Splitting the Uprights: An Analysis of Loss Aversion and Reference-Dependent Preferences in the NFL
Patrick Burke - 11373350 August 2017
Abstract
Critics of behavioral literature maintain that behavioral biases and nonstandard preferences disappear in competitive markets. In this analysis1, the goal is to assess whether loss aversion and reference dependent preferences persist in the National Football League (NFL). Models with reference-dependent preferences around a fixed reference point imply that players would exert additional effort to avoid falling into the loss domain. This paper provides suggestive evidence that NFL kickers exhibit these tendencies, using field goal attempts in the 2015 and 2016 seasons. Kickers improve their performance when they are kicking while their team is losing. As their teams fall further behind in a game, kicker performance increases - suggesting that they slightly exhibit reference-dependent preferences around a tie game.
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University of Amsterdam MSc Economics: Behavioural Economics and Game Theory MSc ECO - 15 ECTS
1, I would like to thank my thesis supervisor Arthur Schram for guidance and support during the process of writing this thesis. Statement of Originality
This document is written by Patrick Burke who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document are original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Table of Contents
1. Introduction 2. Institutional Background 2.1 American Football and the National Football League 2.2 Field Goals 3. Literature Review 3.1 Prospect Theory and Loss Aversion 3.2 Prospect Theory in Sports 4. Methodology 4.1 Conceptual Framework 4.2 Data 4.3 Empirical Strategy 5. Results 5.1 General Results 5.2 Initial Regression 5.3 Second Regression 5.4 Third Regression 5.5 Discussion of Results & Limitations 6. Conclusion 7. Appendix 8. References
Summary of Tables and Figures
Figures Figure 1: NFL Field Dimensions (Appendix) Figure 2: Prospect Theory – Loss Aversion around Reference Point Figure 3: Prospect Theory – Field Goals applied to Reference Point Figure 4: Field Goal Attempts by Quarter (Appendix) Figure 5: Predicted Probabilities of Field Goals by Distance Figure 6: First Regression - Margins – Predicted Probability by Score Margin Figure 7: Second Regression – Margins – Predicted Probability: Winning Vs. Losing Figure 8: Third Regression – Margins – Predicted Probability: Score Margin (5 Groups)
Tables Table 1: NFL Field Rules (Appendix) Table 2: Predicted Probability of Field Goals by Distance (Appendix) Table 3: Summary Statistics Table 4: Regression and Score Margin Variable Summary Table 5: Field Goals by Distance Table 6: Field Goal Success and Average Distance – Winning, Losing, Tied Domains Table 7a: Initial Regression – Score Margin as Continuous Variable (Odds Ratios) Table 7b: Initial Regression – Score Margin as Continuous Variable Coefficients (Appendix) Table 8: Second Regression – Effect of Score Margin on Likelihood of Field Goal – Winning vs. Losing – Odds Ratios (Appendix) Table 9: Third Regression - Effect of Score Margin on Likelihood of Field Goal – Five Groups - Odds Ratios (Appendix) Table 10: Summary of Predictions
"I hate to lose more than I love to win." -Jimmy Connors, former #1 ranked tennis player in the world
1. Introduction
Behavioral economics has provided researchers the tools to analyze markets, games, and models while taking psychological underpinnings into account. Due to the emergence of this field, a substantial amount of literature has demonstrated ways in which agents consistently violate classical economic theory (Camerer, Loewenstein, and Rabin, 2004). While these violations may have implications in the real world, most behavioral anomalies have been observed solely in laboratory experiments. Even the most renowned behavioral economists believe that the primary challenge for the field is extending these findings from the lab to the field (Levitt and List, 2008).
Critics of the decision bias literature argue that behavioral biases are likely to disappear in competitive markets with high stakes, and experienced agents. “Individual behavior converges to the neoclassical prediction as experience intensifies...These results provide initial evidence consistent with the notion that market experience eliminates market anomalies” (List, 2003). However, recent works have shown that there is now considerable evidence that biases persist in the field outside of the laboratory (Goldman, 2016; Lien and Zheng, 2015). In their paper, Prospect Theory: An Analysis of Decision Under Risk, Kahneman and Tversky (1979) propose a recurring violation of the normative model of rational choice: loss aversion2. Simply put, losses loom larger than gains.
Taking elements from Prospect Theory and extending them to the National Football League (NFL), this paper examines whether reference-dependent preferences and loss aversion persist in this competitive market. The core focus of the analysis is on field goals, a set kicking play that earns the offensive team three points if the kick is successful. Kickers would perform better when they are losing compared to winning if they are loss averse. Also, their performance would vary around the reference point of a tie game - better kicking as their team falls further behind,
2 Many would argue that loss aversion is not a bias but a characteristic of one’s preferences. This paper does not take sides on such issue – and views loss aversion as an element of Prospect Theory. and worse kicking as their team’s lead increases. Studying reference points and loss aversion in sports has advantages. The highly quantified nature provides clear reference points to observe behavior.
Using nearly 2,000 field goals attempts from the 2015 and 2016 seasons, this study found for every 1-point increase in the score margin (the score difference between the kicking team and the opponent), the probability that a field goal is successful decreases by 1.5%. Also, kicks attempted when losing the game had up to a 4.2% higher predicted probability of success than a similar field goal while winning. Kickers performed best when losing by a large margin3. The probability of a successful kick was lowest when the team was winning by a large margin, consistent with loss aversion and reference-dependent preferences. Overall, the results suggest that with a larger dataset - and minor adjustments to the model - there may be concrete evidence that loss aversion and reference-dependent preferences persist in the NFL.
