In the Olympics, we’ve seen quite a few ties, and each has been resolved in a different way. They have mostly been quite different than what we see in professional sports in the US, where there are a potentially infinite number of overtimes or extra innings until a winner is established (However, the NFL does allow for ties after the first overtime, except in playoff games).
Gymnastics (women’s all around)
The tie-breaker is to choose whoever does the best on their three best events, thus rewarding the person. HT to Jeffrey Herrmann, who writes:
The tiebreaker for the women’s gymnastics all-around is a nice example of rewarding “compensating” solutions (those in which a strong performance in one attribute offsets – compensates for – a weak performance in another) over “non-compensating” solutions. Mustafina’s scores included both a very good one (bars = 16.100) and a poor one (beam = 13.633), and she won the bronze. Raisman’s scores did not have these extremes, and she lost the tiebreaker.
Gymnastics (men’s single event)
Gymnasts are scored in two ways: for their execution and for the difficulty of their routine. The tie-breaker is to side with whoever had the best “execution” score, thus rewarding the person who performed better on an easier routine.
There was a three way tie for the last place to advance in the semi-final heat in one of the races. The three swimmers re-raced for the last qualifying spot in the final race.
The tied swimmers share a medal. Two silvers were awarded in one race this Olympics. This seems reasonable in a sport where winners and losers are determined by as little as a hundredth of a second.
Soccer, field hockey, and others
Soccer has an overtime period and is then decided by penalty kicks. I hear that field hockey is the same.
Volleyball, tennis, badminton
These sports have a best-of-3 or best-of-5 format to prevent ties. However, any set/game in the match can be “tied” if a team does not win by 2, and the games are played until this happens. This has led to matches of epic length, including Federer’s semi-final win that lasted ~5 hours.
As Paul Rubin noted, it’s decided by coin flip. The unfairness of NFL overtimes pales in comparison to fencing!