c 2014

RYAN XAVIER PINHEIRO

ALL RIGHTS RESERVED EFFICIENT FREE AGENT SPENDING IN MAJOR LEAGUE BASEBALL

A Thesis

Presented to

The Graduate Faculty of The University of Akron

In Partial Fulfillment

of the Requirements for the Degree

Master of Science

Ryan Xavier Pinheiro

May, 2014 EFFICIENT FREE AGENT SPENDING IN MAJOR LEAGUE BASEBALL

Ryan Xavier Pinheiro

Thesis

Approved: Accepted:

Advisor Dean of the College Dr. James P. Cossey Dr. Chand Midha

Faculty Reader Dean of the Graduate School Dr. Stefan Forcey Dr. George R. Newkome

Faculty Reader Date Dr. Nao Mimoto

Department Chair Dr. Timothy Norfolk

ii ABSTRACT

During the 2012-2013 offseason, the Cleveland Indians signed a number of high pro-

file free agents. Our goal is to determine whether these free agent signings were efficient. We will focus on Michael Bourn and Mark Reynolds for our paper, though our methods would easily apply to other free agents. To accomplish this we use statis- tical analysis to predict each player’s future performance, and we use game theoretic models to analyze the efficiency of the contracts.

iii ACKNOWLEDGEMENTS

Thanks to my parents, my sister, my extended family around the world, and all my friends for the support and encouragement you have always offered me. I would never have made it to this point without you. Thanks to my professors as well for challenging me intellectually and inspiring with your passion. A special thanks to my advisor, Dr. Cossey, for mentoring me throughout the process of this thesis. It has been more than a pleasure. Lastly, thank you to the Cleveland Indians. Following the

Tribe has been my greatest hobby ever since I was a kid, and I was able to discover my passion for numbers and statistics through this hobby. I hope that the research done in this study will be useful to the organization at some point in time.

iv TABLE OF CONTENTS

Page

LISTOFTABLES...... vii

LISTOFFIGURES ...... viii

CHAPTER

I. INTRODUCTION...... 1

II. USING PLAYER FAMILIES TO PREDICT FUTURE PERFORMANCE 8

2.1 Player Families and Similarity Scores Discussion ...... 8

2.2 BuildingofRegressionModel ...... 16

2.3 Obtaining Predictions Based on Regression Model ...... 23

2.4 Comparing Our Technique with Standard Similarity Scores Technique 26

III. PLAYER VALUE AND GAME THEORY ...... 29

3.1 EarnedValueFormula...... 29

3.2 ModificationtoEarnedValueFormula ...... 31

3.3 DevelopmentofGameTheoryModel...... 32

3.4 Applying Double Auction Model to MLB Free Agency ...... 35

IV.CONCLUSION ...... 42

4.1 ActualResultsfrom2013 ...... 42

4.2 Limitations...... 45

v 4.3 FutureResearch ...... 45

BIBLIOGRAPHY ...... 47

vi LIST OF TABLES

Table Page

2.1 Vladimir Guerrero Similarity Score Example ...... 11

2.2 MichaelBournFamilyWARAverages...... 18

2.3 MarkReynoldsFamilyWARAverages ...... 19

vii LIST OF FIGURES

Figure Page

2.1 MichaelBourn’sRegressionModel...... 20

2.2 MarkReynolds’sRegressionModel ...... 21

2.3 Michael Bourn’s Regression Model Residual Plots ...... 22

2.4 Mark Reynolds’s Regression Model Residual Plots ...... 23

2.5 Michael Bourn Family Autocorrelations ...... 24

2.6 Mark Reynolds Family Autocorrelations ...... 25

2.7 Mark Reynolds Family Regression Model Using Our Approach . . . . . 27

2.8 Mark Reynolds Family Regression Model Using Standard Similarity ScoresApproach ...... 28 3.1 MichaelBournSalaryHistory ...... 37

viii CHAPTER I

INTRODUCTION

The salary disparity between small-market and large-market teams in Major League

Baseball is a major issue that has come under the spotlight in recent years. General

Managers and the front offices of every Major League Baseball team are constantly looking for new ideas and ways to improve the efficiency of how to run their ballclub.

This ranges from selecting the right players in the MLB First-Year Player Draft, to making smart trades that improve the team, to spending the right amount of money on the right players during free agency. We will be focusing on free agency and the efficient spending of money as our topic under investigation.

We will perform a case study to help us examine this topic. Specifically, we will be examining the 2012-2013 offseason of the Cleveland Indians in terms of free agency. The Cleveland Indians are a small-market ballclub that has been notoriously stingy in recent years when it comes to spending money on free agents. However, during the 2012-2013 offseason this perception started to change as the Indians added four free agents to their squad on major league contracts. Two of these free agents were big-name players and signed to large multi-year deals. The major question we will be trying to answer in our study is whether or not the Cleveland Indians efficiently and effectively spent their money during the 2012-2013 offseason.

1 This is an important question to answer because small-market teams have very little room for error when they spend their money. Small-market teams simply cannot afford to tie down large amounts of money to the wrong players for a prolonged period of time. This puts the organization as a whole in a bind, and makes it difficult to make subsequent transactions to significantly improve a team’s roster. The past 10 to 15 years have seen a significant shift in the approach of MLB front offices. Teams have shifted to mathematics and analytics to help determine the optimal ways to add the right players and spend an efficient amount of money. There are many different approaches that teams have attempted in order to determine whether or not money has been efficiently and effectively spent. We will be developing our own approach to determine this by using the Cleveland Indians as a medium for our analysis. The approach that we will take can be applied to any team, and can be a valuable analysis tool for small-market teams in particular.

The new-age thinking of using mathematics to run a MLB front office has led to many publications regarding player valuation and front office efficiency over the past couple of decades. One of the more well-known books describing the mindset and thought process behind using analytics in baseball is Moneyball by Michael Lewis.

Lewis describes the story of Oakland Athletics General Manager Billy Beane trying to build a successful small-market baseball team while spending the least amount of money possible. Beane decided to invest in cheaper players that were able to play good defense and post a high on-base percentage [1]. While Moneyball does not show the in-depth mathematics behind front office decision-making, it does provide

2 some framework for our study and some motivation as well. The ideas of efficient spending and optimization of player value while minimizing costs were two driving factors behind many of the decisions that Beane and the Oakland Athletics made.

