Friday, February 22, 2013

Equitable Stroke Control - A Postmortem

          The previous post, What Does the New Equitable Stroke Control Mean for Golf?,  struck a responsive chord at Golf Digest.  Jerry Tarde, editor of Golf Digest, lifted many parts of the paper for his column entitled “A formula for slow play and higher scores.”[1]  Tarde used the material without either attribution or compensation (if you don’t count an autographed copy of a book he had ghost written for Sam Snead).
I did not take offense, however, for here was a major golf publication at least questioning the USGA.  I was further encouraged when Golf Digest implied it would be conducting an independent review of ESC93.  In a letter to Mr. Tarde, I urged:

"Golf Digest to demand to see the three studies the USGA claimed supported  ESC93. I believe you will find that while the USGA will trumpet the findings of studies supporting their position, they will not make these studies available for peer review.  I hope you will be made privy to these studies so Golf Digest could come to an independent and informed evaluation of the new ESC procedure. [2]

            I was to be greatly disappointed by Golf Digest’s effort.  Golf Digest assigned the story to Guy Yocom.  Yocom did not review any USGA study, and his article was simply a listing of unchallenged replies by Dean Knuth, Director of Handicapping for the USGA.[3]  Yocom’s article lacked insight and demonstrated an interview style better suited to People Magazine.  An example will demonstrate Yocom’s deficiency:

                        Question by Yocom: Since even low handicappers now can score a 6 on a par 3, won’t this be a boon for sandbaggers?

                        Knuth: No…Par 3s are where golfers receive the fewest number of handicap   strokes, where 99 percent of players are likely to try their hardest to score well.

            Now there are two problems with Knuth’s reply.  First, there is no reason to believe a golfer’s effort on a hole is related to whether he gets a handicap stroke.  Second, and more importantly, Knuth is charging that 99 percent of all golfers are unethical since they violate the basic precept of the handicap system that “every player will endeavor to make the best score he can at each hole.”[4]  Knuth’s defense rests on the lack of integrity of the American golfer.  Poor Yocom did not identify either problem.[5]
            Yocom did a follow-up article that was equally unsatisfactory.  He even made a small technical error in trying to explain why the ESC was changed.  When I pointed out the error, it was apparently too much for Yocom.  He wrote:

                       "Just a crazy, giddy guess on my part, but is there a Dealey Plaza (where you live)…Has your company conducted any research on the subject (ESC).  I would be glad to look it over, as you must have some basis for your persistent opinion opposing the new ESC."[6]

I had to respond in a letter to Jerry Tarde:[7]

                       "Though he does not write with great clarity, Yocom seems to liken me to Lee  Harvey Oswald.  I find this personally offensive and beyond the pale of  responsible debate.  It is unfortunate when a representative of your fine magazine does not bring intelligence or insight to an issue, but only personal invective.

                        Mr. Yocom claims that I oppose ESC.  He apparently had not read my paper.  What I oppose is change that does not represent progress.  I do not believe the USGA proved any substantive benefit from the new ESC…The new ESC may be a success as (the USGA) claims.  That judgment, however, should have rested upon an independent evaluation of the data Yocom had promised in March.  Yocom’s January article failed to deliver on that promise."

            The net result was Yocom still gets a press pass to the U.S. Open, and I canceled my subscription to Golf digest.
Despite the USGA claims of great support, ESC93 did not make many people happy.  The USGA was to try a different ESC in 1997.  Currently, ESC consists of a hybrid system.  Low handicappers are now allowed to take a double bogey on any hole.  This was in answer to the many complaints that taking a 6 on a par five was not equitable.  All handicaps of ten and above have the same limits as ESC93.  No research was ever published on why low handicap players should be treated differently than high handicapped players.  It appears the USGA Handicap Procedure Committee was only responding to a political problem with low handicapped players rather than seeking a just solution to the ESC problem.  Given the track history of this most political institution, such a resolution should come as no surprise.

