Friday, May 15, 2015

Handicap Allowances for Eclectic Golf Tournaments

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
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 four-ball eclectic the percentage allowance should be even lower.  Assume that both players get an identical partner for a four-ball 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 Four-Ball


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 four-ball, 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 four-ball eclectic were derived using a very simple model.  To see if these allowances are reasonable, a small four-ball 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
(t-statistic = -.01).  This finding implies taking 90 percent of each player’s handicap led to a fair competition.  A one-day four-ball competition is similar to a two-day 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 t-statistic (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 (t-statistic = -.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 Results-Flight 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 Results-Flight 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.

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