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