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
|
No comments:
Post a Comment