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The Strike Zone as a Probability Density Function

2014 July 8
by Phil Miller

The strike zone in baseball is supposed to be the area above the plate and between the batter’s armpits and knees.  Seems simple enough, but  John Palmer suggests that it is more complicated than that.

Several days go, while watching some very erratic home-plate umpiring, I posted a short item to Facebook quoting myself, “The strike zone is a probability density function.” I had originally made the statement during a play-by-play radio broadcast of a London Tigers baseball game over 20 years ago. The Tigers were a AA minor league team. The radio station manager asked me not to do that again and not to try to explain it on air.

This is an interesting way to think about the strike zone.  What John is implying is that the probability of a strike being called within the strike zone is determined by where it is in the strike zone.  In particular, John suggests that the closer the pitch is to the center of the zone, the more likely it will be called a strike.  He links to this post at FiveThirtyEightSports written by Etan Green that supports John’s claim.  The following graph from the article illustrates John’s belief about the strike zone.


Over at Facebook, King Banaian commented

(T)here is not just one strike zone. The dimensions are not just spatial but umpire specific. So even your three dimensional depiction of the PDF isn’t quite right, and I question whether summing up along the umpire dimension gives us anything really meaningful about the strike zone.”

The strike zone may also be situation-dependent.  For example, an umpire may slightly expand his strike zone if he realizes that he called an earlier ptch a ball when he should have called a strike.

Lastly, in watching my sons play baseball over the years, there is definitely a difference in strike zones between youth baseball and pro baseball.  In youth ball, the strike zone tends to move around more.  In the pros, the strike zone seems more consistent to my untrained eye.

Superb World Cup ‘Selection Bias’ Example

2014 July 4
by Liam Lenten

For all those economics professors and tutors out there who struggle to explain the crucial concept of ‘selection bias’, a nice illustration can be found in FIFA World Cup finals records. With students (at least those who do not loathe sport) currently in soccer-crazy mode, they may be more motivated to understand this concept through the following trivia question:

Q: In World Cup (finals) history, which team has the highest goal-scoring ratio (goals scored divided by games played)?

Scroll below for the answer, which may be surprising to many, except the amateur World Cup historians among you.

Most people would instinctively say Brazil; however, they appear second on this list at 2.16 per game (218 from 101 games, inclusive of the second-round of the 2014 edition). Germany follows at a very-close third with 2.15 (221 from 103).

The record-holders are…wait for it…Hungary! Yes, those ‘Mighty Magyars’ top the list and (get this) by a comfortable margin, too – indeed a chasm – their 87 goals in 32 games comes in at an astonishing 2.72 goals per game.

If you don’t believe me (and you’re more than entitled not to), check the figures here. Hungary has never won the World Cup, but have twice reached the final: in 1938, when they lost to Italy; and again in 1954, with legends Puskás and Kocsis (et al.) in their ‘Golden Team’, which came into that World Cup undefeated in more than 4 years, only to squander a two-goal lead (which they had after only 8′) to (the then-West) Germany, who incidentally they had annihilated in the first round by the incredible scoreline of 8-3.

OK, so what is the selection bias here? Well, look at the chart below (from statista), which displays average goals per game by World Cup. The flags I added at the top of the bars denote the World Cup finals that Hungary both entered and qualified for.

goals by world cup

From this, it is easy to see that scoring outcomes were lower from 1962 compared to earlier, with a further decline (albeit slight) since then. Hungary is but one of a number of national football teams that were among the best handful in the World for considerable periods at any time since the inaugural World Cup in 1930 (according to retrospective Elo ratings, they were ranked number one as late as 1965). However, of all national teams in this category, Hungary is the one that played the highest proportion of its matches in higher-scoring World Cups.

For all you Magyars out there lamenting your boys’ extended absence from the big stage (28 years now and counting), rest assured that (since it’s unlikely that Brazil and Germany will ever get anywhere near 2.72) the only way to guarantee holding this highly-prestigious record in perpetuity is to continue to NOT qualify for the finals – proof that there is indeed success in failure!

