## 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