Monday, May 14, 2012

Pluck* or Luck

*pluck -  definition 12. noun. courage or resolution in the face of difficulties.

College is a good time for students to confront the meaning of life questions.  Those questions take on different forms, depending on the personality and experience of the person posing them.  For shy people those questions may translate into asking whether by searching inward one can find a source of confidence.  To address this question the person is likely to reconsider past successes.  Were they attributable to luck or to force of will?  If the latter, then confidence can emerge from this realization.

Alas, this question is remarkably difficult to answer.  Conceptually what you'd like to do is go back in time to before the event unfolded and then replay it again, but this time as an alter ego as well as participant.  Does the success repeat?  Or is there failure this time around?  Observing failure would connote that luck caused the original success.  Repeated play of this Gedankenexperiment that leads to success each time makes for the inference that force of will determined the outcome.

I believe the issue is quite different for physical performance, playing tennis for example, than for pure thinking, such as solving a hard math problem. With physical performance there is some internal monitoring and control during the activity- get the racket back early,  lean into the swing, follow through.  Too much of that can limit function, the play becomes mechanical as a consequence, but some is necessary to stay in the point.  With thinking however, a big key in arriving at a solution is to become entirely absorbed into the world of the math problem and, consequently, to become totally oblivious to other things that are going on.  That oblivion includes your own thinking.  So there is no monitoring then.  There is only the problem itself and finding a path toward the solution.  This remains true as long as there is another possibility to try.  Having such possibilities encourages motion in thought.  If, however, you get stuck and have no more possibilities, then self-awareness and monitoring return. 

Getting unstuck clearly entails force of will, especially when giving up is a viable alternative.  But fierce determination in solving a hard problem in itself doesn't rule out that serendipity too lent a hand in finding another viable path. What does rule out luck?  Or, on the contrary, what says that luck played a significant role?  I'm neither a probabilist nor a philosopher, so I'm going to try to answer that question as pragmatically as I can.  I'll start with math problems, provide an answer for that case, and iterate from there. 

For math problems there is little or no luck involved if the person working the problem finds it easy.  At this juncture the reader will be apt to challenge that conclusion.  A couple of paragraphs earlier I said the problem was hard.  How can a problem be both hard and easy at the same time?  The SAT offers an answer to that question.  Some practice problems are available online from the College Board.  If you try one, you'll see that they rate it by difficulty.  In other words, the difficulty is measured by aspects of the problem itself, independent of the person who is doing it.  Are there people who find problems easy even when "authorities" like the College Board say they are hard?  Of course there are.  In this case, that the problem is hard means there are comparatively very few of these people in the entire test taking population.  The vast majority of the potential test takers would find it difficult. 

Are there problems that only a few people on the globe can do and even they would say the problems are not easy?  In that case what is it about the problems that make them hard?  A different sort of test, the Putnam Exam, targeted at college students with an interest in math, is a better place to look to consider this question.  I took it twice, I believe in December 1972, when I was at MIT, and in December 1973, after I had transferred to Cornell, though that second time might have been 1974, my memory is not that sharp on some of the details.  That second time I scored in the high 20s out of a possible 120, and that got me ranked around 300 out of the 2000 or so students who took the exam.  Since past exams are available as far back as 1975 and mock exam questions are available as well, one can get a sense of what the exam is like.  In my view, there are two different things that make these questions  hard.  Some require a fair amount of math background, just to gain an understanding of what the question is asking.  If you don't have the background, the question isn't just hard, it's impossible.  More importantly, generating a solution to a Putnam question requires a fair amount of invention.  Since they want proofs, as well as answers, you have to come up with an argument that proves the conclusion.  In contrast, for the SAT the path to the answer is typically immediate, especially when considered from the perspective of somebody who will sit for the Putnam.

I want to illustrate with a particular problem that I remember from taking it the second time.  Anybody should be able to understand what the question is asking.  It is not hard that way.  I'm offering it up so you can ask yourself whether you could find the answer and the proof. The explication will also help provide a sense of what I mean by "a fair amount of invention."

Problem:  A circle with center at C resides in a vertical plane.  The circle touches the ground at point B.  There is a point A, above the ground and outside the circle.  (See Figure 1 for this setup.)  A line segment is constructed from point A to the circle in such a way that an object slides without friction along the segment under the force of gravity till it reaches the circle.  Characterize the particular line segment  that gets the object to the circle in the least time.

