The Nation Has Problems 5
Posted by GreekHouse on Tuesday, November 13th, 2007 at 6:59 pm
Last week I talked about expectation with a short note at the end about sports and expectation. Whether or not you realize it, expectation is at the heart of every strategic decision that a team makes. If a team signs a big name free agent, it is in an attempt to maximize their expected number of wins (or World Series titles, or division titles or whatever). Over the course of a season, a team tries to manage its club in a way that maximizes its number of wins.
Over the course of a particular game, people have different opinions on how to maximize their chances of winning. Typically, I advocate a greedy strategy for trying to win. My goal is to basically try and score as much as possible, while conversely preventing my opponent from scoring as much as possible. There are clearly times when this strategy should be altered, but for the most part I think that you should just try to bludgeon your opponent to death.
A Basketball Example
Basketball is perhaps the most obvious example of a sport where my greedy approach would be successful. Both teams have plenty of opportunities to score over the course of a game (and both teams will have approximately the same number of opportunities), so the one that maximizes their scoring expectation will win much more frequently.
Suppose an NBA team has a player shooting one free throw. There are only two possible strategies here. You can attempt to make the free throw, or you can attempt to miss it and get the offensive rebound. If a player makes the free throw, his expectation for scoring on this play is exactly 1. On the other hand, if he misses the free throw his expectation is the percentage of the time his team will get the rebound times the expected number of points a team will score on a single possession. In the NBA, the expectation of a single possession is almost exactly 1. This means that for the team to benefit from missing the free throw intentionally, they would need to get the rebound every time. Since teams can't possibly do this, it is clearly better to try and make the free throw.
Obviously there are times in the game that a team would want to miss the free throw on purpose, but those opportunities represent a very small minority of free throw attempts. A team that adopted a strategy of intentionally missing the second free throw every time would sacrifice a lot of points and lose a lot of games because of it.
Expectation and Football
I read an interesting article a few years ago about optimal football strategy in terms of expected number of points. There are many questions about what strategy a team should employ. For instance, if a team has fourth and goal on the opponents 1-yard line should they try for a field goal or try to score a touchdown? If a team can score a touchdown more than 3/7 of the time, they clearly they should go for it. But the question is more complicated than that. If you go for the touchdown and miss it, your opponent gets the ball on the one yard line. If you kick the field goal, they will end up (on average) with much better field position than that. This would suggest that you can go for it with a success rate much lower than 3/7.
I haven't been able to track down the original article, but if you google "football expected value" you will find numerous similar studies. The results of all of them is basically the same as the original piece I read. The consensus is that when it comes to questions of kicking (either FGs or punting) or going for it on 4th down, NFL coaches kick way to much. The original one I read suggested that a team can go for it almost anywhere on the field as long as they're within 5 yards of a first down. As you get closer to the end zone, you can be more liberal than that.
It is really frustrating to watch a team like the Vikings. Childress is a conservative coach even by NFL standards and makes horrendous kicking decisions. Attempting long field goals and punting anywhere on the opponent's end of the field is terrible general strategy. In particular a team like the Vikings who have a good defense but terrible offense should be taking more risks to try and score and relying on their good D to bail them out when they fail.
Expectation and Baseball
Because of the relative scarcity of run scoring in baseball, there is perhaps the greatest division between fans of the game in terms of what will maximize your chances of winning. Again, I propose a greedy strategy. When hitting, you basically want to score as much as possible. Other people contend that a good team will be able to manufacture runs using a variety of techniques in order to win close ball games. This topic has been discussed before, so I will leave it to people who are better at verbalizing it to explain why the second school of thought is wrong.
Problems
1. After scoring a touchdown a certain football team can convert an extra point p percent of the time and can convert a 2-point conversion q percent of the time. Under what conditions is it better for the team to attempt a 2-point conversion rather than an extra point?