Accurate player valuation plays a large part in achieving efficiency while spending money. The Book on The Book by Bill Felber develops a formula for a player’s Earned Value. The formula simply indicates how much the player is worth in a dollar amount. We will be discussing and using Earned Value more thoroughly later. The player valuation study completed by Felber simply compares players’

Earned Value to what they really earned. The conclusion was that the vast majority of big-name free agents are highly overpaid, while young players who have yet to experience free agency can be a major bargain for teams [2]. The study does not discuss the negotiation process or efficient spending in depth.

Lewis, Sexton, and Lock (2007) did a thorough study of organizational effi- ciency in the front office. Data envelopment analysis was used to help measure the efficiency of each MLB team. Next, the authors used logistic regression to determine whether teams were competitive or noncompetitive, and finally used the Gini Index to determine the minimum Total Player Salary (TPS) to be competitive and the max- imum TPS a team could spend without overspending. The authors find that teams with a large market size have rarely ever been noncompetitive due to low TPS. The authors also find that teams that play in large markets were more likely to overspend on free agents than small-market teams. Again, this study does not consider the bargaining process during free agency which does have an effect on whether a team

3 overpays for a particular player along with the amount spent on the player as well.

One of the goals of our study will be to take this bargaining process into account. This way we will be able to simulate the entire free agency process with more accuracy.

Rockerbie (2009) focuses on the negotiation process that takes place during free agency in baseball. Rockerbie studies the impact of supply and demand of free agents in Major League Baseball. The author uses an auction model of free agency with multiple bidders to examine the impact of supply and demand. The author also uses a regression model to help determine a player’s salary. The author finds that a greater supply of free agents at a given position can lead to a lower salary in general.

Thus, it might be beneficial for potential free agents to defer free agency until the supply is lower at their given position. The author assumes a simple auction in this study, which limits the effect of negotiating from the free agent’s party. We will be using a double auction model in our study.

Baseball Prospectus published their book Baseball Between the Numbers in

2006. A chapter in this book is devoted to a case study analyzing the free agent moves of a front office, as we will be doing later on. Jonah Keri, the author of this chapter, examines the transactions made by Wayne Huizenga, the General Manager of the 1997 World Champion Florida Marlins during the Marlins’ rise to the top.

The author focuses on Huizenga’s strategy of going “all-in” for the 1997 season by spending a significant amount of money for a small-market team. The author also talks about Huizenga’s decision to dismantle the team immediately after the 1997 season. The author concludes that while the success of Huizenga’s Marlins was short-

4 lived, Huizenga ultimately accomplished the two most important things in Major

League Baseball: to win a championship and make a profit. Huizenga was able to accomplish both by deciding to go “all-in” during the 1996 offseason. While the study does not show the in-depth mathematics behind the decision, it does provide similar analysis to what we will be performing with the 2013 Cleveland Indians.

We will be combining concepts of a few of the mentioned publications to help us analyze whether the Cleveland Indians efficiently spent their money during the 2012-2013 offseason. The Cleveland Indians signed four players to major league contracts during the offseason: Michael Bourn, Brett Myers, Mark Reynolds, and

Nick Swisher. Our approach will be demonstrated by performing a double auction model on the negotiation process of Michael Bourn and Mark Reynolds to determine what game theory indicates is the best price for a deal to take place. We will then analyze the actual amount of money spent on each player and determine how much the Indians overpaid for each player.

We will also need to predict each individual free agent’s future performance in order to properly perform a game theory double auction model. The standardized statistic we will be using to evaluate player performance is Wins Above Replacement

(WAR). WAR is a measure of how many extra games a player’s team won having the player in the lineup instead of an average replacement player. Thus, the expected

WAR of an average replacement player should be zero. We use the concept of families to predict our players’ future performance. The concept of families is discussed in

The Book by Tango, Lichtman, and Dolphin. It is also a widely used tool to help

5 predict a player’s future performance. A family is a group of players that have very similar attributes. For example height, weight, position, and skill set are attributes that are comparable among each member of the family. Thus, each player has his own unique family of players. We will be using a regression model on the WAR of each of our player families to help predict our individual players’ future WAR in upcoming seasons. Next, we will use our predicted WAR and enter it into the equation of Earned Value to determine our valuation of the individual player. Also, we will be estimating the players’ valuation of themselves by performing a multiple regression using their past salaries and past WAR statistics. The determination of these valuations will then allow us to perform our double auction analysis.

The approach that we use is different from past studies in a couple of ways.

First, most other publications that have studied this topic ignore the individual player’s say in the bargaining process. Usually, these studies simply analyze the deals that have taken place in retrospect, thus ignoring the negotiation process alto- gether. We recognize that this process is crucial in determining an individual player’s

final salary and also determining whether a player is overpaid or not. We also use the statistic of WAR as our statistic to evaluate players. Since WAR is a standard- ized statistic, it allows us to compare the WAR statistics of players from different generations without adjusting for different eras.

Our study was limited by the fact that the numbers cannot measure every- thing a player brings to a team. The mathematics and analysis we perform ignore intangibles such as clubhouse presence, veteran leadership, and hustle. These are

6 important player attributes that play a large role in the chemistry and success of a team. We are also limited by the fact that our study only looks at free agency and ignores other avenues of acquiring talent such as trades and the MLB Draft. Organi- zations can certainly use their money wisely by making smart trades and by drafting well. Thus, it is important to realize that the results of our study do not imply that the most efficient spending team in free agency is the best team talent-wise as well.

This study can certainly be extended in many ways. We are looking to expand our game theory double auction model to including multiple bidders, which makes the auction more dynamic. We also would like to expand our study and run our model on every single MLB team to help determine an overall efficient spending threshold for the entire MLB. As we do this, we can look at past multi-year contracts that are still in effect today. Lastly, we would like to further extend our study to trades and the draft to include these transactions in our analysis. Thus, we would be more likely to determine a team’s overall efficiency rather than just efficiency in free agency.

7 CHAPTER II

USING PLAYER FAMILIES TO PREDICT FUTURE PERFORMANCE

2.1 Player Families and Similarity Scores Discussion

The concept of player families that we will be using is not a new one. Sabermetricians- baseball analysts who use advanced mathematical and statistical concepts to help teams make decisions- have been using this idea for the past couple decades, primarily to compare players in modern day Major League Baseball to players from other eras.

Sabermetrics and baseball analytics pioneer Bill James was one of the first people to introduce the idea of similarity scores, which uses the same concepts that we will be using in our player families.

2.1.1 Similarity Scores for Hitters According to Baseball Reference

We describe similarity scores in such a way that each player is given a set number of points to start out with. Points are then subtracted based on statistical differences when comparing the reference player to other players. We look to Baseball Reference for an example of how similarity scores are used with batters:

To compare one player to another, start at 1000 points and then subtract points based on the statistical differences of each player.

