Introduction - The handicap system is
designed to bring equity to the competition between individual
competitors. Many competitions, however,
are based on team performance (four-ball, foursome, scramble). The problem then is how to combine individual
handicaps into team handicaps to make for fair competition.
The United States Golf Association
(USGA) has made recommendations on handicaps for different formats.[1] The USGA, however, has not published any
empirical research validating its recommended team handicaps. Moreover, USGA recommendations are not
internally consistent. In four-ball
match play, the USGA recommends teams play at full handicap. In four-ball stroke play for men, however,
handicaps are reduced by 10 percent. For
women, the recommended reduction for four-ball stroke play is only 5
percent. It would be difficult to
justify on theoretical grounds why four-ball match play should be played at
full handicap while four-ball stroke play should be played at 90 percent of
course handicap.[2] There is also nothing inherent in the
handicap system that could explain why men and women should have different
handicap adjustments in four-ball stroke play competition.
This paper presents a methodology
for the empirical verification of the equity of a handicap procedure. The methodology is applied to the case of
four-ball stroke play.[3] This examination is structured along the
lines of the questions it intends to answer.
First, is the 90 percent allowance recommended by the USGA equitable? Second, do teams with larger differences in
handicaps between partners have an advantage?
And third, what is the optimal handicap allowance for four-ball stroke
play?
Is the 90 Percent Allowance Equitable? -
USGA recommendations on handicap allowances appear to based in part on the work
of F. Scheid of the USGA Handicap Research Team. In a 1971 study, Scheid examined the net
scores of players with various combinations of handicaps.[4] His study was based on 50 players from one
club. Scheid did not have data from a
four-ball event. Instead he took the
scores of these 50 players and simulated matches between them. Based on these simulations, Scheid claimed
players should be allowed 107 percent of their handicap in four-ball stroke
play competition. In 1971, however, a
player’s handicap was computed as 85 percent of the average of his ten best
scores. Today, that percentage has been
increased to 96 percent. Sheid’s
recommendation in terms of the current handicap system would be to allow
players roughly 95 percent of their handicap.
Sheid’s
argument that handicaps should be reduced proportionately for four-ball stroke
play competition can be represented by Model I below:
Net Score =
a + b•H
where
H
= Average handicap of the two players
If
high handicappers had an advantage in four-ball competitions a team’s net score
would be expected to decrease with its handicap. A plot of tournament net scores illustrating
such a bias is shown in the figure below.
The
b-coefficient in Model I is a measure of the inequity of the handicap. This coefficient is the slope of the line
drawn in the figure above. If b were to
equal -.1, for example, the model would predict that for every ten stroke
increase in average handicap, net score would be expected to decrease by one
stroke. If the coefficient were zero,
then there is no correlation between handicap and net score which is the ideal
situation. Estimates of this coefficient
using data from various tournaments would indicate the magnitude and direction
of the bias.
The value of the b-coefficient of
Model I was estimated with the data from the 1997 Southern California Golf
Association (SCGA) Four-Ball Stroke Play Championship. The competition was conducted at four
qualifying sites followed by two rounds at the championship site (i.e., six
sample sites). Each player’s course
handicap was computed using the 90 percent allowance.
Five
of the six estimates of the model coefficient are negative indicating a likely
negative relationship between net score and average handicap (see Table 1).
That is, even with a 10 percent reduction, higher handicap players had an
advantage. In two cases (Menifee Lakes and Vista Valley), the
hypothesis that the b-coefficient was equal to zero could be rejected at the 95
percent level of confidence.
This research only raises the
suspicion that the 90 percent allowance may not be equitable. Any conclusion has to be tempered by the data
limitations which are serious and are discussed in the concluding section.
