Up by 7, late in the game - 1 or 2? A simple economic model

Up by 7, late in the game - 1 or 2? A simple economic model

Submitted by Jivas on October 25th, 2009 at 4:38 AM
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Here’s the situation: Your team leads 21-20 with 2 minutes left in the game, has just scored a touchdown to go up 27-20, and your head coach kicks an extra point to take a 28-20 lead.  Seemingly every coach kicks this extra point in all similar game situations that we’ve witnessed – it’s a no-brainer, right?

Back in my video game-playing days – it’s been a few years, but I’d bet that I’ve played football video games for over 1,000 hours of my life – I used to always go for 2 points in this situation in an effort to build an insurmountable 9-point lead.  My logic was this: in practice, at least in the NFL, no team that scores a touchdown to put them down 1 point in a game-ending situation goes for 2 points to win the game.  (I believe this has happened less than 10 times in the (brief) history of the 2-point play in the NFL, which represents a negligibly small percentage of similar game-end situations).  Therefore, the difference between a 7- or 9-point lead to me was far greater than the difference between a 7- or an 8-point lead; at 7 or 8, the other team has a chance to tie in regulation (but not win, given my assumption), but at 9 the game is effectively over.

In the intervening years I’ve gone along my merry way just assuming that all coaches were making suboptimal decisions with respect to this situation.  Now in my first year as a PhD student in a business program, my brain is starting to work a little bit differently.  Thinking of this situation earlier this week, I developed a simple model to help infer whether either strategy here is dominant.


  • Your team is Team A, the opponent is Team B
  • Team A has just scored a TD with 2 minutes left in regulation to take a 27-20 lead; PAT/conversion pending
  • There is only one meaningful possession remaining in regulation, for Team B, starting with Team A’s kickoff to Team B
  • We assign a probability of β to represent the likelihood that Team B scores a TD on their possession (0 ≤ β ≤ 1)
  • The probability of either team successfully converting a 2-point conversion is 44% (I believe this is the NCAA historical average conversion rate)
  • The probability of either team making an extra point is 100%
  • If Team B scores a TD on their possession to reduce Team A’s lead to 1 point, they will kick the extra point 100% of the time*
  • If the game goes to overtime, both Team A and Team B have an equal 50% chance of winning the game

    * - I expect this to be the most controversial assumption, as in college there is always some consideration with respect for going for 2 in this situation (e.g. the Michigan-Michigan State game this year).  I submit that this is a very matchup-specific assumption at the college level – a heavy underdog is more likely to take their chances on a conversion attempt than on overtime – but as noted above, the assumption should be uncontroversial for the NFL, where going for 2 and the win is a nonfactor.

    Probability of Winning – Go for 2

    There is a 44% likelihood of making the conversion, which makes the score 29-20 and results in a win likelihood of 1 given our assumptions (i.e. one possession remaining in the game).  If the conversion attempt is missed (56% likelihood), we consider that Team B will score a TD with β probability.  If they score, this results in a 50/50 chance to win in overtime; so, in this state, Team A will win with (1 – β/2) probability.  Therefore, the Total Win likelihood is (.44)(1) + (.56)(1- β/2), which reduces to: 1 - 0.28 β.

    Probability of Winning – Kick extra point

    There is a 100% likelihood of making the extra point, giving Team A a 28-20 lead.  In order to lose the game at this state, the following has to happen: (1) Team B scores – β probability; (2) Team B makes a 2-point conversion (44% likelihood); (3) Team B wins the game in overtime (50%).  The total loss likelihood is therefore 0.22 β, meaning that the Total Win likelihood is: 1 – 0.22 β.

    So what?

    Umm…Brian’s bolded alter-ego, is that you?

    No.  Brian’s bolder alter-ego has long, curly hair; I’m bald.  Get it?


    Well then.

    Well then.  What’s next is that we start playing with β.

    Sounds kinky.

    It’s not.  We can now calculate the β at which these two decisions provide an equal probability of winning, which is clear from looking at the formulas:  only when the other team has a 0% likelihood of scoring a TD are these two strategies equal.

    How, exactly, does this help us?

    What this tells us is that, given these assumptions, we have a dominant strategy.  If we set β equal to 1 – that is, there is a 100% likelihood that Team B will score a TD on their drive – we find that going for the 2-point conversion in this situation provides for a 72% probability of winning, whereas kicking the extra point provides for a 78% chance of winning.  Lowering the β into a more realistic region – for convenience, say 0.5 (i.e. 50%) – we find that that going for the 2-point conversion provides for a 86% chance of winning, while kicking the extra point provides for an 89% chance of winning.

    It’s important to not dismiss this difference out of hand and treat the strategies as equal – if you told a coach that a particular decision would increase the chance his team loses from 11% to 14%, I’m quite certain that the difference would be meaningful to him.  And it’s also important to keep in mind that these are just fun game theory assumptions that would need to be modified for each specific scenario; for example, I might have had a play on Madden that I knew would work on a 2-point conversion 80% of the time given the poor game AI.  In that situation, my decision to go for two was probably rational.

    Which leads me to the following conclusion: given that real game situations will have realities that diverge quite a bit from the basic assumptions in this model, over the course of thousands of games there must have been individual circumstances where teams were at least as well off attempting a 2-point conversion in this situation as kicking the PAT.  In fact, it seems likely that there would have been at least a few instances where they would have been better off attempting a 2-point conversion – say, in the college football fringes where PATs are not ~100% propositions and where weak kickoffs will lead to greater βs .  However, in my football-watching experience, I can’t recall ever hearing a discussion as to whether a team should go for the deuce in this situation.

