Violent Strategies

VIOLENT STRATEGIES
AN INQUIRY INTO THE EFFECTS OF
PENALTIES IN THE NFL
____________________________________
A Thesis
Presented to the
Faculty of
California State University, Fullerton
____________________________________
In Partial Fulfillment
of the Requirements for the Degree
Master of Arts
in
Economics
____________________________________
By
Dana Shapiro
Approved by:
Dr. David Wong, Committee Chair Date
Department of Economics
Dr. Andrew Gill, Member Date
Dr. Robert Mead, Member Date

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ABSTRACT
The actions taken by the National Football League (NFL) to mitigate the threat of injuries to quarterbacks were constructed using an inappropriate framework. By failing to take into account the incentives of the offense, policy makers are exacerbating the problem–altering the game in such a way that increasingly violent strategies prevail as optimal. It appears both theoretically and empirically that defensive yardage penalties, assessed for fouls typically committed during passing plays, augment payoffs in such a way that the intention of decreasing injuries is offset by increased passing frequency. In contrast, offensive yardage penalties assessed for feigning illegal contact or being intentionally reckless decrease the probability of injury. Finally research and education regarding the long-term effects of injuries, which allow players to more accurately project the expected costs of violence, have the potential to reduce the probability of injury, and depend on the relative magnitudes of the players’ reactions.

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TABLE OF CONTENTS
ABSTRACT ................................................................................................................... ii
LIST OF TABLES ......................................................................................................... iv
LIST OF FIGURES ....................................................................................................... v
LIST OF EQUATIONS ................................................................................................. vi
ACKNOWLEDGMENTS ............................................................................................. vii
Chapter
1. INTRODUCTION ................................................................................................ 1
2. RELEVANT LITERATURE ................................................................................ 4
3. THEORETICAL MODEL .................................................................................... 9
Players ................................................................................................................... 10
Strategy Sets ......................................................................................................... 10
Payoffs .................................................................................................................. 11
Penalties ................................................................................................................ 13
4. EMPIRICAL MODEL .......................................................................................... 21
5. DATA ................................................................................................................... 26
6. RESULTS ............................................................................................................. 28
7. CONCLUSIONS AND CAVEATS ..................................................................... 33
REFERENCES ............................................................................................................. 38

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LIST OF TABLES
Table Page
1. Summary of the Theoretical Model ..................................................................... 20
2. Summary Statistics .............................................................................................. 27
3. Estimation of Equation 12 ................................................................................... 29
4. Estimation of Equation 13. .................................................................................. 30
5. Joint Significance F Tests for Select Dummies and Interactions. ....................... 32

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LIST OF FIGURES
Figure Page
1. Summary of Game ............................................................................................... 10

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LIST OF EQUATIONS
Equation Page
1. P* Derivation of probability of a pass ............................................................... 13
2. Q* Derivation of probability of a blitz ............................................................... 13
3. Effect of changes in defensive payoff on probability of a pass .................... 13
4. Effect of changes in offensive payoff on probability of a blitz .................... 13
5. Pʀ(Injury) Probability of Injury .......................................................................... 14
6. Effect of changes in defensive yardage penalties on injury .......................... 15
7. Effect of changes in offensive yardage penalties on injury .......................... 17
8. Effect of changes in education on probability of injury ............................... 18
9. Ω Offensive reaction to education .................................................................... 18
10. Δ Defensive reaction to education .................................................................... 18
11. Effect of changes in education of probability of injury ................................ 18
12. Passoiws Empirical estimation of probability of a pass ........................................ 22
13. Sackoiws Empirical estimation of probability of a sack ........................................ 24

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ACKNOWLEDGMENTS
Indispensable to the completion of this analysis were: Dr. David Wong for his attention to eloquence, Dr. Andrew Gill and the creative process which he employs, and Dr. Robert Mead for his awareness of, and ability to account for, the countless nuances of football.

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CHAPTER 1
INTRODUCTION
The problem of players sustaining career-threatening injuries while playing professional football has become an issue of growing concern for both players and league officials. Rules such as defensive yardage penalties have been employed with the intent of protecting quarterbacks who have been deemed defenseless and thus more susceptible to injury. Often, the intentions of rules that attempt to regulate risk, such as seatbelt laws, are offset by individuals acting more carelessly, such as driving more recklessly. The questions become: are the regulations imposed by the NFL susceptible to what game theorists refer as strategic offsetting behavior due to the increased expected profitability of passing, and to what degree will this offsetting change the probability that the players in question sustain injuries.
These questions are important because a 2011 report by the NFL Players Association (NFLPA) found that players’ injuries are becoming both more frequent and more severe. In particular, the number of players on the injured reserve list increased dramatically from 250 to 350 between the 2009 and 2010 seasons. 1 In addition, the percentage of players who sustained at least one concussion rose 300% between 2006 and
1 National Football League Players Association, “Dangers of the Game,” Edgeworth Economics, accessed December 12, 2012, http://www.esquire.com/cm/esquire/data/Dangers-of-the-Game-Report- Esquire.pdf .

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2010.2 Moreover, the medical literature is beginning to show that concussions pose a far greater long-term threat than previously believed. Specifically, findings suggest that the onset of dementia-related syndromes may be initiated by repetitive cerebral concussions. 3 Further compounding the problem has been the perceived innocuousness of minor head trauma by NFL players exemplified by the decision made by Peyton Manning, then the quarterback of the Indianapolis Colts, to openly test low on baseline concussion diagnosis tests in order to be cleared to play after sustaining mild concussions. 4
Recently, the NFL made it public that it intends to discourage violent behavior in the hope of reducing its comparative negligence in lawsuits brought by players who suffer from the long-term effects of concussions. Increases in defensive yardage penalties, implemented as a means of increasing the marginal cost of risky play, are thought to impose a tax on the supply of violent behavior thus reducing the equilibrium quantity. However, this analysis fails to account for the reaction of the offense. The policies being employed by the NFL to regulate the risks of the game are being strategically offset in the same way that seatbelt laws, which attempt to reduce fatalities, cause drivers to drive less carefully and offset the intention of reducing risk. The aim of this study is to analyze the effects that the actions taken by the NFL have on the
2 National Football League Players Association, “Dangers of the Game,” Edgeworth Economics, accessed December 12, 2012, http://www.esquire.com/cm/esquire/data/Dangers-of-the-Game-Report- Esquire.pdf .
3 Kevin Guskiewicz, et al., “Association between Recurrent Concussion and Late-Life Cognitive Impairment in Retired Professional Football Players,” Neurosurgery. 57, no. 3 (2005): 719.
4 Mike Florio, “Peyton Manning Admits to Tanking Baseline Concussion Tests,” NBC Sports, last modified April 27, 2011, http://profootballtalk.nbcsports.com/2011/04/27/peyton-manning-admits-to- tanking-baseline-concussion-tests/.

