# Tag Archives: Olympics

## predicting Olympic medal counts per country

Fast Company recently ran an article [Link] on regression models used to predict medal counts. The modeling was performed by  Dan and Tim Graettinger, two brothers who work for Discovery Corps, Inc. The Graettinger brothers discuss more modeling details on their blog post [Link].

They predicted medal counts in two steps. First, they used logistic regression (which is useful for modeling events with binary outcomes) to predict which countries would medal and which would not. They found that performance medaling in the prior Summer Games was the strongest predictor of whether a country would medal in the Winter Games:

At the last two Winter Games, no nation won a medal without having won at least one medal in the preceding Summer Olympics.  I never expected that!  Our predictive model would ultimately fill in a zero for the anticipated medal count in Sochi if the nation did not win a medal in London.  Also during the profiling stage, we saw other variables rise to the top:  migration rate, doctors per thousand people, latitude of the capital city, value of the nation’s exports, and some measures of gross domestic product.  Ultimately, once we built our logistic model, it had a 96.5% correct rating.

I suppose this makes sense. Big countries all medal in the Summer Games, so the profiling stage essentially picks “small” countries that will medal and that will not. The underlying mechanism for why this works might reflect government and social support for Olympic training programs of any kind. Good support would make medaling in both the Summer Games and Winter Games more likely. But this also implies that the Jamaican bobsled team has a chance due to the excellent Jamaican sprinters.

The next step was to take the output of the logistic regression model–the countries that would medal–and use linear regression to predict how many medals they could expect to walk away with. Because the number of events (and medals) changes every four years, the number of medals is somewhat meaningless. They had to rescale to rescale the output to make consistent predictions using historic data.

The four variables used to predict medal counts (for countries expected to medal) are:

1. geographic size (Russia, China, US). This was somewhat surprising, but could reflect geographic diversity: big countries probably have mountains somewhere where athletes can train.
2. GDP per capita (it’s the economy, stupid)
3. the value of its exports  (it’s the economy, stupid, part 2)
4. the capital city’s latitude (Norway, Sweden, Finland!)

Here is the data used by the linear regression and the output:

Regression output predicting medal counts. I’d love to see confidence intervals.

Linear regression is a simple model that makes a lot of assumptions that may not be valid. Something like negative binomial regression may be more accurate (it is often used to model call counts). In my experience, other more appropriate regression models don’t improve upon linear regression by a whole lot, so I expect that their results and insights would not be greatly affected by this modeling choice. But I’m also not a regression guru, so other comments here would be appreciated (leave a comment!)

I’m going to mention validation in case any of my students are reading (I’m teaching undergraduate simulation this semester). It’s easy to build a bad model, and validation is useful for avoiding a model that spits out nonsense.

The modeling is interesting and fun to do, but nearly all of the work involved collecting and assembling the data.  This will not be a surprise to you if you have worked on a project with real data. I have also emphasized this point in the course I’m teaching this semester.

“If he had known how long it would take to assemble the data,” Dan tells Co.Design, “maybe Tim would’ve told me to work on something else.”

## Wall Street Journal model

The Wall Street Journal also has a model for predicting medal counts [Link]. They use Monte Carlo simulation to generate medal counts. They are light on methodology details, but it looks like they use complete different data than the Graettinger brothers: they mention success probabilities at the individual athlete level, which are then probably aggregated into country medal counts. Here is what they predict.

The Wall Streat Journal’s medal predictions. Again, confidence intervals would be appreciated, but I suppose confidence intervals don’t make for good news.

After the Olympics, we will have to revisit these predictions and see which did better.

For more reading on Punk Rock OR:

## subjective scoring in Olympic sports drives me a little crazy

The Olympics are beginning. When I think of the Olympic sports, I think of a lot of sports that scored subjectively. Not so much stronger, faster, and more goals, more of panels of judges picking winners amid controversy. I prefer number crunching and objective scoring. A New York Times article by John Branch [Link] overviews the changes to the winter Olympic sports in the last two decades. In summary, the new sports are mostly those with  subjective scoring (halfpipe, snowboard cross).

A good run early in the contest might receive an 80. A slightly better run might earn an 83. A brilliant run, one that seems unbeatable, might score 95. All of the others are slotted around them. It can frustrate athletes, who ask why their second-place score was 10 points below that of the winner. They struggle to understand that the value means nothing; what matters is how it ranks.

I’ve noticed this, too, and it’s frustrating. Some sports like figure skating and gymnastics have well-established rubrics for scoring, but they are not perfect. On the positive side, the judges do a fairly good job of recognizing the best performances.

