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# game theory

## Calling the Mathlete: Zook's Decision

I had a brief discussion about this in the Opponents comments thread, but I'm wondering if anyone can provide some advanced metrics about whether Zook made the right call last week to go for it on 4th and 3, down 10 with a little over a minute left. Heiko (as well as the announcers in the game) tore Zook a new one for this, but it certainly seems defensible to me, and probably the right call. My point from the comments discussion:

The way I see it, you need a touchdown and a field goal either way. They were in fourth and makeable, pretty close to the OSU goal line--that might be the best opportunity they have to convert the fourth down and score. If they kick the field goal there, they might face down a 4th and 10 from the 33 on the next drive, but they would have to go for that much-more- difficult-to-convert play. it seems like whenever you're down 10 and you know you're going to have to recover the onside kick, you should take the 4th down opportunity to go for it when it seems like you won't get a better one. It's not like Illinois was moving the ball up and down the field. If they kick the field goal and recover the onside kick, seems likely they're going to face something worse than 4th and 3 and be forced to go for it.

To take it to the extreme, if it were 4th and goal from the 2, no one would criticize going for it--that's your best chance to get the touchdown you're going to need anyway. By kicking the field goal, you're "wasting" all of those yards you got within field goal range. I don't see the benefit of doing the field goal first just because it's the easier one--you might only have one chance in the two possessions to get the TD, so when you have that chance, you need to take it. You're going to have to convert the onside kick either way.

It seems like we should be able to calculate a "right" answer for this, but that's much too complex for my simpleton mind. Mathlete, do you have a chart that can enlighten us?

## Going For Two in OT

I was at the game on Saturday with one of my friends from college, who is a pretty sharp guy. As Michigan scored their second TD in overtime, he made a case for going for two right there instead of kicking an extra point and forcing a third period.

His reasoning was that both offenses were likely to score on the next possession so we should try the 2pt conversion now, when Illinois would not have a chance to answer. At the time this line of reasoning sounded okay; however, I decided that it was somewhat unconvincing. The fact is that as long as our win percentage is higher by kicking an extra point than by going for two, we should, quite obviously, kick the extra point.

The question then becomes, is it possible that our chances of converting the 2pt conversion are higher than our chances of winning in a third overtime? In order to determine the answer to this question, I had to consider a few different factors:

1. We were going to be playing offense first, which carries with it a strategic disadvantage. What is the inherent disadvantage that we’d have in the next overtime?

2. What are the chances of our team converting a 2pt conversion? How much more likely are we to convert than an “average” team?

3. How likely is it that the kick to force a third overtime will be successful?

I did a bit of research and found a study that showed that the team that starts on defense wins about 52.25% of the time in the third overtime and later. You can find the study here. And, looking at M’s kicking statistics I’ve found that the team is 46/47 on extra point attempts, 98%. I used that for our success rate in this spot. So when we kick the extra point we’ll win .4775*.98= .468. So if we can convert the 2pt conversion 47% of the time, we should go for 2.

How often should we expect to make a two point conversion? Advanced NFL Stats says that the conversion is good, on average, 44% of the time. So obviously, if we had an average chance of converting, we should kick the extra point. But our offense is significantly above average.

In order to decide how much more often our 2pt conversion would be successful than an average team’s conversion, I divided our total offense in terms of yards/game by the national average. The result is a multiplier which I applied to the average 2 pt conversion percentage. Our total offense per game is 536 and the national average is 384 giving us a multiplier of 1.39 (our multiplier is similar when considering scoring offense). Applied to the average conversion rate of 44%, our new conversion rate should be 61%.

Now this seems pretty high to me, but given the things we’ve seen our offense do this year, I’d be surprised if we didn’t fall somewhere above the 47% necessary to make going for two at the end of overtime correct.

## Should We Have Onside Kicked after Koger's TD (The Answer is Yes)

I hadn't seen this discussed anywhere on the boards, but I was drunk for 48 hours after the PSU game (because of the PSU game) so I may have missed it.

I've been thinking about this for a little while now, and my gut tells me that after Kevin Koger's TD catch (and the subsequent facemask penalty on PSU), we should have gone for the onside kick.

**Hypothesis: **A normal kickoff will result in the opponent starting, on average, about the 25 yard line. Because we got to kick from the 45 rather than the 30, Penn State's expected starting field position from a normal kickoff (touchback) would be only about 5 yards worse. However, an onside kick from the 45 would probably result in PSU's ball at about the 50. Or we get it back, and the chances of us getting it back are actually greater than the increase Penn State gets from 25 free yards.

In order to test my theory, I'm willing to do some math. I'm going to be using the expected points charts found at Advanced NFL stats. I'll be assuming that we'll always force a touchback if we kick off and that whether we're successful or unsuccessful when we onside kick, the ball will be placed at the PSU 45. My goal is to find how often an onside kick needs to be successful to be better than kicking off.

First thing's first: 1st and 10 from the PSU 20 is worth approximately -.5 points to us. It's obviously worth more to any offense facing our defense and thus the negative number would actually be bigger, but for the sake of the argument I'm going to be as conservative as possible.

1st and 10 from the PSU 45 is worth about -1.7 points when PSU recovers. If M gets the ball it's worth 2.2. So we can represent the equilibrium (i.e. the point where kicking away and onside kicking are equal in value) like so: -.5 =2.2y - 1.7(1-y), where y is the likelihood that the onside kick succeeds.

