FORMULATING OPTIMAL BETTING STRATEGIES

THROUGH A STATISTICAL PREDICTION OF THE

OUTCOME AND MARGIN OF VICTORY OF NFL GAMES

A THESIS

Presented to the University Honors Program

California State University, Long Beach

In Partial Fulfillment

of the Requirements for the

University Honors Program Certificate

Ranil Weerackoon

Spring 2015

I, THE UNDERSIGNED MEMBER OF THE COMMITTEE,

HAVE APPROVED THIS THESIS

FORMULATING OPTIMAL BETTING STRATEGIES

THROUGH A STATISTICAL PREDICTION OF THE

OUTCOME AND MARGIN OF VICTORY OF NFL GAMES

BY

Ranil Weerackoon

____________________________________________________________

Olga Korosteleva, Ph.D. (Thesis Advisor) Department of Mathematics and Statistics

California State University, Long Beach

Spring 2015

ABSTRACT

Billions of dollars are bet on sports each year, with no sport receiving more wagers than

football. The purpose of this thesis is twofold: to figure out which football team to bet on and

then how much to bet on that team. In this thesis, we construct a model that can be used to

predict the outcome of NFL games as well as the margin of victory. Tested over the course of the

2014 NFL season, the model constructs a rating for each NFL team by using variables such as a

team’s strength of schedule and adjusted margin of victory. After calculating the probability of a

team winning as well as its predicted margin of victory, a half Kelly staking strategy is used to

determine the optimal fraction of one’s bankroll that should be used for betting on the outcome

of games. Considering only games with an estimated positive expected value, our model

accurately predicts 70.4% of games and 55% of games against the Vegas point spread and

returns an overall profit.

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TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................................... iv

LIST OF FIGURES .........................................................................................................................v

CHAPTER

1. INTRODUCTION ...............................................................................................................1

Sports Betting and Terminology...................................................................................1

Strategies Associated With Betting ..............................................................................6

Fixed Staking ...............................................................................................7

Percentage Staking .......................................................................................7

Kelly Staking ...............................................................................................8

2. STRATEGIES USED FOR PREDICTIVE PURPOSES ..................................................10

3. DATA DESCRIPTION .....................................................................................................11

Strength of Schedule (SOS) and Simple Rating System ...........................................11

Adjusted Margin of Victory (AMOV) .......................................................................12

Team Rating ...............................................................................................................14

4. METHODOLOGY ............................................................................................................17

5. RESULTS ..........................................................................................................................19

Statistical Model ........................................................................................................19

Betting Strategies .......................................................................................................21

6. DISCUSSION ....................................................................................................................23

REFERENCES ..............................................................................................................................26

iv

LIST OF TABLES

Table Page

1. List of NFL Statistics .........................................................................................................16

2. List of Variables used each week in Statistical Calculations .............................................18

v

LIST OF FIGURES

Figure Page

1. Comparison of Statistical and Intuitive Analyses ..............................................................20

2. Accuracy in Predicting the Outcome of Games .................................................................21

3. Betting Profits by Week .....................................................................................................22

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CHAPTER 1

INTRODUCTION

Sports betting has grown into a marketable industry for bookmakers. Perhaps no sport has

gained greater profits for these industries than football, where a staggering $1.34 billion was bet

solely on football games in the state of Nevada in 2011 (Spear, 2013). However, while casinos

and Vegas have hauled in exorbitant profits, American gamblers lost $119 billion in 2013 betting

on sports (Aziz, 2014).

Unlike traditional gambling, in which the probability of an event occurring is known, the

probability of an outcome in sports can only be estimated. The purpose of this research study is

to ascertain ways to bet efficiently on NFL games. Football was chosen as the sport of interest

due to the high popularity of the game and because an overwhelming 65% of sports betting is

directed at football alone (McIntyre & Sauter, 2011). This study aims to identify a model that

will predict both the outcome and the point differential of a game so that one will have a higher

probability of making a profit by wagering on football. In addition to providing an analysis as to

which team to bet on in a particular game, this thesis also illustrates an optimal strategy as to

how much to bet on a particular team in order to maximize profits.

Sports Betting and Terminology

This paper will focus on two different forms of betting. One type of betting is predicting the

outcome of a game outright (i.e., betting on which team will win). For this form of betting,

bookmakers use odds that can be used to bet on a team for each individual game. The bookmaker

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will denote one team as a favorite and assign lower money odds to the bettor should that team

win while the team designated as the underdog will receive higher odds. For instance, an odds

line of -340 on the favorite and +280 on the underdog indicates that a bettor would make a profit

of $100 per $340 bet when the favored team wins the game. If a bet is placed on the underdog, a

bettor would earn $280 per $100 bet. In other words, the negative odds line (e.g., -340) indicates

how much the bettor would need to bet on the favorite while the positive odds line (e.g., +280)

indicates the winnings in the event the underdog wins the game. As is clear from the money

odds, a bet on both teams results in a profit for the bookmaker. In the example above, if a bettor

were to place a bet on both the favorite and the underdog, the bettor would lose $340 if the

favored team lost and win back only $280 when the underdog wins, resulting in a net profit of

$60 for the bookmaker. If the favorite won the game, the bettor would earn no profit (the $100

bet on the underdog would negate the $100 earned from the favored team winning). These odds

are thus biased towards bookmakers to ensure they have an edge over the bettor. The exception

to this occurs when a football game ends in a tie, in which the money used to bet is returned back

to the bettor and no profit is earned for either the bookmaker or the bettor. However, these

instances are extremely rare in the NFL as only five games out of 4,345 (0.11%) have ended in a

tie since the start of the 1998 season through the conclusion of the 2014 season.