Does subjective scoring bother you?

~

Look for more Olympics posts from me in the next couple of weeks.

I’ve been blogging for almost 7 years, so I have a few old posts about the Olympics. Here are a few that I recommend reading:

## how to predict how many world records will be broken in the 2012 Olympics

I found a paper online by Elliott Holli eld, Victoria Trevino, and Adam Zarn (link) that uses survival analysis to estimate how many Olympic records would be broken in the 2012 Olympic games. Here is the paper abstract:

We use recurrent-events survival analysis techniques and methods to analyze the duration of Olympic records. The Kaplan-Meier estimator is used to perform preliminary tests and recurrent event survivor function estimators proposed by Wang & Chang (1999) and Pena et al. (2001) are used to estimate survival curves. Extensions of the Cox Proportional Hazards model are employed as well as a discrete-time logistic model for repeated events to estimate models and quantify parameter signi cance. The logistic model was the best t to the data according to the Akaike Information Criterion (AIC). We discuss, in detail, covariate signi cance for this model and make predictions of how many records will be set at the 2012 Olympic Games in London.

Here’s what they predicted:

After prleminary measures, we estimated ve di erent models with signi cant covariates. Four of these models were extensions of the Cox Proportional Hazards model while the other was a discrete-time logistic model for repeated events. In the end, the logistic model was the best t to our data based on the Akaike Information Criterion, and it depended primarily on the following covariates: SameAthlete, WorldRecord, PCC, MRI, and TRI. We also used survival estimates from three di erent recurrent event survivor function estimators to determine the number of new records that will be set in the 2012 Olympics. In 51 of the 63 events we considered in track & field, canoeing, cycling, and swimming, the estimated number of Olympic records that will be broken is somewhere between 20.12 and 31.14.

(Note: I added the part in italics)

This site lists the Olympic and world records broken in the Olympics:

Number of world records broken = 13 (4 in Track&Field, 3 in Cycling (Track), and 6 in Swimming)

Number of Olympic records broken = 29 (10 in Track&Field, 5 in Cycling (Track), and 14 in Swimming). This estimate is within the interval found in the paper.  Even more impressive: the acknowledgments indicate that at least one of the authors was an undergraduate. Kudos!

## how to resolve ties in Olympic sports

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.

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!

## when is the optimal strategy to crash in track cycling race?

Apparently as soon as you know you are not going to win and the race rules allows a team to “restart” if there is a crash.

Thanks to @fbahr and @parubin for bringing this to my attention.

Related blog post:

## how to maximize the probability of getting a medal in diving

In some Olympic sports, such as diving, the athlete receives scores based on several trials. In diving, each trial is a separate dive. A diver’s score is the sum of the different dive scores in different trials. What is the best way to maximize the chance of getting a medal?

Women’s springboard diving rules (link):

• Women must complete five dives.
• There is no limit on the total degree of difficulty for these dives.
• At least one dive during the contest must come from each of five different categories – forward, back, reverse, inward, and twisting.
• No dive can be repeated in a list of dives.
• Divers must select dives ahead of time and cannot change the order

Men’s platform diving rules (link):

• Men must complete six dives.
• There is no limit on total degree of difficulty for these dives.
• For the men, at least one dive during the contest must come from each of six different categories – forward, back, reverse, inward, twisting and armstand.
• No category can be repeated in a list of dives.
• All dives must be competed from the 10-meter platform.
• Divers must select dives ahead of time and cannot change the order

Let’s make three assumptions:

1. Divers can select their dives and their order on the fly (clearly not true, but makes it more interesting).
2. A diver’s performance on each dive is independent of his/her performance in other dives.
3. From experience, divers know the distribution of points based on each of their dives.

Model 1: Divers know the “threshold” for medaling ahead of time (dubious!)

Let’s solve this problem using a Markov decision process, where the stage here is the dive (1-5 for women or 1-6 for men).

Let

Vt(S(t)) = value of being in state S(t), where S(t) = the total number of points.

and let

M = point threshold for medaling.

The rewards are Rt(S(t-1),P) = 1 if the diver moves from state S(t-1)<M to S(t)>=M (i.e., the diver moves into medal contention if the P points from the dive at time t-1 moves them above threshold M). All other rewards are zero. The diver wants to maximize the probability of medaling, which is equivalent to maximizing the total value (V1(0)) or the expected value of the 0-1 indicator variable indicating whether the diver medals or not.