Conjectured Solution:  Draw the line from A to B.  Call the point where it first crosses the circle T.  The segment AT is the optimal path. 

Sketch of Proof:  There is a bit of grind work that needs to be done that I will not provide here to reach the following preliminary result.  If you consider an iso-time curve, the locus of all points reached in a fixed amount of time as the path varies, this is a circle whose circumference goes through A and A is the highest point on that circle.  Armed with this result, the rest of the problem requires only high school geometry.

In Figure 2, a possible though not optimal path, AX, is implicitly given. The circle that has both A and X on its circumference is drawn.  The reason AX is not optimal is that one can find a circle through A with smaller radius (less time) that still intersects the circle with center at C.  However, this non-optimal path is useful for constructing some other segments that aid in the proof.

A line segment is drawn from B to X and then extended to the vertical line through A.  It touches that vertical line at B'.  Another line segment is drawn from C to X and extended to touch the vertical line through A.  It touches that vertical line at C'.  Since both BC and CX are radii of the original circle, triangle BCX is an isosceles triangle.  Triangle B'C'X is similar to triangle BCX, so it too is an isosceles triangle.

The optimal path, AT, is such that the iso-time curve through A is tangent to the original circle at T.  This is depicted in Figure 3.  Then it must be the case that C' is the center of the circle through A and T.   And since B'C'T is isosceles, it must be that B'C' is a radius of that circle.  In other words, B' must coincide with A.  That completes the argument.


Let's now return to the role luck plays in working such a hard problem or set of problems.  At first, you might guess at the conjectured solution and guess at a means of proof.  If both prove correct, you are lucky indeed.  You solve the problem and do so quickly.   In writing up the proof yesterday, I was not so lucky.  I stumbled before coming up with a good argument.  What I've written above represents the third thing I tried.  (I also tried using the highest point in the original circle instead of using point C.  But I couldn't get the argument to work with that construction.  Then I focused on point C, but only considered the optimal point T, not an arbitrary point X.)  And I was stuck for a while between these.  As I'm stuck I start to lose confidence that I'll find a solution and go into semi-panic mode.   But I had a need to make this work, as I hope to show later in the post.  That need contributed to the pluck.

In an actual test taking setting, even for somebody who has the ability to do a problem like this, the time to when the solution is complete has to be random.  The problem offers sufficient complexity that it is not possible to forecast the time to completion with any precision because what needs to be done is not well understood at the outset.   The Putnam exam gives two blocks of 3 hours with six questions in each block.  If you're lucky and on the first problem or two that you work on you find the solution quickly, you've now increased the chances that you'll get another problem in that block as well.  Conversely, and I believe more often the case, you fritter away a good chunk of time on a problem without making much headway on that one.   With little to show for that effort you have to decide whether to cut your losses and move to something else or keep on working on that same problem.  Early failure can then make it less likely you'll  succeed later on.

There is another source of randomness in the actual testing situation.  At the beginning you glance through all six questions and make a quick ranking of their difficulty.  The easiest among these is what you will work on first.  However, this ranking depends on first impressions only.  It therefore may be wrong.  The first question you choose to work on may not in fact be the problem you'd have found easiest.  Another problem may seem off putting, but after only a few minutes of thinking about it you might penetrate it.  Then it's your hard luck for not having chosen that one at first.

In the non test taking scenario the issue becomes the opportunity cost of your time and whether you should abandon the problem outright after having worked on it for a while or perhaps not to try it altogether to protect your ego from getting a bruising. Absent getting keyed up to work the problem, a positive effect from doing it in the test environment, tossing it in may appear a good choice.  But it is a choice made in ignorance since it is impossible to know what the consequence from taking the other path would be without having taken it.

The above discussion focuses on internal (to the person) randomness - uncertainty in outcomes leads to uncertainty in how to proceed.  I've argued elsewhere that it is not correct to characterize the process as trial and error.  There is more intelligence to it than that.  The intuitive sparks matter in a good way in the process.  But those sparks can also compound the uncertainty.  Of course there is also environmental uncertainty, the type we usually consider.  As any student who prepares for an exam knows, while the preparation is obviously helpful for the ultimate outcome it still matters which questions appear on the test.  With the Putnam exam, the range of possible questions is huge.  So the environmental uncertainty is large.  Nevertheless, it is a mistake to consider only that uncertainty as the source of possible variation in outcomes.  The internal uncertainty matters a great deal.  Perhaps there are a handful of college students throughout the U.S. and Canada who can get very high scores on the Putnam every time out.  The test is easy for them.  For the rest of us, if we got a good score part of the explanation is that we had a good day. 

