Tuesday, March 4, 2014

The Problem with Section 3-5


A recent article in FORE Magazine (“A Perfect Match,” November/December 2013) argues the Handicap System makes for “perfect “competition even when players compete from different tees. You can trace the push for Sec. 3-5 (Sec. 8-4f back in 1990) to Alice Dye and her article “Tees for Two or More” in the July 1990 PGA Magazine.  Ms. Dye, like the article, used an example of individual stroke play to demonstrate the equity of Sec. 3-5.  If a competition is social (e.g., husband v. wife, buddy v. buddy) stroke play, Sec. 3-5, while not perfect, is good enough.  The problem comes when the competition is more serious and when other forms of competition are introduced.  Articles like this can unintentionally mislead the reader into believing Sec. 3-5 cures all of the equity problems of such competition.   I believe the use of Sec. 3-5 should be discouraged and only used as a last resort.  Here are some reasons why:

1. Biased Handicaps - Rounding errors and the Bonus for Excellence can give one player an advantage when competition is from different tees.  The Appendix presents an example of two players with 13.5 indexes.  When they play from different tees, one player has an expected net score 1.7 strokes below that of his competitor.  Based on certain assumptions, that player should win 65 percent of the time.  Had they played from the same tees, neither player would have an advantage.

2. Equity of Competition – Sec. 3-5 depends upon the accuracy of the Course and Slope Ratings. The standard error of the estimate of Course and Slope Ratings is much larger than the USGA wants to admit.  Given these error, the efficacy of Sec. 3-5 becomes in doubt.  Significant errors can occur in assigning the proper handicap when a player establishes his index playing one set of tees, but plays another set of tees in a competition.

 3. Misuse of Sec. 3-5 – I have seen Sec. 3-5 applied to various forms of competition –scrambles, Chapman, foursome, and four-ball.  I have not seen any evidence indicating Sec. 3-5 makes for equitable competition in these formats.  I have conducted small sample studies that show Sec. 3-5 did not always equalize competition in scramble and four-ball events.  To be fair, the USGA does not recommend using Sec. 3-5 for scramble competitions, but it also does not explicitly prohibit its use.  When Sec. 3-5 is used in a four-man scramble, the handicap allowances for the C and D players are often the same from both the front and back tees. The team that recognizes this anomaly, and places it C and D players on the front tees has a significant advantage.

4. Handicap Stroke Allocation Issue – All the theory behind stroke allocation goes out the window when players compete from different tees.  This is not a great concern, because I do not believe stroke allocations are a major determinant in who wins a match.  Advocating competition from different tees, however, sends the message that Section 17 (Allocation of Handicap Strokes) of the USGA Handicap System isn’t all that important.

5.  Pace of Play – One of the arguments for “Play It Forward” is that rounds would be played faster.  When players compete from different tees in the same foursome, it takes longer to play a round and any time benefit from Play it Forward could be lost.

6. Smaller Scoring Variance from the Front Tees - In some four-ball competitions, the second ball is used as a tie-breaker.  It is proable the scoring variance is smaller from the front tees. This should give the team playing the front tees a small edge when the second ball is used as a tie-breaker. 

Appendix
 Section 3-5 (Players competing from different tees) argues competition would be equitable if the difference in Course Ratings is added to the handicap of the player competing from the tees with the higher course rating.  Applying Section 3-5, however, introduces three possible errors
1. Rounding the Difference in Course Rating Error To demonstrate the equity of Sec. 3-5, examples are manufactured to give both players the same net score. The example shown in Table 1 is taken from the USGA explanation of Sec. 3-5.  Both players have an expected net score of 71, and equity is allegedly assured.
Table 1
USGA’s 3-5 Procedure

Gary (Gold Tees)
Vs.
Bob (Blue Tees)
10.4
Index
10.4
130/113
X Slope Rating/113
140/113
12
Course Handicap
13
71.1
Course Rating
73.2
83
Target Score
86
12
Adjusted Handicap
13 + 2 =15
71
Net Score
71

The findings, however, depend upon the choice of the Course Ratings.  Table 2 demonstrates that changing the Course Ratings can lead to different results.
Table 2
3-5 with Different Course Ratings

Gary (Gold Tees)
Vs.
Bob (Blue Tees)
10.4
Index
10.4
130/113
X Slope Rating/113
140/113
12
Course Handicap
13
71.4
Course Rating
73.8
83
Target Score
87
12
Adjusted Handicap
13 + 2 =15
71
Net Score
72


In Table 2, the Net Scores of the two players are not equal after making the Sec. 3-5 adjustment.  The difference is due to rounding the difference in Course Ratings.  Therefore, competition is not as perfect as the article implies.

