Building a winning team (Transitivity assumption) – Focus on the NFL

Hi Everyone,


I am back on the topic of ‘Game Theory’. In this post, I want to look at the possible application of the ‘transitivity’ assumption to sports. In my post ’Game Theory #7– Transitivity’, I explained the ‘transitivity’ assumption and the possible circumstances where this assumption does not hold true. In short, transitivity occurs when the relationships between phenomena remain consistent. For example, if A>B and B>C, therefore A>C. However, under certain circumstances these relationships do not hold true. I strongly recommend reading the transitivity post to get a better idea of the ‘transitivity’ assumption before reading this post.

In this post, I want to describe how non-transitive relationships can be applied to sports. In this post, I am focusing on strategies applied to exploit the flaws in the transitivity assumption by National Football League (NFL) teams.

My interest in the NFL


I have been a fan of American Football since I was 7 years old. I found the game fascinating and exciting. The teams had cool uniforms and fun mascots. The first time I watched the NFL was on the UK’s Channel 4. This was a one-hour show, which contained extended highlights from one game and short highlights from the others. I did not realise at the time that full games went for almost 4 hours. I eventually found out that games were significantly longer but I obtained a greater appreciation for both the tactics and strategy applied to the game.

Transitivity apparent in results


In the 53 seasons since the first Super Bowl was held in 1967, only one team has completed a season undefeated and only two teams have lost every single game in a season. Over the past 20 years, the team with the most regular season victories, wins the Super Bowl about 30% of the time, see Table 1 below.

Table 1: Best regular season team vs. Super Bowl winner

Note: Teams in green won the Super Bowl and had the best regular season record. Teams in yellow won the Super Bowl and had the joint best regular season record.

The team that has won the Super Bowl has only achieved an outright best regular season record twice in the past 20 years (1999 – 2018).

Some of these inconsistencies can be blamed on things such as injuries, venue, weather, stress from fear of elimination or just luck. However, some teams just consistently play badly against particular teams.

Bogey Teams


An opponent that a team plays particularly bad against is often called a ‘bogey team’. Bogey teams are usually discussed in the context of soccer rather than American Football. For soccer, bogey teams can be more easily identified, as teams play each other more frequently. NFL teams play the teams in their own division twice every season. Teams outside the division are played every 3 to 4 years based on the rotation of the schedule. It is possible 2 teams may only meet twice in one decade. The turnover of players for teams is very high because of injuries and free agency. Therefore, very different players are likely to play each other over a 4-year gap.

There are many one-sided rivalries between teams in the same division. A team normally dominates another team because they are stronger. However, there are a few rivalries where the teams have proven to be of very similar strength when pitted against other opponents. Such rivalries include the Panthers-Falcons, Eagles-Cowboys, and 49ers-Seahawks.

From 1995 to 2018, the Falcons and Panthers had very similar win-loss records (both teams winning approximately 50% of their games). During that stretch, the Falcons and Panthers played each other 48 times. The Falcons won 30 of those games (62.5%). If the Falcons and Panthers have equal chances of winning each game, the probability of the Falcons winning 30 or more games from 48 games is less than 3%. This is a strong indication that the Falcons dominance is more than just luck.

For the Eagles-Cowboys rivalry, the Cowboys have been predominantly the stronger team and have mostly dominated the rivalry because of ability. However, from 2012 to 2018, a period where the teams have been performing roughly on par, the Cowboys have won 9 out of 14 games (64.2%). There appears to be more to the Cowboys dominance than just having better players.

The Seahawks have been the stronger team over the 49ers for the majority of the rivalry. However, between 2011 and 2014, the Seahawks won several games by very large margins when both teams were considered strong Super Bowl contenders. Most of these big victories were in home games where the 49ers team struggled to cope with the noise from the Seahawks fans.

Teams that regularly appear in the playoffs often play each other quite frequently. The Patriots and Colts have played each other 22 times from 2000 to 2018. The Patriots have won 16 of those encounters. Five of the 22 encounters were in the playoffs; the Patriots won four of those games.

