# Pareto frontier and non-dominated quarterbacks

The quarterback position might be the single most important position in all sports (maybe closely followed by the hockey goalie) – even though the Denver Broncos have set to dispute this claim this year! As such, sports analysts, coaching and scouting staff, and the casual fans are trying to come up with ways to evaluate and rank quarterbacks.  This has been – imo – still an open problem with the passer rating being maybe the most widely used metric to evaluate and rank quarterbacks.  Of course, this does not come without drawbacks, since it essentially only accounts for passing statistics and not for example fumbles lost, rushing yards etc. ESPN has developed its own formula for accounting for some of these factors.  However, all these metrics try to collapse multiple dimensions of a quarterbacks performance to a single value! This is then guaranteed to have some information lost in the process.

Here we propose a different approach that tries to avoid reducing the dimensionality of the information. More specifically, we put forward the usage of the notion of Pareto efficiency and non-dominated points.  We will present this idea using as an example two of the dimensions that play a role in the performance of a QB (and the passer rate computation), namely the number of interceptions per attempt $I$ and the number of touchdowns per attempt $T$. The performance of a quarterback $u$ is said to be dominated by that of quarterback $v$, if:

$T_u \le T_v$ and $I_u \ge I_v$

Now a point $(I,T)$ is said to be non-dominated if there is not any other point in the set under examination (in our case the quarterbacks performance on the plane I-T) that dominates $(I,T)$. All these non-dominated points are called Pareto optimal points and form what is called the Pareto frontier.  The above definition of dominating points can of course be extended to multiple dimensions, where the direction of the inequality will depend on whether the corresponding dimension is expected to be minimized or maximized when a quarterback performs well.  Using data from the past regular season we were able to create the following figure, where each point corresponds to a quarterback and captures the corresponding performance with regards to his interceptions and touchdowns per attempt.

The green and red points correspond to quarterbacks whose performance is dominated by at least one other quarterback. On the contrary, the blue points correspond to quarterbacks with Pareto efficiency (with respect to the two variables examined). As we can see, this Pareto frontier includes the 2015 MVP, Cam Newton, as well as Tom Brady and Russell Wilson.  The surprise entry in the Pareto frontier is Case Keenum (maybe this is why he is still the starting QB for the Rams (?))! This shows the potential of the notion of Pareto optimality to reveal quarterback prospects beyond the obvious. In the specific set another interesting point to make is that if we remove the 4 QBs that form the Pareto frontier, then there is another – “second level” – Pareto frontier that appears, with three QBs this time, namely, Aaron Rodgers, Andy Dalton and Carson Palmer.  In the current season, there is a Pareto Frontier that appears that includes 3 QBs, Aaron Rodgers, Matt Ryan and Jimmy Garoppolo.  Of course, we are still only at week 4 and hence, this frontier is more likely going to change by the end of the season!

This notion of Pareto optimality/efficiency can be very important for both the scouting staff (e.g., to identify the unexpected prospect) as well as the perfomance evaluation of QBs. Of course the example mentioned above includes only two dimensions for visualization purposes.  One can imagine including more dimensions, ones that are not included in the passer rating as well, such as russing yards, rushing TDs etc.

Similar ideas can be put forward when evaluating teams.  For example, by considering the points scored and allowed per game one can identify the Pareto efficient teams.  The following figure presents the results for the current season through week 5, where we can see that there are 3 Pareto efficient teams, Vikings, Eagles and Falcons.  When removing them, we have another Pareto frontier that appears, which includes more teams (Seahawks, Patriots, Broncos, Cowboys, Raiders and Chargers).