2. Bonus for Excellence Error -For equity, however, it is not the target score that is important, but a player’s expected net score.  Ideally, the expected net score of both competitors should be the same.  For simplicity, we assume a player’s expected gross score is the average of his ten best out of twenty scores.[1]  A player’s expected gross score is given by the handicap formula:
                Expected Gross Score = (Index · (Slope Rating/113))/.96 + Course Rating
The .96 in the equation is the Bonus for Excellence.
A player’s handicap is:
                Handicap = Index · (Slope Rating)/113
A player’s Expected Net Score is:
                Expected Net Score = Expected Gross Score – Handicap
                                 = (Index· (Slope Rating/113)/.96 + Course Rating - Index · (Slope Rating/113)
                                 = 0.04· (Index · Slope Rating)/113 + Course Rating
When the Slope Ratings of the two tees are fairly close, the Bonus for Excellence introduces only a minor error.  This is not always the case, however.  As an example, assume two payers have an index of 13.5.  One player plays the gold tees with a Slope Rating of 88.  The other player plays from the blue tees with a Slope Rating of 146. The player from the gold tees would have an expected net score .4 strokes over the Course Rating.  The player competing from the blue tees would have an expected net score .7 strokes over the Course Rating.   The player competing from the gold tees will have a slight edge (.3 strokes) that will not be corrected by Section 3.5. (Note: This bias is due to the USGA’s Bonus for Excellence which is yet another argument for its elimination.[2]  The bias favors the player with the lower handicap, and not necessarily the player with the lower index.  This is contrary to the intent of the Bonus for Excellence that supposedly rewards the better player.)

3.  Handicap Rounding Problem – When players with the same index play from the same tees any rounding error is the same for both players.  When they play from different tees, rounding can give one player up to a 1-stroke advantage.  Let’s use the example above  The unrounded handicap of the player playing the gold tees would be 10.51.  The unrounded handicap for the player competing from the white tees would be 17.44.  The difference in unrounded handicaps is 6.93 strokes or approximately 7 strokes.  The difference in their rounded handicap, however, is 6 strokes.  In this case the player competing from the gold tees gains a 1-stroke advantage because of rounding. 


4. Total Error - Table 3 shows how all of these errors can add-up to give the player from the gold tees an advantage  He gains approximately 1-stroke when his handicap is rounded up while his competitor’s handicap is rounded down.  If we assume the difference in Course Ratings is 2.4 strokes, he gains .4 strokes when the difference in Course Ratings is rounded down.  He gains another .3 strokes because of the Bonus for Excellence.  His total advantage is 1.7 strokes.  This is about as large as an advantage that can occur.  All of the rounding errors were in favor of the player competing from the gold tees.  In most cases the bias will be smaller.  The point is, however, is that these biases disappear when these competitors play the same tees.[3]

Table 4 also shows how the errors can affect the probability that a player wins a match. It is assumed that each player’s scoring record has standard deviation of 3.0. In the extreme case, the probability of winning can reach .65.   In this instance, playing from different tees has made the competition less equitable. 
 
Table 4
Probability of Gold Tee Player Winning[4]

Bias
Net Score Advantage
Probability of Winning
Bonus for Excellence (BE)
.3
.53
BE + Course Rating  Rounding(CRR)
.3 + .4 = .7
.57
BE + CRR + Handicap Rounding
.3 +  .4 + 1.0 = 1.7
.65









[1] A player’s actual expected score, under certain assumptions, is the average of his ten best scores plus .8 times a player’s scoring variance.  If both player’s have the same variance, the average of a player’s ten best scores is an adequate proxy for his actual expected score for the purposes here.
[2] Dougharty Laurence, “The USGA’s Bonus for Excellence Ruse,” www.ongolfhandicaps.com, January 15, 2013
[3] There will still be rounding problems when players play from the same tees.  The example was a special case where the indexes of both players were the same. 
[4] The methodology behind the estimates can be found in “A Sandbagger’s Guide to Winning,” www,ongolfhandicaps.com, December 19, 2013

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