Electronic copy available at: http://ssrn.com/abstract=1916234

Tuck School of

Business at Dartmouth

Tuck School of Business Working Paper No. 2011-96

Tournament Qualification, Seeding

and Selection Efficiency: An Analysis of the

PGA TOUR's FedExCup

Robert A. Connolly

University of North Carolina (UNC) at Chapel Hill – Finance Area

Richard J. Rendleman

Tuck School of Business at Dartmouth

July 15, 2011

This paper can be downloaded from the

Social Science Research Network Electronic Paper Collection:

http://ssrn.com/abstract=1916234

Electronic copy available at: http://ssrn.com/abstract=1916234Electronic copy available at: http://ssrn.com/abstract=1916234

Tournament Qualification, Seeding and Selection Efficiency: An

Analysis of the PGA TOUR’s FedExCup1

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

July 15, 2011

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 ShotLink data used in connection with this study, Pranab Sen forsuggesting the Spearman Footrule analysis, and Nicholas Hall and Dmitry Ryvkin for helpful comments.Please address comments to Robert Connolly (email: robert connolly@unc.edu; phone: (919) 962-0053) orto Richard J. Rendleman, Jr. (e-mail: richard rendleman@unc.edu; phone: (919) 962-3188).

Electronic copy available at: http://ssrn.com/abstract=1916234Electronic copy available at: http://ssrn.com/abstract=1916234

Tournament Qualification, Seeding and Selection Efficiency: An Analysis of

the PGA TOUR’s FedExCup

Abstract

Analytical descriptions of tournament selection efficiency properties are elusive for realistic

tournament structures. For example, with more than four competitors, there are very few robust

analytical tournament selection efficiency measures. Combining a Monte Carlo simulation with

a statistical model of player skill and random variation in scoring, we estimate the seeding and

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 along with several new measures

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. Using

these measures, we also estimate the relative efficiency of several alternative formats for the FedEx-

Cup Finals. We conclude that despite its lack of transparency compared with alternative Finals

formats, the present Finals format appears to be the most efficient using all selection efficiency

measures.

Electronic copy available at: http://ssrn.com/abstract=1916234

1. Introduction

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

petition involving a regular season followed by a series of playoff rounds and a “finals” event, where

an overall champion is crowned. We focus on two distinctly different aspects of the competition.

First, we assess how well the initial playoff rounds perform in getting the right players into the

Finals and placing them into appropriate seeding positions, a process we call “seeding efficiency.”

Second, once the Finals begin, we estimate how well its structure (and several alternatives) per-

form in placing players in proper finishing positions, a process termed “selection efficiency” in the

tournament literature. We believe this is the first study to examine simultaneously the efficiency of

a tournament qualification and seeding process and the selection efficiency of the tournament itself

in an actual competitive setting.

Existing research on the effects of tournament seeding schemes appears to generate few results

that generalize to typical, real-world tournament settings. For example, Horen and Reizman (1985)

study the properties of seeding mechanisms in a setting where they place only loose restrictions on

the matrix of winning probabilities. Their paper evaluates the impact of seeding mechanisms with

four criteria: 1) does the mechanism maximize the probability that the best team wins? 2) is it

fair in the sense that a better team has a higher probability of winning? 3) does it maximize the

probability that the two best teams meet in the finals? and 4) does it maximize the expected value

of the winning team? When there are only four teams, there is an unique seeding mechanism that

satisfies all four criteria, the traditional (and very familiar) 1 vs. 4 and 2 vs. 3 seeding arrangement.

They show that when there are eight teams, there is no matrix of winning probabilities consistent

with a single, best seeding mechanism, even when relying only on the first two criteria.

Groh, Moldovanu, Sela, and Sunde (forthcoming) study the impact of different seeding mech-

anisms in an elimination tournament with endogenous effort. Using criteria similar to Horen-

Reizman for assessing the performance of the combined seeding-tournament structure, they confirm

that the traditional 1 vs. 4 and 2 vs. 3 seeding performs best in a four-person tournament. They

also find that there are no general results on seeding efficiency for tournaments with more than four

participants, and express the view that while it is possible in principal to study more complicated

structures, “the exponentially growing number of seedings and the complexity of the fixed-point

1

arguments suggest that analytic solutions are difficult to come by (pg. 14).”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 tourna-

ment 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 rank-

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

tournaments.

While we use their new efficiency measures in our work, we also propose several new measures of

overall tournament selection efficiency. We broaden the focus of the tournament selection efficiency

literature by studying the ability of tournament formats to identify properly the relative skills of all

participants and to distribute efficiently all of the tournament’s prize money.3 We apply a subset of

these measures to estimate whether the initial FedExCup playoff rounds preceding the FedExCup

Finals add incremental value to player ordering as determined at the end of the regular PGA

TOUR season and, therefore, improve player qualifying and seeding for the Finals. In addition to

our analysis of selection efficiency of the FedExCup Finals as presently structured, we also extend

our analysis to consider four additional tournament formats.

Our paper contributes to the literature on tournament efficiency on several other dimensions.

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 where we can explore the effects

1There is another dimension of the seeding efficiency problem that arises in multi-stage tournaments. Hwang(1982) shows that reseeding competitors in a multi-stage elimination tournament will maximize the likelihood ofthe best players ultimately winning the tournament. The new seedings are to be based on rankings of the playersactually in the round, not on the seedings used in previous rounds. A version of this reseeding approach is used insome professional sports (e.g., the National Hockey League playoffs), but this is not a universal approach, and itsproperties don’t appear to be been explored systematically in real-world tournaments. Premised on a set of axiomsdescribing “fair seeding,” Schwenk (2000) suggests use of a cohort-based randomized seeding that distributes collegebasketball teams randomly to each side of the NCAA tournament bracket based on their position in ranking cohorts.While this system satisfies the axioms, there is no additional analysis of the properties of this seeding system.

2See Ryvkin and Ortmann (2008) for an excellent recap of existing work along these lines.3As we discuss in the next section of the paper, only 28.6% of the FedExCup prize money goes to the winner, so

it doesn’t follow that the tournament sponsors would be obviously indifferent about the distribution of the remaining71.4% of the prize money.

2

of both the tournament qualification (seeding) process and the structure of the tournament playoff

and finals. 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.

Beyond the analysis of this specific tournament structure, we believe that a major contribu-

tion of this paper lies in integrating the analysis of seeding and selection efficiency of a complex,

real-world tournament setting together with direct empirical modeling of the underlying skill dis-

tributions that describe the probability basis of tournament outcomes. As such, we believe the

general approach pursued here may prove of value for other real-world tournament problems where

analytical solutions aren’t readily available.

The remainder of our paper is organized as follows. In the next section, we provide an overview

of the FedExCup competition, including a discussion of criticisms of the tournament structure

and several proposals for revamping the tournament structure. We evaluate these alternatives

later in the paper. In Section 3, we describe the data and statistical method we employ to model

player skill and random variation in scoring. We provide a broad view of our simulation design in

Section 4, supplemented by an appendix, which provides the details, beginning with the FedExCup

qualification process and going through the Playoffs. In Section 5 we summarize our simulation-

based findings as they relate to the qualifying and seeding efficiency of the events leading to the

FedExCup Finals. In Section 6, we evaluate the selection efficiency of alternative FedExCup Finals

formats using measures proposed by Ryvkin and Ortmann as well as new measures of efficiency,

which we develop. Finally, we evaluate the efficiency of the entire FedExCup competition as a

large-scale tournament, as summarized in Section 7. Summary and conclusions follow.

2. FedExCup Competition

In 2007, the PGA TOUR began regular season and playoffs competition for the FedExCup, a

season long event that distributes a total of $35 million in prize money to 150 players, with those

in the top five finishing positions earning $10 million, $3 million, $2 million, $1.5 million and $1

million, respectively.4 Under current FedExCup rules, similar in structure to NASCAR’s Sprint

4See http://www.pgatour.com/r/stats/info/?02396.

