At the moment, the Olympics are in full swing in Rio de Janeiro. I happened to be in Rio about six weeks ago for a conference in maths. Besides from being a lot of fun, the location was on the Copacabana. Having strolled around the city quite a bit, I feel a strange connection to this particular edition of the Olympic games and I’ve been watching all sorts of sports fervently.
Two days ago was also the day the Dutch women’s field hockey team would have to face the opponent they fear most, Argentina. Before the tournament, they expressed confidence in being able to face any team. Except for Argentina, they seemed a bit apprehensive on that one. They beat them with with a 3 – 2 victory, but still it makes me wonder.
This example illustrates a desire to not face certain opponents, especially in the knock-out phase the tournament. Other athletes have expressed similar concerns looking at the tournament schedule. This, to a mathematician, smacks of nontransitivity.
Transitivity and nontransitivity
Transitivity, besides from being a good scrabble word, is a property of operators in mathematics such that they “pass things along.” A basic example is the equals-sign: if A = B and B = C, then A = C. The equality of A and B can be passed along to C. If the equals-sign would be nontransitive, the equation A = B = C would be correct, but it wouldn’t imply A = C.
Transitivity is also the thing that allows ordering of elements, say numbers. We know that the number three is greater than two and less than four, which also implies that two is less than four. Likewise, we can order any sequence of numbers and the result would be a list where each element is smaller than its right neighbour but larger than its left. Except for the largest and smallest numbers, of course, that have only one neighbour.
Simple examples of nontransitivity are also available, take for instance rock-paper-scissors. Rock beats scissors, scissors beat paper and paper beats rock. So we can’t really order them from weak to strong. If we were to take a slightly more abstract idea of ordering, we might count the number of wins-versus-losses, but then we find that they are exactly equal. Each object is better than one and worse than one.
The night before last, the Dutch women’s field hockey team beat Argentina, to great joy of the Brazilians present. The Dutch media described Argentina as the “angstgegner” of the Dutch team. By the way, I think it’s rather funny that we have to borrow a word for opponent we really fear to face, as we don’t have a word for that ourselves.
The Dutch team has alway been, of course, a contender for the gold, winning medals since Atlanta in 1996 and winning the gold since Beijing in 2008. Also this time around they are expected to do well and they seemed very confident of bringing home the gold. Until they saw the schedule and had to face Argentina. All of a sudden, they were not so confident any more and became slightly more reserved. The golden promise turned into a golden hope.
Before the knock-out phase, they were hoping that some other team would beat Argentina so that they wouldn’t have to. They were fairly confident they could beat anyone, as long as it wasn’t Argentina. But they were also fairly confident that other teams could beat Argentina. This is the nontransitivity of the ability to beat other teams.
This seems weird to me, to be able to call yourself the best, you don’t have to be the best, you just have to beat all the opponents you face. Due to nontransitivity, you might just end up facing only opponents you can beat. In this way, the random lottery of the knock-out schedule has some influence on the eventual outcome of the tournament.
I suppose due to time and budget restrictions this is preferred over a full round-robin event, as the combinatorics stacks really quickly. To reduce the effect of the nontransitivity to a certain degree, a poule phase is used before the elimination phase, where each poule does have a round-robin structure.
Regardless of any reservations due to the added randomness of the elimination phase, the performance of any athlete during the Olympics is highly admirable. Whether or not they actually win a medal due to a lucky break, it is a world-class performance nonetheless.