• One point for each difference of 20 games played

8 • One point for each difference of 75 at bats

• One point for each difference of 10 runs scored

• One point for each difference of 15 hits

• One point for each difference of 5 doubles

• One point for each difference of 4 triples

• One point for each difference of 2 home runs

• One point for each difference of 10 RBI

• One point for each difference of 25 walks

• One point for each difference of 150 strikeouts

• One point for each difference of 20 stolen bases

• One point for each difference of .001 in batting average

• One point for each difference of .002 in slugging percentage

To this there is a positional adjustment. Each position has a value, and we subtract the difference between the two players position.

• Catcher: 240

• Shortstop: 168

• Second Base: 132

• Third Base: 84 9 • Outfield: 48

• First Base: 12

• DH: 0

2.1.2 An Example of Similarity Scores Calculation

We now demonstrate how similarity scores are used according to the above criteria.

We will use former outfielder Vladimir Guerrero as the player under observation.

Thus, all similarity scores are used to compare the similarity of Vladimir Guerrero to any other player. The player with one of the highest similarity scores to Guerrero is former first baseman Jeff Bagwell with a similarity score of 859. We show the relevant statistics of these two players below.

We calculate the total points to subtract from 1000 by taking the sum of the similarity scores differences and subtracting from 1000. Therefore, the sum of our differences is 141 and after we subtract that from 1000, we obtain a similarity score of 859 for Jeff Bagwell in comparison to Vladimir Guerrero.

Therefore, similar players should have similarity scores that are very close to each other, while players that are different should have significantly different similarity scores. While there are many variations of similarity scores, the above example used by Baseball Reference gives us an idea of how the technique works. The major benefit of using player families or similarity scores is that these comparisons help to give us insight on how a particular type of player has performed based on past data. These comparisons can help to tell give us an idea of when a right-handed power hitter

10 Table 2.1: Vladimir Guerrero Similarity Score Example

Statistic Guerrero Bagwell Similarity Score Difference

GamesPlayed 2147 2150 0

At-Bats 8155 7797 4

Runs 1328 1517 18

Hits 2590 2314 18

Doubles 477 488 2

Triples 46 32 3

HomeRuns 449 449 0

RBI 1496 1529 3

Walks 737 1401 26

Strikeouts 985 1558 3

StolenBases 181 202 1

Batting Average .318 .297 21

Slugging Percentage .553 .540 6

Positional Adjustment 48 12 36

11 starts to encounter a drop-off in power, or when a speedy left-handed leadoff hitter might start to lose his ability to steal bases. Therefore, this is a useful idea to help forecast a player’s future performance.

The standard technique of similarity scores is a very useful tool to compare similar players over the course of many different generations. However, similarity scores are not always best for modeling the behavior of a particular type of player.

Similarity scores use a standardized formula that pertains to all players, as denoted by the above table. While this makes the standard technique of similarity scores easier and more convenient, it does not always lead to the best results.

We can think of similarity scores as a technique in which all players are first considered, and then the players with the weakest comparisons are then eliminated.

This is relatively analogous to a backward elimination regression technique in which all variables are first considered for a regression model, and then the weakest variables are iteratively eliminated. We decided to use more of a “forward regression” technique in approach when creating comparable player families. Instead of considering all players and then eliminating the least comparable players, we considered all players and then included the most comparable players in our families.

Also, the standard similarity scores technique is not entirely relevant for our purposes either. We are considering whether or not it makes sense for a team to pursue certain free agents or not. While there is a positional adjustment with similarity scores, we still see that Vladimir Guerrero-an outfielder- and Jeff Bagwell-a

first baseman- are considered to be comparable players using similarity score analysis.

12 It does not make sense to compare players of two different positions for free agent analysis since teams are only concerned with the position of the individual free agent they are thinking of pursuing. Also, since we are using WAR as our statistic of evaluation, defense is also taken into consideration. Thus, comparing two players at different positions does not make sense for our purposes.

2.1.3 Michael Bourn Family

We begin to demonstrate our technique by explaining the criteria we used to de- termine Michael Bourn’s family of comparable players. The first question we ask ourselves is, “What characteristic of Michael Bourn makes him unique from the ma- jority of other players?” Michael Bourn’s biggest asset over the course of his career has been his speed and his ability to steal bases. We decided to first sort all the players over the course of major league baseball history by stolen bases, starting with the all-time leader in stolen bases, Rickey Henderson. The following is an example of the criteria used to determine whether to include the player in Bourn’s family or not.

All of the following conditions must be satisfied to be included in Michael

Bourn’s family:

• Player must be within 30 percentage points of Michael Bourn’s career batting

average

• Player must be within 30 percentage points of Michael Bourn’s career on base

percentage

13 • Player must be within 30 percentage points of Michael Bourn’s career slugging

percentage

• Player must have less than 100 home runs in his career

• Player must have more than 225 career stolen bases

• Player must be an outfielder

Prior to the 2013 season, Michael Bourn averaged approximately 4.2 HR per season. We rationalize our cutoff point of “less than 100 career home runs” by hypothesizing that a player who is truly similar to Bourn would have to play 20+ seasons while maintaining this same HR rate in order to bypass this threshold. Also, odds are that after they are past their prime, maybe around year 10, their home run totals will even further decrease. Thus, we do not believe there is any player with 100 or more home runs that will significantly change the results of our study. The reason we use “more than 225 career stolen bases” as a criteria is to ensure some degree of longevity with the players we are putting in Bourn’s family. Therefore, we eliminate players that did not have long careers, but might have fit the other criteria necessary to be included in Bourn’s family. Some notable players in Michael Bourn’s family include Vince Coleman, Juan Pierre, and Scott Podsednik.

2.1.4 Mark Reynolds Family

We continued our study by looking at another free agent signing, Mark Reynolds.

Standard similarity score techniques would use the same algorithm altogether to

14 distinguish the player families of two very different players such as Michael Bourn and Mark Reynolds. However, we do not use the same algorithm to construct player families for different players. We decide to first ask ourselves the question, “What characteristic makes this player unique?” Stolen bases were Michael Bourn’s unique characteristic. We decide to use strikeouts as Mark Reynolds unique characteristic to sort by. Here is an example of the criteria used to determine whether or not to include the player in Mark Reynolds’s family.