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incentives of the players and how one might expect the probability of injury to change as a result. The study employs a simplified model of the game and narrows the scope of the analysis to a specific type of risky play in order to anticipate the effects that various regulations will have on the probability that the quarterback sustains an injury. Specifically, it will be shown theoretically and empirically that increased severity and enforcement of defensive yardage penalties, for fouls typically committed during passing plays, leads to an increase in the probability of injury.

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CHAPTER 2
RELEVANT LITERATURE
As mentioned above, the present analysis belongs to the subcategory of Game Theory that is concerned with the strategic offsetting of policies that are intended to regulate risk by the perverse incentives they create. This effect was first rigorously studied by Peltzman who found that following the implementation of seat belt laws increased accident frequency fully offset the effect of increased safety–resulting in no net change to the death rate associated with car accidents. 5 More recently, work has been done studying the effect of a policy instituted by the NCAA in men’s collegiate basketball which moved the three-point line further from the hoop in an attempt to spread out the defense, and thus increase two-point shooting percentage. It is hypothesized that because three-point shots are made harder, offenses tend to value them less–causing the defense to defend more against two-point shots, thus partially defeating the intention of the policy to spread out the defense. 6 This phenomenon appears in football in much the same way as in other contexts with the intent of the defensive yardage penalties, to
5 Sam Peltzman, “The Effects of Automobile Safety Regulation,” Journal of Political Economy 83, no. 4 (1975): 677.
6 Bryan McCannon, “Strategic Offsetting Behavior: Evidence From National Collegiate Athletic Association Men’s Basketball,” Contemporary Economic Policy 29, no. 4 (2011): 550.

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reduce injuries, being offset by the offense putting itself in precarious situations more often due to the increased profitability of these situations.
The problem that the NFL and its players face is similar in nature to that which all employers and employees face in hazardous occupations. However, there is a stark asymmetry between how the NFL attempts to regulate risks and how other hazardous occupations, such as boxing, attempt to regulate risks. Arguments have been advanced contending that football should be considered more of a combative sport than it has traditionally been considered. For example, Daniel Goldberg notes that the model of health delivery currently endorsed by every NFL team is ethically suboptimal. 7 Specifically, there exists a principal-agent problem between the team owners, medical staff, and the players stemming from the asymmetry between their respective utility functions. While attempting to maximize profits, owners coerce medical staff to return injured players to the game prematurely, effectively disregarding the disutility that such premature clearance will ultimately have on players. Due to this asymmetry, Goldberg argues that prevention of, and liability for, injury should be dealt with in a way similar to that of boxing with more objective return to play guidelines. This would then reduce the asymmetry between the utility of the player and the payoff to the owner by more tightly tying the profitability of players to their health.
7 Daniel Goldberg, “Concussions, Professional Sports, and Conflicts of Interest: Why the National Football League’s Current Policies are Bad for Its (Players’) Health,” HEC Forum (2008): 337.

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The decision to commit a foul that would result in a penalty can be analyzed in light of the Becker Model of Crime and Punishment. 8 In the arena of professional sports, fouls may be regarded as akin to criminal activity. Because the sanctioning bodies of professional sports saliently assign various penalties to fouls and players are able to estimate their potential gains, it follows naturally that players will allocate time between “criminal” and reputable behavior in the form of a mixed strategy of lawful and unlawful plays. Authors have used a similar line of reasoning to estimate a demand curve for physically aggressive behavior in the Barclays Premier League of soccer. By considering aggressive behavior as an input into the production of potential league points, an empirical estimation of the demand for aggression yields own price elasticities for all teams in the sample. The findings suggest that although entirely inelastic, the demand curves of the best and worst teams in terms of skill are proportionately less responsive to changes in the price of misconduct as measured by an index of consequences, with certain teams having an infinitely inelastic demand for violent play. 9
Professional sports and the myriad data they produce also lend themselves naturally to game theoretic and applied econometric analysis. Competitive games of skill can serve as analogues for other, less accessible, yet nontrivial economic phenomena. Once the immaterial has been discarded from the structure of both the sport and the phenomenon, similarities to classic game theoretic situations are often discovered. For
8 Gary Becker, “Crime and Punishment: An Economic Approach,” Journal of Political Economy 76, no. 2 (1968): 169.
9 Todd Jewell, “Estimating Demand for Aggressive Play: The Case of English Premier League Football,” International Journal of Sport Finance 4, no. 3 (2009): 192.

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example, the decisions to call a passing or a running play, given that the star quarterback has sustained an injury, has been modeled in the framework of a zero-sum matching pennies game. Findings suggest that given a decrease in the productivity of an input (in this case passing), there exist situations where the players will have no incentive to alter their strategies. 10 That is, if the second-string quarterback is as equally lacking in passing ability as he is in his capacity to capitalize on the defensive error, the players will adopt the same mixed strategies as before the injury occurred. The inadequacy of data on infrequent yet significant events typified by classic games makes it difficult to infer whether or not players actually follow mixed strategies, however it may be argued that if those who play the pass/run game adopted a mixed strategy, then similar behavior may occur in comparable circumstances that are less easily illuminated empirically.
The contributions of the present study to the existing body of literature, which uses professional sports to analyze economic phenomena, are three-fold. First, it shows that, the regulations employed by the NFL are flawed. Gaining a better perspective on the way strategic interactions evolve in response to costs and benefits of potentially injurious behavior may allow policies to be put in place that more efficiently curtail violence. Second, it shows that by understanding the way penalties affect the players in football, one may more accurately predict how the regulation of risk will affect other pseudo-zero-sum games (games where reductions in the payoff to one player amount to increases in the payoff to the other). Finally, this analysis contributes to the growing
10 Joseph McGarrity and Brian Linnen, “Pass or Run: An Empirical Test of the Matching Pennies Game Using Data from the National Football League,” Southern Economic Journal 76, no. 3 (2010): 791- 810.

