Tournament Selection Efficiency: An Analysis of the PGA TOUR’s

FedExCup1

Robert A. Connolly and Richard J. Rendleman, Jr.

June 27, 2012

1Robert A. Connolly is Associate Professor, Kenan-Flagler Business School, University of North Carolina,Chapel Hill. Richard J. Rendleman, Jr. is Visiting Professor, Tuck School of Business at Dartmouth andProfessor Emeritus, Kenan-Flagler Business School, University of North Carolina, Chapel Hill. The authorsthank the PGA TOUR for providing the data used in connection with this study, Pranab Sen, Nicholas Halland Dmitry Ryvkin for helpful comments on an earlier version of the paper and Ken Lovell for providing com-ments on the present version. Please address comments to Robert Connolly (email: robert connolly@unc.edu;phone: (919) 962-0053) or to Richard J. Rendleman, Jr. (e-mail: richard rendleman@unc.edu; phone: (919)962-3188).

Tournament Selection Efficiency: An Analysis of the PGA TOUR’s

FedExCup

Abstract

Analytical descriptions of tournament selection efficiency properties can be elusive for realistic

tournament structures. Combining a Monte Carlo simulation with a statistical model of player

skill and random variation in scoring, we estimate the selection efficiency of the PGA TOUR’s

FedExCup, a very complex multi-stage golf competition, which distributes $35 million in prize

money, including $10 million to the winner. Our assessments of efficiency are based on traditional

selection efficiency measures. We also introduce three new measures of efficiency which focus on

the ability of a given tournament structure to identify properly the relative skills of all tournament

participants and to distribute efficiently all of the tournament’s prize money. We find that reason-

able deviations from the present FedExCup structure do not yield large differences in the various

measures of efficiency.

1. Introduction

In this study, we analyze the selection efficiency of the PGA TOUR’s FedExCup, a large-scale

athletic competition involving a regular season followed by a series of playoff rounds and a “finals”

event, where an overall champion is crowned. FedExCup competition began in 2007. Each year,

at the completion of the competition, a total of $35 million in prize money is distributed to 150

players, with those in the top three finishing positions earning $10 million, $3 million and $2 million,

respectively.1

Research into selection efficiency highlights the importance of the criterion for assessing tour-

nament properties.2 Most who study tournament competition emphasize the probability that the

best player will be declared the winner (“predictive power”) as the critical measure of tournament

selection efficiency. Largely maintaining the focus of the selection efficiency literature on a single

player, Ryvkin and Ortmann (2008) and Ryvkin (2010) introduce two additional selection efficiency

measures, the expected skill level of the tournament winner and the expected skill ranking of the

winner. They develop the properties of these selection efficiency measures in simulated tournament

competition.

While we use these efficiency measures in our work, we also develop three new measures of

selection efficiency that evaluate the overall efficiency of a tournament structure, not just the the

mean skill and mean skill rank of the first-place finisher and the expected finishing position of

the most highly-skilled player. Much of the existing literature (e.g., Ryvkin (2010), Ryvkin and

Ortmann (2008)) assumes a specific set of distributions (e.g., normal, Pareto, and exponential)

to describe competitor skill and random variation in performance. In this paper, we integrate an

empirical model of skill and random variation in performance with a detailed tournament simulation

to explore the selection efficiency of FedExCup competition. We do not specify the matrix of

winning probabilities as in some studies; instead, it is generated naturally from the underlying

estimated distributions of competitor skill and random variation and the tournament structure

itself.

In the next section of the paper we describe the characteristics of FedExCup competition. We

develop tournament selection efficiency measures in Section 3. We present an overview of the

1See http://www.pgatour.com/r/stats/info/?02396.2See Ryvkin and Ortmann (2008) for an excellent recap of existing work along these lines.

1

statistical foundations of our work in Section 4, describe our simulation methods in Section 5, and

present results and a discussion of practical implications of our work in Section 6. We summarize

our findings in the final section. Appendix A describes the details of our simulation.

2. Characteristics of FedExCup Competition

2.1. Structure of FedExCup Competition

Under current FedExCup rules, similar in structure to NASCAR’s Sprint Cup points system, PGA

TOUR members accumulate FedExCup points during the 35-week regular PGA TOUR season.3

As shown in the “Regular Season Points” portion of Table 1, points are awarded in each regular

season PGA TOUR-sanctioned event to those who make cuts using a non-linear points distribution

schedule, with the greatest number of points given to top finishers relative to those finishing near

the bottom. At the end of the regular season, PGA TOUR members who rank 1 - 125 in FedExCup

points are eligible to participate in the FedExCup Playoffs, a series of four regular 72-hole stroke

play events, beginning in late August.

In the Playoffs, points continue to be accumulated, but at a rate equal to five times that of

regular season events. The field of FedExCup participants is reduced to 100 after the first round

of the Playoffs (The Barclays), reduced again to 70 after the second Playoffs round (the Deutsche

Bank Championship), and reduced again to 30 after the third round (the BMW Championship).

At the conclusion of the third round, FedExCup points for the final 30 players are reset according

to a predetermined schedule, with the FedExCup Finals being conducted in connection with THE

TOUR Championship. The player who has accumulated the greatest number of FedExCup points

after THE TOUR Championship wins the FedExCup.4

2.2. FedExCup Competition Objectives

It is clear that the objectives of FedExCup competition are multidimensional and complex. From

the November 25, 2008 interview with PGA TOUR Commissioner Tim Finchem (PGA TOUR

3The rules associated with FedExCup competition have been changed twice. Detail about the revisions is presentedin Hall and Potts (2010).

4A primer on the structure and point accumulation and reset rules may also be found athttp://www.pgatour.com/fedexcup/playoffs-primer/index.html.

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(2008)), it is possible to identify a number of these dimensions.

1. The points system should identify and reward players who have performed exceptionally wellthroughout the regular season and Playoffs. As such, among those who qualify for the Playoffs,performance during the regular season should have a bearing on final FedExCup standings.

2. The Playoffs should build toward a climactic finish, creating a “playoff-type feel,” holding faninterest and generating significant TV revenue throughout the Playoffs.

3. The points system should be structured so that the FedExCup winner is not determinedprior to the Finals. (In 2008, Vijay Singh only needed to “show up” at the Finals to winthe FedExCup. This led to significant changes in the points structure at the end of the 2008PGA TOUR season.)

4. The points system should give each participant in the Finals a mathematical chance of win-ning. We note that Bill Haas, the 2011 FedExCup winner and lowest-seeded player to everwin, was seeded 25th among the 30 players who competed in the Finals.5

5. The points system should be easy to understand. Under the current system, any player amongthe top five going into the Finals who wins the final event (THE TOUR Championship) alsowins the FedExCup. Otherwise, understanding the system, especially during the heat ofcompetition, can be very difficult.

We do not attempt to quantify the PGA TOUR’s objectives, as summarized above. Instead, we

evaluate the optimal selection efficiency of FedExCup competition based on two decision variables.

The first is the Playoffs points multiple. Presently, Playoffs points are five times regular season

points. This has a potential impact on Commissioner Finchem’s objective points 1 and 2 above.

Talking with PGA TOUR officials, we understand that the TOUR reassesses the FedExCup points

structure at the end of every season and that this multiple is an important part of the discussion.

Reflecting these discussions, we vary the multiple between 1 and 5 in integer increments. Our

second decision variable is whether or not to reset accumulated FedExCup points at the end of

the third Playoffs round. The present reset system is structured to satisfy objectives 3 and 4 and

guarantee that any player among the top five going into the Finals who wins the final event will

win the FedExCup (objective 5, at least in part).

Although we are able to identify optimal competition structures evaluated in terms of our six

efficiency measures, we find that the cost of deviating from optimal structure appears to be small.

This finding suggests that the costs of the implicit constraints associated with the objectives listed

above may not be high.

5Although confusing, we adopt the convention used throughout sports competition that a “low” seeding or finishingposition is a higher number than a “high” position. For example, in a 10-player competition, the “highest” seed isseeding position 1, while the lowest seed is position 10.

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3. Measures of Efficiency

In order to measure the selection efficiency of various FedExCup competition structures, we sim-

ulate entire seasons of regular PGA TOUR competition followed by four Playoffs rounds. In each

simulation trial, we begin with a set of “true” player skills, or expected 18-hole scores. Throughout

the regular season and Playoffs competition, each simulated score for a given player equals his ex-

pected score, as given by his true skill level, plus a residual random noise component. As the season

progresses, and throughout the Playoffs, each player accumulates FedExCup points according to a

defined set of rules as described in Section 4.1. We then estimate the efficiency of the FedExCup

points system using the criteria described below.

