Justin Verlander just threw a no-hitter for the Detroit Tigers. On May 3rd, Francisco Liriano threw one for the Minnesota Twins.
There have only been 271 no-hitters in one hundred years of Major League Baseball, so two separated by four days seems unusual.
But two no-hitters within four days of each other has happened several times before. From Wikipedia, there have been two no-hitters within four days of each other on the following dates:
August 19 and 20, 1880
September 19 and 20, 1882
two on April 22, 1898
September 18 and 20, 1908
August 26 and 30, 1916
May 2, 5, and 6, 1917
September 4 and 7, 1923
June 11 and 15, 1938
June 26 and 30, 1962
September 17 and 18, 1968
September 26 and 29, 1983
June 1 and 2, 1990
two on June 29, 1990
September 4 and 8, 1993
May 11 and 14, 1996
Is this list surprisingly long? If you assume that baseball has been played 180 days a year for 130 years, then that's 23,400 days on which baseball has been played. There have been 271 no-hitters, so on an average baseball-playing day there are 0.01158 no-hitters. After any given no-hitter there's a four-day window in which the list I gave above could be added to. So you'd expect (271)(0.01158)(4) = 12.5 pairs in that list. There are 17 pairs on the list. (I'm counting the 1917 triplet as three pairs. I'm not counting today's no-hitter.) So there doesn't seem to be particularly strong evidence for no-hitters somehow causing more no-hitters in their wake. (Although my model of the baseball schedule is, I admit, ridiculously crude. In particular I have ignored the fact that the number of teams isn't constant.)
Showing posts with label baseball. Show all posts
Showing posts with label baseball. Show all posts
29 April 2010
A simple model for baseball
From the April Notices of the AMS, John D'Angelo writes Baseball and Markov Chains: Power Hitting and Power Series. Consider the following simple model of baseball. Players only hit singles; three singles score a run. That is, the third and every following player to get a hit in a given inning score a run. This can either be interpreted as that, say, all runners score from second on a single or all runners go from first to third on a single -- but not both! -- or that every third hit is actually a double. (And I do mean exactly every third hit, not some random one-third of hits, so this is a bit unnatural.) Then the expected number of runs per half inning is p3(3p2-10p+10)/(1-p). For real baseball the average number of runs per half-inning is around one half, which corresponds to p = 0.361.
D'Angelo gives this as an exercise, but I independently came up with this model a while ago and can't resist sharing the solution. Let q = 1-p. The probability of getting k hits in an inning is pk q3 -- that's the probability of getting those hits in a certain order -- times the number of ways in which k hits and 3 outs can be arranged. Since the last batter of an inning must get out, the number of possible arrangements is the number of ways to pick 2 batters out of the first k+2 to get out, which is (k+2)(k+1)/2.
The probability of getting k runs, if k is at least 1, is just the probability of getting k+2 hits, which is pk+2q3(k+4)(k+3)/2. Call this f(k); then
f(1) + 2f(2) + 3f(3) + ... = p3(3p2-10p+10)/(1-p)
by some annoying algebra. I'm pretty sure I came up with this exact model while procrastinating from some real work a couple years ago; it's probably been independently reinvented many times.
With p = 0.361, the probabilities of scoring 0, 1, 2, 3, 4, 5 runs in an inning are .748, .123, .066, .034, .016, .008 (rounded to three decimal places). (Probabilities of larger numbers of runs can also be calculated; together they have probability around .006.)
Assuming that each half-inning is independent, the probability G(k) of a team scoring k runs in a game is, for each k,
with probability about 0.0006 of scoring 18 runs or more. (This seems a bit low to me -- three times a season in the major leagues -- but after all this is a very crude model!) But one interesting thing here is that the distribution of the number of runs per game, which is a sum of nine skewed distributions, is still skewed; the mode is 3, and the median 4. Recall that I chose p so that the mean would be 4.5. And the actual distribution is similarly skewed.
Of course a more sophisticated model of baseball is as a Markov chain. There are twenty-five states in this chain -- zero, one or two outs combined with eight possible ways to have runners on base, and three outs. We assume that each hitter hits randomly according to his actual statistics, and the runners move in the "appropriate" way. Of course determining what's appropriate here would be a bit tricky. How do runners move? A runner is probably more likely to take an extra base when a power hitter is hitting, but the sample size for any individual is fairly small. But one could probably predict from some measure of the hitter's power (say, the number of doubles and home runs, combined appropriately) the chances of a runner taking an extra base on a single. Something similar is necessary for sacrifice flies (which have to be deep enough to score the runner), grounding into double plays, etc. I'm not sure if the Markov models that are out there, such as that by Sagarin, do this. Sagarin computes the (offensive) value of a player by determining how many runs per game a team composed of only that player would score.
D'Angelo gives this as an exercise, but I independently came up with this model a while ago and can't resist sharing the solution. Let q = 1-p. The probability of getting k hits in an inning is pk q3 -- that's the probability of getting those hits in a certain order -- times the number of ways in which k hits and 3 outs can be arranged. Since the last batter of an inning must get out, the number of possible arrangements is the number of ways to pick 2 batters out of the first k+2 to get out, which is (k+2)(k+1)/2.
The probability of getting k runs, if k is at least 1, is just the probability of getting k+2 hits, which is pk+2q3(k+4)(k+3)/2. Call this f(k); then
f(1) + 2f(2) + 3f(3) + ... = p3(3p2-10p+10)/(1-p)
by some annoying algebra. I'm pretty sure I came up with this exact model while procrastinating from some real work a couple years ago; it's probably been independently reinvented many times.
With p = 0.361, the probabilities of scoring 0, 1, 2, 3, 4, 5 runs in an inning are .748, .123, .066, .034, .016, .008 (rounded to three decimal places). (Probabilities of larger numbers of runs can also be calculated; together they have probability around .006.)
Assuming that each half-inning is independent, the probability G(k) of a team scoring k runs in a game is, for each k,
| k | 0 | 1 | 2 | 3 | 4 | 5 |
| G(k) | .073 | .108 | .129 | .133 | .124 | .108 |
| k | 6 | 7 | 8 | 9 | 10 | 11 |
| G(k) | .088 | .069 | .052 | .038 | .026 | .018 |
| k | 12 | 13 | 14 | 15 | 16 | 17 |
| G(k) | .012 | .008 | .005 | .003 | .002 | .001 |
with probability about 0.0006 of scoring 18 runs or more. (This seems a bit low to me -- three times a season in the major leagues -- but after all this is a very crude model!) But one interesting thing here is that the distribution of the number of runs per game, which is a sum of nine skewed distributions, is still skewed; the mode is 3, and the median 4. Recall that I chose p so that the mean would be 4.5. And the actual distribution is similarly skewed.