While this study only finds suggestive evidence of loss aversion in kickers, it does contribute to the existing literature on loss aversion around a reference point in a high stakes market. Some recent theoretical work on reference points has conceptualized expectations as reference points - however, given the difficulty of testing these preferences, there has not been extensive work to test this theory (Abeler, 2011). Outside of the laboratory, it is hard to make a reliable inference about loss aversion since pinning down a reference point can prove to be problematic (Goldman, 2016). This study extends research on reference points and loss aversion by utilizing the zero- sum nature of the NFL to establish a fixed reference point where it is easy to split the space of outcomes into losses and gains.
The rest of this thesis proceeds as follows: Section II provides background on the NFL and the role of field goals. Section III introduces previous literature on Prospect Theory, loss aversion, reference points and applications of these elements in the world of sports economics. Section IV provides the conceptual framework, predictions, and defines variables within the data set. Section V gives the results - from initial summary statistics of the kicks to testing the three
3 Large margin considered to be 11 or more points in this paper. 10 points is a scoring margin that is within a field goal and touchdown. 11 points or more implies that more than one touchdown is needed. predictions. Section VI concludes. The appendix contains materials cited throughout the main paper, followed by the references.
2. Institutional Background
2.1 American Football and the National Football League
The National Football League (NFL) is the elite men's professional American football league in the world. It is made up of 32 clubs which are split into two conferences: the American Football Conference (AFC) and the National Football Conference (NFC). Each of these conferences has four divisions -North, East, South West - consisting of four teams. Teams are allowed a maximum of 46 active players on the roster. These 32 clubs play against each other for a total of 16 games during the 17-week regular season which spans from the first week of September to the week after Christmas. The 12 teams with the best record advance to the post- season tournament to compete for the Super Bowl - the championship game (NFL).
The NFL is a competitive, high stakes market with experienced agents. Players are some of the top performing athletes in the world and competition is fierce as players vie for positions on the rosters and depth charts. Although kickers are not involved in the physicality of the game as much as other players - their performance is vital to winning games. Each team typically has one field goal kicker. Therefore, there is a fine line between job stability and being replaced by another kicker. As NFL coaching legend Buddy Ryan put it, “Football kickers are like taxi cabs. You can always go out and hire another one.” Incentives are high in the NFL; the average salary of kickers is about $1.4 million dollars per year4.
NFL games are divided into four 15-minute quarters. The team with the most points at the end of regulation is the winner. If the score is equal at the end of the 4th quarter, the two teams will play an additional 10-minute period. If no team is winning at the end of this overtime period, the game ends in a tie. Each club fields 11 players at one time - one team on the offensive side of the
4 Source: Sportrac- the largest online sports team, & player contract resource on the internet. ball and the other on defense. There is a designated area at each end of the field called the end zone. The goal of the offense is to move the ball down the field against the defense in increments of 10 yards until they reach the end zone, awarding them six points (touchdown). The offensive team has four chances (“downs”) to move the ball 10 yards so that they may receive a new set of four downs. The offense can attempt to gain 10 yards through a combination of running or passing plays. The defensive team can stop the offense by forcing a fumble, an interception, or by not allowing the offense to make it ten yards within the four downs. If the defense stops the offense on 4th down, the ball is turned over, and the defensive side of the ball takes over possession at that spot on the field. On a fourth down play, the offense will typically punt the ball to avoid a turnover or attempt a field goal. Figure 1 depicts dimensions of the field and Table 1 provides the official NFL field rules5.
2.2 Field Goals
A field goal is a set play in which the offensive team attempts to kick the ball through two posts (uprights) - located at the back of the end zone. A successful kick will earn the kicking team three points, while a missed field goal results in zero points and a loss of possession. The coach's choice to kick a field goal is dependent on field position, situational factors of the game, and environmental conditions. Most kicks occur on a fourth down play in situations where attempting an offensive attempt is risky, but the team is far enough down the field to kick for points. Field goals are a critical component of the NFL game - about four field goals are attempted per game with wins and losses often being determined by their outcomes (Clark et al., 2013). Before analyzing kicker performance, it is important to establish the behavioral foundations linked to loss aversion, reference points, and their applicability to the NFL.
3. Literature Review
3.1 Prospect Theory and Loss Aversion
5 Full NFL Rulebook available on official website - http://operations.nfl.com/the-rules/2017-nfl-rulebook/ Neoclassical economic theory postulates that preferences are consistent regardless of current wealth state. However, behavioral economists and psychologists contested this belief and ascertained that these preferences and decisions are not stable over a spectrum of situations. Kahneman and Tversky (1979) created groundbreaking work to challenge expected utility theory which had dominated the analysis of decision making under risk. They observed an element of nonstandard preferences in the laboratory - which has come to be known as loss aversion. Loss aversion became a centerpiece of their work in Prospect Theory. Using an internal threshold - a reference point - they reported that losses loom larger than gains: the disutility from losing a sum of money is greater than the pleasure associated with obtaining that amount (1979). One’s value for changes in wealth is concave above the reference point, and convex under this reference point - an idea that contested the standard continuous concave utility function over wealth. In simple terms, the adverse emotional effect of losing $5 is larger than the positive emotional effect of finding $5. Figure 2 depicts the value function over a space of outcomes.
Since behavioral economics has gained traction, the applicability of loss aversion around a reference point in different markets has helped explain a variety of behaviors in many economic domains. While most of the literature has been documented in laboratory settings (Camerer et al. 1997), there has been extensive work in the field as well (Camerer, 2000; Lien and Zheng, 2015). With continued research, loss aversion may become more apparent in real markets.