3

Cup points system, PGA TOUR members accumulate FedExCup points during the 35-week regular

PGA TOUR season.5 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 18-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 Barclays,

the first event in the Playoffs, reduced again to 70 after the Deutsche Bank Championship, and

reduced again to 30 after the BMW Championship. At the conclusion of the BMW, 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, which is otherwise a regular

18-hole PGA TOUR event. The player who has accumulated the greatest number of FedExCup

points after THE TOUR Championship wins the FedExCup.6

2.1. Alternative Structures for the FedExCup Finals (THE TOUR Champi-onship)

The present FedExCup system, especially as applied to the final stage of competition, is often

criticized for its lack of transparency. This lack of transparency is evident from the comments of

2010 FedExCup winner Jim Furyk, who upon completing THE TOUR Championship, knew he

had won the tournament itself but was not sure if he had actually won the FedExCup. “‘Did I win

the bonus?’ he asked his wife, Tabitha, as they hugged on the 18th green. ‘I have no idea,’ she

said” (Micheaux (2010)).7 Obviously, there is a fundamental problem with a system of competition

that awards $10 million to the winner when the competitors themselves have only a vague idea

where they stand or what they need to do to win during the heat of competition. Although the

final points reset guarantees that any player among the top five in FedExCup points who wins

THE TOUR Championship will also win the FedExCup, when a top-five player does not win THE

5The rules associated with FedExCup competition have been changed twice. Detail about the revisions is presentedin Hall and Potts (2010).

6A primer on the structure and point accumulation and reset rules may also be found athttp://www.pgatour.com/fedexcup/playoffs-primer/index.html.

7See Micheaux (2010) for further commentary on the inability of players to determine their status and theirpotential effect on the performance of others during the 2010 FedExCup Finals.

4

TOUR Championship, as in 2010, it is very difficult to determine who will actually win the Cup.

Beyond the transparency problems associated with the present TOUR Championship structure,

the more fundamental issue is whether there might be an alternative structure that would be more

“selection efficient” from a purely mathematical standpoint. To this end, we study four alternative

Finals formats, where, for each format, it is perfectly clear what any player in the competition

would need to do to win. Our first alternative structure has been suggested and studied by Hall

and Potts (2010). Rather than conduct THE TOUR Championship with the top 30 players in

FedExCup points, Hall and Potts propose a four-day competition limited to the top 28 players.

The competition would involve eight rounds of 18-hole match play, with players seeded according

to accumulated FedExCup points, with one round conducted each morning and another in the

afternoon.8 Unlike conventional match play competition, the first round would be limited to those

seeded 21-28. The 21st seeded player would be paired against player 28, the 22nd seeded player

would play number 27, etc. At the completion of the first round, the lowest seeded remaining player

would be paired against player number 17 in round 2.9 The next lowest seeded remaining player

would compete against player number 18, etc. This process would continue through round 5, with

players seeded 1-4 entering the competition in round 6. The final eight players would continue

match play for three more rounds, with the winner of the match play competition being declared

the FedExCup winner. We approximate Hall-Potts match play by assuming that the player with

the lower 18-hole score in any head-to-head matchup is the winner.

The second alternative structure for the FedExCup Finals involves four rounds of stroke play,

where the winner of THE TOUR Championship wins the FedExCup. The analysis of stroke play

enables us to determine the extent to which the pre-loading of FedExCup points going into THE

TOUR Championship affects the ultimate FedExCup outcome relative to the outcome associated

with the actual TOUR Championship competition itself.

The third alternative format, which we refer to as binary 1, involves five rounds of 18-hole

binary competition among 32 TOUR Championship qualifiers. Players are selected and seeded for

binary competition based on FedExCup points accumulated prior to THE TOUR Championship,

8In golf, the winner in match play competition between two players is the player who wins the greater number ofholes after 18 holes of play. If the players are tied after 18 holes, they enter into a sudden death playoff.

9Throughout, we will refer to a player in a low-numbered seeding position as a “high seeded” player and one in ahigh-numbered seeding position as a “low seeded player.” Although this terminology is confusing, it conforms withthe actual usage of these terms.

5

but otherwise, FedExCup points do not come into play. In each round of binary competition, the

highest seeded remaining player is paired against the lowest seeded remaining player and similarly

for other seeding positions. The winner of each match is the player with the lower 18-hole score. We

refer to binary 1 as a “reseeding” competition, since pairings in round t+ 1 cannot be determined

until round t is completed. We note that the Hall-Potts proposal also involves reseeding. The

final alternative format, binary 2, is identical to binary 1, except players are placed into permanent

tournament brackets. (With permanent brackets, in the first round player 1 is paired against

player 32, 2 against 31, etc. In the second round, the winner of the 1 vs. 32 match plays the 16

vs. 17 match winner, etc.) The binary 2 format parallels conventional match play in golf, except

in conventional match play, the winner is the player who wins the greater number of holes rather

than the player who shoots the lower 18-hole score.

We note that in all of the alternative Finals formats, each player controls his own destiny.

However, under the present FedExCup Finals format, only the top five players going into the finals

control their destiny.

For analytical purposes, we adjust the number of participating players for the Hall-Potts, present

and stroke play formats so that they all involve 32 players, the same as in binary competition. As

such, the adjusted Hall-Potts format involves nine rounds of binary play rather than eight rounds

of match play as actually proposed. The adjustments eliminate problems in comparing efficiency

measures across tournament formats that do not involve the same number of players. We assign 205

and 200 FedExCup reset points to the 31st and 32nd seeded players, respectively, leading into THE

TOUR Championship. Given the schedule of points awarded to TOUR Championship finishing

positions 1-32, this reset adjustment has the effect of giving the last two players some chance, albeit

a very small chance, to win the FedExCup.10

2.2. Analytical Approach

Compared with straight binary competition and other structures that have been studied in the

tournaments literature, FedExCup competition from beginning to end is orders of magnitude more

complex. The PGA TOUR employs a priority system that determines the eligibility of TOUR

10The Playoffs points distribution schedule can be obtained athttp://www.pgatour.com/2008/fedexcup/11/25/2009changes.chartplayoff/index.html.

6

members to compete in its various events. Moreover, the four “majors,” and the tournaments in the

World Golf Championship series, employ strict eligibility requirements that reflect player positions

in Official World Golf Rankings, positions on the Official PGA TOUR Money List, and similar

criteria. There are also a handful of high-profile invitational events, such as The Arnold Palmer

Invitational (Bay Hill) and The Memorial (sponsored by Jack Nicklaus) that limit participation to

golf’s most elite players.

During the regular PGA TOUR season, no individual player would ever play in all PGA TOUR

sanctioned events. For top-tier players such as Tiger Woods and Phil Mickelson, the tournaments

in which they compete is a matter of choice. By contrast, those among the PGA TOUR lower-

ranked players will typically attempt to compete in any tournament in which they are eligible. For

those in the middle, participation in low-profile tournaments tends to be a matter of choice, while

participation in high-profile events tends to be a question of eligibility.

Although there are exceptions, a typical PGA TOUR event involves 156 players competing in

four rounds of stroke play, with a cut after the second round, where only the players in positions

1-70 (and ties) after round two get to continue for the last two rounds. The tournaments in the

first two stages of the FedExCup Playoffs employ cuts, whereas the last two events, which include

a total of 70 and 30 participants, respectively, do not employ cuts. Whether a regular season or

playoff event, only those who make cuts earn FedExCup points.

The schedule of FedExCup points applied during the regular season and Playoffs is highly non-

linear. Moreover, points earned for Playoffs events are five times those earned for regular season

tournaments. The mathematics of FedExCup points accumulation is made even more complex by

the points reset that occurs immediately before the FedExCup Finals.

3. Statistical Foundations

3.1. Data

Our data, derived from ShotLink, and provided by the PGA TOUR, covers the 2003-2009 PGA

TOUR seasons. It includes 18-hole scores for every player in every stroke play event sanctioned by

the PGA TOUR for years 2003-2009 for a total of 133,645 scores distributed among 1,731 players.

As in Connolly-Rendleman (2008), we limit the sample to players who recorded more than 90

7

scores. The resulting sample consists of 119,060 observations of 18-hole golf scores for 354 active

PGA TOUR players over 321 stroke-play events. As we describe in Connolly and Rendleman

(2008), most of the omitted players are not representative of typical PGA TOUR players. For

example, 643 of the omitted players recorded only two 18-hole scores.11 By excluding players with

90 or fewer scores, we maximize the power of our methodology for estimating player skill and

minimize potential distortions in estimating the statistical properties of scoring for regular PGA

TOUR players.

3.2. Statistical Model

We employ the model of Connolly and Rendleman (2008) to estimate a cubic spline-based skill

function for each of the 354 players in our sample, while simultaneously estimating random round-

course effects, random player-course effects, and first-order autocorrelation in residual player scores.

