Tuesday, October 22, 2019

Eliminating the Blind Draw

(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 ho

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 


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

Course 1(CR=71.2)
Course 2(CR=71.7)
2 or More Under Par
1 Under Par
Even Par
1 Over Par
2 or More Over Par

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

Course 1
Course 2

These probabilities result in the expected hole scores shown in Table 4 for each method.

Table 4

Expected Hole Scores

Method 1
Method 2
Course 1
Course 2

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

Expected Hole Score
Method 1
Method 2

            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 (1/31 + 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?

Tuesday, October 15, 2019

How Will the World Handicap System Affect Your Index?

A player’s Handicap Index is based his adjusted scores and the Course and Slope Ratings.  The USGA Course Rating is based on a model which predicts the better half of scores of the 288 competitors in the U.S. Amateur Championships.  The Bogey Rating is equivalent to the average of the better half of a bogey golfer’s scores under normal playing conditions.[1]  Under the World Handicap System (WHS) that will become effective January 1, 2020, however, a player’s Handicap Index will be based on the better 40 percent of a player’s differentials (i.e., best 8 of 20).  The size of any reduction in a player’s Handicap Index depends on whether the Course and Slope Ratings are changed to reflect the drop in the number of best scores used—i.e., the current Course Rating is an estimate of the performance of a scratch handicap when his best 10 scores are used and not his best 8 scores.

It is unlikely the Scratch and Slope Ratings will be changed to reflect the changes under the WHS.   The cost of changing USGA models and the Course and Slope Ratings at courses is prohibitive.   So what will be the effect on a player’s Index given the Course and Slope Ratings stay the same? Any change in the Slope Rating due to the WHS is negligible and is neglected here.[2]  The change in a players Index will be determined by the standard deviation of the distribution of his scoring differentials and the Bonus for Excellence (BFE) – currently .96.

Assuming a normal distribution, the average of a player’s ten best differentials is .8∙σ below his mean differential. The average a player’s 8 best differentials is .95 below his mean differential.   Therefore, a players Handicap Index should be reduced by .15∙σ multiplied by the BFE.   If a player’s standard deviation is 3.5, the reduction in Handicap Index would be 0.5 ( i.e., .15∙3.5∙.96) .  The impact on a player’s Course Handicap would be would be none or one stroke.  Given all of the other complexities of the WHS (Daily Handicap Index, Daily Course Rating based on weather conditions,  limits on Handicap Index reductions,  possible reduction of the BFE, and changes in Equitable Stroke Control) a player may not notice a change in his Index nor understand the cause for the change if he did notice.  A typical player’s reaction to all of this might be “Whatever.”

[1] Stroud, R.G., Riccio L.J.,” Mathematical Underpinnings of the slope handicap system,” Science and Golf, E & FN Spon,  London, 1990, pp. 141-146.
[2]  The decrease in the Course Rating based on 8 out of 20 differentials is likely to be small.  To estimate the revised Course Rating assume a player’s differentials are normally distributed with a standard deviation of σ.   The average of a player’s ten best differentials will be approximately .8∙σ below his mean differential.  The average of a player’s eight best differentials will be approximately .95∙σ below his mean differential.  So to keep a “scratch player” a “scratch player” the Course Rating should drop by .15 ∙σ.   Assume the distribution of a scratch player’s differential has a standard deviation of 2.5.  In that case, the new Course Rating should be 0.4 strokes (.375 strokes rounded up)
Similarly, if the average Bogey player has a standard deviation of 3.5 strokes, the Bogey Rating should be reduced by 0.5 strokes.
The men’s revised Slope Ratings will then be:

              Revised Slope Rating = 5.381 ((Old Bogey Rating -0.5) – (Old Course Rating -0.4))
                                                     = Old Slope Rating – 5.381∙(-0.1)
                                                     = Old Slope Rating -1.0 (rounded up)

Wednesday, January 9, 2019

Should a Club Adopt the Local Rule on Lost and Out-of-Bounds Balls?

In the new Golf Rules for 2019, the USGA will permit committees to adopt a local rule on Balls Lost or Out-of-Bounds:

Balls Lost or Out-of- Bounds: Alternative to Stroke and Distance: A new Local Rule will now be available in January 2019, permitting committees to allow golfers the option to drop the ball in the vicinity of where the ball is lost or out of bounds (including the nearest fairway area), under a two-stroke penalty. It addresses concerns raised at the club level about the negative impact on pace of play when a player is required to go back under stroke and distance. The Local Rule is not intended for higher levels of play, such as professional or elite level competitions. (Key change: this is a new addition to support pace of play.)

This local rule should pass three tests before it is adopted by a club.  First, is it needed?  Second, does it actually increase the pace of play?  Third, is the cost of administering the rule worth any perceived benefit?

Need – Typically, local rules should only be adopted when abnormal conditions make it impractical to abide by the Rules of Golf.  Examples would be prohibiting play from environmentally sensitive areas, allowing the removal of stones from bunkers, and allowing preferred lies.  This local rule does not meet the requirement of abnormal conditions and is promoted solely to increase the pace of play.  If a ball being hit out-of-bounds or lost is a common occurrence, then the local rules would pass the needs test.  If not, this local rule is a worthless addendum to the rules sheet. 

