Friday, January 30, 2015

What is the Value of a Handicap Stroke?

Attempts to put a value on a handicap stroke fall within the set of problems that are not worth solving.  There is no general solution since the value will depend upon the playing characteristics of both yourself and your opponent.  In this post, only the specific case presented in Section 17 of the USGA Handicap System is examined.   The value of a handicap stroke is evaluated using 1) a continuous probability function associated with average scores and 2) a discrete probability function describing hole scores.

Continuous Probability Function - The value of a handicap stroke depends on the average difference in score between players and their respective standard deviations of scoring.  In the general case, assume there are two competitors, Player A and Player B.  Player A has the lower hole scores.  They have an average score of ASa and ASb, respectively.  Their standard deviations of scoring are σa and σb.  The distribution of the difference in scores is normally distributed with the following mean and standard deviation:

                D = ASb  - ASa = Mean of the difference in average hole scores
And,
                σd = (σa2  + σ b2)1/2  = standard deviation of the distribution of the difference in average scores

As an example, assume the mean difference in average holes scores 2.38 strokes (i.e., hole 18 in the USGA example).  Further assume each player has a standard deviation of 0.7 strokes.  The standard deviation of the difference in hole scores, σd, is then approximately 1.0.  For Player B to win a hole, the difference in average hole scores must be zero or less than zero.  The probability of that happening is the probability of drawing an average difference more than 2.38 (2.38/1.0) standard deviations from the mean.  That probability is .01.  If Player B is given a stroke on the hole, he must now draw an average difference more than 1.38 standard deviations from the mean.  The probability of that occurrence is .08.  Table 1 below shows how getting a handicap stroke affects a player B’s probability of having a lower average score on each hole.

If a player had the opportunity to buy a handicap stroke, he would look for the hole where the stroke increases his winning by the largest margin.   If the prize for winning a hole was $1, then the stroke on hole 6 would be valued at $0.38.  A handicap stroke on hole 18, on the other hand, would only be worth $0.07. In summary, the USGA’s recommended procedure assigns the higher-handicap player strokes where they are of minimal value (i.e., holes 4 and 18).

Table 1
Probability of Winning a Hole


Hole

Avg. Difference
Probability of Winning the Hole

Gain
Without Stroke
With Stroke
1
1.13
.13
.45
.32
2
1.41
.08
.34
.26
3
0.75
.23
.60
.37
4
2.10
.02
.14
.12
5
0.74
.23
.60
.37
6
0.73
.23
.61
.38
7
0.93
.18
.53
.35
8
1.22
.11
.41
.30
9
0.88
.19
.58
.39
10
1.05
.15
.48
.33
11
1.45
.07
.33
.26
12
2.04
.02
.15
.13
13
1.39
.08
.35
.27
14
0.80
.21
.58
.37
15
1.22
.11
.41
.30
16
0.75
.23
.60
.37
17
1.84
.03
.20
.17
18
2.38
.01
.08
.07

Discrete Probability Function - The first case examined is when there is a large difference in average scores between a high handicapper and a low handicapper on a hole.  One set of probabilities that make for a 2.0 stroke difference in average scores is shown in Table 2 below.
Table 2
Discrete Probabilities When Average Difference in Scoring is Equal to 2.0

Low Handicapper Probabilities and Avg. Hole Score
High Handicapper Probabilities and Avg. Hole Score
4
5
6
7
Score
4
5
6
7
Score
0.55
0.45
0.00
0.00
4.45
0.05
0.10
0.20
.65
6.45

Now let’s calculate the probabilities of a win, halve, and loss for the high handicapper with and without a stroke. 
Without a stroke the probabilities are:
                P(win) = .45 ·.05 = .0225
                P(halve) = .55 · .05  + .45·.10 = .0725
                P(lose) = .55· 95 + .45·.85 = .905

With a stroke the probabilities for the three outcomes are:
                P(win) = .05 + .10·.45 = .095
                P(halve) = .10·.55 + .20·.45 = .145
                P(loss) =  .20·.55 + .65 = .76

