A Game Theoretical Analysis of Overtime Proposals for the NFL

By AnaMaria Perez


Introduction:

Patriots vs Falcons – it’s 2017, and they are in the final minutes of the Super Bowl game. With only seconds left to play, Tom Brady ties up the score 28-28, and the game goes into overtime. The Patriots win the coin toss, and soon after, the game (Patriots). Their historic comeback from 3-28 to winning the game is legendary, but the results could have been vastly different, defined, quite literally, by the flip of a coin.


For years, a coin flip has influenced the results of NFL (National Football League) football games that go into overtime. Although the methodology seems fair – each team has a 50/50 chance of winning the coinflip and gaining possession of the ball – the results of the coinflip can hardly be classified as independent from the results of the game. The outcomes of football games should not be so heavily based on a coinflip - if they were, why don’t they flip a coin to choose the Super Bowl winner? And yet we find that out of 52 NFL games that have gone into overtime, the winners of the coin flip have won 28, and tied 4. By having first possession of the ball in overtime, a team has the advantage and is more likely to win the game. Would the Patriots have had such a legendary win if the coin had been flipped with the slightest change in force? While we cannot answer this question, we can all acknowledge the dilemma this question proposes.


Seeing as the NFL is a billion-dollar industry, one can see how this probabilistic overtime procedure would be cause for much discussion and debate, and it is acknowledged that they must look for alternatives that solve this problem. However, the question remains: how should overtime be reconstructed? Any alternative to the coin flip must meet four main objectives:

  1. Be fair: one team should not have an advantage over the other based on any factor uncorrelated to overall team skill.

  2. Be efficient: longer games are both problematic for players’ health and broadcasting windows.

  3. Avoid a tie: this criterium is self-explanatory. The purpose of overtime is to settle ties.

  4. Have flair: the NFL is in the business of entertainment, and thus, an exciting overtime is expected by fans.

The current method meets all but the first objective. As established, the current method gives advantage to the winner of the coin flip. Consequently, several alternate overtime rules have been proposed in an effort to meet each of these objectives; however, of particular interest are two auction-based methods. one proposed in 2003 (Bialik) and one in 2021, and the “spot and choose” method proposed just this year (Beaton). These two methods shall be explained and analyzed from a game theoretical perspective to determine the merit of each method.


Auction-Based Methods:

Auctions can be designed with a variety of rules and mechanisms. When hearing the word “auction,” the mind often goes to the classic English auction: an auctioneer standing in front of a crowd accepting bids of increasing size until the infamous words are spoken, “going once… going twice… sold to the man in the polka dot tie!” First-price sealed-bid auctions (FPSBA), unlike English auctions, require bidders to write down a single bid, and once all bids are collected, the winner is the highest bidder. And while English auctions and FPSBAs take the votes of many bidders, Dutch auctions simply decrease in price until someone agrees to the price.


The two auction-based methods proposed for overtime follow the rules of FPSBAs and Dutch auctions respectively. Both methods use yardage as the currency for the bid, and the team who agrees to start the farthest from their end zone wins the auction. Believe it or not, though these two auction types seem vastly different and may encourage different psychological behavior, they share the same optimal strategy. But how would teams decide what yardage to bid? Consider a football field. If a team is right next to their endzone, they would rather be offense. If they are on the opposite end of the field, they would rather be defense. But if they choose a yardage too close to their endzone, the other team will be on offense, and they will be on defense. If their opponent is too close to their endzone, it will be too easy for this other team to win. Thus, a team must choose a point on the field somewhere in the middle: not too close to their endzone, but also not too far--right in the goldilocks zone, so to say. At some point along the field, there is a point such that the team choosing the point will be indifferent to being offense or defense. We name this point the point of indifference. This, however, is not necessarily the yardage they will bid. The task of pinpointing this exact location, however, is fairly complicated.


In a typical auction, the value of winning is based on the winner’s value for the object and how much they paid for it. The value of losing is 0. However, in the context of football, this auction produces what is called a “zero sum game” meaning the value gained by one player is lost by the other. Therefore, the problem becomes much more complicated than a simple auction. A team’s derived value relies on their bid, and the other team’s bid; therefore, a team would like to figure out the optimal bid as a function of their point of indifference. However, they must also factor in a probabilistic distribution of where the other team might bid and how that affects their value. The other team would be trying to do the same as well. Since teams are bidding simultaneously and not sequentially, the problem would become large and hard to solve, potentially requiring extensive models.


Spot and Choose Method:

The point and choose method, much like the auction method, involves teams choosing a yardage for overtime to start at. However, unlike the auction method, yardage bids are not submitted by both teams. Rather, the coin is flipped, and the winner of the coin flip decides the location of the kickoff. After the “spot” team decides this, the “choose” team gets to decide whether they would prefer to be offense or defense (Beaton). The sequential nature of this game makes it much simpler to solve.


The team picking the location of the ball knows that once the location is picked, the “choose” team will pick offense or defense based on what will make the “spot” team worse off. Knowing this will be the strategy for the “choose” team, the “spot” team will attempt to lessen any advantage the “choose” team might be able to gain, and thus, the optimal yardage for them to choose would be at the point of indifference for the other team. This is a technique called backwards induction that is used to find the best strategy in games such as this.


Discussion and Conclusion:

Based on the ease of calculating optimum strategy, we immediately see that the spot and choose method is far better than either auction method. The auction methods, while fun and entertaining to watch, would require extensive research and modeling to figure out optimal strategy, and as a result, overtime would become a contest of who has the better model. The spot and choose method, as we saw, can be easily solved by backward induction and thus makes it a viable alternative for NFL overtime that solves the problem of fairness.


In terms of efficiency and avoidance of a tie, this method fares similarly to the coin toss, as the rules change would not affect the number of kickoffs or the ability to tie. As far as flair goes, it could be added to make either of these methods much more exciting than a simple coin toss. The suspense to see where the ball will be placed for kickoff are enough to create quite the event (especially for the economically inclined!).


Although the spot and choose method seems to be an improvement to the current method, whether we will actually see the change is unlikely. Some people argue that these game theory based methods are too revolutionary and sophisticated for the NFL. Furthermore, coaches are already under a lot of pressure for each decision they make, and to assume they are perfectly rational game theorists would be a false assumption. Adding a game changing decision like this to their plate would only cause unwarranted scrutiny from fans (Bialik). Is this added stress worth it to offset the biased advantage the coin flip provides? Although this is another discussion in and of itself, the tradeoff is important to keep in mind. Who knows whether or not the “spot and choose” method described will be adopted, but as of right now, we can only hope for overtime rules that rely less on probability and more on skill.


Acknowledgements:

I would like to offer special thanks to my game theory teacher this semester, Dr. Shengwu Li, Assistant Professor of Economics at Harvard University. He not only brought the concept of game theory applied to overtime football to my attention, but advised on the game theoretical analysis done in this article.