3.1. Ryvkin/Ortmann Selection Efficiency Measures

We use the following three measures of tournament selection efficiency, examined in detail by Ryvkin

and Ortmann (2008) and Ryvkin (2010).

1. The winning (%) rate of the most highly-skilled player, also known as “predictive power.”

2. The mean skill level (expected 18-hole score) of the tournament winner.

3. The mean skill ranking of the tournament winner.

Note that these three criteria focus on a single player, either the most highly-skilled player

(predictive power) or the tournament winner. No weight is placed on the finishing positions of

other players other than through their effect on the finishing position of the most highly-skilled

player or the mean skill ranking or skill level of the tournament winner.

We propose three new measures of selection efficiency that capture the ability of a given tour-

nament format to properly classify all tournament participants according to their true skill levels,

not just the player who is the most highly skilled, and to properly allocate tournament prize money.

Even if the most highly-skilled player in FedExCup competition wins most of the time, the FedEx-

Cup would surely lose credibility if the worst players in the competition could frequently finish

near the top and win a significant portion of the prize money. Ideally, the FedExCup design would

not only identify the single best player in the competition with high probability but would also

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place players in finishing positions relatively close to their true skill rankings. As such, tournament

prize money would generally be the highest for the most highly skilled and lowest for the lowest

skilled and, therefore, players would be rewarded in relation to their true skill levels. Our final

three measures of selection efficiency take the form of loss functions that reflect these tradeoffs.

3.2. Mean Squared Rank Error (LRE)

Consider a tournament of N players, i = 1, 2, ..., N , ordered by true skill (or expected score) Yi,

with Y1 < Y2, ... < YN . Let j(i) denote the tournament finishing position of player i. For example,

if the most highly-skilled player finishes the tournament in 5th position, j(1) = 5. Then Yj(i) is

the inverse transformation of true skill implied by player i ’s tournament finishing position, j(i),

which we, henceforth, refer to as “implied skill.” Finally, let Mj(i) denote the monetary prize to

player i if he finishes the tournament in position j(i), with M1 > M2, ... > MN . Thus, Mi denotes

what player i ’s prize would have been if his tournament finishing position had equalled his true

skill ranking and Mj(i) denotes player i ’s actual prize.

Our first loss function, the mean squared ranking error, LRE , measures the extent to which the

tournament errs in identifying the true skill rankings of the N tournament participants.

LRE =1

N

N∑i=1

(i− j (i))2

= 2σ2i(1 − ρi,j(i)

), (1)

where σ2i =(N2 − 1

)/12 is the variance of the ranking positions, i = 1, 2, ..., N , and ρi,j(j) is the

Spearman rank order correlation of the true skill ranks, i, and tournament finishing positions, j(i).

Thus, a tournament scheme that maximizes the Spearman rank, ρi,j(j), will minimize the mean

squared ranking error, LRE .6

We note that LRE weights all ranking errors equally, regardless of the actual skill differences of

the players who have been miss-ranked. Our final two efficiency measures reflect these differences.

6We note that Spearman’s footrule, another measure of ranking error, is equivalent to minimizing the sum ofabsolute ranking errors rather than squared ranking errors.

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3.3. Mean Squared Skill Error (LSE)

The mean squared skill error is defined as follows:

LSE =1

N

N∑i=1

(Yi − Yj(i)

)2= 2σ2Y (1 − βY ) . (2)

Here, σ2Y is the variance of true true player skill, and βY is the OLS slope coefficient associated

with a regression of true player skill Yi on implied player skill, Yj(i), or vice versa. When true skill

rankings and tournament finishing positions are perfectly aligned, βY = 1, and LSE = 0. Note that

if Y is linear in skill rank, LSE = LRE . The mean squared skill error takes the form of a quadratic

loss function, equivalent to the loss function underlying OLS regression and Taguchi’s (2005) loss

function used in quality control.

3.4. Mean Money-Weighted Squared Skill Error (LWSE)

Here we weight each value of(Yi − Yj(i)

)2in (2) by wi = Mi/

N∑i=1

Mi, where Mi is the dollar

tournament prize to the player among N participants who finishes the tournament in position i.

Thus, the mean money-weighted skill error is computed as follows:

LWSE =N∑i=1

(Yi − Yj(i)

)2wi. (3)

In this form, the greatest weight is given to implied skill errors for which the most money is on the

line. Unlike the previous two loss functions as expressed in Equations (1) and (2), this expression

cannot be simplified further without substantial restrictions on the functional form of the weighting

function (and the consequent loss of generality).

3.5. Mean Squared Error Deflators

Inasmuch as the value of each mean squared error is difficult to interpret without a reference point,

we deflate each by the corresponding variance of the variable whose error we are attempting to

estimate (i.e., true skill rankings, true player skill and money-weighted true player skill.) Thus,

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the deflators for the ranking error, skill error and weighted skill error are, respectively, DRE =(N2 − 1

)/12, DSE = σ2Y , and DWSE =

N∑i=1

Y 2i wi − (

N∑i=1

Yiwi)2.

4. Optimizing FedExCup Competition

4.1. FedExCup Points Distribution and Accumulation

The “Regular Season Points” section of Table 1 shows the distribution of regular season FedExCup

points. WGC and Majors are allocated slightly more points than regular PGA TOUR events.

“Additional events,” which are events held opposite of some WGC events and majors, are allocated

half the points associated with each regular event finishing position.

During the regular PGA TOUR season, players accumulate FedExCup points based on the

regular season points schedule. At the end of the regular season, the top 125 players in accumulated

FedExCup points qualify to participate in the Playoffs. Each participant in the Playoffs carries his

accumulated FedExCup points into the Playoffs, but once in the Playoffs, FedExCup points are

awarded and accumulated according to the schedule shown in the “Playoffs Points” section of the

table. Note that the points distribution schedule for the first three rounds of the Playoffs is exactly

five times the points distribution for regular PGA TOUR events conducted prior to the Playoffs.

At the end of the first Playoffs round (The Barclays), only the top 100 players in accumulated

FedExCup points are eligible to continue to the second round. After the second round (The

Deutsche Bank Championship), only the top 70 players are eligible to continue to the third round.

After the third round (The BMW Championship) only the top 30 players qualify for the FedExCup

Finals (The TOUR Championship). Immediately prior to the Finals, points are reset for each of

the Finals qualifiers according to the schedule shown in the “Finals Reset” column of the “Playoffs

Points” section. Points are awarded during the Finals according to the schedule shown in the last

column of Table 1. (Note that this is exactly the same distribution of points awarded to finishing

positions 1-30 during the first three rounds of the Playoffs.) The points reset was put into place

after the second year of FedExCup competition to ensure that no single player could have won the

FedExCup prior to the Finals event and also to give each participant in the Finals a mathematical

chance of winning the FedExCup.

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4.2. What We Evaluate

We limit our analysis of selection efficiency to the 125 players who qualify for the FedExCup

Playoffs. Using all six efficiency measures, we evaluate the efficiency of the regular season points

distribution system.7 For this same group of 125 players, we then evaluate each of the six efficiency

measures at the end of each round of the Playoffs in an attempt to determine if each successive

round of the Playoffs improves selection efficiency for this group of 125 players.

We also evaluate selection efficiency over all six measures at the end of every Playoffs round, but

only for those players who qualify to play in each round. Our concern is whether the points system

improves efficiency incrementally at the end of each round for remaining participating players.

5. Statistical Foundations

5.1. Data

Our data, provided by the PGA TOUR, covers the 2003-2010 PGA TOUR seasons. It includes 18-

hole scores for every player in every stroke play event sanctioned by the PGA TOUR for years 2003-

2010 for a total of 151,954 scores distributed among 1,878 players. We limit the sample to players

who recorded 10 or more 18-hole scores. The resulting sample consists of 148,145 observations of

18-hole golf scores for 699 PGA TOUR players over 366 stroke-play events. Most of the omitted

players are not representative of typical PGA TOUR players. For example, 711 of the omitted

players recorded just one or two 18-hole scores.8

5.2. Player Skill Estimation Model

We employ a variation of the Connolly and Rendleman (2008) model to estimate time-varying

player skill and random variation in scoring for a group of professional golfers representative of

PGA TOUR participants during the eight-year period 2003-2010. As in Connolly and Rendleman

(2008), we employ the cubic spline methodology of Wang (1998) to estimate skill functions and

autocorrelation in residual errors for players with 91 or more scores. We employ a simpler linear

7We take the regular season points distribution schedule as shown in Table 1 as given as well as the number ofplayers who qualify for the Playoffs and each of its stages.