2. Andrew and Big Mak are playing a game where they take turns removing stones from different piles. They start with piles of 2,3,4,5, and 6 stones. During each player's turn, they must do one of the two following things:
- Remove one stone from a pile containing at least 3 stones
- Remove an entire pile of stones if it contains 2 or 3 stones
Note that if a pile contains 2 stones, the player must take the entire pile. The player who takes the last stone wins. If Andrew goes first, which player has a winning strategy?
Rules
- I don’t care if you cheat, but if you do, don’t post the answer.
- Post all solutions/hints in white text (see below for instructions). Please post some dummy text at the beginning so that your solution doesn't show up on the main page.
- Ask questions and state ideas about the problem to generate discussion.
- I will post the solution eventually (assuming I know how to solve it) if nobody else does.
To post the answer in white use the following template below. If you're posting in a gray box, change "white" to "#f8f8f8".
<font color="white">
before your answer and
</font>
after your answer. You are encouraged to cut and paste that little bit of HTML.
Note to anyone with comment modification privileges: if you see an answer posted and not in all white, please edit the comment to add the whitening HTML.
Loading ...
I have an idea for #1, but I'm not comfortable with posting it right now. Something isn't sitting right.
And I'm honored to be the subject of a TNHP question.
The answer for #1 should be very intuitive. If it seems wrong, it probably is. If it seems right, it probably is.
Your comments about Childress are right on. Moss always thought the last coach didn't have a clue about his own team, either.
If the Vikes were behind by 7 or fewer fairly late in a game, Tice would almost always go ahead and punt the ball back to the other team on 4th down on the theory that his "defense" would hold. This was flat-out wrong and a complete misunderstanding of the nature of his own team.
They had a lousy defense and were very unlikely to ever get their hands on the ball. The team simply wasn't a field-position team -- it didn't really matter where the opponent got the ball, they were likely to score regardless. So why give them the ball? Why would the coach put the pressure on the defense, when the defense never proved that it could withstand the pressure?
Nevertheless, he chose to punt almost every time. Dumb football coaches abound.
If the Vikes were behind by 7 or fewer fairly late in a game, Tice would almost always go ahead and punt the ball back to the other team on 4th down on the theory that his "defense" would hold. This was flat-out wrong and a complete misunderstanding of the nature of his own team.
Right, this type of thinking is completely wrong. When you're down late you can't play a field position game, you need to try and score. If you go for it on 4th down, you actually give your team two chances to score. You could make it on 4th and score or you could fail to make it, then have your defense hold them and get the ball back. If you punt, the only chance you have to score is if your defense holds them. Punting in situations like this is truly horrific strategy. It's the same deal with onside kicking. Even if you fail to get the ball, you can still stop the other team. Late in the game it typically doesn't matter where on the field your opponent is. If they drive 40 yards from the 40 yard line, they will score and win. If they drive 40 yards from the 20 yard line, they will basically run the clock out and win.
Your point is true generally (i.e., for a normal team).
It is even more true in the case of Tice's teams, who had crappy defenses. They should have NEVER, at any point of the game, played the field-position game. It just wasn't an issue.
Here's a real-world problem that calls for an expectation analysis. Why have teams gone away from coffin-corner kicking? Now they try for the deep kick in the hope of downing it inside the 10, while taking a large risk that the ball goes into the end zone. It seems that the punter should be able to get the ball OOB inside the 10 with some consistency, with the same or reduced risk of the ball going into the end zone.
So the question is, are coaches really realizing the better expectation value? Or are they just mirroring their punting strategies after everyone else?
A related question is, are the punters really not good enough to drop the ball at the 10-yard line consistently? If that was your whole profession, it seems like you'd get reasonably good at it after awhile. But so many of them kick it too far. Is it really that difficult?
The punters are probably a bit risk-averse in actuality. You hardly ever see the punt go to the 15 or the 20, but it often goes into the end zone. So they are aiming for the 5 and taking the risk of going long, rather than aiming for the 10 and taking the risk of going short. (So really they are averse to the risk of being embarrassed by a short kick more than anything.)