Table 1
Model I Coefficient
(90% Allowance)
|
Site
|
b-coefficient
(t-stat.)
|
|
Menifee Lakes
|
-0.38
(2.55)
|
|
Sierra La Verne
|
-0.09
(0.38)
|
|
Soule Park
|
0.31
(0.16)
|
|
Vista Valley
|
-0.46
(2.09)
|
|
Bear Creek Day 1
|
-0.11
(0.54)
|
|
Bear Creek Day 2
|
-0.06
(0.25)
|
Do Teams with Large Difference in Handicaps
Between Partners have an Advantage? - Sheid argued that the average
handicap of a team is not an adequate description of their ability in four-ball
competition.[5] He found that teams with spreads in handicap
do better (e.g., a 25 and 1-handicap will team better than two 13-handicap
players even though the average handicap of each team is the same.)
Sheid’s finding
suggests Model II:
Net Score =
a + b•H + c•DIFF
where,
H = Average Handicap of the Two Players
DIFF
= Difference in Handicap between High and Low Handicap
If high
handicappers had an advantage, the b-coefficient should be negative. And if larger differences in handicaps
between players are advantageous, then the c-coefficient should also be
negative.
The model was estimated using
standard multivariate regression techniques.
There was a five-stroke limitation on the difference in handicap. In the tournament, the higher handicapped
player had his handicap reduced so the five-stroke limitation was met. Such teams, however, were eliminated from the
sample used in estimating the model. The estimates of the model coefficients
are shown in Table 2 below:
Table 2
Model II Coefficients
|
Site
|
b-coefficient
(t-stat.)
|
c-coefficient
(t-stat.)
|
|
Menifee Lakes
|
-.403
(2.36)
|
-.203
(0.63)
|
|
Sierra LaVerne
|
-.077
(0.34)
|
.119
(0.27)
|
|
Vista Valley
|
-.489
(2.20)
|
-.456
(0.87)
|
|
Soule Park
|
.025
(0.12)
|
-.038
(0.09)
|
|
Bear Creek - Day 1
|
-.079
(0.37)
|
.340
(0.75)
|
|
Bear Creek - Day 2
|
.048
(0.20)
|
1.00
(1.96)
|
The c-coefficient is never
statistically significant at the 95 percent level of confidence. In three cases the estimate of the
c-coefficient is positive and in three cases it is negative. Within the limitations of the data, it is
concluded that the difference in handicap did not have a significant effect on
the net score of a team.
The evidence for the b-coefficient
is similar to that found in Model I which was expected. In two cases (Menifee Lakes and Vista Valley)
the b-coefficient is both significant and negative. The size of the b-coefficient at these two
sites indicates a large bias favoring the higher handicapped player.[6] In the other four cases, however, the
b-coefficient is not statistically significant.
What is the Optimal Allocation for
Four-Ball Stroke Play Competition? - As shown above, the 90 percent
allowance did not lead to equitable competition at several sites. What allowance would have eliminated the
correlation between net score and average handicap? To explore this question, tournament results
were simulated using actual gross scores but varying the handicap
allowance. An estimate of the
b-coefficient was made for each allowance.
The allowance that yielded an estimate of the b-coefficient nearest to
zero was considered optimal.
The simulations found the estimate
of the optimal allowance differed widely among sites as shown in Table 3.
Table 3
Optimal Handicap
Allowance
|
Site
|
Optimal Handicap
Allowance(%)
|
|
Menifee Lakes
|
55
|
|
Vista Valley
|
56
|
|
Sierra LaVerne
|
79
|
|
Soule Park
|
91
|
|
Bear Creek Day 1
|
83
|
|
Bear Creek Day 2
|
89
|
The optimal allowance is shown as a
point estimate in Table 3. In fact,
however, there is an optimal range of allowances that yield little correlation
between net scores and average handicap.
For example, in Bear Creek Day 2, any allowance between 87 and 103
percent would have led an estimated b-coefficient of less than .1.[7] Therefore, the USGA recommended allowance of
a 90 percent allowance would give equitable results at four of the sites.
At two of the sites (Menifee Lakes
and Vista Valley), however, the optimal allowance is well below the USGA
recommendation. It is impossible from
this data to tell whether the results at these two sites are due to random
variation in the optimal allowance or to anomalies caused by players (e.g.,
sandbaggers). An examination of the
slope rating at these two courses did not reveal the size of error that could
lead to the results presented here (See Appendix B).