* * * * *

I'd now like to extend the discussion by moving away from closed ended math problem solving to more open ended creative endeavors where there is no "right answer" but rather only pleasing outcomes or less satisfactory ones.  But before I do let me briefly take on the shyness issue once again.  I have to laugh at myself now for some of the meaning of life thinking I did in college and why I thought an examination of the math problem solving would help create a sense of confidence for me in other domains, where I had less proficiency. I must have had an impulse that the creative intelligence cuts across domains.  But I also must have believed it can substitute for some of the lack of proficiency.  I no longer believe that.  If you're ignorant about something you need to learn quite a bit about it first before a real creative process can begin.  And if you don't have talent in some area, you can't fake it that you do, at least not for very long.  Yet while this particular investigation went for naught, there are a couple of other things that I've glommed onto over time that have helped.  One is to take satisfaction in personal idiosyncrasies.  This is in the spirit of the who-cares-what-others-think approach, but it it a bit more in that it separates work related things for which the opinions of others should matter from the rest.  The other is to recognize that the nervousness will come in certain situations but not to dread those situations quite so much ahead of time.  Neither offers a perfect solution, by any means.  But together they've allowed me an outgoing style of performance when that is required by the situation.

For my example of open ended creations I will focus on "slow blogging" posts such as this one, since I've got a fairly good grasp of what goes into the writing.  The situation is substantially more complex than with the Putnam exam math problems.  I will delineate some of the additional sources of complexity.

First is the choice of topic.  This comes from issues I've been thinking about or one particular piece I've read recently.  But it is my own spin on the matter.  On the topic of this post, for example, many other people have written on the subject and I will try to get to some of them later in the post.  But this is not a simple regurgitation of their views.  It's my integration and synthesis plus perhaps some novel contribution to the discussion.  Part of that novelty comes from prior thinking on my own about how even very intelligent people are remarkably ignorant when confronted with important decisions and in situations where they hold responsibility they often act based on that ignorance rather than seek additional information.  Topic choice for a blog post is a very open ended matter, even when trying to go with the crowd and write on something that is being batted around, which I used to do with the blog when I held my administrative position in learning technology and would write about issues that were relevant to the field.

Then there is some mental scaffolding done about what you want to say in the piece.  I've written elsewhere that I don't like to make outlines because they tend to block flexibility later and they require that you know more than you do early on.  The mental scaffolding comes from an activity called pre-writing - thinking about the piece before getting to the keyboard.  I learned that term from Donald Murray.  Murray was an early proponent if not the inventor of the concept called, "writing to learn."  This means there is much discovery along the way.  It's tautological to note that if you are about to discover something, then you don't know about it ahead of time.  This means the discovery approach to writing is full of uncertainty.  I believe that all real learning is.  I should also add here, because recently I've tried to use writing as a big component in the course I teach, that pre-writing is not understood as a necessary component to writing.  Most students have the expectation that you just sit down and write.  Where is their thinking in that?

Sometimes I go well beyond scaffolding and try out full arguments and lines I might use in the post.  For me, pre-writing is having a debate with yourself.  Often, however, my conclusion is that I don't know enough to resolve the issue then and there.  I need to do more research/reading to come to a conclusion.  At this point discovery refers to what pieces to read I might find from my searches as well as my take away from having read them.  There is as a consequence an interaction between the environmental uncertainty and the internal uncertainty that doesn't exist in the math problem case. Surely this increases the complexity.

It may very well be that after doing some research in this way that I see the early mental scaffolding is flawed or, even if not, that a different structure to the piece would make the argument more convincingly.  So the scaffolding must be modified based on what has been learned since.  Those modifications are subject to the understandings recently arrived upon.  Even if we can envision environmental uncertainty here, the uncertainty in understanding is due to a mixture of that and my sense of taste.  As disclaimers are wont to say, your mileage may vary.  In other words, this too can contribute uncertainty to what will be seen in the final product.

