Houston vs. Tennessee
In the Houston/Tennessee football game Sunday afternoon there was a situation in which Houston was leading by 7 points, 36-29, having just scored a touchdown and being in position to attempt (a) a regular point after touchdown (kick) or (b) a two-point conversion. Time was running out; Tennessee had. . I don’t know, two minutes to score, or something like that. The obvious thing to do is to kick the extra point and put the pressure on Tennessee to score 8 points in the last two minutes in order to tie, and, for better or worse, that is what I would have done—the obvious thing.
The Houston coach, however, did not do the obvious thing. He decided to go for two. It didn’t work, so Houston had a 7-point lead with basically one possession left in the game. Tennessee came back and scored, and won the game in overtime.
The TV analyst who was working the game second-guessed the Houston decision in intemperate language, and blamed the decision—repeatedly, and in frankly ugly terms, on "analytics". It would be difficult to say whether he sounded more like a moron or an asshole; he sounded like a moron and an asshole. He hit it REALLY hard, laying it in "our" laps, us meaning the analytics community. It seemed to me unlikely that analytics had very much to do with this, so my gut reaction was that it seemed unlikely that this was an analytics-driven decision. This led to discussion, and. . .well, here we are.
The problem of identifying the optimal strategy in a football game can be stated as
a) What is the probability that we will win this game if we do (A), and
b) What is the probability that we will win this game if we do (B).
If (a) is greater than (b), then you do (a); otherwise, you do (b). Not in a dogmatic sense, of course; the coach has to do what he believes is "best", but that comes down to the same thing; what is "best" for the coach is that which gives his team the best chance to win. My point is that the statistical analyst is always dealing with a lot of unknowns which may change the right answer in the real world, and the coach may know things of which we have no knowledge. He may have been convinced that his third blocker on the left side can destroy the man he would have to block. He may have been convinced that he had drawn up a play that Tennessee had never seen before, which would leave his receiver wide open. Since we do not KNOW what it was that he knew (or thought he knew), we have to be cautious in our statements about it (unlike the asshole moron on TV), not out of generosity of spirit but as a consequence of logic. Logically, we have to understand that we do not KNOW what the right answer is.
But doing the best we can with the mathematical puzzle. . .
It would appear to me that if Houston attempts the two point conversion, then there are five scenarios which are reasonably likely from that point on. Those five scenarios are:
(1) Two-point conversion is good; Tennessee does not score subsequently, Houston wins,
(2) Two-point conversion is good, putting Houston 9 points ahead, Tennessee DOES score on next possession, it doesn’t matter, Houston wins anyway,
(3) Two-point conversion fails, Tennessee does not score after that, Houston still wins,
(4) Two-point conversion fails, Tennessee DOES score and kicks the PAT, game goes into overtime, Houston wins it in overtime, Houston wins again, and
(5) Two-point conversion fails, Tennessee does score and kicks the PAT, game goes into overtime, Tennessee wins it in overtime.
Only in scenario (5) does Tennessee win, and that is what actually happened: the two-point conversion DID fail, Tennessee DID score and kick the PAT, the game DID go into overtime, and Tennessee DID win it in overtime. But the issue relevant to us is, what was the LIKELIHOOD that that would happen?
The most critical assumption that we have to make is, "What is the likelihood that Tennessee would score 7 points in those last couple of minutes?" Let’s assume for the sake of argument that it is 50%. If that probability is 50%, then Tennessee’s chance of winning the game was, I think, 13.025%--assuming the Houston did attempt the two-point conversion.
Three things would have to happen in order for Tennessee to win:
1) The two-point conversion would have to fail,
2) Tennessee would have to score and hit their extra point, and
3) Tennessee would have to win the game in overtime.
According to some source I found online, 52.1% of two-point conversion attempts fail, so we will assume that the chance the two-point conversion would fail is 52.1%.
We have assumed for the sake of argument that the chance that Tennesee would score seven on their final possession is 50%.
And, since an overtime is basically a tossup, we will assume that the likelihood that Tennessee would win the overtime period is 50%. Noting again that none of those three probabilities is necessarily RIGHT; that’s just as much as we know. The coaches involved, or some smart person such as yourself, might very probably know more than we do. But that’s the numbers we have to work with.
Working with those numbers, the probability that (1) Houston goes for two and (b) Tennessee wins is .521, times .500, times .500. That’s 13.025%. Houston’s chance of winning the game, IF they go for two, is .86975.
I will note in passing that you CAN divide the possibility of Tennessee scoring seven points into many more probabilities: 1. That Tennessee can score a touchdown, 2. That they decide to go for two points in regulation, rather than risking overtime, 3. That the two-point conversion succeeds, 4. That the two-point conversion fails, 5. That they decide to just kick the PAT, 6. That the PAT kick succeeds, and 7. That the PAT kick fails. All of those things are possible, thus are probable at some level.
But that’s not really a BETTER way to analyze the problem; it is merely a more complicated way to analyze the problem. Since Tennessee did in fact decide to just kick the PAT and did in fact hit it, we can treat that choice as a "known" rather than an "unknown", and spare ourselves more branches breaking off from the tree of probabilities.
So Houston’s chance of winning if they did go for two was 86.975%. Now, what was their probability of winning if they had just kicked the PAT like normal people and myself and the asshole in the TV booth all expected them to do?
Here, it seems to me, we have SEVEN probable scenarios, rather than five as we had before—using the same assumed values. Let me note first of all that if Houston kicks the PAT, Tennessee then HAS to go for a two-point conversion, so that is a "known". The seven reasonably probable scenarios are:
1) Houston’s PAT is good and Tennessee does not score, Houston Wins,
2) Houston’s PAT is good, Tennessee scores and goes for two but misses, Houston Wins.