8Generally, these are one-time qualifiers for the U.S. Open, British Open and PGA Championship who, otherwise,would have little opportunity to participate in PGA TOUR sanctioned events.

8

representation without autocorrelation, as in Connolly and Rendleman (2012), for players with 10 to

90 scores over the full sample period.9 Simultaneously, we estimate fixed course effects and random

round effects. We note that the model does not take account of specific information about playing

conditions (e.g., adverse weather as in Brown (2011), pin placements, morning or afternoon starting

times, etc.) or, in general, the particular conditions that could make scoring for all players more

or less difficult, when estimating random round effects. Nevertheless, if such conditions combine

to produce abnormally high or low scores in a given 18-hole round, the effects of these conditions

should be reflected in the estimated round-related random effects.10

When estimating player skill functions, we also obtain sets of player-specific residual scoring

errors, denoted as θ and η. The θ errors represent potentially autocorrelated differences between a

player’s actual 18-hole scores, reduced by estimated fixed course and random round effects, and his

predicted scores. The η errors represent θ errors adjusted for estimated first-order autocorrelation,

and are assumed to be white noise. We refer to a player’s skill estimate at a given point in time

as an estimate of his “neutral” score, since estimated fixed course effects and random round effects

have been removed.

6. Simulation of FedExCup Competition

6.1. Simulation Design

We structure each of 40,000 simulation trials so that the composition of the player pool is similar to

what one might observe in a typical PGA TOUR season. As such, we do not include all 699 players

from the statistical sample in each trial. Instead, the number of players per trial varies between

9We established the 91-score minimum in Connolly-Rendleman (2008) as a compromise between having a samplesize sufficiently large to employ Wang’s (1998) cubic spline model (which requires 50 to 100 observations) to estimateplayer-specific skill functions, while maintaining as many established PGA TOUR players in the sample as possible.The censoring of a sample in this fashion will have a tendency to exclude older players who are ending their careersin the early part of the sample and younger players who are beginning their careers near the end. If player skill tendsto vary with age, such a censoring mechanism can create a spurious relationship, where mean skill across all playersin the sample appears to be a function of time. (Berry, Reese and Larkey (1999) show that skill among PGA TOURgolfers tends to improve with age up to about age 29 and decline with age starting around age 36. Thus, ages 30-35tend to represent peak years for professional golfers.) To eliminate any type of age-related sample bias arising from acensored sample, we employ a 10-score minimum, rather than a 91-score minimum, and use simpler linear functionsto estimate skill for those who recorded between 10 and 90 scores.

10Interacted random round-course effects, with similar justification, are also estimated in Berry, Reese and Larkey(1999) and Berry (2001). We also estimate random round-course effects in our original 2008 model. However, webelieve that the course component of a potential round-course effect is better modeled as fixed than random.

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415 and 459 and reflects the actual number of players in the sample in each year, 2003-2010. We

also structure the simulations so that the simulated distributions of player skill (mean neutral score

per round), scoring, and player tournament participation rates during the simulated regular season

closely approximate those observed in the actual sample. Simulation details are provided in the

Appendix.

6.2. Simulation Results

Table 2 summarizes the simulation sample mean value of each of six efficiency measures at the end of

the regular PGA TOUR season and at the end of each round of the Playoffs evaluated with respect

to the 125 players who qualify for the Playoffs in simulated competition. Each of the six panels of

Table 2 represents one of the six selection efficiency measures. For the efficiency measure shown in

Panel A, “First Place Rate of Best Player,” higher values are better. In the remaining five panels,

lower values indicate greater efficiency. Each efficiency measure is evaluated using a Playoffs points

to regular season points weighting ratio that varies from 1 to 5. All efficiency measures shown in

Panels D-F are the values computed from Equations 1-3, deflated by the corresponding variance of

the variable whose error we are attempting to estimate. Efficiency for Playoff round 4 is evaluated

without a points reset (NR) and with a reset (R). The points reset schedule is that given in Table 1

times weighting ratio divided by 5. We denote the end of the PGA TOUR regular season as “Stage

0” and the end of Playoffs rounds 1-4 as Stages 1 through 4, respectively.

In each panel, the best efficiency value is shown in bold for each stage 1-4. Except for the

few efficiency measures shown in italics, the measures shown in bold are statistically superior in a

one-sided test at the 0.05 level relative to all other values shown for the same stage.11

Regardless of the points weighting or efficiency measure, efficiency improves during each stage

of competition during Playoffs rounds 1-3 and from round 3 to round 4 when there is no points

reset. Only in Panel C (mean skill rank of player in first place) and Panel D (mean squared rank

error) are there any entries where the efficiency measure improves from round 3 to round 4 when

points are reset after round 3. Thus, from the standpoint of pure mathematical efficiency, without

regard to non-mathematical objectives that might lead to a reset being optimal, the 125 players in

11We estimate statistical significance using 10,000 bootstrap samples drawn from the simulated data generated by40,000 trials.

10

the Playoffs are generally ordered more efficiently after the third round of the Playoffs than after

the final round with a reset.

The optimal Playoffs points weight seems to vary by selection efficiency measure; weights of 3

and 4 generally provide the best efficiency. Although this suggests that the present Playoffs points

weight of 5 may be too high, we argue in Section 6.3 that these differences may have little practical

significance.

Table 3 is organized similarly to Table 2. In contrast to Table 2 where the focus is on the

125 Playoffs participants, the focus in Table 3 is on selection efficiency computed incrementally

for each round of the Playoffs for just those players participating in a specific round. Each value

shown in the table reflects the mean value over 40,000 simulation trials of the ratio of the efficiency

measure computed at the end of the stage to the efficiency measure computed at the beginning

of the stage for stage participants only. In all but Panel A, a ratio less than 1 represents an

improvement in efficiency from one stage of Playoffs competition to the next. Again, values shown

in bold correspond to the best values per stage. Unless shown in italics, all other values in the

same stage are significantly inferior at the 0.05 level than the best value shown in bold.

Again, the points reset after the third Playoffs round tends to decrease selection efficiency;

players who participate in the finals with a points reset tend to be ordered less efficiently after the

final stage of competition than they were ordered prior to the final stage. However, a decrease in

selection efficiency is not indicated for all measures. For example, there is unambiguous improve-

ment in the mean squared rank error (Panel D) and an indication of improvement, depending upon

the Playoffs points weighting, in Panel C (mean skill rank of player in first place) and in Panel E

(mean square skill error). However in only one case (a weight of 2 in Panel D) is the efficiency

value with a points reset better than that without.

As in Table 2, optimal points weightings tend to vary by efficiency measure. Nevertheless, a

weight of around 3 appears to produce the best efficiency measures, but in some cases, the current

weight of 5 appears to be optimal.

6.3. Practical Significance

Despite finding optimal values for the Playoffs points weighting and the decision whether to reset

FedExCup points going into the final Playoffs round, we believe that the practical differences are

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insignificant among efficiency outcomes based on optimal tournament design and those based on

non-optimal design over the range of possible Playoffs design schemes that we consider. The entries

in Table 4, which show the best and worst efficiency outcomes from the corresponding panels of

Table 2 along with efficiency outcomes where the outcomes in each regular season and Playoffs event

are determined randomly, provide support for this view.12 (We show results for random outcomes

using a Playoffs weight of 3 only. With random tournament outcomes, the weight has essentially

no impact on any of the efficiency measures.)

The outcomes in Panels A-C, based on the efficiency measures of Ryvkin and Ortmann, are

the most straightforward to interpret. Panel A shows the rate at which the best player in the

competition wins. At the end of the competition, the best and worst outcomes associated with

regular (non-random) tournament competition fall between 63% and 44%. By contrast, with ran-

dom tournament outcomes, the best player wins less than 1% of the time. Panel B shows the mean

skill level (mean neutral score per round) of the first-place finisher. The best and worst outcomes

at the end of regular tournament competition fall between 68.51 and 68.78 compared with 70.65

when tournament outcomes are determined randomly. Panel C shows the mean skill rank of the

player who finishes the competition in first place. Here the best and worst outcomes at the end of

regular tournament competition fall between 3.17 and 4.58 compared with 64.80 when outcomes

are determined randomly. Clearly, on the basis of these three measures, (non-random) regular tour-

nament competition dramatically improves each of the three efficiency measures over what might

have otherwise been obtained with random tournament outcomes. Whether tournament design is

technically optimal appears to be of second-order importance relative to the general structure of

the competition itself.