Random round-course effects reflect variation in scoring due to the relative difficulty of specific

courses on the days tournament rounds are played. Random player-course effects reflect variation

in scoring due to courses playing favorably or unfavorably for individual players. We refer to

residual scores adjusted for first-order autocorrelation as η errors and unadjusted residual scores

as θ errors.

In fitting the model, the pseudo adjusted R2 equals 0.293, compared with 0.296 in the original

Connolly-Rendleman (2008) sample covering 1998-2001.12 Estimated random round-course effects

range from −4.46 to 7.70 strokes per round (−3.92 to 6.95 strokes per round in the original sam-

ple) and by construction, sum to zero.13 By contrast, estimated random player-course effects are

essentially zero across-the-board in the 2003-2009 sample (−0.065 to 0.044 strokes per round in the

original 1998-2001 sample) and contribute very little to variation in scoring.14

11Generally, 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.

12This is computed as 1 −mean square error/mean square total.13Inasmuch as we model round-course effects as random effects, rather than fixed effects, we make no claim that

the range of these effects is statistically significant. Instead, we present this range as an indication of the variationin scoring associated with the round-course interactions in our sample.

14We estimate the Connolly-Rendleman (2008) model in two stages. In the first stage, we assume no first-orderautocorrelation in residual errors about each player’s estimated cubic spline. At the end of this stage, estimatedplayer-course effects range from −0.334 to 0.247 strokes per round in the 2003-2009 sample. The second stage beginswith solution values from the first stage and then assumes that residual errors about each player’s fitted spline followan AR(1) process. In this stage, all estimate player-course effects converge to zero in the 2003-2009 sample. First-order autocorrelation in individual player residual errors appears to subsume any temporal patterns in individualplayer-course effects. This does not occur in the 1998-2001 sample, nor does it occur in a separate 2004-2008 sample,

8

Throughout, we focus on the properties of FedExCup competition as conducted with “neutral”

player scores – scores reduced by estimated round-course and player-course effects. As such, neu-

tral scores provide an estimate of what a player’s score would have been after adjusting for the

relative difficulty of the round in which the score was recorded as well as any personal advantage

or disadvantage the player might have had when playing the course.

Figure 1 shows neutral-scoring-based spline fits for Tiger Woods and Aaron Baddeley. Actual

18-hole scores, reduced by random round-course and player-course effects, are plotted about the

smooth spline fits. The spline fit for Tiger Woods is slightly concave, reaching its minimum of 67.52

strokes per round at the end of the 2003-2009 sample period after having reached a maximum of

68.25 strokes approximately 30% of the way into the 2003-2009 sample period. The spline fit for

Baddeley has two distinct sections characterizing his estimated skill level over time, with maximum

and minimum estimated scores reaching 71.56 and 70.12, respectively. (Over all 354 players in the

2003-2009 sample, 162 spline fits are exactly linear.) “Scaled golf time,” specific to each player,

represents the chronological sequence of rounds for the player scaled to the {0, 1} interval. Although

both players participated on the PGA TOUR during roughly the same period of time, the scaled

golf times for each player represent the sequencing of their own individual scores rather than the

joint sequencing of scores.

4. Simulation Design

4.1. Overview

We structure each of 28,000 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 354

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

between 278 and 300 and reflects the actual number of players in the sample in each year, 2003-

2009. We also structure the simulations so that the simulated distributions of player skill, scoring,

and player tournament participation rates during the simulated regular season closely approximate

those observed in the actual sample. In each simulation trial, the five alternative structures for the

which we have used in other projects prior to obtaining the ShotLink data. Therefore, we believe the convergence ofplayer-course effects to zero reflects the unique characteristics of the 2003-2009 Shotlink data.

9

FedExCup finals are evaluated using the same simulated scoring sample. Simulation details are

provided in the Appendix.

4.2. Distributions of Player Skill and Scoring

Figure 2 shows distributions of simulated player skill and scoring applicable to the regular season

portion of the FedExCup simulations. Each distribution is plotted against a normal distribution

with the same mean and variance. Although the distribution of player skill is reasonably symmetric

and of the same general shape as a normal distribution, it is more peaked in the middle and has

a small peak in the extreme left tail near a mean skill level of 68 strokes per round. The left-side

peak generally represents the portion of the skill distribution pertaining to Tiger Woods. The mean

of Tiger Woods’ spline-based scoring estimates is 68.00, 1.19 strokes lower than that of the next

most highly-skilled player, Vijay Singh. A total of 13 players from the original 354 player sample

have mean spline-based scoring estimates less than 70, 94 have mean spline-based scoring estimates

between 70 and 71, while 187 have mean estimates between 71 and 72. We note that standard

goodness of fit tests reject player skill as being characterized by a normal distribution.

The distribution of player scores reflects the distribution of skill and player-specific random

variation in scoring. Although its general appearance is closer to that of a normal distribution

than the distribution of player skill, standard goodness of fit tests reject the overall distribution of

18-hole scores as being characterized by a normal distribution.

4.3. Treatment of Tiger Woods

One might reasonably argue that having a player as dominant as Tiger Woods on the PGA TOUR

is the exception rather than the rule and that “normal times” might be more characteristic of the

TOUR without Woods. We do not make that claim ourselves but offer an analysis of FedExCup

performance with and without Woods for those who might find the contrast to be of interest. When

Woods is omitted, we treat the simulation sample as if it consists of 353 players rather than 354.

We do not re-estimate our statistical model, as described in Section 3.2, when Woods is omitted.

10

4.4. Summary of Simulation Results for Selected Players

Table 1 summarizes the number of times selected players are included in various stages of FedEx-

Cup competition in 28,000 simulation trials, assuming qualification and seeding for THE TOUR

Championship is based on the accumulation of FedExCup points under the present system. Players

shown in Table 1 include the top ten players based on rankings of average spline-based skill, Aaron

Baddeley, whose spline estimate is illustrated along with that of Tiger Woods in Figure 1, and

several players with lower estimated mean skill (higher estimated mean scores).

The individual player FedExCup Playoff participation and winning rates are generally in line

with what one would expect. Tiger Woods wins the FedExCup in slightly over half of the simulation

trials, almost five times the winning rate of Vijay Singh, the player with the second most wins.

Phil Mickelson qualifies for the Playoffs at the highest rate, even higher than Woods, reflecting his

relatively high skill level and the fact that he plays in TOUR events more frequently than Woods,

thereby giving him an opportunity to earn more FedExCup points. Aaron Baddeley qualifies for

the Playoffs in approximately 73% of simulation trials, which reflects his lower mean skill level

relative to the top ten players and the temporal variation in his mean skill, as illustrated in Figure

1. The lack of success in making the Playoffs among the last five players shown in Figure 1 reflects

a combination of lower skill and less frequent participation in TOUR events (37 to 68 weeks on

Tour per player over the seven years, 2003-2009). It is interesting to note that Woods and Singh

were the only two players to have won the FedExCup during its first three years, 2007-2009. Jim

Furyk, who won the Cup in 2010, is the fifth most successful player in the simulations.

5. Efficiency of TOUR Championship Qualifying and Seeding

The FedExCup points system as structured through the regular season and the first three rounds

of the Playoffs can be viewed as a complex, multistage qualification and seeding process leading up

to the FedExCup Finals (THE TOUR Championship). To what extent does the qualification and

seeding system get the ‘right’ players into the Finals? Do the first three rounds of the Playoffs,

where FedExCup points earned per tournament finishing position are five times those applied per

finishing position during the 35-week regular season, improve the selection of qualifiers and seeding

for THE TOUR Championship or just add noise?

11

Similar to the common use of “predictive power” to evaluate tournament selection efficiency,

optimal seeding is typically viewed in terms of maximizing the probability that the most highly-

skilled player will win the tournament for which seeding is based. (See Ryvkin (2006), Hwang

(1982), Horen and Riezman (1985), Schwenk (2000), and Groh et al. (2010). Other seeding

criteria include maximizing the probability of a tournament final between the top two teams (binary

competition) and maximizing total tournament effort.) Inasmuch as the FedExCup distributes

prize money to 125 Playoffs participants and to the top 25 players who do not qualify for the

Playoffs, we believe more broad-based seeding efficiency measures, which reflect on more than

just the best player, are appropriate. More specifically, we view qualifying and seeding efficiency

as a qualifying scheme’s ability to order players according to true skill prior to entering actual

tournament competition. (Of course, if qualifying and seeding were perfect, there would be no

need to hold a tournament.) As such, we propose the following three measures of tournament

qualifying and seeding efficiency, which relate seeding position to player skill.