Let the set of dives in each category be captured by D1, D2,…,D5 for women or D1, D2,…,D6 for men. For each d in Di, there are known probability distributions for the point totals P.

Now the Bellman equations are

Vt(S(t)) = max_{d in Dt} E{Rt(S(t),P) + V[t+1](S(t)+P)}

Here, the expectation is taken over the points distribution for dive d in Dt. The probability of medaling is found by V1(0), the value before dive 1 starting with 0 points. The boundary conditions are V[T+1](S(T+1)) = 0 for all values of S(T+1) since rewards are accumulated once.

The optimal policy indicates what dive should be chosen in each trial (MDP stage) based on the total number of points that have been accumulated thus far. If a diver is successful with tough dives early on, the diver can choose easier dives later on (and vice versa).

Model 2: Divers do not know the “threshold” for medaling ahead of time so they maximize the total number of points

The model dynamics here are identical to those of the model above except for the rewards. The random rewards Rt(S(t-1),P) = P(d,t) if the diver gets P points for their dive d. The rest is the same yielding Bellman equations of

Vt(S(t)) = max_{d in Dt} E{P(d,t) + V[t+1](S(t)+P(d))}

They look the same as above, but the rewards in boldface are different. The expected number of points is captured by V1(0) and the boundary conditions are V[T+1](S(T+1)) = 0 for all values of S(T+1) as before.

Here, the diver doesn’t really need to solve an MDP. He or she can simply select the dive that yields the most points (on average) in each category, since the choices will not depend on the number of points accumulated thus far. Let EP(d,t) denote the average points from dive d. Then, the policies depend on the stage t rather than the full state variable S(t), yielding Bellman equations of

V[t] = max_{d in Dt} (EP(d,t) + V[t+1]).

The expected number of points is captured by V(1)  and the boundary condition is V[T+1] = 0. Note that we lose the expectation here. The optimal solution isn’t rocket science: it is to select the dive with the largest EP(d,t) for each t.

Why these models are different

On face value, it may not be obvious why these models could yield different solutions. They are both used to identify dives that yield many points. The second model maximizes the expected number of points whereas the first model maximizes the point total distribution above a threshold (which can be thought of as moving as much of the tail of the distribution past a fixed threshold M). The first model could lead to “riskier” dive strategies for a diver who is not a favorite to win: the diver has a chance of being on the podium but could also go down in flames. For the gold medal favorite, the first strategy might lead to a conservative strategy that weeds out all of the dives that have a chance of a disastrous result.  The second model leads to more conservative dive selections that yield more points (on average) but that might yield the most points but would almost certainly not lead to a medal.

Lift the first assumption: divers must make their selections ahead of time

If we lift the first assumption, the choice of dives cannot depend on the point total thus far to identify the best set of dives regardless of what happens with the earlier dives.

The answer to the second model is still obvious. The optimal solution is never change dives on the fly, even when one has that choice.

The first model, however, can be examined by looking at the joint probability distribution in the points that would be expected.

Let the decision variables be captured by

x(d,t) = 1 if dive d is selected in stage t and 0 otherwise.

Let P(d,t)

Then our stochastic optimization model is

max Pr( P(1,1)x(1,1) + P(1,2)x(1,2)+…+P(|D1|,1)x(|D1|,1)+…+P(|D5|,5)(|D5|,5) > M )

subject to x(1,t)+x(2,t)+…+x(|Dt|,t) = 1 for all t=1,2,3,4,5

x(d,t) \in {0,1} for t=1,2,…,|Dt|, t=1,2,3,4,5.

I’ll stop here because I’ve been up late watching the Olympics. I’ll leave the solution as an exercise for the reader. Leave feedback and corrections in the comments.

If you’ve ever dove from 10m, you rule. I jumped from 7m a few times and am unwilling to go any higher.

## when is the optimal strategy is to throw a badminton match?

I’ve been following the badminton scandal in the news. First, let me say that I never thought I’d hear the words “badminton” and “scandal” in the same sentence. I thought that this was one of those rare cheating events that sometimes happens. The evidence suggests that the optimal strategy may sometimes be to lose on purpose.

In previous Olympics, the entire set of badminton matches were in a tournament with knockout rounds. A team could apparently not chance their seed or path to gold. In this Olympics, there was a new preliminary pool round with the performance in the pool leading to seeds in the later single elimination tournament with 16 teams. This introduced a wrinkle to the optimal strategy. Now a team might want to think about how they could optimize their seed/path in the single elimination rounds. Obviously, introducing a path to the gold by doing something other than winning every possible match potentially invites trouble. However, the badminton rounds of play with a preliminary pool and later tournament is like what happens in volleyball, basketball, soccer, and other sports. I haven’t seen “lazy” players obviously throwing a match. Usually, winning more in the preliminary round only makes it easier later on.

