A model-based approach to football strategy.
|September 15, 2008|
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With 0:24 left in regulation, Denver scored a touchdown to trail 38-37 pending the try. Denver coach Mike Shanahan then made the remarkable decision to attempt a two-point conversion rather than kick the extra point.
Had Denver's touchdown come on the final play of regulation, it would be right for Shanahan to go for two if the probability p of success exceeds the probability of winning the game in overtime (presumably about 0.5). In other words, the required success probability on the two-point conversion would be 0.5. (This ignores the small chance of missing a kicked extra point.) However, with time remaining we must recognize that even if the try fails, Denver can still win by recovering an onside kick. This possibility lowers the required success probability on the two-point conversion.
There is an additional—and more important—effect that comes into play when there is time left in regulation. If the Broncos kick the extra point to tie, then following the ensuing kickoff, the Chargers can't risk a turnover by being too aggressive. However, if the Broncos take the lead with a successful two-point conversion, the Chargers can go for broke to score. They have no downside risk. As we showed formally in the appendix to a 2005 strategy analysis, this effect raises the required success probability on the two-point conversion.
Actually, since Denver scored with just 0:24 left, it's reasonable to suppose that if the Broncos kick the extra point to tie the game, the Chargers will elect to run out the clock and play for overtime rather than risk a turnover in their own end. In this case the analysis becomes simple. As before, let p denote the probability of a successful two-point conversion. Assume that kicked extra points succeed with probability 0.985, and that the kicking team recovers an anticipated onside kick with probability 0.12. Our model for the two-minute drill suggests that if the Broncos go for two and miss, but recover an onside kick, their win probability is 0.21. Using the same model we estimate that if the Denver goes for two and succeeds, San Diego still has a 0.1 win probability. Finally, assume each team's win probability in overtime is 0.5. (Since kicked extra points rarely miss, the scenarios in which Denver misses a kicked extra point and still wins have such low probability that we can ignore them.)
With these assumptions, if the Broncos attempt to tie the game with a kicked extra point, their win probability is 0.985(0.5)=0.4925. If instead they attempt a two-point conversion, their win probability is
|p(1 − 0.1) + (1 − p)(0.12)(0.21).|
Solving, we find that going for two gives Denver a higher win probability only if
Copyright © 2008 by William S. Krasker