The related point on the Tice strategy discussion is that the Vikes had some measure of control when they had the ball, and none when they did not. You would think that a rational coach would prefer to have some control, rather than cede control to the opposition.
Nowadays, it is much more neutral. The Vikes can't do anything with the ball (unless AP breaks one) so they don't have a whole lot of control even when they have the ball. And the defense is improved in some respects.
I think what drives this strategy is that kickers are better at controlling the distance of their punt than the angle of their punt. (Or at least more confident in their ability to control the distance.) By punting towards the center of the field, a small deviation in angle makes a small difference in the placement of the ball. By punting towards the edge of the field, a small deviation in angle could make a big difference in the placement of the ball.
Plus, by kicking to the sideline you give the officials more power to screw up, and they usually take every opportunity they can to make a poor decision.
Your point is true generally (i.e., for a normal team).
It is even more true in the case of Tice's teams, who had crappy defenses. They should have NEVER, at any point of the game, played the field-position game. It just wasn't an issue.
If Tice had thought about it rationally for a minute, he might have figured this out. The whole point of punting the ball away is that you're trying to avoid giving your opponents a good situation to score when you don't have much of a chance to score yourself. Most NFL coaches completely ignore that this is the reason that they're punting. Instead, they give too much weight to what will happen if they fail to get a first down. In reality, I think this has a lot to do with risk aversion.
Here's a real-world problem that calls for an expectation analysis. Why have teams gone away from coffin-corner kicking? Now they try for the deep kick in the hope of downing it inside the 10, while taking a large risk that the ball goes into the end zone. It seems that the punter should be able to get the ball OOB inside the 10 with some consistency, with the same or reduced risk of the ball going into the end zone.
If you ask me, your punter should rarely be punting in situations where this is even an issue. If I were going to start a football team, I would look for a punter who could kick the ball as far as possible and not really care about his accuracy.
Dumb football coaches abound.
It's the culture of dumb. They routinely "work" until midnight or later every night. Having a job that requires that I think, I know that I can only think so long before my brain is mush. The same has to be true with football coaches. Yet, they stay at the office until all hours. Dumb.
North Dakota State athletic director Gene Taylor said the chances of the Bison football team playing a game after the regular season ends remains slim to none. But he said he’s also going to look into the issue this week.
The 10-0 Bison play South Dakota State for the Great West Football Conference title Saturday in Brookings. Both teams are ineligible for the playoffs because of a five-year Division I reclassification.
NDSU, which plays in the Division I Football Championship Subdivision (formerly I-AA) does not meet the criteria for a Division I Football Bowl Subdivision (formerly Division I-A) bowl game. NDSU has two wins over an FBS team and to be bowl-eligible you need six.
But Taylor said he wonders if a 10-1 or an 11-0 team would make for a better story and be more attractive to a bowl committee than a 6-6 team that finished in the lower half of its league. He’s doing it under the “can’t-hurt-to-ask” department.
Here's the answer to the first question. If 2q > p, then go for two. Of course, in a game where you need one point to win, you go for one if p > q.
Yes, this is right of course.
On the one hand, I'm actually kind of glad my "WRONG!" detector still works. On the other hand, I am ashamed that I missed it.
But to complicate things a little bit more,
Do we allow for conversion attempts recovered by the defense and returned to their own endzone for a score? Is that available in the NFL? And what are the rules? Is a blocked kick returned to the endzone worth one or two points? I'd assume a returned-for-score 2-pt attempt would be worth 2 pts if allowed.
So, we'd have to reflect the probability of the other team scoring from the two options in the non 0-to-2 pt margin at the end of the game situation.
This is the last time I give you guys a question about an actual sport. You're all too good at analyzing all the particular details and cases for me to make a question seem "simple". From now on, all sports related questions will be about blurnsball, quidditch or some other made up sport.
Feel free to answer the question any way you want and add any generalizations you want. I don't think you're allowed to score on the other team's conversion attempt in the NFL, but if you want to try and generalize the question further, be free to add new probabilities at will.
We are nothing if not trouble makers.