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body of empirical work that seeks to expand the implications of game theoretic constructs by lending evidence in favor of the hypothesis that players follow mixed strategies.

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CHAPTER 3
THEORETICAL MODEL
The game of football as whole is far too intricate to analyze in any one all- encompassing model. The sheer magnitudes of the different strategies sets that are available to the players make a comprehensive model of the entire game impossibly complex. A solution to the game of football would be so nuanced and would require such an intimate knowledge of every possible detail that it quickly becomes unattainable. However, upon narrowing the scope of analysis and constructing a set of plausible assumptions, it becomes possible to analyze the specific situations that are relevant to the concerns that the NFL has regarding injuries to its quarterbacks.
Because the aim of the analysis is to determine how quarterbacks and defenders will respond to regulation, the theoretical model focuses on the decision that the offense faces of how frequently to pass, the decision that the defense faces of how often to defend against a pass, and how regulations will augment the payoffs of both sides in this context.
Unlike basketball and soccer, which are played in an essentially continuous fashion, football is naturally broken up into non-arbitrary, discrete time intervals, or plays. It seems natural therefore, to model each play as a separate stage game.

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Players
Let the Home and Away teams be Players 1 and 2, respectively. The team as a whole is considered one strategic decision-making unit. At any stage one player is of the type–Offense and the other Defense. Players make decisions simultaneously.
Strategy Sets
The offensive strategy set is partitioned by passing plays and running plays; kicking plays can safely be omitted due to their being strictly dominated or dominating in most stages.
The defensive strategy set is more complex. Given that the offense will attempt to advance the football by either passing or running (rushing) down the field, it can be assumed that the defense will focus on either defending against a pass or a run. One effective strategy to defend is with the use of a blitz. There are broadly three types of blitzes: pass, run, and zone. In a pass-blitz, the defense devotes more than the usual amount of defenders to targeting the quarterback, increasing pressure with the intent of tackling (sacking) the quarterback, or causing an interception or fumble. The run-blitz is designed in much the same way as the pass-blitz, but with the intent of devoting more than the usual amount of defenders to tackling a runner. The zone-blitz is different in that it seeks to cause an offensive error with the use of confusion rather than with the use of increased pressure. For the purpose of this analysis it may be instructive to partition the defensive strategy set between the pass-blitz, and other forms of defense, as the pass- blitz poses the greatest risk of injury to the quarterback. In addition to reduce confusion, a pass-blitz will henceforth be referred to simply as a blitz.

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Payoffs
Each team seeks to win a championship. The result of each game alters the probability of winning a championship. Each play changes the context of the game, and thus changes the probability of winning. Payoffs are therefore defined as the marginal change in the probability of winning a championship given: any yards gained or lost on each play, game context, which teams are playing, at what point in the season the game is taking place, etc. In other words, players are always thinking about how each action they take will change the chance that they win the championship. For example, the expected utility from attempting a pass that wins the Super Bowl is large compared to one that simply gets a first down in an otherwise uneventful blowout during the season.
Although the payoffs differ among stages, their relative magnitudes are assumed to remain constant throughout all stages. It is assumed that the offense would rather run against a blitz, and the defense would rather blitz against a pass. The game is summarized in Figure 1.
Where: A< B
C > D
E > F G < H
Figure 1. Summary of Game
Defense
Q
(1 – Q)
Blitz
Other
Offense
P
Pass
(A,E)
(C,F)
(1 – P)
Run
(B,G)
(D,H)

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The elements {A,…,H} of the bi-matrix are the payoffs to the offense and the defense from each of the four strategy combinations or states of nature. By convention the first element of the parenthetic pair in each square is the payoff to the offense, and the second is the payoff to the defense. For example, given that the offense attempts a pass while the defense attempts a blitz, the payoffs will be A and E to offense and defense, respectively.
The bold and underlined payoffs are the highest payoff that can be expected given the other players choice of action. In other words, given a blitz, the payoff to the offense from passing (A) will always be lower than that of running (B). In addition, given a pass, the payoff to the defense from a blitz (E) will be higher than other forms of defense (F).
The assumptions imply that no pure-strategy Nash equilibrium exists, i.e. no strategy is dominant in all situations. Thus, it may be assumed that the players mix their play between their respective strategies according to a probability distribution that makes their opponent indifferent between their choices, a method known as employing mixed strategies. Given this indifference, there will be no incentive for either player to alter their chosen probability distribution, and equilibrium will be reached. The mixed strategies are calculated in Equations 1 and 2, where the + and – indicate the sign of the terms below them. Subsequent equations are numbered in ascending order.
E(Blitz) = E(Other) E(Pass) = E(Run)
PE + (1 – P)G = PF + (1 – P)H QA + (1 – Q)C = QB + (1 – Q)D
PE + G – PG = PF + H – PH QA + C – QC = QB + D – QD
PE – PF – PG + PH = H – G QA – QB – QC – QD = D – C
P(E – F – G + H) = H – G Q(A – B – C + D) = D – C

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+ –
0 < P* = < 1 (1) 0 < Q* = < 1 (2)
+ + – –
where P* is the probability that the offense will pass, (1 – P*) is the probability that the offense will run, Q* is the probability that the defense will blitz, and (1 – Q*) is the probability that the defense will not blitz. Note:
+
= − < 0 (3)
+
and,
–
= − > 0 (4)
+
Equation 3 implies that decreases in the defensive payoff of the pass/blitz situation (E↓) cause the offense to pass more often (P*↑). Equation 4 implies that increases of the offensive payoff of the pass/blitz situation (A↑) cause the defense to blitz more frequently (Q*↑).
Penalties
The aim of this theoretical analysis is to determine how penalties change the probability that the quarterback sustains an injury. To do this, it must first be established how penalties change the payoffs of each player, then a link between the payoffs and the probability of injury can be formed to shed light on the consequences of specific policies.

