The National Football League (NFL) is a multi-billion dollar business whose goal is to satisfy its fans. In order to achieve optimal satisfaction, the NFL adopted the sudden-death overtime period in 1974 because fans were not pleased with the conclusions of games ending in a tie. Fans wanted a winner to be declared for each game and in hopes of fulfilling this request, NFL management implemented the sudden-death overtime system, which states if a game is tied at the end of regulation then one fifteen minute period will be played. The first team to score will be declared the victor and if no team scores during the overtime period then the game will end in a tie. For decades this system has produced many exciting finishes and a highly satisfactory product. However, in recent years, the sudden-death overtime system has been scrutinized due to lack-luster outcomes that leave fans, players and the overall product of the NFL displeased. The game of football has changed significantly since the installment of the sudden-death in 1974. Rules have been implemented to protect players’ health, and the players themselves are much larger, stronger and faster. Hence, many fans feel that the sudden-death overtime system leaves the outcome of a game to be decided not by its entirety but instead by the result of an overtime coin toss. In today’s game kickers are making field goals from much farther distances with greater accuracy than ever before. Therefore, the sudden-death overtime system is setup for dull finishes in which the team who wins the coin toss kicks a long field goal to win the game while the other team does not get the opportunity to respond. With an increase in fan dissatisfaction the NFL has recently changed its overtime policy, such that if the receiving team in overtime scores a touchdown, then they will win the game. Otherwise the opponent will have an opportunity to possess the football and win the game. This paper will discuss a few proposed overtime policies, but mainly will focus on the first-to-six system investigated through Markov chain analysis. It has been hypothesized that the first-to-six system would give the team who lost the coin toss a better chance of winning. Will the sudden-death overtime system be more or less balanced in terms of game outcomes compared to the first-to-six system?
In order to determine whether the proposed system will be more desirable, it is necessary to establish a baseline for the sudden-death overtime system using Markov chain analysis. The article Win, Lose, or Draw: A Markov Chain Analysis of Overtime in the National Football League (2004) analyzes the impact of the coin toss on the outcome of the game and the efficiency at eliminating ties. A Markov chain model to analyze the sudden-death overtime system was developed using the following notation: [a b T], where a is the number of points scored by team A in overtime, b is the number of points scored by team B in overtime, and T is the team that has possession of the football (Jones, 2004, p. 331). In this system, a team can win by scoring in any manner. The most typical methods of scoring are to either score a touchdown or kick a field goal. Let the probability of a given team scoring a touchdown be represented by α, such that α ≥ 0, and the probability of kicking a field goal be represented by β, such that β ≥ 0, on a possession. Notice that α + β < 1 since it does not take into consideration a team scoring a safety. Thus γ = α + β will represent the probability that a team scores provided they have possession of the football (Jones, 2004, p. 331).
In Table 1, the possible states of the system have been represented in which state one is the point in overtime where team A has possession of the football, state two is the point in overtime where team B has possession of the football, state three denotes team A winning the game, and state four denotes team B winning the game (Jones, 2004, p. 331). Notice that states three and four are absorbing since upon entering either of these states the game will be over. A directed graph of the Markov process appears in Figure 1 (Jones, 2004, pp. 331-332).
Now the transition matrix Psd can be formed such that Psd = 
. Once again notice that states three and four are absorbing which means that the transition matrix can be rearranged as Psd = 
(Jones, 2004, p. 332). If c represents the number of absorbing states and m represents the number of non-absorbing states then I is a c x c identity matrix, 0 is a c x m zero matrix, R is a m x c matrix that gives the probabilities of going from non-absorbing to absorbing states, and Q is a m x m matrix that gives the probabilities of going from non-absorbing to non-absorbing states. With transition matrix Psd rearranged in the above form, (I – Q)-1 can be calculated which yields the expected number of times the Markov process will be in a given state before being absorbed. Furthermore, (I – Q)-1 R = 
provides the probability of absorption by any given absorbing state. Thus, if team A receives the football in the overtime period then team A wins with probability of 
which means that team B wins with probability of 
(Jones, 2004, pp. 332-333). In theory, the transient states will eventually enter one of the absorbing states but in reality this may not be the case since a game can end in a tie due to no team scoring within the fifteen minute period. Since there is a time constraint this implies that there will be a finite number of possessions in the overtime period, which will be represented by n. If team A receives the football to begin overtime then they will have n/2 possessions and team B will have n/2 possessions. Thus the probabilities for the possible outcomes of the game can be determined as the product of [1 0 0 0] 
(Jones, 2004, p. 333). Now that a general solution set has been formed for the sudden-death overtime system a general solution set must be calculated for the first-to-six system.
