Equitable handicapping of scramble golf tournaments

This lack of effort from participants is not limited to amateur golf. In a study of professional golfers, Brown (2011) found that the performance of many professionals decreased when a superstar, in this case Tiger Woods, was in the field.

Another possibility, provided the number of teams is relatively large, is to divide the teams into “flights” based on skill level, so that each team is only competing against teams of similar ability. However, even when flights are used there will still be disparities in the abilities of competing teams, so handicapping is still useful.

For brevity, we provide a basic overview of the relevant features of course rating and slope rating here. For a much more detailed discussion by the USGA’s Director of Handicapping, we refer the interested reader to Knuth (2010).

All quotes in this paper of the USGA Handicap Manual are taken from the online version of that document, which is available online at http://www.usga.org/Rule-Books/Handicap-System-Manual/Handicap-Manual.

The slope augments the course rating by incorporating the concept of a “bogey golfer” as well as a scratch golfer. The approach to calculating the “bogey rating” is essentially the same as that for the course rating, except that it is constructed from the perspective of a bogey golfer. The slope is determined by computing the difference between the bogey rating and the course rating, then multiplying this by 5.381. For further details, we refer the interested reader to the USGA Handicap Manual, Section 2.

Equation (3) is used when 20 scores are available. The number of scores used to compute the handicap index can range from a minimum of 5 to a maximum of 20 scores. As the number of scores in the playing record decreases, the percentage of scores used decreases from 50% and the handicap formula above changes appropriately. Please see Section 10 of the USGA Handicap Manual for additional information on this and on other features of the handicap index.

If some team members do not have a handicap, the common practice is to estimate a handicap based on the typical score recorded by the player. This estimate, while potentially imprecise, is generally considered acceptable and preferable to excluding the player from participation.

Based on our analysis of a large set of individual golf scores (described in Section 4), we find that their distribution resembles a normal distribution, except that the right (left) tail is longer (shorter) than expected. Thus, we use a gamma distribution to model G_{S, i}. This distribution closely approximates the normal distribution when the shape parameter is relatively large, but still enables us to capture the asymmetric nature of the tails of the distribution of golf scores. Note that also follows a G(α_i, β_i, γ–72) distribution. For the purpose of exposition we concentrate on the distribution of gross score in the initial discussion of the model.

These findings are consistent with those found using a smaller data set (20 rounds of golf for 49 golfers) by Bingham and Swartz (2000). They found, using least-squares regression, that The paper also developed an estimator for average differential score of the i^th golfer; and, although not based on least-squares, their estimator is implicitly a linear function of HI_i equal to

We remark that the USGA system is considered to be superior to simpler scramble handicapping methods such as the “Ambrose” format, where the team handicap is simply one half of the average handicap of the four team members. Analysis using our empirical data confirms that such an approach, which ignores the outsize importance of better players to team performance, provides highly biased outcomes.

The larger magnitude of those negative coefficients for higher skilled team members reflects the fact that higher skilled players have a larger impact on team performance.

A study by Kupper et al. (2001) suggests that the USGA handicap index favors better golfers in individual contests between two players. Although their study did not consider a scramble competition, they suggest that “our findings can be reasonably extrapolated to other types of golfing competitions in which USGA handicaps are used” (p. 35).