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A typical way penalties are given is by taking yards from the team that committed a foul and awarding them to the other team in a zero-sum fashion. 11 However, since payoffs are in terms of changes in the probability of winning a championship, the yards gained and lost do not necessarily increase and decrease the payoffs symmetrically. Therefore, changes in payoffs from penalties are pseudo-zero sum in that they necessarily cause an opposing award, but not necessarily of the same magnitude. For instance, if the top ranked team were to be penalized for a foul committed against a team that is in need of a win to remain in contention, the decrease in the probability of the top ranked team winning the championship would be much smaller than the increase in the probability of the mid-level team winning the championship.
Let ϑ represent a penalty such that > 0 and < 0.
characterizes the typical defensive yardage penalty that only affects the payoffs in the pass/blitz situation, such as roughing the passer, as defenses cannot commit such a foul when the offense attempts a running play. This penalty accrues as a net loss to the defense (E↓), and a net gain to the offense (A↑).
To establish a link between the players’ strategies and injuries, let the probability that the quarterback sustains and injury be given by,
Pʀ(Injury) = ƒ(P*,Q*, X) (5)
where X is a vector of exogenous determinants of Pʀ(Injury).
11 National Football League, “Official Playing Rules.” 2012 Rule Book, http://www.nfl.com/rulebook (accessed Dec 12, 2012).

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Assume , > 0. In other words, if the offense attempts more passing plays (P*↑) or the defense attempts to blitz more frequently (Q*↑), then the probability that the quarterback will sustain an injury will increase (Pʀ(Injury) ↑) due to his exposure to situations conducive to injury.
Consider the marginal effect of a penalty, ϑ, on the probability that the quarterback sustains an injury, where the signs are indicated above each term:
+ + – – + + +
= • • + • • > 0 (6)
Note that if other factors are held constant, increases in the severity of this type of defensive penalty (ϑ↑) unambiguously increases the probability of injury (ƒ↑).
To show the merit of this conclusion, consider a stylized rule change that increases the severity of the penalty for roughing the passer from 5 to 15 yards, in a situation where the expected payoff gained from 15 yards is high, but that gained from 5 yards is low–i.e. when a first down is crucial. Due to the pseudo-zero-sum nature of the game, the increased severity of the penalty (ϑ↑) raises the expected payoff from drawing a foul (A↑), causing the offense to value passing more through , which in turn makes the defense attack the quarterback more frequently (Q*↑) through . Moreover, the penalty (ϑ↑) decreases the expected value of attempting a blitz (E↓), causing the defense to value attacking the quarterback less through , which in turn makes the offense able to pass more (P*↑) through . Since both the offense is passing more (P*↑) and the

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defense is blitzing more (Q*↑), the probability that the quarterback will get injured increases ( ↑) due to his exposure through and respectively.
Since payoffs are simply expectations, decreases in the variability of expectations act in much the same way as increases in severity. For instance, in 2005 the NCAA removed all references to the word “intent” from the definition of helmet-to-helmet contact in an effort to reduce traumatic head injuries. 12 This omission reduces the variability of attempting to draw a helmet-to-helmet foul, making plays where such fouls may be drawn more valuable to the offense. This causes a string of events similar to that described above, resulting in an increase in the likelihood that the quarterback sustains an injury.
Consider now an offensive yardage penalty.
Let Γ represent a penalty such that < 0 and > 0.
Γ characterizes the typical offensive yardage penalty that accrues as a net loss in expected utility to the offense (A↓) and as a net gain in expected utility to the defense (E↑), and typically only effects the pass/blitz situation. In addition to penalties for offensive misconduct typical to this state of nature such as holding, other types of offensive penalties exist. For example, in soccer, and more recently in basketball, the offense has been penalized for attempting to draw a foul by feigning illegal defensive contact, colloquially known as flopping. Although this exact type of conduct does not yet
12 National Collegiate Athletics Association, “A Primer on NCAA Rules for Football Safety,” Latest News, last modified October 20, 2010, http://www.ncaa.org/wps/wcm/connect/public/NCAA/Resources/Latest+News/2010+news+stories/October/A+primer+on+NCAA+rules+for+football+safety.

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exist in football, a penalty that awards yards to the defense in response to an offensive attempt to initiate contact would act in much the same way.
Consider the marginal effect of Γ on the probability that the quarterback sustains an injury.
– + – + + + –
= • • + • • < 0 (7)
The Γ penalty works in the opposite direction as the ϑ penalty. An increase in Γ decreases the expected value of attempting a pass (A↓) through , thereby making the defense necessitate the blitz less often (Q*↓) through . In addition, due to the pseudo- zero-sum nature of the game, the Γ penalty increases the expected value of attempting a blitz (E↑) through , thereby making the offense less able to attempt passing plays (P*↓) through . Since both the defense is blitzing less (Q*↓) and the offense is passing less (P*↓), the probability that the quarterback will sustain an injury decreases ( ↓) through and .
This type of offensive yardage penalty manages violent behavior by reducing the exposure of the quarterback to potentially injurious situations (P*↓), while reducing the frequency with which the defense attempts to blitz (Q*↓), thus reducing the probability of injury ( ↓).
Finally, consider how making players aware of the research being done regarding the long-term effects of the injuries in question lowers the expected payoff from plays where said injuries are most likely to occur.

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In this specific context, the research and education proposed pertains to injuries typically suffered when the offense attempts to pass and the defense attempts to blitz, such as head trauma to the linebackers and quarterbacks. The education affects the offense and the defense in the same way, by decreasing the payoff of attempting a pass and blitz respectively, but not necessarily to the same degree.
Let Ʈ represent an increased awareness by players of the long term effects of injuries such that < 0 and < 0.
Consider the marginal effect of Ʈ on the probability that the quarterback sustains an injury:
+ – – + + –
= • • + • • (8)
+ – –
Let Ω = • • > 0 (9)
+ + –
and Δ = • • < 0 (10)
Ω and Δ can broadly be interpreted as the offensive and defensive reactions to the penalty, and the effect those reactions have on the probability of injury respectively. Then,
< 0 if Ω < Δ (11)
In contrast to the defensive yardage penalty, the penalty (Ʈ↑) decreases the payoff expected by the offense from passing (A↓) through , which encourages the defense to blitz less frequently (Q*↓) through , ultimately decreasing the probability of injury

