The Four-Ball Stroke Play Adjustment Under Section 3-5

The USGA recommends different procedures for adjusting handicaps under Section 3-5 depending upon the format of the competition.  In foursomes and Chapman competitions, the team playing the course with the higher Course Rating adds d-strokes (d being the difference in Course Ratings) to its team handicap.  In four-ball competitions, however, each player has d-strokes added to his handicap.

In foursome and Chapman formats, the effect of the adjustment is certain and known in advance.  The teams playing from the tees with the higher Course Rating will have their net scores reduced by  d‑strokes.  The effect in four-ball stroke play, however, is not certain.  If the two players have the same handicap, the effect will be to reduce the team score by d-strokes, the same as in foursome and Chapman competitions.  If the difference in handicaps between partners is equal to or greater than d, the expected reduction in team score will be d-strokes but can range from zero to 2d-strokes.

The probability that a team will gain more than d-strokes increases with the value of d.  As shown in the Appendix below, if d is equal to 3, a team has a 34 percent chance of reducing its team score by more than 3-strokes.  In a large field competition, the overall winner will most likely come from the teams playing the tees with the higher course rating.[1]   To eliminate this inequity and to make the handicap adjustment under Sec. 3-5 consistent over all forms of competition, it is suggested the adjustment for four-ball should also be a reduction in team score by d-strokes.

Appendix

Expected Reduction in Net Score

To simplify the model, it is assumed that only five scores on a hole are possible—eagle. birdie, par, bogey, and double bogey.  The probability of making each score for the two players is shown in Table A1 below.

Table A1

Probability of Making Various Scores

Score	Player 1	Player 2
Eagle	a1	b1
Birdie	a2	b2
Par	a3	b3
Bogey	a4	b4
Double Bogey	a5	b5

Now on any hole there are 25 possible outcomes as shown in Table A2.  Assuming the scores by each player are independent, the probability of each outcome is shown column 3.  Assume that d is equal to 1 so that each player gets an additional stroke, and the handicap of Player 1 is at least one stroke lower than the handicap of Player 2.  Column 4 shows the results for when Player 1 gets an additional stroke.  As an example, if both players have eagled the hole, the additional stroke does not result in a reduction in the total score (i.e., player 2 by definition already has a stroke on that hole).  That is why zero is shown in Column 4 for the eagle-eagle outcome.

Similarly, Column 5 shows the reductions when Player 2 gets an additional stroke, but Player 1 does not.  The eagle-eagle outcome results in a reduction of -1 since Player 2 now strokes on the hole.

Table 2A

Reduction in Net Score for Possible Outcomes

(1) Player 1	(2) Player 2	(3) Probability	(4) Player 1 Gets +1 Stroke	(5) Player 2  Gets +1 Stroke
Eagle	Eagle	a1·b1	0	-1
Eagle	Birdie	a1·b2	-1	0
Eagle	Par	a1·b3	-1	0
Eagle	Bogey	a1·b4	-1	0
Eagle	Double Bogey	a1·b5	-1	0
Birdie	Eagle	a2·b1	0	-1
Birdie	Birdie	a2·b2	0	-1
Birdie	Par	a2·b3	-1	0
Birdie	Bogey	a2·b4	-1	0
Birdie	Double Bogey	a2·b5	-1	0
Par	Eagle	a3·b1	0	-1
Par	Birdie	a3·b2	0	-1
Par	Par	a3·b3	0	-1
Par	Bogey	a3·b4	-1	0
Par	Double Bogey	a3·b5	-1	0
Bogey	Eagle	a4·b1	0	-1
Bogey	Birdie	a4·b2	0	-1
Bogey	Par	a4·b3	0	-1
Bogey	Bogey	a4·b4	0	-1
Bogey	Double Bogey	a4·b5	-1	0
Double Bogey	Eagle	a5·b1	0	-1
Double Bogey	Birdie	a5·b2	0	-1
Double Bogey	Par	a5·b3	0	-1
Double Bogey	Bogey	a5·b4	0	-1
Double Bogey	Double Bogey	a5·b5	0	-1

The probability that Player 1 successfully uses an additional stroke on a hole to lower the team score is:

1)            p = a1·b2+a1·b3+a1·b4+a1·b5+a2·b3+a2·b4+a2·b5+a3·b4+a3·b5+a4·b5

The probability that Player 2 successfully uses an additional stroke on a hole to lower the team score is:

2)            q = a1·b1+a2·b1+a2·b2+a3·b1+a3·b2+a3·b3+a4·b1+a4·b2+a4·b3+a4·b4 +a5

Assume the difference in Course Ratings is “d” strokes.  The probability that Player 1 lowers the team score on “n” holes is:

3)            P1(n) = (d!/(n!(d-n)!) )· pⁿ·(1-p)^d-n

Similarly, the probability that Player 2 lowers the team score on “n” holes is:

4)            P2(n)  = (d!/(n!(d-n)!))·qⁿ·(1-q)^d-n

To evaluate these probabilities, it is necessary to know the likelihood of making each score for both players.  In previous posts, reasonable estimates of these likelihoods have been used and are presented in Table 3A below.

Table 3A

Probabilities of Scoring

Score	5-Handicap	10-Handicap
Eagle	.005	.003
Birdie	.140	.090
Par	.450	.350
Bogey	.310	.380
Double Bogey	.095	.177

Based on the assumptions in Table 3A, the estimates of p and q are shown in Table 4A.

Table 4A

Probability of a Reduction in Team Score for One Additional Handicap Stroke

Player	Probability	Estimated Probability
Player 1	P	.44
Player 2	q	.56

Two cases are examined to demonstrate how the competition could be affected by using Sec. 3-5.  In the first case “d” is one stroke.  The probability of each player lowering the team score is shown in Table 5A below:

Table 5A

Probability of Team Score Outcomes When d =1

Outcome	Formula	Estimated Probability
0	(1-p)·(1-q)	.25
-1	p·(1-q) + q·(1-p)	.50
-2	p·q	.25

The expected reduction in team score is -1 stroke.  There is a 25 percent chance, however, a team will lower its score by -2 strokes.

Table 6A shows the probability of various outcomes when the difference in course ratings is 3 strokes and the difference in handicaps is 3 or greater.

Table 6A

Probability of Team Score Outcomes When d=3

Outcome	Formula	Estimated Probability
0	P1(0)·P2(0)	.015
-1	P1(1)·P2(0) + P1(0)·P2(1)	.092
-2	P1(2)·(P2(0) +P1(1)·P2(1) + P1(0)·P2(2)	.235
-3	P1(3)·P2(0) + P1(2)·P2(1) + P1(1)·P2(2) +P1(0)·P2(3)	.315
-4	P1(3)·P2(1) +P1(2)·P2(2) + P1(1)·P2(3)	.235
-5	P1(3)·P2(2) + P1(2)·P2(3)	.092
-6	P1(3)·P2(3)	.015

Table 6A indicates the probability of reducing a team score by the full 2d strokes declines as d increases.  The probability of reducing a score by more than d-strokes, however, increases with d.  In this case, a team has a 34 percent chance of reducing its net score by more than 3-strokes.

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[1] The winner will most likely come from the teams playing the tees with a lower course rating if those teams have their handicaps adjusted downward by d-strokes.