Please note: all future TSE contributions from Liam Lenten will also be cross-posted on his new personal blog, see:

Introduction to Neymar

2014 May 31
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by Skip Sauer

This week’s WSJ Magazine has an entertaining story by Jason Gay on “Neymar da Silva Santos Júnior, the 22-year-old prince of Brazilian soccer.” He’ll be bigger than Beckham if Brazil win the forthcoming World Cup (current odds: 3-1).

Gay’s story is full of good lines. My favorite is on the scale of the stage for Neymar and company: “The last World Cup final, in 2010 between Spain and the Netherlands, was watched by an estimated 700 million people, which makes the Super Bowl seem like open-mike night at a coffee shop.” On Neymar’s fame, Gay relays Roger Bennett’s observation that Neymar’s jaw-dropping skill set made him into “‘a footballer for the YouTube age’ … Bennett believes that his worldwide reputation grew principally through short video clips, perfect for dissemination on social media.” That’s an interesting point. Here’s a startling fact: Nike signed Neymar at the age of 15 (he’s now 22). They were on to him early.

One thing you won’t find much of in Gay’s story is a discussion of Neymar’s stats, but these are easily found at ESPNFC. This year he’s been scoring at a rate of one goal every two games, with two assists every five, all while undergoing the transition into an established Barcelona team. His strike rate for Brazil has been a degree or two higher. Gay’s story paints a picture of an athlete who won’t shrink from the spotlight of the World Cup. He’s clearly capable of the spectacular, so I expect we’ll see a “Nike moment” or two in the next month of games.

Two Weeks Out From the World Cup, a Familiar Picture Emerges

2014 May 28
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by Skip Sauer

The opening match between Brazil and Croatia will kick off two weeks from tomorrow. But there seems less anticipation leading up to the Cup on whether Brazil will reign supreme on its home soil, or on the fitness level of stars like Messi and Neymar. Rather, the attention in the business press is on Brazil’s investment in stadiums (12!) and transportation infrastructure. In years past, the sales pitch that such investment would spur economic development was repeated ad infinitum in the press. Events finally seem to have caught up with the fourth estate on this topic, as most stories now point to the follies and failures of mega-event motivated investment.

Today brings two informative (and ungated) stories from the Wall Street Journal, “Hope Fades … for an Economic Boost,” and “A Dozen Stadiums, a Million Problems.” Reading these, it seems clear that Brazil’s $11.5 billion investment in the World Cup was pitched to the public as being “transformative” (once again), with the follow-on economic boost from new facilities justifying the massive expenditures. This tune has been played many times and has been chronicled here at TSE since 2004. The evidence on the ground, then as now, is much the same: once the show leaves town, the super-sized facilities stand as symbols of political ambition, inflated expectations, waste, corruption, and ultimately as organizational failure at the highest level of international sport. Last week, HBO’s Real Sports with Bryant Gumbel had a piece on the “Olympic White Elephants” that dot the landscape of Athens, 10 years after the 2004 Games. It’s “must see tv” and quite sad. Greece would have been so much better off had these investments been made in sustainable enterprises.

In Minnesota, all the stadiums are above average

2014 May 20
by Victor Matheson

Minneapolis was awarded the 2018 Super Bowl today beating out competing bids from Indianapolis and New Orleans. The conventional wisdom seems to be that this is a surprising move on the part of the NFL and that Minneapolis was “the supposed underdog city with the frigid temperatures in the winter.” The New Orleans Times-Picayune called it “a stunning upset” likening it to Joe Namath’s Jets back in Super Bowl III.

At least one economist, however, nailed Minneapolis‘ selection. Choosing Minneapolis as the odds-on favorite really wasn’t that hard when one looks at the history of the game. First of all, despite New Orleans’ claim that Minneapolis is “a relative neophyte in the big-event hosting game,” the Twin Cities are hardly amateurs at hosting big events. Minneapolis hosted the Super Bowl back in 1992. In fact, I ate lunch right next John Madden at Minneapolis’ finest Chinese restaurant, the Village Wok, while in grad school in the city at the time. MLB is sending the All-Star Game to Target Field this July, and the city has hosted the World Series and the NCAA Final Four on multiple occasions. St. Paul hosted the 2004 Republican National Convention, a much bigger organizational and security undertaking than a simple football game. And let’s also just remember that experience isn’t everything. New Orleans, despite hosting more Super Bowls than any other city, couldn’t even manage to keep the lights on during their last attempt at hosting.