[1] Jerry Tarde, “A formula for slow play and higher scores,” Golf Digest, Trumbull, CT, June 1993.
[2] Letter from the author to Jerry Tarde, January 15, 1993.
[3] Guy Yocom, “Why your handicap will change this year,” Golf Digest, Trumbull, CT, March 1993
[4] USGA Handicap Manual, p. 1
[5] Yocom has no technical background and could not review any studies if the USGA gave them to him.  His principle background is that of ghost writer.  He wrote a book with Corey Pavin on shotmaking shortly before Mr. Pavin’s career started to ebb.  Therefore, the lack of analysis and evaluation is the fault of those at Golf Digest who assigned Yocom to this inquiry.
[6] Letter to the author from Guy Yocom, December 15, 1993
[7] Letter from the author to Jerry Tarde, December 27, 1993.

Wednesday, February 13, 2013

What Does the New Equitable Stroke Control Procedure Mean for Golf?

(Note: This post is a reprint of an article first published in 1993.  It is posted here because 1) It will be referenced in an upcoming post,  and 2) It iilustrates the political nature of decisions on handicapping at the USGA--a recurring theme of this blog.)
Introduction - The United States Golf Association (USGA) adopted changes in the Equitable Stroke Control (ESC) procedure for adjusting scores in 1993.  The USGA maintained that the new procedure would increase the equity of competition, and should have no effect on playing time or the prevalence of sandbagging.[1]

The USGA's conclusions, however, appear to be based on questionable analysis and an unrealistic view of the golfing world.  A more probable outcome is that the new ESC (termed ESC93 in this paper) will not lead to any significant improvement in the equity of competition and might actually lessen it.  ESC93 procedures will undoubtedly increase the problem of slow play, and make life easier for the sandbagger.  To see why all of this will happen, this paper examines the old ESC and ESC93 in light of four criteria mentioned by the USGA in defending the change: equity of competition, simplicity, speed of play, and the effect on the practice of sandbagging.

Equity of Competition - The purpose of ESC is to prevent unduly high scores from having an abnormal effect on a player's Handicap Index.  The USGA presented three arguments for why the ESC93 procedure will increase the equity of competition.  Each is examined in turn:

1. Under the Old ESC (see Table 1 below) high-handicap players frequently were not able to post their average hole-score on difficult holes. 

An example of the inequity ESC93 was designed to correct would be a player with a thirty-four handicap who averages 8 strokes on a par four, but can only post a maximum of 7 strokes (i.e., a triple bogey) under the old ESC.  Under ESC93 he would be allowed to post a maximum hole-score of 9 (See Table 2).


Table 1


Course Handicap
Limitation on Hole Score
Plus or Scratch
Limit of one over par on any hole.
1 through 18
Limit of two over par on number of holes equal to Course Handicap.  Limit of one over par on balance of holes.
19 through 36
Limit of three over par on as many holes as course handicap exceeds 18 strokes.  Limit of two over par on balance of holes.
37 through 54
Limit of four over par on as many holes as the Course Handicap exceeds 36 strokes.  Limit of three over par on balance of holes.

Table 2


Maximum Score Posted on Any Hole
9 or less

            The size of any correction to equity will depend in part on the frequency with which a player's average score exceeds the maximum score permitted by ESC.  In our example, a thirty-four handicap player averages approximately two over par on the holes he plays.  A hole that would require 4 or more strokes over par on average should be a statistical rarity.  Therefore ESC93 would not appear to have a large impact on equity because it corrects for a problem that seldom occurs.[2]

            Even the small contribution to equity of ESC93 in stroke play may be outweighed by the inequity it could cause in match play.  The player in our example is now allowed to post a quintuple bogey on a par four hole.  To demonstrate a possible inequity created by ESC93, assume the player in the example has two holes where he always posts the ESC limit of 9-strokes, and that he is still a thirty-four handicap.  He will then average approximately 1.6 strokes over par on the remaining 16 holes (i.e., he plays like a 28 handicap).  Our player should have a big advantage over the player who earns a thirty-four handicap with a more even scoring pattern.

            In match play a score of 5 strokes over par should be considered "abnormal."  Any strokes over a triple bogey are irrelevant since the hole most probably would have been lost by then.  Allowing a player to post a nine would seem to be just the thing that ESC was designed to eliminate.  In effect, our player is rewarded because of his abysmal play on two holes.  This is not only inequitable, but it sets up an incentive system that is detrimental to promoting good golf. 