Solve for y to get: .307 so we'd only have to be successful a little over 30% of the time with these parameters to make kicking an onside kick correct. Given that surprise onside kicks are successful 60% of the time in the NFL, it seems like a pretty large mistake not to onside kick in that situation.

In fact, it's pretty easy to imagine a scenario where a team has a very good offense and a very bad defense (just try to imagine such a thing) where we'd only need to be successful 25% of the time or less. For example, if receiving the ball at their 20 is worth a full point for PSU and recovering an onside kick is worth 2.5 for them and 3.5 for us*, the equation would look like: -1=3.5y-2.5(1-y). Then we'd only need to be successful 25% of the time to make going for the onside kick correct.

Add to all this the fact that in this particular game we were down by multiple scores and would have wanted to increase variance, onside kicking in that spot is an absolute no brainer.

*Numbers pulled from my ass

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

**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.

**Assumptions**

- 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?**

Yes.

**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.*

## Going for 2 is NOT a "no-brainer"

One of my pet peeves when discussing college football is the conventional "wisdom" that there are supposedly set-in-stone rules about going for 2. Broadcasters constantly say that "coaches have a chart" that tells them when they should go for 2, based simply on the score.

I doubt this is the case, because I imagine coaches know that going for 2 is a decision that must be made based on the context of the situation. It's complicated and risky, and it's not always clear what the right thing to do is. No scoring chart will be able to take all the important factors into account.

What are those factors? Aside from the obvious factor of...

**1. What's the score?**

...there are, in my opinion, at least three other questions that must be asked, all with the assumption of a missed 2 point conversion:

**2. How much time is left in the game?**

**3. Do I believe the opponent will score again?**

**4. Do I believe my team will score again after that?**

My philosophy on 2 point conversions: assume you'll miss, and only go for 2 if it could make a difference in the score, there is little time left in the game, and/or there won't be any more scoring done by my team. In other words, I believe the 2 point conversion should be put off until the last score. That way, missing the 2 point conversion doesn't unnecessarily hurt your team's chances of victory.

Questions 2, 3, and 4 are related, of course- with little time left in a game, for example, there will most likely be little to no more scoring done by anyone. But again, the context matters- a lights out offense like Texas Tech might believe they'll do more scoring with only 3 minutes left in the game, whereas a slug offense like Virginia Tech might feel they are done scoring with 6 minutes left in the game.

With all this in mind, let's turn to this past weekend's game: should Michigan have gone for 2 after the Thompson interception?

Reading through the liveblog transcript, there seemed to be universal and instinctive agreement that yes, Michigan should go for 2. Everyone quickly came to that decision and expressed confusion about RichRod's decision to take a time out to think about it. Based on that, I assume everyone was looking at the scoreboard as the *only* factor in the decision. But let's look at the context:

1. The score was 20-19. A successful 2 point conversion prevents a subsequent Wisconsin field goal from winning the game. The score says "Go For It."

2. But there was over 10 minutes left in the game. Each team has at least 2 more possessions coming. Time left says "Go For It Later, Not Now."

3. The Wisky offense was not doing much, kicking FGs off of turnovers in the first half. One big run in the first half set up their lone touchdown. They had yet to score in the second half. Their quarterback was not showing himself to be anything special, having just thrown a Pick 6. Was Wisconsin done scoring for the day, even with more than 10 minutes left? It appeared so. A Wisky score appeared unlikely, but even on the off chance that they pull something together, that was offset by the off chance of a *successful* 2 point conversion by Michigan. Wisky offense says "Go For It and Ensure Overtime, Just In Case."

4. But the Michigan offense wasn't really a solid bet either way. The offense had been a little, uh, erratic. 21 total yards and multiple turnovers in the first half was just plain awful. A solid drive for a TD and a big TD run by Minor in the second half, however, had raised hopes. Was Michigan done scoring for the day, even with more than 10 minutes left? Maybe, maybe not.

That's why I believe RichRod took a timeout- it wasn't clear whether Michigan was done scoring or not, and he needed a moment to think about it.

It was a gamble, and as it turned out it hurt the team. If he ended up deciding to just take the PAT, then the ensuing touchdown would have put Michigan up by more than one score (28-19), and the game would have effectively been on ice. Michigan scoring again wasn't out of the realm of possibility- the Wisky defense had been on the field for a LOT of the second half, and they were starting to get pushed around. There was at least a *hint* of foresight that indicated another Michigan score.

Instead, RichRod gambled on the thought that they wouldn't get another chance to score, so he may as well get some while the getting's good. Michigan did score again, though, and the chance to ice game had already been lost in the previous 2 point conversion. Because of the failed 2 point conversion, Wisconsin was only down by one score and still had a chance to tie. As we all saw, everything worked out, but when Beckum caught the 2 point conversion pass (and before we saw the flags), for a moment Michigan's failed 2 point conversion loomed large.

Ultimately, I agreed with the call to go for 2- I was still unsure about Michigan's ability to score again, despite the gobs of time left on the clock and the tired Wisky defense. My point, however, is that this was NOT an easy call. This was NOT a "no-brainer." And in hindsight, it was the wrong call.

These decisions are NOT easy, and they depend on more nebulous things than just the score.