Figure 2: Prospect Theory – Loss Aversion around Reference Point
-The asymmetrical value function that passes through the reference point is S- shaped. The value function is steeper for losses than gains, which shows that losses loom larger than gains.
There have been important studies on reference points since Prospect Theory first introduced the concept. Preferences shaped by a reference point are called reference-dependent preferences. Agents split decision making into two domains: a loss and a gain domain. Koszegi and Rabin (2006) modeled these preferences: “By directly constructing reference-dependent utility from consumption utility and assuming that the reference point is endogenously determined as rational expectations about incomes, our theory provides an algorithm for translating a classical reference independent model into the corresponding reference dependent on." This work theoretically showed that expectations about outcomes could act as an internal threshold. However, it's hard to measure one's expectations in a field setting.
The research into reference-dependent preferences has encountered the difficulty in establishing reference points especially in field studies - since they are often unobserved, heterogeneous and non-stationary (Pope and Schweitzer, 2011). While work using Koszegi and Rabin’s (2006) model has started to become more common, there has not been extensive work using these types of preferences outside of laboratory experiments. In his paper Reference Points and Effort Provision, Abeler (2011) used field data on worker effort choices to test whether the response of effort to changes in incentives. He found that the exertion of effort depends highly on expectations. Reference points have been shown to be malleable in the field studies based on adjustments to future earnings for a variety of workers (Fryer et al., 2012). Some evidence suggests that reference-dependent preferences also shape our choices around a static point (Crawford and Meng 2011).
3.2 Prospect Theory in Sports
The quantified setting of sports is a model opportunity to test for loss aversion in the field since a fixed reference point easily divides outcomes into losses and gains. In their study Is Tiger Woods Loss Averse? Persistent Bias in the Face of Experience, Competition, and High Stakes, Pope and Schweitzer (2011) showed that biases persist in competitive markets. They did this by analyzing over 2.5 million putts in the game of golf. The reference point of par influenced the behavior and performance of golfers during the PGA tour6. Their findings revealed that golfers performed
6 Golf is a sport in which shooting ‘below par’ is good. The lowest score in golf wins. better to avoid losses. When golfers are under par (gain domain), their putting is significantly less accurate than putts with similar characteristics for par or over par scores (loss domain).
There is evidence that players in the National Basketball Association (NBA) also exhibit loss aversion and reference-dependent preferences. (Lusher and He, 2016). Their results suggest that players increase effort and focus when shooting for a round number, to avoid falling below round numbers. In his study, Goldman (2016) found evidence that a losing team performs better when trailing by ten points. This work was an important departure from the notion of expectations as a reference point (demonstrated in Lusher and He) and implements a fixed reference point at a tie game. Since the game of basketball is zero-sum, a team trailing by five points faces the same marginal returns to scoring a point as the other team who is leading by five points. The neoclassical framework would suggest that there is perfect symmetry around a fixed reference point, but this paper tells a different narrative of player performance in a competitive market.
While not pervasive, behavioral economic literature on the NFL has emerged within the past decade. David Romer (2006), known for his work in macroeconomics, examined strategic decisions made throughout an NFL game to address whether firms (teams) maximize profits. Romer found that coaches’ decisions “show systematic, clear-cut, and overwhelmingly statistically significant departures from the decisions that would maximize teams’ chances of winning” (Romer, 2006). Additional studies (Urschel and Zhuang, 2011; Kovash and Levitt, 2009) on American football have built upon Romer’s work by showing extensive evidence that coaches are risk and loss averse. While previous literature examined coaching decisions, this paper analyzes player performance in the NFL.
Some of the more recent analyses of field goals have concentrated on environmental factors such as weather, precipitation and the wind. Clark et al. (2013) studied these factors- but went beyond distance and environment to introduce psychological factors such as ‘icing' the kicker and high situational pressure. Despite conventional wisdom, these psychological effects do not affect kicker performance (Dubner, 2012). While their analysis proves to be instrumental in setting up a binomial logit regression model for field goals, their focus did not address whether kickers respond to different score margins.
This paper combines elements of Prospect Theory, the sports world, and player performance. Establishing a reference point at a tie game creates a simple model to observe preferences around that point. While there is evidence of loss aversion and reference-dependent preferences in other sports, there is no significant research into whether players exhibit these tendencies in the NFL. The focus pertains to on-field performance of kickers, rather than coaching decisions. Some analyses have determined key factors in the likelihood of a successful field goal attempt. NFL kickers may adjust their preferences based on the score margin at the time of the kick – which has not been analyzed.
4. Methodology
4.1 Conceptual Framework
This paper develops a simple theoretical framework and incorporates a logit regression method model to test the effect that loss aversion may have on kicking performance (Pope and Schweitzer, 2011). The first assumption is that the probability of making a kick is a function of effort (e) and other characteristics of the field goal - distance, environmental conditions, score margin, situational variables and other factors (z) plus a random error (ε). Equation 1) depicts the probability. The second assumption is that exerting additional effort raises the likelihood that the field goal is good (f'(e) ≥ 0 and f'' (e) ≤ 0; f (e) is weakly concave).
1) P(Make = 1) = f (e, z) + ε
A Kicker earns his team 3 points when a field goal is successful and zero with a miss. A 'miss' results in a loss of expected points which is 3 points multiplied by the predicted probability of success at that distance7. The effort in taking the kick is shown by cost(e). Each kicker's utility equals the value placed on making and missing the kick by the probabilities and subtracting the effort costs (Pope and Schweitzer, 2011).