2. The old ESC depends upon the par of each hole, and par ratings are often set higher than they should be. 

To make the argument, the USGA gives the example of a hole under 401 yards for women that is set as a par five at one course but as a par four at another course.  This would result in different equitable stroke control limits because of the difference in par ratings. If this is a problem, it does not appear to be a compelling reason to change the old ESC system.  A simpler solution would be to require governing golf associations to approve the par rating of each hole in accordance with USGA guidelines.[3]

3. The old ESC causes unfairly low course handicaps just below the break-points.

            In fact, the ESCC93 procedure has the same problem, if not worse, as the old ESC procedure at the "break-points."  Let's examine the breakpoint problem under the old ESC procedure first.  Assume we have an eighteen and a nineteen-handicap -- one player above and below the breakpoint under the old ESC system.  The nineteen-handicap can post one triple bogey while the eighteen-handicap cannot post any triple-bogeys.  Now assume the two players have identical eighteen-hole scores and each player has a triple bogey.  The handicap difference of one stroke would remain even though the players had identical rounds.  While this is inequitable, it is also fairly insignificant.  Even in the unlikely scenario of our example, the handicap inequity is at most one-stroke.  In trying to correct this inequity, however, the USGA has only made matters worse.

            Assume a nine and a ten-handicap as an example of players at the breakpoint under ESC93.  If each player has the same score which includes a double-bogey on a par five, the nine-handicap would post a score one-stroke lower than the ten-handicap.  If they both scored the same but each had a triple bogey on a par four, the same one-stroke disparity would appear in their adjusted scores.  Just as with the old ESC, the handicap of the player below the breakpoint is underestimated.

            While the old ESC limited the adjusted score difference of identical scores to one-stroke, under ESC93 the difference in adjusted scores could be much larger.  If both players double-bogeyed all of the par fives, for example, then the difference in adjusted scores would be four-strokes.  In the words of the USGA, the nine-handicap would receive an "unfairly low handicap."  Therefore, ESC93 does not eliminate the breakpoint inequities as the USGA maintains, but actually increases the possibility and the size of such breakpoint inequities.

            Additionally, the ESC93 procedure exacerbates the problem by adding more breakpoints.  Under the old ESC procedure, only the breakpoints between 18-19 and 36-37 affected a significant number of golfers.  The ESC93 procedure includes four breakpoints (9-10, 19-20, 29-30, and 39-40) for this same group.  In summary, the ESC93 procedure makes the problem at each breakpoint more severe and adds more breakpoints.

ESC93 also has the problem of not treating all players evenhandedly.  While the ESC93 tends to raise the index of most players, it will reduce the indexes of high single-digit handicap players.  Based on a simple probabilistic model, a high single-digit player will be giving his low double-digit handicap competitor one more stroke when the ESC93 procedure is fully implemented. [4] 

Simplicity - The USGA argued that "clubs have difficulty teaching golfers to use the old ESC procedure correctly."  The argument that a growing innumeracy among the golfing public requires a switch to a mathematically less demanding ESC formula seems strange coming from an organization that just foisted the Slope System on that same public.  Certainly more golfers understand the old ESC system than can correctly explain the Slope System.  Does this mean the Slope System should be abolished?  While simplicity is a virtue, it should not be the primary criterion for selecting appropriate handicapping procedures.

Pace of Play - The USGA states that the pace of play has nothing to do with ESC.  The USGA's position simply is not logical. ESC93 will require players to play more strokes.  That more strokes require more time seems irrefutable.  Under the old ESC, a thirty-handicap could pick up after five strokes on a par three (scoring a 6-x).  Under ESC93, the player picks up after eight strokes (scoring a 9-x).  Watching a player play out for his nine will be as painful as it is time consuming. 

            The USGA has defended its position on the specious argument that high handicappers can play as fast as low handicappers.  While that assertion may be true under certain assumptions, it avoids the appropriate research question.  The USGA should have asked, "Does it take the same player longer to play 8 strokes than 5?"  Framed in this way, the direction, if not the size, of the effect of the ESC93 on the pace of play is obvious.

