Wednesday, October 31, 2012

Eliminating the Blind Draw


Introduction - Many tournaments consist of a format where foursomes compete against other foursomes in the field.  When the field cannot be divided evenly into foursomes, threesomes are created.  The threesome is then allowed a “blind draw” for the fourth player (i.e., the score of another player in the field is drawn and his score becomes that of the missing fourth player.)

While the “blind draw” is equitable it has several problems.  First, a team’s performance is determined in part by luck rather than on how well the team played.  Second, if the blind draw played well, his performance can help the threesome and therefore hurt the chances of his own team. Third, it is more difficult for the player in a threesome to evaluate risk/reward decisions when the performance of the fourth player is unknown.

This paper evaluates two methods around this problem:
 

·         Method 1: The threesome is allowed to use one player’s score twice on a hole.  The chosen player is rotated each hole so that each player’s score can be used twice on six holes.  A typical rotation would have the lowest handicap player take the first hole, the second lowest handicap the second hole, and the third lowest handicap player the third hole.  This rotation would be repeated every three holes.

·         Method 2: The threesome is assigned a player who always has a net par on each hole.


The evaluation proceeds in four steps.  First, the basic probability model for the evaluation is described.  Second, probability values are estimated using data from two courses.  Expected hole scores for various methods are then computed to determine the preferred method for threesome competition.  Third, a sensitivity analysis is performed to see over what range one method is preferred over the other.  Fourth, conclusions are drawn as to the best method for achieving equitable competition.   
 

1. The Probability Model -  Assume a player has three different outcomes when playing a hole.  A net birdie is assigned the value of 0, a net par is assigned the value of 1, and a net bogey is assigned the value of 2.  For demonstration purposes, probabilities are assigned to each outcome as shown in Table 1:
 

Table 1

Probability of Scoring

 
Score
Probability
0
.25
1
.50
2
.25


The criterion for measuring equity is the expected hole score for each team.  The method that yields an expected score for the threesome closest to that of the foursome would be preferred.  

The foursome has 81 different scoring combinations as shown in Table A-1 of the Appendix.  (Note: The Appendix has not been included here for space considerations.)  Each combination has a team score and a probability of occurrence.  The expected score is the product of the team score and the probability of occurrence summed over all outcomes.  The expected two-best ball score of the foursome is 1.14.

For Method 1 where the threesome can use one ball twice, there are 27 different scoring combinations.  Those combinations and their associated probabilities of occurrence are shown in Table A-2 of the Appendix.  The expected two-best ball score on each hole for the threesome would be 1.25.  In an eighteen-hole competition, the foursome would have a two-stroke ((1.25-1.14)·18=1.98) advantage over the threesome.

Under Method 2, the probabilities of each outcome for the three players is the same as in Method 1.  The value of the outcomes may differ, however, as shown in Table A-3.  The expected hole score under Method 2 is 1.28.  The foursome has a 2.5 stroke advantage over a threesome competing with Method 2.

 

2. An Empirical Test - The selection of the best method will depend upon the  player’s probability function at a course.  The probability function was estimated for two courses using the same 88 players.  The net scores for each player were sorted into five categories as shown in Table 2.  The estimated probabilities are the number of hole scores in each category divided by the total number of hole scores.  These probabilities are presented in Table 2. 

 
Table 2

Estimated Probability Functions
 

 
Probability
Score
Course 1(CR=71.2)
Course 2(CR=71.7)
2 or More Under Par
.024
.027
1 Under Par
.191
.178
Even Par
.333
.319
1 Over Par
.307
.308
2 or More Over Par
.145
.168

 

Table 2 shows there is a significant probability that a player will have a net score of 2 over par or more.  The three-score model (0,1,2) used here does not take into account such high scores.   To have a score of two over par used in a foursome event, however, three players must have that score.  The probability of that outcome is small, so the bias introduced by the three-score model should not be large.

            To evaluate the expected scores under each scoring alternative, the probabilities of 2 under and over are combined with the probabilities for 1 under and 1 over, respectively, as shown in Table 3.   (Note: Par is considered “1” in the three-score model.)

 

Table 3

Estimated Probabilities
 

 
Probability
Score
Course 1
Course 2
P(0)
.215
.205
P(1)
.333
.319
P(2)
.452
.476

 

These probabilities result in the expected hole scores shown in Table 4 for each method.
 