Now let’s change the probabilities on the hole for each player so the average difference in scores is only 1.0 stroke.  One set of probabilities that meet this criterion is shown in Table 3.
Table 3
Discrete Probabilities When Average Difference in Scoring is Equal to 1.0

Low Handicapper Probabilities and Avg. Hole Score
High Handicapper Probabilities and Avg. Hole Score
4
5
6
7
Score
4
5
6
7
Score
.1
.7
.1
.1
5.2
.01
.20
.36
.43
6.2

Without a stroke the probabilities for the three outcomes are: 
                P(win) = .01·.9 + .2·.2 +.36·.1 = .085
                P(halve) = .1·.01 + .7·.2 +.1·.36 + .1·.43 = .220
                P(lose) = .1 ·.99 +.7·.79 +.1·.43 = .695

With a stroke the probabilities of three outcomes are:
                P(win) = .01 + .2·.9 + .36·.2 + .43·.1 = .305
                P(halve) = .2·.1 +.36·.7 + .43·.1 = .315
                P(lose) = .36·.1 +.43·.8 = .380

Table 4 shows the change in probabilities when a handicap stroke is applied.  As expected, the stroke applied when there is a 2.0 stroke difference in average score is less helpful to the high handicapper than where there is a 1.0 stroke difference.  For example, a stroke increases his winning percentage by .15 when applied to the 1.0 hole rather than the 2.0 hole.  Interestingly, the smallest impact is on the number of halved holes though this may be an artifact of the selected probabilities.

As in the continuous probability analysis, assume that a player receives $1 for winning a hole.  If the average difference in holes scores is 2.0, a handicap stroke would only increase the player’s expected winning by $0.07.  On a hole with a 1.0 difference in average score, a handicap stroke would increase the player’s expected winning by $0.22.   Given a choice, the high handicapper would take his stroke on the 1.0 hole.  Given a choice, the USGA procedure would assign the player a stroke on the 2.0 hole.

Table 4
Change in Probabilities

Average Score Difference
Change in P(win)
Change in P(halve)
Change  in P(lose)
2.0
.0725
.0725
-.145
1.0
.2200
.0950
-.315



Saturday, January 17, 2015

Problems with the USGA’s Stroke Allocation Procedure

Introduction and Summary – For 30 years the United States Golf Association (USGA) has recommended the following handicap stroke allocation procedure for match play:[1]

A handicap stroke should be an equalizer rather than a winning stroke and should be available on a hole where it most likely will be need (sic) by the higher-handicapped player to obtain a half (sic) in singles… [2]

The USGA has never demonstrated its recommended procedure does in fact assign strokes where they are most needed, nor has it demonstrated the procedure is superior to other procedures in ensuring the equity of competition.  This post attempts to evaluate the efficacy of the USGA procedure by asking and answering a series of questions about handicap stroke allocation.

1.     Does the USGA procedure actually identify where a high-handicap player will most likely need a stroke to obtain a halve?  It does not.  As shown in the USGA’s example in Section 17, the procedure assigns the first handicap stroke to a hole where it will be of the least use to the higher-handicap player. 

2.     Is there a better procedure for allocating handicap strokes for match play?  Probably not.  If the USGA wanted handicap strokes to serve as equalizers, allocations should be made on the basis of how close the average difference in hole scores between higher and lower handicap players is to 1.0.  This average difference, however, will change with each handicap pairing.  Even if this equalizer objective is adopted—and there is no evidence it should be—there would not be one best stroke allocation suited for all matches.  Golf Australia recognized this when it recommended the same match play stroke allocation for all courses.  

3.     Is the USGA procedure defined with sufficient specificity to be implemented?  No.  The USGA advocates that the first handicap stroke should be allocated so that this stroke is most useful in a match between players of almost equal ability.   The USGA, however, does not identify any process for determining where a stroke would be most useful in such a match.

