certainty only as to what is not true).  Thank you, Karl Popper. In our business, as in science generally, we need to build our investment processes from the ground up, with hypotheses offered only after a careful analysis of all relevant facts and tentatively held only to the extent the facts and data allow. Yet the markets demand action.  There is nothing tentative about them. That’s the conundrum we face.

Even after 100 heads in a row, the odds of the next toss being heads remains one-in-two (the “gambler’s fallacy” is committed when one assumes that a departure from what occurs on average or in the long-term will be corrected in the short-term). We look for patterns (“shiny objects”) to convince ourselves that we have found a “secret sauce” that justifies our making big bets on less likely outcomes. In this regard, we are dumber than rats – literally.

In numerous studies (most prominently those by Edwards and Estes, as reported by Philip Tetlock in Expert Political Judgment), the stated task was predicting which side of a “T-maze” holds food for the subject rat.  Unbeknownst both to observers and the rat, the maze was rigged such that the food was randomly placed (no pattern), but 60 percent of the time on one side and 40 percent of the time on the other.

The rat quickly “gets it” and waits at the “60 percent side” every time and is thus correct 60 percent of the time.  Human observers kept looking for patterns and chose sides in rough proportion to recent results.  As a consequence, the humans were right only 52 percent of the time – they (we!) were much dumber than rats.  Overall, we insist on rejecting probabilistic strategies that accept the inevitability of randomness and error.

As I described yesterday, the great gambler Billy Walters uses a probabilistic sports betting model that is correct roughly 57 percent of time.  He expects and plans for being wrong 43 percent of the time.  Since he can’t predict the timing of his successes and failures, he has to be prepared for long losing streaks (although he obviously hopes that none are long as Guildenstern’s).  Common gambling practice had been (and often still is) to make fewer bets – to bet only on those games one is most sure of.  But that approach is not thinking probabilistically.  Walters makes as many bets as he can within the confines of his model (when he thinks the point spread is off by at least one-and-one-half points). 

For investors, the lessons to be gained here relate to diversification, a carefully delineated and bounded process, clear execution rules, and stick-to-itiveness over the long haul.  This doesn’t mean that quants should control everything.  Old school analysis and judgment still matter, perhaps more than ever since the pile of available data has gotten so large.  But it does mean that our conclusions need to be consistent with and supported by the data, no matter how bizarre the numbers or how long the streak.

Even 100 in a row.

Models.Behaving.Badly

Emanuel Derman’s new book, Models.Behaving.Badly, is a cautionary tale.  Derman, a former Goldman Sachs quant, examines why confusing illusion with reality can lead to disaster on Wall Street and in life.  You can read a bit about Derman (in his own words) here and here; his occasional blog is here. He is currently a professor at Columbia University, where he runs the financial engineering program, and is also a principal at Prisma Capital Partners, where he co-heads risk management.

As Derman puts it, the book is “about metaphors and analogies, about the nature of modeling and theorizing, the difference between them, why financial models will intrinsically and always at best be very limited approximations to reality, and what to do as a consequence.”

I had this general point driven home to me over the week-end in a surprising way.  On Friday, my better half and I went to see Moneyball.  I have been a big fan of the Michael Lewis book since it first came out back in 2003 (see my comments here) and was looking forward to the movie.  I was not disappointed.  It’s a great yarn with interesting applications to business and to life.

A small but significant role in the film was the Yankee cast-off, David Justice, played by Stephen Bishop (the actor, not the musician), a former minor league baseball player turned actor.  Bishop and Justice look a lot alike (see below) and Bishop had spent many hours as a kid idolizing Justice and mimicking his swing.   

Since Bishop and Justice had a bit of a relationship before the movie, Bishop drew on that experience and called on Justice for help.  No less an authority than Justice’s wife (not ex-wife Halle Berry) approved of the result, acknowledging that Bishop “nailed it.”

Happily, Moneyball the movie doesn’t deviate from reality nearly as much as the typical “based on a true story” Hollywood blockbuster.  But it is still a far from perfect representation of reality.  Obviously, movie-makers have much more of an interest in telling a good story than in scrupulously sticking to the facts.  But the limits of film vis-à-vis reality are far greater than those relating to telling a good story. 

On Saturday, I went to a youth football game near my home.  One of my wife’s students had written the best persuasive essay in her class.  Since the essay argued why his teacher should go watch him play football, we were in the stands.  David Justice was one of the coaches roaming the sidelines. I’m not interested in violating anyone’s privacy here, so suffice it to say that the real David Justice was significantly different from the David Justice character I had seen portrayed on screen the night before, at least in that instance.*  I shouldn’t have been surprised by that, but I was.  

Derman argues that in finance, models can only hope to provide a simplistic and very limited approximation to reality.  Movies — as in the portrayal of David Justice — cannot aspire to more and do not.  Indeed, they do not even aspire to that much. We would all do well constantly to bear in mind the limits of financial modeling in general as well as the limits of any specific model.  They aren’t representational; at best they are illustrative.

The resemblance of the Moneyball David Justice to the real David Justice is more than coincidental.  But it was still a long ways from reality — just like economic models.

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* Obviously, people are more variable and complex than any model, which means that it would be easy to push this metaphor too far.  However, one instance of inaccuracy is sufficient to falsify a model, yet it is not enough to conclude that it is altogether useless.  Indeed, it might be highly accurate overall, just mistaken in the particular instance I witnessed.  That said, based upon what I saw, “highly accurate overall” seems unlikely.