Apparently, there is a huge incentive in badminton to throw matches. The badminton magazine Badzine conducted a study about throwing matches, since it is common knowledge that the Chinese throw matches. They found that matches were won by “walkovers” or early retirements in 20 of 99 (20%) of all-Chinese matches [link to article]. Walkovers/retirements occured in 5.3% of matches when non-Chinese athletes played their compatriots and in 0.74% of the matches between Chinese athletes and non-Chinese athletes. So the Chinese “throwing” rate is an outlier, which suggests that the Chinese badminton players are all in cahoots with each other. The Olympics scandal, however, included two Chinese teams, one Indonesian team, and one South Korea team.

Here is why the players’ strategy changed from winning all preliminary games to throwing a match:

The attempt to throw the two matches in the women’s doubles had it roots in a surprise result earlier in the day when the Chinese second seeds, Tian Qing and Zhao Yunlei, were beaten by a Danish pair. The result meant that Yang and Xiaoli had to lose to avoid meeting their compatriots before a potential showdown in the final.

A similar desire to avoid the hot favourites seemed to have cross-infected the second match, between  South Korean third seeds Ha Jung-Eun and Kim Min-Jung and Indonesian pair Meiliana Juahari and Greysia Polii [link]

And yes, the cheating was so obvious that people in the audience booed the athletes’ lack of effort:

Lori Halford, 35, who had paid £40 along with her husband for tickets to the evening session, told The Independent: “It was immediately clear that something wasn’t right. The first shot went into the net, then the next shot went into the net. The next return went under the net. There was no speed or strength to their play. There was a complete contrast to the others playing in different games who were giving it everything they could.” [link]

Throwing a badminton match is not dissimilar to sumo wrestlers throwing matches, which was discussed in detail in Freakonomics.

## beach volleyball and game theory

I played volleyball from junior high until when I finished my PhD.  I mostly played traditional 6-on-6 inside volleyball, but I was happy to play just about any type of pickup game. In all those hours on the court, I learned a few things about volleyball strategy.  6-on-6 volleyball highly specialized. A team usually has a setter, two outside hitters, two centers, a weak side hitter, and a defensive specialist that plays in the back row usually for the centers. The team uses their three hits to set up an offense, and no player can touch the ball twice in a row.  Usually, the setter is shielded from the first hit so that she can handle the second hit, accurately passing the ball to a hitter for the best possible offensive attack.

Beach volleyball is a bit different, and the strategy is worth discussing. Clearly, both players have to be good at everything (defense, passing, hitting, blocking, and anticipating the other team’s move).

Less obvious is the role of game theory.  Here, there are two players per team, and a team has three hits to get the ball over the net. It is almost always best to use all three hits to properly set up the offense to maximize the chance of winning the point.

The opposing team, however, wants to minimize the chance of this team getting the point. Every team has a stronger hitter. Usually it’s the tallest person on the team. A team’s goal is to get the best hitter the ball on the third hit. This means that the best hitter needs to get the ball on the first shot. The strategy here is less complicated with a deterministic route of hits between the two players (i.e., if player A gets the ball first, then only player B can get the ball next).

The opposing team is aware of who should not get the ball first, and generally serve or hit to the shorter player. This is like a zero-sum game with perfect information. During the US beach volleyball games with Misty May-Treanor Kerri Walsh Jennings, I noticed that teams almost always served to Misty (the shorter of the two) than to Kerri, despite how good Misty is at serve receiving and defense. This was not a coincidence. This is also not the same strategy as in indoor volleyball, where tall players are usually less skilled defensively, and as a result, often are the target of serves and spikes.

Have you noticed other Olympic sports that have a strong game theory aspect? I’m sure there are lots!

## Are you looking forward to the Olympics?

I am looking forward to the Olympic games starting tomorrow.

I am happy to announce that there will be at least one operations research major at the Olympics: Quanitra Hollingsworth, who has a B.S. in Mathematical Sciences/Operations Research from VCU will be competing in the Olympics for the Turkish women’s basketball team. I am so proud of her!

Aside from watching Quanitra compete, I am most looking forward to seeing sports that I don’t normally follow in between Summer Olympics, including:

• men’s and women’s gymnastics
• all swimming events