## Follow-Up on Game Theory and The Passing Premium

The strategy of play selection on both offense and defense is one of the most compelling aspects of football. Most other sports don't feature such a strategic battle on every play. The balance between running and passing has long been at the center of football strategy debate. Research on the run/pass balance has mostly focused on the "passing premium"--the apparent difference in average expected yards yielded by a pass compared to a run.

Here is a great discussion of the importance of game theory in understanding run/pass balance at Tom Tango's sabermetrics site. "MGL's" comments are particularly insightful. In this post, I'll try to summarize some of the discussion's main points (in bold) and offer my own thoughts (in normal type).

1. It's true that according to game theory, when a game is in equilibrium between two strategy options (like run and pass), the expected outcomes of both strategies will be of equal value. Recent research papers use expected yards (sometimes accounting for risk premiums) as the measure of value. They expect average yards gained by running and passing to be equal at the optimum proportion of each strategy. But the assumption that yards = value in football is mistaken. Often, an offense faces 3rd and 8, and 7 yards just won't help any more than 3 yards. Also, time has to play a factor. It may be better for a winning team to trade yards for time off the clock. The entire analysis hinges on a valid utility function for football.

2. It would be extremely difficult to actually chart the utility matrix and solve the equilibrium between running and passing. This is undoubtedly true. We would need to have good data for when a defense is 100% sure that the play will be a pass and for when a defense is certain the play will be a run. There are some situations which may be close enough to "certain pass" or "certain run" to shed some light. We would need such data to pin down the normal form of the game, i.e. the payoff matrix. This assumes we have a valid utility function.

3. The '10 yards in 4 downs' rule severely complicates the analysis. This is true, as the utility of 3 yards is very different on 3rd and 1 than it is on 3rd and 10. However, we can isolate the analysis at specific down and distance situations. We could start with 1st and 10s as commenter Chris suggested. (He also suggested charting 3rd downs as binary--success or failure.) I'd also want to limit it to ordinary game situations such as between the 20-yard-lines and prior to the 4th quarter.

4. There is a difference between the "optimum" strategy mix and the "equilibrium" strategy. In game theory, the Nash equilibrium (NE) is the strategy mix of both opponents at which neither would benefit from unilaterally changing his own mix. Let's say on 1st and 10 in ordinary situations, the NE solution is for the offense to play 60% run/40% pass and for the defense to play 60% run D/40% pass D. The offense can guarantee itself a minimum average payoff by playing the 60/40 NE, and if the defense played its NE mix, the "equilibrium" mix would indeed be "optimum" for both teams. Neither team would benefit from altering its mix.

However, if the defense failed to play the correct mix, and played 50/50 for instance, equilibrium and optimum are no longer the same. The offense can continue to guarantee itself a minimum average payoff by playing the 60/40 NE mix, but it could get more by taking advantage of the defense's mistake. The offense's optimum mix might now be 70/30 or so. Because the defense is overplaying pass D, the offense should run more often. (In theory, the offense should run 100% of the time to optimize. But in repeated iterations, this would too obvious and even the most oblivious defense would adjust.)

There are several other insightful comments there. The baseball guys are a little further along thinking about these kinds of things. For example, there is tremendous similarity between the run/pass strategy game between offense and defense, and the fastball/curve game between pitcher and batter.

I realize the game theory aspect of the run/pass balance question is very obvious in a way. Even the dimmist TV commentators talk about the run setting up the pass. But I think it's useful to think about the topic in a more formal way, and let the math inform us.

### 3 Responses to “Follow-Up on Game Theory and The Passing Premium”

1. Anonymous says:

Also recommend footballoutsiders.com. The entire concept of their analysis revolves around a utility function -- offensive success is roughly
- 1st and 2nd down-- get 50% of the needed yards; 3rd/4th down get 100% of needed yards. They have modifiers for red zone plays; longer plays; turnovers...

2. Brian Burke says:

Miles--Yes, you're getting a little ahead of me though. I'm drafting a post that discusses some of the alternate methods out there including DVOA.

As you point out, Football Outsiders' DVOA is a type of utility function. It's actually not theirs, but a system invented by the authors Carrol, Palmer, and Thorn in The Hidden Game of Football. Unfortunately it's not valid in a strict utility sense. For example, it's not linearly proportional. (A DVOA of 4 is not "twice as good" as 2, at least as far as I can tell.) I don't think it's terribly useful beyond its main application as a post-hoc ranking system, but it was a major advance in thinking when it was published in '88 and it definitely points in the right direction.

3. Anonymous says:

Brian and Miles --

The question is whether FO's DVOA utility function employs weights that profile the intrinsic ability of the team well. We should give them credit for employing a fairly sensible set of weights, even if they are a bit idiosyncratic. I myself am interested in seeing if one can empirically determine a better set of weights.

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