Each of the efficiency measures in Panels D-F are the values computed from Equations 1-3,

respectively, deflated by the corresponding variance of the variable whose error is being estimated.

If we further divide the values in Panel D by 2, we obtain(1 − ρi,j(i)

), where ρi,j(i) is the Spear-

man rank order correlation of the true skill ranks and tournament finishing positions. Best and

worst values from regular tournament competition fall between 66% and 71%, which correspond to

Spearman rank correlations of 0.673 and 0.644. By contrast, the 1.975 value for the same efficiency

12We maintain exactly the same simulation design as described in the appendix, but instead of basing tournamentoutcomes on scores, outcomes are based on random orderings of tournament participants, both before and after cuts.

12

measure corresponds to a Spearman rank correlation of 0.013, essentially zero. Clearly the tour-

nament competition, whether optimally designed in terms of Playoffs point weights and the reset,

significantly improves the rank ordering of participating players.

If we divide the values in Panel E by 2, we obtain (1 − βY ), where βY is the OLS slope co-

efficient associated with a regression of true player skill on skill implied by tournament finishing

position. Best and worst values in Panel E fall between 0.561 and 0.613, which correspond to

slope coefficients of 0.720 and 0.694. With random ordering, the 1.975 value for the same efficiency

measure corresponds to a slope of 0.013, essentially zero. As in Panel D, the efficiency values from

non-random competition, whether or not they reflect optimal tournament design, are substantially

better than that obtained by a random ordering of players.

The values for the money-weighted squared skill error, shown in Panel F, are not as readily

interpreted. Nevertheless, best and worst values associated with regular competition fall between

0.273 and 0.394 compared with 3.245 with random tournament outcomes, suggesting that achieving

exactly optimal tournament design is not critical.

Finally, it is clear that the reset substantially reduces efficiency as measured by the winning

rate of the best player as summarized in the Panel A sections of Tables 2-4. With a points reset,

efficiency, as measured at the end of the competition, is actually worse than at the end of the regular

season (Stage 0). Although it is not so easy for the average person following professional golf to

appreciate all the dimensions of the other efficiency measures, we suspect that most would have an

intuitive feel for the skill levels of the best players in golf.13 If the best players are not winning the

FedExCup at a reasonably high rate, and in particular Tiger Woods over the 2003-2010 period of

our study, it isn’t unreasonable to expect that the competition could lose credibility among those

who follow professional golf. Otherwise, we see little cost in changing Playoffs points weights or

the reset to satisfy PGA TOUR objectives that might not be easy to quantify.

13In fact, the TOUR publishes player scoring averages and scoring averages adjusted for field strength throughoutthe PGA TOUR season. Although neither of these measures corresponds exactly to our mean neutral score, theseaverages, along with Official World Golf Rankings and other performance measures, make it relatively straightforwardto identify the best golfers.

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6.4. Competitiveness and Excitement

It is clear from PGA TOUR Commissioner’s November 25, 2008 interview that the PGA TOUR

strives to create a competitive and exciting playoffs system, building toward a climactic finish,

that will hold fan interest throughout. While aiming to reward players who have performed ex-

ceptionally well throughout the regular season, the TOUR does not want the FedExCup winner to

be determined prior to the Finals. Thus, the TOUR is seeking to achieve a fine balance between

player performance during both the regular season and Playoffs. This balance may not be easily

quantified in terms of the tournament selection efficiency measures we have considered thus far.

By construction, the points reset ensures that the ultimate winner of the FedExCup cannot be

determined until the completion of the final Playoffs event. In the same simulations that underlie

the results summarized in Tables 2 and 3, the winner of the competition would be determined prior

to the Finals with probability 0.337 under the present Playoffs points weighting scheme (weight =

5) if there were no reset. With Playoffs points weights of 1, 2, 3 and 4, the probabilities would

be 0.143, 0.200, 0.252, and 0.299, respectively. Clearly, if the FedExCup winner were determined

prior to the FedExCup Finals, there would be little fan interest in the final event, THE TOUR

Championship, which occurs during the middle of the professional and college football seasons. As

such, we believe that the PGA TOUR would view these probabilities as being unacceptably high.

(If Tiger Woods, or equivalently, a player with his scoring characteristics, is excluded from the

simulations, the probabilities for Playoffs weights of 1-5 would be 0.039, 0.050, 0.066, 0.087, and

0.113, respectively.14 More detailed results for simulations that exclude Woods are provided in the

online appendix.)

Table 5 shows FedExCup winning percentages for 20 of the 125 players in the Playoffs. In Panel

A, the players are ordered by their seeding positions, 1-20, at the beginning of the Playoffs. In Panel

B, players are ordered by their skill rankings, 1-20, in relation to the field of Playoffs participants.

We include an online appendix as a supplement to this paper, which shows results in both panels

for players in all positions, 1-125.

Table 5 shows that without a reset and without giving more weight to FedExCup points earned

14Since Woods was such a dominant player during the 2003-2010 period on which our simulations are based,projections based on the inclusion of a player of Woods’ skill may be misleading, at least at the top position, forfuture periods where there may be no equivalently dominant player.

14

during the Playoffs relative to the regular season, the player who finished the regular season in first

place would win the FedExCup 81.8% of the time. (If Woods is excluded from the simulations, this

estimate falls to 78.6%.) Moreover, without a reset and with a Playoffs weight of 1, all players in

seeding positions 5-125 have less that a 1% probability of winning. (This is also the case if Woods

in not included.) Clearly the winning rate of 81.9% for the top-seeded player and the very low

winning rates associated with players seeded beyond position 4 are inconsistent with the TOUR’s

objectives. Even with a Playoffs weight of 5, the top-seeded player going into the Playoffs wins the

FedExCup over 50% of the time with no reset, and no player beyond seeding position 9 has over

a 1% chance of winning. (Without Woods, 33.5%, and no player past position 14 has more than

a 1% chance of winning.) By contrast, regardless of the weight, the winning percentage rate of

the top-seeded player is substantially lower with a reset, 35.6% to 39.2%, but not so low that his

performance during the regular season goes unrewarded, and many more players have a legitimate

chance to win. (Without Woods, 21.4% to 29.9%.)

From the “Stage 3” column of Table 2, we see that the most-highly skilled player in the com-

petition is the number 1 Playoffs seed 56% to 60% of the time. Therefore, ignoring the remote

possibility that this player would not make the Playoffs, the number 1 seed would be the most

highly-skilled among the 125 Playoffs participants 56% to 60% of the time.15 When this player is

not the top seed, he is very likely be near the top. Panel B shows that with a reset, the winning

percentage rate of the most highly-skilled player in the playoffs is greater than that of the number

1 seed. This suggests that even if the most highly-skilled player is not the number 1 seed, he still

has a reasonably high chance of winning. By contrast, without a reset and with a Playoffs weight

of 1, the most highly-skilled player does not win as often as the number 1 seed, but he does win

more often with a Playoffs weight of 5. (These same relationships also hold when Tiger Woods is

excluded from the simulations.)

Table 6 shows percentage rates per FedExCup finishing position (through finishing position 10)

for the top-10-seeded players going into the Finals. (The online appendix shows the same results

for all 30 players in the Finals over all 30 possible finishing positions.) Panels A and B indicate

that the percentage rates per finishing position are hardly affected by the Playoffs weighting scheme

when there is a points reset going into the Finals. With either a weight of 1 or 5, the top 5 seeds all

15Tiger Woods misses the Playoffs in 39 of 40,000 simulation trials.

15

have a reasonable chance to win, ranging from 5.5% to 45.8%. Although not shown, winning rates

for players in seeding positions 11-30 are all less than 1%. Also, we estimate that a player seeded

in position 25 or worse, the same as Bill Haas’ position going into the 2011 Finals, would win the

FedExCup only 0.26% of the time under the present system with a reset and Playoffs weight of 5.

Thus, Haas’ win was clearly a very rare event.