            The USGA has also argued "ESC does not tell a player when to quit hitting a ball."  Again, the USGA misses the point.  While ESC does not prohibit a player from exceeding the limit, it does act as an informal barrier to such action.  If a player is not involved in stroke play competition, he will often quit hitting the ball when he reaches the ESC limit.  To quit sooner than the ESC limit would cause an unjustifiable low handicap.  To quit after the limit is reached would serve only to irritate his playing partners.  The USGA has chosen not to recognize the role of ESC in shortening playing time because such recognition would not support their change in policy on ESC. 

Effect on Sandbagging - Under the old ESC, the unethical player (assuming a ten-handicap) was limited to two-strokes over par on any one hole.  ESC93 allows him to take four-strokes over par on a par three and three-strokes over par on a par four.  Under the ESC93 procedure, he can have fewer bad holes and still maintain his handicap.  This increases the strategic advantage the sandbagger already holds over the ethical player.  The USGA recognizes ESC93 makes it easier for the sandbagger to ply his trade, but makes the argument "that it would be a step backward to write rules or to develop a handicap system solely to trap sandbaggers."

            Of course no one has argued that trapping sandbaggers should be the sole purpose of the handicap system. The system should have the dual objectives of estimating a player's ability and making it difficult for the unethical player to manipulate the system to his advantage. 

            Having two objectives can often force tradeoffs.  For example, in calculating a player's handicap only the ten best scores out of the last twenty are used.  This is inequitable to the player who has large variation in his scoring.  His handicap will overestimate his actual ability, and he will be at a disadvantage to the steadier player.  By selecting only ten scores, however, the USGA was making it difficult on the sandbagger to artificially inflate his handicap.

            While no system can be devised that is "sandbagger-proof," any new system should make it tougher for the sandbagger to ply his trade.  ESC93 does not meet this simple test.

Conclusions - ESC93 has the advantage of being simpler, but will not make the contribution to equity the USGA claims.  Simplicity may lead to a more uniform and consistent application of ESC.  This benefit may be outweighed, however, by the detrimental effects on the pace of play and the prevalence of sandbagging.

            ESC93 is also an example of the type of tinkering with the Handicap System that should be avoided.  The conservative administration of the game of golf should be the guiding philosophy of the USGA.  Changes should only be undertaken if there is documented evidence that new policies will make significant contribution to the objectives of the USGA.  The USGA has a good record in that regard with their equipment testing program, rules and competitions, and turf grass research.[5]  It is only in the area of handicap administration where too often the USGA has confused motion with progress. 


[1]  See Position Paper on the New Equitable Stroke Control Procedure, United States Golf Association, Far Hills, NJ, 1992. 
[2] And if it does occur, it reduces the probability that the round would be in his lowest ten of twenty differentials.
[3]  The USGA went back to par-based ESC in 1997 for players with handicaps under ten. Consistency in reasoning has never been a USGA strongpoint.
[4] This is a rare time the Handicap System was changed to benefit the high-handicap player.  A more complete history of the discrimination against the high-handicap player will be documented in a future post, In Defense of Sandbagging, forthcoming.
[5] The USGA’s record in equipment testing is open to debate.  Geoff Shackelford makes a strong argument that the USGA was too late in stopping the increased distance due to changes in clubs and the ball.  See Shackelford, Geoff, The Future of Golf, Sasquatch Books, Seattle, WA, 2005.

Monday, February 4, 2013

The Mathematical Underpinnings of the "New" Slope Rating

To see  how the “new” Slope Rating is related to the USGA’s Slope Rating, it is necessary to review the origins of the USGA’s Slope System.  The theory behind the Slope System was first put forward by R.C. Stroud and L.J. Riccio in an article entitled Mathematical underpinning so the slope handicap system.[1]  Stroud starts with Figure 1 below that purports to show how the average unadjusted differential of players from Perfect Valley vary by course.[2]


Perfect Valley is depicted as a course of moderate difficulty.  Stroud assigns Perfect Valley a slope of 1.13.    The slope of 1.13 is due to three factors embedded in the handicap system: 1) using only the best 10 of the last 20 differentials, 2) Equitable stroke control, and 3) the .96 bonus for excellence factor. This gives the illusion that an average course would have a Slope Rating of 1.13.  This would not be correct.  Whatever course Stroud selected as his reference course would have a Slope Rating of 1.13.  For example, assume Stroud picked Panther Mountain as his reference course.  The slope of the line relating Average Differential and Panther Mountain Handicap would still be 1.13.  The slope relating the Average Score at Perfect Valley with Panther Mountain Handicap would be 0.92  (i.e., 1.13x1.13/1.39) and the slope at Open Flats would be 0.74.[3]