This local rule will come into play primarily in singles stroke play events.  In match play or better-ball events a player who believes his ball is out-of bounds will usually play a provisional ball (more on this later).  If he does not, and finds his ball out-of-bounds, he often does not go back to the tee, but disqualifies himself from the hole.  In either of these cases, the local rule is of no use.

There are courses where balls can be easily lost.   The new rules permit the committee to declare areas where balls can easily be lost (e.g., gorse, woods, and raw desert) penalty areas.  If many lost balls are now in what can be considered penalty areas, the need for this local rule is greatly reduced if not eliminated.

Pace of Play – The USGA argues this local rule would increase the pace of play.  There are counter arguments that contend it will not increase, and possibly decrease, the pace of play.  A player who finds his ball out-of-bounds still has the option to return to the tee.  For example, assume a player hooks his ball out-of-bounds on a par 3 hole.  He finds it (Point A on the figure below) some 60 yards from the hole.  He can now drop the ball on the fairway 60 yards from the hole and be hitting four or return to the tee hitting three.  After a few minutes of calculating his odds of making a five, he decides to re-tee.  In this case, the new local rule actually decreases the pace of play.  

While some of the new rules are intended to diminish conflict among competitors, this local rule does not.  The question of whether a player hit the ball twice, for example, does not need be resolved under the new rules.  This local rule, however, can lead to disagreements among competitors.  Assume a player hits the ball out-of-bounds on long par four.  The ball is not found, but all participants’ believe the ball went out-of- bounds.  The player who hit the ball believes the ball traveled twenty yards further than his opponent believes.  The Committee is called and it decides that without any contrary evidence, the player’s judgment should prevail.  The player drops his ball at point B.  His opponent argues the ball is now closer to the hole than point A.  Both players draw their range finders and conclude the ball must be dropped five yards further back.  In this case, the local rule increases both the time of play and the enmity among players.

There are also cases where the local rule is difficult to apply.  One example would be when a player’s ball is out-of-bounds behind he green.  A player must first estimate how far the out-of-bounds boundary is from the flagstick (i.e., Point A).  Then he may have to walk back to the fairway to find the nearest point no closer to the flagstick (i.e., Point B).  The player now has a large swath of “general area” created by an arc connecting Point A and Point B.   If the player takes anytime deciding the best place to drop, the local rule will not increase the pace of play.

Another example would be on a hole with a sharp dogleg to the right.  If a player hooks his ball, there could be no Point B that is not closer to the hole.  The USGA suggests in such cases:

If a ball is estimated to be lost on the course or last crossed the edge of the course boundary short of the fairway, the fairway reference point may be a grass path or a teeing ground for the hole being played cut to fairway height or less.

If a player goes back to a tee (either the one played from or a forward tee) under the local rule, he would be hitting four without a tee.  A better option might be to abandon the original ball and be hitting three with a tee.  Again, the local rule does not increase the pace of play.

This local rule option is not available if the player hits a provisional ball.   A player is well advised not to hit a provisional ball, but proceed to the out-of-bounds ball and analyze his options.  If he finds, for example, his Point B is further from the hole than his typical drive, he can then return to the tee and hit his third stroke. Again, whether this local rule increases the pace of play is questionable.

Cost of Administration – Members at clubs will eventually understand the geometry of this local rule if it is adopted.  Visitors unfamiliar with the rule could struggle.  The club could publish a lengthy rules sheet explaining how to determine Points A, B, and C.  Such a sheet is unlikely to be read and would obscure other local rules that are more likely to come into play.   

Clubs have the option to suspend this local rule for some tournaments.  If the rule is not suspended for the club championship, players could be insulted that their tournament is not a “higher level of play.”  If it is suspended the Committee must make sure the suspension is included on the tournament rules sheet.  The "on again off again" nature of a local rule leads to confusion and should be avoided.

These are not large administrative problems.  A club must assess whether even these minor hassles are worth the limited benefit of the local rule.

Conclusion – The USGA’s objective was to make the game simpler and faster for the recreational player.  This local rule will not meet these two objectives at most clubs.  The local rule is not simpler than having a player hit a provisional ball.   And if the Committee is worried about the pace of play, it has better weapons in its arsenal such as conducting tournaments with Stableford scoring.  

Another argument against the local rule is it violates the principle that everyone plays by the same rules. The USGA followed this principle when it allowed all golfers, regardless of ability, to drop outside of a bunker with a two stroke penalty.  To be consistent, the USGA could have made the relief procedure in this local rule an alternative to stroke and distance relief found in the Rules of Golf.   It did not.  It could be the USGA believed the local rule option would affect the equity of the game by favoring the long hitter.  Or it could have believed having Tiger Woods drop a ball in the fairway after an out-of-bounds shot would be too big a departure from the history and traditions of the game. 

While this local rule looks like it should be rejected on its merits, many clubs will adopt it merely because it comes with the imprimatur of the USGA.  This will allow for an evaluation of the local rule based on empirical evidence and not the speculation.  Any comments by readers on how this rule works in practice would be greatly appreciated.