Panels C and D, show percentage rates per finishing position without a reset. With a Playoffs

weight of 1, there is little remaining uncertainty about the ultimate winner and other top finishers;

all are very likely to finish in the positions in which they started. This problem is mitigated

somewhat with a Playoffs weight of 5. Nevertheless, the number 1 seed wins almost four out of

five times. Interestingly, these same results tend to hold, but just to a slightly lesser extent, when

Tiger Woods is not included in the FedExCup competition.

Although not entirely evident, since not all players and finishing positions are shown, in panels

A and B, except for the first and last seeds, each player’s most likely finishing position is worse

than his initial seed. In Panel C, where there is no points reset and points per Playoffs event are

the same as those of regular season events, the most likely finishing position for almost every Finals

participant is his Finals seeding position. (The only exceptions are for seeding positions 16-20.)

This suggests that a competition scheme with no reset and no differential weighting of Playoffs and

regular season events leaves very little drama and potential for position changes in the Finals. By

contrast, even without a reset, but with a Playoffs points weight of 5, players in Finals seeding

positions 4-28 are most likely to finish worse than they started (positions 10-28 not shown in the

table).

Taken as a whole, we believe that the reset and the weighting of Playoffs points more heavily than

those for regular season events plays a critical role in maintaining drama and potential fan interest

throughout the Playoffs. From a pure efficiency standpoint, the reset tends to be suboptimal.

Nevertheless, it is clear that without a reset, the PGA TOUR could not satisfy its objectives of

conducting a meaningful regular season leading to playoffs with a climactic finish that both holds

fan interest and has the potential to generate significant TV revenue.

16

7. Summary and Conclusions

In this paper we introduce several new tournament selection efficiency measures and apply these

measures and several existing measures in a systematic evaluation of the selection efficiency of

the FedExCup competition run by the PGA TOUR. Our new measures are defined on the full

range of tournament outcomes, not just the characteristics of the top finisher or most highly-

skilled player. Using simulation, we evaluate the efficiency characteristics of specific alternative

tournament structures.

Our simulations show that relative to random selection, every variation on the FedExCup tour-

nament selection method that we consider produces significant improvements in selection efficiency.

Beyond this result, perhaps the most important regularity is that the points reset impairs tourna-

ment efficiency. On the other hand, one important aim of the points reset is to ensure that the

competition is in doubt until the last moment. We show that the reset and weighting of Playoffs

points more heavily than those of regular season events are critical elements in creating an exciting

and dramatic set of Playoffs events. We acknowledge that our analysis of excitement and drama is

much less scientific than our more direct mathematical assessment of tournament selection efficiency

and believe that a more formal development of this aspect of competition could be an interesting

area for future research.

17

AppendixSimulation Methodology

A. FedExCup Regular Season and Playoffs Competition

In simulating the accumulation of FedExCup points during the regular PGA TOUR season andPlayoffs, we make the following assumptions.

1. Between 415 and 459 players participate for a full “regular season” prior to the FedExCupPlayoffs in 35 4-round stroke play events.16 144 players participate in each event. There is no“picking and choosing” of tournaments nor any qualifying requirements.17 The probabilitythat any single player participates in a regular season event reflects his actual participationfrequency on the TOUR.

2. After the first two rounds of each regular season event, the field is cut to the lowest-scoring70 players who then continue for two more rounds of tournament play.18

3. FedExCup points are awarded for each tournament using the “PGA TOUR Regular Seasonevents points distribution” schedule shown in Table 1, assuming each of the 35 tournamentsis a regular PGA TOUR event rather than a “major,” a World Golf Championship event oran “alternate” event held opposite tournaments in the World Golf Championship series.

4. At the end of the 35-event regular season, the Playoffs begin with the top 125 players inFedExCup points participating in The Barclays, the first of four Playoffs events. The Barclaysemploys a cut after the first two rounds, with the lowest-scoring 70 players advancing to thefinal two rounds. At the completion of play, FedExCup points are added to those previouslyaccumulated for each of the 125 Playoffs participants according to the schedule of Playoffspoints shown in Table 1.

5. After The Barclays, the top 100 players in FedExCup points advance to the Deutsche BankChampionship. The Deutsche Bank employs a cut after the first two rounds, with the lowest-scoring 70 players advancing to the final two rounds. FedExCup points are added to thosepreviously accumulated for each of the remaining 100 Playoffs participants according to theschedule of Playoffs points shown in Table 1.

6. After the Deutsche Bank Championship, the top 70 players in FedExCup points advanceto the BMW Championship, where there is no cut. FedExCup points are added to thosepreviously accumulated for each of the remaining 70 Playoffs participants according to theschedule of Playoffs points shown in Table 1.

1635 regular season events reflects the number of weeks of regular season PGA TOUR competition prior to theFedExCup Playoffs during 2010. In three of the 35 weeks, two PGA TOUR sanctioned events were played simulta-neously, but no single player could have participated in the two events at the same time. Therefore, to simplify thesimulations, we treat these weeks as if a single event were held.

17A standard PGA TOUR event consists of 144 players. In the early and late parts of the PGA TOUR season,regular events tend to be reduced in size to 144 players due to limited daylight hours. The TOUR also conducts a few“invitationals” with smaller fields, along with a few smaller field select events, including tournaments in the WorldGolf Championship series. In addition, the Masters, one of the four “majors,” is a small field event, with 97 playersparticipating in 2010.

18Generally, the lowest-scoring 70 players and ties make the cut in regular PGA TOUR events. It is almost certainthat no ties will occur with our simulation methodology, but in the unlikely event that a tie does occur, the tie isbroken randomly.

18

7. After the BMW Championship, the top 30 players in FedExCup points advance to THETOUR Championship.

8. When simulating the present TOUR Championship structure, the number of FedExCuppoints for the 30 participating players is reset according to reset schedule shown in Table1. Players are then awarded additional FedExCup points according to their finishing posi-tion in THE TOUR Championship, a four-round stroke play event with no cut, using thepoints distribution schedule for the Finals as shown in Table 1. The FedExCup winner is theplayer who has earned the most FedExCup points, not necessarily THE TOUR Championshipwinner.

B. Player Selection

Players are selected for regular season tournament participation using the following procedure.

1. A single year from our statistical sample, 2003-2010, is selected, with each year being selectedexactly 40, 000/8 = 5, 000 times.

2. All players who actually participated in the selected year become the regular season playerpool.

3. Players from the regular season pool are selected randomly for participation in each of the35 regular season events, where the probability of any player being selected among the 144tournament participants is equal to the proportion of total player weeks in which he actuallyparticipated in the year selected, assuming sampling without replacement.19

C. Simulated 18-Hole Scoring

The following procedure is used to generate 18-hole scores for players who could potentially competein a given randomly selected PGA TOUR season.

1. A single mean skill level (mean neutral score) for each player is selected at random fromthe portion of his estimated spline-based skill occurring in the selected PGA TOUR season,2003-2010. This becomes the player’s mean skill level for the entire season.20

2. For each player k, a single θ residual is selected at random from among the entire distributionof nk θ residuals estimated in connection with his cubic spline-based skill function.

3. For each player k, 166 η residuals are selected randomly with replacement from among theentire distribution of nk η residuals estimated in connection with his cubic spline-based skillfunction.

4. Using the initial randomly selected θ residual, the vector of 166 randomly-selected η residuals,and player k ’s first-order autocorrelation coefficient as estimated in connection with his cubicspline fit, a sequence of 166 estimated θ residuals is computed.

19In determining the extent of individual player participation on the TOUR, we use weeks played rather thantournament played, since, in a few weeks each year, two PGA TOUR-sanctioned events are held simultaneously.

20We assume that the level of effort for each player throughout the entire regular season and Playoffs is the sameas that reflected, implicitly, in his estimated skill function.

19

5. The 166 θ residuals are applied to player k ’s skill estimate to produce 166 simulated random18-holes scores. The first 10 scores are not used in simulated competition but, instead, aregenerated to allow the first-order autocorrelation process to “burn in.” The next 156 are thescores required for a player who might be selected to play in every regular season tournamentand who misses no cuts during the regular season (35 × 4 = 140) or during the four roundsof the Playoffs (4 × 4 = 16). We note that it is highly unlikely that all 156 scores would beused for any single player.