Of course a more sophisticated model of baseball is as a Markov chain. There are twenty-five states in this chain -- zero, one or two outs combined with eight possible ways to have runners on base, and three outs. We assume that each hitter hits randomly according to his actual statistics, and the runners move in the "appropriate" way. Of course determining what's appropriate here would be a bit tricky. How do runners move? A runner is probably more likely to take an extra base when a power hitter is hitting, but the sample size for any individual is fairly small. But one could probably predict from some measure of the hitter's power (say, the number of doubles and home runs, combined appropriately) the chances of a runner taking an extra base on a single. Something similar is necessary for sacrifice flies (which have to be deep enough to score the runner), grounding into double plays, etc. I'm not sure if the Markov models that are out there, such as that by Sagarin, do this. Sagarin computes the (offensive) value of a player by determining how many runs per game a team composed of only that player would score.
22 August 2009
Two baseball games the same?
The Yankees and Red Sox have played each other two thousand and something times, as the good folks on Fox told us, so I started to wonder: which baseball teams have played each other the most times? If I had to guess it's Giants-Dodgers; Wikipedia agrees with me but gives no source. Apparently Cardinals-Cubs, Cardinals-Pirates, and Cubs-Pirates are all a close second. These are the ones you'd expect if you take into account when teams changed divisions, etc.
Anyway, trying to find this out I found a blog post entitled Have Two Baseball Games Ever Played Out Identically?. The answer is no, but "identically" is defined a bit too strictly; (say) a groundout to second and a groundout to shortstop are counted as different. And the metric that the author uses for similarity of two games A and B is, I think, the number of times where the nth plate appearance in games A and B had the same outcome. Intuitively I think you'd want to line up innings with each other. Two "most similar" games should at least have similar-looking line scores. I think what one wants is some notion of "edit distance" between games, and defining that is hardly trivial.
I've sort of poked at this before: in 2007 I asked what's the most common line score in connection with a promotion that MLB did for that year's all-star game.
There's a nice combinatorial/probabilistic question hiding here; I've seen results on, say, the probability that two randomly chosen permutations of [n] have the same cycle type, or the probability that two binary trees with n labelled nodes have the same shape. Baseball games are combinatorial structures, and I'm not just saying that to justify the fact that I'm probably going to spend six hours today watching baseball. (The Yankees and Red Sox are on TV now, the Phillies and Mets later.)
Anyway, trying to find this out I found a blog post entitled Have Two Baseball Games Ever Played Out Identically?. The answer is no, but "identically" is defined a bit too strictly; (say) a groundout to second and a groundout to shortstop are counted as different. And the metric that the author uses for similarity of two games A and B is, I think, the number of times where the nth plate appearance in games A and B had the same outcome. Intuitively I think you'd want to line up innings with each other. Two "most similar" games should at least have similar-looking line scores. I think what one wants is some notion of "edit distance" between games, and defining that is hardly trivial.
I've sort of poked at this before: in 2007 I asked what's the most common line score in connection with a promotion that MLB did for that year's all-star game.
There's a nice combinatorial/probabilistic question hiding here; I've seen results on, say, the probability that two randomly chosen permutations of [n] have the same cycle type, or the probability that two binary trees with n labelled nodes have the same shape. Baseball games are combinatorial structures, and I'm not just saying that to justify the fact that I'm probably going to spend six hours today watching baseball. (The Yankees and Red Sox are on TV now, the Phillies and Mets later.)
15 July 2009
Batting under .200
Stat of the day (from baseball-reference.com) has a list of players who went an entire season, had enough at bats to qualify for the batting title (I forget the statistics for this, but this basically means they have to play regularly), and are batting under .200.
Most of them are from a long time ago. Why? Because .200 is well below average and always has been (which is why the list was worth compiling) and the variance in batting averages has gone down as the standard of play has improved. Stephen Jay Gould wrote about this in Full House: The Spread of Excellence from Plato to Darwin
; the argument is roughly that as baseball scouting and training has gotten better, there are not as many bad pitchers in the major leagues as there were in the past, so players can't inflate their batting average that way. (I'm in Ithaca and my copy of the book is in Philadelphia, so I can't check if I'm stating this correctly.)
Most of them are from a long time ago. Why? Because .200 is well below average and always has been (which is why the list was worth compiling) and the variance in batting averages has gone down as the standard of play has improved. Stephen Jay Gould wrote about this in Full House: The Spread of Excellence from Plato to Darwin
; the argument is roughly that as baseball scouting and training has gotten better, there are not as many bad pitchers in the major leagues as there were in the past, so players can't inflate their batting average that way. (I'm in Ithaca and my copy of the book is in Philadelphia, so I can't check if I'm stating this correctly.)
03 November 2008
A thought on expectation
Political campaigns should not campaign in such a way as to maximize their expected number of votes. Rather, they should campaign in such a way as to maximize their probability of winning.
Early in a campaign, these are pretty much the same thing, because one doesn't know how things are going to play out; late in a campaign they diverge. The candidate that's behind in the polls should, perhaps, start doing things that are, in expectation, Bad Ideas. If there were something that McCain could do that had a 1% chance of swinging 10% of the vote to him in the next 24 hours, and a 99% chance of scaring all the voters away, he should do it. (For example, let's say McCain turns out to secretly be an extraterrestrial; we probably don't want to be ruled by extraterrestrials, but who knows, we might change our mind? Of course, I'm being deliberately silly here.)
This is somewhat analogous to what happens in sports; strategies change late in a game. In the early innings of a baseball game you play, basically, in such a way as to maximize the difference between the expectation of your number of runs and your opponents' number of runs. In the late innings, when you have a better idea of how many runs you need, you change your strategies. See, for example, the bottom of the ninth inning of Game 3 of the World Series; tie game, bases loaded, nobody out. The Rays bring in one of their outfielders to create a fifth infielder; the idea is essentially that they want to maximize the probability of the Phillies scoring zero runs, and if the ball gets into the outfield a run is scoring anyway. (As it turns out, the Phillies won that game -- on a hit in the infield.) Of course, nobody saw that because it happened at two in the morning. Things are different when you're playing for one run.