All of the following conditions must be satisfied to be included in Mark Reynolds’s family:

• Player must be within 30 percentage points of Mark Reynolds’s career batting

average

• Player must be within 30 percentage points of Mark Reynolds’s career on base

percentage

• Player must be within 30 percentage points of Mark Reynolds’s career slugging

percentage

• Player must be either a third baseman or a first baseman

• Third basemen must have at least 690 career strikeouts

• First basemen must have at least 1000 career strikeouts

We rationalize the number of strikeouts used to again ensure longevity with the players we are including in Mark Reynolds’s family. Thus, as we did with Michael

15 Bourn’s family, we eliminate players that fit the rest of the criteria, but did not have long enough careers to be included in our model. Since Mark Reynolds has been primarily a third baseman throughout his career, we make it easier for third basemen to be included in our model than first basemen. Hence, we have the difference in our career strikeout threshold used in our criteria for Mark Reynolds’s family. Some notable players in Mark Reynolds Family include Gary Gaetti and Dean Palmer.

2.2 Building of Regression Model

We were able to gather a significant amount of data using our technique for building player families. The vast majority of our data was obtained from the website Fan-

Graphs, a website that serves as a database of advanced statistics for all players in

Major League Baseball history. We ended up having 55 players in Michael Bourn’s family and 40 players in Mark Reynolds’s family after finding all the players that met our criteria.

The main statistic we gathered from each one of these players was Wins

Above Replacement, or WAR. This statistic tells us how many wins each individual player earns his team over a replacement player. Thus, if Player A has a WAR of

2.0, we interpret this by saying that this player earned his team 2.0 more wins over the course of the season than if a replacement player had been playing in Player A’s place. It should also be noted that the WAR statistic is time dependent as well.

Thus, over the course of a season, a player with 500 at-bats is much more likely to have a higher WAR that season than a player with only 100 at-bats. Also, a player 16 that misses a significant amount of time due to injury will also have a lower WAR than a comparable player that was healthy for the entire season.

We looked up the WAR for every individual season of every player in each of our player families. The WAR statistic will serve as the dependent variable in our model. We will use the number of years the player played in the league as our independent variable. Thus, the player’s rookie season will be “Year 1”, the player’s second season will be “Year 2” and so on. We used the qualification of rookie status in a season in order to officially be categorized as a “season” in our player family data. That is, the player must have exceeded 130 at bats for the season to qualify for our study.

We decided to use the number of years in the league as our independent variable as opposed to age, because we believe that experience in Major League

Baseball is a common and essential part of the success of any Major League Baseball player. Thus, we believe that it would be unfair and inaccurate to compare a player at the age of 23 with 3 years of MLB service and a player at the age of 23 in his rookie season on the same level. The WAR data we obtained for each player was averaged for each number of years served in the Major Leagues. Thus, for our regression model, our data consisted of the number of years spent in the league, along with the average

WAR for all players in each respective family. Using the average WAR for each one of our respective families as our dependent variable and the corresponding year spent in the league as our independent variable, we were able to build our regression models for each of our families of interest. The following are examples of the data we used

17 Table 2.2: Michael Bourn Family WAR Averages

Year WAR

1 1.26

2 1.84

3 2.32

4 2.41

5 2.78

6 2.50

7 2.16

8 2.18

9 1.80

10 1.67

11 1.81 after averaging all the WARs in our player families.

The reason we stop after 11 years for Michael Bourn is because there are a lot of missing values for player in Bourn’s family after year 11. Thus, the small sample size for year 12 would give us misleading results. Also, the Indians were likely not looking to sign Michael Bourn to a contract for more than five years. Thus, the data from year 12 and above is not relevant to us in this situation. We stop after 10 years for Reynolds for the same reasons we stopped Bourn’s family after 11 years.

18 Table 2.3: Mark Reynolds Family WAR Averages

Year WAR

1 1.11

2 1.98

3 2.87

4 3.07

5 2.67

6 2.27

7 2.26

8 2.05

9 2.18

10 1.97

19 2.2.1 Regression Model for Michael Bourn Family

Figure 2.1: Michael Bourn’s Regression Model

The above figure shows us the regression model for Michael Bourn’s family given the data we used. We see that there is a strong cubic shape to our data, and the corresponding cubic polynomial regression model gives us a regression equation of

W AR =0.1471 + 1.229 ∗ Y ear − 0.1909 ∗ Y ear2 + .008407 ∗ Y ear3. (2.1)

Also, please note the strong R-squared value of 93.9 percent for our model.

2.2.2 Regression Model for Mark Reynolds Family

Now we take a look at Mark Reynolds’s regression model. Similar to our previous case,

Mark Reynolds’s regression model shows a strong cubic shape. The corresponding

20 Figure 2.2: Mark Reynolds’s Regression Model

cubic polynomial regression model for Mark Reynolds gives us a regression equation of

W AR = −0.5274 + 1.954 ∗ Y ear − 0.3479 ∗ Y ear2 +0.01789 ∗ Y ear3 (2.2)

Again, note the strong R-squared value of 90.2 percent for this model.

2.2.3 Check of Regression Assumptions

We now take a look to make sure that all regression assumptions are satisfied in our model. Most importantly, we believe that we have identified a correct model function for our data, as our data does seem to follow a cubic pattern. We also precisely know the values of our independent variables, satisfying another assumption. We take a look at the histograms and the normal probability plots to decide whether our resid- uals seem to be normally distributed. While neither case gives us a perfectly normal 21 Figure 2.3: Michael Bourn’s Regression Model Residual Plots

distribution, it does not appear that there is enough evidence to violate our assump- tion either. The residual versus fits plot also shows nothing that would violate our assumption of constant variance. The last assumption we must check is uncorrelated errors.

Since our independent variable is a unit of time, we must take a closer look at our data to ensure that our residuals are not correlated. We can not clearly determine whether or not serial correlation is a problem simply by observing the order plot of the residuals in our above figures. We must take a look at the autocorrelations at each lag to determine whether or not serial correlation will be an issue.

We observe in both cases that our residuals are not significantly correlated with each other. Therefore, our assumption of uncorrelated residuals is not violated.

22 Figure 2.4: Mark Reynolds’s Regression Model Residual Plots

We have now checked and satisfied all regression assumptions, and we can continue with our regression analysis.

2.3 Obtaining Predictions Based on Regression Model

We were then able to construct our regression model to give us a prediction of each player’s future performance. While our player predictions are based on the regression models we have built using our player families, there is more to our prediction than simply plugging the corresponding year in the league into our model. As we have mentioned above, our regression model is based on average WARs of all the players in our respective player families. Thus, in order to use our model most effectively, we

23 Series resid(y) ACF −0.5 0.5 0 2 4 6 8 10 Lag

Figure 2.5: Michael Bourn Family Autocorrelations

must measure how Michael Bourn and Mark Reynolds have performed throughout their careers thus far, in comparison to the average of all other players in their families.

