When Winning Leads to Winning: A Response

Here are two interesting follow-ups to Tuesday’s post, in which I described how basketball teams who are behind at half-time fare a bit better than might be expected.

First, my friend Lionel Page points me to a related study of his, which analyzes tennis. Lionel uses a similar approach to arrive at a different conclusion, but I think his results point to the same psychological factors. Let me explain.

Lionel analyzes over half a million tennis matches, but focuses on that subset of matches where the first set is played to a tie break (there are about 70,000 of them). In fact, he focuses on those matches where the first set is awfully close to a draw — a long and drawn-out tiebreak, lasting more than 20 points. But eventually someone wins, and it turns out that player is much more likely to win the second set. And so, winning leads to winning in tennis, but losing leads to winning in basketball.

Are these results in conflict? I don’t think so: the effect of falling a little behind in a tennis tiebreak is very different to basketball. In tennis, falling behind in a tiebreak means losing the entire set, which is a big hurdle to overcome. Recall that Berger and Pope ran laboratory experiments in which they tried to figure out the (de-) motivating effect of being a little behind, a long way behind, tied, a little ahead, or a long way ahead. They found that being a little behind led to a burst of extra motivation, which describes the basketball finding nicely. But being a long way behind led to no extra motivation, and so the tennis player who just lost the first set doesn’t get these motivating benefits. In fact, Lionel’s finding suggests that it might actually be de-motivating. Because basketball scores are continuous, and tennis outcomes are so sharply discrete, the psychological impacts of small differences in early play are very different.

Second, Freako commenters showed that they are among the most demanding consumers of statistical analyses, and were pretty vocal in what they perceived to be shortcomings in the Berger-Pope analysis (and see more from Andrew Gelman, here). I passed on your comments to the authors, and here’s their response:

In our sample of N.C.A.A. basketball games, teams that were losing by one point at halftime were more likely to win the game than teams winning by one point. This is indisputable. However, is this finding statistically significant given the noise in the data?

The key issue here is understanding exactly what hypothesis we are testing to see whether losing by a small amount increases motivation. Directly comparing the winning percentage of teams down by one with teams up by one is problematic because these are different types of teams in different situations. Teams are not randomly assigned to be up or down by one. On average, teams down by one tend to be worse (they have a lower season winning percentage). Furthermore, it is mechanically harder for teams down by one to win. They have to score at least two more points than their opponent to win, while their opponents can win even if the teams trade baskets the rest of the game.

This means that we shouldn’t expect teams down by one to win 50 percent of games. What should be expected? This is where reasonable people may begin to differ on the right way to construct a counterfactual. Many different curves can be fitted to the data. One may argue (as many did in the comments) that a linear line should be fitted; Andrew Gelman suggests a logistic function. It ends up that it doesn’t really matter what curve is fit.

For example, consider the figure below (the exact figure requested by Andrew Gelman) which indicates the winning percentage for the home team as the halftime point difference for the home team ranges from -10 to 10. Also, note the inclusion of standard error bars for sophisticated readers. The dotted line represents the fitted curve from a simple logistic function when including the halftime score difference linearly. Focus on the winning percentage when either the away team was losing by a point, or the home team was losing by a point. In both of these situations, the losing team did better than expected. For example, when the home team is ahead by one point, they end up only winning 57.5 percent of games while we would have expected them to win 65.6 percent of games. This difference in actual versus expected performance (8.1 percent) is statistically significant at conventional levels and provides evidence in favor of our hypothesis that losing can be motivating.

This difference persists when controlling for home-team advantage, possession arrow to start the second half, prior season winning percentage, and team fixed effects (see Table 1 of the paper).

Further, supplementary analyses show that teams losing by one point closed the gap the most in the first few minutes after halftime (supporting our motivation hypothesis). Laboratory studies, using random assignment, also demonstrate that merely telling people they are slightly behind halfway through a competition leads them to exert more effort.

Taken together, these findings indicate that being slightly behind motivates people to work harder and be more successful.

Finally, let me say just why I like this paper. It’s easy to mine sports data to find interesting anomalies, but sometimes it is hard to see what it means. And it is easy to get students in an experimental setting to do weird things that have no relevance to the real world. It is the juxtaposition of suggestive data from the field with a well-designed experiment that leads me to conclude there’s some interesting social science here.

My friends at the Association for Professional Basketball Research have collected some interesting discussion threads on this controversy, here, including data suggesting a similar pattern in the NBA.