Basically, what's happening as I write this is that Obama is running out the clock. (Yes, baseball doesn't have a clock. But Obama's a basketball player, so I think he'd like this metaphor.)
What politics and sports have in common, of course, is that there's a huge difference between second place and first place. If you're a company of some sort, does it matter if you sell 1% more or 1% less than your competitor? Not really, although it might have meaning for your pride. But if you're a politician, 1% of the vote makes a big difference.
Early in a campaign, these are pretty much the same thing, because one doesn't know how things are going to play out; late in a campaign they diverge. The candidate that's behind in the polls should, perhaps, start doing things that are, in expectation, Bad Ideas. If there were something that McCain could do that had a 1% chance of swinging 10% of the vote to him in the next 24 hours, and a 99% chance of scaring all the voters away, he should do it. (For example, let's say McCain turns out to secretly be an extraterrestrial; we probably don't want to be ruled by extraterrestrials, but who knows, we might change our mind? Of course, I'm being deliberately silly here.)
This is somewhat analogous to what happens in sports; strategies change late in a game. In the early innings of a baseball game you play, basically, in such a way as to maximize the difference between the expectation of your number of runs and your opponents' number of runs. In the late innings, when you have a better idea of how many runs you need, you change your strategies. See, for example, the bottom of the ninth inning of Game 3 of the World Series; tie game, bases loaded, nobody out. The Rays bring in one of their outfielders to create a fifth infielder; the idea is essentially that they want to maximize the probability of the Phillies scoring zero runs, and if the ball gets into the outfield a run is scoring anyway. (As it turns out, the Phillies won that game -- on a hit in the infield.) Of course, nobody saw that because it happened at two in the morning. Things are different when you're playing for one run.
Basically, what's happening as I write this is that Obama is running out the clock. (Yes, baseball doesn't have a clock. But Obama's a basketball player, so I think he'd like this metaphor.)
What politics and sports have in common, of course, is that there's a huge difference between second place and first place. If you're a company of some sort, does it matter if you sell 1% more or 1% less than your competitor? Not really, although it might have meaning for your pride. But if you're a politician, 1% of the vote makes a big difference.
30 October 2008
Congratulations to the Phillies
Really, one World Series win in twenty-eight years is pretty close to average; over the last twenty-eight years there have been between twenty-six and thirty teams in Major League Baseball, so the average team should have won the World Series about once in that time span.
Still, this blog congratulates the Philadelphia Phillies on their winning the World Series in five and a third games. Not that any of them are reading it.
(If I cared about sports other than baseball, things would be different; Philadelphia has four major professional sports teams, and until last night none of them had won the championship of their sport since the 1983 Sixers; that's about 100 sports seasons without a championship, which is noteworthy. At that time I was a small ball of undifferentiated cells. As was Cole Hamels.)
Still, this blog congratulates the Philadelphia Phillies on their winning the World Series in five and a third games. Not that any of them are reading it.
(If I cared about sports other than baseball, things would be different; Philadelphia has four major professional sports teams, and until last night none of them had won the championship of their sport since the 1983 Sixers; that's about 100 sports seasons without a championship, which is noteworthy. At that time I was a small ball of undifferentiated cells. As was Cole Hamels.)
23 October 2008
The curse of Al Polaneczky
The little-known curse of Al Polaneczky. Apparently the Phillies' 1964 collapse (they were six and a half games up with twelve to play, lost ten in a row, and ended up finishing in a tie for second place) can be blamed on Al Polaneczky, a statistician at the Franklin Institute, who used the institute's computers to determine the odds of the Phillies winning the pennant that year.
Of course, by that logic, all baseball teams are cursed now, because people are crunching numbers constantly. Maybe all the curses cancel out.
Of course, by that logic, all baseball teams are cursed now, because people are crunching numbers constantly. Maybe all the curses cancel out.
21 October 2008
Two sorts of predictions
Lately, there are two things that a lot of people searching for this blog seem to want to know: predictions about the election, and predictions about the World Series.
Baseball Prospectus says the Phillies have a 51.7% chance of winning, and the Rays 48.3%. That seems a bit surprising, though; I would have thought (although I don't like to admit it) that the Rays would be a slight favorite. I suspect there's some difficulty in incorporating the fact that the American League (in which the Rays play) has a higher quality of play that the National League (in which the Phillies play).
And Fivethirtyeight.com says Barack Obama has a 93.8% chance of winning. (The election, not the World Series.) The prediction there is based on a Monte Carlo method which simulates the election results; there's a plot of the number of electoral votes received in the simulations, and it gets spikier and spikier as more and more states become settled one way or the other. (The site's model assumes that states may move between now and Election Day, and as "now" gets closer to Election Day, that effect diminishes.)
What do these two predictions have in common? Fivethirtyeight.com is Nate Silver's web site; he works for Baseball Prospectus.
By the way, various people have said that Fivethirtyeight.com doesn't take into account correlations between states. The most common misconception seems to be that the site assumes the results in each state are independent. This is not true; if you start from the state win probabilities displayed there and combine them under the assumption of independence, you get a distribution much different than the one currently displayed on the site.
Baseball Prospectus says the Phillies have a 51.7% chance of winning, and the Rays 48.3%. That seems a bit surprising, though; I would have thought (although I don't like to admit it) that the Rays would be a slight favorite. I suspect there's some difficulty in incorporating the fact that the American League (in which the Rays play) has a higher quality of play that the National League (in which the Phillies play).
And Fivethirtyeight.com says Barack Obama has a 93.8% chance of winning. (The election, not the World Series.) The prediction there is based on a Monte Carlo method which simulates the election results; there's a plot of the number of electoral votes received in the simulations, and it gets spikier and spikier as more and more states become settled one way or the other. (The site's model assumes that states may move between now and Election Day, and as "now" gets closer to Election Day, that effect diminishes.)
What do these two predictions have in common? Fivethirtyeight.com is Nate Silver's web site; he works for Baseball Prospectus.
By the way, various people have said that Fivethirtyeight.com doesn't take into account correlations between states. The most common misconception seems to be that the site assumes the results in each state are independent. This is not true; if you start from the state win probabilities displayed there and combine them under the assumption of independence, you get a distribution much different than the one currently displayed on the site.
01 October 2008
Who knew high school geometry was good for something?
