## Thursday, July 18, 2013

### The Equity of Stableford Scoring

Introduction -Stableford scoring assigns points based on a player’s score relative to some fixed score.  Choosing the fixed score can affect the equity of competition and the pace of play.  Table 1 below demonstrates the Stableford point system for three different fixed scores.  The fixed score is assigned 2 points, and more than one over the fixed score receives zero points.
Table 1
Stableford Points
 Player’s Net Score Fixed Score Birdie Par Bogey Double Eagle 4 5 6 Eagle 3 4 5 Birdie 2 3 4 Par 1 2 3 Bogey 0 1 2 Double Bogey 0 0 1 Triple Bogey 0 0 0

The pace of play is fastest with the lowest fixed score (i.e., birdie). After a player has hit a number of strokes equal to a net par without holing, he picks up and moves on to the next hole—in theory, but there are some players who will not pick up unless they are out of balls.  But how does the fixed score affect equity?  A 25-handicap player may want a fixed score of double bogey so he earns points on his less than stellar holes.  Is he making the right choice?
Equity -To examine the question of equity, the probabilities of a 5-handicap and a 25-handicap player making various hole scores used in previous posts are adopted for this study.[1]   It is also assumed that the probability function is equal across all holes.  For example if the 5-handicap had a .45 chance of scoring par, then 5/18th of the time it will be on a stroke hole, and 13/18th of the time it will be on a non-stroke hole. Given these assumptions, the probabilities and Stableford Scores for a 5- and 25-handicap player are shown in Tables 2 and 3.
Table 2
Stableford Scores for 5-Handicap
 Net Score to Par Probability Fix Score = Birdie Fixed Score =  Bogey Points Avg. Hole Pts. Points Avg. Hole Pts. -3 .001389 4 .005556 6 .008334 -2 .042500 3 .127500 5 .212500 -1 .226111 2 .452222 4 .904444 0 .411111 1 .411111 3 1.233333 1 .243333 0 .000000 2 .486666 2 .056112 0 .000000 1 .056112 Total .996389 Total 2.901389

Table 3
Stableford Scores for 25-Handicap
 Net Score to Par Probability Fix Score = Birdie Fixed Score =  Bogey Points Avg. Hole Pts. Points Avg. Hole Pts. -3 .003889 4 .015556 6 .023334 -2 .064444 3 .193332 5 .322220 -1 .239445 2 .478890 4 .957778 0 .348889 1 .348889 3 1.046667 1 .222222 0 .000000 2 .444445 2 .084441 0 .000000 1 .084441 Total 1.036667 Total 2.878885

Comparing Tables 3 and 4, the 25-handicap player has a small advantage when the lower fixed score (birdie) is used, but loses the advantage when the higher fixed score (bogey) is used. The tendency for the high-handicap player to benefit from low fixed scores is shown in Table 4.  The high-handicap player has an advantage for eagle and birdie fixed scores, has no advantage for a fixed score of par, and is at a disadvantage for bogey and triple bogey fixed scores.
Table 4
Average Stableford Points per Hole
 Handicap Fixed Score Eagle Birdie Par Bogey Triple bogey 5-Handicap .31 1.00 1.92 2.90 4.90 25-Handicap .38 1.03 1.92 2.88 4.88

To see why this advantage for the high-handicap occurs at low fixed scores, Table 5 show the probability of scoring points on each hole when the fixed score is eagle.  (Note: This is not a realistic fixed score that would actually be used in competition.  It is presented here to demonstrate how high fixed scores benefit the high-handicap player.)
Table 5
Probability of Scoring Points with an Eagle Fixed Score
 Points 5-Handicap 25-Handicap 3 .001389 .003889 2 .042500 .064444 1 .226111 .239445

Neither player has much of a chance to score 3 points.  The 25-handicap player can score 2 points (i.e., a net eagle) by making a gross par with two strokes or a gross birdie with one stroke.  To score 2 points, the 5-handicap player faces a tougher test.  He must either score a gross eagle with no stroke, or a gross birdie with one stroke. In essence, the high-handicap player has an advantage since he has a better chance of making a gross par than the low handicap player has of making a gross birdie.
Table 6 shows why the advantage disappears when bogey becomes the fixed score. The 5-handicap player has a much larger probability of scoring 3 points (i.e., scoring a net par) than a 25-handicap.  He does this by either scoring a gross par without a stroke or a gross bogey with a stroke.  A 5-handicap scores a gross par or bogey 76 percent of the time.  The 25-handicap player scores 3 points by either making a gross bogey with a stroke or a gross double bogey with two strokes. A 25-handicap only makes gross bogey or double bogey 68 percent of the time.  This edge in 3-point scoring is what levels the competition for the 5-handicap player.
Table 6
Probability of Scoring Points with a Bogey Fixed Score
 Points 5-Handicap 25-handicap 6 .001389 .003889 5 .042500 .064444 4 .226111 .239445 3 .411111 .348889 2 .243333 .222222 1 .061116 .084444

Conclusions –The narrow focus of this study on just two players and their probability functions, preclude it from claiming any universal truths about Stableford competitions.  Two general guidelines, however, do flow from the study.
·        Using a fixed score of par appears to be the most equitable consistent with increasing the pace of play. No fixed score studied here gave either the high- or low-handicap player a significant advantage.
·        Modified Stableford completion where the points system is non-linear (e.g., double eagle = 8, eagle = 5, Birdie =3, and par =1) would favor the high handicap player. (Remember, from Table 5 the 25-handicap player has a higher probability of making net eagle than the 5-handicap player.  Moreover, the relative value of a net par (the 5-handicap player’s strength) is diminished, leaving him at a disadvantage.

[1] See Why You Lose (or Win) at Skins, www.ongolfhandicaps.com, June 25, 2013.