After scoring a touchdown a certain football team can convert an extra point p percent of the time and can convert a 2-point conversion q percent of the time. Under what conditions is it better for the team to attempt a 2-point conversion rather than an extra point?
This is a trickier question than I think you were intending it to be. This probability clearly depends on the game situation.
For instance, what if you just tied the score with a touchdown as time expired. Now you are given the chance between an extra point or a two-point conversion, either of which will win the game. Clearly you choose based on whichever is greater: p or q, so obviously you will kick the extra point.
This is obviously a bit of an extreme case, but I would argue that touchdowns are a rather infrequent event during football games and that it can be a good strategy to hedge risk during the game even if it doesn't lead to the most points in the long-run. There have been 642 extra points attempted this season so far, so say there have been about 660 touchdowns. This gives a touchdown scoring rate of about 2.3 touchdowns per team per game. (And they say baseball is a low-scoring game.) Even if you can get a 2-point conversion 2 out of every three attempts, you run a 2/9ths risk of scoring fewer points wtih 2-point conversions than with extra points (assuming, say, p=1.) In general, I don't know that that's a risk you want to take. (Or if that's even an attainable conversion rate.)
Now, if football games were 8 hours long and they played 40 games each season, then I think you'd see teams more interested in the expectation value. But I think that risk and variance play a big role in this particular question. (Sort of like how early in a baseball game you are concerned with scoring the most runs, but late in the game the optimal strategies depend more on what the game score is.)
This is obviously a bit of an extreme case, but I would argue that touchdowns are a rather infrequent event during football games and that it can be a good strategy to hedge risk during the game even if it doesn't lead to the most points in the long-run. There have been 642 extra points attempted this season so far, so say there have been about 660 touchdowns. This gives a touchdown scoring rate of about 2.3 touchdowns per team per game. (And they say baseball is a low-scoring game.) Even if you can get a 2-point conversion 2 out of every three attempts, you run a 2/9ths risk of scoring fewer points wtih 2-point conversions than with extra points (assuming, say, p=1.) In general, I don't know that that's a risk you want to take. (Or if that's even an attainable conversion rate.)
This question was asked in the original question I asked with regard to comparing 3-points 100% of the time with 7 points 3/7ths of the time. The conclusion was that the former was better, but just marginal. Suppose in the example that you gave that you're in a game where both teams score two touchdowns and suppose the other team will always kick the extra point with 100% accuracy. Now if you go for two every time, you will have the following:
If you had gone for 1 every time, you would have won 1/2 of the games (assuming that in the event of a tie, each team wins half the time). In the latter case, you will win 80% of the games of the 5/9 of the games that would have been ties anyway. If your team employs this strategy, they will win 2/3 of the games. I would say that this is much more than a marginal gain.
I think NFL average is around 50% for extra points, but imagine that you have a tremendous offense that could make it with around even a 60% rate. This is part of the reason I wanted to post this question. You could significantly increase your win percentage. However, due to the way the NFL works, I doubt any coach would actually have the balls to try and do this since the first time they miss all their conversions the press would be all over them and if it happened repeatedly they would probably get fired.
If you had gone for 1 every time, you would have won 1/2 of the games (assuming that in the event of a tie, each team wins half the time). In the latter case, you will win 80% of the games of the 5/9 of the games that would have been ties anyway.
This is specious argument, though. Order matters here, too. Even a traditional team won't necessarily keep kicking extra points if is down 16-0 late in the game, and a team will definitely change its strategy late in a game if it is down by just one point rather than tied. (On offense, teams will play more conservatively in tied games than in games where they are down by one point, increasing the probability that they score more than one point late in the game. Your one-point lead could wind up enticing the other team into employing a more aggressive strategy.) By picking their spots the traditional team will move that 2/3 edge closer to 1/2, which also only applies when teams score precisely the same number of touchdows, which is a small subset of total games. And as you say, you would need a tremendous offense just to get a 60% success rate, for most teams the 2/3rd edge is going to be smaller. At which point you the edge becomes marginal. I don't think you're going to boost your winning percentage significantly, especially when compared to the variance in a 16-game season.