The first-to-six system states that the first team to score six points in overtime wins the game. In this system the sample space of possible events is larger compared to the sudden-death system. For example, if team A wins the coin toss and elects to receive the football then they can win the game by scoring a touchdown, but if they kick a field goal team B will get possession of the football. If team B scores a touchdown they win the game, otherwise the game will continue, within the fifteen minute period, until either team A or B has at least six points. If the overtime period expires, then the team with the most points will win or the game will result in a tie.
Table 2. The states for a finite Markov chain representing the first-to-six overtime rule.
In Table 2 the possible states of the system have been represented with respect to the previously defined notation [a b T] that was used during the sudden-death overtime procedure (Jones, 2004, p. 333). Certainly this policy will decrease the chances that the receiving team wins the game on the first possession, but the question that this paper focuses on is how will this new system change the probability of winning the game for the team that kicks the football to start the overtime period? To determine an answer to this question the transition matrix for the first-to-six system is constructed as 
. Once again states nine and ten are absorbing since upon entering these states the game is finished which means that the transition matrix can be rearranged such that Pft6 = . Assuming that team A receives the football, it follows that (I – Q)-1 R can be calculated and the probability that team A wins is 
and the probability that team B wins is 
(Jones, 2004, p. 334). This is the general solution set for the first-to-six system that will be used to answer whether the first-to-six system will affect the likelihood of team A winning the game. In order to get numerical solutions to compare, data from the 2002 NFL football season will be used.
In the 2002 NFL regular season there were 6049 total possession in which 1270 possessions resulted in a touchdown, 737 possessions resulted in a field goal and the average number of possession per period were approximately 6. So it follows that α = 
and β = 
, which means that γ = α + β = 0.210 + 0.122 = 0.332. Using the sudden-death overtime system if team A receives the football then theoretically team A will win with probability 
= 0.599 , team B wins with probability of 
= 0.401, but these probabilities assume that a tie cannot occur. So to take the result of a tie into account the initial state of the system [1 0 0 0] must be multiplied with our nth transition matrix, 
, where n = 6. Therefore [1 0 0 0] 
= [0.089 0 0.546 0.365], which means that for the sudden-death system approximately 8.9% of games end in a tie, team A wins 54.6% of the games, and team B wins 36.5% of the games. The probabilities of game outcomes for the first-to-six policy can be calculated in a similar manner. Assuming that team A receives the football it follows that team A wins approximately 49.1% of the games; team B wins is 39.3% of the games and 11.6% of the games end in a tie (Jones, 2004, p. 335). Based on the above data, it is clear that the first-to-six policy does in theory decrease the probability that receiving team will win the game in overtime.
To provide additional statistical support for the thesis that the first-to-six policy does decrease the probability of the receiving team winning in overtime, I decided to carry out my own research. In order to create a larger statistical model, I accumulated data that ranged from the 2002 NFL season till the 2010 NFL season. After gathering this statistical data I used the general solution sets for the sudden-death system and the first-to-six system that were derived above and graphed the results in Table 3.
Table 3. The statistical NFL overtime analysis from the 2002 season till 2010 season.
Assuming that team A receives the football to start the overtime period, the first row of the chart represents that theoretical probability that team A wins for the given season using the current sudden-death system and the second row represents the theoretical probability that team B wins for the given season using the sudden-death system. Rows three and four represent the theoretical probabilities that team A or team B wins, respectively, for the given season using the proposed first-to-six system. From Table 3 it has been shown that the probability of team A winning the game was not only approximately 3.53% lower per year for the first-to-six system but in fact the first-to-six system produced more desirable outcomes for every season when compared with the sudden-death system.
With the increase in demand for an exciting overtime system, NFL management has several types of systems that they could implement which will provide an overall satisfactory product for the players, owners, and most importantly fans. The question of whether or not the proposed first-to-six system would be more balanced in terms of victor in comparison with the sudden-death system was answered. After reviewing the article, Win, Lose, or Draw: A Markov Chain Analysis of Overtime in the National Football League, the general solution sets were found for both systems. Next actual data from a football season was entered in the previous formulations, the numerical results were compared, and it was concluded that the first-to-six system does decrease the probability that the team who receives the football to start the overtime period will win with respect to the sudden-death system. Further research was executed and involved expanding the data set by using data from multiple seasons. The larger research did support the hypothesized thesis that the first-to-six system would make the outcomes of overtime games fairer.
References
Jones M. (2004) Win, Lose, or Draw: A Markov Chain Analysis of Overtime in the National Football League. The College Mathematics Journal, 35 (5). 330-336.