The following concept is pretty different from what I normally write about, but I think it’s really neat, and hope some people will find topics like this quite interesting!

The Most Useful Course in High School (in my opinion)

If you were to ask me today what my most useful high school course was, I would have to say Data Management. Ironically, Data Management is not really a prerequisite for any but a few programs. For those of you who haven’t taken Data Management, it was a course that covered basic probability, statistics, distributions, and other things of the like. And although it is a mathematics course, I would have to say that many concepts it covered are more useful to the way in which we live than the concepts taught in any other high school course. There are far too many to count and explain, but there’s one in particular I’d like to write about today.

Frequency Distributions

One of the really neat concepts you would learn in Data Management is the idea of a frequency distribution. Many of you probably know this, and if not, it’s a pretty simple idea. A frequency distribution is basically a mathematical representation of the frequencies for possible outcomes of an event. For example, a sample frequency distribution for the grades in your 30 student class could be 10 A’s, 10 B’s, 5 C’s, and 5 D’s. These frequencies could also be represented as fractions or percentages of the total class.

Simple enough, right? In spite of its simplicity, the powerful of this concept is in recognizing that thinking in terms of frequency distributions can apply to so many more things than just the examples about classroom grades in your Data Management or Statistics textbook. One of the really cool things is that you can use the concept of frequency distributions to solve problems and make better decisions.

Rock, Paper, Scissors

A really good example for better understanding this is the game of Rock, Paper, Scissors (RPS). At this point, I’d like to introduce another term for frequency distribution that I like using better: range. Range can either refer to just the possible set of outcomes, or the outcomes along with their frequencies. So, if I were to ask you what the range of possible actions you could take in Rock, Paper, and Scissors was, you would answer: throw rock, throw paper, or throw scissors.

Now imagine you were playing a friend in RPS. And let’s say that your friend was the World RPS Champion, and that for whatever reason, he could often predict your actions (maybe there are some patterns in the way you play) and was constantly beating you. You know that you’re not as observant or as smart in RPS as he is, so you will never be able to pick up on his throwing habits: what should you do?

Unexploitable Strategy: Balancing Your Range Perfectly

Well, one thing you could do is perfectly balance your range. By this, I mean making the range of your actions perfectly balanced between rock, paper, and scissors: that is, in the long run, throw Rock 33% of the time, throw Paper 33% of the time, throw Scissors 33% of the time – and in a random manner. Now, being able to randomize your actions is a completely other topic (ways you can do this is using a random number generator, among other things) – the important thing is the concept behind this. By completely randomizing your actions, and balancing your range perfectly, your World Champion RPS friend cannot gain an edge over you.

How is this so? Well, if your range here is perfectly balanced and randomized, your friend will never be able to predict your actions. And if he can’t predict your actions, then it doesn’t matter what actions he chooses for himself: you will always win half of the time. Maybe before, you were doing something strange like always betting 2 Rocks, then 3 Scissors, and then 1 Paper, over and over again, and your friend picked up on this pattern – well, if you completely randomize your play, there’s no pattern to pick up on anymore. With your perfectly balanced range, your strategy is unexploitable. You can’t ever lose.

But Sometimes it Makes More Sense to Use an Exploitive Strategy

Initially, you might think to yourself: wow, that’s awesome! Now no one can beat me at RPS. Well, if you think about it a bit more, you’ll realize something else: while you can’t ever lose in the long run, you can’t ever win in the long run either – because with a perfectly balanced strategy, you are guaranteed to win half the time, but also lose half the time in the long term. Essentially, you have a net-gain of zero.

Now, if your opponent is indeed the World RPS Champion, and you know you can’t beat him or her, then it makes sense to use a perfectly balanced, unexploitable strategy.

But what if you could pick up on the patterns of your opponent? In this case, it would not make sense to have a balanced range. In fact, in such a case, it makes sense to create an exploitive strategy: that is, adjust your RPS range in such a way that you are exploiting, or taking advantage of the patterns in your opponent’s RPS range. Let’s do an example so that this idea is clear.

Suppose you are very observant, and you notice that your opponent always picks Rock. In this case, it makes more sense to just always pick Paper and win 100% of the time. Keep in mind that we are assuming that our opponent is incapable of realizing our strategy, and keeps picking Rock no matter what. That is, while it’s true that our exploitive strategy is in itself exploitable, that is okay as long as our opponent isn’t doing anything to take advantage of that.

Now, this situation where our opponent always picks Rock is a pretty simple case, but the concept still holds even if our opponent’s range is not 100% predictable. For example, let’s say that our opponent’s RPS range is 70% Rock, 15% Scissors, and 15% Paper, but you can’t predict exactly what he will throw at any one point. All you know is that in the long run, he’s going to make these actions with these frequencies. You still have a huge edge by simply picking Paper every time. Think about it: Your paper beats his rock 70% of the time, you tie his paper 15% of the time, and you lose to his scissors 15% of the time.

At the Highest Level, the Player Who Wins is the One who Adjusts Faster

Now, recall that it’s better to use an exploitive strategy as long as your opponent doesn’t realize what’s going on. But what if your opponent is good, and starts adjusting to your strategy? For example, let’s say that your opponent was initially playing only Rock, so you started playing only Paper. You win initially, but say your opponent catches on, and starts playing only Scissors – now he’s exploiting your strategy! Now you’re going to lose every time, unless you yourself catch on, and start playing Rock yourself! And so on, and so on…

The idea here is that when both players are competent and know what’s going on, the winner is going to be the player who adjusts faster than the other.

Probably an even more important concept is that in anything in life where success depends on a possibly changing environment, the most successful person is the one who adjusts to the environment the fastest, and the best.

Example Application of these Concepts: Tennis

Rock, Paper, Scissors might not seem really important, so where else are these ideas applicable? One example is tennis. One of the things that separates Roger Federer as arguably the best male tennis player ever, is how good he is at adjusting to his opponents. In fact, I really doubt you could ever be the best tennis player in the world at some point without being very good at exploiting your opponent’s weaknesses and mistakes – the talent at the highest levels of tennis is just too good to be sticking to a single, non-changing game plan. Remember, the optimal strategy depends on who you are playing, and depends on the adjustments you make to your opponent.

For instance, let’s say that Roger Federer is serving. To simplify this example a bit, let’s say that he has 3 types of serves he can do: serve to his opponents left, right, or body. If he knows that his opponent’s backhand is very weak, Federer will just keep serving to his backhand. Unless the opponent adjusts to try and compensate, Federer will just keep picking on his backhand.

On the other hand, what if Federer was playing his rival, Rafael Nadal, who recently dethroned Federer as the number 1 male tennis player in the world? Nadal is obviously a smart, thinking player capable of making adjustments in his return game. Even if he has a weaker backhand, he is probably smart enough to adjust his pre-serve positioning to improve his ability to return from the backhand, at a frequency close to optimal versus the frequency that Federer is serving to his backhand (okay, maybe this is a bit of a stretch, but I think you get the idea).

Does this Mean You are Always Thinking in Numbers?

I get this question a lot: how can you extrapolate this idea to more complex everyday life situations where you can’t always know the numbers of the frequencies? That’s a really good question, and the fact is, I don’t use numbers really that often – at least no exact numbers, that’s really too hard.

If I do use frequency distributions to solve problems or make real life decisions, I’m going to be approximating and guesstimating a lot of the time, just because that’s probably good enough for most things.

Another way to look at frequency distribution is the idea of a model. Like, say I know your RPS range, I’ve basically created a model for how you play the game. It should become pretty clear that even beyond games or sports, being able to create models for other situations using frequency distributions can be quite useful.