3) Houston’s PAT is good, Tennessee scores and goes for two, makes the two points to put the game in overtime, but Houston wins in overtime,
4) Houston’s PAT is good, Tennessee scores and goes for two, makes the two points to put the game in overtime, and Tennesee wins the game in overtime.
5) Houston’s PAT is NOT good, but Tennessee does not score, Houston wins.
6) Houston’s PAT is not good, Tennessee scores, games goes into overtime, Houston wins, and
7) Houston’s misses the PAT kick, Tennessee scores, game goes overtime, Tennessee wins.
There are seven scenarios there, but the only two that we actually need to worry about are the ones in which Tennessee wins, which are numbers 4 and 7—the ones which are in bold face. (Well, I had to worry about all seven, just to be sure that my math added up to 1.0000. But to reach the result, we only have to worry about the two in which Tennessee might win.)
4) Houston’s PAT is good, Tennessee scores and goes for two, makes the two points to put the game in overtime, and Tennessee wins the game in overtime.
That depends on five things happening: Houston hits their PAT, Tennessee scores, Tennessee goes for two, Tennessee makes the two-point conversion, and Tennessee wins in overtime.
HOUSTON HITS THEIR PAT is assumed to be a 94.4% probability, since 94.4% of all kicks were successful after the distance was increased last year.
TENNESEE SCORES we have assumed to be a 50% probability.
TENNESEE DECIDES TO GO FOR TWO is certain, since they would be 2 points behind with seconds left in the game,
TENNESSEE MAKES THE TWO-POINT CONVERSION is assumed to be a 47.9% probability, consistent with the other scenario, and
TENNESSEE WINS IN OVERTIME is assumed to be a 50% probability, consistent with the other scenarios.
So the probability of all five of those things happening is .944, times .500, times 1.000, times .479, times .500. That works out to 11.3044%. But there is a second way that Tennessee can win, which is #7 on the list above:
7) Houston’s misses the PAT kick, Tennessee scores, game goes overtime, Tennessee wins.
Three things have to happen there: Missed PAT attempt, Tennessee scores, Tennessee wins in overtime.
The probability that the kick will be missed, consistent with earlier analysis, is .056.
The probability that Tennessee will score 7 to tie is assumed to be 50%, and
The probability that Tennessee will win in overtime is assumed to be 50%.
So the probability of ALL THREE of those things happening appears to be .014--.056, times .500, times .500.
So if Houston just decides to kick the PAT like a normal person, the probability that Tennessee will win anyway appears to be .127 044. That means the probability that HOUSTON will win is .872 956. So the probability that Houston will win appears to me to be:
.872 956 if they just kick the PAT, and
.869 75 if they go for two.
So far I don’t see the analytical argument for going for two in that situation. I acknowledge again that this analysis is not absolutely persuasive; no analytical weighing and measuring of the percentages CAN be absolutely persuasive. But when I suggested earlier on that I was more or less on the side of the asshole (apart from the fact that he was blaming this on analytics), people wrote to tell me that analytically, going for two WAS the better choice. So far, I don’t see it.
The critical questionable assumption here is that Tennessee’s chance of scoring 7 in the closing two minutes is 50%. I mean, ALL of the assumptions are somewhat questionable, but that’s the big one. There are six key assumptions in this analysis:
1) That a team’s chance of scoring on a 2-point conversion is .479,
2) That a team’s chance of kicking a field goal is .944,
3) That there was only time for ONE remaining drive,
4) That Tennessee’s chance of scoring 7 on that drive was 50%,
5) That an overtime result would be 50/50, and
6) That nothing weird was going to happen like Tennessee fumbling, Houston picking it up, throwing an interception, the man who intercepted it has his pants fall down, he stops to pick them up, the ball is taken away from him but then lateraled to a member of the opposite team, and Houston winds up winning 51-49. We all know that a lot of weird stuff can happen in two minutes of football.
But the key questionable assumption there is #4. We really don’t KNOW whether Tennessee’s chance of scoring 7 there was 50% or 40% or 30% or 80%. It’s not an easy thing to research. It’s not an easy thing to put an accurate number on.
So let us assume, to make a more fulsome analysis, that it was not 50%, but 40%.
Then, repeating the entire, analysis but in quicker form, Tennessee’s chance of winning is 10.42% if Houston goes for two, but 10.16% if Houston just does the expected thing, kicks the PAT. The 10.42% is .521, times .400, times .500. The 10.16% is .0904 (.944, times .400, times .479, times .500), PLUS .0112 (.056, times .400, times .500. I probably shouldn’t give you those numbers, because they probably don’t make any sense to you, but I just have a compulsion to make sure you can track my work if you’re a mind to do so.) So if we assume that Tennessee’s chance of scoring 7 is 40%, then Houston’s chance of winning is:
.8984 if they kick the PAT, and
.8958 if they go for two.
Again, analysis does NOT seem to support the notion that going for two is the smart move here. But let’s assume, instead, that Tennessee’s chance of scoring 7 on their last-minute drive is 60%, rather than 50% or 40%.
Same result. Different numbers, but the same result. If we assume that Tennessee’s chance of scoring 7 on that drive was 60%, then Houston’s chance of winning the game is:
.8745 if they kick the PAT, and
.8437 if they go for two.
What about 70?. What if we assume that Tennessee’s chance of scoring on the last drive is 70%?
Still doesn’t work. If we assume that Tennessee’s chance of scoring is 70% and work the same math, then Houston’s chance of winning the game is
.8221 if Houston kicks the PAT, and
.8177 if Houston goes for two.
So. . .I don’t see it. Not saying my analysis is perfect; not even necessarily saying that my math is right; feel free to run the numbers yourself. But I don’t see the argument in favor of going for two in that situation.