4.     Can a case be made for eliminating the stroke allocation procedure for match play?  Yes.  The USGA’s recommended method yields some strange results that are not easily understood by the average player—e.g., why don’t I get a stroke on a difficult hole?  Moreover, given the random nature of scoring it does not appear to make a significant difference to the equity of competition where strokes are given.  Given the many failings of its recommendation, the USGA should eliminate its match play stroke allocation procedure and replace it with general guidelines for ranking holes. 

Each question is discussed in turn:

1.  Does the USGA procedure actually identify where a high-handicap player will most likely need a stroke to obtain a halve?

The USGA’s recommended procedure for allocating handicap strokes does not ensure a player will get a stroke where it is most needed to ensure a halve.  This can be seen in the example presented in Sec. 17 of the Handicap System which is reproduced in part in the Table 1 below.[3]

Table 1
Stroke Allocation Based on Average Score Difference

(1)

Hole
(2)
Average  Score Difference
(3)

Rank
(4)

Strokes
(5)
Net Average Score Difference
(6)
Match
Standing
1
1.13
8
1
0.13
-1
2
1.41
4
1
0.41
-2
3
0.75
14
1
-0.25
-1
4
2.10
2
1
1.10
-2
5
0.74
16
1
-0.26
-1
6
0.73
18
0
0.73
-2
7
0.93
10
1
-0.07
-1
8
1.22
6
1
0.22
-2
9
0.88
12
1
-0.12
-1
10
1.05
13
1
0.05
-2
11
1.45
7
1
0.45
-3
12
2.04
3
1
1.04
-4
13
1.39
9
1
0.39
-5
14
0.8
15
1
-0.20
-4
15
1.22
11
1
0.22
-5
16
0.75
17
0
0.75
-6
17
1.84
5
1
0.84
-7
18
2.38
1
1
1.38
-8

The USGA recommends the stroke allocation for match play be determined by the average difference in score between low- and high-handicap players.  The average difference is used as a proxy for where the high handicap player needs a stroke.   To demonstrate the computation procedure, the USGA used two groups of players.  The first group had an average handicap of 6.  The second group had an average handicap of 22.  The difference in average score between the two groups for each hole is shown in Column 2 of the Table.

Within each nine, the hole with the largest average difference gets the lowest stroke allocation.  For example, hole 4 has the largest average difference among the holes on the first nine.  Therefore, it is assigned a stroke allocation of 2.  (Note: The second nine is considered more difficult so it is assigned the odd-number strokes.)  The next largest difference (i.e., 1.41) is assigned a stroke allocation of 4 and so on as shown in Column 3.

Does this stroke allocation promote equity?  Let’s assume a match between a 6- and a 22-handicap with the same average score differences shown in Column 2 of the Table.  Column 5 presents the net average difference in score after the 16 handicap strokes are applied according to the recommended stroke allocation.  The last column shows the standing in an average match.  The 22-handicap loses 5 and 3 and would lose 8 down if the match continued.[4] 

The problem with the USGA’s method is that it does not automatically assign strokes to where they would be of most use to the high-handicap player.  In the USGA’s example, the stroke allocation method ensures that a high-handicap player will get a stroke where it probably will be of little use.  The high-handicap player gets a stroke on hole 18, but the average difference in scores will still be 1.38 strokes even after the handicap stroke is applied.  With such a large difference in scores, the high-handicap player can be expected to have a higher average score than his competitor 92 percent of the time.   In summary, the USGA procedure assigned the first handicap stroke to where it will likely be of least use to the higher-handicap player.        

2. Is there a better procedure for allocating handicap strokes for match play?

The USGA procedure allegedly assigns handicap strokes where they most likely will be needed to obtain a halve.  The USGA measures “need” by the difference in scores between low- and high- handicap players.  As discussed above, the average difference in score is not a good proxy for “need.” 

