Making Mistakes—And Quickly Correcting Them

A couple days ago on my math blog, I talked about my interview experiences with a certain trading firm. I would normally write about job or life experiences on this current blog, but given the amount of mathematics in those interviews, I wrote it over there instead.


An Interview Mistake

One of the things I did not mention in that post was a particular chip betting situation during one of the on-site interviews. I do not want to give away their on-site questions on the web, but I can say enough of it to make a point here.

The situation was a game where I had positive expected value. That is, if I played it again and again with my strategy, then over the long run I would gain chips.

My interviewer added a new rule to the game which did not affect the expected payoff of the game given that I kept the same strategy. However, the new rule was psychologically intimidating enough that I changed my strategy, and after a couple of plays, I realized I was now losing chips on average, instead of gaining.

My old strategy would have kept gaining chips, but the new strategy that I switched to was losing chips. I only realized this after 3 rounds, and just before the interviewer started the 4th round, I interjected, saying I realized my new strategy was a bad strategy, and I stated what the new (negative) expected value was for this strategy.

At this moment I felt that I had made a fatal error that would be reflected on in the decision. But instead of giving me a stern look for my mistake, my interviewer suddenly became really happy that I had corrected it! In fact, he said that almost everyone they interviewed had done the same thing, by switching their good strategy to a bad strategy when that new rule was added.

Acknowledging Mistakes

The first and most important part of correcting a mistake—and eventually benefiting from one—is to acknowledge the mistake. This most likely sounds trite, but acknowledging a mistake really is the most significant step of this process.

In matters involving numbers, it can usually be very easy to acknowledge a mistake. In my interview, all I had to do was to sense something fishy about the bet, and then recalculate the expected value to see that I had made a mistake. Since numbers don’t lie (and since I had chips on the table), I acknowledged the mistake as quickly as possible.

It can be much tougher, however, when the mistake is on some emotionally vested or less clear-cut issue. We’ve all had arguments with someone where we were totally sure we were correct, and only much later, we realized we were flat-out wrong.

And then sometimes we still maintain our original position even though we know we are completely wrong. This can lead to strange effects, but often, a person in such a state of mind is difficult to convince otherwise. Anyone who has tried arguing on the internet can give testament to this phenomenon.

Looking at the Evidence

Someone in such a state undergoes several cognitive biases:

  • Refusal to look at opposing evidence.
  • Cherry-picking evidence to only consider supporting evidence.
  • Blaming something else for opposing evidence, and waving it off.
  • Etc.

Let’s say that during my interview, I was adamant that my new strategy was good. After I start losing chips for a while, I might explain away losing streaks as bad luck, while at the same time explaining winning streaks by superior choice of strategy. I might complain that the coin was unevenly weighted, that the die was rigged, or that the deck had been stacked.

While these are somewhat reasonable conclusions to make, the problem would be if I were confronted with the fact that my strategy was bad. For instance, if I knew I was losing chips (say I lost 20% of them), but I believed in my mind that my strategy was still winning chips, then suppose the interviewer informed me that my strategy was losing chips. My first reaction, in this state, would be to reject this information and maintain that my loss of chips was due to bad luck or to unfair conditions. Of course, this behavior would be disastrous in an interview, and I would probably be rightfully rejected right there.

In the real scenario, I had some intuition about the probabilities involved, so I realized after 3 rounds that my strategy was flawed. But even if I had no intuition about what the probabilities were, after I played say 10 rounds, I would have seen the evidence and realized I was losing chips, and would have begun to start questioning my strategy.

Catching Mistakes and Learning From Them

Sometimes you are not afforded enough time to completely think something through. In this case, you need to give a most likely answer, but the important part is to keep thinking about the answer even after you have stated it. Sometimes, you might be given additional time to reanalyze it, other times what you state is final. This can be the worse feeling, when you catch a mistake only after making a final decision.


I used to play chess competitively, and while at the high levels winning often requires outsmarting your opponent, at the lower levels a win is typically achieved simply by making fewer mistakes than your opponent. If I were ever to get back into chess, my #1 area of improvement would be to reduce the number of blatant mistakes. I have turned many equal or favored positions into hopelessly lost positions by accidentally dropping a piece.

It is often psychologically damaging in chess because sometimes you know you’ve made a mistake after you made your move but before your opponent makes a decision. At this point you could hope your opponent doesn’t see your mistake. Or, you could think about how to avoid that mistake in the future. I think in the latter part of my chess playing, I dwelt too long on the first option and didn’t spend enough time on the second, and as a result, my chess rating hit a plateau.

Lies, Damned Lies, and Statistics

In addition, in 9th/10th grade I went through a phase where I thought global warming was not a well-founded theory. I subscribed to the solar cycle explanation for the “recent” warming, and thought that was more significant than the greenhouse effect contribution. I do have to add one caveat though for the record: even with that position on global warming, I still considered myself an environmentalist—I thought there were many issues with the environment, some which were far more urgent than global warming, and that global warming shouldn’t have eaten up all the priority and public interest. However, as debates go, my opposition always were able to label me as a “denier” of a sort, even though I never really denied it.

Anyways, I think the evidence since then has put a nail in the coffin. I knew the burden of proof was on the solar cycle model, and I waited to see if the temperature would drop back down. But it kept going up (in fact, even it had just stayed constant, it would have contradicted the solar cycle model). Moreover, one of the leading advocates of the solar cycle model abandoned it a couple years later. As a result, sometime during 12th grade, I went back to the scientific consensus view on this.

The Portals of Discovery

Realizing a mistake can be a rewarding experience. There is a quote by Donald Foster:

No one who cannot rejoice in the discovery of his own mistakes deserves to be called a scholar.