Let me turn to the goal, or the set of goals I've got with the writing.  I believe strongly that in all open ended creations the creator's first and foremost responsibility is to please himself or herself.  Taking that as a given, doing so requires a sense of taste regarding what might be so pleasing.  That sense of taste is necessarily idiosyncratic, typically learned from several prior masters through previous reading and finding those pieces that were adored.  It is learned too from other writers, where their work is appreciated though not loved.  And it probably is learned in the negative too, from pieces that weren't liked at all and maybe weren't completely read as a result.  The sense of taste is also modified through my own prior efforts in constructing posts and reflecting on the better ones and the attributes they had that put them in that category.  

The audience matters too, of course.  Does what pleases me also please my readers? How does one get to know the answer to that question?  Once in a while I get a comment with a positive reaction from a reader.  (I re-post these blog essays in Facebook.  They are publicly viewable there, but only Facebook friends get an indicator that a new post is available.  Nowadays, more comments come in from that venue than through the blog itself.  Still, the entire stream is only a trickle.)  The post meant something to that particular reader.  In turn, this typically means it helped the reader think through an issue that either wasn't previously considered at all or which had been thought about but not to a fully satisfactory conclusion.  What does the comment say about the reactions of other readers?  And what, if anything, can be inferred when I get no comments? 

Then there is the issue of whether potential readers actually end up reading the post.  Most of them lead very busy lives.  I have a reputation for writing longish meandering posts that might require some effort to read through.  (See here and here, for example.)   The potential reader might see the post title and the first sentence or two and based on that make a judgment whether to read the rest.  I do exactly the same thing for my own reading, for example in choosing pieces in the New York Review of Books to read.   It is a hit and miss proposition by its very nature.  I might write a wonderful piece yet it remains totally obscure.  Once in a while the opposite occurs.  An uber blog may post about it.  Or the Google search engine ranks the piece highly for certain keywords.  Then the readership can expand much beyond the usual.

At a minimum there are two distinct vantages to consider the success of a post.  One is the about the journey I've taken in writing it.  What have I learned?  Where did I compromise in order to produce something palatable?  Have my inner demons on that issue been expunged so I can move onto something else?  The other is about the produced object itself.  Hit counters and related metrics give some indicators of success this way.  But I want deep more than I want wide.  How does the object do on that score?  Alas, that remains a vast unknown.

* * * * *

Ever since having finished reading Mary Parker Follett's Creative Experience and trying to apply the lessons in it to our present times, I've felt an obligation to engage Conservative thinkers in their policy views and do so on their own terms.  I tried to do this in my previous post.  But it is difficult if not impossible to refrain from being mocking or abrasive, since the world views are so far apart.  Arguing in a mocking or abrasive tone, however, produces gridlock, not a better understanding.  Two distinct positions remain without any convergence between them.

In the process of playing out one such policy debate in my head, making an argument that I took as obviously true but that a conservative thinker might disagree with, it occurred to me to take on something more basic first, the pluck or luck question.  So I did a Google search on "the role of luck in new business success" but without the quotes.  From that I found this essay in the New York Times by Jim Collins and Morten Hansen.  I had seen Collins on the Charlie Rose show, so I paid attention to this piece.  The essay argues for pluck, but of a certain kind - when good fortune comes your way you need to know to "go all in," while when bad fortune occurs you need to make dramatic reforms in what you're doing.  The people who do that are wildly successful.  The rest of us are more wishy-washy in the face of uncertainty and hedge our bets.  There seemed to me something to this.  Michael Lewis in The New New Thing made essentially the same point about Jim Clarke.

Collins and Hansen tell the story of a young Bill Gates.  Was Gates unduly lucky or did he have the je ne sais quoi that allowed him to outgun all his contemporaries?  Collins and Hansen say that Gates's advantages were also there for many others at that time.  So while Gates was positioned to succeed that wasn't sufficient to determine his subsequent success.  I wasn't completely convinced by their story.  Flipping coins, it is possible to get 10 heads in a row right off the bat.  It's not likely, to be sure, but it's possible.  One should not attribute a special skill to the coin flipper when that happens. If the population Collins and Hansen are referring to is on the order of magnitude of a few thousand people, how do they rule out in their work that Gates was the one who kept on getting heads?  Switching to their actually story, I was kind of amazed that IBM didn't appear in it.  Had IBM decided to develop in house the operating system for the early PCs, we'd have never known who Bill Gates is.  So I had my doubts that the Collins and Hansen story was very convincing.