Groundskeepers Display Artistry on the Diamond, September 30 (?) New York Times. David Mellor, the current head groundskeeper at Fenway Park, is the one who started the trend of mowing the grass in such a way as to create interesting-looking patterns; apparently he first did it in order to camouflage some damaged grass in Milwaukee in the early 90s. Roughly speaking, the mowing action bends the grass in one direction or another, creating contrasts of light and dark; you can see a similar effect when you vacuum your carpet. (I really mean "you" here. I can't; I have wood floors.)
And we're told that "High-school geometry classes visit [Mellor] at Fenway Park to study ways that an odd-shaped field can be divided and subdivided by straight lines and sharp angles."
And we're told that "High-school geometry classes visit [Mellor] at Fenway Park to study ways that an odd-shaped field can be divided and subdivided by straight lines and sharp angles."
04 August 2008
Continued fractions and baseball hitting streaks
The title of this post is misleading, because you might think there's a substantial connection between its two halves. The connection is only that I happened to come across both of these things this morning and it seemed silly to make separate posts about them.
Todd Trimble gives two proofs of the continued fraction expansion for e, namely e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, ...]. This is in the usual notation for continued fractions, so it actually means
e = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + ...)))))
but all the extra 1s obscure the pattern. This is one of those things that it's much easier to state than it is to prove. And I must admit it's always seemed a bit strange to me that e has such a nice continued fraction expansion, while π doesn't -- e has a special place in continued-fraction land.
From the arXiv, via the physics arXiv blog: A Monte Carlo Approach to Joe DiMaggio and Streaks in Baseball, by Samuel Arbesman and Steven Strogatz, which is what it sounds like. This expands upon this piece in the New York Times on the same subject, which I wrote about back in March. And no, I'm not sure why it's in a physics category at the arXiv.
Todd Trimble gives two proofs of the continued fraction expansion for e, namely e = [2, 1, 2, 1, 1, 4, 1, 1, 6, 1, 1, ...]. This is in the usual notation for continued fractions, so it actually means
e = 2 + 1/(1 + 1/(2 + 1/(1 + 1/(1 + 1/(4 + ...)))))
but all the extra 1s obscure the pattern. This is one of those things that it's much easier to state than it is to prove. And I must admit it's always seemed a bit strange to me that e has such a nice continued fraction expansion, while π doesn't -- e has a special place in continued-fraction land.
From the arXiv, via the physics arXiv blog: A Monte Carlo Approach to Joe DiMaggio and Streaks in Baseball, by Samuel Arbesman and Steven Strogatz, which is what it sounds like. This expands upon this piece in the New York Times on the same subject, which I wrote about back in March. And no, I'm not sure why it's in a physics category at the arXiv.
02 August 2008
Fox Sports' World Series odds
At this article: Fox Sports expects at least 1.5 teams to win the World Series this year.
No, I'm not making this up.
They give odds for fifteen Major League Baseball teams to win the World Series, ranging from the Marlins at 40 to 1 to the Angels at 5 to 2. I'll just give the top five teams here: Angels 5:2, Yankees 3:1, Cubs 7:2, Red Sox 4:1, Mets 8:1. So they figure that the Angels has probability 2/7 of winning the World Series, the Yankees 1/4, and so on.
Add up their probabilities for the top five teams: 2/7 + 1/4 + 2/9 + 1/5 + 1/9 = 449/420, which is greater than one. If you add up the probabilities for the fifteen teams they give probabilities for, you get about 1.506.
I'm not arguing with their ranking of the teams. (There might be things to argue with, but contrary to what you might come to believe in the next two months, or to what longtime readers might remember from last August and September, this isn't a baseball blog.) But a bookie that offered these odds would be broke pretty quickly.
Also, about the Phillies (to whom they give 16 to 1 odds to win at all), we're told that "Odds say they can't mash their way in two years in a row." Okay, it may be true that they can't rely on just their offense. But the fact that they did it last year really has no bearing on whether they can do it this year. If anything, the fact that they did make the playoffs last year is evidence that their offense is enough.
No, I'm not making this up.
They give odds for fifteen Major League Baseball teams to win the World Series, ranging from the Marlins at 40 to 1 to the Angels at 5 to 2. I'll just give the top five teams here: Angels 5:2, Yankees 3:1, Cubs 7:2, Red Sox 4:1, Mets 8:1. So they figure that the Angels has probability 2/7 of winning the World Series, the Yankees 1/4, and so on.
Add up their probabilities for the top five teams: 2/7 + 1/4 + 2/9 + 1/5 + 1/9 = 449/420, which is greater than one. If you add up the probabilities for the fifteen teams they give probabilities for, you get about 1.506.
I'm not arguing with their ranking of the teams. (There might be things to argue with, but contrary to what you might come to believe in the next two months, or to what longtime readers might remember from last August and September, this isn't a baseball blog.) But a bookie that offered these odds would be broke pretty quickly.
Also, about the Phillies (to whom they give 16 to 1 odds to win at all), we're told that "Odds say they can't mash their way in two years in a row." Okay, it may be true that they can't rely on just their offense. But the fact that they did it last year really has no bearing on whether they can do it this year. If anything, the fact that they did make the playoffs last year is evidence that their offense is enough.
28 June 2008
Baseball bats
Oh, apparently baseball people use "length-to-weight ratio" to describe a bat, as I learned from the people who talk too much before the Saturday afternoon game on Fox today. This is calculated by taking the weight (in ounces) minus the length (in inches), and in the major leagues can't be less than -3.5.
Of course, it's actually a difference, not a ratio.
It looks like some people call it the "differential", though, which is fine with me -- to me "differential" has other connotations, but expecting mathematical terminology not to collide with terminology used in other things is a Bad Idea. (Although why not just call it the "difference"?)
Of course, it's actually a difference, not a ratio.
It looks like some people call it the "differential", though, which is fine with me -- to me "differential" has other connotations, but expecting mathematical terminology not to collide with terminology used in other things is a Bad Idea. (Although why not just call it the "difference"?)
A thing I'm tired of hearing
"X does especially well/badly in interleague play."
Small sample size, people.
Small sample size, people.
27 June 2008
A variant of the traveling salesman problem
Josh Robbins attempts to see a baseball game in all thirty major league baseball parks in 26 days.
Yes, you read that right.
And Major League Baseball doesn't make that easy. As you can guess, he has to see two games in one day four times -- but in markets with two teams (New York, Chicago, San Francisco/Oakland, Los Angeles) they try to schedule the two teams to be on the home at opposite times. That makes sense, because that way if you think "I want to see a baseball game today" you've got a good chance.