7 Predicted Probabilities at each distance can be seen in Table 2 within the Appendix.
2) U = (f(e,z) + ε)(V) + (1 - f(e,z) - ε)(V) − cost(e)
This paper also assumes cost of effort to be strictly increasing: cost’(e) > 0 and convex: cost’’(e) < 0. Using Kahneman and Tversky’s work on loss aversion (1979), the value function depends on the situation in which the field goal is attempted. Also, the value of the kick denoted, V is dependent on the score margin at the time of the kick, x8. The value of the field goal is in expected points.
ƛ is the degree of loss aversion which is ≥ 1. Based on this simple value function, we can see that losses loom larger than gains for kickers. Due to the degree of loss aversion (ƛ), when the kicking team is losing by a certain margin, the loss in value is greater than the same kick missed when winning. Also, the increase of value is higher for a successful field goal while losing. Figure 3 depicts the value function regarding field goal kicking.
Figure 3: Prospect Theory – Field Goals applied to Reference Point
-As Figure 2, this kick shows potential outcomes split into gains and loss domains. Field goals attempted on the left of the reference point (tied game) are within the loss domain. The increase of value is higher for a successful field goal while losing.
8 x≥0: team is winning; x<0: team is losing First order conditions are in equation (3). These mean that a kicker sets his marginal cost of effort equal to the marginal benefit of effort when they are winning (x ≥ 0). When kicking in the loss domain, the kicker exerts additional effort, which is equal to their degree of loss aversion, (Pope, 2011).
훿푐표푠푡′(푒) = 1 푖푓 푥 ≥ 0 훿푓(푒, 푧)/훿푒 3) 훿푐표푠푡′(푒) = ƛ 푖푓 푥 < 0 훿푓(푒, 푧)/훿푒
Kickers use higher levels of effort when losing which will increase the likelihood of a successful kick. Assuming the above statements are true, this paper predicts:
Prediction 1: Kickers exhibit loss aversion. At a given distance, score margin and the probability of a successful field goal have a negative relationship.
Prediction 2: Controlling for kick characteristics, a field goal attempt when losing has a higher predicted probability compared to a field goal with similar characteristics when winning by the same margin.
Prediction 3: Kickers exhibit reference-dependent preferences. Controlling for kick characteristics, the predicted probability of a field goal will increase as the kicking team's score margin decreases. Conversely, the predicted likelihood of a successful field goal will decrease as the kicking team's score margin increases9.
4.2 Data
This study uses a data set of all field goal attempts (1,996) over the course of the 2015 and 2016 regular seasons. This paper assumes homogeneity between the two years. A query tool extracted the all field goal attempts from www.pro-football-reference.com. This site provides
9. Field goals attempted while losing by a large margin will be the most accurate. Kicks attempted while winning by a large margin will be the least accurate. detailed information across numerous sports. Analysts, media companies, and economists use this site for a variety of projects.
The initial data extraction included: kick number (total per year), date of game, kicking team, opponent, the result of that match, kick distance, and whether the kick was successful or not. However, this data does not provide all the information needed to test the researcher's predictions. It was necessary to investigate each kick in the play-by-play tables to collect environmental and situational factors. Among the additional variables collected were: Score margin, game week, quarter, windy game, precipitation, cold temperature, kicker name. Table 3 summarizes the variables used in the model.
Table 3: Summary Statistics
Except for distance and scoring margin, all the above factors are binary (0 or 1) variables, dependent on whether it meets the condition. If it is different than the baseline, then that variable is equal to 1.
For this model, specific environmental variables would be ideal. However, some of these variables were reduced into simpler binary variables for analysis. Clark et al. (2013) observed that the make probability of a field goal increased with rising temperature in a nonlinear fashion. Field goal chances in temperatures below 50 °F were similar. There is a considerable jump in field goals percentages in temperatures above 50 ° F- therefore, this paper classifies temperature into two groups (coded as a ‘1’ for cold temperature). The same method sorts wind speed into two bins as well - coded as '1' if the wind speed is greater than or equal to 10 mph, and 0 if less than 10 mph. This paper did not record wind direction due to data availability. The precipitation variable indicates whether there was rain or snow at the time of kickoff - which has been found to affect many aspects of the game, especially field goals (Clark, 2013). Simplifying the environmental variables allows for easier interpretation of the model easier.
This paper includes dummy variables indicating if the kicking team or opponent made the playoffs to control for the quality of the kicking team and opponent. These variables account for the strength of the offensive and defensive lines (the blocking and defending units on kicks) and offensive productivity. There is also a variable for home/away games. Playing on one's home turf may have an impact on the success of a kick (Dohmen, 2005). Also, this paper includes a variable to control for the psychological pressure that may occur due to high-pressure situations. The pressure situation variable in Table 3 is ‘1’ if an attempt is within the last five minutes of the game and the scoring margin is within -7 and +7 - since 7 points are equal to one touchdown and point after touchdown (PAT).
One consideration regarding the environmental data is that all information is recorded at the beginning of the game, and not at the time of the kick (Clark et al., 2013). Weather conditions may fluctuate throughout the game, but if a match begins with specific environmental conditions (precipitation, wind, temperature) this paper assumes those are consistent throughout the contest. Another potential setback in the data is that it that there is not an indication of which hash mark. On an NFL field, two hash marks denote each yard line 70 feet, 9 inches from the sidelines. These marks are about 18.5 feet away from one another (Figure 1). When the previous play ends between the hash marks, the subsequent play will start at the position. However, if the previous play ends outside of the hash marks, the following play will start from the closest hash mark.