6. Starting with the 11th score, scores for each player k are applied in sequence as needed tosimulate scoring during the regular season and Playoffs.21

21Suppose player 1 makes the cut in the first regular season event and player 2 missed the cut. If both are selectedto play in the second regular season event, then simulated scoring in the second event will start with scores 15 and13 for players 1 and 2, respectively.

20

Table 1: FedExCup Points Distribution and Reset Schedule

Regular Season Points Playoffs PointsFinishing Regular WGC Additional FinalsPosition Events Events Majors* Events Rounds 1-3 Reset Finals

1 500 550 600 250.0 2,500 2,500 2,5002 300 315 330 150.0 1,500 2,250 1,5003 190 200 210 95.0 1,000 2,000 1,0004 135 140 150 70.0 750 1,800 7505 110 115 120 55.0 550 1,600 5506 100 105 110 50.0 500 1,400 5007 90 95 100 45.0 450 1,200 4508 85 89 94 43.0 425 1,000 4259 80 83 88 40.0 400 800 40010 75 78 82 37.5 375 600 37511 70 73 77 35.0 350 480 35012 65 69 72 32.5 325 460 32513 60 65 68 30.0 300 440 30014 57 62 64 28.5 285 420 28515 56 59 61 28.0 280 400 28016 55 57 59 27.5 275 380 27517 54 55 57 27.0 270 360 27018 53 53 55 26.5 265 340 26519 52 52 53 26.0 260 320 26020 51 51 51 25.5 255 310 25521 50 50 50 25.0 250 300 25022 49 49 49 24.5 245 290 24523 48 48 48 24.0 240 280 24024 47 47 47 23.5 235 270 23525 46 46 46 23.0 230 260 23026 45 45 45 22.5 225 250 22527 44 44 44 22.0 220 240 22028 43 43 43 21.5 215 230 21529 42 42 42 21.0 210 220 21030 41 41 41 20.5 205 210 205. .. .

66 5 5 5 2.5 2567 4 4 4 2.0 2068 3 3 3 1.5 1569 2 2 2 1.0 1070 1 1 1 0.5 5

71-75 576-85 4

* = includes THE PLAYERS Championship. Points for regular events during regular seasondecreased by 0.02 points per finishing position past 70.

21

Tab

le2:

Effi

cien

cyM

easu

res

Com

pu

ted

for

all

125

Fed

ExC

up

Pla

yoff

sQ

ual

ifier

s

Panel

A:

Fir

stP

lace

Rate

of

Bes

tP

layer

Panel

D:

Mea

nSquare

dR

ank

Err

or

(Defl

ate

d)

Wei

ght

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

10.4

92

0.5

14

0.5

37

0.5

60

0.5

80

0.4

37

0.7

96

0.7

57

0.7

30

0.7

15

0.7

13

0.7

12

20.4

92

0.5

30

0.5

65

0.5

97

0.6

24

0.4

49

0.7

96

0.7

41

0.7

03

0.6

83

0.6

81

0.6

80

30.4

92

0.534

0.573

0.606

0.633

0.4

54

0.7

96

0.7

35

0.6

93

0.6

71

0.6

68

0.6

68

40.4

92

0.5

22

0.5

62

0.5

99

0.6

28

0.458

0.7

96

0.735

0.691

0.668

0.665

0.665

*5

0.4

92

0.4

94

0.5

45

0.5

85

0.6

18

0.459

0.7

96

0.7

39

0.6

95

0.6

71

0.6

68

0.6

68

Panel

B:

Mea

nSkill

of

Pla

yer

inF

irst

Pla

ceP

anel

E:

Mea

nSquare

dSkill

Err

or

(Defl

ate

d)

Wei

ght

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

168.8

23

68.7

61

68.7

04

68.6

54

68.6

11

68.7

77

0.6

89

0.6

49

0.6

23

0.6

06

0.5

98

0.6

13

268.8

23

68.7

21

68.6

41

68.5

75

68.5

24

68.7

38

0.6

89

0.6

34

0.5

99

0.5

76

0.5

67

0.5

87

368.8

23

68.707

68.620

68.555

68.507

68.7

26

0.6

89

0.631

0.594

0.570

0.561

0.581

468.8

23

68.7

16

68.6

31

68.5

64

68.5

13

68.721

0.6

89

0.6

36

0.5

99

0.5

74

0.5

63

0.5

82

568.8

23

68.7

47

68.6

56

68.5

84

68.5

27

68.722

0.6

89

0.6

45

0.6

08

0.5

82

0.5

70

0.5

88

Panel

C:

Mea

nSkill

Rank

of

Pla

yer

inF

irst

Pla

ceP

anel

F:

Money

-Wei

ghte

dSquare

dSkill

Err

or

(Defl

ate

d)

Wei

ght

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

17.8

14

6.3

42

5.3

69

4.7

66

4.4

55

4.5

82

0.5

08

0.4

47

0.4

03

0.3

69

0.3

47

0.3

94

27.8

14

5.6

86

4.5

23

3.8

76

3.5

90

3.8

71

0.5

08

0.4

17

0.3

57

0.3

15

0.2

88

0.3

57

37.8

14

5.3

64

4.1

88

3.5

56

3.2

87

3.5

93

0.5

08

0.4

03

0.340

0.298

0.273

0.3

44

47.8

14

5.2

02

4.0

69

3.463

3.1

85

3.4

85

0.5

08

0.401

0.340

0.298

0.273

0.341

57.8

14

5.134

4.046

3.459

3.170

3.452

0.5

08

0.4

08

0.3

48

0.3

06

0.2

78

0.3

43

Sta

ge

0=

end

of

regula

rse

aso

n;

Sta

ge

1=

end

of

firs

tP

layoff

sro

und

(Barc

lays)

;Sta

ge

2=

end

of

seco

nd

Pla

yoff

sro

und

(Deu

tsch

eB

ank);

Sta

ge

3=

end

of

thir

dP

layoff

sro

und

(BM

W);

Sta

ge

4N

R=

end

of

final

Pla

yoff

sR

ound

(TO

UR

Cham

pio

nsh

ip)

wit

hno

poin

tsre

set;

Sta

ge

4R

=en

dof

final

Pla

yoff

sR

ound

(TO

UR

Cham

pio

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wit

hp

oin

tsre

set.

“W

eight”

isth

ew

eighti

ng

of

Fed

ExC

up

poin

tsaw

ard

edp

erto

urn

am

ent

finis

hin

gp

osi

tion

duri

ng

the

Pla

yoff

sre

lati

ve

toth

ose

award

edduri

ng

the

regula

rse

aso

n.

Each

effici

ency

mea

sure

reflec

tsth

em

ean

valu

eco

mpute

dov

er40,0

00

sim

ula

tion

tria

ls,

5,0

00

per

yea

rfo

rea

chyea

r2003-2

010.

For

each

effici

ency

mea

sure

exce

pt

the

firs

t,a

low

erva

lue

isb

ette

r.T

he

bes

tva

lue

per

stage

issh

own

inb

old

.V

alu

esin

each

stage

that

are

not

signifi

cantl

yin

feri

or

stati

stic

ally

ina

one-

sided

test

at

the

0.0

5le

vel

rela

tive

toth

eopti

mal

valu

efo

rth

esa

me

stage

are

show

nin

italics

.T

he

opti

mal

stage-

4va

lue

wit

hno

poin

tsre

set

isalw

ays

signifi

cantl

yb

ette

rat

the

0.0

5le

vel

than

the

opti

mal

valu

ew

ith

ap

oin

tsre

set

exce

pt

wher

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dw

ith

an

ast

eris

k,

inw

hic

hca

seth

eopti

mal

valu

ew

ith

rese

tis

bet

ter.