5.1. Qualifying and Seeding Efficiency Measures

Consider z1, a vector of sequential integers from 1 to 125, denoting true skill rankings among the

125 players who qualify for the FedExCup Playoffs and z2, a vector of true skill rankings among the

M = 125 qualifiers ordered by FedExCup point standings. Then the simple correlation between z1

and z2, equivalent to the Spearman rank order correlation, is our first measure of qualifying and

seeding efficiency.

Our second measure, Spearman’s footrule, is a commonly used measure of disarray. As applied

to skill rankings and the same rankings ordered by standings in FedExCup points, it provides

a measure of distance between the two sets of rankings: footrule = M−1M∑i=1|z1,i − z2,i|. Lower

footrule values imply greater qualifying and seeding efficiency.

Our third qualifying and seeding efficiency measure, the skill-based regression slope, is similar

to the Spearman rank order correlation, except it takes account of the degree to which player

skill differs across skill rankings. Let y1 denote the vector of ‘true’ mean player skill ordered

by skill level and y2 denote a vector of mean player skill ordered by FedExCup point standings.

Then, the skill-based regression slope is simply the slope of the OLS regression of y1 on y2 (or

vice versa). It can be shown that the sum of squared differences between y1 and y2 is given by

12

SS =∑

(y1 − y2)2 = 2 (1− β) var (y). Thus, when β = 1, SS is zero, implying that FedExCup

points order players perfectly by skill ranking. When β = 0, SS = 2var (y), implying there is no

linear statistical association between y1 and y2.

5.2. Performance of the Qualifying and Seeding Process

In each simulation trial, we evaluate the three efficiency measures for all 125 Playoffs participants

at the end of the regular season and, again, at the end of the third Playoff round, in an effort to

determine whether the first three rounds of the Playoffs improve player selection and seeding for

THE TOUR Championship. Panel A of Table 2 shows that the efficiency of qualifying and seeding

for the FedExCup Finals indeed improves over the first three rounds of the Playoffs. The Spearman

rank correlation improves in approximately 85% of simulation trials, Spearman’s footrule improves

in all trials, and the skill-based regression slope improves in approximately 74% of simulation trials,

with little difference when Tiger Woods is included and excluded from the analysis.

Panel B shows median values of each of the efficiency measures at the end of the regular

season and at the end of three subsequent playoffs rounds, immediately prior to THE TOUR

Championship. Median Spearman rank correlations increase by approximately 0.04 from values

of 0.63-0.64 at the end of the regular season. Median values of Spearman’s footrule show that

the mean absolute difference between player positions in the FedExCup standings and player skill

rankings is approximately 26 positions at the end of the regular season, improving to approximately

17 positions going into the FedExCup Finals. Median skill-based regression slopes also improve –

by a little more than 0.02.

On the basis of these tests, we conclude that the first three rounds of the FedExCup Playoffs

improve the efficiency of selecting the 32 qualifiers for the FedExCup Finals and determining their

seeding positions. Despite the fact that FedExCup points awarded during the Playoffs are five

times those awarded for regular season tournaments, the initial playoff rounds appear to improve

the selection and seeding of TOUR Championship qualifiers rather than make things worse.

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6. Tournament Selection Efficiency

6.1. Measures of Tournament Selection Efficiency Proposed by Ryvkin and Ort-mann

In this section we examine the selection efficiency of the 32-contestant TOUR Championship, the

final event in the FedExCup Playoffs, evaluated in terms of the present FedExCup system, the

Hall-Potts proposal, four rounds of stroke play and reseeded and bracketed binary competition.

We begin by focusing on 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.

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

3. The mean skill ranking of the FedExCup winner.

Note that two of the three efficiency measures make reference to player skill rankings. Since we

are evaluating the selection efficiency of the TOUR Championship, rather than the entire FedExCup

competition, the skill ranking to which we refer in the third of the Ryvkin-Ortmann measures is the

relative skill ranking among the 32 players in THE TOUR Championship. Table 3 summarizes the

Ryvkin-Ortmann efficiency measures as applied to the five tournament formats for the FedExCup

finals and a sixth format that produces a random ordering among FedExCup finalists.

The present FedExCup format produces the most favorable outcome for each of the Ryvkin-

Ortmann measures of efficiency evaluated with Tiger Woods included and excluded from the FedEx-

Cup competition. Moreover, for each efficiency measure, the relative ordering among the remaining

formats for the FedExCup finals is Hall-Potts, followed closely by four rounds of stroke play, and

not as closely by reseeded binary competition, bracketed binary and a random ordering of finalists.

Note that each of the efficiency measures is affected significantly by the inclusion or exclusion of

Tiger Woods from the competition, especially the winning rate of the most highly-skilled player,

falling from approximately 51% with Woods in the competition to approximately 25% when Woods

does not compete.

We are interested, not only in the values of each of the Ryvkin-Ortmann measures as computed

for each TOUR Championship format, but also the probability that any of six tournament structures

14

will produce a more favorable Ryvkin-Ortmann efficiency value than the others. Note that each

of the Ryvkin-Ortmann efficiency measures represents a mean value computed over all 28,000

simulation trials; we do not obtain a value of each measure on a trial-by-trail basis. Therefore,

we employ the bootstrap, applied to the 28,000 trials, to produce sample values of each Ryvkin-

Ortmann efficiency measure for all six tournament formats. Each bootstrap sample with Woods

included and excluded from the competition consists of 5,000 observations.

Consider the six tournament formats listed in the same order as in Table 3: present FedExCup

format, Hall-Potts, stroke play, binary 1, binary 2 and random ordering. Although not shown in the

table, with only one minor exception, there is no bootstrap sample in which a Finals format listed

on the right within the ordered list produces a more favorable Ryvkin-Ortmann efficiency measure

than any format on the left. This is the case with or without Tiger Woods in the competition.

Therefore, we conclude that the order of tournament selection efficiency based on the Ryvkin-

Ortmann measures is the present FedExCup format followed by, Hall-Potts, stroke play, binary 1

and binary 2.

6.2. New Measures of General Tournament Selection Efficiency

In this section we develop five new measures of general tournament selection efficiency that capture

the ability of a given TOUR Championship format to properly classify all tournament participants

according to their true skill levels, not just those who are the most highly skilled or highly seeded,

and to properly allocate tournament prize money. Since we are concerned with the classification

of all tournament participants, we must determine how to deal with ties when determining TOUR

Championship finishing positions associated with players who are eliminated prior to the final round

under the Hall-Potts and binary formats. For example, in binary competition involving 32 players,

16 will be eliminated in the first round. Should all 16 players be assigned the same tournament

finishing position or, almost equivalently, should their finishing positions be determined randomly?

Or, as a completely different alternative, should we employ additional information gathered in

connection with the competition to break ties among those who are eliminated in the same round?

We take the later approach by breaking ties on the basis of player seeding positions at the beginning

of the FedExCup Finals, equivalent to breaking ties on the basis of accumulated FedExCup points.

We also note that prize money is distributed to all participants in the FedExCup Finals, as

15

well as to all others who participate in the Playoffs and to the 25 players with the highest point

totals who do not qualify for the Playoffs. Thus, it is important not just to determine the overall

FedExCup winner as efficiently as possible but also to assign appropriate finishing positions to all

players who participate in the FedExCup Finals.

The first three general selection efficiency measures are the Spearman rank correlation, Spear-

man footrule and the skill-based regression slope, all evaluated with respect to the M = 32 TOUR

Championship participants. These measures are defined in Section 5.1 and, therefore, we do not

redefine them here.

Our final two measures of general tournament selection efficiency reflect, simultaneously, the

proper skill ordering of tournament finishing positions and the distribution of tournament prize

money. Suppose a given tournament structure properly classifies the most highly-skilled player,

Tiger Woods, relative to the second most highly-skilled, Vijay Singh, where there is roughly a

1.2 stroke per round difference in mean player skill, but does a poor job classifying players in the

right tail of the distribution, where differences in mean player skill might be only 0.02 strokes

per round. Does miss-classifying players for whom there is essentially no difference in mean skill

really matter? Moreover, in the FedExCup, the difference in prize money awarded to the first

and second-place finisher ($7 million) is far greater than the money difference for the next-to-last

and last-place finisher ($10,000, assuming 32 participants). Clearly, classifying the top tournament

finishers correctly would have far greater economic consequences than getting the bottom finishers

in correct order.