Table 4: Regression and Score Margin Variable Summary
Regression Predictions Score Margin Variable Kickers exhibit loss aversion. At a given distance, score margin and Continuous 1 the probability of a successful field goal have a negative relationship. -41 ------30 Spread (3 groups) Controlling for kick characteristics, a field goal attempt when losing 2 has a higher predicted probability compared to a field goal with 1) Losing similar characteristics when winning by the same margin. 2) Tied 3) Winning Score Margin (5 groups)
Kickers exhibit reference-dependent preferences. Controlling for kick 1) Losing by 11 or more 3 characteristics, the predicted probability of a field goal will increase 2) Losing by 1 to 10 pts as the kicking team’s score margin decreases. 3) Tie game (Ref. Pt.) 4) Winning by 1 to 10 pts. 5) Winning by 11 or more -It is necessary to run three different regressions, to test all three predictions. In the first regression, score margin is a continuous variable. In the subsequent regressions, score margin breaks into three and five groups.
4.2 Empirical Strategy
This paper uses a binomial logistic regression model to test predictions. The dependent variable in this model is a binary dichotomous mutually exclusive variable: 1 if the kick is ‘good' and 0 if the kick is ‘not good.' Considering situational, psychological and environmental factors that may influence a field goal outcome, this paper analyzes whether scoring margin has a significant impact on kicker performance, and how kickers perform in a variety of different situations. This analysis can lead to answering the question whether NFL kickers exhibit loss averse tendencies and reference-dependent preferences.
Logistic regression models in this study will give outputs shown in odds ratios as opposed to coefficients (apart from the first regression). Odds ratios describe how much the odds of the dependent variable change for each unit change of the independent variable. An odds ratio less than 1 mean that the odds of the dependent variable occurring (field goal is good) decrease as the independent variable increases. If the odds ratio is equal to 1, there is no relationship between the independent and dependent variable. If the ratio is greater than 1, then the odds of the dependent variable occurring increase as independent variable increases. The coefficients for variables are the log of odds ratio between the groups. Exponentiating the coefficient gives the odds ratio.
This paper compares the probability of a field goal with different values of the score margin that are similar in other ways to test all three predictions. Our main specifications take the following form:
푀푎푘푒 퐾푖푐푘푖 = 훽0 + 훽1(푆푐표푟푒 푀푎푟푔푖푛 푅푒푙푎푡푖푣푒 푡표 푇푖푒푖) + 푍푖 + 휀푖
Since this paper assumes homogeneity over all kickers, and years – the controls in place allow the study to compare kicks(i) from the same distance and weather conditions at different scoring margins.
In the first of the three regressions to test the predictions, score margin is a continuous variable. For example, if a team is losing by 5 points, the score margin is coded as -5. A tie game is 0, and winning by 10 is +10. In the subsequent regressions, this paper categorizes score margin into three bins and five different bins respectively (Table 4). Categorization allows analysis of the probability of a successful field goal at various scoring margins - which is necessary to test the second and third predictions.
5. Results
5.1 General Results
In the 2015 and 2016, there were 512 regular season games played. During this time, there were 1,996 field goals attempted, 84.37% (1684) of these were good. The average distance of all attempts was 37.95 yards - the shortest was from 18 yards and the longest from 66 yards. Field goal percentages fall below the overall average at the 40-44-yard range. The largest amount of field goals was attempted from the 45-49-yard range (534); kickers converted on only 73% of these. The most substantial amount of successful field goals were from the 30-34-yard range (271) at a rate of 94%. Table 5 shows field goals by distance. Kickers attempted considerably more field goals in the second and fourth quarter of the game than the first and third quarters. The difference is due to several reasons: 1) possession is lost at the end of the second and fourth quarters and 2) they do not start with a kickoff - therefore, a team may already be able to kick a field goal when the second or fourth quarter start. Teams attempted field goals from shorter distances in the first and third quarter. The shorter field goals may be due to a lack of ‘desperation' field goals that would otherwise occur at the end of the second and fourth quarters. Figure 4 in the appendix depicts kicks taken throughout the game.
Table 5: Field Goal Attempts by Distance
Distance Field Goal Field Goals Field Goal (yards) Attempts Made Percentage
18, 19 19 19 100.00%
20-24 236 232 98.31%
25-29 250 239 95.60%
30-34 288 271 94.10%
35-39 287 263 91.64%
40-44 287 239 83.28%
45-49 319 232 72.73%
50-54 247 161 65.18%
55+ 63 28 44.44%
Total 1996 1684 84.37%
-The left column shows the range of kicks taken in yards. As attempts increase in yards, the field goal percentage of these attempts decreases. In the 40-44 range, field goal percentage falls below 84% which is overall average.
The number of kicks taken when winning and losing is approximately the same amount. Interestingly, when a team is losing, kickers tend to perform better while kicking from a farther distance. There were 819 kicks attempted when the kicking team is losing. They made these 84.86% (694) of these kicks. When teams were winning, they kicked 781 field goals and made 83.61% (653). The remaining 396 kicks occurred during a tie game. Table 6 (appendix), column e shows that teams attempted field goals from farther distances when losing e). In the next section, blocked kicks have been dropped out of the model, since the fault of these blocked kicks is not solely on the kicker, but also on the players blocking during the set play. Taking out the 43 blocked kicks brings the total amount of kicks to 1,953.