The probability of a team winning a game can be calculated through the bookmaker’s

odds by taking the quotient of the amount bet and the potential amount received in revenue. For

example, the aforementioned -340 odds used for the favorite indicates that the favorite has a

𝟑𝟒𝟎

𝟒𝟒𝟎= 77.3% chance of winning the game while the +280 odds used for the underdog indicates

that the underdog has a 𝟏𝟎𝟎

𝟑𝟖𝟎= 26.3% chance of emerging victorious. The probabilities sum to

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103.6%, which accounts for the vigorish (also known as the vig or the overround) used by the

bookmaker to create an edge over the bettor and thus ensure a profit for the bookmaker, granted

the game does not end in a tie. However, the probability of a team winning after accounting for

the vigorish can be calculated by taking the denoted probability and divided it by the overall sum

that includes the vigorish (Sillanpää & Heino, 2013). Using the example from above, the favored

team has a 𝟎.𝟕𝟕𝟑

𝟏.𝟎𝟑𝟔× 100% = 74.6% chance of winning. Similarly, the underdog’s probability of

winning can be depicted as 𝟎.𝟐𝟔𝟑

𝟏.𝟎𝟑𝟔× 100% = 25.4%. The odds are designed such that the expected

value for the bettor is always negative and bets on the favorite assume greater risk. From the

example above, the expected value on a bet placed on the underdog is (-100) (0.746) + (280)

(0.254) = – $3.48 while the expected value on a bet placed on the favorite is (100) (0.746) + (-

340) (0.254) = – $11.76.

The second form of betting involves the point spread, or the line, for games. The point

spread is devised such that the team favored to win the game is assigned a handicap. For

instance, a bet on a team that is designated as a seven point favorite (i.e., a line of -7) needs not

only for the team to win the game, but for the team to win by more than seven points in order to

make a profit. In other words, the margin of victory for the favored team must be bigger than

seven points. If the margin of victory is identical to the point spread, it is denoted as a push and

no money changes hands (i.e., no profit is attained for either the bookmaker or the bettor). To

avoid a push, half point spreads (i.e., a line of 7.5) are often devised so that a win or a loss on a

bet is guaranteed. If the team favored wins by a margin greater than the line or if the underdog

“wins” the game after the handicap is assigned to the final score, the team is denoted as having

covered the spread.

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Most bookmakers used 11-10 odds for betting on a point spread (Winston, 2012). More

specifically, an $11 bet on a team would result in a profit of $10 if the team covers the spread.

For example, in Super Bowl XLVIII, the Denver Broncos were a 2.5 point favorite over the

Seattle Seahawks. Bettors who bet on the Broncos -2.5 would only receive a $10 profit per $11

bet if the Broncos won by three or more points. Bettors who bet on the Seahawks +2.5 would

win $10 per $11 bet if the Seahawks lost by less than three points or if the Seahawks won the

game. Thus, if a bet is placed on each team, the bookmaker receives a total of $22 and only has

to pay out $21 to the bet that results in a win (the $11 used to bet and the $10 profit), thus

guaranteeing the bookmaker a profit of $1 (Winston, 2012). Thus, the bookmaker is assured a

𝟏

𝟐𝟐= 4.5% profit, which accounts for the bookmaker’s vigorish. If a bettor has a probability 𝒑 of

winning a bet, then one will break even when betting on the spread when 𝟏𝟎𝒑 − 𝟏𝟏(𝟏 − 𝒑) =

𝟎, indicating that p = 𝟏𝟏

𝟐𝟏= 52.4%. Therefore, a bettor must win at least 52.4% of the time when

betting on the point spread in order to make a profit.

A common misconception as it pertains to sports betting is that the point spread is

devised in such a manner as to get an equal amount of money bets on both sides to ensure the

bookmakers’ 4.5% profit. However, Steven Levitt showed that bettors have a propensity to bet

on favorites even though home favorites and away favorites cover the spread only 49.1% and

47.8% of the time, respectively, while home underdogs and away underdogs cover the spread

57.7% and 50.4% of the time, respectively (as cited in Winston, 2012). Thus, bookmakers can

make a greater profit by increasing the point spread due to bettors’ inclination to place money on

the favorite. For example, assume the true point spread is Team A –6 over Team B (i.e., Team A

is a six point favorite). The bookmaker may opt to increase the point spread to seven because of

5

the bias towards the favored team, thus resulting in a situation where Team A no longer has a

50% chance of covering the spread. This gives the bookmaker an opportunity to earn a profit

exceeding 4.5%. In Levitt’s sample, bettors won 49.45% of their bets and lost 50.55% of the

time, thus creating 0.4945(-10) + 0.5055(11) = 61.56 cents profit for the bookmaker per $10 bet.

This exceeds situations where bettors win 50% of the time, in which bookmakers would make a

0.5(-10) + 0.5(11) = 50 cent profit, resulting in a 61.56/50% = 23% increase in profit for the

bookmaker (as cited in Winston, 2012).

