Introduction  Many
tournaments consist of a format where foursomes compete against other foursomes
in the field. When the field cannot be
divided evenly into foursomes, threesomes are created. The threesome is then allowed a “blind draw”
for the fourth player (i.e., the score of another player in the field is drawn
and his score becomes that of the missing fourth player.)
While the “blind draw” is equitable
it has several problems. First, a team’s
performance is determined in part by luck rather than on how well the team
played. Second, if the blind draw played
well, his performance can help the threesome and therefore hurt the chances of
his own team. Third, it is more difficult for the player in a threesome to
evaluate risk/reward decisions when the performance of the fourth player is
unknown.
This paper evaluates two methods
around this problem:
·
Method 1: The threesome is allowed to use one
player’s score twice on a hole. The
chosen player is rotated each hole so that each player’s score can be used
twice on six holes. A typical rotation
would have the lowest handicap player take the first hole, the second lowest
handicap the second hole, and the third lowest handicap player the third
hole. This rotation would be repeated
every three holes.
·
Method 2: The threesome is assigned a player who
always has a net par on each hole.
The evaluation proceeds in four
steps. First, the basic probability
model for the evaluation is described.
Second, probability values are estimated using data from two
courses. Expected hole scores for
various methods are then computed to determine the preferred method for
threesome competition. Third, a
sensitivity analysis is performed to see over what range one method is
preferred over the other. Fourth,
conclusions are drawn as to the best method for achieving equitable
competition.
1. The Probability
Model  Assume a player has three
different outcomes when playing a hole.
A net birdie is assigned the value of 0, a net par is assigned the value
of 1, and a net bogey is assigned the value of 2. For demonstration purposes, probabilities are
assigned to each outcome as shown in Table 1:
Table 1
Probability of Scoring
Score

Probability

0

.25

1

.50

2

.25

The criterion for measuring equity
is the expected hole score for each team.
The method that yields an expected score for the threesome closest to
that of the foursome would be preferred.
The foursome has 81 different
scoring combinations as shown in Table A1 of the Appendix. (Note: The Appendix has not been included here for space considerations.) Each combination has a team score and a
probability of occurrence. The expected
score is the product of the team score and the probability of occurrence summed
over all outcomes. The expected twobest
ball score of the foursome is 1.14.
For Method 1 where the threesome
can use one ball twice, there are 27 different scoring combinations. Those combinations and their associated
probabilities of occurrence are shown in Table A2 of the Appendix. The expected twobest ball score on each hole
for the threesome would be 1.25. In an
eighteenhole competition, the foursome would have a twostroke ((1.251.14)·18=1.98)
advantage over the threesome.
Under Method 2, the probabilities
of each outcome for the three players is the same as in Method 1. The value of the outcomes may differ,
however, as shown in Table A3. The
expected hole score under Method 2 is 1.28.
The foursome has a 2.5 stroke advantage over a threesome competing with
Method 2.
2. An Empirical Test
 The selection of the best method will depend upon the player’s probability function at a
course. The probability function was
estimated for two courses using the same 88 players. The net scores for each player were sorted
into five categories as shown in Table 2.
The estimated probabilities are the number of hole scores in each
category divided by the total number of hole scores. These probabilities are presented in Table
2.
Table 2
Estimated Probability Functions

Probability


Score

Course 1(CR=71.2)

Course 2(CR=71.7)

2 or More Under Par

.024

.027

1 Under Par

.191

.178

Even Par

.333

.319

1 Over Par

.307

.308

2 or More Over Par

.145

.168

Table 2 shows there is a
significant probability that a player will have a net score of 2 over par or
more. The threescore model (0,1,2) used
here does not take into account such high scores. To have a score of two over par used in a
foursome event, however, three players must have that score. The probability of that outcome is small, so
the bias introduced by the threescore model should not be large.
To evaluate
the expected scores under each scoring alternative, the probabilities of 2
under and over are combined with the probabilities for 1 under and 1 over,
respectively, as shown in Table 3.
(Note: Par is considered “1” in the threescore model.)
Table 3
Estimated Probabilities