Data Limitations and Future Research -
The conclusions of this research are far from definitive because of the
following data limitations:
1. Narrow
Range of Handicaps - Handicaps in the championship flight at Bear Creek varied
over the narrow range of between 4 and 20.
Only about 14 percent of the field had single digit handicaps. A broader and more evenly distributed sample
of handicaps is needed to have better estimates and to extend the results to
the general population of golfers.
2. Error
in the Slope Rating - An error in the slope rating could bias the estimate of
the optimal allowance. For example, if
the slope rating at Bear Creek was 128 rather than 136, handicaps would be
overestimated by 6 percent. Players
would actually be playing at 96 percent of their true handicap rather than the
90 percent allowance.
3. Limited
Difference in Partnership Handicaps - The SCGA limited the difference in
handicaps (after taking 90 percent) to five strokes. To better explore the effect of the
difference in handicap between partners a much more varied sample is required.[8]
4. Small
Sample Size - Estimates of the b-coefficient were obtained at only six
sites. To get a better understanding of
the mean and variance of this coefficient, many more sites need to be examined.
To mitigate these
limitations, data should be collected from a number of four-ball tournaments
held at clubs. These tournaments should
be member tournaments and not member-guests or invitationals. This will minimize the slope rating error
problem since handicaps would be based on scores (for the most part) from the
same course where the tournament was played.
These tournaments should also attract a greater diversity in handicaps
and in differences between partners than the SCGA tournament.
With data from say
20 clubs, estimates of the model coefficient can be made for a variety of
handicap allowances. If a small range of
allowance is consistently best across the clubs, then a recommendation on the
allowance to be used in four-ball tournaments can be made with some confidence.
APPENDIX A
Changing Index Restrictions in the SCGA Four-Ball Tournament
The
SCGA places two restrictions on the handicaps of competitors that do not
contribute to the equity of competition.
This appendix examines those restrictions and makes recommendations for
change.
Five Stroke Difference in Handicap Between Partners - The stroke
limitation means that some qualifying teams had their handicaps reduced when
they played the championship course (Bear Creek) where the slope rating was
136. Assume a team had indices of 9.5
and 15.4. At a qualifying site with a
low slope rating, they would have handicaps of 9 and 14 (90 percent of their
course handicap). At Bear Creek 90
percent of their course handicaps would be 10 and 16. The 16-handicap would have to be reduced to
15 to meet the five stroke differential.
In essence, this player is playing at 83 percent of his course handicap,
while most of his fellow competitors are playing at 90 percent of their
handicap.
To
eliminate this inequity, the SCGA should place the restriction on the
difference in indices rather than the differences in course handicaps. For example, the maximum difference in
indices between partners could be set at 5.0.
This would be easier for the players to understand, be administratively
simple for the SCGA tournament staff to compute course handicaps, and would
eliminate the inequity of players competing at different percentages of their
handicaps because of the slope rating.
Computing the Reduced Handicap - The USGA recommends taking 90 percent
of a player’s course handicap for four-ball stroke play. It is suspected, however, that this
recommendation was written before the advent of the Slope System and has never
been revised.
The USGA’s
method contains two rounding errors.
First, an error occurs when a player’s index is converted to an integer
course handicap. A second error results
when 90 percent of the course handicap is rounded to the nearest integer for
the four-ball handicap.
It
is suggested that the SCGA compute a player’s four-ball handicap based on 90
(or whatever allowance is used) percent of his index and not his course
handicap. This method is easier to compute and yields handicaps as close or
closer to a player’s ability (as measured by the non-rounded course handicap)
than the USGA method.[9]
Appendix B
Testing the Accuracy of the Slope Rating
The
methodology described in this paper can also be used to make an assessment of
the accuracy of the slope rating of a course.