What should a player’s Standard Course Handicap (i.e., Perfect Valley Handicap or what the USGA now terms a player’s "Index") be at other courses?[4] For generality, let us name the course in question “Any Course.”  The handicap formula tells us:

1)    Standard Handicap(Any Course) = ATBD(Any Course) x .96
ATBD(Any Course) = Average of Ten Best Differentials at Any Course                                  
The player’s Course Handicap at Any Course can be written as,

2)    Course Handicap(Any Course) = ATBD(Any Course)/1.13

The ATBD(Any Course) is related to his Standard Handicap by eq. 3,

3)    ATBD(Any Course) x .96 = Slope Rating(Any Course) x Standard Handicap
Slope Rating(Any Course) = Slope of the line relating average differentials at Any  Course with Standard Course handicaps
Substituting eq. 3 into eq. 2, the Handicap(Any Course) becomes

4)    Handicap(Any Course) = Slope Rating(Any Course) x Standard Handicap/1.13
In summary, a player’s handicap at Any Course is given by the product of his Standard Course Handicap and Slope Rating(Any Course) divided by 1.13.

The Slope Rating, however, has little intuitive meaning.  A more informative measure of a course is what is termed the “new” Slope Rating. The new Slope Rating is simply the percentage of your Standard Handicap that is allowed at a course.  In essence,

5)    Handicap(Any Course) = New Slope Rating(Any Course) x Standard Handicap
Substituting from eq. 4 into eq. 5, the New Slope Rating can be defined as:

6)    New Slope Rating(Any Course) = Slope Rating(Any Course)/1.13
If the New Slope Rating of a course is 120, for example, a player gets to play at 120 percent of his Standard Handicap.[5] 

[1] Stroud, R.C., Riccio, L.J., “Mathematical underpinnings of the slope handicap system,” Science and Golf, E & F Spon, London, 1990, p. 136.
[2] Stroud actually plots average score versus handicap in his paper.  But he also claims (op.cit., p. 140):
So the better half of average differentials (ATBD) divided by average score (for any handicap level) is 1.04/1.13 = 0.92. 
This is not correct.  A ten-handicap, for example, would have an ATBD of 10.4 (ATBD= Handicap/.96).  His average score would be the Course Rating plus 11.3 (Handicap x 1.13).   If the Course Rating was 72.0, the ratio of ATBD to average score would be 10.4/83.3 = 0.12.  What Stroud probably meant was that the ratio of ATBD to Average Differential (Average Score – Course Rating) is 0.92.  Therefore, his figure has been changed to reflect this correction. The figure also assumes the average differential of a scratch player is zero.  This assumption is not correct, but is built into the Slope Handicap System.
[3] The myth that 113 is the Slope Rating of an average course was probably started by Dean Knuth, former Director of Handicapping for the USGA, when he wrote:
“The slope of the scores line of an average course has been observed to be 1.13 and USGA Slope Rating is referenced as 113 to deal with whole numbers.” (See Knuth D., “A two parameter golf course rating system,” Science and Golf, E & F Spon, London, 1990, p. 143.)
 Knuth’s explanation is either wrong or possibly just not stated with precision.  The Slope Rating is the slope of the regression line of total score versus USGA handicap for players from the reference course (named Perfect Valley in USGA research papers).  The slope of the regression line of score versus handicap for players at their home course will be 1.13 for all courses according to the USGA. 
[4] The handicap of a player at the Standard Course is measured in tenths and is identical to what is now termed a player’s “index.” See “Simplifying the Slope Handicap System,” www.ongolfhandicaps, 2/1/2013.
[5] The new Slope Rating has the same validity problems as the old Slope Rating.  Both the USGA’s old Slope Rating and the New Slope Rating are based on the same two assumptions: 1) There is linear relationship between average differentials at Any Course and the handicaps at the Standard Course, and 2) Average differentials are 1.13 times a player’s handicaps.  The validity of these assumptions has not been tested in the peer reviewed literature.