Mathematics is the last thing that should determine a winner in sports. Sports is visual. Sports is art. We want to see the winner run across the finish line and strike a pose. We want to see the winner jump and twist and defy gravity. We want to see the winner swim like a dolphin (I like dolphins) and then get of the pool majestically. We want to watch and know that what we saw was the best we would see that day and maybe ever. So it is unfortunate that we need to resort to math, statistics, and complex formulas to identify a winner, but in Heptathlon and Decathlon, that has become the most important event.
A few days ago, Jessica Ennis-Hill of Britain lost to the Belgian athlete, Nafissatou Thiam, narrowly missing gold and thus not being able to defend her Olympic title from the 2012 London games. Narrowly is used unscientifically or visually here, 35 points, which without perspective is meaningless. Heptathletes are scored by a system defined by Viennese mathematician, Karl Ulbrich and I might need a Ph.D or way more time than I have to figure that out and there is the first fault. When I googled Heptathlon scoring, websites with worksheets showed up, where I could enter my times, heights and distances and it would tell me my overall score. I have to trust it with all math hidden behind. A good scoring system should be easy to understand. An analogy at work is when I built a complex risk calculator for losing a client and started scoring our delivery teams on their likelihood of loss, every client delivery manager was up in arms saying it was too complicated and he/she did not trust it. Though I stood by my math, I had to reconfigure for the realities of the business.
So what are the best ways to score a multi-sport event? Like any business problem, before we jump into the math and analysis, let’s figure out what the score/ performance measure is supposed to accomplish.
The score should tell us who is the best athlete of that day.
Is that enough? Do we need the score to inherently do more?
- The score should account for how much better you are than the others.
- The score should be advantageous to an athlete who is reasonably good in all the sports and not just great in 1 or 2 and terrible in the others.
- The score should allow us to compare across years and identify athletes who were the best ever.
- The score should be your own, non-manupilatable by any other individual athlete.
- The score should be on visual glance, tell you how much better you are than the next competitor.
Method 1: Sport Placement: Simply rank each athlete from 1 to last for each event and then add up the scores. The person with the minimum wins. If an athlete won all seven events in a Heptathlon, she would get 7 points and win.
Method 2: Top Score Deviation: An athlete gets 0 score for any sport that they were the best in. The other athletes score for each sport is calculated as percentage difference from the Top Score. The sum of all individual scores are added and the athlete with the lowest score wins.
Method 3: World Record Deviation: Same as Method 2, except we use the World Record for each of the sports and then calculate the athletes percentage difference from the World Record. The sum of all individual scores are added and the athlete with the lowest score wins.
Method 4a,b,c: World Record Deviation Points: An extension of Method 3, but 10 points are deducted for each percentage point an athlete is away from the world record. An athlete who breaks the WR gets 1000 points and then every percentage point is 10 points less and thus if you are 100% worse than the WR, then for example in the throwing events, throwing 0m will get you 0 points. 4B and 4C are same methodology but rounding up or rounding down your percentage difference to a whole number and calculating the score, both methods yielding same final ranks.
Method 5: WR to Bottom Performer Spread: In this method, the spread between the WR and the last athlete’s sport score is calculated and then divided by 1000 to get the spread per point. So if the WR is 12 seconds the lowest athlete ran 15 seconds, the spread per point would be (15-12)/1000 = 0.003. Then the best athlete who might have run 12.2 seconds gets (15-12.2)/0.003 = 933 points for the sport. The sum of all individual scores are added and the athlete with the highest score wins.
Method 6: WR to 25th percentile Spread: Same as Method 5, but instead of taking the last athlete’s sport score, we take the 25th percentile.
Method 7: WR to Min Qualification: Same as Method 5, but instead of taking a score from an athlete in the competition or the 25th percentile, we take the minimum times, heights and distances that an international body deems necessary to score in a sport. For this exercise we used the minimum from the USATF that would have got 0 scores in the current system.
It can be seen that Method 1 is simple, and meets a lot of the criteria but has a big drawback in accounting for differences in level in each sport. An athlete could beat another by just 0.1 seconds in 200m and lose out by 10 seconds in the 800m and that difference in skill is not being accounted for. Method 2 and 3 are inelegant in that the top scores are 0 and we always want the winner to have big scores. Method 4 is one of my favorites. It is simple in that it is tied to the world record, if you beat it you get 1000 points and thus it is easy to understand. Method 5 and 6 fall prey to losing control of your score by someone else’s bad performance. Method 7 might give Method 4 a run, jump, throw for its money and is very similar except the bottom is fixed by an understanding of the sports and the top is fixed by the world record. *A tweak might be to measure everyone to the WR at the time of calculation instead of to the new records that might be broken thus athletes might be able to score higher than 7000.
Here’s how the scores and ranks would play out for the 2016 Women’s Olympics Heptathlon
Conclusion
While the discussion over the last few days has been around Jessica Ennis-Hill narrowly missing out on the Gold and blaming it on the scoring system, I think the competition was extremely tight between Jessica and Nafissatou but Nafissatou did deserve to win. Sport is cruel in that the smallest margins can determine a winner when the effort to get there is indistinguishable. The real victim I believe is Laura Ikauniece-Admidiņa who by many methods of calculations should have ended up 3rd, ahead of Brianne Theisen-Eaton. But this is just math.
Karthik Raja 