Monday, September 15, 2014

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.

[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.