Nevertheless, bookmakers constantly adjust the lines for games to take into account the

public perception of the teams and to ensure that not an overwhelming amount of bets are placed

on one team. For example, assume Team A starts off as a three point favorite over Team B. If the

public believes Team A is more than three points better than Team B and a large influx of bets

come in on Team A, the bookmaker may opt to change the odds to designate Team A as a five

point favorite. While the people who have already placed bets on Team A as a three point

favorite retain that point spread, future bettors are now forced to bet with the new point spread. If

an overwhelming amount of bets are placed on Team B (now a five point underdog), the

bookmaker may again adjust the line to Team A –4 to create a situation where the amount of

money placed is more evenly distributed.

While the bookmaker will seek to inflate the line so that it is biased towards the favorite

to increase the probability of a profit, the bookmaker also needs to make sure there are enough

bets placed on the underdog to reduce the chance of a loss should the favorite cover the spread.

For instance, in the event the favorite covers the spread when a total of $770 is bet on the

underdog and a total of $1320 is bet on the favorite, the bookmaker would retain the $770 bet on

6

the underdog but would have to pay bettors the $10 profit per $11 bet ($1320 ×𝟏𝟎

𝟏𝟏 = $1200) and

thus lose $770 – $1200 = $430. Thus, the bookmaker would look to increase the point spread in

this scenario to encourage more bets on the underdog and ensure a profit regardless of the

outcome.

This point adjustment on the part of the bookmaker in turn creates an inherent advantage

towards the bettor. As the bookmaker is implementing a point spread to forecast the behavior of

the bettor rather than the most accurate forecast of the true point spread, it is possible for the

bettor to formulate profitable betting strategies (Warner, 2010). Even though in the long run

bookmakers design odds and lines to make successful bets increasingly difficult, this bias can be

exploited if bets are optimally placed on the hidden true point spread of the game rather than the

line established by bookmakers.

While there are many other forms of betting (i.e., parlays, progressive parlays, teasers,

etc.), this thesis will focus exclusively on betting on the outcome of games and the point spread

as these are the most common forms of betting.

Strategies Associated with Betting

While it is clearly important to determine on who to bet, it is imperative to ascertain how

much to bet in order to ensure maximum profits. For instance, it may not be advantageous to bet

the same amount on a team that has an estimated 60% chance of winning compared to a team

that has an estimated 80% chance of winning. In situations where there is a positive expected

value, it may initially seem that the goal of the bettor should be to maximize the amount he/she

currently possesses. However, the expected payoff is only maximized when the bettor gambles

7

her/his entire bankroll (Moya, 2012). This is disadvantageous as the probability of losing

approaches one after a number of bets, thus resulting in bankruptcy for the bettor. Conversely, a

bettor may choose to minimize the risk of total ruin by minimizing the amount bet on each game.

However, this too is unfavorable in that it leads to smaller expected profits (Moya, 2012). Thus,

staking plans are used to maximize profit, minimize losses, and minimize overall variance of

one’s bank payroll. A discussion of various staking plans follow.

Fixed Staking

Fixed, or level, staking assigns a certain value for each bet (Moya, 2012). In other words,

the same amount is placed on each bet, regardless of the odds or percentage of winning the bet.

This staking plan has obvious drawbacks in that the betting amount doesn’t change when the

bankroll changes and the odds of success are not taken into account. Thus, there is a greater risk

of bankruptcy when the current bankroll is smaller than the initial bankroll as the staking plan

encompasses a greater proportion of the bank. Additionally, the opportunities to maximize

profits are mitigated when the current bankroll has expanded from its initial state, as a smaller

proportion of the bank is being used for betting purposes. Despite its inherent flaws, level staking

is used as the basis for other staking plans.

Percentage Staking

Percentage staking assigns a certain percentage of the current bankroll for each bet

(Moya, 2012). Thus, percentage staking addresses the weaknesses of fixed staking in that it calls

for larger bets when the current bankroll is larger than the initial bankroll, allowing for greater

profits, as well as calling for smaller bets upon a reduction from the current to the initial

bankroll, thus avoiding a further precipitous decline. The main drawback with percentage staking

8

is that it takes longer to recover losses than fixed staking as a smaller bet is assigned. However,

this form of staking is theoretically safer and sacrifices a short term profit with a more long term

approach (Moya, 2012).

Kelly Staking

Kelly staking assigns a variable percentage to be used for each bet. Rather than

maximizing the expected value for each bet, Edward Kelly posited that the expected log of the

payoff should be maximized as this maximizes the growth rate of the profit. (Moya, 2012). For a

given estimated probability of winning a bet, the optimal fraction to bet is:

𝒇 =𝒑×𝑾𝑰𝑵−𝒒×𝑳𝑶𝑺𝑬

𝑾𝑰𝑵×𝑳𝑶𝑺𝑬=

𝒑

𝑳𝑶𝑺𝑬 −

𝒒

𝑾𝑰𝑵 (1)

where 𝒇 = the fraction of the bankroll that should be used to bet, WIN = profit made per $1 bet,

LOSE = loss per $1 bet, 𝒑 = probability of winning the bet, and 𝒒 = 1 – 𝒑 = probability of losing

the bet (Winston, 2012). Thus, the Kelly betting strategy calls for high bets when the probability

of the team winning or the WIN odds are high. Conversely, if a team’s probability of winning are

low or if the LOSE odds are high, the amount bet should be lowered. Given the probability of a

team winning, the optimal betting fraction can be used to calculate the log of the expected

growth rate of one’s bankroll, as depicted in Eq. (2):