Table 4

Expected Hole Scores
 

Course
Foursome
Method 1
Method 2
Course 1
1.48
1.64
1.46
Course 2
1.55
1.72
1.50

     

The table demonstrates Method 2 is the preferred format at these courses.  The expected differences in hole scores is .02 for Course 1 and .05 for Course 2.  For an 18-hole competition, a threesome would have a small edge of less than one-stroke.  Under Method 1, the threesome has an expected 18-hole score approximately three strokes higher than that of a foursome. 


3. Sensitivity Analysis - The expected value of the score will depend on the probability distribution of individual hole scores by a player.  Table 5 below shows the expected team scores for alternative  probability distributions.


Table 5

 Alternative Probability Distributions


 
Probabilities
Expected Hole Score
Alternative
P(0)
P(1)
P(2)
Foursome
Method 1
Method 2
1
.1
.5
.4
1.85
1.94
1.77
2
.2
.5
.3
1.38
1.46
1.44
3
.3
.5
.2
0.95
1.06
1.14
4
.4
.5
.1
0.62
0.74
0.86

 

            The table demonstrates the preferred method depends on whether a course is relatively easy or difficult.[1]  When net bogeys are likely (i.e., P(2)=.4 or .3) Method 2 is the most equitable format for threesomes.   On an easier course (i.e., P(2)= .2 or .1), Method 1 yields an expected score closer to the foursome expected score and would be the preferred format. 

            Realistically, courses where Method 1 is preferred are rare.  The expected net score of a player with 4th probability distribution, for example,  would be 5.4 under par.   This would imply that the course rating is approximately 9 under par.[2]   A review of the golf courses in Southern California found no golf course with such a wide disparity between par and the course rating. [3] 


4. Conclusion - The research found that Method 1—one player’s ball counting twice—is not an equitable format.  This method was found to be marginally superior only on courses that do not seem to exist.  On most courses, a threesome playing under Method 1 would have an expected score some three strokes more than a foursome (e.g., on Course 1 the difference would be (1.64-1.48)·18=2.88).   Method 2 appears to ensure equitable competition on courses where the course rating is around par.[4]  Since most course fall in this category, Method 2 is the recommended format.

           

 

 

 

 



[1] The best measure of difficulty is the difference between the course rating and par.  If the course rating is much lower than par (e.g., 67 versus 72), the player would be expected to have fewer net bogeys than on a course with a course rating of 73.0. 
[2] A player’s index is determined by the average of his ten best scores out of the last twenty scores.  Depending on the variance in the player’s scoring distribution, the average used for his handicap will be around 3-5 strokes lower than his average for all scores (i.e., the course rating must be 3-5 strokes lower than his expected score).    
[3] Southern California Directory of Golf, Southern California Golf Association, North Hollywood, CA 2006
[4] On courses where the course rating is much higher than par, Method 2 may yield too big of an advantage to the threesome.   When adopting any method, records should be kept so that the equity of competition can be empirically tested.  That is, do threesomes or foursomes win more than their fair share of competitions?

Monday, October 22, 2012

Is Your Tournament Equitable?



Introduction - A golf handicap is a rough measure of a player's ability.  It is not a perfect measure.  It is biased in favor of the low handicap player, biased against a player with a large variance in his scoring, and biased against the player whose scores are trending upward.  Even with these flaws, however, it is probably the best predictor of a player's gross score in a stroke play event.
         Many tournaments, however, are played in a format where no scoring data is available.  In a four-ball event, each player may post an individual  score, but the team score is not posted.  There are clearly practical problems with trying to get a team handicap for four-ball events (e.g., not sufficient scores with the same partnership). To get around this problem, studies have been undertaken to examine the relative performance of generic teams.  For example, the USGA believes the teams with the higher combined handicap will do better in four-ball stroke play than teams with lower combined handicaps.  To correct for this inequity, the USGA recommends each player only receive 90 percent of their course handicap (See USGA Handicap System, Sec. 9-4bii).  The 90 percent figure is called an allowance.   A tournament committee has a great deal of discretion in choosing the allowance for a particular format.  That choice will have a large impact on the equity of the tournament.  Typically, there is no ex post facto examination of equity after a tournament is completed.    This lack of analysis only ensures the same mistakes will be made next year.
         This paper examines the equity of a tournament to provide an example of what should be done to increase equity.  The exemplar was a two-day tournament with a two-man scramble and four-ball format.  Thirty two teams participated and could select to play from any of three sets of tees with their handicaps adjusted according to Section 3-5 of the USGA Handicap System.  Five areas of possible equity problems are studied: 1) Prize format, 2) Scramble handicap allowance, 3) Four-ball handicap allowance, 4) The effectiveness of Sec. 3-5 in ensuring fairness, and 5) The spread in the difference in handicaps between partners.   A concluding section suggests possible policy and research implications for the United States Golf Association (USGA).

