While on vacation, I passed some of my time looking for math on the beach, and I saw a sine wave creeping up the shore.
I thought I might be looking too hard for something mathematical, thereby seeing things that weren’t really there. Thankfully, Geogebra is well-equipped to justify my observations.
Or perhaps it’s more accurate to say that Geogebra helps enable my obsessions.
This is a nice infographic from the NYT highlighting some of the data from the 2010 U.S. Census:
http://www.nytimes.com/interactive/2011/01/07/us/CENSUS.html
Some of the highlights
The accompanying article from the Times can be found here, and the U.S. Census Bureau’s Statistical Abstract of the United States, where this summary information is drawn from, can be found here.
At the beginning of the 2011 playoffs, the NFL implemented a new set of overtime rules that drastically altered how tie games would be decided. The rules entail many significant strategic consequences, and I’m not sure how well-understood those consequences are.
The old overtime rule is straightforward: initial possession of the ball in overtime is determined by a coin-toss, and the first team to score is the winner. Critics of this model argued that since initial possession of the ball was such an advantage, awarding it based on a coin-toss essentially made the outcome of the game equivalent to a coin-toss. The argument seems reasonable, although whether the data backs it up is a story for another time.
The NFL’s new system tries to counterbalance the advantage of initial possession. From here on out, let’s call the team that wins the coin-toss, and therefore begins overtime with the ball, Team A. In the new system, Team A can win the game on its initial possession only if they score a touchdown. If Team A scores a field goal on their initial possession, Team B will receive the ball and have a possession to then try to win or tie the game.
Here is a diagram illustrating the old system.
The basic idea here is if Team A scores, they win. If they don’t, then Team B gets the football and starts at the front of the graph.
Here’s a graph representing the new system.
This system is far more complex. First of all, Team A now has a great incentive to try for a touchdown on its first possession, whereas under the old system, they had virtually no incentive to do so. Furthermore, the value of a field goal for Team A on its initial possession has been greatly decreased. Under the old system, a first-possession field goal won the game. Under the new system, its value is considerably less.
What are the consequences of these new realities?
Consider a long field goal on the initial possession. Make it, and your opponent gets the ball and can beat you with a touchdown or tie you with a field goal. Miss it, and your opponent will have good field position and now can beat you with a field goal. Perhaps punting is the better option: you gain field position and if you defend successfully, you get the ball back and now all you need is a field goal to win.
Even more interesting is a short field goal attempt on the initial possession. Make it, and your opponent still gets field position and a chance to beat or tie you. Why not go for a touchdown? You win outright if you succeed, and if you don’t, not only does your opponent get poor field position, but if you defend successfully you’ll only need a field goal to win.
Because of the altered values of touchdowns and field goals mentioned above, I think 4th down strategies on the initial possession should be modified as well. As complex as it is, at least the new system reverts to the old ruleset after (at most) two possessions.
As interesting as the many consequences of these rule changes are, I still think the “first to six” idea makes more sense. We’ll see what happens this weekend!
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This article in the New York Times discusses a controversy surrounding a recent ranking of colleges that put Egypt’s University of Alexandria among the top 200 universities in the world. An informed observer describes the school as “not even the best university in Alexandria“.
http://www.nytimes.com/2010/11/15/education/15iht-educLede15.html
How did this school achieve such a high ranking? In a subcategory measuring the impact of research, which counted for approximately 33% of the school’s overall score, the University of Alexandria placed fourth in the world, ahead of Harvard and Stanford. Sound fishy? Seems as though most of the research citations came from one professor who published 320 articles in a journal he, himself, was in charge of.
Trying to come up with quantitative measures for colleges (or high schools, or teachers) is tricky business. Not only is it hard to agree on what to measure, but it’s tough to figure out how to measure it.
And once the rating culture sets in, gaming of the system, as seen in this particular case, will inevitably follow. “Tell me how I will be measured, and I will tell you how I will behave“. I’m not sure who said this originally, but an engineer friend shared it with me many years ago, and it always comes to mind in these situations.
Hopefully in the future, more schools will follow the example of Reed College and refuse to participate in these rankings.
The following article from the New York Times discusses the recent fare increase for public transit in NYC:
http://cityroom.blogs.nytimes.com/2010/10/07/m-t-a-meets-to-increase-transit-fares/
The base fare for a ride on a NYC subway or bus is now $2.25, and price changes have been made both to the monthly unlimited-ride cards and the bulk-purchase bonus rate.
For those of us who live in New York and frequently use public transit, we are faced with the following optimization problem: is it better to buy the monthly unlimited-use card for $104, or purchase some amount of “single-rides” in bulk and receive a 7% bonus?
The above Times article discusses some of the strategies and factors to consider in making this decision. In addition, it points the reader to the website www.metrocardbonuscalculator.com that allows the user to experiment with various purchasing scenarios.
Fare hikes have become more commonplace over the past few years, but reliable public transit is still a bargain. And I’d pay an extra few cents for some additional cleaning crews.