Probability


Score

Course 1

Course 2

P(0)

.215

.205

P(1)

.333

.319

P(2)

.452

.476

These probabilities result in the
expected hole scores shown in Table 4 for each method.
Table 4
Expected Hole Scores
Course

Foursome

Method 1

Method 2

Course 1

1.48

1.64

1.46

Course 2

1.55

1.72

1.50

The table demonstrates Method 2 is
the preferred format at these courses.
The expected differences in hole scores is .02 for Course 1 and .05 for
Course 2. For an 18hole competition, a
threesome would have a small edge of less than onestroke. Under Method 1, the threesome has an expected
18hole score approximately three strokes higher than that of a foursome.
3. Sensitivity
Analysis  The expected value of the score will depend on the probability
distribution of individual hole scores by a player. Table 5 below shows the expected team scores
for alternative probability
distributions.
Table 5
Alternative Probability
Distributions

Probabilities

Expected Hole Score


Alternative

P(0)

P(1)

P(2)

Foursome

Method 1

Method 2

1

.1

.5

.4

1.85

1.94

1.77

2

.2

.5

.3

1.38

1.46

1.44

3

.3

.5

.2

0.95

1.06

1.14

4

.4

.5

.1

0.62

0.74

0.86

The table
demonstrates the preferred method depends on whether a course is relatively
easy or difficult.[1] When net bogeys are likely (i.e., P(2)=.4 or
.3) Method 2 is the most equitable format for threesomes. On an easier course (i.e., P(2)= .2 or .1),
Method 1 yields an expected score closer to the foursome expected score and
would be the preferred format.
Realistically,
courses where Method 1 is preferred are rare.
The expected net score of a player with 4^{th} probability
distribution, for example, would be 5.4
under par. This would imply that the
course rating is approximately 9 under par.[2] A review of the golf courses in Southern
California found no golf course with such a wide disparity between par and the
course rating. [3]
4. Conclusion 
The research found that Method 1—one player’s ball counting twice—is not an
equitable format. This method was found
to be marginally superior only on courses that do not seem to exist. On most courses, a threesome playing under
Method 1 would have an expected score some three strokes more than a foursome
(e.g., on Course 1 the difference would be (1.641.48)·18=2.88). Method 2 appears to ensure equitable
competition on courses where the course rating is around par.[4] Since most course fall in this category,
Method 2 is the recommended format.
[1]
The best measure of difficulty is the difference between the course rating and
par. If the course rating is much lower
than par (e.g., 67 versus 72), the player would be expected to have fewer net
bogeys than on a course with a course rating of 73.0.
[2]
A player’s index is determined by the average of his ten best scores out of the
last twenty scores. Depending on the
variance in the player’s scoring distribution, the average used for his
handicap will be around 35 strokes lower than his average for all scores
(i.e., the course rating must be 35 strokes lower than his expected
score).
[3]
Southern California Directory of Golf,
Southern California Golf Association, North Hollywood, CA 2006
[4]
On courses where the course rating is much higher than par, Method 2 may yield
too big of an advantage to the threesome.
When adopting any method, records should be kept so that the equity of
competition can be empirically tested.
That is, do threesomes or foursomes win more than their fair share of
competitions?
Eliminating the Blind Draw.
ReplyDeleteMy club had been messing around with options to handle this problem until I (a high school mathematician only) suggested that we should try using the second best score as the score of the third player.
EG Stableford scoring.
A  2 points
B  2 points
C = 1 point
Team gets 6 points.
This has now been accepted by all the informal groups in the club as results seems to suggest that teams of 3 win a prize in proportion to the number of teams of 3 or 4 entered. The no of prizes is in proportion to the size of the field.
I would be interested to know if you thought there was any mathematical support for this.
I posted ("Eliminating the Blind Draw Continued") a quick analysis of your method on December 2, 2015. Your method does indeed appear to be equitable. The acceptance of your method by other club members should be proof enough. Good job!
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