As expected, distance is the determining factor in the success of a field goal. Figure 5 shows the predicted probability of field goal success by distance. The probabilities curve is an inverted S- curve - since the difficulty of a kick increases as distance increases. A margin is a statistic based on the fitted model calculated over a dataset in which some or all covariates are fixed at values. The margins reported in a binomial logistic model are the average predicted probabilities – in this case, the likelihood of a successful field goal. Based on the model, field goals under 20 yards are almost sure to be good - with predicted probabilities near .99. At a point of about 56 yards, the likelihood of making a kick dips below .5 which means that a miss becomes more likely than a 'make.' Table 2 in the appendix shows the predicted probabilities from 18 to 66 yards.
Figure 5: Predicted Probability of Field Goals by Distance
-These predicted probability values stem from the model using all the control variables introduced in Table 3. The horizontal line depicts the predicted probability of .5.
5.2 Initial Regression
This paper incorporates the explanatory variables addressed in Table 3 to observe the factors influencing the likelihood of a field goal. Table 7a below shows the results of logit regressions in which score margin is a continuous variable. This same model was used to fit Figure 5 above. The odds ratio for Distance is .881 and statistically significant at the 5% level. This odds ratio means that an increase of one unit in the distance (yard) decreases the chances of a successful kick by 11.9%. When odds ratios are less than 1, the corresponding coefficients table should show a negative coefficient for that variable. The related table 7b supports the odds ratios in these regressions in the appendix, which lists the coefficients of the same variables. A one-unit increase in distance means a .127 decrease in the log odds of a field goal.
Table 7a: Initial Regression – Score Margin as Continuous Variable (Odds Ratios)
(1) (2) (3) Distance .881*** .881*** .881*** (.008) (.008) (.008)
Score Margin .989 .986** .985** (.0077) (.008) (.008)
Precipitation .539** .512** .504*** (.187) (.179) (.175)
Wind .708 .715 .723 (.159) (.161) (.163)
Cold Weather 1.23 1.20 1.21 (.227) (.224) (.225)
Playoff Team? - 1.31** 1.306** (.200) (.199)
Opponent Playoff Team? - .909 .902 (.136) (.136)
Home Game - - 1.03** (.149)
Pressure Situation - - .756 (.181)
Constant 1361.3 1263.5 1276.08 (580.1) (544.7) (559)
Pseudo R2 .1790 .1813 .1822
Logit estimation - Dependent variable equals 1 if kick was made
***Statistically significant at the 5% level. **Statistically significant at the 10% level.
However, to determine whether kickers in the NFL exhibit loss aversion, the variable of interest in this paper is score margin. In the same regression, the odds ratio (Table 7a) of score margin in column (3) of is slightly below 1, at .985. Since the ratio is .985, the odds of a successful kick decrease by 1.5% with every one-point increase in the scoring margin. Controlling for other variables, the relationship between score margin and field goal success is statistically significant at the 10% level.
It is possible to look at predicted probabilities of field goals across all score margins using the margins command (like the method of predicted probabilities by the distance in Figure 5). The lowest scoring spread in the data set is -41 and the highest is 30. Holding field goal distance at the mean of 38 yards (controlling for weather and other factors) a kick attempt when losing by 25 points has a .949 predicted probability of success. When losing by ten points, this likelihood drops to .927 and eventually to .898 when a team is winning by 25. The ‘Margin’ column in Figure 6 shows the relationship between the marginal predicted probability of an average kick's success across all score margins. One can observe the negative trend as the spread increases past the reference point of a tie game. The predicted probabilities are all significant at the 1% level.
Figure 6: First Regression - Margins – Predicted Probability by Score Margin
-The left and the right side is the same, one in graphical form and the other in a table. Holding explanatory variables at the mean, one can observe the negative. When losing by 25 points, a field goal has a 5.1% higher chance of success than the same field goal when winning by 25 points.
These results confirm Prediction 1, score margin and the probability of a successful field goal have a negative relationship. As the score margin increases, the likelihood of a field goal decreases.
5.3 Second Regression
To test the second prediction using marginal effects it is necessary to categorize field goal attempts into three distinct groups: losing, tied, and winning. At each field goal distance, there are two predicted probabilities of a successful kick, one probability when the kicking team is losing and one when the kicking team is winning. This paper analyzes the difference by holding other control variables at their mean. For a given kick, do kickers perform better when they are losing or winning? Figure 7 depicts the relationship between the two groups on distance. This prediction appears to be true for distances of more than 35 yards. At distances from 43 to 55 yards, kickers have anywhere from a .022 to .041 higher probability of making a kick when losing during the game.
Figure 7: Second Regression – Margins – Predicted Probability: Winning Vs. Losing
In Table 8 (Appendix), odds ratios for kicks taken when winning are less than one compared to a tie game. Column (3) shows this relationship - a kick when losing a team is losing, the odds of making the field goal 18% (odds ratio = 1.18) than the kick in a tied game. That same kick would have a decrease in odds of -9.4% (odds ratio =.906) if the kicking team were winning.
These results appear to confirm prediction 2, but this is only suggestive evidence considering statistical significance of the variable diminished after categorization into the three groups. The statistical differences between kicks in the losing, tied, and winning domain is not significant. Although the statistical power of the explanatory variable drops, it is still interesting to observe the marginal probabilities, which are significant at the 1% level. Kicks taken when losing are predicted to be more accurate. However, this paper cannot conclude that they are different from those taken during a tie game. An additional regression is required to answer the third prediction.
