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.
Swimming (heat)
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.
Swimming (final)
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.
Fencing
As Paul Rubin noted, it’s decided by coin flip. The unfairness of NFL overtimes pales in comparison to fencing!
Punk Rock Operations Research 
August 7th, 2012 at 7:47 am
Interesting. We were trying to figure out why Mustafina got the bronze over Raisman. Thanks!
August 7th, 2012 at 9:18 am
In weightlifting, if two competitors lift the same weight amount, the competitor who weighs less wins. Imagine using this to break ties in the NFL: the team with the lowest payroll wins the game. This appeals to me as an OR guy: doing more with less.
August 7th, 2012 at 10:54 am
Apparently execution score is also the tiebreaker in women’s gymnastic apparatus finals (http://sports.yahoo.com/news/olympics–aly-raisman-wins-bronze-medal-in-balance-beam-after-successful-protest-to-recalculate-scores.html).
In the Olympics, a tie score in Tae Kwon Do is resolved by a “sudden victory” round where first point wins. At lower level tournaments, I think there are multiple tie-breakers including which fighter was more aggressive and/or which fighter threw the greater array of techniques.
There are no forms or board-breaking competitions in the Olympics, but I’ve seen multiple tie-breaking rules at lower level tournaments. Interestingly, one of them is the exact opposite of the rule used in the women’s all-around: in the event of a tie, the competitor whose scores had the narrowest range wins. I’m not sure if that is intended to reward consistency on the part of the competitor or screen out bipolar judges. I think I prefer that to the Olympic rule that cost Raisman the bronze. The all-around medalists should do a good job all around, not a great job on some apparatuses and a poor job on others.
August 16th, 2012 at 1:29 pm
I was going through some old papers today and found an article in the October, 2008, issue of OR/MS Today by Adrian Lee and Sheldon Jacobson on Olympic tiebreakers (http://www.orms-today.org/orms-10-08/frforum.html). For gymnastics apparatus finals, they mention the first tiebreaker (the execution score) and then a second tiebreaker (which is equivalent to averaging the three highest execution scores). The article states that the first tiebreaker favors the gymnast who had the more difficult (riskier) exercise, but that contradicts Laura’s statement above.