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( ↓) through . The effect of the penalty (Ʈ↑) on the offense is exactly the same as above, reducing the expected payoff for the defense (E↓) through , thus increasing the frequency of passing (P*↑) through and the probability of injury ( ↑) through .
To speculate about the magnitudes of Ω and Δ is not a simple task as each has three components that change with context of the game and the abilities each team. They are, (1) the degree to which the education reduces the payoff of the opposing team, (2) the degree to which that reduction increases passing and decreases blitzing, and (3) the degree to which the increase in passing and decrease in blitzing change with the probability of injury. For example, consider an offense that specializes in passing against a low ability defense. The change in the frequency that the offense passes in response to the change in payoff, , may be large because of their ability to capitalize by passing more. However, the change in the frequency that defense blitzes in response to the change in payoff, , may be small, because they may blitz very infrequently to begin with due to their low ability. This could result in an increase in the probability of injury because the increased passing drowns out the effect of the decreased blitzing. It is important however, to keep in mind that within this framework the defense does not reduce the frequency with which they blitz out of concern for their own players’ wellbeing, but instead do so in response to the decreased value placed on the pass by the offense.
Changing the perception of injuries, that have traditionally been considered benign in the long-term, effectively increases expected costs, thereby lowering the net

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expected payoff for both offense (A↓) and defense (E↓). A net reduction in the probability of injury ( ↓) will occur when the total effect of the penalty on the defense (Δ) is greater in magnitude than the effect on the offense (Ω).
By using a simplified model of a generic situation encountered by teams in an average football game, combined with a set of plausible assumptions, the following conclusions have been derived. First, defensive yardage penalties, assessed for fouls typically committed within the pass/blitz situation, unambiguously increase the probability that the quarterback sustains an injury. Second, offensive yardage penalties, assessed for fouls typically committed within the pass/blitz situation, unambiguously decrease the probability that the quarterback sustains an injury by decreasing the quarterback’s exposure and by reducing the probability that the defense attempts to blitz. Finally, research and education of the long term effects of concussions, that increase the marginal cost of the pass/blitz situation for both offense and defense, can decrease the probability that the quarterback sustains an injury if the way that the defense responds to the penalty outweighs the way that the offense responds, as summarized in Table 1.
Table 1. Summary of the Theoretical Model
E
P*
A
Q*
Pr(Injury)
Defensive Penalty
↓
↑
↑
↑
↑
E
P*
A
Q*
Pr(Injury)
Offensive Penalty
↑
↓
↓
↓
↓
E
P*
A
Q*
Pr(Injury)
Research and Education
↓
↑
↓
↓
↓ or ↑

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CHAPTER 4
EMPIRICAL MODEL
The primary theoretical conclusion that: defensive yaradge penalties–assessed for fouls typically commited during the pass/blitz situation–unambiguously increase the probability that the quarterback sustains an injury, hinges on the comparative statics of the mixed strategies. Specifically,
< 0 and > 0.
The first comparative static result reached in the theoretical model deals with how the offense will react to reductions in the defense’s payoffs. Specifically, < 0. Once again this implies that decreases in the payoff to the defense of a pass/blitz situation cause the offense to pass more frequently. In the theoretical model each play is modeled individually, so it follows that an emprical model designed to test the implications of the theory should be modeled at the same level. The following linear probability model of an offenseo passing on a given playi in a given weekw during a given seasons, is proposed:

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Passoiws = α + β1ln(average sacks per gameos) + β2field goal rangeoiws + β3can’t puntoiws + β4score differentialoiws + β5yards to gooiws + β6yard lineoiws+ β7quarteroiws + β8downoiws + β9defenseows + β10offenseows + β11weekows + β12biseasonos + β13offenseows*weekows + β14offenseows*biseasonos + β15weekows*biseasonos + β16offenseows*weekows*biseasonos + uoiws (12)
where uiows is assumed to be normally distributed with zero mean, and bold face parameters and variables denote vectors.
Equation 12 is an attempt to model empirically the choice faced by the offense of how frequently to pass. Since this choice is in theory dependent on the payoffs to the defense, which are in turn dependent on such things as: game context, which teams are playing, at what point in the season the game is taking place, etc., variables to account for such variation have been included in the empirical model. The score differential, the number of yards to go for a first down or touchdown, the yard line where the play begins, whether the offense is in field goal range, and whether punting is a dominated strategy have been included to account for how the game context will affect the probability that the offense attempts a pass. The average number of sacks per game for the defense in a given play, as well as a dummy for each defense have been included to account for variation due to which teams are playing. In addition dummy variables for the quarter and down have been included as passing is thought to bear some correlation to both. Offense dummies are included to account for the time invariant team characteristics. Week dummies are included to account for the point during the season at which the game is played. Dummies indicating every other season (biseason) are included to account for

23
season to season variation such as rule changes. Finally, a set of interactions are included to account for the variation in team makeup and thus ability over the course of, and between seasons. This set of interactions is crucial to account for the most classic example of omitted variables, that which is due to ability.
In the empirical analysis, the parameter of interest is β1, which can be interpreted as the change in the probability that the offense will pass (P*), given a one percent increase in average number of sacks per game held by the defense against which they are attempting to pass. The main point of the analysis is to test the null hypothesis that β1 < 0. In other words, the average number of sacks per game is used as a proxy for variation in the expected payoff to the defense of a pass/blitz situation (E). It is assumed that defenses with more sacks on average will expect higher payoffs from blitzing (E↑). Therefore, β1 approximates , qualitatively. From this estimate, inferences can be made regarding what effect a penalty to the defense will have on the offensive strategy.
From the theory and our understanding of the game, a priori predictions as to the signs of select coefficients in the model can be made. Specifically, it is expected that increases in the number of yards to go for a first down or touch down, and positive score differentials will increase the probability of a pass on any given play. In addition it is expected that situations where punting is dominated would increase the likelihood of passing, while situations where attempting a field goal dominates would lead to decreases in the probability of passing. For the other parameters in this model, no a priori predictions are made.

