With the Redskins on a bye last week, I thought it might be helpful to take a step back and look at what analytics can and can’t do in football. Of all the professional sports, football may be the toughest to crack analytically. The 16-game season doesn’t offer the same sample sizes that other sports do, and the game itself is exceptionally complex. Despite these limitations, we can still learn a remarkable amount about the sport.

The lowest hanging fruit are team-level statistics. We can see which statistics correlate best with winning and rank teams according to those numbers. We can look at which teams succeed more often in various situations and rank them. Adjustments for categories such as opponent strength can enhance the accuracy of team rankings. Unfortunately, that’s almost all we can do with this type of approach. It’s great fodder for Internet message board arguments, but it doesn’t help teams gain an edge.

To really help coaches, a deep understanding of how teams come to win games is required. Advanced models such as Win Probability and Expected Points are very powerful tools, because of what an economist would call “linear utility functions.” The term linear refers to the quality of proportional desirability. For example, it’s exactly twice as good to have a 40% chance of winning a game as a 20% chance of winning a game. You can’t say that about total yards or simple points. Toward the end of a game, leading by eight points is much more than twice as good as being up by four points. In the first case, an opponent needs a touchdown and a two point conversion just to tie; in the second, a touchdown could steal the game.

Linear utility functions are special because they allow us to analyze very complex questions with relatively simple arithmetic. Consider a fourth down situation in which  attempting a field goal would result in a 40% chance of winning. If a failed conversion attempt would result in a 20% chance of winning, and a successful conversion would result in a 60% chance of winning, we can calculate the conversion probability that would be needed to justify going for the first down. [It would be 50%.]

Another advantage of special frameworks like Win Probability and Expected Points is that they operate in terms of universally meaningful units. The units of Win Probability are ‘chances to win’ and the units of Expected Points are in ‘net point differential.’ This allows us to directly compare the impact of all events in a game. What’s better to have–a kicker who produces a touchback on nearly every kickoff or a quarterback who throws one fewer interception a season? Analytics provides the currency to value

Almost any in-game decision can be analyzed with a linear utility framework. Fourth downs are the decisions with the biggest potential impact, but other decisions such as play selection, run-pass balance, clock management, play-action frequency, blitz frequency, coaches’ challenges, penalty acceptance and onside kicks are all suitable for expected utility analysis.

Football analytics does have its limitations. The most glaring is the inability to accurately separate a player’s individual contributions to winning. Football truly is the ultimate team sport, so there is no mathematical way to isolate a player’s value from that of his teammates. When a running back makes a long touchdown run, how much of that play was due to the back’s efforts and how much was due to his blockers? Or when a defensive end sacks the quarterback, how much of the play was caused by good pass coverage?

Despite this weakness, we can still learn much about the relative contributions of positions in general. Based on the Win Probability and Expected Points frameworks, we can estimate the relative value of, say, running backs and quarterbacks. It’s clear, for example,  that teams greatly overpay for top runners. We can also measure the relative value of draft picks and compare them to the value of free agents.

Player skill and talent will always be the primary driver of wins and losses, but insights like the ones analytics provide have the potential to turn an 8-win team into a wildcard team, or a wildcard team into a top playoff seed. But once a critical mass of teams are using these tools, the advantage is lost.

Brian Burke is the creator of Advanced NFL Stats, a Web site about football, statistics and game theory.