More importantly, however, is the fact that the NFL wants to reward cities that build new stadiums, especially those that shower their franchises with lots of taxpayer subsidies. The NFL constantly dangles the carrot of a Super Bowl in front of otherwise reluctant taxpayers in order to receive public handouts. Put  in $498 million, like the citizens of Minnesota did, and the NFL will send the Super Bowl and its supposed $498 million in economic impact  your way. It’s almost like getting a stadium for free. Of course, the bribe only works if the NFL is actually seen coming through with the big game.

Indeed the NFL has come through. Of the 16 Super Bowls hosted between 2001 and 2016, over half were held at newly constructed stadiums hosting Super Bowls for the first time including games in Tampa, Glendale, AZ, Detroit, Indy, New York/New Jersey, Jacksonville, Santa Clara, CA, Arlington, TX, and Houston. And now add Minneapolis to list.

Minneapolis will be a great host, but if you are planning on going to the game in 2018, I recommend you bring a really good coat and that you keep warm with some hot and sour soup at the Village Wok.

Team Valuations – Milwaukee/Alabama edition

2014 April 17
by Victor Matheson

While the Philadelphia 76ers got all of the press this year with their record-tying 26-game losing streak, the Milwaukee Bucks quietly ended up blowing out the competition for worst team in the NBA finishing 15-67, fully 4 games worse than the hapless 76ers.

So what kind of reward does owner Herb Kohl get for guiding his team to NBA ignominy? Try $550 million, the price tag announced today as Kohl sold the team to New York investment firm executives Marc Lasry and Wesley Edens.

In other news, it was widely reported last week that in the latest government data on college athletic programs, the University of Alabama athletic department generated more revenue than every NHL team and 26 of 30 NBA teams.

So, here is an interesting thought experiment:  if the University of Alabama football team, which finished with only 4 fewer wins than the Bucks despite playing 68 fewer games, were a regular sports franchise, what would it sell for?

Of the $143 million in revenues generated by the athletic department, the football team is directly responsible for $88.7 million. The Bucks just sold for 5.05 times their annual revenue. Using this multiplier would put a value of roughly $450 million on the Crimson Tide football team. Revenue to market value numbers for other teams in the the NBA and NFL run between 3.2 and 5.2.

Of course, Alabama has another thing going for it as well. While Kohl had to pay his players about $54 million this season to generate those 15 wins and their corresponding $109 million in revenue, the Crimson Tide only incurred $2.2 million in player payroll expenses. In total, the school spent only $41.6 million on football, generating a profit of $47.1 million. On average, NBA and NFL teams at valued at 26.6 times their annual profits in the Forbes rankings. Using this multiplier, the Alabama football team would go for $1.253 billion, making it the 13th most valuable franchise in the US. By contrast, the endowment of the entire University of Alabama system in 2013 was $1.055 billion.

(Thanks to Josh Congdon-Hohman for the idea.)

The Economics of a Shot Clock

2014 March 17
by Phil Miller

A recent high school playoff basketball game has kindled a shot clock debate in Minnesota.  Shakopee and Hopkins, both part of the Minneapolis-St. Paul metropolitan area, were playing a semifinal game last week.  The game went to quadruple overtime and Hopkins won the game with a last-second prayer shot by Amir Coffey, a 60-foot heave that got nothing but net.  Here are the details.  Here is a decent video of the shot.

Here in Minnesota, high school basketball does not use a shot clock.  So part of the Hopkins strategy was to get the ball and stall to give it the final shot.  One of the reasons for having a shot clock is to keep teams from stalling and, thereby, to increase the fan interest in the sport.