24
The second comparative static result reached by the theory is that > 0. Once again this implies that increases in the payoff to the offense of the pass/blitz situation cause the defense to blitz more. Following a similar line of reasoning, the following linear probability model of a defensed sacking the quarterback on a given playi in a given weekw during a given seasons is proposed to test this conclusion:
Sackoiws = α + λ1ln(average team incompletionsos) + λ2field goal rangeoiws + λ3can’t puntoiws + λ4score differentialoiws + λ5yards to gooiws + λ 6yardlineoiws + λ7quarteroiws + λ 8downoiws + λ9offenseows + λ10defenseows + λ11weekows + λ12biseasonos + λ13defenseows*weekows + λ14defenseows*biseasonos + λ15weekows*biseasonos + λ16defenseows*weekows*biseasonos + uoiws (13)
where uoiws is assumed to be standard normal and bold face denote vectors.
Equation 13 is an attempt to model the choice faced by the defense of how frequently to defend against a pass. Since this choice is in theory dependent on the payoffs to the offense, which are in turn dependent on such things as: game context, which teams are playing, at what point in the season the game is taking place, etc., variables to account for such variation have been included in the empirical model. Again, the score differential, the number of yards to go for a first down or touchdown, the yard line where the play begins, whether the offense is in field goal range, and whether punting is a dominated strategy have been included to account for how the game. The average number of team incompletions per game for the offense in a given play, as well as a dummy for each offense have been included to account for variation due to which teams are playing. Once again, week dummies are included to account for the point

25
during the season at which the game is played, and the biseason dummies are included to account for season to season variation, and defense dummies are included to account for time invariant team characteristics. In addition a similar set of interactions are included to account for the time varying ability of the defense over time.
Here, the parameter of interest is λ1, which may be interpreted as the effect that an increase of one incompletion per game on average of the offense will have on the probability that the defense accomplishes a sack of the quarterback. The main point of this analysis is to test the null hypothesis that -λ1 > 0. In other words, fewer incompletions are assumed to be indicative of less ineptitude on the part of the offense, causing them to expect higher payoffs from the pass/blitz situation (A↑) due to their increased ability to survive a blitz. If it can be assumed that more sacks are indicative of more attempts to sack (Q*), then, since incompletions proxy decreases in expected payoff (A↓), - λ1 can be used to approximate qualitatively. From this estimate, inferences can be made regarding the effect of a defensive penalty on the defensive strategy. Other than this prediction, no other a priori predictions are made about the parameters.

26
CHAPTER 5
DATA
Play-by-play data from the 2003-2010 regular seasons have been compiled by a team of statisticians and football enthusiasts headed by Brain Burke, founder of Advanced NFL Stats - a website dedicated to analysis of the NFL and vetted by the community of analysts. 13 Each observation includes the following: a unique game identification; the quarter, minute, and second at the start of the play; the teams on offense and defense; the down, line of scrimmage (ydline), and how many yards to go for either a first down or touchdown (togo); the season; the scores of the offense and the defense; and a description of the play. The score differential (diff) is generated by taking the difference between the scores of the defense and offense, with a positive value indicating that the offense is losing. Dummy variables for an attempted pass (pass) or a sack (sack) are set equal to one if somewhere within the description variable the word “pass” or “sack”, respectively, appear. A dummy indicating if the offense is within field goal range (fgrange) is equal to one if it is fourth down and ydline is less than 50. A dummy indicating that the offense can expect little to no value from punting (cantpunt) is equal to one if the play occurs in the last minute of the fourth quarter, it is fourth down,
13 Brian Burke, “Play-by-Play Data,” Advanced NFL Stats, last modified June 6, 2010, http://www.advancednflstats.com/2010/04/play-by-play-data.html.

27
and the score differential is positive. In addition, dummy variables indicating the offense, defense, quarter, down, week and every other season (biseason) were created.
Finally, average team statistics were gathered from Team Rankings, a website composed of various sports data. 14 In particular, average team incompletions (avtminc) and average sacks per game (avskpgm) were collected at the season level, and appended to the dataset. Table 2 provides summary statistics for select variables
Table 2. Summary Statistics
Variable
Observations
Mean
St. Dev.
Min
Max
Natural log of average sacks per game of the defense
310083
0.752
0.237
0.51
1.31
Average team incompletions of the offense
310083
12.92
1.98
0
17.7
Score differential (defense - offense)
310083
0.7
10.67
-59
59
Yards to go for a first or touch down
310083
7.77
4.59
0
90
Yard line
310083
48.92
25.51
0
100
Pass (binary)
310083
0.422
0
1
Sack (binary)
310083
0.028
0
1
4th Down on the 50 yardline or closer (binary)
310083
0.02
0
1
4th Down in 4th quarter with less than a minute left (binary)
310083
0.002
0
1
14 Team Rankings, “Team Stats,” NFL Stats, last modified February 2, 2013, http://www.teamrankings.com/nfl/stats

28
CHAPTER 6
RESULTS
The empirical model seeks to test the validity of the conclusion that defensive yardage penalties, assesed for fouls typically committed in a pass/blitz situation (which amount to A↑ and E↓), tend to increase the probability that quarterbacks will sustain injuries. Equation 6 states mathematically the above hypothesis:
+ – – + + +
= • • + • • > 0
hinges on the two intermediate conclusions: (1) that decreases in the payoff to the defense of the pass/blitz situation (E) increase the probability that the offense passes (P*), or < 0; and (2) that increases in the payoff to the offense of the pass/blitz situation (A) increase the probability that the defense attempts a blitz (Q*), or > 0.
Equation 12 seeks to estimate the change in the probability that the offense attempts a pass (P*) given a one percent increase in the average number of sacks of the defense against whom they are playing. Table 3 presents the results of the estimation of Equation 12, with standard errors clustered by offense.
Table 3 shows that holding week, season, defense, and game context constant, the natural log of the average number of sacks held by the defense has a statistically

29
significant negative effect on the probability that the offense attmepts a pass (P*). This supports the conclusion that < 0 because the ineptitude indicated by fewer sacks
Table 3. Estimation of the Probability of Pass
Independent Variables
Coefficient
P Value
Natural log of average sacks per game of the defense
-0.0295
0.001
4th Down on the 50 yardline or closer
-0.0756
0.000
4th Down in 4th quarter less than a minute and diff > 0
0.5551
0.000
0.0080
0.000
Yards to go for a first or a touch down
0.0129
0.000
Yardline
0.0003
0.000
This supports the conclusion that < 0 because the ineptitude indicated by fewer sacks on average reduces the expected payoff to defense in a similar way to that of a defensive yardage penalty, and causes the offense to pass more. Therefore, one may infer that the defensive yardage penalties would elicit similar behavior, and cause a marginal increase in the probability that the quarterback sustains an injury through his increased exposure, thereby offsetting to some extent the intention of the policy to reduce injuries.
In addition Table 3 shows that being within field goal range on a fourth down decreases the probility of passing. It can also be seen that situations where a punt is a dominated strategy (4th down at the end of the fourth quarter and trailing in points) increase the probability of passing. Finally, in line with a priori expectation, the greater the number of yards to go for a first or touch down, the greater the probability that the offense will pass.
Equation 13 seeks to test the less inuitive portion of the theory, that increases in the offensive payoff in a pass/blitz situtation similar to that of a defensive yardage