2.3.1 Michael Bourn Prediction

We will first make Michael Bourn’s prediction. The 2013 Major League Baseball sea- son was the seventh season for Michael Bourn using our definition of what we classified as an official season. Using regression model we developed for Michael Bourn, we sub- stitute in the value 7 into our x variable. The result gives us an expected WAR of 2.2 based on our regression equation. However, we need to determine how Michael Bourn compares to the rest of the model on average in order to make the best prediction possible for Michael Bourn. We decided to write a program in R to calculate each player’s average deviation from their corresponding family regression equations. The

24 Series resid(y) ACF −0.5 0.5 0 2 4 6 8 Lag

Figure 2.6: Mark Reynolds Family Autocorrelations

result we obtain for Michael Bourn is an additional WAR of approximately 1.1. The value of 1.1 serves as a final adjustment for quality of player in our model, as it tells us that on average, Michael Bourn has performed better than the other players in his family by approximately 1.1 wins in his first six MLB seasons. Therefore we add

1.1 to our regression model prediction of 2.2 to obtain a prediction of a 3.3 WAR for

Michael Bourn in 2013.

2.3.2 Mark Reynolds Prediction

We now take a look at Mark Reynolds’s prediction. The 2013 MLB season was

Reynolds’s seventh season in the Major Leagues, again by our criteria of a season.

Thus, using our regression model for Mark Reynolds, we plug 7 in for x. We obtain a predicted WAR of 2.2 for Mark Reynolds for the 2013 season after using our regression

25 model to predict WAR for Mark Reynolds’s Family in Year 7. Now we must obtain

Reynolds’s average deviation from our regression equation. Using a similar program in R as we did for Michael Bourn, we obtain a negative additional WAR of −1.1 for Mark Reynolds. Thus, on average, Mark Reynolds has performed about 1.1 wins worse than the other players in his family during Reynolds first six years in the

MLB. Hence, we subtract 1.1 from our regression model prediction of 2.2 to obtain a prediction of a 1.1 WAR for Mark Reynolds in 2013.

2.4 Comparing Our Technique with Standard Similarity Scores Technique

We would like to show the difference in results between the technique that we used, and the standard similarity scores technique when comparing players. We will show this comparison by taking 20 players with the closest similarity scores to Mark

Reynolds using the standard similarity scores approach, and we will compare this class of players with the 20 most similar players to Mark Reynolds using our approach.

Please take a look at the following two tables in which the standard similarity scores technique and our technique is shown.

The reason there is a difference between the model we show in (2.7) and in

(2.2) is because we are only looking at the 20 most similar players in Mark Reynolds family using our technique compared to the standard similarity scores technique.

Thus, the sample size is smaller in (2.7) than it is in (2.2). This explains the difference in the two regression models.

26 Figure 2.7: Mark Reynolds Family Regression Model Using Our Approach

Clearly it appears that our model outperforms the standard similarity score model. There does not appear to be much of a trend or pattern in the standard similarity scores technique, while our model again shows a relatively strong cubic trend. Also, note the differences in the R-squared between the two models. Thus, while our method might be more time consuming, we believe that it will yield more accurate results than the standard similarity scores technique.

27 Figure 2.8: Mark Reynolds Family Regression Model Using Standard Similarity Scores Approach

28 CHAPTER III

PLAYER VALUE AND GAME THEORY

We now focus our attention on valuing players based on the predictions we obtained in the previous chapter. We will be looking to convert our predicted WAR values to a reasonable dollar value for each player. Subsequently we will be using the values we derive to simulate a double-auction game theory model. This will help us to determine the amount of money to make the deal at according to game theory. Throughout this chapter we will also be integrating many original ideas and concepts into theories that have already been developed.

3.1 Earned Value Formula

An important part of the game theory model we will be using is being able to identify the player’s valuation of himself and the team’s valuation of the player. Thus, we need to find a way to convert our predictions to a dollar amount. We will be using a formula that was developed in The Book on the Book by Bill Felber. Felber uses

Total Player Rating and Total Pitcher Index as his standardized player statistic of evaluation. The variables Felber uses in his equation are defined as follows:

• A = Adjusted Player TPR or TPI

29 • B = League Average Adjusted TPR or TPI

• C = (League Average Salary at Position)/(League Average Adjusted TPR or

TPI)

• D = League Average Salary at Position

Now here is the formula as Felber presents it in the book:

EarnedV alue =(A − B) ∗ (C)+(D) (3.1)

Here is an example of how the Earned Value formula is used from The Book on The Book. The example calculates the Earned Value from third baseman Scott

Rolen’s 1998 season with the Philadelphia Phillies.

3.1.1 Scott Rolen Earned Value Example

Scott Rolen had an Adjusted TPR of 10.6 in 1998. The average Adjusted TPR of

National League third basemen in 1998 was 4.07. Additionally, the average salary for

National League third basemen was 1.536 million dollars. Thus, in this example:

• A = 10.6

• B =4.07

• C =1.536/4.07 = .377747

• D =1.5364

• EarnedV alue = (10.6 − 4.07) ∗ (.377747) + 1.536 = 3.818

30 This means that Rolen’s Earned Value in 1998 was 3.818 million dollars. This made Rolen the most valuable third baseman in 1998. Additionally, Rolen only made

750,000 dollars in 1998 as this was prior to Rolen experiencing free agency. Therefore, the Phillies got a huge bargain out of Rolen in 1998 as they paid very little for an extremely valuable talent in Rolen [2].

3.2 Modification to Earned Value Formula

The Earned Value formula is valuable to us as it gives us a method to convert player performance into a dollar amount. However, to better fit our study, we will be mod- ifying a couple of components of the Earned Value formula. The first modification is obvious for our purposes as we will be using WAR as our standardized player evalu- ation statistic rather than TPR or TPI. Next, for “League Average Adjusted WAR”, we will use the entire MLB in our data rather than separate into only American

League and only National League as Felber does in the Rolen example.

The most significant change to the formula that we will be making is to the

“League Average Salary at Position” part of the formula. Instead of using the actual league average salary at the position, we will use the average salary of players that have qualified for free agency at our given position under observation. Thus, for

Michael Bourn’s Earned Value we will use the average salary of all outfielders that were under “free agency contracts” in 2012. We will obtain data for all of our player salaries from USATODAY.com.

