(Note: This is a corrected version of a post of the same name from 2012. The previous post omitted the Appendix. The Appendix is shown in this version)
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 A-1 of the Appendix. 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 two-best ball score of
the foursome is 1.11.
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 A-2 of the Appendix. The expected two-best ball score on each hole
for the threesome would be 1.25. In an eighteen-hole competition, the
foursome would have a two and a hall stroke ((1.25-1.11)·18=2.52) 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 A-3. 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 three-score 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 three-score 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 three-score 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 18-hole
competition, a threesome would have a small edge of less than one-stroke.
Under Method 1, the threesome has an expected 18-hole 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 Distribution
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 4th 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
(1.64-1.48)·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.
Appendix A
Table A-1 presents the possible combinations of scores for a foursome (0 = Birdie, 1 = Par, 2=Bogey). Column 2 shows the probability of each combination. Column 3 presents the frequency of each combination. That is, how many different ways can a foursome make two bogeys and two birdies for example? As shown in the table, there are 6 ways that combination can occur. The probability of having two birdies and two bogeys is 0.003906. Since this combination can occur in six different ways, the probability of this outcome is.0234375 as shown in column 4. The 2-best ball score for each combination is shown in column 5. In the example there are two birdies so the two best ball score is zero. The expected team score is the product of the Probability of Occurrence and the 2-Best Score summed over all combinations. In this case, the expected team score for a foursome is 1.11.
The expected score of a threesome
under Method 1 is derived using the same methodology as shown above. The expected score is 1.44 as shown in Table
A-2. The 2-best score is found by taking
the expected value for each combination.
For example, assume a team has scores of 2,1,0. If the player scoring a 2 could be used
twice, the 2-best score would be 1. If
the player scoring 1 could be used twice, the 2-best score would be 1. If the player scoring 0 could be used twice,
the 2-best score would be 0. Since each
player is equally likely to be able to use his score twice, the expected 2-bes
score is .67 (1/3∙1
+ 1/3∙ +1/3∙0). The expected team score under Method 1 is
1.25
Under Method 2 the probabilities
stay the same but the 2-Best Scores are slightly different. Having a guaranteed par on a hole reduces the
size of a bad hole score. The expected
score under Method 2 is 1.28.
[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 3-5 strokes
lower than his average for all scores (i.e., the course rating must be 3-5
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?