5.4 Third Regression
The third prediction states that kickers will exhibit reference-dependent preferences around a tie game to avoid falling into the loss domain. In this context, kickers will be more accurate when they are losing by double digits, than when they are losing by single digits. Kicks attempted while losing by single digits will be more accurate than tied games and so on. The least accurate kicks will occur when a team is winning by double digits. Categorizing the score margin into five groups is necessary. The first group consists of attempted kicks when the score margin is higher than 10 points. The next group consists of kicks that were attempted when the kicking team was winning by 1-10 points. Field goals during a tie game make up the third group (reference point). After this, the next groups are made up of kicks when the kicking team is losing by 1-10 points, and above 10 points respectively (see Table 4).
This paper found that the likelihood of a successful field goal is greater if their team is losing by a large margin, and slowly decreases as the score margin increases. Compared to a tie game (baseline in the categorization), losing by 11+ increases the odds of success by 22% (Table 9 - column 3, Appendix). Conversely, winning by 11+ points decreases the chances of success by 26.5%. As the findings in the previous regression, this is only suggestive evidence since categorizing score margin into groups the variable once again loses statistical significance. The statistical difference between these kicks and a tie game are not significant. However, running the marginal effects on this categorized variable can potentially give a better understanding of kicker performance.
Figure 8 shows a visual interpretation of the marginal effects. At all seven distances, the predicted probability of making a kick increases as the scoring margin decreases. Field goals attempted while winning by a large margin (11+) consistently had the lowest predicted probability of success across all distances over 25 yards. On the other hand, field goals taken while losing by a large margin had the highest predicted probability of success.
The model predicts that probabilities for these kicks are higher as the team falls deeper into the loss domain (Figure 8). This paper concludes that kickers exhibit evidence of reference- dependent preferences. Since this is the case, future studies with more than two seasons of data may provide more confirmation of these non-standard preferences.
Figure 8: Third Regression – Margins – Predicted Probability: Score Margin (5 Groups)
5.5 Discussion of Results & Limitations
The first prediction confirmed that score margin has an adverse impact on the likelihood of a successful field goal attempt. As score margin increases by one unit, the odds of a field goal decrease by 1.5%, which is significant at the 10% level. This points towards kicker's tendency to be loss averse. As the score margin moves in favor of their team, kicker performance falls. Kicker performance improves as their team falls further behind to avoid falling into the loss domain. In the second and third regressions, there is suggestive evidence that kicker is more likely to make a kick when losing than winning. Also, there is evidence that kickers appear to exhibit reference-dependent preferences around the reference point of a tie game.
The first regression which used the score margin as a continuous variable accurately predicted 86.28% of the kicks. Using the predict function within Stata, this model stated that all kicks with a predicted probability over .50 were successful field goals. This prediction ability decreased with the second regression - where attempts were split into winning, losing and tied games, as the model accurately predicted 86.23% of the kicks. The final regression correctly predicted 86.33% of the kicks after it split the observations into five distinct domains based on the scoring margin. Table 11 revisits all three predictions and an overview of the results. Table 10: Summary of Predictions
There were several data limitations. Environmental data recorded is at the time of kickoff. Detailed information on the weather conditions for each attempt field goal attempt is unknown, so this paper assumes constant conditions throughout the game. Also, kick placement between the hash marks was not available. Therefore, the coordinates of kicks on an (x, y) plane were not able to be recorded.
Another limitation of this paper is the interpretation of a field goal as a focus-oriented task rather than an effort-oriented task. Ariely et al. (2009) found that increased performance in certain tasks may not mean an increased desire to win. Their findings assume that improved performance in effort oriented tasks is a sign of loss aversion. Improved performance in focus oriented tasks is a sign of positive self-focus – rising to the occasion of the pressure. Ariely connected loss aversion in these settings to effort oriented tasks - so the appearance of loss aversion in field goal kickers may have alternative explanations.
6. Conclusion
Some of the most robust findings in behavioral economics appear in the lab, where it is easier to control for factors to create environments where the behavior of interest can be isolated (Levitt and List, 2008). However, critics of this literature believe that biases may disappear in the field in competitive markets with experienced agents (List, 2003). There is a growing amount of research in field settings to investigate whether biases persist in competitive markets - especially loss aversion around reference points. This paper analyzes field goal kicker performance in the NFL to examine whether loss aversion and reference-dependent preferences endure in these markets,
The abundance of data and zero-sum nature of the NFL provides a field setting which is ideal for testing if salient reference points exist, as well as loss aversion. The set play of a field goal allows researchers to explore these elements through a simple logistic regression model.
Previous analyses have found that kickers performance tends to respond to the environmental and situational factors in a game, and not just the distance of the attempt. This thesis observed that the score margin has a slightly negative relationship with the outcome of a successful kick. As a team falls into the loss domain, kickers performance improves, which may suggest mild loss aversion. The question whether kickers have reference-dependent preferences around requires additional research. The model within this paper predicted that kicks taken within the losing domain have a higher probability of success compared similar kicks taken within the winning domain. This finding appears to show a concentration of performance around a tie game reference point (since kicks while winning are less accurate and kicks while losing are more accurate) which would suggest that kickers have reference-dependent preferences. However, this finding did not have statistical significance when the field goals were split up amongst the various score margins.
Extensions of this paper should involve a larger data set with more than two years of field goals. Future studies should include Additional control variables for environmental factors and field characteristics (grass vs. turf). With a more developed framework and larger data set, researchers will be able to make more accurate inferences about whether loss aversion around a fixed reference point exists in the NFL. While this study only found mild evidence of loss aversion in kickers, it does contribute to the existing literature on loss aversion around a reference point in a competitive market with experienced agents. Observing reference points in the sports world can help future researchers establish reference points that are fixed, rather than based on expectations.