If the USGA wants a stroke to be an equalizer, it should be applied to a hole where it evens the probability of winning between competitors.  Using the average score differences  in Sec. 17, Table 2 shows the probability of winning (i.e., having an equal or lower average score than the lower handicap player) for the higher-handicap player if a stroke was given on a hole.  Holes are then ranked by the closeness of the probability of winning to .50.  These changes would make the match more competitive.   The higher-handicap player no longer wastes a handicap stroke on holes 4 and 18.  The 22-handicap would lose 3 and 1 under the allocations shown in Table 2 rather than 5 and 3 under the USGA’s recommended procedure.
Table 2
Using the Probability of Winning a Hole as a Stroke Allocation Rule

Hole
Probability
Allocation
Hole
Probability
Allocation
1
.45
6
10
.48
1
2
.34
16
11
.33
11
3
.60
10
12
.15
15
4
.14
18
13
.35
9
5
.60
12
14
.58
3
6
.61
14
15
.41
5
7
.53
2
16
.60
7
8
.41
8
17
.20
13
9
.55
4
18
.08
17

To determine the allocations shown in Table 2 it is not necessary to calculate the probabilities of winning for each hole.  Instead, holes can be ranked by how close its average score difference is to 1.0.  For example, if two holes had average score differences of .85 and 1.1, the hole with the 1.1 score difference would receive the lower allocation.  This “proximity to 1.0” rule will yield the same allocations as the probability rule shown in Table 2.

Unfortunately, the “proximity to 1.0” rule gives different results depending upon the difference in handicaps.  If the difference in handicap between competitors is halved, for example, the average score differences are also halved under the USGA assumption.  Different holes would now have an average score difference closer to 1.0.  So even if the “proximity to 1.0” test was the most equitable, it is administratively infeasible and could not be implemented. 

Golf Australia (GA)—Australia’s counterpart to the USGA— recognized that there is no perfect allocation.   It recommended the same stroke allocation for all courses.  GA argued that it disregarded hole difficulties (acknowledging that a 30-marker receives 5 strokes from a 25-marker, which is what a 5- marker receives from a scratch-marker, but that there can be clear differences in the holes a 5-marker and a scratch-marker will find most challenging).[5]  GA’s recommended stroke allocation avoids allocating low-numbered strokes to the last two holes and allocating low-numbered strokes to the first three holes in case a match goes to extra holes.  In essence, GA is arguing that any match play allocation, including a “difficulty” allocation, would be acceptable as long as its recommendations on the opening and closing holes were followed. 
     
The USGA’s recommendation is also alleged to promote equity in four-ball matches.  The USGA, however, has never presented any theoretical or empirical evidence to prove that assertion.  The USGA argues that the optimal allocation in a four-ball match can be determined by the average difference in scores between low- and high-handicap players.  As noted above, however, the best allocation may vary by the difference in the handicaps of the competitors.  When you introduce two more players to the competition, the analytical problem becomes intractable.   Fortunately, there is so much randomness in scoring the actual allocation is probably not determinant of the final outcome of the four-ball match.    

 3. Is the USGA procedure defined with sufficient specificity to be implemented?

 The questionable logic behind the USGA’s recommendation becomes apparent in the allocation criterion presented in Sec. 17-1bii:[6]

The first handicap stroke should be allocated so that this stroke is most useful in a match between players of almost equal ability (e.g., a match involving players with a Course Handicap of 0 and 1, 10 and 11, and 29 and 30).  The second handicap stroke should be allocated so that this stroke is most useful in a match between players having a slightly greater difference in Course Handicap (e.g., a match involving players with a Course Handicap of 0 and 2, 10 and 12, or 29 and 31).  This process should be continued until the first six strokes have been assigned. (Editorial note: The USGA does not delineate how stroke allocations for the remaining 12 holes are to be determined.)

The USGA does not present any procedure to determine where a stroke will be most useful in a match between players of almost the same ability.  The USGA also assumes, without evidence, that a handicap stroke would be most useful over a broad range of ability—i.e., would a 1 handicap and a 30 handicap find a handicap stroke most useful on the same hole? 

When 17-1bii is applied to the rankings shown in the USGA example, the allocations remain the same on the front nine, and only modified on the back nine to eliminate Hole 18 as the number 1 stroke hole.  Stroke allocations were essentially based on the average difference between players of dissimilar ability and not of equal ability.  There is no mention in the example of assigning the first stroke hole to where it would be useful in a match between players of comparable ability.  This criterion is never used, never explained, and should be deleted from the Handicap System.