No, it won't matter that much because the value of the extra points and two point conversions is small compared to the value of the touchdown. If one team scores 2 touchdowns and the other scores 3, it makes no difference how frequently they make either of their conversions or what they try to do. The team that scores 3 TDs will win no matter what.
My point is that a coach who attempted this strategy would likely be crucified by the media if it went wrong, despite employing what might be an optimal strategy. I wouldn't use variance as an excuse either. It's like letting a baseball team bunt a runner from second to third with no outs in the first inning of the first game in a playoff series. It probably won't make a difference in who wins the series, but that's no excuse for doing something retarded. However, if your team routinely makes these suboptimal plays, it could easily make a difference in who wins the series.
It probably won't make a difference in who wins the series, but that's no excuse for doing something retarded.
There are differently levels of poor decision-making, though. So you think that football coaches are pursuing a slightly sub-optimal strategy w/r/t 2-point conversions--just how sub-optimal is it? Some offenses might not even be able to hit the breakeven point. Also, it's tough to tell how the odds of making a 2-point conversion might change as you started to attempt more of them. The more data you give the other team on your 2-point converting strategies, the better they might get at defending them.
I mean, if it's 1 in 100 games, then maybe it's a strategy worth pursuing. If it's 1 in 1,000 games, then it's not even worth mentioning. It'd matter twice a century, and at that point, you're getting to the level where you have to start considering the strategy's effect on team morale and things like that. Just how large the effect is means everything.
For instance, what if you just tied the score with a touchdown as time expired. Now you are given the chance between an extra point or a two-point conversion, either of which will win the game. Clearly you choose based on whichever is greater: p or q, so obviously you will kick the extra point.
Well yes, but this isn't really the point of the problem. The problem is to maximize expectation (which was the point of everything I wrote).
But my whole point here is that in the game of football, the point is to win the game, not to maximize expectation. These are not always the same goal, just like in baseball the strategy for scoring the most runs in the inning is not necessarily the strategy which will lead to victory most often.
Here's another situation. Late in the game an extra point puts the team ahead by nine points where a two-point conversion could give you a ten-point edge or leave you with an eight-point edge. I'd kick the extra point every time in that situation to force the other team to use two possessions to even tie the game rather than leaving the door open for them to tie it on one possession.
These are quotes from my post above. I understand your point and of course there are plenty of times when you want to employ suboptimal strategy (in terms of scoring expectation) in order to maximize your chances of winning the game. My point is that these situations represent a minority of game situations. For the most part, you just want to score as much as possible. Many people--particularly in baseball--are willing to give up tons in expected runs for strategical purposes. In general, I think that people are willing to give up way too much for the sake of consistency or for strategical purposes.
This particular question obviously only has an answer if you assume there is no strategical advantage in either case.
I'm assuming the reason I've been featured in one of the "Nation has Problems" questions is because I've been singled out as having significant problems? (Whoever said 'paranoia', I heard that!)
That being said, I am having problems with the second question.
Andrew (going first) can always win, but I'm having trouble developing a foolproof strategy.
The longest possible game is 15 turns, one stone is taken by both players for 10 turns to give 5 piles of 2 stones each. Then Andrew removes the 1st, 3rd, and final piles of two, winning the game.
The only way to shorten the game is to remove a pile of 3 in one turn. If every time that I remove a pile of 3, Andrew is able to do the same, he will be able to reduce the number of turns in the game by multiples of 2, ensuring his victory.
But I can't get a general strategy for this. If Andrew takes one stone from the pile of 4 in his first turn, then he is set up initially, because there are now two piles of 3. If I remove one, he can remove the other. If I don't remove any piles of 3, Andrew just needs to take the minimum amount of stones possible (without providing another pile of 3 for me).