Let y1 denote a vector of mean player skill among M = 32 TOUR Championship participants

ordered by skill level, and y2 denote a vector of mean player skill ordered by TOUR Championship

finishing position. Consistent with our concern for getting the most highly-skilled players in THE

TOUR Championship classified and paid correctly, our fourth measure of general tournament selec-

tion efficiency, the weighted skill-based regression slope, is an OLS regression of y1 on y2 weighted

by x1, a vector of money payout weights to TOUR Championship participants ordered sequentially

from the highest to lowest weight.

Our final measure of general tournament selection efficiency is the CR statistic, computed as

CR = −M−1M∑i=1

(y1,i − y2,i) (x1,i − x2,i). The y portion of the statistic reflects the extent to which

the FedExCup finals gets player skill levels wrong, while the x portion of the statistic reflects the

16

extent to which the Finals miss-allocates tournament prize money. Noting that (y1,i − y2,i) and

(x1,i − x2,i) will always be of opposite sign, the negative sign preceding M within the equation

defining the CR statistic ensures that the statistic is non-negative. It can be shown that CR =

([M − 1] /M) (βy1,x2 + βy2,x1 − 2βy1,x1) s2x1

, where βy,x is the estimated OLS slope coefficient of a

regression of y on x and sx1 is the sample variance of x1 (or x2, since both variances are the same).

Note that when player skill rankings and tournament finishing positions are perfectly aligned,

βy1,x2 = βy2,x1 = βy1,x1 , and CR = 0.

Figure 3 shows cumulative density functions for all five general efficiency measures applied to

the FedExCup Finals, with Tiger Woods in the competition. Starting at the top and going from left

to right, for the first three efficiency measures, Spearman rank correlation, skill-based regression

slope and weighted regression slope, more favorable values for the cumulative density functions

plot to the right, whereas with the Spearman footrule and CR statistic, more favorable values plot

to the left. Cumulative density functions for reseeded binary competition are not shown, since

they are indistinguishable visually from those associated with bracketed binary competition. With

only one exception, where the cumulative density plots for the weighted regression slope cross at

approximately the 95% level, the plots do not cross, and the order of general efficiency is the present

TOUR Championship format followed by Hall-Potts, binary competition and stroke play. (When

Woods is not part of the competition, the plots do not cross.)

Table 4 summarizes the proportion of simulation trials for which a given tournament format

outperforms the others with respect to each of the five general efficiency measures. For example,

the top panel shows the proportion of simulation trials for which the Spearman rank correlation

for tournament format j, represented in columns, and including a random ordering of finishers in

the FedExCup finals, is higher than that of format i, represented in rows. Note that the present

FedExCup format provides a higher Spearman rank correlation relative to the other formats in most

simulation trials. With Tiger Woods in the field, among the alternative formats, the Hall-Potts

format results in the highest proportion of favorable Spearman rank values relative to the present

system, approximately 34%, followed by four rounds of stroke play, reseeded binary competition

(binary 1) and bracketed binary (binary 1). The present FedExCup format gives a higher Spearman

rank correlation relative to a random ordering of tournament finishers in approximately 99% of

simulation trials. With Woods out of the field, the percentage rates are generally of the same order

17

of magnitude. With just a few exceptions, this same ordering occurs for each of the other four

general efficiency measures.

7. More Detailed Analysis of FedExCup Predictive Power

In this section we examine the predictive power of the entire FedExCup competition, viewed as a

large-scale tournament, rather than a series of qualifying events leading to selection and seeding for

THE TOUR Championship. We compute predictive power, or winning rates for the competition, as

a function of true skill rankings defined in relation to the M = 1 to 278-300 players who participate

on the PGA TOUR in a given simulation trial (PGA TOUR season). As in our previous analyses,

we evaluate predictive power in terms of the five alternative formats for the FedExCup Finals.

In addition to determining qualifiers and the seeding of players for THE TOUR Championship

using accumulated FedExCup points, we consider two additional qualifying and seeding methods.

With the first method, players from the 278 to 300-player regular season pool are selected as

qualifiers and seeded for THE TOUR Championship on a random basis.15 With the second method,

qualifying and seeding for THE TOUR Championship are based on true skill rankings among the

278-300 pool players. At the one extreme, by selecting qualifiers and seeding positions randomly,

we can determine the extent to which the FedExCup points system adds incremental value in

identifying the “best” players relative to a purely random qualifying and seeding process. At the

other extreme, we can determine the extent to which the points system performs relative to a

“perfect” selection and seeding of qualifiers based on true player skill.

Table 5 summarizes winning rates for the five most highly-skilled players among the 278-300

in the FedExCup competition, when selection and seeding for THE TOUR Championship is 1)

random, 2) based on accumulated FedExCup points, and 3) based on true skill rankings.16 The

top row of Figure 4 supplements Table 5 and shows winning rates for players in skill positions

1-256 when Tiger Woods is in the competition. For ease of reading, skill positions in Figure 4 are

1532 players from the player pool are selected at random without replacement. The probability that any player willbe selected among the 32 is proportional to the total weeks the player actually participated in PGA TOUR eventsduring the randomly selected year, 2003-2009. Seeding for under the current FedExCup format, Hall-Potts and bothforms of binary competition is based on the order selected.

16The winning rates shown in Table 5, when selection and seeding for the TOUR Championship are based onaccumulated FedExCup points, are slightly lower than those shown in Table 3. In Table 3, the most highly-skilledplayer is the most highly-skilled among the 32 players in the FedExCup finals, whereas in Table 5, the most highly-skilled player is the most highly-skilled among those in the 278-300 player pool.

18

expressed in terms of base 2 logs.

Our first observation is that winning rates for all TOUR Championship formats are considerably

higher when qualifying for the Finals is based on accumulated FedExCup points rather than random

selection and seeding. When selection and seeding is random, the winning rates for the five most

highly-skilled players are the highest with four rounds of stroke play. In stroke play competition,

players record four 18-hole scores, and the player with the lowest total, and equivalently, the lowest

mean, wins. Although the two binary formats involve five rounds of competition, most players get

eliminated in early rounds, and as a result, an average of only 3.56 rounds per player is observed,

with the majority of players recording fewer than 3.56 scores over the five rounds. Therefore,

neither of the binary formats is as efficient as stroke play in determining winning rates of the most

highly-skilled players.

We note that when qualifying and seeding for THE TOUR Championship is random, stroke

play is also more efficient than the present FedExCup system. The present system involves four

rounds of stroke play in the Finals. Unlike conventional stroke play, however, the players who are

seeded randomly in the top positions enter the competition with more FedExCup points than those

who are in middle and bottom seeding positions. This random assignment of FedExCup points

makes it more difficult for the most highly-skilled players to win relative to what their winning

rates might have been if all TOUR Championship participants started out with the same number

of points, equivalent to four rounds of straight stroke play.

With random seeding, winning rates for the five most highly-skilled players under the Hall-Potts

format are generally lower than with reseeded and bracketed binary competition. Although a 32-

player Hall-Potts format involves nine rounds of binary play, rather than five, the top four players

get to begin play in round seven, the next four players begin in round six, etc. Since it is likely

that many lesser-skilled players will obtain high seeding positions and get to skip opening rounds

when qualifying and seeding for THE TOUR Championship is random, it may be more difficult for

the most highly-skilled players to win relative to conventional five-round binary play.

It is interesting to note that for all TOUR Championship formats, the most highly-skilled player

wins the FedExCup slightly more often when qualifying and seeding for THE TOUR Championship

is based on accumulated FedExCup points rather than true skill. Not surprisingly, Tiger Woods

is the top TOUR Championship seed in approximately 68% of simulation trials (not shown in the

19

table) and is among the top four seeds in approximately 93% of trials. Otherwise, he qualifies

for the Finals in almost every trial. By contrast, the other top players do not qualify for the

Finals as often. As a result, when qualifying and seeding for the Finals is based on accumulated

FedExCup points, Tiger Woods is less likely to be competing against top players like Vijay Singh,

Phil Mickelson and Jim Furyk than if selection and seeding were based on true skill. Hence, the top

seed, typically Woods, wins THE TOUR Championship more often when qualifying and seeding is

based on FedExCup points.