And a good one by James Joyce:

Mistakes are the portals of discovery.

Wile E Coyote(Well, not if you keep making the same mistakes.)

Talent Is Overrated

“Talent” is a word that is tossed around all too often, whether for top musicians or businessmen, or even just a person who creates popular Youtube videos. The idea of talent is in nearly every case taken for granted. As a young member of a very supportive family and community, I had heard the saying myself many times. But is talent a correct or even useful explanation for high-level performance?

Talent Is Overrated

I recently read a very intriguing book by Geoff Colvin. It was really a lucky buy—I was actually reading through reviews of Josh Waitzkin’s The Art of Learning, when the ever-so-omniscient Amazon Recommendations pointed me to a bizarre and blatantly absurd statement: Talent is Overrated.

With a plethora of examples, data, accumulation of research, and forcible writing, Colvin argues convincingly that the source of great performance in just about every field is best explained not by reference to the mysterious force known as talent, but by sheer amount and direction of deliberate practice.

My Personal Experience

First, a line from Colvin (193):

Their parents made them practice, as parents have always done, though it’s interesting to note that in these cases, when push came to shove and parents had to make a direct threat, it frequently played off the student’s intrinsic motivators. So it wasn’t “If you don’t do your piano practice we’ll cancel your allowance,” but rather “we’ll sell the piano.”  Not “If you don’t go to swimming practice you’ll be grounded Saturday night,” but rather “we’ll take you off the team.” If the child truly didn’t care about the piano or swimming, the threats wouldn’t have worked.

I was one of those kids who was, regarding the piano, totally immune to such a threat. As I wrote earlier, I absolutely dreaded playing the piano, and would have loved to see the piano disappear and find a bunch of cash in its place. But what I lacked in interest for the piano I made up for in my interest in chess. From 2003 to 2010, I competed in more than 70 rated chess tournaments. But looking back at the distribution of tournaments, I found that the majority of them occurred between 2003 and 2006, with one resurgence in 2008 [data]. It would be accurate to say that my tournament frequency was very closely correlated to how much time I spent on the game outside of tournaments in practice. As if to confirm Colvin’s thesis, here are my regular and quick rating graphs:


When the frequency of tournaments, and thus training, increased, my rating climbed. And when the frequency of tournaments and training decreased, my rating stagnated or declined. This seems to support the dedicated practice model argued in Colvin’s book. The performance in a given time period seemed to be determined by the amount of training in the same time period.

But what about compared to others? I am hardly an expert player, but my very first rating  after my first tournament, 1372, was in the 96-97th percentile of scholastic players at the time. By contrast, the current US chess champion Hikaru Nakamura, whose current USCF rating is a whopping 2834, started at a provisional rating of 684, an unimpressive statistic. However, he has played in 439 rated events over a period of 17 years, which is a hell of a lot more effort than I had ever thought about spending on the game. Thus even when you have an “advantage,” such as having a starting rating of 1372 versus 684, thinking of it in terms of talent is useless. If you do not follow it up with the necessary amount of work, the advantage will assuredly disappear.

There is a third point, to truly put the nail in the coffin of the talent model. In a two year span from 2006 to 2008, my rating stopped improving in the 1700s. Excuses aside, I simply didn’t practice the game much. But one thing I think could have happened is what Josh Waitzkin described, from Colvin (197):

The most gifted kids in chess fall apart. They are told that they are winners, and when they inevitably run into a wall, they get stuck and think they must be losers.

I don’t think it takes a gifted kid to run into the wall and get stuck (the 1372 initial rating was actually in part due to luck, as my first few tournaments were counted out of order, and a tournament that I had done really well in was incidentally the first one counted). For those two plateau years, I did feel the way that Waitzkin forewarned. I thought the high initial rating meant something special, i.e. talent, and that the 1700 plateau meant I was doomed. This thought process in terms of talent condemned me mentally to not advance. Even though I was still fairly high rated in my age group, I stopped practicing and reading as much, and as a result did not prepare myself adequately for tournament events. This caused my rating to drop.

How to Be a World-Class Performer

Colvin’s thesis works for far more than just chess. He applies it to the violin, piano, football, players, business, investment, management, art, teamwork, and just about anything, all while citing tremendous amounts of evidence for his claims. For music, the obvious counterexample is Mozart, yet early in the book Colvin disposes of this myth, as well as that of Tiger Woods. Mozart, for instance, had my years of intense, expert training starting at an early age, and Tiger Woods swung his first club at age seven… months, also trained by his father.

Another result of years of deliberate practice is the ability for an expert to see complex patterns that would completely elude an average person. A professional tennis player can return a serve of a ball traveling at a speed so high that a normal human should not even have time to react. Yet they are normal in this sense. But they don’t watch the ball, they watch their opponent’s body movements instead, and know approximately where the serve is going to land (or whether it will fault) before the racket even hits the ball. Similarly, a top stock trader can see signs that the average trader does not even consider to be relevant. A top manager sees the critical signs more so than an average one. And a master chess player can memorize an entire chess position in seconds and reproduce it perfectly, while the average person can recall only five or seven pieces. Most notably, this is not from better general memory, but by extensive training to be familiar with certain positions and patterns, so that they read a position by words instead of letters.

I would most certainly recommend this book to anyone. It breaks the shackle of “talent,” which although is a warm, comforting hope, it is no more than that, a beloved superstition with little evidence, and which discourages so many from even attempting something because they believe they “don’t have talent” or “divine spark” for it. But as it has repeatedly occurred, looking back at the backgrounds of top performers give little or no indication of any talent early on, but rather, what is common to all of them is an immense amount of training and dedicated practice. Perhaps this is the even more fascinating hope, that the world is within reach to everyone.