## 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.

1. Hi Laurence Dougharty,
I 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

2. 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.
The 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.

3. simply! should the competition be played over 12 months would the players handicap be his full handicap on the last round?

1. As shown in the article, playing to full handicap would give the higher handicap players a big advantage. Using the handicap as of the last round is also subject to manipulation. There are apps (e.g., Golf Genius) that can keep a player's eclectic score based on some percentage of his current handicap. I'm afraid I cannot be of much help as there has been no research in this area.

4. Thank you for this analysis, it has helped me to understand more fully how playing handicaps impact outcomes in multi round eclectic competitions. Can I ask you to provide a breakdown of the calculation used to generate the high handicapper expected outcome of 12.24 (strokes over par average) given in the paragraph which follows table 2.
Thanking you in anticipation.

5. With regard to my request above dated 7thOctober2022 please disregard my request for more detail - I’ve reviewed the process and have arrived at the figure you quote.

1. Thanks for saving me the trouble. I was just about to test my memory about the expected outcome. Since things are kind of slow, I might do it anyway.

6. Thanks for replying Laurence. Perhaps you could advise on a related matter. My club, not unlike others I’m sure, currently runs an eclectic in association with a series of 8 Monthly Medals throughout the main golfing season. Can you suggest a suitable handicap reduction factor used in the final calculation of the Final Eclectic Nett score. Currently no such factor is used, full h’cap allowances are employed in evaluation of the Nett winner. Thanks again.
Barry

1. This comment has been removed by a blog administrator.

2. This comment has been removed by a blog administrator.

3. I didn't like my two previous answers. I will work on a better one.

7. I think I’m correct in using appropriate terms from the expansion of a 4 term squared bracket to establish probability values for 4 possible outcomes (B,P,Bo&DBo) you list in Table 2. My calculations of the high h’cap probabilities give the same as you quote, but those for the low h’cap do not concur. In the order given : B,P,Bo&DBo I get 0.172, 0.656, 0.0001 and 0.
The fact that the high h’cap value are the same suggests I’m going about it in the right way? Any comment would be helpful. Thanks.
Barry

1. There 7 ways the best score of two low handicap players is a Birdie --BB, BP, PB, BBo, BoB, BDBo, and DBoB. The probabilities for those events are .01,.08. .08,.01, .01, .00, and .00 which add up to .19. There are five ways the best score is a par--PP, PBo, BoP, PDBo, and DBoP. The probabilities of those events add up to .80. There are three ways the best score is a bogey. Hope this helps.

8. I found one example you might try. The Committee tracks a player's gross eclectic score. At the end of the season, 75 percent of a player's handicap is deducted from his gross eclectic score to find his adjusted score. Low adjusted score wins. For handicap you can use ending, starting or some average handicap, The Committee should plot adjusted score versus handicap. If high handicap players are dominating, the percentage should be lowered.

9. Thank you for your note of 16th Oct which confirmed my thinking on how to achieve certain outcomes. It led me to double check my arithmetic and find the error - prob(Bo)=0.1 not 0.01.
Also for the advice (Oct 15th) concerning appropriate handicap ‘adjustment’ factors - a pragmatic approach which accounts for course conditions and can be justified with real local data.