Table 6 summarizes winning rates for the five most highly-skilled players when Woods is not

in the competition, with the bottom row of Figure 4 giving a more complete picture. As noted

previously, when Woods is not in the competition, the winning rate for the most highly-skilled player

drops dramatically, from approximately 51% under present FedExCup rules to approximately 25%,

and winning rates for the other four TOUR Championship formats are much lower as well. With

Woods out of the competition, the most highly-skilled player wins the FedExCup more often under

present rules when qualifying and seeding for THE TOUR Championship is based on true skill.

This is also the case under the Hall-Potts format, but not with stroke play and the two forms of

binary competition.

Interestingly, if one’s objective in structuring tournament competition is to maximize the proba-

bility that the most highly-skilled player will win, an imperfect noisy qualifying and seeding system,

such as that used for the FedExCup, may be preferable to a more perfect system that identifies and

properly seeds the very best players, especially if the best player is a dominant player like Tiger

Woods. If the imperfect system is sufficiently perfect to ensure that the dominant player qualifies

almost all the time but eliminates some among the next-best competitors due to unfavorable ran-

dom variation in scoring, the best player could win with greater probability than with a selection

and seeding process that perfectly identifies and seeds the most highly-skilled. Of course, such a

system may not efficiently identify and reward players beyond the most highly-skilled.

8. Summary and Conclusions

In this study we make two contributions to the tournaments literature. First, we evaluate the

efficiency of the FedExCup, one of the PGA TOUR’s premiere events, which pays out $35 million

20

in total prize money to 150 players, including $10 million to the winner. Second, we advance the

methodology for measuring tournament selection efficiency as well as tournament qualifying and

seeding efficiency.

Our primary focus in evaluating FedExCup efficiency is on THE TOUR Championship, the

final event in the FedExCup Playoffs. We evaluate the selection efficiency of each of five possible

formats for THE TOUR Championship using the three measures of efficiency studied by Ryvkin

and Ortmann (predictive power, the mean skill level of the winning player and the mean skill

ranking of the winner) as well as five new measures of general tournament selection efficiency that

we develop. The first of the new measures, the Spearman rank order correlation, is applied to true

skill orderings and player finishing positions and captures the ability of a tournament to properly

order players. The second measure, Spearman’s footrule, provides an estimate of the degree of

disarray between the same two sets of orderings. The third measure, the skill-based regression

slope, is computed as the slope of an OLS regression of player skill ordered by skill on player skill

ordered by tournament finishing position. Although similar to Spearman’s rank correlation, the

regression slope captures the extent to which player skill might be miss-estimated using tournament

finishing positions. The final two efficiency measures capture the extent to which a tournament

distributes prize money efficiently. The first is the weighted skill-based regression slope, computed

as a weighted OLS regression of the same skill levels reflected in the computation of the skill-based

regression slope but weighted by the vector of money payout weights ordered sequentially from the

highest to lowest weight. The final efficiency measure, the CR statistic, reflects simultaneously, the

potential miss-ordering of player skill and money payouts, giving greater weight, for a given payout

miss-ordering, when there is a large difference in misaligned skill and greater weight to a given skill

miss-ordering, when there is a large difference in misaligned prize money.

From a purely mathematical standpoint, we find that the present structure of the FedExCup

Finals is more efficient than the four alternative formats when evaluated relative to each of the

Ryvkin-Ortmann measures and the five new general efficiency measures. Among the alternative

Finals formats, the format proposed by Hall and Potts appears to be the most efficient, but far less

efficient than the present system. For example, in evaluating selection efficiency in terms of “pre-

dictive power,” defined as the probability that the most highly-skilled player in the competition will

win, we estimate that the most highly-skilled player in the FedExCup Finals will win approximately

21

51% of the time under the present system compared with only 31% under Hall-Potts. Therefore,

if one had to make a choice between the present system and Hall-Potts, one would have to weigh

the disadvantages of lower selection efficiency against the advantages of transparency and potential

fan, player and sponsor interest that, most certainly, would weigh in favor of Hall-Potts.

The process of qualifying and being seeded for the FedExCup Finals is complex process and

involves the accumulation of FedExCup points during the 35-week regular PGA TOUR season

plus three additional playoff events, where FedExCup points earned are five times those earned in

regular season events. Due to the heavy weighting of points for the final three qualifying events,

it is possible that these events simply add noise to player orderings and that players might have

been more ordered more efficiently for THE TOUR Championship at the end of the regular season.

Using the first three new general efficiency measures, we show, however, that this is not the case;

the first three playoff events improve the ordering of players relative to the ordering established

before the Playoffs.

We also study the predictive power of the entire FedExCup competition, viewed as a large-scale

tournament, rather than a series of qualifying events leading to selection and seeding for THE

TOUR Championship. We find that the FedExCup, played under all five tournament structures,

produces substantially greater predictive power relative to outcomes where players are selected and

seeded for the Finals on a random basis. However, if the best players on TOUR are selected and

seeded for the Finals based on their true skill levels, predictive power decreases; the most highly-

skilled player, generally Tiger Woods, wins less often. This is not the case, however, for the present

TOUR Championship and Hall-Potts formats if Woods is excluded from the FedExCup field. (It

remains the case for the other three TOUR Championship formats.)

Finally, we point out that the strength of our results is highly dependent on whether Tiger

Woods is included or excluded from the analysis. We offer this contrast, not because we believe

that Woods’ days of dominating the TOUR are over but, instead, to illustrate the impact that a

single dominant player could have on measures of tournament selection efficiency. In the case of

golf, the effect is substantial.

22

AppendixSimulation Methodology

A. FedExCup Qualifying, Playoffs and TOUR Championship Seed-ing

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

1. Between 278 and 300 players participate for a full “regular season” prior to the FedExCupPlayoffs in 35 4-round stroke play events.17 156 players participate in each event. There is no“picking and choosing” of tournaments nor any qualifying requirements.18 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.19

3. FedExCup points are awarded for each tournament using the “PGA TOUR Regular Seasonevents points distribution” schedule, assuming each of the 35 tournaments is a regular PGATOUR event rather than a “major,” a World Golf Championship event or an “alternate”event held opposite tournaments in the World Golf Championship series.20

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 “PGA TOUR Playoffsevent points distribution” schedule.21

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 lowestscoring 70 players advancing to the final two rounds. FedExCup points are added to those

1735 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.

18A standard PGA TOUR event consists of 156 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.

19Generally, 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.

20The points schedule was obtained from http://www.pgatour.com/2008/fedexcup/11/25/2009changes.chart/index.html.World Golf Championship events earn 10% more FedExCup points for the highest finishers. “Majors” and thePlayers Championship earn 20% more points for high finishers. Alternate events earn only half the number of regularevent FedExCup points for all finishers.

21The schedule was obtained from http://www.pgatour.com/2008/fedexcup/11/25/2009changes.chartplayoff/index.html.

23

previously accumulated for each of the remaining 100 Playoffs participants according to the“PGA TOUR Playoffs event points distribution” schedule.

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 the“PGA TOUR Playoffs event points distribution” schedule.

7. After the BMW Championship, the top 32 players in FedExCup points advance to THETOUR Championship. (As we mention in Section 2.1, we standardize all alternative TOURChampionship formats to include 32 players. THE TOUR Championship, as it is currentlyconducted, consists of the top 30 players, and in the actual Hall-Potts structure, the 28 topplayers advance.)

8. When simulating the present TOUR Championship structure, the number of FedExCuppoints for the 32 participating players is reset according to “Playoffs Information: How thereset works after BMW,” for the first 30 players, with 205 and 200 FedExCup points assignedto players in positions 31 and 32, respectively.22 Players are then awarded additional FedEx-Cup points according to their finishing position in THE TOUR Championship, a four-roundstroke play event with no cut, using the “PGA TOUR Playoffs event points distribution”schedule. The FedExCup winner is the player who has earned the most FedExCup points,not necessarily THE TOUR Championship winner.

9. When simulating the Hall-Potts proposal and alternative TOUR Championship structuresinvolving binary competition, players are seeded according to the number of FedExCup pointsaccumulated prior to the Championship. The winner of each head-to-head match is the playerwith the lower 18-hole score rather than the player who won the most holes as in conventionalmatch play competition.

10. When simulating a 4-round stroke play alternative, players are selected for stroke play basedon accumulated FedExCup points, but seeding and accumulated FedExCup points do notcome into play during the competition. Essentially, the player who wins THE TOUR Cham-pionship as currently structured becomes the FedExCup winner.