The world of sports is a highly competitive marketplace where interactions and behavior can be easily quantified. This saliency of this setting is uncommon in many other aspects of markets - which is why future research can leverage sports to make conclusions on human behavior and biases.
7. Appendix
Figure 1 - Dimensions of NFL Field
Source: American Football Field Layout and Dimensions - SportsCourtDimensions.com
Table 1 – Official NFL Field Rules
Sidelines and end lines are out of bounds. The goal line is actually in the end zone. A player with the ball in his possession scores a touchdown when the ball is on, above, or over the goal line.
The field is rimmed by a white border, six feet wide, along the sidelines. All of this is out of bounds.
The hashmarks (inbound lines) is 70 feet, 9 inches from each sideline.
Goal posts must be single-standard type, offset from the end line and painted bright gold. The goal posts must be 18 feet, 6 inches wide and the top face of the crossbar must be 10 feet above the ground. Vertical posts extend at least 30 feet above the crossbar. A ribbon 4 inches by 42 inches long is to be attached to the top of each post. The actual goal is the plane extending indefinitely above the crossbar and between the outer edges of the posts.
The field is 360 feet long and 160 feet wide. The end zones are 30 feet deep. The line used in try-for-point plays is two yards out from the goal line.
Chain crew members and ball boys must be uniformly identifiable. All clubs must use standardized sideline markers. Pylons must be used for goal line and end line markings.
End zone markings and club identification at 50-yard line must be approved by the Commissioner to avoid any confusion as to delineation of goal lines, sidelines, and end lines.
Table 2 – Predicted Probabilities of Field Goals by Distance
Figure 4: Field Goal Attempts by Quarter
-Percentages marked within each bar indicate the share of the total field goal attempts (y-axis)
Table 6 – Field Goal Success and Average Distance - Winning, Losing and Tied Domains
-This chart depicts the different field goal percentages split into winning, losing and tied areas. Each domain shows the magnitude of the score margin. Amounts are bolded to illustrate the higher values between winning and losing areas. -Column d is the field goal percentage which is column c/column d.
Table 7b: Initial Regression – Score Margin as Continuous Variable (Coefficients)
(1) (2) (3)
Distance -.127*** -.127** -.127*** (.009) (.009) (.009)
Score Margin -.011 -.014** -.014** (.0078) (.008) (.008)
Precipitation -.617** -.669** -.684*** (.348) (.349) (.349)
Wind -.345 -.335 -.324 (.224) (.225) (.226)
Cold Weather .206 .186 .188 (.185) (.186) (.186)
Playoff Team? - .27** .267** (.153) (.153)
Opponent Playoff Team? - -.094 -.103 (.150) (.150)
Home Game - - .027** (.144)
Pressure Situation - - -.279 (.239)
Constant 7.21 7.14 7.15 (.426) (.431) (.438)
Pseudo R2 .1790 .1813 .1822
Logit estimation - Dependent variable equals 1 if kick was made
***Statistically significant at the 5% level. **Statistically significant at the 10% level.
Table 8: Second Regression – Effect of Score Margin on Likelihood of Field Goal - Winning vs. Losing (Odds Ratios)
(1) (2) (3)
Kick Attempt Losing 1.19 1.21 1.18 (.239) (.245) (.363)
Kick Attempt Winning .986 .941 .906 (.197) (.189) (.248)
Distance .880*** .881*** .881*** (.008) (.008) (.008)
Precipitation .540** .513** .504** (.188) (.179) (.176)
Wind .708 .715 .724 (.158) (.160) (.163)
Cold Weather 1.22 1.19 1.20 (.226) (.223) (.224)
Playoff Team? - 1.29** 1.29** (.198) (.198)
Opponent Playoff Team? - .914 .906 (.137) (.136)
Home Game - - 1.03** (.149)
Pressure Situation - - .747 (.180)
Constant 1284.2 1207.4 1244.7 (568) (539.7) (567.5)
Pseudo R2 .1788 .1809 .1819
Logit estimation - Dependent variable equals 1 if kick was made
***Statistically significant at the 5% level. **Statistically significant at the 10% level.
Table 9: Third Regression - Effect of Score Margin on Likelihood of Field Goal – Five Groups - (Odds Ratios)
(1) (2) (3)
Kick Attempt Losing by 11+ 1.26 1.29 1.22 (.366) (.379) (.363)
Kick Attempt Losing by 1-10 1.17 1.18 1.17 (.246) (.251) (.248)
Kick Attempt Winning by 1-10 1.07 1.02 .995 (.230) (.223) (.218)
Kick Attempt Winning by 11+ .841 .779 .735 (.212) (.199) (.192)
Distance .880*** .881*** .881*** (.008) (.008) (.008)
Precipitation .546** .518** .510*** (.187) (.181) (.175)
Wind .702 .708 .716 (.157) (.159) (.162)
Cold Weather 1.23 1.21 1.21 (.229) (.225) (.227)
Playoff Team? - 1.32** 1.32** (.203) (.203)
Opponent Playoff Team? - .911 .903 (.137) (.136)
Home Game - - 1.03** (.149)
Pressure Situation - - .732 (.179)
Constant 1292.42 1211.7 1250.4 (573) (542.0) (570.5)
Pseudo R2 .1795 .1819 .1829
Logit estimation - Dependent variable equals 1 if kick was made
***Statistically significant at the 5% level. **Statistically significant at the 10% level. -Kick Attempt (rows 1-4) odds ratios are relative to a tie game. An odds ratio over 1 shows that a kick taken with that score spread is more likely to go in than a tie 8. References
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