Here's a hypothesis: Teams succeed more often when the game is tied than when they're down one, because they underestimate the value of first downs. That is, the very phenomenon which people above have suggested should result in them scoring more often when trailing, actually results in their play calling being sub-optimal on first through third downs, because they're thinking "well, we have four downs to work with."
In 189 tied games, there has not been a single turnover on downs, which is pretty amazing if you ask me. There have been:
- 47 punts (24.9%)
- 54 made field goals (28.6%)
- 19 missed field goals (10.1%)
- 23.44 yards/drive
- 4.2 plays/drive
- Average start: Own 29.3
- Pass: 72.4%
- Rush: 22.2%
In 147 games down by 1-2 points, there have been:
- 22 turnovers on downs (15.0%)
- 24 made field goals (16.3%)
- 14 missed field goals (9.5%)
- 20.36 yards/drive
- 3.9 plays/drive
- Average start: Own 27.3
- Pass: 92.7% (!)
- Rush: 7.5%
So, some things that stand out about that are:
- Starting field position is non-trivially different (2 yards). Is that random variation, or something tactical about how teams approach the kickoff or the kickoff return?
- Tied teams have more effective drives, on top of their better field position (combined difference +5 yards per drive)
- The pass/run ratio when trailing by less than a FG is astonishing. 532/574 plays were passes. That's probably sub-optimal play calling when you just need a FG.
Maybe the difference in results is that those who are playing for a tie game keep the offense more balanced, because they are willing to run time off the clock, and don't want their opponent to get a chance to score. Teams which are trailing are more likely to panic and abandon the run game even if there's plenty of time to use it.
That's the best I've got.