𝐥𝐧(𝒆𝒙𝒑𝒆𝒄𝒕𝒆𝒅 𝒈𝒓𝒐𝒘𝒕𝒉 𝒓𝒂𝒕𝒆) = 𝒑 × 𝐥𝐧(𝟏 + 𝑾𝑰𝑵 × 𝒇) + 𝒒 × 𝐥𝐧(𝟏 − 𝑳𝑶𝑺𝑬 × 𝒇) (2)

Eq. (1) and (2) can be used for determining the most efficacious amount of a bankroll that should

be used to bet on both the outcome of a game and the point spread. For example, assume that a

team has a 70% of chance of covering the spread, where one earns $1 for a successful $1 bet and

9

loses $1.10 per $1 losing bet. Then the Kelly betting strategy would suggest to bet 𝒇 =

(.𝟕×𝟏)−(.𝟑×𝟏.𝟏)

𝟏×𝟏.𝟏 = 33.64% of the amount of money one has available to bet. Over time, the log of

one’s bankroll would be expected to grow 𝟎. 𝟕 × 𝐥𝐧(𝟏 + 𝟏 ×. 𝟑𝟑𝟔𝟒) + 𝟎. 𝟑 × 𝐥𝐧(𝟏 − 𝟏. 𝟏 ×

. 𝟑𝟑𝟔𝟒) = . 𝟎𝟔𝟒𝟒, indicating that the expected growth rate is 𝒆.𝟎𝟔𝟒𝟒 =1.06647, or a 6.647%

increase in one’s bankroll if one wins 70% of her/his bets.

The biggest failure of the Kelly approach is that, unlike traditional gambling, the true

probability of success is unknown and can only be estimated. Thus, it is not definitive that a

maximum profit will be attained (Sillanpää & Heino, 2013). Perhaps more significantly, the

Kelly approach calls for large bets where the estimated probability of success is high. In the case

a bet on a far statistically superior team fails (a non-atypical occurrence in the NFL due to

injuries, overall parity, and upsets), a large proportion of the bankroll is lost due to that one bet,

thus resulting in a high risk of bankruptcy.

The half Kelly staking plan, in which only half of the optimal Kelly fraction is used, has

often been suggested to address this shortcoming. This method has 75% of the growth rate of the

Kelly staking plan but significantly reduces the probability of a damaging loss and thus decreases

the risk of bankruptcy (Moya, 2012). Due to the randomness of NFL games that prevent an

ability to calculate an exact probability of success, this thesis adopts the half Kelly staking plan

as the optimal betting strategy for maximizing profits while also minimizing potential losses.

10

CHAPTER 2

STRATEGIES USED FOR PREDICTIVE PURPOSES

In order to predict accurately the outcome of games and the margin of victory, two

separate statistical models must be determined for each type. With regards to predicting the

outcome of games, in which the dependent variable is binary as it only has two outcomes (win or

loss, excluding games that end in a tie), a logistic regression analysis can be performed (Bi &

Jeske, 2010). For predicting the point spread, a multiple linear regression analysis is preferred as

the dependent variable (margin of victory) is a continuous variable. The results from the

formulation of these models will be compared to other statistical models, such as the NFL ELO

ratings designed by Nate Silver and picks from the Microsoft Cortana system.

This study will also make use of predictions based on intuition in order to compare how

an intuitive analysis compares with a statistical analysis. As there are numerous variables in

football that are difficult to quantify and account for (i.e., injuries, momentum, weather), there

may be some benefits to going with intuition. Thus, a comparison from an intuitive perspective

to what the statistics suggest regarding the game’s outcome or point spread may be noteworthy.

Along with 13 NFL experts and analysts from ESPN, we will use our own intuitive analysis as a

means of comparison to the statistical model.

11

CHAPTER 3

DATA DESCRIPTION

The data used for this research include information from weeks 5-17 of the 2014 NFL

season, a sample size of 195 games. The first four weeks of the season (61 games) will be used

as the training data set to predict the outcome of the games in week five. The data accumulated

from the first five weeks will then be used to predict the outcomes in week six. It will continue in

an iterative process so that outcomes from the last week of the NFL season (week 17) will be

predicted using data from the first 16 weeks (240 games). The exclusion of the first four games

for predictive purposes mirrors the approach of Warner’s statistical formulation (Warner, 2010).

A number of variables exist that can be used for predictive purposes. The model developed in

this thesis incorporates some basic NFL statistics, such as points per game and turnovers, as well

as a few more complex analytical features, described in detail below.

Strength of Schedule (SOS) and Simple Rating System (SRS)

The strength of schedule feature is designed to take into account the quality of opponent a

team has faced. Each team is assigned a variable rating, thus forming a system of 32 equations

with 32 unknowns overall, with each unknown value representing each of the 32 teams in the

NFL. The simple rating system is the sum of a team’s average strength of schedule and the

team’s average margin of victory. For instance, if Team A plays Team B, Team C, and Team D

and has an average margin of victory of five points per game, then Team A’s SRS can be

denoted as 𝑻𝒆𝒂𝒎 𝑨 = 𝟓 + 𝟏

𝟑 (𝑻𝒆𝒂𝒎 𝑩 + 𝑻𝒆𝒂𝒎 𝑪 + 𝑻𝒆𝒂𝒎 𝑫). In order to determine Team

12

B’s SRS, a separate equation needs to be set up for Team B based on Team B’s strength of

schedule. This simple formulation provides an intuitive approach to designing a team’s rating.