30
penalty, cause the defense to blitz more often. Specifically it estimates the effect that an increase of one team incompletion per game on average has on the probability that the defenses sacks the quarterback. The results are presented in Table 4, with standard errors clustered by defense.
Table 4. Estimation of Probability of Sack
Independent Variables
Coefficient
P Value
Average team incompletions of the offense
-4.2 x 10-4
0.30
4th Down on the 50 yardline or closer
0.0110
0.00
4th Down in 4th quarter with less than a minute and diff > 0
0.0362
0.09
5.5 x 10-4
0.00
Yards to go for a first or a touch down
0.0014
0.00
Yardline
5.6 x 10-5
0.00
Table 4 shows that the average number of incompletions of the offense has no statiscally significant effect on the frequency that the quarterback gets sacked that. If it can be assumed that sacks are indicative of blitzes, then since the high quality of a team with low incompletions raises the expected payoff to the offense of attempting a pass in a similar manner as a defensive yardage penalty, it may be inferred that such penalties would have no effect on the number of sacks as well, and therefore no effect on the number of blitzes, or that
= 0.
Recall the expression for Equation 6 is,
+ – – + + +
= • • + • • > 0

31
if we can assume that the more often the offense chooses to pass, the more often quarterbacks will get injured, then it has been shown empirically that decreases in the expected defensive payoff of a pass/blitz situation similar to those created by a defensive yardage penalty raise the probability that the offense will pass, thus increasing the marginal probability that the quarterback will sustain an injury. In addition, if we can assume that the more often defenses choose to blitz, the more often they will get a sack, then it has been shown empirically that variation in the expected offensive payoff of a pass/blitz situation similar to that created by a defensive yardage penalty has no effect on the marginal probability that the quarterback sustains an injury. It should be noted that, although the defense does not contribute to the problem by blitzing more frequently, the probability of injury still increases due to the actions of the offense.
To test whether the interaction terms are necessary in the two preceding estimations, joint significance tests were performed on the team, week, and season dummies, and all of the interactions. The results of these tests are presented in Table 5, and lend overwhelming evidence for their inclusion in the model.
First, the joint significance of the offense and defense in the pass and sack models respectively suggests that team specific fixed effects are important. In other words, there exists some time-invariant heterogeneity between teams that significantly affects both

32
Table 5. Joint Significance F Tests for Select Dummies and Interactions
Pass
Significance
Sack
Significance
Offense
0.000
Defense
0.000
Week
0.000
Week
0.000
Biseason
0.000
Biseason
0.000
Offense*Week
0.000
Defense*Week
0.000
Week*Biseason
0.000
Week*Biseason
0.000
Offense*Biseason
0.000
Defense*Biseason
0.000
Offense*Week*Biseason
0.000
Defense*Week*Biseason
0.000
passing and sacking. In addition the week and biseason variables have been shown to be jointly significant. This suggests that the point during the season (earlier or later) and the season play a significant role in both passing and sacking. Also, all the interactions in both models exhibit joint significance. This suggests that the changing make up of the team over the course of, and between seasons plays a role in passing and sacking repsectively. Finally the joint significance of the triple interactions imply that the match up between two teams, which two teams are playing against each other, is an important factor to include in the model.

33
CHAPTER 7
CONCLUSIONS AND CAVEATS
It has become apparent that the argument regarding what actions should be taken by the NFL to reduce the number of injuries suffered by its players is being conducted within an inappropriate framework. By failing to account for the perverse incentives that defensive yardage penalties assessed for violence create, policies are being employed that act in the exact opposite direction to that intended.
Defensive yardage penalties for fouls typically committed while defending against a pass, are intended to curb injury, and are designed to decrease the payoff that the defense expects from blitzing. Theoretically, offenses will capitalize on this by increasing the frequency with which they pass and by doing so, increase the probability that quarterbacks sustain injuries, thus offsetting the intention of the policy. In addition, due to the pseudo-zero-sum nature of the game, these penalties increase the payoff that the offense expects from passing. Theoretically, defenses will react to this by increasing the frequency with which they blitz, and by doing so, increase the probability that quarterbacks sustain injuries, further offsetting the intention of the policy.
Alternatively, offensive yardage penalties, that decrease the expected value of passing, which have typically not been implemented to protect quarterbacks from injury, do just that. In theory, defenses will react to the decreased value placed passing by the offense by decreasing the frequency with which they blitz, while offenses will react to the

34
increased valuation place on the blitzing by decreasing the frequency with which they pass. Since both players are engaging in potentially injurious behavior less frequently, the probability that the quarterback sustains an injury is reduced. Although this result is appealing, it must be noted that it pushes the game to be more one dimensional, centered on running plays, a far less exciting game to watch.
On the other hand, research and education, regarding the unknown effects of injuries typically sustained in the pass/blitz situation effectively raise costs for both players. Theoretically, offenses will capitalize on this in the same way as a defensive yardage penalty by increasing the frequency with which they pass, thus increasing the probability that the quarterback sustains an injury. Defenses however, will react to the decreased value placed on passing by the offense by reducing the frequency with which they blitz, thus reducing the probability that the quarterback sustains an injury. If the total effect of the increased awareness of the defense (the degree to which the education augments the payoff to the offense, how that changes the frequency of blitzing, and how it all taken together reduces the probability of injury) outweighs the total effect of the increased awareness of the offense (the degree to which the education augments the payoff to the defense, how that changes the frequency of passing, and how it all taken together increases the probability of injury), there will be a net reduction in the probability that quarterbacks sustain injuries.
Empirically, it has been show that there is a statistically significant negative relationship between the probability that the offense will pass and the relative sacking ability of the defense against whom they are playing. Since relative sacking ability changes the expected value of a blitz in a similar way as the possibility of a defensive