In fact, his four doubleheaders are Dodgers-Padres (which apparently was a bit of a tight squeeze, since the Dodgers went into extra innings), Yankees-Mets (that one should be easy; every few years the Mets and the Yankees play a game at one park in the afternoon and at the other park the same night; they're doing it today); Phillies-Nationals (which will be tight even if the games go the ordinary length; they start six hours apart, average game length is three hours or so, and the cities are two and a half hours apart with no traffic -- oh, and he's doing it on a Thursday); Cubs-Brewers.
My point is that 25 days might be possible -- but probably not. Most baseball games are scheduled for around 1 PM or around 7 PM, and games last three hours, to see two in one day requires the sites to be no more than three hours apart. The pairs that are doable in one day are probably:
Mets-Yankees, Mets-Phillies, Yankees-Phillies
Phillies-Orioles, Phillies-Nationals, Orioles-Nationals
Dodgers-Padres, Padres-Angels, Angels-Dodgers
White Sox-Cubs, White Sox-Brewers, Cubs-Brewers
Giants-A's
but of course one can do only one from each row, so it's only possible to double up on five days. Basically, this is the problem of looking for the largest matching in the graph that I defined above, where the edges are teams within about three hours' driving distance of each other.
(Oddly enough, each two-team market (and yes, I know, Baltimore and Washington may or may not be the same market) seems to have another team a couple hours away. In two cases that team is the Phillies. As you may know, this blog likes the Phillies.)
So 25 is theoretically possible, if the Scheduling Gods worked in one's favor -- but I'd be scared to even look at the schedules to try and figure it out. And what happens if there's a rainout?
As a problem in actually scheduling things, the other tricky part is that Denver really isn't near any other team. And Robbins' schedule had him at a 7:05 game in San Diego, followed by a 1:05 game in Denver the next day -- but Denver's a time zone to the east of San Diego, so that's seventeen hours between starts. Fourteen hours driving time. For 1,078 miles.
For some other variants of the traveling salesman problem which involve the road network, see Barry Stiefel's 50 states in a week's vacation (driving, with flights to Alaska and Hawaii) and 21 states in one day. The last one cheats a bit -- it's a 26-hour day, since he started in the Eastern time zone during daylight savings time (GMT-4), and did the trip on a day when we went back to standard time (GMT-5) and then crossed into the Central time zone (GMT-6). The difference here is that you only have to enter each state instead of reaching a point.
Oh, and I feel obliged to point out that I find the meme of going on a long road trip this summer because "this is the last summer it'll ever be possible" kind of stupid. (Not that anybody here brought it up.)
Edited (Saturday morning): Google Maps says Cleveland to Detroit can be driven in 2:46. I didn't realize they were that close together. They'd be even closer if someone built a bridge across Lake Erie.
(Saturday afternoon): Cleveland to Pittsburgh in 2:18. I'll admit the reason I forgot this one is that mentally I think of Pittsburgh as being in the same state as me and Cleveland as not being in it, so they must be far apart. This is despite the fact that I live about five miles from New Jersey.
Anyway, you could shave off yet another day by combining the Indians with either the Pirates or the Tigers.
Yes, you read that right.
And Major League Baseball doesn't make that easy. As you can guess, he has to see two games in one day four times -- but in markets with two teams (New York, Chicago, San Francisco/Oakland, Los Angeles) they try to schedule the two teams to be on the home at opposite times. That makes sense, because that way if you think "I want to see a baseball game today" you've got a good chance.
In fact, his four doubleheaders are Dodgers-Padres (which apparently was a bit of a tight squeeze, since the Dodgers went into extra innings), Yankees-Mets (that one should be easy; every few years the Mets and the Yankees play a game at one park in the afternoon and at the other park the same night; they're doing it today); Phillies-Nationals (which will be tight even if the games go the ordinary length; they start six hours apart, average game length is three hours or so, and the cities are two and a half hours apart with no traffic -- oh, and he's doing it on a Thursday); Cubs-Brewers.
My point is that 25 days might be possible -- but probably not. Most baseball games are scheduled for around 1 PM or around 7 PM, and games last three hours, to see two in one day requires the sites to be no more than three hours apart. The pairs that are doable in one day are probably:
Mets-Yankees, Mets-Phillies, Yankees-Phillies
Phillies-Orioles, Phillies-Nationals, Orioles-Nationals
Dodgers-Padres, Padres-Angels, Angels-Dodgers
White Sox-Cubs, White Sox-Brewers, Cubs-Brewers
Giants-A's
but of course one can do only one from each row, so it's only possible to double up on five days. Basically, this is the problem of looking for the largest matching in the graph that I defined above, where the edges are teams within about three hours' driving distance of each other.
(Oddly enough, each two-team market (and yes, I know, Baltimore and Washington may or may not be the same market) seems to have another team a couple hours away. In two cases that team is the Phillies. As you may know, this blog likes the Phillies.)
So 25 is theoretically possible, if the Scheduling Gods worked in one's favor -- but I'd be scared to even look at the schedules to try and figure it out. And what happens if there's a rainout?
As a problem in actually scheduling things, the other tricky part is that Denver really isn't near any other team. And Robbins' schedule had him at a 7:05 game in San Diego, followed by a 1:05 game in Denver the next day -- but Denver's a time zone to the east of San Diego, so that's seventeen hours between starts. Fourteen hours driving time. For 1,078 miles.
For some other variants of the traveling salesman problem which involve the road network, see Barry Stiefel's 50 states in a week's vacation (driving, with flights to Alaska and Hawaii) and 21 states in one day. The last one cheats a bit -- it's a 26-hour day, since he started in the Eastern time zone during daylight savings time (GMT-4), and did the trip on a day when we went back to standard time (GMT-5) and then crossed into the Central time zone (GMT-6). The difference here is that you only have to enter each state instead of reaching a point.
Oh, and I feel obliged to point out that I find the meme of going on a long road trip this summer because "this is the last summer it'll ever be possible" kind of stupid. (Not that anybody here brought it up.)
Edited (Saturday morning): Google Maps says Cleveland to Detroit can be driven in 2:46. I didn't realize they were that close together. They'd be even closer if someone built a bridge across Lake Erie.
(Saturday afternoon): Cleveland to Pittsburgh in 2:18. I'll admit the reason I forgot this one is that mentally I think of Pittsburgh as being in the same state as me and Cleveland as not being in it, so they must be far apart. This is despite the fact that I live about five miles from New Jersey.
Anyway, you could shave off yet another day by combining the Indians with either the Pirates or the Tigers.
