An Unprecedented Situation at the Olympic Trials Resulted in Game Theory Being Implemented in the Rule Books
http://olympics.time.com/2012/06/27/dead-heat-how-the-u-s-track-and-field-trials-got-all-muddled/
https://www.nytimes.com/2012/07/03/sports/olympics/tarmoh-withdraws-from-100-meter-runoff.html
Game theory is an interesting concept, particularly because it is applicable in virtually every aspect of life. Sports are a common place where game theory is practiced, but we usually see this analysis done in team sports such as football, baseball, and soccer. But for a sport that rarely sees true head-to-head matchups, U.S. Track & Field fans were treated to a unique situation when Allyson Felix and Jeneba Tarmoh finished in a dead heat for third place (the last qualifying spot) in the 100 meter dash at the 2012 Olympic Team Trials. Photo-finish cameras literally could not separate the two sprinters as they finished with the same exact time down to the thousandth of a second (ties cannot be broken past the third decimal place according to competition rules). Unfortunately, the U.S. could not simply send both athletes to the Olympics since each team can only send three athletes per event, and there were no written procedures by the national governing body on how to break a tie since ties are not only extremely rare, but acceptable. This is where game theory comes into play.
During the week of the Trials, USA Track & Field outlined a procedure for resolving ties when selecting international teams. The resulting rules ended up describing a two-player game like the ones discussed in lecture.
| Jeneba Tarmoh (Player 2) | |||
| Coin Toss | Run-Off | ||
| Allyson Felix
(Player 1) |
Coin Toss | 1/2, 1/2 | 4/5, 1/5 |
| Run-off | 4/5, 1/5 | 4/5, 1/5 | |
In the first matrix, we know based on the new rules that if neither athlete declines their position on the team, each player will receive the option to choose either a coin toss or a run-off to determine who gets selected. If the two players agree, the method they agreed upon will be used. If the two players disagree, a run-off will be used as the tiebreaker. In this scenario, we assume that a coin toss is fair, and each outcome is equal to 0.5 probability if they agree to a coin toss. All other strategy combinations would result in a run-off, in which Allyson Felix, who at the time owned 13 Olympic and World Championship medals compared to Tarmoh’s 0, would most likely win in a high-pressure situation. With these payoffs assumed, there are two pure strategy Nash equilibria: (Run-off, Coin Toss) and (Run-off, Run-off). However, this game assumes that neither athlete declined their spot on the team, which leads us to our next game.
| Jeneba Tarmoh (Player 2) | |||
| Declines Position | Does Not Decline | ||
| Allyson Felix
(Player 1) |
Declines Position |
0,0 |
0,1 |
| Does Not Decline |
1,0 |
29/40, 11/40 | |
If at least one of the athletes wanted to avoid a run-off, they could elect to instead decline their position on the Olympic team and give the third and final spot to the other athlete. You might ask yourself, “Why would anybody decline a spot on the Olympic team?” All we can assume is that the option is there, and confounding variables should always be considered when analyzing players’ decisions. After calculating the payoffs for the bottom-right box based on the assumed probabilities in the first game, we find that (Does Not Decline, Does Not Decline) is the pure strategy Nash equilibrium. This makes sense given that a rational player would not simply decline a spot on the Olympic team. However, this is not what occurred. In this scenario, Tarmoh declined her position by refusing to participate in a run-off, saying she felted wronged by the ruling of an initial tie and that racing Felix again would imply that she acknowledged the dead heat happened. Fortunately, Tarmoh was still named to the Olympic team as a member of the 4×100 meter relay, but the sport was robbed of what could have been an epic head-to-head race for Olympic glory. It has been 7 years since this anomaly occurred, and to my knowledge this is still the only instance in which this section of the rule book has been used. It would certainly be interesting to see another case of this, considering the payoffs for each athlete would differ based on how closely matched they are.