# Adjusting Priors in Science, Basketball, and Life

*The following is from Allan Maymin, Philip Maymin, and Eugene Shen. This trio recently published “How Much Trouble is Early Foul Trouble?” (International Journal of Sport Finance 7:4 (2012), 324-33.”). What follows is a brief post detailing the research reported in this article. *

A few years ago, the three of us met in a private room at a local library with only a whiteboard and a dream. We wanted to take our academic and practical finance backgrounds and apply them to basketball research. To us longtime fans, it seemed clear from afar that there should be a lot of low-hanging fruit to be picked using our tools and techniques, ranging from high frequency algorithmic trading to dynamic portfolio construction to derivatives valuation and hedging. We decided to start with a simple question: should players in foul trouble be benched?

We unanimously felt that the answer would turn out to be a resounding NO! Benching was a mistake. We cringed whenever we watched seemingly dogmatic coaches yank players early in the first quarter after their second foul. In fact, we almost chose not to do the research because it was just so obvious that benching was a bad idea.

Here was our intuition. Players are like stocks: they have a drift, and they have volatility. Put LeBron James on the court with nine average players and — though there will be randomness along the way — his team will dominate over time. Since there are only 48 minutes in regulation, you want to play your starters as long as you can. Sure, you may want to limit their minutes to minimize injury risk, but beyond that, if you’ve decided to play LeBron for 40 minutes a night, then you should try to play him for 40 minutes a night! And benching him when he is in foul trouble does not maximize his minutes. In other words, it’s better for LeBron to foul out early in the game and play, say, 30 minutes total, then for him to not foul out and end up playing only 24 minutes total.

We were so sure that we would find this result that we were already planning to explore what behavioral biases best explained coaches’ reluctance to let their players play. Do they suffer from status quo bias? Loss aversion? Overconfidence?

It turned out the answer was: the coaches were basically right, and our priors were wrong. Despite our suggestive intuition and the evidence of our eyes, the numbers told a different story.

And the story was more interesting and subtle than just Yes or No. It turns out the optimal decision of whether to bench a player in foul trouble depends on how good the player is relative to his replacement on the bench, and how early in the game it is.

We can measure the skill of each player on the court or on the bench with their Wins Produced per 48 minutes. And we of course know what time remains in the game.

If we just do a standard regression, looking at the change in net points conditional on the number of players on the court in foul trouble, as well as other variables, then it seems as if locally our intuition was right! Players should play even when in foul trouble.

But focusing on winning locally is like seeing the trees but not the forest. The only thing that matters in basketball is the final outcome, not the path along the way. A coach’s objective function is not to maximize the minutes played by his starters, but to win the game. The fact that a player may not be available in the important end-of-game situations may have an effect, and potentially enough of an effect to offset the loss of his contribution earlier in the game. The only way to measure that is to discard all notions of local regressions and focus solely on the probability of winning a game.

So that’s what we did. Introducing an extensible and general win probability framework, we looked at the impact on victories from playing vs. benching starters in foul trouble. We looked at various definitions of foul trouble and the best one seemed to be the one that coaches tend to use, called “Q+1.” Under this definition, a player is in foul trouble once he has more fouls than the current quarter; so, two fouls in the first quarter, three fouls in the second, and so on. Once the quarter ends, the player may no longer be in foul trouble if “Q” is now equal to the number of fouls.

It is from this win probability framework that we got our counter-intuitive results.

It turns out that the optimal strategy looks something like this: bench starters in foul trouble only if it is early in the game and the player has a strong backup on the bench. The specifics are of course in the paper but the basic idea can be explained using option pricing.

Option pricing is the most common example of derivatives valuation in the field of finance. It is also the most notable, being the source of the famous Black-Scholes pricing formula under specific assumptions about the underlying process for which Robert C. Merton and Myron S. Scholes won their Nobel prizes in 1998 (two of us worked with them at Long Term Capital Management, the hedge fund where they were partners).

In finance, thanks largely to the pioneering work of Black, Merton, and Scholes, and the research field they launched, we have a lot of intuition about option pricing. We know, for example, that in relatively general conditions, the drift of the underlying process does not matter for the price of the option; only volatility matters. You and I may disagree about the expected returns of Apple stock but still agree on the price of an Apple option, so long as we agree about the volatility we expect Apple will exhibit.

In the context of basketball, benching a player creates an option for the coach. The coach has the choice to put him back in the game at a critical juncture later in the game. In options terminology, the coach has the right to exercise his option early, by letting the player play through his foul trouble, but he risks that the player fouls out and will be unavailable at the end of the game, when matchups become important.

Another surprising option pricing result from finance is this: the right to exercise an option early is quite often worthless. In basketball terms: it can pay to wait. The option the coach creates by benching his foul troubled starter is valuable. In another paper we authored, “NBA Chemistry: Positive and Negative Synergies in Basketball”, we show that a player’s value is context dependent, and depends on the other nine players on the court. At the end of a close game, we often see coaches aggressively playing matchups, inserting their best defenders when the other team has the ball, and playing their best offensive players when they have possession. If a player has fouled out, then he is unavailable during the most critical moments of the game. A coach should maximize the number of favorable match-ups over the course of the game, and may need to bench a foul troubled player to do so.

Finance theory also tells us that options are more valuable when there is more time left until their expiration. Similarly, coaches should yank more aggressively early in the game than later, because the option value they create deteriorates with time. On the other hand, the cost of the option matters too. By benching his starter, the coach may be playing an inferior player, and the drop-off in talent may be large enough to completely offset the gains from the option.

To say we were surprised by the result would be quite an understatement. We spent much of our time trying to find the problem with our analysis. Fortunately, that made the resulting research stronger. But the important personal lesson to us was: free your priors.

We had prior beliefs that benching was universally bad; we were wrong. Option traders may have had priors about the importance of knowing the drift to valuing options, or the value of the right to exercise early. Coaches and GMs have priors about the potential of their young players.

Having priors is good. But releasing them in the face of overwhelming data is even better. When the Houston Rockets watched Jeremy Lin develop into a star just a few months after they had waived him, they released their priors and signed him to a substantial long term contract. When options traders were convinced of the merit of academic pricing results, they adjusted their trading strategies. And when the data said that what coaches did was more or less the right thing to do, well, we had no choice but to change our mind too.