Playing to Win, or Playing for OT? - Rams vs. 49ers, November 2, 2014

In Week 9 of the 2014 NFL season, San Francisco 49ers were losing to the St. Louis Rams 10-13 with just seconds left to go in the game. The 49ers could have taken the "safe" play and kicked a field goal, or taken a more risky play to win the game right away by going for a touchdown. With time running out, most people would have taken the field goal try, gone to overtime, and taken their chances, but what's really the right move here? It turns out it depends, but not necessarily on what you think.

The Factors

The real question to ultimately answer is: Which choice gives the 49ers the best chance to win? Many people say the game isn't played by math, but by players which is of course true. However, if coaches can accurately estimate the chances of success in given situations, it would give them better data with which to make decisions.

Since the game is near the end, we only really have to analyze a few factors: 1) the chance of making the 20 yard field goal try, 2) the chance of making the 3rd and 1 attempt, and 3) the chance of winning the game, if the game goes to overtime.

Kicking the Field Goal

The chance of making a 20 yard field goal is about the same as making a point after touchdown, which is close enough to 100% it can be rounded up. That leaves only the chance of winning in OT that we have to consider. Nominally, this is a 50% chance, assuming we have 2 evenly matched teams, since they are tied after all. However, one team may still be clearly better than the other, allowing for fluke plays or bad calls to lead to the tie.  

In this case, Vegas had the 49ers as 10 point favorites going into the game. Clearly they are not going to win by 10 points in this spot, but if we are to believe the pundits, they are still the better team. 

You can argue the opposite way, with the Rams having dominated the line of scrimmage during the game, sacking QB Colin Kaepernick 8 times. The claim that the Rams have just simply been playing better football for that game can certainly be made here.

Which ever way you think about this, there is a non-intuitive rule of thumb at work here. The worse team should take more chances, to give them more of a chance to win. The better team should take fewer chances, allowing their skill to prevail in the long run.  Taking more chances increases variance, which gives an underdog a better chance of winning, with a higher chance of also getting blown out. Taking fewer chances will be more conservative, with teams scoring more consistently with their skill level.

If you believe the 49ers to still be playing better football than the Rams,  it gives them greater than 50% chance to win the game in overtime. The opposite is true if the Rams are playing better football.

Going for the Touchdown

If the 49ers go for the touchdown (which they chose to do), they will only get a maximum of 2 plays to win the game and that is only true if the first play stops the clock (either by a score which renders the second play moot, or by an incomplete pass or a penalty). The official NFL play-by-play lists the play having started with 9 seconds left. Ultimately, coach Jim Harbaugh needs to call the plays which gives his team the best chance to score a touchdown. 

Analyzing Win Probabilities

Since there is very little time left, the analysis comes down to this; The 49ers should go for the touchdown in this spot if:

P(Scoring a TD) > P(Winning in Overtime)

The probability of scoring a TD can be broken down into the 2 possible plays to run, but this is up to the coach to call the best plays which give them a chance to score. This number can be different depending on the different players, the teams' tendencies, and the actual play call.

The probability of winning in overtime also depends on how well the different teams are playing, what the injury status is of certain players, and how worn out the teams are.

Though we can't perfectly figure out the percentages, we can conclude this:

The 49ers should kick the field goal if the chances of winning in overtime are better than that of these next 2 plays.

OR

The 49ers should go for the touchdown if they're chances of winning in overtime are worse than that of the next 2 plays.

The Outcome

The play which was called was a QB sneak, meaning they only have ONE play with which to score the touchdown. On the play, QB Colin Kaepernick fumbled, giving them no chance at a 2nd play. The Rams ultimately knelt out for the win.

(Edit: The 49ers had a time out left, which makes running the ball an option here)

What would I have done? I probably would have taken one shot at a pass play into the end zone, then kicked the field goal to go to OT.