Misix Library

Standard Deviation and Aggressive Strategies: Why Bad Teams Should Play Risky (But Don’t)

November 4, 2013

It was arguably the biggest upset in the history of college football. The fifth-ranked Michigan Wolverines hosted the relatively unknown Appalachian State Mountaineers in both teams’ first game of the 2007 season. The Wolverines were heavily favored, so much so that Vegas didn’t even offer a betting line on the game. Of the 100,000+ people in attendance, nobody expected the Mountaineers to win except their parents—and they would have been forgiven for having their doubts. Then, a couple failed two-point conversions and a blocked kick later, the Mountaineers stunned Michigan, 34-32.

Underdogs need to have a lot of breaks go their way to beat a favorite: lucky bounces, opponent’s mistakes, unlikely plays and questionable officiating, to name a few. Appalachian State got plenty of these breaks en route to their victory. Almost every game has some of these occurrences, though they are few and far between. But, with the strategies they employ, teams can also create situations that lead to more of these occurrences. In particular, playing more aggressively and taking more chances makes the game much less predictable, giving rise to opportunities for lucky breaks to change the game.

Think about a great team (Team A) and a not-so-great team (Team B) that are set to play one another. Team A scores an average of 35 points per game and Team B an average of only 20 points. Though we know Team A is better, this information alone won’t tell us how likely it is each will win.

Consider two different scenarios:

  1. Team A scores exactly 35 points in every single one of its games, and Team B scores exactly 20 points in every single one of its games. Team A will beat Team B every time they play (the probability of victory for Team A is 100%).
  2. Team A scores exactly 35 points in every single one of its games, and Team B scores three points in half of its games and 37 points in the other half of its games. Team A will blow out Team B in half of the games they play but lose by two points in the other half (the probability of victory for Team A is 50%).

It is clearly in Team B’s best interest to play a game that reflects the second scenario, giving it a 50% chance of winning instead of no chance. Obviously, this is an oversimplified example with only two scenarios. But if you extend it to many scenarios, you can see how the distribution of possible points scored for each team can affect their probabilities of winning.

Figure 1 shows the first scenario, with the assumptions regarding the number of points scored relaxed a bit. This is the scenario in which both teams play conservatively—neither is trying to increase the variability in the number of points they score. You see Team A is still more likely to score 35 points than some other total, but there is also a chance it scores slightly more or less than 35 points. The same goes for Team B; it is more likely to score 20 points than any other total, but it also could score slightly more or less. In this scenario, Team B has an outside chance of winning if it hits the far right end of its curve on the same day Team A hits the far left end of its curve. But Team A wins the vast majority of the games—typically by 10 to 20 points.

Figure 1

 

distribution1

Figure 2

distribution2

 

Figure 2 shows a scenario in which Team B plays aggressively—for example, going for it on fourth down instead of kicking the field goal, trying onside kicks or lunging for the interception instead of making a sure tackle. The variability in the number of points it scores is now much higher. It is still most likely to score 20 points, but it has a higher chance of scoring much more. On the flip side, it also has a much higher chance of scoring very few or no points. In other words, it has a better chance of beating Team A than if it employed a conservative strategy, but it also has a better chance of getting blown out.