In an eclectic golf tournament, players play the same course
more than once and select the best score on each hole as the score for the
competition. If played a full handicap,
the competition would favor the high handicap player. To demonstrate this, assume two players have
the probabilities for scoring shown in
Table 1.
Table 1.
Table 1
Probability of Scoring Relative to Par on
Each Hole
Handicap

Score
Relative to Par


1

0

+1

+2


Low Handicap

.1

.8

.1

.0

High Handicap

.0

.2

.6

.2

Over eighteen holes, the low handicap player would average
even par. The high handicap player would
average 18 over par for a difference between players of 18 strokes. There
handicaps would be approximately 18 strokes different.[1] Now assume each player is allowed to play two
rounds and take his best score. The new
probabilities for each hole are shown in Table 2.
Table 2
Probability of Scoring Relative to Par on
Each Hole after Two Rounds
Handicap

Score
Relative to Par


1

0

+1

+2


Low Handicap

.19

.80

.01

.00

High Handicap

.00

.36

.60

.04

The low handicap player would average 3.24 strokes under
par while the high handicap player would average 12.24 strokes over par. The difference between players is now 15.48
strokes. To make the competition fair,
the handicaps should be multiplied by a factor of .86 (i.e., 15.48/18).
For a fourball eclectic the percentage allowance should
be even lower. Assume that both players
get an identical partner for a fourball eclectic. The probabilities of each score after two
rounds for both teams are shown in Table 3.
Table 3
Probability of Scoring Relative to Par on
Each Hole after Two Rounds of FourBall
Handicap

Score
Relative to Par


1

0

+1

+2


Low Handicap Team

.3439

.6560

.0001

.0000

High Handicap Team

.0000

.5904

.4080

.0016

The low handicap team would average 6.19 strokes under
par and the high handicap team would average 7.37 strokes over par. The difference in team scores would be 13.56
strokes. To make the competition fair
for fourball, the allowance should be 75 percent (i.e., 13.56/18).
The allowances of 86 percent for a stroke play eclectic and
75 percent for a fourball eclectic were derived using a very simple
model. To see if these allowances are
reasonable, a small fourball eclectic tournament is analyzed. The allowance for each player in the
competition was 90 percent of his handicap.
The tournament handicaps and results from the first and second day for
two flights are presented in the Appendix.
In a perfectly equitable competition, scores should not be
correlated with handicaps. When Day 1
Scores were regressed against Total Handicap, the estimated equations were:
Flight 1 Day 1 Score = 64.7  .004·Total
Handicap
Flight 2 Day
1 Score = 72.1  .250·Total Handicap
In Flight 1, the coefficient of Total Handicap variable was
both small and not statistically significant
(tstatistic = .01). This finding implies taking 90 percent of each player’s handicap led to a fair competition. A oneday fourball competition is similar to a twoday stroke play eclectic. Basically, one best score out of two is chosen. So the theoretical allowance of 86 percent is clearly in the ball park. In Flight 2, both the coefficient of the Total Handicap variable and its tstatistic (t=.9) were larger. While the coefficient is not statistically significant at the 95 percent level of confidence, it does indicate equity could be improved if the allowance was less than 90 percent.
(tstatistic = .01). This finding implies taking 90 percent of each player’s handicap led to a fair competition. A oneday fourball competition is similar to a twoday stroke play eclectic. Basically, one best score out of two is chosen. So the theoretical allowance of 86 percent is clearly in the ball park. In Flight 2, both the coefficient of the Total Handicap variable and its tstatistic (t=.9) were larger. While the coefficient is not statistically significant at the 95 percent level of confidence, it does indicate equity could be improved if the allowance was less than 90 percent.
The second day results were:
Flight 1 Day
2 Score = 60.8  .086·Total Handicap (tstatistic = .59)
Flight 2 Day
2 Score = 67.1  .257·Total Handicap (t=statistic = 1.30)
The equations indicate the higher handicap teams are
favored, but neither coefficient is significant at the 95 percent level of
confidence. The equation for Flight 1
predicts the highest handicap team should have a score 1.3 strokes lower that
the lowest handicap team (i.e., the difference in team handicaps multiplied by
.086). If the 75 percent allowance was
used, the high handicap team would lose three more strokes than the low
handicap team. On average, this should
lead to a reduction in the highest handicap team’s score of 1.5 strokes. In essence, the advantage of the high
handicap team is wiped out when the 75 percent allowance is used. In Flight 2, the regression equation predicts
a 2.5 stroke advantage for the highest handicap team. Using the 75 percent allowance would reduce
the score of the highest handicap team by 1.7 strokes on average. While the highest handicap team would still
have an advantage, it would be less than one stroke.
Of the top three finishers in each flight, only one came
from the top half (i.e., the lower handicap teams) the flight. Clearly, 90 percent did not produce an
equitable competition. The lesson here
is Tournament Committees should study the effect of allowances on tournament
outcomes. Experiments with different
allowances should proceed until equitable competition is achieved.
Appendix
Flight Scores and
Handicaps
Tournament ResultsFlight
1
Team

Total Handicap

Day 1

Day 2

1

11

67

62

2

12

60

57

3

16

67

60

4

21

64

59

5

21

69

63

6

23

63

56

7

24

68

59

8

24

62

58

9

25

62

57

10

26

65

60

Tournament ResultsFlight
2
Team

Total Handicap

Day 1

Day 2

1

26

62

59

2

26

65

61

3

28

67

60

4

29

66

58

5

30

69

63

6

30

63

60

7

34

61

56

8

33

65

57

9

35

66

61

10

36

60

57

[1]
The USGA’s “bonus for excellence” and possible differences in scoring variances
would have small effects on the handicaps.
The effects, however, are small and are neglected in this analysis.
Hi Laurence Dougharty,
ReplyDeleteI appreciate your articles on the handicap for golf but in this case, I have some difficulty in understanding how the probabilities appear in percentage for the players of high and low handicap to reach birdies, pars, bogeys and 2 bogeys.
How are they collected for eclectic two rounds?
If you can explain in advance, thank you.
I am a Portuguese golfer  hcp 18
Thanks and regards
JosÃ© Rocha
The probabilities for the low and high handicap players were derived from a sample of scorecards. A scratch player averages about 2 biridies a round. This equates to a probability of .11 (2/18) for making a birdie. The probability is rounded down to .1 for simplicity. The probabilities are meant to be illustrative only as probabilities can vary for players with the same handicap.
ReplyDeleteThe probabilities for two rounds stem directly from the probabilities for one round. For example, the low handicap players has three outcomes: Birdie, Par, and Other. There are five different ways his best score on a hole would be Birdie: BB,BP,PB,BO, and OB. The probability of the event BB is .1 x .1 = .01. The probability for events BP and PB is .8 x .1 =.08 for each. The probability for the events BO and OB is .1 x .1 = .01 for each. Therefore, the probability of having a best score of Birdie is .01 + .08 + .08 + .01 +.01 = .19 as shown in Table 2.