James Surowiecki writing about Mark Sanchez at the New Yorker:
After a farcical 2012 season, in which the New York Jets invented ever new ways to lose games (thus the “butt fumble”), the team’s general manager, offensive coördinator, and quarterback coach are all gone. Yet Mark Sanchez, the starting quarterback, remains. He has played poorly for two seasons in a row, and has now thrown more interceptions in his career than touchdowns. But the Jets have invested an enormous amount of energy and money in Sanchez, and, assuming that no one will trade for him, they are contracted to pay him $8.25 million next year, whether he plays or not. So figuring out what to do with Sanchez will be trickier than you might think.
The Jets have stumbled into a classic economic dilemma, known as the sunk-cost effect. In a purely rational world, Sanchez’s guaranteed salary would be irrelevant to the decision of whether or not to start him (since the Jets have to pay it either way).
Here and here are two definitions of sunk costs, both of which IMHO don't go far enough because they only consider past costs sunk. Generally speaking, I think Surowiecki has it more accurately. A cost is sunk if it cannot be avoided. Whether the cost has already been paid is irrelevant. What matters is can it be avoided if a different choice is made. With guaranteed money committed to a player, whether that player sees the field in meaningful duty, that money is essentially gone/kaput/sunk.
Why then would franchises play players based on supposedly sunk costs? For one, there is the desire not to waste.
Hal Arkes, a psychologist at Ohio State University who has spent much of his career studying the subject, explains, “Abandoning a project that you’ve invested a lot in feels like you’ve wasted everything, and waste is something we’re told to avoid.” This means that we often end up sticking with something when we’d be better off cutting our losses—sitting through a bad movie, say, just because we’ve paid for the ticket. In business and government, the effect pushes people to throw good money after bad.
Surowiecki notes that the more someone has invested in a project, the greater the belief that it will work out for the best. If you wonder why your struggling team stays with it's high draft pick when there is a competent back up already on the team, this may be why.
Then there is the concern for reputation.
“Giving up on a project, though, means that somebody has to admit that he shouldn’t have done it in the first place,” Arkes says. “And there are lots of executives who would rather be tortured than admit that they’re wrong.”
This reason is interesting because it suggests that by going through with a decision, the person is avoiding an expected reputational hit. Even if the costs are sunk, the hit to the reputation is not. Depending on the circumstances, sticking to one's guns may be fully rational.