B. Player Selection

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

1. A single year from our statistical sample, 2003-2009, is selected, with each year being selectedexactly 28, 000/7 = 4, 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 156

22See http://www.pgatour.com/2009/fedexcup/09/10/reset explainer/index.html.

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tournament participants is equal to the proportion of total player weeks in which he actuallyparticipated in the year selected, assuming sampling without replacement.23

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 for each player is selected at random from the portion of his estimatedspline-based skill occurring in the selected PGA TOUR season, 2003-2009. This becomes theplayer’s mean skill level for the entire season.24

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, 171 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 171 randomly-selected η residuals,and player k ’s first-order autocorrelation coefficient as estimated in connection with his cubicspline fit, a sequence of 171 estimated θ residuals is computed.

5. The 171 θ residuals are applied to player k ’s skill estimate to produce 171 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 160 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 first threerounds of the Playoffs (3×4 = 12) and makes it to the finals of the Hall-Potts version of THETOUR Championship, which requires nine rounds total. We note that it is highly unlikelythat all 161 scores would be used 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.25

23In 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.

24We 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.

25Suppose 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.

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Scaled Golf Time

0.0 0.2 0.4 0.6 0.8 1.0

6065

7075

80

Woods, Tiger

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Scaled Golf Time

0.0 0.2 0.4 0.6 0.8 1.0

6065

7075

80

Baddeley, Aaron

Figure 1: ‘Neutral” 18-hole Scores for Tiger Woods and Aaron Baddeley, 2003-2009.

Plots show 18-hole scores reduced by random round-course and player-course effects along withcorresponding spline fits (smooth lines). Scaled golf time, specific to each player, represents thechronological sequence of rounds for the player scaled to the {0, 1} interval.

26

68 69 70 71 72 73 74

0.0

0.1

0.2

0.3

0.4

0.5

SIMULATED SKILL VS. NORMAL DIST.

18−Hole Score60 65 70 75 80 85

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

SIMULATED SCORES VS. NORMAL DIST.

18−Hole Score

Figure 2: Distributions of Neutral Player Skill and Scoring, 2003-2009.

The two plot show the distributions of neutral player skill and scoring in 1,000 simulation trialsplotted against normal distributions with the same mean and variance.

27

Present FedEx Finals Hall−Potts 4 Rounds Stroke Play Binary: Bracketed

−0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Spearman Rank Correlation

−0.2 0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Skill−Based Regression Slope

−1.0 −0.5 0.0 0.5 1.0

0.0

0.2

0.4

0.6

0.8

1.0

Weighted Regression Slope

4 6 8 10 12

0.0

0.2

0.4

0.6

0.8

1.0

Spearman's Footrule

0.00 0.02 0.04 0.06 0.08

0.0

0.2

0.4

0.6

0.8

1.0

CR Statistic

Figure 3: Cumulative Density Functions for Five Measures of General Tournament Selection Effi-ciency: Tiger Woods in Field

28

0 2 4 6 8

02

46

810

RANDOM SELECTION & SEEDING

Log2 SKILL RANK (Woods In)

Present FedEx Finals Hall−Potts 4 Rounds Stroke Play Binary: Bracketed

0 2 4 6 8

010

2030

4050

60

FedEx−BASED SELECTION & SEEDING

Log2 SKILL RANK (Woods In)0 2 4 6 8

010

2030

4050

60

TRUE SKILL SELECTION & SEEDING

Log2 SKILL RANK (Woods In)

0 2 4 6 8

02

46

810

RANDOM SELECTION & SEEDING

Log2 SKILL RANK (Woods OUT)

Present FedEx Finals Hall−Potts 4 Rounds Stroke Play Binary: Bracketed

0 2 4 6 8

010

2030

4050

60

FedEx−BASED SELECTION & SEEDING

Log2 SKILL RANK (Woods OUT)0 2 4 6 8

010

2030

4050

60

TRUE SKILL SELECTION & SEEDING

Log2 SKILL RANK (Woods OUT)

Figure 4: FedExCup Finals Winning Percentages as a Function of True Skill Ranking with TigerWoods in and out of the competition.

29

Tab

le1:

Su

mm

ary

Sta

tist

ics

for

Sel

ecte

dP

laye

rs

Num

ber

of

Tim

esin

28,0

00

Sim

ula

tion

Tri

als

Avg

Spline

Pla

yoff

sIn

TO

UR

Fed

ExC

up

Rank

Pla

yer

Avg

Spline

Max

Spline

Min

Spline

Part

icip

ant

Cham

pio

nsh

ipW

inner

1W

oods,

Tig

er68.0

068.2

567.5

227,9

62

27,9

55

14,1

33

2Sin

gh,

Vij

ay69.1

970.5

067.9

527,9

20

24,0

85

3,1

38

3F

ury

k,

Jim

69.2

769.7

868.2

027,9

68

25,8

26

1,1

56

4E

ls,

Ern

ie69.3

570.3

268.3

827,9

03

23,9

15

1,3

89

5M

ickel

son,

Phil

69.3

869.7

569.2

227,9

97

26,5

12

1,3

81

6G

oose

n,

Ret

ief

69.6

970.3

568.4

627,2

88

18,6

76

635

7G

arc

ia,

Ser

gio

69.7

570.0

969.6

527,8

18

19,6

79

239

8D

onald

,L

uke

69.8

670.7

369.4

827,4

11

18,4

21

276

9H

arr

ingto

n,

Padra

ig69.8

970.2

368.8

926,7

11

17,3

03

307

10

Cin

k,

Ste

wart

69.9

370.4

269.4

427,9

04

18,4

10

93

88

Baddel

ey,

Aaro

n70.9

071.5

670.1

220,4

79

5,8

28

45

342

Boro

s,G

uy

72.6

172.7

372.4

9106

00

343

Magin

nes

,John

72.6

374.5

970.6

7999

15

0344

Park

,Jin

72.6

976.7

271.3

5309

10

345

Goss

ett,

Dav

id72.7

775.9

970.2

83,2

65

577

1346

Allre

d,

Jaso

n72.8

073.2

472.3

5129

10

Part

icip

ati

on

inth

eF

edE

xC

up

Pla

yoff

s,se

lect

ion

and

seed

ing

for

TH

ET

OU

RC

ham

pio

nsh

ip,

and

the

Fed

ExC

up

win

ner

are

all

det

erm

ined

usi

ng

the

pre

sent

Fed

ExC

up

poin

tssc

ori

ng

syst

em.

Aver

age,

maxim

um

and

min

imum

spline

valu

esare

the

aver

age,

maxim

um

and

min

imum

valu

esof

mea

npla

yer

skill,

aft

erre

mov

ing

esti

mate

dra

ndom

round-c

ours

eand

pla

yer

-cours

eeff

ects

,as

esti

mate

dov

erth

e2003-2

009

per

iod.

30

Table 2: Qualifying and Seeding Efficiency

Panel A: Proportion of Efficiency Measures that ImproveWith First Three Rounds of Playoffs

Efficiency Measure With Woods Without WoodsSpearman Rank Correlation 0.853 0.847Spearman’s Footrule 1.000 1.000Regression Slope 0.746 0.740

Panel B: Median Values of Efficiency Measures

After Regular SeasonEfficiency Measure With Woods Without WoodsSpearman Rank Correlation 0.640 0.631Spearman’s Footrule 26.4 26.7Regression Slope 0.713 0.665

After First Three Playoff RoundsEfficiency Measure With Woods Without WoodsSpearman Rank Correlation 0.682 0.673Spearman’s Footrule 17.4 17.7Regression Slope 0.737 0.690

31

Tab

le3:

Ryvkin

-Ort

man

nE

ffici

ency

Mea

sure

sas

Ap

pli

edto

the

Fed

ExC

up

Fin

als

(TO

UR

Ch

amp

ion

ship

)

Form

at

for

Fed

ExC

up

Fin

als

Pre

sent

4R

ou

nd

sR

esee

ded

Bra

cket

edR

an

dom

Ryvkin

-Ort

man

nE

ffici

ency

Mea

sure

Wood

sF

edE

xC

up

Hall

-Pott

sS

troke

Pla

yB

inary

Bin

ary

Ord

erin

gW

inn

ing

%R

ate

ofH

igh

est

Skille

dP

laye

rIn

51.0

630.6

428.6

118.9

918.1

73.0

9M

ean

Skil

lof

Win

nin

gP

laye

rIn

68.5

668.9

969.1

369.3

869.4

270.0

2M

ean

Skil

lR

ank

ofW

inn

ing

Pla

yer

In3.7

36.0

67.7

49.6

810.0

316.4

8

Win

nin

g%

Rat

eof

Hig

hes

tS

kille

dP

laye

rO

ut

24.7

417.5

011.5

39.6

79.0

53.1

8M

ean

Skil

lof

Win

nin

gP

laye

rO

ut

69.4

269.5

569.6

969.7

769.7

970.0

9M

ean

Skil

lR

ank

ofW

inn

ing

Pla

yer

Ou

t6.5

78.1

810.0

011.3

111.5

916.4

5T

he

ran

dom

ord

erin

gfo

rmat

pro

du

ces

ara

ndom

ord

erin

gam

on

g32

pla

yers

wh

oqu

ali

fyfo

rth

eF

edE

xC

up

Fin

als

.

