How Mathematics Can Predict March Madness
As a mathematician, I’ve long been interested in the science of gambling. From lotteries to sports leagues, what is the best strategy? Where are the loopholes? How is it possible to beat the house? With NCAA Men’s Division I Basketball Championship — otherwise known as March Madness — starting this week, I’ve collected together some of the key ideas involved in scientific betting to show how these concepts can lead to better predictions when picking a bracket.
1. Mapping the sample space
Choosing a bracket means predicting the result of every game in the March Madness tournament. This is really a question of probability, and the first thing to do when faced with a probability puzzle is to look at the set of possible outcomes. Mathematicians call this the “sample space” and it was one of the first probability concepts ever devised (by a gambler, as it happens).
In total, 68 teams are involved in March Madness. Once the four play-in games are out of the way, there are 64 teams left to compete in the main tournament. This means there are 63 games to predict (as all teams except the winner will lose a game). That’s a lot of games. If you flipped a coin to decide the result of each match-up, you’d have a 1 in 9,223,372,036,854,775,808 chance of picking the correct bracket (i.e. 2 multiplied by itself 63 times).
Unfortunately, it’s far more likely you’ll fail to predict any games correctly. This is because tournament predictions are “path dependent”: If you fail to get any of the first round games correct (which has about a 1 in 4 billion chance of happening), you won’t have any correct teams in the second round. Which means you’ll automatically have predicted all the subsequent matches wrong too.
But surely there’s a way to improve on guesswork alone?
2. Getting the fundamentals right
Whether looking at hockey or basketball, you need to measure teams’ ability if you’re going to work out their chances of winning. The successful sports bettors I’ve interviewed all follow a crucial rule when it comes to predictions: Come up with you own analysis before looking at what other people are doing. This is known as fundamental analysis. They gather data, decide what information they need to consider, then use it put together an opinion.
One way to measure team ability is to rank them in some way — perhaps by recent performances, or league standing, or a combination of different statistics. At the start of the tournament, the teams are seeded, and so one option is to use these seedings to make bracket choices. For each game, you could assume that the team with the higher ranking will win, and use this method to pick your bracket.
Ideally, you’d also get hold of pre-tournament data from past years, and test your ranking system against actual performance. Based on the results, you could tweak your method so it can predict past games as well as possible. Scientific betting teams generally go one step further and also test their strategy on “out-of-sample” data: If you used data from last year to hone your method until it could predict last year’s outcomes, you should check that it could predict other years’ outcomes as well.
3. Wisdom of the crowds
Although betting syndicates don’t initially look at what other people are doing, they often incorporate betting odds later on. This is because betting markets can produce a “wisdom of crowds” effect: If lots of people are betting on the same game, the odds may contain useful information about who might win.
For instance, you could start off by ranking the teams by yourself (perhaps using the seeding to guide your choices) and seeing what the bracket looks like under the assumption that the higher ranked team will always win. You could then find a website that displays each team’s odds of winning the tournament, and use these to rank the teams again. If the brackets don’t match, it suggests that public opinion disagrees with your ranking — which means you might be missing a crucial detail.
4. Finding a new dimension
There is a downside to making predictions based on simple rankings, however. Rankings are one-dimensional, with team ability is summarized by a single value. This means that predictions are “transitive”: If team A is ranked above team B, and team B above team C, then by definition team A must be above team C.
But sport is not transitive. One obvious example is the home-court advantage. Team A might generally beat team B at home, and team B might usually win against team C at home. But then team C might be expected to win if they host team A. So it doesn’t really make sense to use a single measurement when comparing teams. Teams typically play on neutral courts during March Madness, removing the home advantage, but some may still have to travel further than others. Other “non-transitive” factors may also influence results, with some teams particularly vulnerable to certain tactics or players.
One way to account for this would be to assume that the outcome of each game depends on your ranking (perhaps tweaked after looking at overall odds of winning), and predict a set of winners. You’d then compare the resulting predictions with the betting odds for each first-round game. If rankings are good at identifying the winning team, the predictions should be fairly consistent with the favourite as suggested by the betting odds for the individual games.
This might seem like a lot of work to predict the early games, when it is the later ones that generate the most points. But it’s worth putting effort into getting the early games correct, because as mentioned earlier, “path dependence” in tournaments means that early mistakes will have an impact on subsequent predictions too.
The Perfect Bet: How Science and Math Are Taking the Luck Out of Gambling is published by Basic Books. It is available to buy on Amazon and in local bookstores.
