Endnotes:

1. To properly adjust for course difficulty and to estimate player ability, we roughly follow the method used in Connolly and Rendleman (2008). To build intuition about this method, consider the following linear regression: $$ S_{ij} = \mu_{i} + \delta_{j} + \epsilon_{ij} $$ where \( S_{ij} \) is golfer i's score at tournament-round j, \( \mu_{i} \) represents the fixed player skill level of player i, and \( \delta_{j} \) represents the course difficulty for tournament-round j. The latter two sets of terms are regression coefficents from indicator variables for players (indexed by i) and for tournament-rounds (indexed by j). The intuition behind the estimation of player abilities in this setup is that we are comparing the performance of golfers on the same course; similarly, to obtain estimates of course difficulty, we are comparing the same golfer's performance on different courses. One restrictive feature of this regression setup is that a player's skill level is defined to be fixed over time. We augment this specification by allowing \( \mu_{i} \) to vary over “golf time” (this is the chronological sequence of rounds the golfer plays). Consider the following regression: $$ S_{ij} = \mu_{i}(t_{i}) + \delta_{j} + \epsilon_{ij} $$ where \( \mu_{i}(t) \) is now a function of time (where time is specific to each player, and represents their sequence of rounds). The notation used does not specify the functional form: in practice, we use second or third degree polynomials depending on the number of rounds in the golfer's sample. The intuition behind the estimation of player ability here is a little harder to see but is fundamentally the same as the case with fixed player ability: here we model a player's ability over time with two or three parameters, whereas before we only used one. With these estimates in hand, a golfer's adjusted strokes-gained in a given round is defined as \( S_{ij} - \delta_{j} \). This is a relative measure: differences in adjusted strokes-gained are meaningful, but the level is not (e.g. values of 100 and 101 contain the same information as values of 0 and 1). We decide to make the average adjusted strokes-gained on the PGA TOUR in 2017 equal to zero for our reference point.

2. From the estimation procedure above, we have an estimate of each player's ability at each point in time. Let's focus on how to simulate a single tournament. Each of the golfer abilities are expressed in adjusted strokes-gained per round; further, ability level should be thought of as an expectation (i.e. their mean score). We need to account for the fact that an individual golfer's scores vary across rounds. This is accomplished by assuming that a golfer's scores follow a normal distribution with a mean equal to our estimated ability, and a standard deviation which can be estimated from the data. Then, in a single simulation of the tournament, we "pull" a value from each golfer's posited distribution (i.e. Normal with a given mean and standard deviation). The golfer with the highest "pulls" from their distribution wins the tournament. Do this many times (recognizing that each simulation produces a different "pull" from each golfer's distribution), and we are able to estimate win probabilities for each player.