Some tactical discussions in our current D&D game brought to mind the debate over ending a contest quickly vs. playing optimally for the duration, one of my favorite game theory discussions.
The basic idea is that there is value in taking some risk in order to gain the chance to force your opponent to lose earlier than he normally would, even if taking this risk is not optimal play in a vacuum. That is, even if in an arbitrary game state you would be more likely to ultimately win the game by making a conservative play now than by making a risky play now, it may be in your best interest to make a risky play anyway in some actual game states.
Every post is more fun with examples, and here’s the best one I know of:

(American) Football:
It comes up every so often in football games that one team will be up by 2 points near the end of the game, and have to make a kick vs. throw* decision, deciding between 3 sure and 7 risky points. A lot of statistics have been done on whether it is better to kick or throw, and in general it depends on the yard line, the score, the amount of time remaining, and many intangible factors. In a few rare cases though, this balance can be tipped immensely in favor of ‘throw’ by the possibility of a forced decision.
For example, suppose the Aardvarks have the ball on the Badgers’ 13 yard line, and it’s 4th down and 4 to go, with 1:36 remaining in the game. The Aardvarks are winning by a score of 25-23. They have to decide whether to kick an easy 30-yard field goal for 3 points, or attempt to get at least 4 yards to keep their possession alive and ultimately score a touchdown for 7 points.
Kicking the field goal is more likely to succeed – it’s an easier play to make. It would leave the Aardvarks up 28-23 when they turn the ball over; enough that a Badgers field goal would not be enough to tie. The Badgers’ chances of answering with a winning touchdown, by marching the entire length of the field in a minute and a half, are low (much lower than the chance of them getting into field goal range and scoring 3), so this seems like the most statistically sound choice.
However, kicking in this situation is usually wrong, and here’s why: if the Aardvarks score a touchdown instead, the game ends immediately. They would gain 7 points and be up 32-23, which is a 9 point margin. It’s impossible to score more than 8 points on a single possession in football. Thus, even if the Badgers had an amazing offense that could drive the field in seconds, it wouldn’t matter; they’d have to turn the ball over to the Aardvarks’ offense after bringing the game to 32-31, at which point the Aardvarks could trivially kneel the ball to run out the clock.
This is a great example of a game state where taking the play with lower expected value (making a risky attempt to go for it on 4th down) generally actually increases your odds of winning the game. I’m not sure that it happens all that often; football creates the situation neatly because of the artificial limitation on possible points in a single possession, but maybe other games have it too.
*Yes, I am simplifying here and ignoring the option of running the football. This is because running is generally a silly option anyway. This the 21st century after all.