One of the reasons why high schools up here don’t use the shot clock is because of the expense.  Here is some information from Reg Chapman at WCCO.

Newer baskets with shot clocks can cost upwards of $5,000, and that’s a lot for smaller boosters.

“There’s a cost of putting it in, there’s a cost of having an operator and there’s a lot of little rules that go along with that,” Merkle said. “The operator has to know what they’re doing. It brings in a whole other level of officials.”

It makes no sense to install a sh0t clock if the marginal revenue from doing so won’t cover the marginal cost.  Chapman notes that some of the bigger Minnesota high schools have installed shot clocks at their courts and have used them on an experimental basis.   But the smaller schools in lower divisions find the cost too prohibitive.  These , schools sometimes don’t even have enough players to fill the rosters(as Chapman notes) let alone buy expensive equipment.

Like it has done with so many other things, perhaps technology will improve enough to where the marginal cost of installing and operating  a shot clock will stop being cost prohibitive.  Until then in Minnesota at least, one can get a feeling for what basketball was like before the shot clock by going to a high school basketball game.

CFP NAASE Sessions at the Southern Economic Association Conference

2014 February 24
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by Phil Miller

Brad Humphreys sent along a call for papers for North America Association of Sports Economists sessions at this year’s SEA meetings in Atlanta Nov. 22-24.  Note that a session on the teaching of sports econ is of particular interest.  Here’s the CFP

The North American Association of Sports Economists (NAASE) is again organizing sports economics sessions at the Southern Economic Association (SEA) conference.  The conference will be held November 22-24, 2014 in Atlanta, Georgia.  The basic format of the NAASE sessions will be the same as in the past.  I hope to have several complete sessions to take to the SEA organizers to assure us of a good room and good session times.

*Deadline for submissions is April 1.*

We are trying to put together a session on teaching sports economics.  Anyone with a paper idea that would fit a session on teaching sports economics is especially encouraged to submit their paper.

If you have a sports economics paper you would like to present in Atlanta, please email Dennis Coates (see below for contact information) a title and full contact information including post and email addresses. Your submission must contain full contact information for all co-authors.

Here is the contact information for the session organizer:

Dennis Coates

Southern Finance Association CFP

2014 February 21
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by Phil Miller

Jason Berkowitz, Assistant Professor of Finance at St. John’s, is putting together a proposal for a session or two at this year’s Southern Finance Association Meeting in Key West, Fl (a location that, to this northerner enduring another honest to goodness blizzard, sounds really, really appealing).  Here’s the skinny on what Jason is looking for.

Hello everyone, I am reaching out to you all as I would like to organize a sports betting session (possible two) at the Southern Finance Association Annual Meeting in Key West, Florida from November 19-22.  If you have a paper you would like to present please e-mail me your abstract or completed paper by February 25th <> so I can have a few days to put the strongest possible proposal together.

Call for Sports Econ Papers – 2014 Southern Economic Association Annual Conference

2014 February 4
by Phil Miller

Dennis Coates of the University of Maryland – Baltimore County is organizing sports economics sessions for the North American Association of Sports Economists (NAASE) to be held at this years Southern Economics Association (SEA) conference in Atlanta.  Here is the CFP:

Call for Papers – NAASE Sports Economics Sessions
84th Annual SEA Conference
Atlanta, Georgia

The North American Association of Sports Economists (NAASE) is again
organizing sports economics sessions at the Southern Economic
Association (SEA) conference.  The conference will be held November
22-24, 2014 in Atlanta, Georgia.  The basic format of the NAASE
sessions will be the same as in the past.  I hope to have several
complete sessions to take to the SEA organizers to assure us of a good
room and good session times.

If you have a sports economics paper you would like to present in
Atlanta, please email Dennis Coates (see below for contact
information) a title and full contact information including post and
email addresses. Your submission must contain full contact information
for all co-authors.

Here is the contact information for the session organizer:

Dennis Coates

Here is the NAASE website for anyone interested in joining this association of sports economists.  Here is information about the SEA’s 2014 conference.