22 May 2008
Phillies radio silliness
There's a commercial which airs on Phillies radio broadcasts. Since I dislike Comcast, but am addicted to the Internet, I'm willing to pay them for high-speed Internet access but not for cable TV. This means I listen to Phillies games on the radio.
Anyway, there's a commercial for Citizens Bank (who own the naming rights to Citizens Bank Park, where the Phillies play) that regularly airs during these broadcasts. It goes something like this (I'm paraphrasing, but the "math" is correct):
The voice then proceeds to tell us how Citizens Bank is open seven days a week. Which I suppose would be nice to know if I didn't do basically all my banking at ATMs anyway.
But I think the commercial is intended to make fun of people who jump through hoops like this to make the numbers work out. And seven is a small number and therefore easy to manufacture by the manipulation of other small numbers, or so I noted on July 7, 2007.
Anyway, there's a commercial for Citizens Bank (who own the naming rights to Citizens Bank Park, where the Phillies play) that regularly airs during these broadcasts. It goes something like this (I'm paraphrasing, but the "math" is correct):
Did you ever notice how important the number seven is to the Phillies? They've won seven National League East division titles. There are seven letters in Ashburn, Schmidt, and Carlton. Thirty-two Hall of Famers played for the Phillies. Divide by the number of runs scored in a grand slam and, yes, you get eight. Subtract the number on Whitey's back and what do you get. Seven!
The voice then proceeds to tell us how Citizens Bank is open seven days a week. Which I suppose would be nice to know if I didn't do basically all my banking at ATMs anyway.
But I think the commercial is intended to make fun of people who jump through hoops like this to make the numbers work out. And seven is a small number and therefore easy to manufacture by the manipulation of other small numbers, or so I noted on July 7, 2007.
30 March 2008
It's almost baseball season again...
A Journey To Baseball's Alternate Universe, in today's New York Times, by Samuel Arbesman and Steven Strogatz.
Arbesman and Strogatz ask the question: how likely is it that some major league baseball player, at some point, would have had a 56-game hitting streak, as Joe DiMaggio did in 1941?
I've seen attempts to determine this before, but they're usually handwaving things that start out by saying "assume everybody bats .266 and gets 3.83 at-bats a game" (actual averages for the 2007 National League), and then let's compute the probability that such a player has a 56-game hitting streak in any given sequence of 56 games. In this case that's easy; the average player gets has a probability (1-.266)3.83 = 0.306 of not getting a hit in any given game, thus a probability 0.694 of getting a hit in any given game; raising this to the 56th power tells you that the average player has a probability of 1.31 in a billion of getting hits in, say, the 56 games starting tomorrow and ending sometime in early June.
So what's the expected number of 56-game hitting streaks this season, according to this model? There are 107 ways any given player could get a 56-game streak -- starting in game 1, 2, ..., 107. So the expected number of 56-game streaks for Joe Qankee (yes, I'm reviving the Qankees) is this probability times 107, or 1.41 × 10-7. Now, assume there are eight Qankees that play every day. (The Qankees are an extraordinarily healthy National League team. The fact that their name rhymes with that of the American League team that DiMaggio actually had his streak with is purely coincidence.) The expected number of 56-game hitting streaks by Qankees this season is thus 1.13 × 10-6. (Note that this is not the probability that one of them has such a streak. A 57-game streak would get counted twice here, a 58-game streak three times, and so on. However, it is an upper bound for the probability of a Qankee having a streak of at least 56 games.)
Now, there have been something less than three thousand "team seasons" in Major League Baseball (one team playing for one season). So the expected number of 56-games streaks is bounded above by (1.13 × 10-6) × 3000 = 0.0338, or about one in 300.
But we've had one. That seems like a lot.
What's the problem here? Well, the average player isn't the one that's going to have that streak. A .280 hitter will put together a streak in 7.40 56-game frames out of every billion. A .300 hitter, in 69 out of every billion. A .320 hitter, in 498 out of every billion. (And I'm still assuming such a player only gets 3.83 at-bats a game; that's probably not true, because the player who hits well will lead his team as a whole to have more at bats.) But an equally bad hitter doesn't drag down the expectation nearly as much. I've ignored batting order (which Arbesman and Strogatz did take into account, implicitly; their inputs for each player are the total number of hits, number of games played, and number of plate appearances, and number of plate appearances varies with position in the batting order).
Rather than making some assumptions on how batting averages are distributed (which would probably be wrong, and even if they were right in the peak of the distribution would still be wrong because what really matters is the tails), I'll defer to Arbesman and Strogatz. Their method is to simulate the entire history of baseball 10,000 times, which is enough to get a nice basically-smooth curve for the distribution of the length of the longest streak. The median length of the longest streak, in their simulations, is 53 games.
Simulation might not be entirely necessary, though. It's routine to calculate the distrbution of the length of the longest streaks in sequences of biased coin flips; aggregating that information together is a little harder. But I don't care enough to do it, so I'll stop here.
Arbesman and Strogatz ask the question: how likely is it that some major league baseball player, at some point, would have had a 56-game hitting streak, as Joe DiMaggio did in 1941?
I've seen attempts to determine this before, but they're usually handwaving things that start out by saying "assume everybody bats .266 and gets 3.83 at-bats a game" (actual averages for the 2007 National League), and then let's compute the probability that such a player has a 56-game hitting streak in any given sequence of 56 games. In this case that's easy; the average player gets has a probability (1-.266)3.83 = 0.306 of not getting a hit in any given game, thus a probability 0.694 of getting a hit in any given game; raising this to the 56th power tells you that the average player has a probability of 1.31 in a billion of getting hits in, say, the 56 games starting tomorrow and ending sometime in early June.
So what's the expected number of 56-game hitting streaks this season, according to this model? There are 107 ways any given player could get a 56-game streak -- starting in game 1, 2, ..., 107. So the expected number of 56-game streaks for Joe Qankee (yes, I'm reviving the Qankees) is this probability times 107, or 1.41 × 10-7. Now, assume there are eight Qankees that play every day. (The Qankees are an extraordinarily healthy National League team. The fact that their name rhymes with that of the American League team that DiMaggio actually had his streak with is purely coincidence.) The expected number of 56-game hitting streaks by Qankees this season is thus 1.13 × 10-6. (Note that this is not the probability that one of them has such a streak. A 57-game streak would get counted twice here, a 58-game streak three times, and so on. However, it is an upper bound for the probability of a Qankee having a streak of at least 56 games.)