35
yardage penalty, it may be reasonable to expect that offenses will react to changes in penalties similarly to how they react to changes in sacking ability. Additionally, the data show that the probability that the defense will sack the quarterback is unaffected by the average number incompletions of the offense. Since average incompletions changes the expected value of a pass in a similar way as the possibility of a defensive yardage penalty, it may be reasonable to expect that defenses will react to changes in penalties similarly to how they react to changes in the ineptitude of the offense.
In total, even though the defensive reaction to the change in offensive payoff, , has been shown to be nearly zero, if it can be assumed that by passing more frequently the quarterback is more prone to injury, then the empirical evidence supports the hypothesis that the offenses are offsetting the intention of the policy to reduce injuries by passing more frequently. In other words, in the absence of a defensive reaction, the probability of injury still increases due solely to the choice made by the offense to pass more often.
Since the coefficients of both empirical models were estimated using a linear probability model, and such models do not allow the probabilities in question to be nonlinear functions of the independent variables, it may be unwise to use them to make quantitative predictions, since both models fail to account for nonlinearities such as diminishing marginal returns. Moreover, the use of a linear probability model poses problems because the values that the dependent variables (pass and sack) can take are limited to the interval [0,1], thus predictions outside this interval make no sense. Since the dependent variables in Equations 12 and 13 are not limited to this interval, the above models may result in such erroneous predictions. Although the use of probit estimation

36
may be more appropriate to predict marginal effects in a limited dependent variable model, given the nature of the question (to sign coefficients), using a linear probability model adequately answers this question.
Another caveat to the empirical estimation of Equation 13 is that regarding the asymmetry between the value of losses and gains. Since average team incompletions proxy ineptitude which amounts to a net decrease in expected utility of a pass (A↓), and the theory deals with an increase in expected utility of a pass (A↑), quantitative approximations may be misleading. In other words, the players may react more dramatically to decreases in their expected payoff than they do to increases. It is however, not the intention of this analysis to make quantitative predictions regarding the effect of penalties, but to show that they are working in the wrong direction. The qualitative conclusions derived from the theory and shown in the empirics can be expected to be consistent in the face of diminishing return and the asymmetry of the value function.
In addition, it must be noted that educating players about the long term effects of their choices to engage in potentially injurious activity changes the present value of certain costs as they will not come to light for many years. It is therefore worthwhile to note that such awareness has the potential to change not only perceived cost, but also players’ discount rates, and the time frame that the consequences will manifest within. For example, game situations create very high discount rates, but by making players aware that concussions pose a higher long term threat than previously believed, players may weigh such costs more heavily in the decision making process, effectively lowering their discount rates. Moreover, the discovery that such costs may be felt sooner in the

37
long term than previously believed may decrease the number of periods that players discount such costs back. It must be noted however, that the qualitative conclusions derived in the theory are robust to this deconstruction of the cost function, and the reader is urged to perform the comparative statics for themselves, for they are beyond the scope of the present analysis.
To conclude, football is an inherently violent game. If violence is considered to be an economic good, efficient allocations are possible given the right market conditions. Attempts to regulate risk, however, which distort the incentives of the players, only lead to inefficiency. A wider understanding of the long-term effects of concussions and other injuries can in theory reduce the frequency of injury while not materially altering the game. Further lines of research may include the effect that such education will have on the demand for quality medical diagnoses by players in the NFL. An increased awareness of costs will presumably make more apparent the principal-agent problem stemming from a doctor who is paid by an owner and not the player. In other words, it is plausible that encouraging players to employ their own medical staff may also decrease the probability that players will allow themselves and others to be put in situations conducive to injury.

38
REFERENCES
Becker, Gary. “Crime and Punishment: An Economic Approach.” Journal of Political Economy 76, no. 2: 169-217.
Florio, Mike. “Peyton Manning Admits to Tanking Baseline Concussion Tests.” NBC Sports. Last modified April 27, 2011. Accessed December 12, 2012. http://profootballtalk.nbcsports.com/2011/04/27/peyton-manning-admits-to- tanking-baseline-concussion-tests/.
Goldberg, Daniel. “Concussions, Professional Sports, and Conflicts of Interest: Why
the National Football League’s Current Policies are Bad for Its (Players’) Health.” HEC Forum 2008: 337-355.
Guskiewicz, Kevin , Stephen Marshall, Julian Bailes, Michael McCrea, Robert C. Cantu, Christopher Randolph, and Barry Jordan. “Association between Recurrent Concussion and Late-Life Cognitive Impairment in Retired Professional Football Players.” Neurosurgery 57, no. 4: 719-726.
Jewell, R. Todd. “Estimating Demand for Aggressive Play: The Case of English Premier League Football.” International Journal of Sport Finance 4, no. 3: 192-206.
McCannon, Bryan. “Strategic Offsetting Behavior: Evidence From National Collegiate Athletic Association Men’s Basketball.” Contemporary Economic Policy 29, no. 4: 550-563.
McGarrity, Joseph P. and Brian Linnen. “Pass or Run: An Empirical Test of the Matching Pennies Game Using Data from the National Football League.” Southern Economic Journal 76, no. 3: 791-810.
National Collegiate Athletics Association. “A Primer on NCAA Rules for Football Safety.” Latest News. Last Modified October 20, 2010. Accessed December 12, 2012. http://www.ncaa.org/wps/wcm/connect/public/NCAA/Resources/Latest+News/2010+news+stories/October/A+primer+on+NCAA+rules+for+football+safety.
National Football League. “Official Playing Rules.” Rulebook, 2012. Accessed Dec 12, 2012. http://www.nfl.com/rulebook.

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National Football League Players Association. “Dangers of the Game.” Edgeworth Economics, January 26, 2011. Accessed December 12, 2012.
http://www.esquire.com/cm/esquire/data/Dangers-of-the-Game-Report- Esquire.pdf
“NFL Stats.” Team Rankings. Accessed November 22, 2012.
http://www.teamrankings.com/nfl/stats.
Peltzman, Sam. “The Effects of Automobile Safety Regulation.” Journal of Political Economy 83, no. 4: 677-725.
“Play-by-Play Data.” Advanced NFL Stats. Accessed November 22, 2012. http://www.advancednflstats.com/2010/04/play-by-play-data.html.

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