32

Tab

le4:

Pro

port

ion

ofS

imu

lati

onT

rials

for

wh

ich

aG

iven

Fed

ExC

up

Fin

als

Tou

rnam

ent

For

mat

Pro

du

ces

mor

eF

avor

able

Ou

tcom

esfo

rF

ive

Mea

sure

sof

Gen

eral

Tou

rnam

ent

Sel

ecti

onE

ffici

ency

Wit

hT

iger

Wood

sW

ith

ou

tT

iger

Wood

sT

OU

RC

ham

pio

nsh

ipF

orm

atj

TO

UR

Ch

am

pio

nsh

ipF

orm

atj

Effi

cien

cyS

troke

Ran

dom

Str

oke

Ran

dom

Mea

sure

Form

ati

Hall-P

ott

sP

lay

Bin

ary

1B

inary

2O

rder

ing

Hall-P

ott

sP

lay

Bin

ary

1B

inary

2O

rder

ing

Sp

earm

an

Ran

kP

rese

nt

Fed

ExC

up

0.3

412

0.1

483

0.0

903

0.0

834

0.0

120

0.3

559

0.1

800

0.1

118

0.1

074

0.0

271

Hall-P

ott

s0.2

565

0.1

531

0.1

447

0.0

185

0.2

856

0.1

783

0.1

726

0.0

367

Str

oke

Pla

y0.3

811

0.3

725

0.0

664

0.3

815

0.3

736

0.0

948

Bin

ary

10.4

543

0.1

120

0.4

703

0.1

467

Bin

ary

20.1

146

0.1

498

Footr

ule

Pre

sent

Fed

ExC

up

0.3

201

0.1

346

0.0

770

0.0

708

0.0

129

0.3

403

0.1

729

0.1

024

0.0

993

0.0

285

Hall-P

ott

s0.2

429

0.1

394

0.1

321

0.0

203

0.2

749

0.1

666

0.1

585

0.0

395

Str

oke

Pla

y0.3

552

0.3

480

0.0

752

0.3

611

0.3

531

0.1

054

Bin

ary

10.4

295

0.1

406

0.4

388

0.1

766

Bin

ary

20.1

444

0.1

797

Slo

pe

Pre

sent

Fed

ExC

up

0.2

844

0.0

809

0.0

586

0.0

513

0.0

053

0.3

305

0.1

578

0.0

988

0.0

911

0.0

222

Hall-P

ott

s0.2

867

0.1

766

0.1

622

0.0

166

0.2

869

0.1

821

0.1

697

0.0

347

Str

oke

Pla

y0.3

396

0.3

189

0.0

506

0.3

771

0.3

594

0.0

870

Bin

ary

10.4

663

0.1

061

0.4

713

0.1

375

Bin

ary

20.1

085

0.1

380

Wei

ghte

dS

lop

eP

rese

nt

Fed

ExC

up

0.3

938

0.2

567

0.2

567

0.2

386

0.1

445

0.4

185

0.2

561

0.2

547

0.2

344

0.1

261

Hall-P

ott

s0.4

229

0.3

729

0.3

529

0.2

018

0.4

131

0.3

445

0.3

270

0.1

700

Str

oke

Pla

y0.4

294

0.4

073

0.2

468

0.4

067

0.3

870

0.2

274

Bin

ary

10.4

740

0.2

980

0.4

792

0.2

936

Bin

ary

20.3

088

0.2

959

CR

Sta

tist

icP

rese

nt

Fed

ExC

up

0.3

223

0.0

597

0.0

894

0.0

749

0.0

153

0.3

884

0.1

183

0.1

409

0.1

262

0.0

496

Hall-P

ott

s0.3

415

0.2

402

0.2

246

0.0

385

0.3

021

0.2

427

0.2

219

0.0

694

Str

oke

Pla

y0.3

539

0.3

341

0.0

903

0.4

190

0.3

981

0.1

638

Bin

ary

10.4

696

0.1

591

0.4

780

0.2

077

Bin

ary

20.1

637

0.2

149

Th

era

nd

om

ord

erin

gfo

rmat

pro

du

ces

ara

nd

om

ord

erin

gam

on

g32

pla

yer

sw

ho

qu

alify

for

the

Fed

ExC

up

Fin

als

.

33

Table 5: FedExCup Finals (TOUR Championship) Winning Rates of the Five Most Highly-SkilledPlayers: Woods in Competition

Present System

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 2.87 51.03 49.99 48.15 1.032 2.62 13.44 14.06 10.82 -0.623 1.71 7.73 9.52 6.02 -1.794 1.62 5.13 6.79 3.51 -1.665 1.62 4.06 5.51 2.44 -1.45

Hall-Potts

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 2.38 30.61 28.93 28.23 1.692 1.92 13.70 15.79 11.78 -2.093 1.46 8.96 13.27 7.50 -4.314 1.24 6.60 11.47 5.35 -4.875 1.20 5.28 5.52 4.08 -0.24

Stroke Play

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 4.68 28.59 26.53 23.91 2.062 3.71 8.80 8.03 5.09 0.783 2.25 6.84 6.35 4.59 0.494 2.23 5.56 5.15 3.33 0.425 2.13 4.66 4.49 2.53 0.17

Reseeded Binary

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 2.66 18.98 17.56 16.31 1.422 2.46 8.53 7.81 6.07 0.713 1.65 6.80 6.60 5.14 0.194 1.40 5.35 5.61 3.95 -0.255 1.52 4.67 4.86 3.15 -0.20

Bracketed Binary

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 2.66 18.16 16.96 15.50 1.202 2.41 8.16 7.51 5.75 0.653 1.57 6.39 6.10 4.82 0.294 1.46 5.31 5.06 3.85 0.255 1.43 4.48 4.75 3.05 -0.27

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Table 6: FedExCup Finals (TOUR Championship) Winning Rates of the Five Most Highly-SkilledPlayers: Woods not in Competition

Present System

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 2.71 24.61 32.28 21.89 -7.682 1.79 13.64 16.11 11.86 -2.463 1.66 9.78 11.40 8.11 -1.624 1.64 7.69 9.02 6.05 -1.345 1.37 5.47 6.96 4.10 -1.49

Hall-Potts

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 2.17 17.44 18.73 15.26 -1.292 1.26 11.98 16.30 10.72 -4.313 1.19 8.85 13.75 7.66 -4.914 1.18 7.09 12.91 5.91 -5.825 1.06 5.49 6.45 4.43 -0.96

Stroke Play

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 3.87 11.46 10.12 7.59 1.342 2.48 8.91 7.99 6.43 0.933 2.24 7.49 6.88 5.25 0.604 2.20 6.41 5.99 4.22 0.435 1.89 5.23 5.20 3.34 0.02

Reseeded Binary

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 2.40 9.63 9.27 7.23 0.362 1.56 7.48 7.28 5.91 0.203 1.41 6.40 6.24 4.99 0.164 1.48 5.74 5.47 4.26 0.275 1.28 4.53 5.08 3.25 -0.55

Bracketed Binary

Skill TOUR Champ Seeding FedEx vs. FedEx vs.Rank Random FedEx True Skill Random True Skill

1 2.42 9.00 7.98 6.58 1.012 1.58 7.14 6.98 5.56 0.163 1.49 6.23 5.98 4.74 0.254 1.41 5.36 5.43 3.96 -0.075 1.28 4.53 4.62 3.25 -0.09

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