Now, there have been something less than three thousand "team seasons" in Major League Baseball (one team playing for one season). So the expected number of 56-games streaks is bounded above by (1.13 × 10-6) × 3000 = 0.0338, or about one in 300.
But we've had one. That seems like a lot.
What's the problem here? Well, the average player isn't the one that's going to have that streak. A .280 hitter will put together a streak in 7.40 56-game frames out of every billion. A .300 hitter, in 69 out of every billion. A .320 hitter, in 498 out of every billion. (And I'm still assuming such a player only gets 3.83 at-bats a game; that's probably not true, because the player who hits well will lead his team as a whole to have more at bats.) But an equally bad hitter doesn't drag down the expectation nearly as much. I've ignored batting order (which Arbesman and Strogatz did take into account, implicitly; their inputs for each player are the total number of hits, number of games played, and number of plate appearances, and number of plate appearances varies with position in the batting order).
Rather than making some assumptions on how batting averages are distributed (which would probably be wrong, and even if they were right in the peak of the distribution would still be wrong because what really matters is the tails), I'll defer to Arbesman and Strogatz. Their method is to simulate the entire history of baseball 10,000 times, which is enough to get a nice basically-smooth curve for the distribution of the length of the longest streak. The median length of the longest streak, in their simulations, is 53 games.
Simulation might not be entirely necessary, though. It's routine to calculate the distrbution of the length of the longest streaks in sequences of biased coin flips; aggregating that information together is a little harder. But I don't care enough to do it, so I'll stop here.
29 January 2008
Prediction markets do measure something!
So I've talked on and off before about prediction markets. One of the questions one wants to ask is -- are they actually measuring something? I asked this last week.
In a comment to that post, John Armstrong has informed me of this post from the Volokh Conspiracy that shows that yes, they are. At least in the case of certain prediction markets dealing with Major League Baseball, that is. The standard contract here pays $10 if a team wins a particular game. A plot is provided which aggregates all the trades for contracts on MLB games in each ten-cent interval.
Now, let's say the Phillies and the Qankees are playing each other. (Longtime readers may recall that the reason for the Qankees' existence is because their name starts with Q, which is the letter after P. I haven't talked about them for a while, because it's not baseball season.) And let's say that I think the Phillies have a probability of 62.5% of beating the Qankees. Then I will be willing to pay up to $6.25 for the aforementioned contract, since that's the expected payout.
Now, what does it mean that this probability is 62.5%? Well, it means that if lots and lots of games like that one were played, then the Phillies would win about 62.5% of them. (The meaning of "lots and lots" and "about" can be made precise via the law of large numbers.) But that particular game will never be played again, so we can't check if my intuition is right. But we can do the next best thing -- look at all the games where people paid $6.25 for a contract for some team to win $10, and ask if that team won 62.5% of the time.
It turns out that, basically, they do. That's the point of the chart over at Volokh, which is due to Michael Abramowicz, author of the book Predictocracy: Market Mechanisms for Public and Private Decision Making. It's nice to see some evidence that at least in the world of sports -- which seems to be a good test bed for a lot of statistical and economic work, because it's possible to collect basically all the relevant data -- these things appear to actually be measuring probabilities.
In a comment to that post, John Armstrong has informed me of this post from the Volokh Conspiracy that shows that yes, they are. At least in the case of certain prediction markets dealing with Major League Baseball, that is. The standard contract here pays $10 if a team wins a particular game. A plot is provided which aggregates all the trades for contracts on MLB games in each ten-cent interval.
Now, let's say the Phillies and the Qankees are playing each other. (Longtime readers may recall that the reason for the Qankees' existence is because their name starts with Q, which is the letter after P. I haven't talked about them for a while, because it's not baseball season.) And let's say that I think the Phillies have a probability of 62.5% of beating the Qankees. Then I will be willing to pay up to $6.25 for the aforementioned contract, since that's the expected payout.
Now, what does it mean that this probability is 62.5%? Well, it means that if lots and lots of games like that one were played, then the Phillies would win about 62.5% of them. (The meaning of "lots and lots" and "about" can be made precise via the law of large numbers.) But that particular game will never be played again, so we can't check if my intuition is right. But we can do the next best thing -- look at all the games where people paid $6.25 for a contract for some team to win $10, and ask if that team won 62.5% of the time.
It turns out that, basically, they do. That's the point of the chart over at Volokh, which is due to Michael Abramowicz, author of the book Predictocracy: Market Mechanisms for Public and Private Decision Making. It's nice to see some evidence that at least in the world of sports -- which seems to be a good test bed for a lot of statistical and economic work, because it's possible to collect basically all the relevant data -- these things appear to actually be measuring probabilities.
14 December 2007
Why mathematics doesn't need a Mitchell Report
Yesterday, the Mitchell Report, on steroid usage in baseball, was released. A large number of players were named as users or potential users.
The media coverage of this has routinely mentioned that amphetamines seem to be more common in baseball than steroids; one source I ran into said it's believed that half of baseball players use amphetamines regularly.
Now, when I think of amphetamines, I think of Paul Erdos, and the following story: Ron Graham bet Erdos that he couldn't quit amphetamines for a month, cold turkey. Erdos did. Graham paid up. Erdos said "you've set mathematics back a month".
As far as I know, we mathematicians don't have a drug problem. (Unless you count coffee. I'll freely admit I have a coffee problem.) But I don't think that anybody would say that Erdos should be stripped of any award he won because he was using drugs. The difference is that in sports, the players are competing against each other; thus a level playing field is essential. But in mathematics, we like to believe that we are not competing against each other but cooperating to discover truth; thus we may gladly accept all the chemical help we can get.
The media coverage of this has routinely mentioned that amphetamines seem to be more common in baseball than steroids; one source I ran into said it's believed that half of baseball players use amphetamines regularly.
Now, when I think of amphetamines, I think of Paul Erdos, and the following story: Ron Graham bet Erdos that he couldn't quit amphetamines for a month, cold turkey. Erdos did. Graham paid up. Erdos said "you've set mathematics back a month".
As far as I know, we mathematicians don't have a drug problem. (Unless you count coffee. I'll freely admit I have a coffee problem.) But I don't think that anybody would say that Erdos should be stripped of any award he won because he was using drugs. The difference is that in sports, the players are competing against each other; thus a level playing field is essential. But in mathematics, we like to believe that we are not competing against each other but cooperating to discover truth; thus we may gladly accept all the chemical help we can get.