For instance, an average team would have a rating of zero, and a team that has faced an average

strength of schedule would have a rating exactly equal to its average margin of victory. The

major benefits to this system are that the rankings are easy to interpret; for example, a team with

an SRS of 5 indicates the team is five points better than an average team while a team with an

SRS of -3 is three points worse than an average team.

Adjusted Margin of Victory (AMOV)

Margin of victory can often be viewed as more informative in ranking a team’s true

performance than winning percentage. However, one of the main drawbacks in simply using

margin of victory is that blowouts can oftentimes overstate a team’s rating. For instance, a team

that wins a game by 20 points as compared to 5 points (a 15 point difference) is significant in

that the former would represent a convincing win while the latter would seem to indicate a close

win. However, if a team wins a game by 35 points as compared to 20 points (still a 15 point

difference), the difference is negligible as both games represent decisive victories. To

compensate for this, Keener (1993) adjusted the margin of victory in the following fashion:

𝒂𝒊𝒋 =𝑺𝒊𝒋+𝟏

𝑺𝒊𝒋 + 𝑺𝒋𝒊+𝟐 (3)

where 𝒂𝒊𝒋 depicts Team 𝒊′s adjusted margin of victory (AMOV) against Team j, 𝑺𝒊𝒋 represents

the amount of points scored by Team 𝒊 against Team 𝒋, and 𝑺𝒋𝒊 represents the amount of points

scored by Team 𝒋 against Team 𝒊. The 1 and 2 on the numerator and denominator, respectively,

are included to ensure that a team doesn’t take all the credit in the case of a shutout. This

13

approach ensures that a team’s margin of victory for a given game falls between 0 and 1. Warner

(2010) developed a similar system, but in a manner such that the AMOV was not distributed in a

linear fashion:

𝒂𝒊𝒋 = 𝑨𝑴𝑶𝑽 = 𝟎. 𝟓 + 𝟎. 𝟓𝒔𝒈𝒏 (𝑺𝒊𝒋+𝟏

𝑺𝒊𝒋 + 𝑺𝒋𝒊+𝟐−

𝟏

𝟐) √|𝟐 (

𝑺𝒊𝒋+𝟏

𝑺𝒊𝒋 + 𝑺𝒋𝒊+𝟐) − 𝟏| (4)

where 𝒔𝒈𝒏(𝒙) = {

−𝟏 𝒊𝒇 𝒙 < 𝟎𝟎 𝒊𝒇 𝒙 = 𝟎𝟏 𝒊𝒇 𝒙 > 𝟎

} denotes the standard sign function.

The use of the square root function allows 𝒂𝒊𝒋 to approach 0 or 1 quickly to compensate for

blowout wins. One of the drawbacks of Warner’s method for adjusting the margin of victory is

that it can rank a team with a close win in a low-scoring game over a bigger win in a high-

scoring game. For example, a 6-3 win by Team A would be considered a close win, but the

AMOV = 𝟎. 𝟓 + 𝟎. 𝟓√|𝟐 (𝟔+𝟏

𝟔+𝟑+𝟐) − 𝟏| = 𝟎. 𝟕𝟔, thus indicating a rather significant win for

Team A. However, a 38-24 win would be considered by many to be a more impressive win, and

yet the AMOV = 𝟎. 𝟓 + 𝟎. 𝟓√|𝟐 (𝟑𝟖+𝟏

𝟑𝟖+𝟐𝟒+𝟐) − 𝟏| = 𝟎. 𝟕𝟑. This thesis compensates for this by

taking into account both the margin of victory and the AMOV, as described in the section below.

The model used in this thesis also simplifies Warner’s formulation such that the adjusted margin

of victory falls between -1 and 1 rather than 0 and 1 to create a significant difference between a

win and a loss, as depicted in Eq. 5:

𝒂𝒊𝒋 = √|𝟐 (𝑺𝒊𝒋+𝟏

𝑺𝒊𝒋 + 𝑺𝒋𝒊+𝟐) − 𝟏| (5)

14

The AMOV is kept between -1 and 1 so that a loss produces a negative AMOV while a win is

designated by a positive AMOV as opposed to the formulation designed by Warner in which a

loss has an AMOV less than 0.5 and a win produces an AMOV greater than 0.5. The approach

used in this thesis is designed in a manner such that for each game, the loser’s AMOV is the

negative value of the winner’s AMOV.

Team Rating

Each team’s rating is composed of the sum of the simple rating system (which takes into

account strength of schedule) and a team’s average adjusted margin of victory. Since the AMOV

falls between -1 and 1, the simple rating system was adjusted to fall between 0 and 1, as depicted

in Eq. 6:

𝒔𝒊 =𝑺𝑹𝑺𝒊−𝐦𝐢𝐧(𝑺𝑹𝑺𝒋)+𝟏

𝐦𝐚𝐱(𝑺𝑹𝑺𝒋)−𝐦𝐢𝐧(𝑺𝑹𝑺𝒋)+𝟐 (6)

where 𝑺𝒊 is a team’s adjusted SRS, 𝑺𝑹𝑺𝒊 is a team’s SRS, and 𝑺𝑹𝑺𝒋 is the rating for the 𝒋𝒕𝒉team

in the NFL that Team 𝒊 has faced. In using the SRS rather than SOS, the actual margin of victory

is still taken into account to ensure that low scoring wins don’t bolster a team’s rating (as

described in the above section), but is combined with the AMOV to ensure that blowouts don’t

overstate or understate a team’s rating. Thus, a team’s rating, 𝒓𝒊 can be denoted as:

𝒓𝒊 =𝟏

𝒏𝒊∑ (𝒂𝒊𝒋+𝒔𝒊)

𝟑𝟐𝒋=𝟏 (7)

15

where 𝒏𝒊 is the number of games Team 𝒊 has played up to the point in the season the rating is

calculated. This rating system ensures that a big win against a mediocre team doesn’t buffer a

team’s rating as much as a smaller win against a good team.

