by Rifath Rashid
It’s a hot and humid summer’s day in New York, and you’ve just entered the Belmont racetrack for the Belmont Stakes, the final leg of the prestigious Triple Crown in thoroughbred horse racing. Today, there are five horses on the track, and you have $100 in your pocket that you’re looking to spend on bets. As the tumultuous crowd cheers and the announcer’s voice crackles overhead, you wonder, how should you bet?
This is arguably the most important question in a gambling decision. You might imagine that there are infinitely many scenarios here, all dependent on too many factors to count. What horses are racing? Who are the jockeys? What was each horse fed in the morning? What will the weather be like in an hour and which horses are best equipped for that weather?
One way to make your decision, it turns out, is to turn to a concept that you might have heard in your middle school science classroom. Remember the phrase “the entropy of the universe is always increasing?” This statement, which derives its truth from the second law of thermodynamics, has likely passed through one ear and out the other for generations of students in chemistry courses, many of whom don’t end up in careers in science. Yet, the notion of entropy crops up in many fields and situations outside the walls of the 7th grade physical science classroom.
Take, for instance, the role of entropy in the field of investment theory. Entropy serves as an underpinning of many studied and successful betting strategies, like Kelley proportional investing, and also serves as critical element in understanding modern decision making algorithms, some of which have shaped the field of Artificial Intelligence. In the real world, most investment strategies depend on exploiting side information in some useful way, and understanding entropy provides us with a framework for figuring out how to value this information.
Entropy is most commonly explained as the level of unpredictability of something you’re trying to measure. In the case the racetrack, what you’re trying to gauge is the outcome of the race. If you walk in and all the horses appear to be in perfect shape, rearing at the sound of the crowds as their jockeys hype them up for a win, it might be hard to tell which horse is a winner. In investment theory, this situation is said to have high entropy: given that all horses have a good chance of winning, the outcome of the race is hard to predict. Now, imagine that moments after you walk in a freak lighting strikes every horse but one. The organizers of the Belmont racetrack—aware of where they placed their own bets—yell, “The race must go on!” A situation like this, where it’s easy to predict the winner, is said to have low entropy.
Now imagine you’re somewhere in between the two extreme scenarios. That is, some horses look healthy and ready to win, while others seem not so interested in the race. How will you place your bets? Your first instinct might be to take a hard look at each horse and their past races to figure out which horse has the best chance of winning. From looking at the results of the last three races, you might convince yourself that the horse “Hailey’s Charm” is most likely to win, and you decide to place all your money on her. If you lose, you lose. If you win, you really win.
This strategy makes sense on an intuitive level, but investment schemes like this are often considered sub-optimal. Although you’ll get a bigger payout on some races (when you get lucky), you might get lucky on fewer races, resulting in an overall middling winnings over a number of races. A good investment scheme accounts for the fact that outcomes are probabilistic events, and helps you accrue the biggest gain over a large number of events.
In order to come up with a good investment scheme that beats the odds, it’s important to consider something called a doubling rate. The doubling rate of an investment scheme is the rate at which a betting strategy grows exponentially given all possible outcomes of the event you’re betting on. For instance, if your doubling rate is 1, then with each race you bet on using the same investment strategy, you’re expected to double your investment. But a caveat to the terminology: “doubling” doesn’t always refer to gains. If your betting strategy is sub-optimal, your doubling rate will be negative, which actually means that your investment is shrinking with every race.
In 1956, John Kelley, a researcher at Bell labs, showed that the optimal investment strategy in situations like our horse race would be something called proportional investing. That is, the best doubling rate you can get is when you place bets on each horse in proportion to the probability that it will win. Originally, you might have ignored some horse because it didn’t look so peppy, and so you decided to skip putting any money on it at all. What Kelly showed is that in the long run, your best option is to still place some money on that horse, even if you personally don’t think it is going to win.
The doubling rate of a Kelley proportional investing scheme ultimately boils down to two things: expectation of return and the entropy of the underlying event. Here, the expectation of return is simply how much you expect to make on the horse race as a multiple of the wealth you invested on each horse. But the doubling rate is actually negatively correlated with the entropy of the underlying event. In other words, if every horse has the same chance of winning (high entropy situation), then we probably won’t get that great of a return. This relationship between doubling rate and entropy means that before you can head to the racetrack and start proportionally investing all of your money, you need to know the probability that each horse will win, because these probabilities define the entropy of the race’s outcome.
The difficult reality is that it’s impossible to know exactly what each horse’s winning probability is. This is because there are simply too many uncertainties at play and too much information to possibly consider, like the horse’s morning regimen, the horse’s focus at the start of the race, the current weather, and so on. But we can try to focus on the most relevant information. How might we understand which pieces of information are useful and which are not? Consider weather information. If our favorite meteorologist just called and told us that it is almost certain to rain today, it might not be obvious whether this information gives us a better or worse understanding of which horses will win. But if we know that some horses perform very well in rain while others simply can’t function in rain, then knowing rain is incredibly valuable because it helps us better understand the outcome of the event. In this case, having additional, or “side” information about the weather decreases the entropy of our situation since we have a better idea of which horse might win. Conversely, if all horses become unpredictable in the rain, then basing our decisions on the weather might adversely affect our understanding of the race. Either way, it’s clear that weather could be an important factor. It turns out that there’s a special type of entropy, called conditional entropy, which can tell us how much a piece of side information—like weather—is actually useful for our task of betting on horses.
In 2011, researchers at Stanford generalized this paradigm to the multivariate case where we have many outcomes we care about and multiple sources of side information. A big takeaway from their work is the realization that strength of the relationship between our sources of side information and our events are an upper bound on the growth of our investments. Upper bounds, such as this one, are interesting because they give us reasonable guarantees to aim for.
Entropy is integral to our understanding of how probabilistic systems and events work. Every day, we make judgements and bets on outcomes that we may not completely understand, but advancements in information theory have helped to delineate the underlying mathematical nature of these decisions.
I pursued two projects this quarter, and focused my outreach efforts on my other group project “On Entropy in NBA Basketball.” Please check it out!