22

Tab

le3:

Effi

cien

cyM

easu

res

Com

pu

ted

Incr

emen

tall

yp

erP

layo

ffs

Sta

ge

Panel

A:

Fir

stP

lace

Rate

of

Bes

tP

layer

Panel

D:

Mea

nSquare

dR

ank

Err

or

(Defl

ate

d)

Wei

ght

Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

11.0

44

1.0

45

1.0

42

1.0

35

0.782

0.9

51

0.9

47

0.9

44

0.9

42

0.919

21.0

77

1.0

64

1.0

58

1.0

46

0.7

51

0.9

31

0.9

26

0.9

23

0.9

31

0.9

26

31.085

1.0

73

1.0

57

1.0

44

0.7

51

0.9

24

0.9

18

0.9

14

0.9

27

0.9

30

41.0

60

1.0

75

1.0

67

1.0

49

0.7

64

0.923

0.915

0.9

10

0.9

25

0.9

31

51.0

04

1.101

1.074

1.057

0.785

0.9

28

0.915

0.908

0.923

0.9

30

Panel

B:

Mea

nSkill

of

Pla

yer

inF

irst

Pla

ceP

anel

E:

Mea

nSquare

dSkill

Err

or

(Defl

ate

d)

Wei

ght

Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

10.9

99

0.9

99

0.9

99

0.9

99

1.002

0.9

43

0.9

37

0.9

31

0.9

28

0.987

20.9

99

0.9

99

0.9

99

0.9

99

1.0

02

0.9

21

0.9

15

0.9

07

0.9

11

1.0

11

30.998

0.9

99

0.9

99

0.9

99

1.0

02

0.916

0.909

0.9

01

0.9

08

1.0

10

40.9

98

0.9

99

0.9

99

0.9

99

1.0

02

0.9

23

0.9

10

0.8

98

0.9

06

0.9

99

50.9

99

0.999

0.999

0.999

1.0

02

0.9

37

0.9

11

0.896

0.902

0.986

Panel

C:

Mea

nSkill

Rank

of

Pla

yer

inF

irst

Pla

ceP

anel

F:

Money

-Wei

ghte

dSquare

dSkill

Err

or

(Defl

ate

d)

Wei

ght

Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

10.8

12

0.8

39

0.8

68

0.896

0.937

0.8

81

0.8

82

0.8

84

0.8

80

1.003

20.7

28

0.7

90

0.8

40

0.895

0.9

97

0.8

21

0.8

36

0.8

43

0.852

1.0

72

30.6

87

0.775

0.837

0.9

00

1.0

12

0.7

94

0.822

0.836

0.854

1.0

92

40.6

66

0.777

0.8

40

0.898

1.0

07

0.790

0.8

29

0.838

0.855

1.0

80

50.657

0.7

83

0.8

45

0.897

0.9

97

0.8

03

0.8

34

0.8

42

0.854

1.0

56

Sta

ge

1=

end

of

firs

tP

layoff

sro

und

(Barc

lays)

;Sta

ge

2=

end

of

seco

nd

Pla

yoff

sro

und

(Deu

tsch

eB

ank);

Sta

ge

3=

end

of

thir

dP

layoff

sro

und

(BM

W);

Sta

ge

4N

R=

end

of

finalP

layoff

sR

ound

(TO

UR

Cham

pio

nsh

ip)

wit

hno

poin

tsre

set;

Sta

ge

4R

=en

dof

final

Pla

yoff

sR

ound

(TO

UR

Cham

pio

nsh

ip)

wit

hp

oin

tsre

set.

“W

eight”

isth

ew

eighti

ng

of

Fed

ExC

up

poin

tsaw

ard

edp

erto

urn

am

ent

finis

hin

gp

osi

tion

duri

ng

the

Pla

yoff

sre

lati

ve

toth

ose

award

edduri

ng

the

regula

rse

aso

n.

Each

effici

ency

mea

sure

reflec

tsth

em

ean

valu

eof

the

rati

oof

the

effici

ency

mea

sure

com

pute

dat

the

end

of

the

stage

toth

eeffi

cien

cym

easu

reco

mpute

dat

the

beg

innin

gof

the

stage

for

stage

part

icip

ants

only

over

40,0

00

sim

ula

tion

tria

ls,

5,0

00

per

yea

rfo

rea

chyea

r2003-2

010.

For

each

effici

ency

mea

sure

exce

pt

the

firs

t,a

low

erva

lue

isb

ette

r.T

he

bes

tva

lue

per

stage

issh

own

inb

old

.V

alu

esin

each

stage

that

are

not

signifi

cantl

yin

feri

or

stati

stic

ally

ina

one-

sided

test

at

the

0.0

5le

vel

rela

tive

toth

eopti

mal

valu

efo

rth

esa

me

stage

are

show

nin

italics

.T

he

opti

mal

stage-

4va

lue

wit

hno

poin

tsre

set

isalw

ays

signifi

cantl

yb

ette

rat

the

0.0

5le

vel

than

the

opti

mal

valu

ew

ith

ap

oin

tsre

set

exce

pt

wher

enote

dw

ith

an

ast

eris

k,

inw

hic

hca

seth

eopti

mal

valu

ew

ith

rese

tis

bet

ter.

23

Tab

le4:

Effi

cien

cyM

easu

res

wit

hR

and

omT

ourn

amen

tO

utc

omes

Panel

A:

Fir

stP

lace

Rate

of

Bes

tP

layer

Panel

D:

Mea

nSquare

dR

ank

Err

or

(Defl

ate

d)

Met

hod

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Bes

t0.4

92

0.5

34

0.5

73

0.6

06

0.6

33

0.4

59

0.7

96

0.7

35

0.6

91

0.6

68

0.6

65

0.6

65

Wors

t0.4

92

0.4

94

0.5

37

0.5

60

0.5

80

0.4

37

0.7

96

0.7

57

0.7

30

0.7

15

0.7

13

0.7

12

Random

0.0

06

0.0

08

0.0

08

0.0

08

0.0

08

0.0

08

1.9

69

1.9

69

1.9

73

1.9

75

1.9

75

1.9

75

Panel

B:

Mea

nSkill

of

Pla

yer

inF

irst

Pla

ceP

anel

E:

Mea

nSquare

dSkill

Err

or

(Defl

ate

d)

Met

hod

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Bes

t68.8

23

68.7

07

68.6

20

68.5

55

68.5

07

68.7

21

0.6

89

0.6

31

0.5

94

0.5

70

0.5

61

0.5

81

Wors

t68.8

23

68.7

61

68.7

04

68.6

54

68.6

11

68.7

77

0.6

89

0.6

49

0.6

23

0.6

06

0.5

98

0.6

13

Random

70.6

64

70.6

54

70.6

53

70.6

52

70.6

51

70.6

51

1.9

67

1.9

69

1.9

74

1.9

75

1.9

75

1.9

75

Panel

C:

Mea

nSkill

Rank

of

Pla

yer

inF

irst

Pla

ceP

anel

F:

Money

-Wei

ghte

dSquare

dSkill

Err

or

(Defl

ate

d)

Met

hod

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Sta

ge

0Sta

ge

1Sta

ge

2Sta

ge

3Sta

ge

4N

RSta

ge

4R

Bes

t7.8

14

5.1

34

4.0

46

3.4

59

3.1

70

3.4

52

0.5

08

0.4

01

0.3

40

0.2

98

0.2

73

0.3

41

Wors

t7.8

14

6.3

42

5.3

69

4.7

66

4.4

55

4.5

82

0.5

08

0.4

47

0.4

03

0.3

69

0.3

47

0.3

94

Random

65.3

62

64.8

81

64.8

86

64.8

00

64.8

01

64.8

01

3.2

69

3.2

47

3.2

48

3.2

46

3.2

45

3.2

45

Sta

ge

0=

end

of

regula

rse

aso

n;

Sta

ge

1=

end

of

firs

tP

layoff

sro

und

(Barc

lays)

;Sta

ge

2=

end

of

seco

nd

Pla

yoff

sro

und

(Deu

tsch

eB

ank);

Sta

ge

3=

end

of

thir

dP

layoff

sro

und

(BM

W);

Sta

ge

4N

R=

end

of

final

Pla

yoff

sR

ound

(TO

UR

Cham

pio

nsh

ip)

wit

hno

poin

tsre

set;

Sta

ge

4R

=en

dof

final

Pla

yoff

sR

ound

(TO

UR

Cham

pio

nsh

ip)

wit

hp

oin

tsre

set.

“M

ethod”

isth

em

ethod

by

whic

hva

lues

are

com

pute

d,

wit

h“O

pti

mal”

den

oti

ng

the

opti

mal

valu

efr

om

the

corr

esp

ondin

gpanel

of

Table

2,

“W

ors

t”den

oti

ng

the

wors

tva

lue

from

the

corr

esp

ondin

gpanel

of

Table

2and

“R

andom

”den

oti

ng

the

valu

ew

hen

all

tourn

am

ent

outc

om

esare

det

erm

ined

random

lyusi

ng

aP

layoff

sw

eight

of

3.

Each

effici

ency

mea

sure

reflec

tsth

em

ean

valu

eco

mpute

dov

er40,0

00

sim

ula

tion

tria

ls,

5,0

00

per

yea

rfo

rea

chyea

r2003-2

010.

