By Brian Burke
I think this is irrational. Football is a zero-sum game. Whatever is good for me is equally bad for you, and vice versa. So if stopping the clock right now is not what you want, then it must be what I want. It can't be possible for both teams to benefit from allowing the clock to run down. One or the other team derives an advantage, however small, from stopping the clock.
The only plausible exception I can think of is when the possibility of either team scoring is so remote that the cost of potential for injury on the remaining plays exceeds the value of whatever advantage could be squeezed from trying to pursue a score. In this sense, the game becomes non-zero-sum.
But I think it's more likely that one or both of the teams are excessively pessimistic. The punting team is worried that the receiving team might have enough time to put together a scoring drive, and the receiving team is worried they might turn the ball over or be forced to punt again from deep in its own territory.
This phenomenon might be related to Prospect Theory, which says people weight potential losses more heavily than equivalent gains. In this model, both coaches would be overly gun-shy about stopping the clock because their pessimistic mental calculations tell them both that stopping the clock is bad.
But I'd guess it has more to do with the a large degree of uncertainty surrounding the probabilities. As the Ellsberg Paradox demonstrates, people prefer known risks to unknown uncertainties, even when the more uncertain option has higher expected value. Rather than pursue the overall optimization of their chances winning, coaches seek out the choice with highest minimum plausible chance of winning, within the band of uncertainty. In other words, they are selecting the option with least-bad worst case scenario. In this situation, the highest minimum plausible expected value for both coaches is to allow time to run out. After all, letting time run out is the least uncertain option of all.
In 'normal' situations when time is not a factor, if the punt can be downed inside the 15 yard line, the punting team will probably be the next to score. Outside the 15, it's probably the receiving team who will be the next to score. But in this particular situation near the end of the half, the numbers aren't so clear. The one thing we do know is that the answer isn't none of the above.