It's Ryder Cup week and we are going to be providing live probabilistic forecasts Friday through Sunday! In this blog we are going to (very briefly) outline how the predictions work, and then highlight two interesting things that have to do with choosing optimal pairings.

To predict match play, we are going to simply adapt our round-level predictions to provide a hole-level scoring distribution for each player (that is, the probability of making eagle, birdie, par, etc.). Here is how we currently estimate the ability level of each player on the American and European sides (in strokes per round):

The Americans are stronger at every position except for the bottom 3 players, where the Europeans have the edge. In the team formats on Friday and Saturday, where only 8 golfers play each session, the lower-end quality of each team may be less important. The average ability of the US team is 1.55 strokes per round above the PGA Tour average, while for the European team it is 1.43.

Next I'll outline how predictions are formed. To predict singles match outcomes, the process is simple given that we have a hole-level scoring distribution for each golfer. For foursomes matches, keeping with our M.O., we do not consider "pairings interactions" which means that all that matters for our predictions are the individual estimates of ability. That is, we don't allow for certain players to be complements when they play the alternate shot format together. These complementarities may exist, but to include them in a model will require some assumptions. There is simply not enough data on team golf to draw strong conclusions, and so you must appeal to a theory of how individual golf relates to team golf. We abstract from this and model foursomes as a singles match, where the scoring distribution for each team is an average of the two team members. Finally, we model a fourball match as the minimum score on each hole from the two respective scoring distributions of a team's golfers. This is quite nice as it will pick up any complementarities that could exist in this format (e.g. pairing a player who makes a lot of birdies / bogies with one who makes a lot of pars).

To put everything together and predict the Ryder Cup, we have to make some assumptions about how specific pairings will be formed. Luckily for us, team captains submit their lineups without knowledge of the other side's picks. This solves part of the problem, but we still need to choose which 8 players will participate in each team session. For this, we give higher ability players a higher probability of being selected: for example, on the American side, Dustin Johnson has an 85% chance of playing in any given team session, while Bubba Watson has just a 35% chance. These are arbitrary choices, but ones that have to be made. Once we select the 8-man squad on each side for a given session, matches are set randomly (as they are in actuality). With these assumptions in hand, it is straightforward to simulate the Ryder Cup using the scoring distributions we have for each player and the adjustments we have described for the team formats.

Our start-of-tournament estimates give the American team a 54% chance of winning the Ryder Cup outright and an 8% chance of tieing and retaining the cup. This leaves a 38% probability that the European side wins outright. The only real arbitrary choices we are making is how pairings are selected. If lineups were set completely randomly, so that every player had an equal chance of playing in any given match, the relevant probabilities are 52%, 8% and 40%. This makes sense, given that the American team is relatively weaker in the lower ranks of their team.

Now let's consider a couple interesting exercises. One question we had while thinking about predicting the better-ball format was whether it was better to have a team with one good player and one bad player, or a team with two mediocre players. The figure below answers this question.

The left figure is just shown for reference, and indicates the expected points in a singles match for the better player as a function of how big their skill advantage is. The right figure is the interesting one. Here, we fix one team's ability levels at 0 for both team members. Then, for the other team ("team A"), we keep their average ability at 0 but vary the gap between individual abilities (i.e. +1 and -1, +2 and -2, etc.). What we see is that it's (slightly) better to have one good player and one bad player than two mediocre ones. I suppose this is intutive, especially when you consider the extreme case: a +10 ability player and -10 player would certainly have a big advantage over two 0 ability players in the fourball format.

Now let's talk about optimal pairings. Normally the popular conversation on this topic is centered around which players "fit" together well - either because of personality or due to specific attributes of their golf games. We are going to abstract from this and take a pure data approach. The first thing to note is that it is not possible to solve for optimal pairings. This is an incredibly complex game theory problem. Here's why. By my rough calculations there are about 52,000 unique teams that can be chosen for a team session from EACH of the 12-man teams. Now let's think about team captain strategy. It would be (somewhat) feasible for us to determine the best American team assuming the European side is forming teams randomly. We could do the same for Europe. However, this is not optimal (or, not an equilibrium you could say). If Furyk knows that Europe is picking their best team assuming the Americans are forming teams randomly, then he should take this into account and tailor his team to Europe's "optimal" team (the one they would choose if Americans were picking randomly). Of course, Europe should also take this into account about the American team, and so on. Long story short, there will be an incredibly complex equilibrium that we can't begin to solve for.

What we are going to do is try to come up with the best teams for each format assuming the other side is forming their pairings randomly. While this may not be the most realistic, it is a useful benchmark and should highlight which sorts of teams yield the highest expected points. It is interesting to think about what will make for the best teams in this exercise. Let's think about the better-ball format first as it has more nuance to it. Obviously, good teams will have players with higher predicted abilities. Further, we saw in the exercise above that it is slightly advantageous to pair a skilled player with a less skilled player, all else equal. But, there is also the question (in either team format) of whether the two best players should play together to "guarantee" a point, and if the two worst players should play together to "sacrifice" a point. Again, it's not at all clear to me what the optimal strategy would be. Let's head to the data.

We take a brute force approach and loop through as many combinations of teams as we can (subject to our computation power limitations), calculating the expected points for each team under consideration. For example, we would form a possible American team, and then do many simulations where that same American team competes against a randomly formed European team in each simulation. Average points over all the simulations, and we can rank teams based off their expected points. Here are the top 10 teams on the American side for each format (we don't do the optimal European teams, as the exercise takes quite a long time):

Not surprisingly, our "optimal teams" for the US are basically just comprised of their top 8 players. It's worth noting that the expected points for the US team obtained using the selection algorithm that we described above for our Ryder Cup predictions is 2.11 for a fourball session, and 2.06 for a foursomes session. This, in addition to the optimal team results above, indicates that the fourball session provides a greater advantage to the better team - this is because there are roughly twice as many shots being hit, which makes the outcome slightly less influenced by randomness. The differences in these tables may seem small, but a 0.1 difference in expected points translates to about 2% greater win probability for the overall match. Unfortunately, I don't think there is much else to read into for these "optimal teams" - because we only ran 2000 simulations, some of these differences could just be simulation error. It is hard to determine whether any of the mechanisms mentioned above are at work here (e.g. pairing your very top guys together to try and "guarantee" a point). It is interesting to recognize that if the American team played their top 8 guys in every session this would, in theory, result in an overall win probability that is about 3-6% higher (depending on what strategy the Europeans are assumed to adopt) than where we currently have it. However, this would be a very bold strategy for a captain to take - one that would garner a lot of criticism if it did not work out in their favour. It is hard to ignore the numbers though: is there really a justification for playing Phil or Bubba in place of any of DJ, JT, Fowler, or Tiger (guys we estimate to be more than a shot better than Phil)? We don't think so. In any case, it should be an exciting week.