For

each

effici

ency

mea

sure

exce

pt

the

firs

t,a

low

erva

lue

isb

ette

r.

24

Tab

le5:

Fed

ExC

up

Win

nin

gP

erce

nta

ges

by

Pla

yoff

sSee

din

gP

osit

ion

and

Rel

ativ

eS

kil

lR

ank

Panel

A:

Posi

tion

Base

don

Pla

yoff

sSee

din

gP

osi

tion

Panel

B:

Posi

tion

Base

don

Rel

ati

ve

Skill

Rankin

gs

Wit

hR

eset

Wit

hout

Res

etW

ith

Res

etW

ithout

Res

etP

osi

tion

Wei

ght

=1

Wei

ght

=5

Wei

ght

=1

Wei

ght

=5

Wei

ght

=1

Wei

ght

=5

Wei

ght

=1

Wei

ght

=5

139.2

35.6

81.8

52.8

43.7

45.9

58.0

61.8

216.6

14.5

10.8

14.4

14.7

13.1

20.1

14.4

39.8

8.0

3.1

6.7

8.7

8.2

6.6

6.1

46.8

5.5

1.4

4.1

5.8

5.3

3.8

3.5

55.2

4.0

0.8

2.8

4.2

4.0

2.2

2.3

63.9

3.2

0.5

2.2

2.8

2.7

1.4

1.6

73.0

2.4

0.3

1.5

2.3

2.1

1.1

1.2

82.4

2.1

0.3

1.3

1.9

2.0

1.1

1.2

91.8

1.8

0.2

1.1

1.8

1.8

0.8

1.1

10

1.4

1.5

0.1

0.9

1.4

1.4

0.7

0.7

11

1.2

1.4

0.1

0.8

1.1

1.2

0.5

0.7

12

1.0

1.3

0.1

0.7

1.0

1.0

0.4

0.5

13

0.8

1.1

0.1

0.7

0.9

0.8

0.4

0.4

14

0.7

1.0

0.0

0.5

0.7

0.8

0.3

0.4

15

0.7

1.0

0.1

0.6

0.7

0.7

0.3

0.3

16

0.6

0.8

0.0

0.5

0.6

0.6

0.2

0.3

17

0.5

0.7

0.0

0.4

0.6

0.6

0.2

0.3

18

0.5

0.7

0.0

0.4

0.6

0.6

0.2

0.3

19

0.4

0.6

0.0

0.3

0.4

0.5

0.2

0.3

20

0.4

0.6

0.0

0.3

0.5

0.5

0.2

0.2

Base

don

40,0

00

sim

ula

tion

tria

ls(8

,000

per

yea

rfo

ryea

rs2003-2

010).

“W

eight”

isth

ew

eighti

ng

of

Fed

ExC

up

poin

tsaw

ard

edp

erto

urn

am

ent

finis

hin

gp

osi

tion

duri

ng

the

Pla

yoff

sre

lati

ve

toth

ose

award

edduri

ng

the

regula

rse

aso

n.

InP

anel

B,

skill

rankin

gs

are

rela

tive

toth

e125

pla

yer

sin

the

Pla

yoff

s.If

dis

pla

yed

toa

pre

cisi

on

of

0.1

%,

all

entr

ies

bel

owp

osi

tion

91

would

equal

zero

.

25

Tab

le6:

Fin

ish

ing

Pos

itio

nP

erce

nta

geR

ates

by

Fin

als

See

din

gP

osit

ion

Panel

A:

Wit

hR

eset

,P

layoff

sW

eight

=1

Panel

C:

Wit

hout

Res

et,

Pla

yoff

sW

eight

=1

Fin

ishin

gP

osi

tion

Fin

ishin

gP

osi

tion

See

d1

23

45

67

89

10

12

34

56

78

910

143.8

37.0

16.7

2.4

0.1

0.0

0.0

0.0

0.0

0.0

92.7

6.9

0.4

0.0

0.0

0.0

0.0

0.0

0.0

0.0

216.6

21.0

39.0

20.5

2.9

0.1

0.0

0.0

0.0

0.0

5.5

80.8

12.4

1.3

0.1

0.0

0.0

0.0

0.0

0.0

39.8

7.8

18.5

40.0

20.7

3.2

0.1

0.0

0.0

0.0

1.2

7.6

71.1

17.5

2.4

0.2

0.0

0.0

0.0

0.0

47.1

5.2

5.6

21.4

39.5

18.5

2.6

0.1

0.0

0.0

0.4

2.5

8.8

63.3

21.0

3.5

0.4

0.0

0.0

0.0

55.5

3.1

3.1

5.5

21.4

37.2

20.3

3.5

0.2

0.0

0.1

1.1

3.3

8.5

57.2

24.0

5.0

0.7

0.1

0.0

63.7

2.8

2.1

3.0

4.7

15.6

36.7

25.9

5.2

0.3

0.1

0.5

1.6

3.6

8.4

52.4

26.1

6.3

0.9

0.1

72.7

1.3

2.2

1.8

2.9

3.8

14.9

37.8

26.6

5.6

0.0

0.3

0.9

2.2

3.6

7.9

48.1

27.9

7.7

1.3

82.2

1.0

1.1

1.8

1.5

2.3

3.4

13.3

32.4

29.6

0.0

0.1

0.6

1.3

2.1

3.5

8.0

44.8

28.9

8.7

91.8

1.1

0.5

1.4

1.2

1.0

2.0

3.3

9.5

26.5

0.0

0.1

0.3

0.8

1.5

2.2

3.3

7.8

41.2

30.0

10

1.1

1.3

0.2

0.5

1.4

1.1

0.7

1.9

2.9

5.6

0.0

0.0

0.2

0.6

1.0

1.5

2.0

3.2

7.9

38.9

Panel

B:

Wit

hR

eset

,P

layoff

sW

eight

=5

Panel

D:

Wit

hout

Res

et,

Pla

yoff

sW

eight

=5

Fin

ishin

gP

osi

tion

Fin

ishin

gP

osi

tion

See

d1

23

45

67

89

10

12

34

56

78

910

145.8

36.2

15.6

2.2

0.1

0.0

0.0

0.0

0.0

0.0

79.4

16.7

3.4

0.5

0.1

0.0

0.0

0.0

0.0

0.0

214.5

20.8

41.3

20.5

2.7

0.1

0.0

0.0

0.0

0.0

9.6

50.4

28.8

9.1

1.7

0.3

0.0

0.0

0.0

0.0

39.2

6.9

18.2

41.2

21.0

3.3

0.1

0.0

0.0

0.0

3.9

9.6

34.4

33.9

14.2

3.4

0.6

0.1

0.0

0.0

46.8

4.7

5.3

20.8

40.5

19.0

2.8

0.1

0.0

0.0

2.2

5.2

7.7

26.1

34.7

17.9

5.1

1.0

0.1

0.0

55.5

3.3

3.2

5.1

19.7

37.6

21.4

4.0

0.2

0.0

1.3

3.6

4.4

6.6

21.6

33.2

20.2

7.1

1.6

0.3

63.9

3.0

2.0

3.0

4.8

14.2

35.5

27.4

5.9

0.3

1.0

2.7

3.2

3.6

6.1

18.6

30.8

21.7

8.9

2.7

72.7

1.4

2.4

1.9

2.9

3.8

14.0

36.8

27.5

6.2

0.6

2.0

2.4

2.7

3.5

5.7

16.6

28.4

22.6

10.7

82.3

1.2

1.2

1.8

1.5

2.4

3.3

12.4

31.1

30.6

0.5

1.6

1.9

1.9

2.3

3.3

5.9

15.2

26.3

23.1

91.9

1.1

0.5

1.5

1.2

1.0

2.1

3.5

9.1

24.9

0.3

1.3

1.5

1.6

1.9

2.4

3.2

5.9

14.0

24.5

10

1.3

1.4

0.2

0.5

1.5

1.2

0.7

2.0

3.0

5.4

0.2

1.1

1.3

1.3

1.4

2.0

2.4

3.3

5.9

13.4

Base

don

40,0

00

sim

ula

tion

tria

ls(8

,000

per

yea

rfo

ryea

rs2003-2

010).

“P

layoff

sW

eight”

isth

ew

eighti

ng

ofF

edE

xC

up

poin

tsaw

ard

edp

erto

urn

am

ent

finis

hin

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