Understanding transitivity can help a team become more successful


When teams face each other, there are multiple matchups occurring across the field. There are:

  • Defensive Ends facing off against Offensive Tackles
  • Cornerbacks covering Wide Receivers
  • Linebackers and/or Safeties covering Tight Ends and Running Backs
  • Defensive Tackles facing off against interior Offensive Linemen

There are also multiple packages of players with different skill sets and abilities. Different teams favour different types of formations and schemes. All formations and schemes have different weaknesses and strengths. Different schemes and formations are better suited to different player types. Some schemes work better for fast agile players some work better for physically strong players.

Some teams are built around a particular scheme or system. The team selects players that best fit into that system. Some teams adapt their system to the players they have or are available to the team and develop further from that point. Playing a consistent system allows teams to become very good at that particular system, which enables consistent high quality play.

Unfortunately, every system has its weaknesses. Every game is recorded from multiple angles, which enables other teams to closely scrutinise a particular system to identify its weaknesses. The more a system is played, the more video footage there is to be scrutinised by opposing teams. Some teams are more adequately prepared to exploit an opponent’s weaknesses than other teams.

A basic example is to compare player sizes. Small defensive players typically struggle to stop larger stronger offensive players. These bigger players can break tackles or catch balls over the heads of the smaller defenders. Bigger defensive players typically struggle to stop small faster offensive players. The faster offensive players can outrun the bigger players or use their agility to escape man-to-man coverage. The matrix in Figure 1 provides a brief demonstration of the above example.

Figure 1: Big vs. Small


A ‘1’ in the matrix signifies the offense having success over the defence and ‘0’ signifies the defence having success over the offense. This game does not have a dominant strategy if it is played simultaneously. If this is a sequential game, i.e. one player moves before the other player, the second mover has the advantage. In a game (season) with multiple teams, selecting the types of players that matchup best against the most number of teams will produce the most success. Table 2 contains 8 different teams with different sized offensive and defensive players.

Table 2: Results based on size


Team F is most likely to be successful in this league, as the team’s small offensive players will have success against 4 of the opponents big defensive players and the team’s big defensive players will have success against 4 of the opponents big offensive players. Team A is least likely to be successful in this league, as the team’s big offensive players will not have success against 4 of the opponents big defensive players and the team’s small defensive players will not have success against 4 of the opponents big offensive players.

If Teams B and E replace all their big players with small players, Team A will be more likely to succeed than Team F. Table 3 shows the expected results with these changes made.

Table 3: Results based on size when Teams B and E go small


The above tables demonstrate that neither big nor small players are superior but the level of success depends on how these players matchup against their opponents. This is similar to how rock paper scissors game works. For a team to be consistently successful, it needs to constantly adapt and change to meet the changes made by other teams.

How to use transitivity with an NFL team


An NFL team does not need to restrict itself to just small or just big players at particular positions. A team is allowed up to 53 players on its roster. A team needs in the very strictest sense 11 offensive players, 11 defensive players, 3 special team players (kicker, punter, and returner). Approximately half the roster are substitutes when the team is healthy. The substitute players can play the role of reserves to replace injured players, situational players for changes in formations, and potential starters against teams that their skill sets align better against than the original starting players. For all of the above roles to be possible would require a team to have high quality substitute players that have different skill sets, abilities or physical attributes than the starting players.

In 1994, the NFL introduced the salary cap. The salary cap is intended to even the playing field between all teams. Prior to the salary cap, rich teams could acquire all the best players and therefore dominant the league. The NFL salary cap prevents teams from loading up on star players. For a team to be able to obtain high quality substitute players, they will most likely need to forgo obtaining star players. Table 4 compares the highest salaries to the median salaries of starting players for the offense.

Table 4: Offensive player annual salaries in millions $

Source: Madden 18 (PS4 Game)

Note: The Madden Game uses data obtained from the NFL

Table 5 compares the highest salaries to the median salaries of starting players for the defence.