31 The reason we make this change to the salary portion of the Earned Value formula is to eliminate talented young players that have not experienced free agency from our salary data. Young players that are not yet eligible for free agency generally make significantly less money than a player who has experienced free agency. Since we are specifically focusing on free agent players in the middle of their careers, it does not make sense for us to compare these players to younger players with non- comparable salaries. The inclusion of younger players would significantly bring down the salary term in our equation, and thus would not give us optimal results for our purposes.

3.3 Development of Game Theory Model

We will now show the development of the game theory model we will be using to simulate our negotiation process. The model we will be explaining is a double auction model that is developed in the book Game Theory for Applied Economists by Robert

Gibbons [3]. Gibbons bases his model off of prior work done by Chatterjee and

Samuelson in 1983 [4].

The double auction we will be using is a static game of incomplete informa- tion. That is, the valuations of both the buyer and the seller are private. Double auctions work in such a way that the seller names an asking price, Ps, and the buyer will simultaneously give the seller his offer price, Pb. If Pb ≥ Ps, then trade should occur at (Pb + Ps)/2. However, if Pb ≤ Ps, then no trade should take place.

32 We call the buyer’s valuation of the seller’s good Vb and we call the seller’s valuation of his own good Vs. As we said earlier, these valuations are private in- formation to both the buyer and the seller. Also, the valuations are drawn from independent uniform distributions on the interval [0, 1]. Based on the valuations, we can obtain a couple of utility equations that measure the utility change of each group. The buyer’s utility is given by the equation Vb − p, where p is the actual price at which the buyer gets the good. The seller’s utility is given by the equation p − Vs, where p is the price for which the seller sells the good. If no trade takes place, both the buyer and seller utility will be zero.

We will denote the strategy of the buyer as Pb(Vb), a function of the buyer’s valuation. Consequently we will denote the strategy of the seller as Ps(Vs), a function of the seller’s valuation. Our goal in our game theory model is to find the Bayesian

Nash equilibrium. This will help to tell us that amount at which a deal should take place. Two conditions must hold in order for the strategies to achieve the Bayesian

Nash equilibrium we will be looking for. First, for each Vb in [0, 1], Pb(Vb) must solve

maxPb [Vb − (Pb + E[Ps(Vs)|Pb ≥ Ps(Vs)]/2)] ∗ Prob(Pb ≥ Ps(Vs)) (3.2)

where E[Ps(Vs)|Pb ≥ Ps(Vs)] is the expected price given by the seller strategy, condi- tional that the buyer’s offer price is greater than or equal to the price of the seller’s strategy. This condition is constructed based off of the buyer utility. Recall that the buyer’s utility is Vb − p. Also recall that if Pb ≥ Ps, then trade should occur at (Pb + Ps)/2. We are simply implementing this requirement into the buyer’s util-

33 ity in 3.2. Similarly, the second condition that must hold to achieve Bayesian Nash equilibrium is that for each Vs in [0, 1], the seller’s strategy, Ps(Vs), must solve

maxPs [(Ps + E[Pb(Vb)|Pb(Vb) ≥ Ps]/2) − Vs] ∗ Prob(Pb(Vb) ≥ Ps) (3.3)

where E[Pb(Vb)|Pb(Vb) ≥ Ps] is the expected price given by the buyer’s strategy, conditional that the price of the buyer’s strategy is greater than or equal to the price of the seller’s offer price.

We will now follow Gibbons’s model for deriving a linear Bayesian Nash equi- librium of a double auction. As Gibbons notes, “we are not restricting the players’ strategy spaces to only include linear strategies. Rather, we allow the players to choose arbitrary strategies but ask whether the equilibrium is linear.” Though other equilibria strategies do exist, the linear equilibrium strategy is one of the more fa- vorable strategies in terms of properties of efficiency. That is, the linear equilibrium strategies tend to maximize the gains of each player more than any other possible

Bayesian Nash equilibria in a double auction.

We start by supposing that the seller’s straegy is Ps(Vs)= as + csVs. There- fore, Ps is uniformly distributed on [as, as + cs]. Thus, 3.2 will become

maxPb [Vb − (1/2) ∗ (Pb +(as + Pb)/2)] ∗ (Pb − as)/cs (3.4)

and solving 3.4 for Pb, we obtain

Pb = (2/3)Vb + (1/3)as. (3.5)

34 Similarly, if we suppose that Pb(Vb) = ab + cbVb, then Pb is uniformly dis- tributed on [ab, ab + cb]. Therefore, 3.3 will become

maxPs [(1/2) ∗ Ps +(Ps + ab + cb)/2 − Vs] ∗ (ab + cb − Ps)/cb, (3.6)

which consequently gives us Ps = (2/3)Vs + (1/3)(ab + cb).

Therefore, from 3.4 and 3.6 we get cb = (2/3), ab = as/3,cs = (2/3), and as = (ab + cb)/3. Using these values, we can finally solve for our linear equilibrium strategies. Thus,

Pb(Vb)=(2/3)Vb + (1/12) (3.7) and

Ps(Vs)=(2/3)Vs + (1/4). (3.8)

3.4 Applying Double Auction Model to MLB Free Agency

Now that we have successfully developed our double auction model, we can apply it to the free agencies of Michael Bourn and Mark Reynolds. Bourn and Reynolds will be the sellers in our double auction model, and the Cleveland Indians will be the buyers. It will be necessary for us to develop reasonable buyer and seller valuations so that we can perform the double auction as we derived above. We will use our results from our player family regression models along with our Earned Value Formula to develop the buyer valuations in our study. Our seller valuations will be estimated based on each individual player’s past salaries and performance. Again, our goal is to

find out what our linear equilibrium strategies are for the buyer and the seller. Thus, 35 we will be able to estimate an efficient amount of money the deal should take place at according to our game theoretic double auction model.

3.4.1 Michael Bourn Double Auction Model

We will first simulate our double auction with Michael Bourn. The first thing that we will calculate is the buyer’s valuation. The buyer’s valuation will be based on

Michael Bourn’s predicted WAR in our previous chapter, as well as the Earned Value

Formula we previously presented in this chapter. We had obtained a predicted WAR of 3.3 for Michael Bourn using our Bourn’s player family regression model. Therefore we will set A = 3.3 in our Earned Value formula. The league average WAR for outfielders was 1.5 in 2012. We calculated this average by obtaining data for all qualified outfielders in 2012 from FanGraphs, and taking the average WAR of all these players. Thus, B = 1.5. The league average salary for outfielders that were free agents was 6.75 million dollars. We obtained the reports on player salaries from usatoday.com. Therefore D =6.75, and we calculate C =6.75/1.5=4.50. Therefore, we calculate Michael Bourn’s Earned Value to be

EarnedV alue = (3.3 − 1.5) ∗ (6.75/1.5)+6.75 = 14.85, (3.9)

and set Vb = 14.85.

