Monday, September 2, 2013

Rather be Good than Lucky

Coaches often say that "winning is everything."  This has the ring of just another cliche that fills many sports interviews, and it's easy to nod in agreement and move on to the next boilerplate response. But, does this point of view actually make sense?

Think about a sports outcome as a rough equation that takes into account skill, strategy, physical attributes, and all the other similar inputs. In real life, we know that there is a random factor as well. Did a player guess correctly? Did the wind change direction at the wrong second? Did the referee make the right call? And on and on.  In statistics, the above equation would have " +  ε" tacked onto the end to represent an error term to account for what the equation does not get right.

From xkcd:
Also, all financial analysis.  And, more directly, D&D.

How about an example:
Say an American football team has worked hard and is expected to score 30 points in an upcoming game, while its opponent is a bit inferior and is expected to score 26 points. Now the error term is, say, a range of up to three points either way for each team. A statistician might say that the first team is clearly better. If the error term is uniformly distributed (to simplify the example), then there are seven potential point totals for each team and, thus, forty-nine scenarios. 88% of the time, the first team would actually win; 6% of the time the second team would get the upset; 6% would be a tie.

To put this in perspective, imagine a rule change whereby a random number was added to each team's final score to determine the winner. There would be an outcry, and the team with the better original score could claim to be better. This is essentially what happens now, except the random number is hidden in the incalculable seams of wind currents and grass divots and sun glare, etc.

A coach can certainly work on minimizing the chance that an upset might occur based on randomness, but not eliminate it. Thus, it might seem more logical for a coach to focus on improving the team's expected chance of winning without the error term and not worry so much about the random number added to the end of it on any given Sunday. This is because a coach should focus on making the team better going forward; the previous game's outcome cannot be changed.

A statistician would calculate a confidence interval to determine the better team. Not likely to get the same television ratings though!

The University of Alabama football team has enjoyed great success under head coach Nick Saban. A polarizing figure, he seems to understand the above perspective. His comment after winning the National Championship? "My job is to put the players in position to win." Does anyone doubt that Saban would be thankful but scowling and unhappy if his team were to play sub-par yet got some random luck and won? His strategy to maximize the team's success? Follow "The Process," which focuses on improving the team's expected performance while minimizing randomness. Seems to be working pretty well. But given the error term, who really knows?