According to the theory behind the slope system, net score should not be
correlated with handicap if the slope rating is correct.[10]
To examine this premise, the adjusted net score of each player was regressed
against his handicap. The b-coefficient
estimated at each site is shown in the Table below:
|
Site
|
b-Coefficient
(t-stat.)
|
|
Menifee Lakes
|
-0.16
(1.12)
|
|
Sierra LaVerne
|
0.07
(0.50)
|
|
Soule Park
|
0.18
(1.24)
|
|
Vista Valley
|
0.06
(0.40)
|
|
Bear Creek Day 1
|
0.12
(0.72)
|
|
Bear Creek Day 2
|
0.09
(0.50)
|
The results at Bear Creek, Vista
Valley, and Sierra LaVerne indicate the course slope rating is working
according to theory (i.e., the estimated coefficients are not significantly
different from zero). In the other two
cases, Menifee Lakes and Soule Park there is some apparent bias in the
results. High handicappers have an
advantage at Menifee Lakes while low handicappers have an advantage at Soule
Park. The estimates of the coefficients
at these sites are nearly statistically significant at the 90 percent level of
confidence. To lower the estimate of the
b-coefficient to the range of the other sites (i.e., ±.09), the slope rating at
Menifee Lakes would have to be reduced from 115 to 107. Similarly, the slope rating at Soule Park
would have to be raised from 107 to 120.
If Menifee
Lakes was overrated that would help explain the bias favoring high handicappers
at this site. Running a simulation at
Menifee Lakes with a slope rating of 107, however, does not eliminate the bias
as measured by the b-coefficient. The
b-coefficient was still statistically significant.
This
methodology cannot say conclusively that Menifee Lakes is overrated and Soule
Park is underrated. The research does
indicate, however, the Rating Committee should review the slope rating at these
two courses.
[1] USGA, USGA Handicap System, Far Hills, New Jersey, 1994.
[2] An argument could be made for
reducing a player’s handicap for match play.
A high handicapper will have his handicap based in part on hole scores
that would not matter to the outcome in match play (e.g., a 9 on a par
five). The USGA recommendation of
lowering handicaps for stroke play and not for match play, however, does not
appear to have a theoretical basis.
[3] In four-ball stroke play, two
competitors play as partners. The lower
of the partners’ scores is the score for the hole.
[4] Francis Scheid, “You’re not getting
enough strokes,” Golf Digest,
Trumbull, Connecticut, June 1971, p. 52.
[5] Ibid.
[6] There is no obvious explanation for
why scores at Menifee Lakes and Vista Valley behaved differently from other
sites. If the slope rating at these two
courses was overestimated, some of the bias could be explained. It would take an error of over 40 slope
rating points, however, to explain all of the bias at Menifee Lakes. Another possible explanation is that Menifee
Lakes had a disproportionate number of “sandbaggers” who had high
handicaps. Qualifiers from Menifee Lakes
were to take first, third and tie for fourth in the championship
tournament. When these three teams were
excluded from the Menifee Lakes sample, the b-coefficient in both models was no
longer statistically significant.
[7] At Bear Creek, for example, the
same teams finish in the top ten whether 90 or 100 percent of the handicap is
used.
[8] For equity and administrative ease
it is suggested the SCGA : 1) base handicaps on 90 percent of a player’s index
and not 90 percent of the player’s course handicap, and 2) place the limit on
the difference in player’s index and not on the difference in course
handicap. Appendix A details the
reasoning behind these recommendations.
[9]
Assume a player had an index of 22.0 and was to play a course with a slope
rating of 136. His four-ball course
handicap with a 90 percent allowance would be 23.83 before rounding. Using the USGA method his four-ball handicap
would be 23 (i.e. 26 handicap multiplied by .9 and round to 23.) Using the 90 percent of index method, his
four-ball handicap would be 24. The 90
percent of index method yields a handicap closer to the player’s actual ability
(i.e., 23.83).
[10]
Stroud, R.C., Riccio, L.J., “Mathematical underpinning of the slope handicap
system,” in Science and Golf I, E
& FN Spon, London, 1990, pp. 135-146.