Now we must calculate a reasonable estimate of Vs, or Michael Bourn’s val- uation of himself. We come up with this estimate by observing the following graph.

This graph shows a regression model fit to Michael Bourn’s salary history.

Note the high value of R squared in this model. Also, Michael Bourn’s performance 36 Figure 3.1: Michael Bourn Salary History

has either improved or stayed relatively consistent during this period of time. Thus, it would not be unreasonable for this trend to continue for year seven. Therefore, we substitute the number 7 into our regression equation obtained form Bourn’s salary history. Therefore, we obtain

11.78 − 11.77(7) + 0.36(72)=10.44

from our equation. Thus, Vs = 10.44 million dollars.

Now that we have Vb and Vs, we are able to derive Pb and Ps based on our linear equilibrium strategies. However, in order to use our double auction model, all valuations must be drawn from independent uniform distributions on the interval

[0, 1]. Therefore, we need to find a way to reasonably scale our valuations to fit this requirement. 37 First, we need to establish a reasonable interval for possible valuations of

Michael Bourn. We do this by taking a look at the career high and low WARs for

Bourn over the course of his career. Bourn had a career high WAR of 6.2 in 2012 with the Atlanta Braves, and a career low WAR of −0.2 in 2008 with the Houston

Astros. Using our Earned Value formula, the career high year yields an Earned Value of 27.89 while the career low year yields a negative Earned Value. Thus we will set our interval for possible valuations of Michael Bourn at [0, 27.89]. If we scale this to the interval [0, 1], we find that our Vb of 14.85 is approximately 0.53 on the interval [0, 1].

Similarly, our Vs for Bourn of 10.44 is approximately 0.37 on the interval [0, 1]. We

find it interesting that the team’s valuation of Bourn is higher than Bourn’s valuation of himself based on our results.

Thus from 3.7, we obtain

Pb(0.53) = (2/3) ∗ (0.53)+ (1/12) = .44, (3.10) and from 3.8 we obtain

Ps(0.37) = (2/3) ∗ (0.37)+ (1/4) = .50 (3.11)

Therefore, converting back to dollars, we see that Pb = 12.27 million dollars and Ps = 13.95 million dollars. Since Pb ≤ Ps, then trade should not occur at all.

Thus, our double auction linear equilibrium strategies tell us that the it is not possible to achieve a double auction linear equilibrium with the valuations we have obtained.

38 3.4.2 Mark Reynolds Double Auction Model

We now go through a similar process with Mark Reynolds. Reynolds’s predicted

WAR for 2013 using our regression model from the previous chapter was 1.1. Thus

A =1.1. The league average WAR for third basemen in 2012 was 1.5. Therefore we set B =1.5 again. The league average salary for all third basemen who have qualified for free agency was 6.72 million dollars. Again, this was obtained from usatoday.com.

Thus, D = 6.72 and C = 6.72/1.5=4.48. Using our Earned Value formula, we calculate Mark Reynolds’s Earned Value to be

EarnedV alue = (1.1 − 1.5) ∗ (6.72/1.5)+6.72=4.93. (3.12)

Therefore, Vb =4.93 million dollars.

Now we explain how we estimate Vs, or Reynolds’s valuation of himself. How- ever, we must estimate Reynolds’s valuation differently than we estimated Michael

Bourn’s, as Reynolds has not performed with the same consistency that Bourn had performed at throughout his career. It made sense to look at the trend in Bourn’s salary, since his performance either improved or stayed consistent for the majority of his career. Reynolds on the other hand, was coming off of his two worst seasons in terms of WAR. During his first four seasons in the MLB, Reynolds averaged a WAR of 1.95. However, Reynolds’s performance then regressed in 2011 with a WAR of

-0.1. Despite this decrease in performance, the Baltimore Orioles paid Reynolds 7.5 million dollars in 2012, as they likely were counting on Reynolds returning to closer

39 to his past form. Instead, Reynolds turned in his second straight season with a WAR of -0.1.

We calculate Vs by using simple proportions of Reynolds’s past WAR data and salary data. Consider that prior to the 2013 season, Mark Reynolds always had a WAR in the interval [−0.1, 3.2]. We will add 0.1 to this interval to make the minimum value zero. Thus, Reynolds’s season WAR interval ranges from [0, 3.3].

We use Reynolds’s average WAR of 1.95 from 2007-2010, and we relate this WAR to the 7.5 million dollars Reynolds earned in 2012. We then look at Reynolds’s subpar 2011 and 2012 seasons in which he had a WAR of −0.1 in each year. Taking these two seasons in account, Reynolds had an average WAR of 1.27 throughout his whole career. These two poor seasons led to an approximate 21 percent decrease in career average production for the interval we’re looking at. Therefore, we decrease

Reynolds’s 2012 salary of 7.5 million by 21 percent to obtain Vs. Thus we obtain

Vs = (7.5) − (7.5 ∗ .21) = 5.93 (3.13)

Now we calculate Pb and Ps for Mark Reynolds’s double auction. Again, we need to find a reasonable interval for Mark Reynolds’s valuations, and then we need to scale it to the interval [0, 1]. Reynolds’s career high WAR is 3.2 with the

Arizona Diamondbacks in 2009, while his career WAR is −0.1 with the Baltimore

Orioles in 2011 and 2012. Using our Earned Value formula for Reynolds, we see that his career high year yields an Earned Value of 14.34 and his career low year yields a negative Earned Value. Thus, our interval of possible valuations for Mark Reynolds

40 is [0, 14.34]. If we scale this to [0, 1], our Vb of 4.93 is approximately 0.34, while our

Vs of 5.93 for Reynolds is approximately 0.41. Thus, we get

Pb(0.34) = (2/3) ∗ (0.34)+ (1/12) = 0.31 (3.14) and

Ps(0.41) = (2/3) ∗ (0.41)+ (1/4)=0.52. (3.15)

Converting these back to dollars, we get Pb = 4.45 million and Ps = 7.46 million. Again, since Pb ≤ Ps, trade should not occur. As was the case with

Michael Bourn, we conclude that no double auction equilibrium is attainable for

Mark Reynolds.