Vegas point spreads and other statistical models often take into account home-field

advantage by giving the home team 2.5 or 3 points. For example, if Team A has an SRS of 5 and

Team B has an SRS of 3, it can be interpreted that Team A is two points better than Team B on a

neutral field. When Team A hosts Team B, Vegas will give Team A three points to account for

home-field advantage, and thus Team A becomes a five point favorite. If Team B is playing at

home, Team B becomes a 1 point favorite. This system has significant flaws as it assumes every

team benefits by about three points from playing at home. However, some teams may have a

greater home-field advantage and/or play better at home than an average team. Conversely, some

teams may not have as decisive a home-field advantage or a team might play particularly better

on the road, thus not justifying the approach of giving the home team an additional three points.

To compensate for this, a home/road adjusted rating is composed for each team by calculating

the team’s rating in Eq. 7 for (1) home games only and (2) road games only. This provides a

home rating and a road rating for each team. Thus, a home adjusted rating can be developed for

each team by placing greater weight (i.e. three times as much) on home games than road games.

𝒓𝒊𝒉𝒐𝒎𝒆_𝒂𝒅𝒋=

𝟑(𝒓𝒊𝒉𝒐𝒎𝒆)+(𝒓𝒊𝒓𝒐𝒂𝒅)

𝟒 (8)

Similarly, a team’s road adjusted rating can be devised by placing three times as much weight on

road games than home games.

𝒓𝒊𝒓𝒐𝒂𝒅_𝒂𝒅𝒋=

𝟑𝒓𝒊𝒓𝒐𝒂𝒅+(𝒓𝒊𝒉𝒐𝒎𝒆)

𝟒 (9)

16

The complete set of variables considered in this model are listed in Table 1.

TABLE 1: List of NFL statistics

Statistics

1. Win percentage

2. Points per game

3. Points per game allowed

4. Turnovers (giveaways) per game

5. Defensive turnovers (takeaways) per game

6. Net passing yards per attempt (PYA)

7. Defensive net passing yards per attempt (DPYA)

8. Home-field advantage

9. Strength of Schedule (SOS)

10. Simple Rating System (SRS)

11. Adjusted Simple Rating System (ASRS)

12. Adjusted Margin of Victory (AMOV)

13. Team Rating

14. Home/Road Adjusted Rating

In a vein similar to the neural network system developed by Khan, the underdog’s statistics

are subtracted from the designated favorite in order to simplify each variable (Khan, 2013). In

other words, if Team A is favored and averages 25 points per game and Team B averages 22

points per game, the data entry for the teams’ difference in points per game is 25-22=3,

indicating that Team A’s offensive output is three points per game better than Team B. The

analyses will reveal whether these variables are significant predictors in determining the results

of a game and the point spread.

For games that were played in London, the difference in each team’s rating was used in place

of a home/road adjusted rating. Since the designated home team isn’t playing in its home

stadium, it did not warrant using the home/road adjusted rating statistic.

17

CHAPTER 4

METHODOLOGY

The odds of a team winning as well as the point spread was gathered from Football Locks, a

website that accrues the average Vegas team odds and point spreads for each game. In predicting

the outcome of each game for the upcoming week, data is accumulated for the variables listed in

Table 1 for each team from all the previous weeks. A stepwise selection procedure was run prior

to a logistic regression analysis to determine which subset of variables should be used in the final

model. As a result, it is possible that different variables may be used to predict the outcome of

games each week as the significance of a particular variable in the final model can change with

the data accumulated over the previous weeks. Similarly, a stepwise selection procedure was

conducted to determine the subset of variables used in a multiple regression analysis to forecast

the margin of victory. Table 2 details the variables used in both the logistic regression and

multiple regression analyses for each week of the season.

With regards to predicting the outcome of games, a call for a bet on a team will be placed

when there is a positive expectation of winning (i.e., probability of the team winning calculated

through the model multiplied by the odds for a win is greater than the probability of a loss

multiplied by the odds for a loss). The fraction of a bankroll to be used for betting purposes was

ascertained using the half Kelly staking approach. Because there were between 12 and 16 games

per week, the sum of the half Kelly fractions were greater than zero (i.e. if 𝑲𝟏 = . 𝟓𝟓 and 𝑲𝟐 =

. 𝟓𝟑, the sum of the two optimal fractions to be used is 1.08 > 1 = 100% of the bankroll). To

adjust for this, the total bankroll was divided by the sum of the half Kelly fractions to determine

18

the amount of money to be bet on the game, as shown in Eq. 10:

𝑲𝒋 =𝑩

∑ 𝑲𝒊𝒏𝒊=𝟏

(10)

where 𝑲𝒋 represents the amount of money to bet on game 𝒋, 𝑩 is the total bankroll for that week,

and 𝑲𝒊 is the half Kelly fraction. For the purposes of this thesis, a bankroll of $𝟐𝟎𝒏 was

estimated per week, where 𝒏 represents the amount of games for that week. If 𝑲𝒋 < 𝟓, then

𝑲𝒋 was rounded up to 5 since a minimum $5 bet is generally required when betting. Additionally,

𝑲𝒋 was rounded to the nearest dollar for each game to simplify betting procedures.

