One of the most fundamental contributions we can make as consultants is to help our clients ask the right questions of their data. We’re often asked to help solve problems that turn out to be too broadly or too narrowly defined—or are not aligned with achieving business goals or eliminating pain points.
I find we can often draw an example from the world of sports. A question I’ve seen debated frequently is, “Will Tiger Woods beat Jack Nicklaus’s 18 major wins?” This question is especially relevant given Tiger’s recent win at the 2019 Masters, but is the wrong question entirely. What people really want to know, but can’t answer directly, is, “Is Tiger better than Jack?”[1]. Is winning one major worth more than placing second ten times? Do we count major wins just because it is easier? Defining the criteria of merit is important. At Elder Research we work with varied data and modeling techniques across many industries. This variety of experience helps us guide clients to ask better questions and develop analytics solutions to answer them.
Comparing Golfers Across Eras
The standard for evaluating golfers leaves much to be desired. The most-cited metric is the Official World Golf Ranking (OWGR), in which a player’s rank is determined by the average number of “points” they have accrued from tournaments during the past two years. Setting aside the specifics of how points are assigned, this sounds simple: the average is just a player’s total points divided by the number of tournaments they’ve played. However, the OWGR requires a minimum divisor of 40 tournaments. After winning the 2019 Masters, Tiger Woods had played in only 26 tournaments over the most recent two years, but his point total was nonetheless divided by 40, resulting in an OWGR of 6. Were one to have used 26, he would have surged to first in the world (and there’d be even more articles proclaiming his return to prominence)!
But the biggest flaw in how we evaluate golfers is not an arbitrary minimum divisor or the lack of context when counting wins in majors. A Forbes article sums it up nicely, courtesy of the legendary Gary Player:
“It’s impossible to make comparisons across eras. You can’t do it. It’s not even on the table.”
Yet, I respectfully disagree. Besides articles from well-informed but un-quantitative sports writers, there has been at least one serious attempt at tackling this problem. In the paper “Bridging Different Eras in Sports”, Scott Berry et al. attempt to reconcile differences in eras by modeling the distribution of skill levels across different decades while simultaneously estimating how each player’s performance changes as they age. Published in 1999 during the early stages of Tiger’s professional career, it reasonably establishes Jack Nicklaus as the all-time great and dismisses Tiger Woods as a talented young player likely to regress to the mean. Had the analysis been performed just a year later the results may well have been different.
A Graph Network–Based Solution
We propose a simpler, more intuitive method for comparing golfers across eras: We can rely on the multitude of players who bridge the gaps between Jack Nicklaus, Tiger Woods, and still-young phenoms like Rory McIlroy. To do so, we gathered data from the four annual golf tournaments characterized as “majors” from 1970 through the 2019 PGA Championship. Using round-by-round tournament results, we built a large, directed, densely-connected graph network in which every player is represented by a node and each pair of players within one tournament is represented by an edge. A single tournament generates thousands of edges; the player who places first in a tournament has one edge connecting him with every other participant. The weights of these edges scale with the number of strokes separating the two players. The graph below shows just the connections between the top seven finishers in the 2019 PGA Championship. Note that every player points to the winner, Brooks Koepka, while the only players without connections are those that tied for third (Cantlay, Spieth, and Wallace). The full graph contains 1.7 million edges connecting 4,000 golfers.
Figure 1. Connections between the top 7 finishers in the 2019 PGA Championship.
With major golf history now represented as a graph, the PageRank algorithm offers a clever way to determine the most “important” nodes in the network. Originally conceived for ranking web pages for Google[2] , it is also an elegant way to determine which golfers are “pointed to,” and therefore win, most often. There’s more to the graph than just counting the connections: a skilled golfer who has collected more edges (placed ahead of more golfers) is ranked more impactful and then passes that impact when pointing to another golfer who played better.
Applying PageRank to the golfers in our graph ranks Jack Nicklaus as just the sixth-best player. This approach punishes golfers like Nicklaus and Tom Watson for their remarkable longevity; they often qualified for majors well into their 50s but usually missed the cut. This is crucial to note because we are more interested in golfers’ abilities in their prime, not their composite skill level over an impressive —but highly variable— 45-year career.