That strategy isn't foolproof though:
Andrew takes one from pile C - 23356
I take 3 from pile B - 2-356
Andrew takes 3 from pile C - 2--56
I take 2 from pile A - ---56
Andrew and I both take 1 for couple of turns - ---34
If Andrew takes one from pile D (the strategy advocated above) then I win. Andrew must clear the pile of 3 to win.
I get the feeling I'm approaching this the wrong way. Hmmm.
I'll be back.
I'd retrace the steps that lead me here
but nothing lives behind me.
So I lie in this field bathed in the light that loves me,
with nothing left to lose.
Will you be my beloved?
Will you help me to get through?
Will you be my destruction?
Will you help me to be true?
Now that I read my answer, I think that a simple revision should get me to the correct answer.
If Andrew takes one stone from the pile of 4 in his first turn, then he is set up because there are now two piles of 3. If I remove one, he can remove the other. If I don't remove any piles of 3, Andrew just needs to take the minimum amount of stones possible (without providing another pile of 3 for me). If I generate another pile of 3, Andrew can either remove a stone from that pile to reduce the number of piles of 3 back to an even number, or he can reduce a different pile to 3 stones increasing the number of piles of 3 to an even number.
Thus the endgame of the example above is:
Andrew takes one from pile C - 23356
I take 3 from pile B - 2-356
Andrew takes 3 from pile C - 2--56
I take 2 from pile A - ---56
Andrew and I both take 1 for couple of turns - ---34
Andrew take one from pile E - ---33
Then Andrew mirrors my moves and takes the last pile.
I saw a star beneath the stairs
glowing bright before descent
and in the morning there is nothing left
but what's inside of me.
And I don't want to die tonight;
will you believe in me?
And I don't want to fall into the light.
Monroe gives the Twins a bat who is worth potentially 20 runs above replacement level in a corner OF/DH spot--approximately the level he was at from 2004-2006. He also gives the Twins the risk of paying $5M for a sub-replacement level performance.
.222/.264/.373 -- Player A, age 30
.240/.288/.378 -- Player B, age 30
.256/.303/.446 -- Player A, career
.284/.336/.462 -- Player B, career
In his age 31 season, Player B hit .289/.341/.488 for an OPS+ of 120. Two Twins had an OPS+ of 120 or better last season. Two. And one of them is leaving either to play on the truest grass he can find, in a bigger media market, or someplace he can have a big impact on the community, depending on which day you ask him.
I think this is a very hard proof to formalize. Your idea is right, but it needs to be formalized a little bit better. I think you have the general premise of the idea and it is based upon being able to make "safe" moves.
This is an interesting problem where you can use a strategy stealing argument to show that Andrew has a winning strategy without ever demonstrating what that strategy is.
First, lets assume that Mak has a winning strategy. Mak can't ever force Andrew to take any stones from the pile of 3 unless it's the only pile remaining. If it's the only pile remaining, then Andrew can simply take the pile and win.
Thus, there must come a point during the game when Mak will decide to take either 1 stone from the pile of 3 or take all three. Since Andrew goes first and knows Mak's strategy he has a turn before this.
If Mak's strategy involves taking all three, then Andrew takes all three on the turn prior. Otherwise, if Mak's plan is to only take one, then Andrew takes only one. In either case, Andrew has "stolen" Mak's strategy. Since Mak can't follow this strategy, it can't exist and so Andrew must be the player with the winning strategy.
Note that this argument works with any number of piles of stones with any number of stones in them as long as there is only one pile with 3 stones.
After further review, I've decided that my proof above is flawed. Typically, you can only apply strategy stealing arguments in cases where moves can't hurt your position. This isn't really the case here. I think there may be a way to fix it, but it would basically require developing the strategy which seems sort of pointless because it would just be easier to demonstrate a winning strategy in the first place.
Once again, I have let my love of non-constructive existential proofs get the best of me.
Big Mak is the only one who gave any lyrics? What's up with that?
sorry, JeffA. I didn't get around to submitting flawed answers to this one.
You know, we've all got to go anyway
Said we've all got to go anyway
And I'd just as soon be the last in line