As it pertains to predicting the point spread, a call for a bet on a team will be determined

when the predicted margin of victory is significantly higher than the bookmaker’s point spread (a

difference of more than one point from the Vegas spread). When the probability of a team

winning or the predicted margin of victory is not significantly higher, it will be advised that no

bet be placed. Once again, $20 was used as the average bankroll for each game.

TABLE 2. List of Variables used each week in Statistical Calculations

Week Predicting Outcome of Games Predicting Margin of Victory

5 Win Percentage + Home/Road Adj. Rating SRS + Home/Road Adj. Rating

6 Home/Road Adj. Rating Home/Road Adj. Rating

7 Win Percentage + Home/Road Adj. Rating SRS + Home/Road Adj. Rating

8 Win Percentage + AMOV + Home/Road

Adj. Rating

Points per game + Points per game allowed

+ AMOV + Home/Road Adj. Rating

9 Home/Road Adj. Rating SRS + team rating + Home/Road Adj.

Rating

10 Win Percentage + Home/Road Adj. Rating Home/Road Adj. Rating

11 Win Percentage + Home/Road Adj. Rating Home/Road Adj. Rating

12 Home/Road Adj. Rating Home/Road Adj. Rating

13 Win Percentage + Home/Road Adj. Rating Home/Road Adj. Rating

14 Home/Road Adj. Rating Home/Road Adj. Rating

15 Home/Road Adj. Rating Home/Road Adj. Rating

16 Win Percentage + AMOV + Home/Road

Adj. Rating

SRS + team rating + Home/Road Adj.

Rating

17 Home/Road Adj. Rating Home/Road Adj. Rating

19

CHAPTER 5

RESULTS

Statistical Model

A Hosmer and Lemeshow Goodness-of-Fit Test was conducted to test to see if the logistic

analysis was a good fit for the model (Magel & Childress, 2011). The logistic equation was

deemed appropriate with a p-value > 0.6 for all 12 weeks used in the analysis.

Overall, the model predicted 130 out of the 194 games (67.0%) successfully, with one game

excluded due to it ending in a tie. However, when removing games where there was a negative

expected value and thus no bet was advised, the model predicted 112 out of 159 games correctly

(70.4%). This was comparable to other statistical models, such as the ELO system and the

Microsoft Cortana system, which both had a record of 135-59 (69.6%), over the same time

period used by the model developed in this thesis. The ELO ranking system went 176-79

(69.0%) over the course of the entire season, slightly higher than the 169-86 (66.3%) record

compiled by the Cortana system.

The statistical analysis was also compared to intuitive analyses from 13 NFL experts from

ESPN. The 13 experts compiled an average record of 131-63 (67.5%) after the fourth week of

the season, with a 166-89 (65.1%) record over the course of the entire season, comparable again

to the records compiled by the statistical analyses. To estimate how an average NFL fan would

fare predicting games solely on instinct, we also recorded our own intuitive analysis over the

course of the season and finished with a 126-68 (64.9%) record. A graphical representation of

the records for each system can be found in Figure 1.

20

FIGURE 1: Comparison of statistical and intuitive analyses. The analyses are compared for

predicting game outcomes between Weeks 5-17 of the 2014 NFL season. The probability

corresponding to the ESPN experts is calculated from the average of 13 ESPN experts.

With regards to the point spread, no bet was advised on 24 games due to a similar point

spread predicted by Vegas bookmakers and our model. This resulted in a 94-77 (55.0%) overall

record against the spread, which is slightly, but not significantly, higher than the 52.4% needed

to break even when betting against the spread (p = 0.25). The ELO ranking system was the only

other system that predicts against the spread, and it compiled a comparable 104-75 (58.1%)

record between weeks 5 and 17 and a 129-108 (54.4%) record overall.

We were unable to conduct an intuitive analysis using the ESPN experts as they did not pick

against the point spread. However, our own intuitive analysis led to a 98-95 (50.8%) record

between weeks 5 and 17 while we registered a mark of 130-123 (51.4%) over the entire season,

both of which fall below the threshold to break even.

62%

63%

64%

65%

66%

67%

68%

69%

70%

71%

Model Model (Betting) ELO Cortana ESPN Experts Intutiveanalysis

System

21

FIGURE 2: Accuracy in predicting the outcome of games. The graph depicts the accuracy of

predicting games using our statistical model and when using the model only for games with a

positive expectation of winning (games in which betting is advised). Aside from Week 5 and

Week 16, our model performed better when recommending betting only on games with a

positive expectation as compared to betting on every game.