 Prize Format – One of the most common prize formats is “Equal Gross and Net.”  Unfortunately, putting “equal” in the title does not make it so. In the tournament in question, three gross prizes and five prizes were awarded.  The three gross prizes went to the three teams with the lowest combined indices. A conservative estimate of their probability of winning a prize, barring a major health emergency, would be between 80 and 95 percent.  This left the five net prizes to be fought over by the remaining 29 teams (i.e., each team in theory had only a 17% of winning a prize). The disparity in probabilities of winning is one measure of inequity. (Equally annoying, but not a problem in this tournament, is giving a gross prize in the B-flight.  This gives an edge to a team who were sufficiently mediocre to miss the cut for the A-flight)

The argument for “equal gross and net” is the low-handicap player cannot compete against the high-handicap player in a net tournament.  This argument may be valid when there a large number of players and a wide range of handicaps.  Typically, a high-handicap player has a larger variance in his scoring so it is likely one of the high-handicappers has a good chance of winning—as well as finishing dead last.  If the competition is flighted and the range of handicaps within each flight relatively small, the argument that a low-handicap player cannot compete loses much if not all of its strength.  This tournament only had one flight with handicaps ranging from 1 to 28. The low-handicap players, however, did very well.  For example, the team winning low gross also had the lowest net score.  The advantage of the low-handicap player stemmed an inequitable handicap allowance formula which is discussed next.

Scramble Handicap Allowance – In the scramble event, the player with the lower course handicap was allowed 25 percent of his course handicap.  The player with the higher course handicap was allowed 15 percent of his course handicap.  The total was rounded off with fractions of .5 or more rounded up.  This allocation seems inequitable on its face.  A team of ten-handicap players would receive a handicap of 4.  A team of scratch players would receive a handicap of 0.   In essence, the ten-handicaps would have to play even with the scratch players over 14 holes and just lose by a stroke on 4 others just to tie.  This seems unlikely. 

In an ideal tournament, net scores should not be correlated with handicaps.  Fig.1 shows a plot of the net scores versus the scramble handicap of each team.  Net scores and handicaps are highly correlated (R2 = .54).  The linear regression equation predicts that for each one-stroke increase in team handicap, the net score will increase by 1.3 strokes. This explains why the lowest net scores were posted by the three teams with the lowest combined indices. 
  


Clearly, the 25,15 allocation was unfair to the high handicap players.  Is there a more equitable allocation?   That is, is there an allocation that reduces the slope of the regression line to near zero?  The USGA suggests a 35,15 allocation.  Using the USGA allocation, the estimate of the slope was reduced to 0.8 for the full sample as shown in Table 1.  Based on this data set, the “best” allocation would be 50,25.  This allocation produces a minimal slope (.2)and the R2 value indicates a team’s handicap only accounts for 14 percent of the variance in net scores.  Because of the small sample size, however, this result only suggests the USGA recommended allocations may be too low and further study is definitely needed. 

     

Table 1

Bias in Scramble Handicap Allocations 

Allocation
    Slope
       R2
     25,15
      1.3
     .54
     35,15
      0.8
     .46
     45,15
      0.6
     .31
     50,25
      0.2
     .14

 

 Four-Ball Allowance – The net scores of each team are plotted against their average team handicap in Figure 2.  The regression equation indicates average handicap may have a small negative effect (-.1 for every increase in average handicap) on net score.   If there is a wide range in average handicaps, however, even a small effect could be important.  In this tournament there was a 20 stroke difference in handicap between the low-handicap and high handicap teams.  This translates (20 x .1) into a 2 stroke edge for the high handicap team.  Handicaps, however, were not reduced by 10 percent as recommended by the USGA.  If they were, it is likely the effect of average handicap on net score would disappear.
 
 
 

 
The coefficient for the average handicap variable, however, was not significant at the 95 percent level of confidence.  The finding of bias found here is merely suggestive, and a more definitive conclusion waits upon more and larger samples.