    Anything else?

    I discussed my “model” with an experienced PhD student, and his feedback was invaluable.  One major issue that he raised was that there is a covariance between (1) the likelihood of Team B successfully converting a 2-point conversion and (2) β, the likelihood of Team B scoring a TD on their final possession.  The point being, the strength of Team A’s defense (and Team B’s offense) will cause these values to be related.

    There are also economic concepts of utility and risk-aversion which are being ignored here.  And of course, the emotional and psychological implications that any given Result A will have on each team, thereby potentially influencing the outcomes of Result B.

    So, minor quibbles with assumptions aside, through a very, very simple economic model I’ve provided evidence to help answer a question that’s been bugging me for some time.  Unfortunately, the results are inconclusive – while I find no fault in the general strategy of kicking the extra point in this situation (indeed, a dominant strategy in this model), I have to believe that the ingrained nature of this decision and the strict adherence to the conversion chart has caused a few coaches to make suboptimal decisions.  In any event, hopefully this creates some fun discussion, and hopefully a future look at a different question will provide a more conclusive and illuminating result.


    Thanks to my friend Andy for catching an embarrassing error with my initial model and thereby proving the immense value of editors.  If there are any less-than-minor quibbles with the assumptions or any other issues with the model, please let me know – constructive feedback is welcomed.

    Coaching Question: When to play non-starters

    Coaching Question: When to play non-starters

    Submitted by StephenRKass on September 21st, 2009 at 9:38 AM

    I was disappointed that Priestly from EMU kept on gashing the right side for significant yardage Saturday. From what I could tell, Mouton was greatly missed. His replacement, Leach, got a lot of tackles, iirc, but they came after too much yardage.

    Other than the disappointment of the defense giving up the rushing yards, I also was disappointed that the reserves couldn't get to the field sooner. The game was a bit too close from Michigan's perspective, possibly keeping Forcier et al in for a few more series.

    I want to know what kind of margin is needed before coaches put in the 2nd string. I'm sure part of it is getting the first string more reps, especially on the offense. However, I continue to be very concerned about all parts of our defense: the line, the linebackers, and the secondary. Our depth is so bad that if Max Martin, Brandon Graham, or Warren go down, I'm concerned that no one will be able to adequately fill the gap. Any real game experience that subs can get will only help.

    This question goes particularly to Indiana and Delaware State. Should we get a 3 score lead in any game, it seems that we should get the subs as much game experience as possible. Thoughts?

    The comeback, done the right way

    The comeback, done the right way

    Submitted by M Squared on August 25th, 2009 at 8:25 AM

    The notion of the comeback is distinctively unfamiliar to a program that is first in all-time wins and winning percentage.  But as we continue to rebuild in order to regain that top status, we will find ourselves, unfortunately, in the comeback position in many games this coming year.  A lot of comebacks require a 2 point conversion along the way, and that in turn requires some decision making.  Various interesting decisions arise as to 2 point conversions.  I want to address a specific kind here – one that, in my opinion, just about every football coach gets wrong. 


    The Setting: It is the first game of Coach Rod’s era at Michigan.  There are about 9 minutes left in the game, Michigan is down 25 – 10 against the Utes but marching down the field with surprisingly crisp execution.  Then, Steven Threet hits Junior Hemingway in the corner for a 33 yd touchdown.  Michigan has cut the 15 point deficit to 9.  The score is 25 – 16.  What now?  Before the touchdown, everyone’s thinking we need a 7 point conversion, an 8 point conversion, and the defense not to give up any more points.  So far, so good.  But does the order matter?  Yes!  Not only does it matter, almost every coach chooses the wrong order!  In his first game, Coach Rod also chose the wrong order, and went for the 7 point conversion before the 8 point conversion.  Ultimately, it didn’t matter in that game, but that doesn’t mean it won’t matter in other games. 


    The Principle: It’s better to have a small chance to win than no chance to win.  The 8 point conversion attempt must come first. 


    In Coach Rod’s first game, Michigan happened to score the second touchdown with 6:26 left in the game.  So, Michigan had ample time to overcome Coach Rod’s mistake.  As noted, choosing the right order in that game would not have made a difference.  But suppose instead, that Michigan scored the second touchdown with 26 seconds left in the game.  Now, if Michigan fails to convert the 2 points, Michigan is out of luck.  To recap, MI scored TD + the kick after with 9 mins left.  Then, MI scored w/ 26 seconds left, failed the 2 point conversion, game over.  


    Switch up the order.  MI scored with 9 mins left, but now goes for the 2 point conversion right away and fails.  The score is 25-16 with 9 mins left.  The deficit is substantial but not insurmountable.  To compare, 25-23 with 26 seconds left is much, much more substantial (i.e., almost no chance to win).  It is better to have a small chance to win than no chance to win.   


    Essentially, in any comeback where the coach believes that a 2 point conversion is necessary to win the game, it is better to attempt it earlier than later.  Put another way, the likelihood of converting the 7 point conversion and the 8 point conversion is the same in the two scenarios.  It is more probable than not the 2 point conversion will fail.  It is better to give the team more time to address that likely failure than less time. 


    I can think of two major objections to my position.  First, attempting the conversion earlier and failing deflates the team’s momentum.  Second, delaying the conversion is better because maybe the team won’t have to make that decision (i.e., fortuitous events such as a pick 6 make it unnecessary). 


    This post seems ridiculously long already, so I will stop here instead of diving into further analysis.  I will note that I have thought about those two objections, and, one, I do not think they outweigh the force of my argument, and two, I do not think they are the rationale of coaches when they select, what I believe, is the wrong order.