...With a little ingenuity, there is a way for overtime to be both fair and fast. One solution is usually associated with cake-cutting: one person divides, the other chooses which half to take. In a football overtime, the divide-and-choose rule would dispense with the kickoff and just give the ball to one side. The coin-toss loser would decide how far forward the offense would start—say, the 30-yard line. The coin-toss winner would then decide whether to take possession or let the coin-toss loser have the ball at the 30. The nice thing about these rules is that they would naturally adapt to the game's changing dynamics. The current system, by contrast, seems to have been fair when introduced in 1974, but as field-goal kickers became more accurate, possession has become more valuable.
...An even more elegant solution to the overtime problem was proposed in 2002 by Chris Quanbeck, an electrical engineer (and Green Bay Packers fan). Quanbeck's idea was to auction off possession of the ball in the natural currency of the game: field position. The team that was willing to begin closest to its own goal line would receive the privilege of possession.
Harford notes that the proposal was picked up by economist Yeon-Koo Che, an expert in auctions at Columbia University who, with his co-author Terrence Hendershott showed that the auction method was bettrer than both the coin-toss and the "divide and choose" ways of determining who gets the ball when teams had asymmetric priors about what was being divided.
The NFL has a lot more parity between teams than in college football. It would make for an interesting test of interleague comparison of parity if such an auction were used in both the NFL and in college football's various divisions. Of course there is a selection problem that needs to be solved. The teams that end up in over time were more likely to have been evenly-matched to begin with.
*Update: thanks to John Palmer (Eclectecon) for the link.