Sorry for being a little late to the party, but here's a couple of other thoughts on the going for two question. If you're sick of the whole bloody mess, you should probably stop reading about now.
First off, I largely agree with ikestoys's diary (http://mgoblog.com/diaries/down-14-and-going-2
). I have often thought that football is a game that rewards aggressive play calling, like going for two and on fourth down more often, and fake punts from your own 20... Eh...
Anyway, I disagree with a couple of points ikestoys made, both explicitly and implicitly, and I thought I'd chuck 'em out here.Trials are not independent
This point was made by a commenter in the original diary, but the basic idea of treating the different sorts of trials (going for 2, going for 1, overtime) as independent events (and therefore as amenable to the application of the mathematics of garden-variety probability theory) is flawed.
In football the outcome of one trial affects the probability of another trial even occurring, and not in predictable ways. Let's say UM had made the first two-point conversion. Would State have played their next drive differently than they did? Maybe, maybe not. Perhaps they would have come out throwing and scored a field goal to go up by nine. We have no way of knowing how things would have unfolded in that alternative universe.Relative frequencies are not probabilities
Second, and another point made by a commenter, is that ikestoys treats relative frequencies (the proportion of successful two-point conversions) as the same thing as probabilities of success. They are not. That's like saying that because 1% of adults die of lung cancer, you have a 1 in 100 chance of dying of lung cancer. Do you smoke? If so, then your probability is surely higher. If not, it's lower. The point here is that the probability of success of a two point conversion depends on many factors, as various people have noted.
Because relative frequencies =/= probabilities, I thought it would be interesting to see how the probabilities of winning fared if you didn't assume the probability of a successful two-point conversion was 0.44. So, two graphs for your viewing pleasure. The y-axis is the probability of winning the game after all events have unfolded (post-touchdown try after TD 1, TD 2, and possibly overtime). The x-axis is the probability of success of the two-point conversion (I limited the range of this probability to between 20 and 80%).Graph the first
In the first graph, I have plotted the cumulative probabilities of winning for two strategies: going for 2 after scoring a TD to be down by 8 (iketoys's strategy--the black line), and going for 1 (RichRod's strategy--the blue line). The only thing I have allowed to vary is the probability of success of a two-point conversion (on the x-axis).
Graph the second
- Note that I have reproduced the probability ikestoys does, where the dashed red line intersects with the black curve at about 57% when Pr(success) for a two-point conversion is 44%.
- Note also that despite ikestoys's implicit claim that going for two is always the better move, if the probability of success falls below 35.5%, it is better to go for 1, as RichRod did. I'm not suggesting that this is what the probability would have been, though people's comments about a dog-tired Tate, a driving rain, etc., make this idea not too farfetched).
There are two other variables in the process: the probability of a successful PAT (which I held constant at 0.95), and the probability of winning in OT. The latter probability doesn't change the black curve below much, so I left it at 50/50, as did ikestoys.
In the graph below, the three non-black curves represent three different probabilities of winning in overtime: 40% (orange), 50% (blue), and 60% (green).
The only thing to take away here is that if you believe your probability of winning in overtime is high (based on your style of play, being at home, etc.) and if you believe your probability of a successful 2-point conversion is less than 44%(ish), then you should adopt RichRod's strategy. If you believe that your chances of winning in OT is 50/50, and you believe your chances of scoring on a two-pointer are > 35%, then you should follow ikestoys's strategy.In conclusion (I know, finally)
Of course, coaches don't think this way in the heat of a game. Again, I basically agree with ikestoys, but the story is a bit more complex.