4.  Can a case be made for eliminating the stroke allocation procedure for match play?

In matches between players of similar ability (i.e., a 1 to 2 stroke difference in handicap), the difference in average scores is minimal—maybe around .1 strokes at the most.   Therefore, wherever the handicap stroke is assigned, it is expected to be a winning stroke.  If most matches are between players of comparable ability, the USGA’s allocation procedure will not increase the number of halved holes.  Strokes could just as well be allocated randomly and the results of the matches would not be changed significantly.[7]  

The USGA’s match play recommendation creates other problems.   Often the number one stroke hole is a moderately easy hole.  The handicap chairmen must then explain—often and repeatedly—the arcane procedure recommended by the USGA.  In many cases a tough hole will get a high stroke allocation.  This leads to the player with the higher handicap (i.e., he gets a stroke) carrying the load on the difficult holes in a four-ball stroke-play competition.  The USGA states the Committee may develop a separate allocation based on difficulty relative to par to handle stroke-play competitions.   This would be adding more complexity to the scorecard without any benefit. 

The match play allocation procedure recommended by the USGA should be eliminated.[8]  In its place, the USGA should suggest a ranking of holes by difficulty subject to certain guidelines such as spreading low stroke holes evenly over the 18 holes.[9]  This can be done by consensus of the Handicap Committee.  The USGA procedure of evaluating numerous scorecards implies a level of scientific certainty in the outcome that is not justified.  As long as the recommendations on the placement of low strokes are followed (Sec. 17-1bii), one set of allocations should serve both for match and stroke play.[10]




[1] An early description of the procedure was presented in “New Handicap Surveys May Change the Stroke Allocations at your Course,” Golf Digest, Trumbull, Connecticut, July 1985, pp. 32-33.  The allocation procedure was described in the 1994 USGA Handicap System.  The same example used in 1994 appears in the USGA Handicap System of 2012.
[2] The USGA Handicap System, 2012-2015, The United States Golf Association, Far Hills, NJ, 2012, p. 103.
[3] Ibid, p. 107.
[4] The main reason the 22-handicap loses is that the average scores presented in Table 1 are not consistent with the theory behind the Slope System.  The average score of the 6-handicap is 74.56.  The average score of the 22-handicap is 98.37.  If the average score of the 6-handicap is correct, the average score of the 22-handicap should be 93.63 (75.56 + 1.13·16).  As it is, the average score difference is approximately 23 strokes, yet the 22-handicap only receives 16 strokes.  It is also hard to believe a bogey player (a 22 handicap by the USGA’s definition) averages about a triple bogey on hole 18. If this section is kept, the USGA should employ a more realistic example.  
[5] Course Management-Golf Australia Recommendations, Golf Australia, Melbourne, Australia, June 1, 2014.
[6] Op. cit., p. 103.  This recommendation is in conflict with the USGA’s previous assertion that the first handicap stroke should be assigned to a hole where a higher-handicap player most needs a stroke as an equalizer.
[7] Dougharty, Laurence, “One Set of Stroke Allocations is Enough,” www.ongolfhandicaps.com, July 24, 2013.  Dougharty reported that changing the stroke allocations had little effect on the outcome of the match.
[8] If the USGA decides to keep both allocation methods, it should illustrate each method with the same data.  This would give the reader some idea of the differences between methods.  Sec. 17-5 was added to the Handicap System, but used different data than that used to illustrate the allocation method set forth in Sec. 17-2.
[9] A sensible set of guidelines is presented in Appendix G of the Council of National Golf Union’s Unified Handicap System, 2012.
[10] To judge the efficacy of its stroke allocation procedure, the USGA should undertake simulation studies similar to those conducted by Dr. Francis Scheid, a past member of Handicap Research Team.  Scheid took scores from a golf club and used a computer to simulate matches between players.  This technique could be employed to determine if using the “average difference in scores” or the “difficulty to par” for stroke allocation has a significant impact on the outcome of a match.