 
Sec. 3-5 – In this tournament, teams were allowed to complete from any of three sets of tees.  The player’s handicaps were adjusted in accordance with Sec. 3-5.  The assumption was the particular set of tees chosen would not have an effect on the team’s net score.  This assumption is examined for the scramble and four-ball competitions.

Scramble – To test for any effect of tee selection a dummy variable (T) was created.   The variable was assigned a value of 1 if the team played from the longest tees.  The variable was assigned a value of 0 if the team played from the shortest tees. (Note: Only three teams played from the combination tees so they were excluded from the sample.)  The following equation was estimated:

             Net Score = a + b1 · Scramble Handicap + b2 · T

The estimated equation was:


Net Score = 60.4 + 1.3 · Scramble Handicap - 0.2 · T 


 
The coefficient of the T variable was not significant (t-statistic = -0.17).  This would indicate Sec. 3-5 adequately compensates for the differences in tees.   

Four-ball – A similar model was estimated for the four-ball competition.  The estimated equation was:


Net Score = 63.8 – 0.4 · Average Handicap + 1.6 · T


The equation estimates that playing the longer tees results in a 1.6 stroke increase in the teams net score.  Again, the coefficient of the T variable was not significant (t-statistic = 0.93) at the 95 percent level of confidence.  The equation does suggest, however, that Sec. 3-5 has not equalized competition in the four-ball event.

Limitation on the Difference in Handicaps Between Partners - The USGA is convinced, mainly on the basis of research done in the 1970’s, that the spread between handicaps is an important determinant of net score in four-ball events.  The argument is, for example, that a team composed of a 6 and a 12 handicap is better than one composed of two 9 handicaps.  To examine if the USGA’s assertion is correct, the following model was estimated using data from the four-ball event:


Net Score = a +b1 · LH + b2 ·T + b3· Spread


Where,

                       LH = Low Handicap of the two players adjusted in accordance with Sec. 3-5,
                         T = Dummy variable representing tee selection (1= Long tees, 0 = short tees)
               Spread = Difference in handicaps between partners


The estimated equation was:


                Net Score(Four-Ball) = 64.8 + .01·LH + 1.9·T – 0.5·Spread


The coefficient for LH was not significant as before.  The coefficient for the T variable was slightly more significant (t statistic = 1.24), but still did not pass the 95 percent level of confidence.  The coefficient for the Spread variable was significant at the 95 percent level (t-statistic = -2.18).   This confirms the USGA’s recommendation of placing a limit on the difference in handicaps between partners.

USGA research, however, applies to four-ball events and not to scrambles.  To examine if “spread” was important in scramble events a model similar to that above was employed.  The estimated equation was:

                Net Score (Scramble) = 60.4 + 1.3 · H - .3· T - .01·Spread

 Where,

                                H = Scramble Handicap (25,15 allowance)
                                 T = Dummy variable representing tee selection (1= Long tees, 0 = short tees)
                     Spread = Difference in handicaps between partners

 
The coefficients of the T and Spread variable were not significant (t-statistic = -.2 and -.1 respectively).  This would indicate the limitation on the difference in handicap between partners may not be necessary for scramble events.  Given the peculiar nature of this tournament (the use of difference tees and a biased handicap allowance), any finding from the scramble event is not much more than conjecture.
 

 Implications for future Research - The limited purpose of this paper was to demonstrate a methodology for evaluating the equity of golf tournaments.  In the course of this research, however, several policy and research questions surfaced that should be addressed by the USGA. 

 
·        The USGA requires clubs to use Sec. 3-5 when players are competing from different tees.  The USGA, however, has not reported any research that proves Sec. 3-5 provides for equitable competition for different formats.  This should be corrected.  Moreover, the USGA should provide guidance on when Sec. 3-5 should be used and when it should be avoided if possible.  The USGA published "How to Conduct a Competition,"  but it is of little help in selecting formats to help ensure equity or in analysing tournament results. 

·         The four-ball allowance recommended by the USGA was developed around 1978.  That was 34 years ago.  It seems time to revisit the allowance since there have been changes to the handicap system since then.

·         The USGA only gives a soft recommendation on the allowance for scramble events—i.e., 35,15 seems to work, but you can use anything you want.  The USGA could instruct handicap chairpersons on ways to evaluate the equity of tournaments, as done here, so they are in a better position to select the appropriate allowances.