To get a sense of when each golfer was at their peak, we calculated annual PageRanks using a graph built from a rolling 3 years of historical data. Since our data set begins in 1970, our first full graph describes the year 1972. Taking that year as an example, the graph is built by down-weighting results from previous years: edges from 1971 are given half as much weight as those from 1972, and edges from 1970 carry half again as much weight. Figure 2 summarizes this “windowed PageRank” of golfers over time, with special attention paid to Jack Nicklaus and Tiger Woods.
Figure 2. PageRank of golfers over time with a focus on Jack Nicklaus and Tiger Woods.
After studying golfers who played in majors across at least twelve years, we determined that most golfers remain at their “peak” for approximately eight years. We found twelve years to be long enough to make reasonable comparisons but not so long as to exclude a prohibitive number of players. To construct a final graph we only kept golfers with at least twelve years of participation in majors and further restricted the graph to include only those golfers’ eight best years (according to their windowed PageRank). The resulting graph contains 687 golfers and around 260,000 edges. The rationale for such a draconian reduction in graph size is straightforward. First, to compare across eras we only include golfers whose careers spanned a meaningful length of time. A fringe major-qualifier with two lone US Open appearances does not teach our model much, particularly when that type of player may have had a starkly different skill level in the early 1970’s than in the late 1990’s. (This suspicion is validated by the results from Berry, et al.) Second, we only consider an individual player’s best years to keep us from deriving inferences from, for example, an age-21 Tiger Woods beating an age-56 Jack Nicklaus by 29 strokes. Figure 3 shows a small fraction of the resulting players that connect Woods to Nicklaus. Each edge represents one tournament that the two players participated in. Each of these comparisons are made during the golfers’ respective primes.
Figure 3. Sample of players that connect Woods to Nicklaus from our reduced graph network.
After constructing our reduced graph, one final adjustment was necessary to account for the fact that the starting and ending points of our date range are unequally affected by the requirement of twelve years of longevity: more golfers meet the 12-year minimum in the middle of our date range, where the beginnings or endings of careers are not truncated. A foundational principle of PageRank is that being “pointed to” by more edges makes a node more important (indicates a golfer is more skilled). This results in an inherent advantage for players like Woods who peaked in the middle of our time frame. We can adjust for this by using a technique similar to normalizing a graph Laplacian matrix[3]: divide each edge weight by the product of the square roots of the degrees of the connected nodes. This has the effect of reducing the relative edge weights for golfers with more contemporaries in our graph, leveling the playing field for all golfers regardless of when they played.
The Result
The top ten players in the final PageRank results are shown in the table below. Reassuringly, none of the names in the table are surprising.
The result, which you might have scrolled past the methodology to check, is that Tiger Woods beats out Jack Nicklaus! [5] While Nicklaus was more dominant in his time (see the height of his peak in Figure 2), the talent level of second-tier competition has increased. It would be impossible to infer this result with any certainty from just the information shown in Figure 3 (recall that there are hundreds more golfers included in the complete graph). Clearly there is no strict transitive property in golf, but with enough connections you can compare the greats of old with modern players. Since all connections in the graph are between golfers in or near their prime, we can assume that the players that “bridge the gap” provide information with which to compare their predecessors and successors.
Some golf fans may bristle at Dustin Johnson edging out Seve Ballesteros, or Rory McIlroy outshining Nick Faldo. I hope they can recognize that asking—“Who was the better player?”—is better than asking “Who won the most majors?” The answer is harder to find, but the question is worth asking.
[1] Daniel Kahneman talks at length about the widespread phenomenon of substituting difficult questions with easy-to-answer simplifications in his book Thinking Fast and Slow.
[2] The PageRank algorithm is named for Larry Page, co-founder of Google, but also for its role ranking web pages.
[3] Chung, F. R. K. Spectral Graph Theory. Providence, RI: Amer. Math. Soc., 1997.
[4] The score is a player’s final PageRank multiplied by 1,000.
[5] Some will point out that Nicklaus’ career is incomplete. But, since we only keep 8 years in the data set and Nicklaus was most dominant in the 1970’s, the effect of the omission is small. Certainly not large enough to make up the gap between his score and Woods’.