41 CHAPTER IV

CONCLUSION

4.1 Actual Results from 2013

We have gone through the process of predicting the future performances of Michael

Bourn and Mark Reynolds, converting these predictions to dollar amounts, and sim- ulating a double auction to see if a linear equilibrium if possible in either case.

Michael Bourn ended up signing with the Cleveland Indians for 48 million dollars over four years with a vesting option for 2017. This means Bourn is averaging

12 million dollars per year. Since our model results told us that the deal should not have taken place, this indicates that either the Indians overvalued Bourn, Bourn undervalued himself, or a combination of the two occurred in reality in relation to our model. We believe that the most likely scenario was that Bourn undervalued himself more than the Indians overvalued Bourn. Our reasoning for this has to do with the timing of when Bourn was signed. Bourn was signed by the Indians very late in the offseason just prior to Spring Training. Most quality free agents have signed with their new teams weeks prior to this. We hypothesize that Bourn lowered his self-valuation in order to increase the chances that a team would sign him. Using our model valuations of Pb and Vb from last chapter, we find that Bourn’s Vs had to have

42 been around 7.25 million dollars for the deal to take place at 12 million dollars. We simply solved our game theory strategy equation for Vs to obtain this value.

Considering that our Vb for Michael Bourn was 14.85 million dollars, this appears to be a smart deal on the surface for the Indians, as they likely took advantage of Bourn undervaluing himself to get the deal done. The deal works out to a favorable buyer utility of 14.85 − 12 = 2.85 million dollars, which means that the Indians are paying Bourn less than what our model showed he is worth. However we still need to be cautious when interpreting the results. Right now we are assuming that Bourn’s contract is evenly divided each year, and also that Bourn’s performance will stay constant each season. Neither assumption is considered to be true. Bourn actually made 7 million dollars in 2013, and is scheduled to make 13.5 million dollars in 2014,

13.5 million dollars in 2015, and 14 million dollars is 2016. Also remember that

Bourn’s performance is expected to trend downward to some degree in future years based on Bourn’s regression model. The deal might not look so effective for the

Indians in 2016 when Bourn is making 14million dollars, and his on field production has likely further decreased. Nonetheless, it certainly does look like the Cleveland

Indians signed Bourn to reasonable contract based on our Vb, the resulting postive buyer utility, and the likely hypothesis that Bourn ended up undervaluing himself.

Also, for the record, Bourn did end up experiencing a decrease in production in 2013 with a WAR of 2.0. Though our prediction overestimated Bourn’s WAR by 1.1 wins, our prediction was more accurate than the vast majority of expert predictions from professional baseball analysts and complex forecasting systems as well. We do

43 expect Bourn to rebound to some extent in 2014 based on the results of our model, but Bourn’s past season in 2013 might further validate our claim that the deal will not look very efficient nor effective in the later years of Bourn’s contract.

Mark Reynolds ended up signing a one-year deal with the Cleveland Indians for 6 million dollars in 2013. Unfortunately, based on our results, this led the Cleve- land Indians to a negative buyer utility of 4.93 − 6= −1.07 million dollars. As with

Michael Bourn, our results showed that a deal should not have taken place between the Indians and Mark Reynolds. Again, this means that either the Indians overvalued

Reynolds, Reynolds undervalued himself, or a combination of the two occured. Based on our negative buyer utility, it appears that the Indians overvalued Reynolds. If we keep the Ps and Vs the same for Reynolds as we used in our previous chapter, then we find the Indians must have valued Reynolds at around 7.32 million dollars based on our model. Our utility tells us that the Indians overpaid Reynolds by 1.07 million dollars in 2013.

What actually transpired in 2013 for Mark Reynolds and the Indians supports our results. Reynolds posted a WAR of 0.4 in 2013, which was 0.7 wins less than we predicted. However, Reynolds was actually released by the Indians in mid-August, as the Indians could not afford Reynolds’s lack of production as they were vying for a spot in the postseason. We are able to accurately conclude from our results that this was an ineffective contract negotiated by the Cleveland Indians.

44 4.2 Limitations

We would like to keep in mind a few limitations of our study. One of our bigger limitations occured in determining Vs in our double auction model. We used our regression model to develop the Vb in our study, as our perspective was consistent with the team’s perspective. However, we truly do not know the true value of Vs.

Michael Bourn and Mark Reynolds clearly never made their valuations of themselves public. Therefore, we simply had to make our best estimate at reasonable values of

Vs.

We are also limited in our conclusion, as we do not know what a typical buyer’s utility is for a Major League Baseball team. We do not know if it is normal to have a positive utility as in the case with Michael Bourn, or a negative utility as in the case with Mark Reynolds. Are free agents usually underpaid or overpaid according to our model? These questions need to be addressed in order to entirely assess how effectively the Cleveland Indians spent their money. We would need to run our model for every free agent signing for every team to properly compare our results. This is something that can be considered potential future research for our study.

4.3 Future Research

There are a number of different routes we can go in terms of future research. As we just mentioned, it would be wise to run our analysis for all teams to better interpret

45 our results. Also, we could further develop and test additional statistics and cutoff points to be included in our criteria. Our criteria was rather general in nature and included the major time-independent statistics. The major purpose of building our player family model was to show that often times, similar players have similar trends in terms of production throughout their careers. It was not a major goal of ours to test the thousands of possible statistical combinations and cutoff points to show which combination led to the best relationship. This is something that can be explored more in the future.

We could also research ways to extend our study beyond free agency. The techniques and processes used could potentially be extended to evaluate trades, the

MLB Draft, and minor league signings. We would also like to more rigorously extend our study to evaluating multi-year contracts as well. The last major area of future research would be to look into extending our double auction into an auction with one seller and multiple bidders. This would account for competition between teams for an individual player, and the changes that come in the negotiation process with these additional factors taken into consideration. It can certainly still be argued that the double auction is sufficient for our purposes though. Many do not believe that significantly overpaying for a player to prevent competition from obtaining this player is the most effective and efficient way to go. Regardless, we have developed the framework of a process that can be a very valuable tool for evaluating the spending of Major League Baseball teams.

46 BIBLIOGRAPHY

[1] Michael Lewis. Moneyball. W.W. Norton, New York, New York, 2004.

[2] Bill Felber. The Book on the Book. Thomas Dunne Books, New York, New York, 2005.

[3] Robert Gibbons. Game Theory for Applied Economists. Princeton University Press, Princeton, New Jersey, 1992.

[4] K. Chatterjee and W. Samuelson. Bargaining under incomplete information. Op- erations Research, 31.5:835–851, 1983.

47