A little later that same day I experienced a bit of true serendipity.  There was an awards ceremony at the high school.  My younger son is a graduating senior so I went to the school to see the ceremony.  Since my older son had been through this before, I knew the drill.  We'd be sitting in the wooden bleachers in the gym (not my favorite seating) and the entire senior class would be sitting in folding chairs on the gym floor.  A lot of names and awards would be read off.  In some cases the recipient would walk to the front shake hands with the person giving the awards and receive some certificate.  Mostly the recipients were kids I didn't know.  While I did pay attention when my son's name was called, I brought my iPad so during the rest of the time I could think about what I was reading rather than focus on the hard benches we were sitting on.  I've got Kindle for the iPad and I've been reading Daniel Kahneman's Thinking Fast and Slow on it, a few chapters each week. 

I was up to Chapter 19, The Illusion of Understanding.  It is about rationalizing things ex post, after an experience has unfolded, without being able to recall the prior ignorance of what would happen, before the experience has occurred.  Kahneman calls this hindsight bias.   In this chapter Jim Collins appears, the serendipity I mentioned, and not in a favorable light.

The halo effect and outcome bias combine to explain the extraordinary appeal of books that seek to draw operational morals from systematic examination of successful businesses. One of the best-known examples of this genre is Jim Collins and Jerry I. Porras’s Built to Last. The book contains a thorough analysis of eighteen pairs of competing companies, in which one was more successful than the other. The data for these comparisons are ratings of various aspects of corporate culture, strategy, and management practices. “We believe every CEO, manager, and entrepreneur in the world should read this book,” the authors proclaim. “You can build a visionary company.” 

The basic message of Built to Last and other similar books is that good practices can be identified and that good practices will be rewarded by good results. Both messages are overstated. The comparison of firms that have been more or less successful is to a significant extent a comparison between firms that have been more or less lucky. Knowing the importance of luck, you should be particularly suspicious when highly consistent patterns emerge from the comparison of successful and less successful firms. In the presence of randomness, regular patterns can only be mirages.

Immediately after reading this I started to ask myself the following question.  If Kahneman and Collins could get together for several days and talk it through, could they resolve their differences on the role that luck plays?  If so, would it be Collins who ends up seeing the errors in his (former) ways?  Then I started to ask whether Collins, having done that, could be turned into an important proselytizer on the role of luck.  He is somebody who likes to aggressively promote his ideas.  It's probably too much to hope for.  But it's that sort of change in belief that is necessary to get some new synthesis between Conservatives and Liberals, in my view.  

The next day I stumbled onto a Web site that articulates The Just World Theory, and makes reference to a book from 1980 called The Belief in a Just World: A Fundamental Delusion.  That the book is from more than 30 years ago suggests the issue has been with us for quite some time.  The Just World Theory leaves out any role for luck whatsoever.  People get what they deserve.  It is an extreme form of hindsight bias.

Zick Rubin of Harvard University and Letitia Anne Peplau of UCLA have conducted surveys to examine the characteristics of people with strong beliefs in a just world. They found that people who have a strong tendency to believe in a just world also tend to be more religious, more authoritarian, more conservative, more likely to admire political leaders and existing social institutions, and more likely to have negative attitudes toward underprivileged groups. To a lesser but still significant degree, the believers in a just world tend to "feel less of a need to engage in activities to change society or to alleviate plight of social victims."

Since these beliefs are so ingrained, it seems to me the role of luck needs to be confronted first in entirely non-threatening environments.  This is why I led off the post by talking about solving hard math problems.  They are as apolitical as you can get.  They are salient only because they have inherent complexity.  The initial discussion should be about complexity in the world that emerges in our ordinary lives.  That initial discussion should not only talk about prediction in that setting, it should ask for some predictions to be made.  There then needs to be follow up conversations with more on complexity, a review of the prior predictions and the actual outcomes, and the making of more predictions.  The process would deliberately be slow.  The goal would be an evolution in views, not an epiphany obtained in short order.

A healthy respect for the role of luck does not preclude an important role for skill to play.  The word "or" in my title means one-but-not-the-other or both.   The evolution in views is only to get away from one-but-not-the-other beliefs.  Is there a way to steer our society in that direction?   I don't know, but if not then I hope we can luck into it.

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