Betting Strategies

Using the half-Kelly staking strategy and assuming a bankroll of $20 per game, a bankroll of

195 games × $20 = $3900 is required over the course of the season when betting on the outcome

of a game (an actual total of $3924 was used for betting on the outcome of games due to

rounding to the nearest dollar for each game). Overall, the half Kelly staking strategy resulted in

a profit of $107.25 over the season, a 2.73% return on investment. A fixed staking approach

where exactly $20 was bet on each game (and thus a total of $3900 on the season) accumulated a

profit of $154.83, a 3.97% return on investment. Surprisingly, the fixed staking approach also

recorded profits in 10 of the 13 weeks, as opposed to a profit return in only eight out of the 13

weeks when using the half Kelly staking approach.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

5 6 7 8 9 10 11 12 13 14 15 16 17

Acc

ura

cy P

erce

nta

ge

Week

Our Model Our Model for Positive Expectation

22

FIGURE 3: Betting profits by week. The betting profits are operated under the assumption of a

bankroll of $20𝑛 each when betting on both the outcomes and point spread for 𝑛 games per

week.

Since the point spread model is not equated with a probability but only used for betting

purposes when there was a one point differential between our model and the Vegas spread, no

optimal fraction was needed. The $20 bet on games against the spread for each of the 173 games

($3420 on the season) resulted in a total profit of $186 on the season, a 5.44% return on

investment on the season. Figure 3 contains the complete breakdown of the profit accumulated

from betting on both the outcome and point spread by week. For the season, betting on both the

outcome and point spread resulted in a total profit of $107.25 + $186 = $293.25, or a 3.99%

yield rate.

-200.00

-150.00

-100.00

-50.00

0.00

50.00

100.00

150.00

200.00

5 6 7 8 9 10 11 12 13 14 15 16 17Pro

fit

Week

Outcomes Point Spread

23

CHAPTER 6

DISCUSSION

Formulating a successful betting approach against established bookmakers has proven to be a

difficult endeavor. Previous studies, such as the one conducted by Warner, were not able to

develop a model that had a sustainable level of success against Vegas odds. For instance, even

though Warner’s initial feature set included a wide array of variables, such as a ranking system

whereby the margin of victory was modified as well as temperature difference to determine if a

team’s level of play was affected by the temperature at game time, his model only had a 64.36%

success rate in predicting the outcome of games and a 50.90% success rate against the spread

during the course of the 2008 and 2009 seasons (Warner, 2010). Both of these percentages were

below the threshold needed to make money when betting against Vegas.

This model was able to build on the ranking system for each team by taking into account the

adjusted margin of victory as well as a team’s strength of schedule. The calculations in this thesis

correctly predicted 67.0% games and 55.0% games against the spread. However, a sample size of

one season is particularly small and further compounded by the fact that multiple games had to

be excluded because of the similarity in the predictions by both the models of this thesis and the

one developed by Vegas.

One of the significant drawbacks to the model is its inability to take into account a player’s

value or contribution to a team’s success. In other words, an injury to a star player on a football

team would dramatically affect the odds of a team winning a game as well as the point spread in

Vegas but would be unaccounted for in this model. This has frequently been a problem in other

24

statistical models as Vegas has the advantage of heuristics and other real life information that

cannot be explained by statistics only. Thus, investigation into a manner of rating a team’s

players to account for injuries could prove beneficial, such as the implementation of a rating that

incorporates wins above replacement for each player by examining the difference in a team’s

performance with a starter as compared to the use of an average-level replacement player

(Hughes, Koedel, & Price, 2014). A ranking system for each individual player in the NFL is

outside the scope of this thesis. However, accounting for each player’s individual impact on a

team’s overall performance can help in modifying the rating system when a certain player is

unable to play due to an injury or suspension.

While the model had some success in its statistical methodology, which in turn led to a

successful return on investment on both the outcome of games (2.73%) and margin of victory

(5.44%), the appropriateness of a half Kelly staking strategy for predicting games requires

additional testing because a fixed staking strategy resulted in a greater profit (3.97% return on

investment). The failure of the half Kelly approach was largely due to the fact that huge losses

were accrued when large bets were placed on heavy favorites and those favorites were upset.

Because only small profits are attained when heavy favorites win and large losses ensue when

those favorites lose, any one game that results in an upset when bets are solely placed on

favorites can result in an overall deficit. Since upsets are commonplace in the NFL due to overall

league parity and other mitigating factors (i.e. injuries), a fixed staking strategy has some merit

and some balance between the two staking strategies might prove more beneficial. While the half

Kelly staking approach was used to predict games, only a fixed staking approach was used when

predicting margin of victory. It may prove advantageous to determine if there are optimal betting

strategies that can be used when betting against the spread.

25

Additionally, further investigation is warranted into the possible use of the predictive model

for postseason play. This thesis covered regular season games only due to other circumstances

that can affect postseason games. Nevertheless, a test trial was conducted for the 2014 NFL

playoffs and the model recorded a 7-4 overall record in predicting games as well as a 6-4 record

against the spread. When excluding playoff games where no bet was advised because of a

negative expected value in profit, the model registered a 4-1 record against the spread. However,

this is an incredibly small sample size for playoff games and thus an examination into how this

model can predict playoff games across multiple seasons is needed before any definitive

conclusions can be made.

This study provides an innovative examination into how to construct a team rating while

taking into consideration strength of schedule and an adjusted margin of victory in addition to

accounting for home-field advantage. Though this predictive model achieved a certain level of

success in predicting the outcome of games and margin of victory and attained an overall profit

from the half Kelly staking approach, further research needs to be conducted to determine if both

the model and betting strategies are sustainable over time.

26

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