Table 5: Defensive player annual salaries in millions $

Source: Madden 18 (PS4 Game)

For many positions, the average salary of the highest paid players (top 4 highest) is about twice as much as the average starting player (paid between 15th and 18th highest). If a team does not target star players, it will be able to build a stronger all round team and have better quality backup players. However, there are a few positions where star players can be considered necessary for a team to be very successful. The Quarterback is an essential player who will not be substituted regardless of scheme. Other positions that can be considered essential to have star players are Left Tackle, Defensive End, and Cornerback. These positions also tend to be highly paid across all teams as can be seen from the smaller difference in salaries between the highest paid players and average paid starting players. Positions such as Centre and Inside Linebacker can be considered less essential to have star players.

Successful teams are also penalised by the structure of the NFL draft. The draft aims at balancing the league by giving the teams with the worst win-loss records first picks in each draft round. Therefore, it should be close to impossible for one team to consistently have the best players in the league and therefore should not be able to dominate the league for a prolonged period. However, the New England Patriots have been more dominant in the past two decades than any other team since the creation of the modern NFL league (introduction of the Super Bowl).

Bill Belichick Schemes


Bill Belichick has been Head Coach of the New England Patriots from 2000 to present. During this period, the New England Patriots have won 6 Super Bowls, 9 AFC Championship games, and have appeared in the playoffs 16 times. The Patriots have won 74% of games played under Bill Belichick (Wikipedia). All of the Patriots success has occurred during the salary cap era.

The success of Bill Belichick could be attributed to many factors. In my opinion, the most important factor to his success is his ability to adapt his team to his opponents. Bill Belichick often moves away from strategies that have been successful which other teams have learnt to replicate. Belichick predominantly used the 3-4 defence and a two Tight End offence in the 2000s. He has now predominantly switched to a 4-3 defence and rarely uses 2 Tight End formations. The changes have been made so that his team can better matchup against teams that are emulating his earlier formulae (The Guardian).

Bill Belichick also often makes scheme changes to improve a matchup against a particular opponent. A great example of these changes was against the Los Angeles Rams in Super Bowl 53. The Patriots played mostly man-to-man coverage in the season. In the Super Bowl, the Patriots used a mixture of man-to-man and zone coverages as well as rushed more players at the Quarterback than they typically would. Bill Belichick has used a wide variety of defensive formations and schemes over his career as a Head Coach. The wide variety of schemes as well as constant adjustments has made the Patriots a very difficult team to play against.

The Patriots have mostly not been able to acquire the most hyped (highly played) players in the league but the players that they have acquired have been utilised extremely well to ensure the success of the team.



American football is a very complex game. The success of teams relies very heavily on coaching and the team’s front office. The structure of the NFL draft and the salary cap pressures the league towards parity but some teams have still been dominant despite the restrictions. I believe understanding how to exploit opponent’s weaknesses to particular schemes, formations, and player types has played an important role in the success of particular teams.

The teams with the most star players will not necessarily win the Super Bowl or the most games. Instead, the teams that best understand how to matchup favourably against different opponents have the better chances of success. This can be done by acquiring players that enable a team to be flexible to change schemes and strategies that can exploit an opponent’s particular weakness. The large rosters enable teams to add players that have sufficiently diverse skills, abilities, and/or attributes for this to be possible. A team also needs a coaching staff that understands how to exploit other team’s weaknesses as well as to best use their team’s players.

I believe American football is great sport to demonstrate how non-transitive relationships between teams can be exploited so that teams that may not necessarily have the best players can be successful.

Future work

I will revisit the ‘transitivity’ assumption again soon. I will investigate how businesses can exploit the non-transitive relationship between products to gain success without necessarily providing the best product.

I will also revisit the NFL as well. I will discuss the possibility of match fixing. Is it possible that NFL games are rigged? If so, how would it be possible to rig such games? To what extent is rigging likely to occur? If not, what fuels the arguments that games might be rigged?

More posts

If you want to read any of my other posts, you can click on the links below. These links will lead you to posts containing my collection of works. These posts will be updated frequently.






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