11 December 2007
Why don't sports teams use randomization?
Why don't sports teams use randomization?, a guest post by Ian Ayres at Freakonomics. It's well-known that the best strategy in a lot of game-theoretic situations is to choose at random what you'll do at each encounter, where the appropriate probabilities can be calculated. (A sports example would be a pitcher choosing what sort of pitch to make, and a batter deciding whether to swing or not.) So why doesn't anybody (so far as we know) use a random number generator to call pitches?
Although this doesn't seem to be brought up in the comments there (at least not yet), I suspect that the reason for this is that sports people are for the most part resistant to change. But more importantly, they have to answer to the media. And when your closer throws a pitch down the middle of the plate and it gets hit out of the park, and you lose, do you really want to explain to the news people that a random number generator told you to do that? There is no other line of work I can think of where it would be quite so obvious which decision led to the bad outcome; more importantly, there is no other line of work with call-in radio shows devoted to dissecting everything that happens.
Although this doesn't seem to be brought up in the comments there (at least not yet), I suspect that the reason for this is that sports people are for the most part resistant to change. But more importantly, they have to answer to the media. And when your closer throws a pitch down the middle of the plate and it gets hit out of the park, and you lose, do you really want to explain to the news people that a random number generator told you to do that? There is no other line of work I can think of where it would be quite so obvious which decision led to the bad outcome; more importantly, there is no other line of work with call-in radio shows devoted to dissecting everything that happens.
02 November 2007
How long are extra-inning games?
From the Baseball Reference Stat of the Day: the number of Major League Baseball games which went exactly n innings this year are as follows:
(The table in the original post actually gives twice the number of games which have gone at least n innings; I've done the obvious manipulations to get this table.)
If you look at this sequence, one thing that sticks out is that it's roughly geometric. Each number is just about half the number before. What this indicates is that a game has roughly the same probability of ending in any extra inning, namely about one-half. The geometric distribution is "memoryless"; this indicates that at the beginning of the nth inning, for any n ≥ 10, the probability of the game going one more inning is 1/2, two more innings 1/4, three more innings 1/8, and so on. The expected number of additional innings to be played is always the same -- around two.
This isn't all that surprising -- every extra inning "looks the same" when it starts. It wouldn't surprise me to see that things start to change, say, around the fifteenth or sixteenth inning when teams just run out of pitchers -- it wouldn't surprise me to learn that scoring rates go up in, say, the seventeenth inning because whoever is on the mound either is the guy you never let play or the guy who usually pitches an inning or two that's in his fifth inning of work -- but there's not enough data from just a single season to say anything for sure.
But if we look a bit closer at the data, it does look like something like that happens. To be somewhat more precise, there were 220 extra-inning games played this year, and 465 extra innings; thus at the beginning of the tenth, one should expect a game to go 2.11 more innings. There were 123 11-inning-or-more games played this year, and 245 innings numbered 11 or higher; thus at the beginning of the eleventh a game should go 1.99 more innings. Similar numbers for the beginning of the twelfth, ..., seventeenth are 1.88, 1.84, 1.73, 1.83, 1.67, 1.00. The distribution isn't quite geometric, and the expected number of innings remaining in an extra-inning game actually does decrease as time goes on.
(Also, in general the fifth and sixth innings are surprisingly high-scoring. I blame the fact that starters seem to fall apart a lot in the fifth or sixth these days. I'm not saying I could do better, but these guys make a hundred times what I do, so they should be held to higher standards.)
I actually saw this in action on July 25th, when the Phillies played the Nationals and won in fourteen innings, of which I only saw the first nine. My father and I went to the game, and I believe my father had never left a baseball game early in his life. But we left after the ninth, because we had to pick my mother up at the train station. She said that she wouldn't have minded waiting, say, an inning or so. But you never know how long the game is going to last, which is why we left... and if we had stayed around for the rest of the game she would have been waiting at the train station for two hours. You always think the game will end soon. And it usually will. But sometimes it doesn't.
| n | 10 | 11 | 12 | 13 | 14 | 15 | 16 | 17 |
| number of n-inning games | 97 | 58 | 34 | 16 | 9 | 3 | 1 | 2 |
(The table in the original post actually gives twice the number of games which have gone at least n innings; I've done the obvious manipulations to get this table.)
If you look at this sequence, one thing that sticks out is that it's roughly geometric. Each number is just about half the number before. What this indicates is that a game has roughly the same probability of ending in any extra inning, namely about one-half. The geometric distribution is "memoryless"; this indicates that at the beginning of the nth inning, for any n ≥ 10, the probability of the game going one more inning is 1/2, two more innings 1/4, three more innings 1/8, and so on. The expected number of additional innings to be played is always the same -- around two.
This isn't all that surprising -- every extra inning "looks the same" when it starts. It wouldn't surprise me to see that things start to change, say, around the fifteenth or sixteenth inning when teams just run out of pitchers -- it wouldn't surprise me to learn that scoring rates go up in, say, the seventeenth inning because whoever is on the mound either is the guy you never let play or the guy who usually pitches an inning or two that's in his fifth inning of work -- but there's not enough data from just a single season to say anything for sure.
But if we look a bit closer at the data, it does look like something like that happens. To be somewhat more precise, there were 220 extra-inning games played this year, and 465 extra innings; thus at the beginning of the tenth, one should expect a game to go 2.11 more innings. There were 123 11-inning-or-more games played this year, and 245 innings numbered 11 or higher; thus at the beginning of the eleventh a game should go 1.99 more innings. Similar numbers for the beginning of the twelfth, ..., seventeenth are 1.88, 1.84, 1.73, 1.83, 1.67, 1.00. The distribution isn't quite geometric, and the expected number of innings remaining in an extra-inning game actually does decrease as time goes on.
(Also, in general the fifth and sixth innings are surprisingly high-scoring. I blame the fact that starters seem to fall apart a lot in the fifth or sixth these days. I'm not saying I could do better, but these guys make a hundred times what I do, so they should be held to higher standards.)
I actually saw this in action on July 25th, when the Phillies played the Nationals and won in fourteen innings, of which I only saw the first nine. My father and I went to the game, and I believe my father had never left a baseball game early in his life. But we left after the ninth, because we had to pick my mother up at the train station. She said that she wouldn't have minded waiting, say, an inning or so. But you never know how long the game is going to last, which is why we left... and if we had stayed around for the rest of the game she would have been waiting at the train station for two hours. You always think the game will end soon. And it usually will. But sometimes it doesn't.
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