This semester I attended my first county GOP convention. The convention provided a close view of grass roots politics. The goals of the convention in a non-election year were to choose the county GOP leadership for the next two years and to update the rules governing the county party organization.

In our county, the Republican party is a small minority, and it can be difficult to convince people that local participation is worthwhile. About 50% of the eligible delegates from my precinct attended the convention, and that was considered an exceptionally high turn out. When trying to encourage my neighbors to attend, I encountered many excuses for not attending. The prize for the most annoying response goes to those people who told me that they did not want to participate because the people who did take the time to participate were moving the party in directions antithetical to the nonparticipants. This response was particularly nettlesome this year because our leadership race pitted our politically inoffensive current chairman against a newcomer with a record of attacking opponents on religious grounds. Note that I am not complaining that he attacked policies on religious grounds but that he impugned the faith of the proponents of liberal policies. In fact, on one occasion he even made public remarks that appeared to me and my neighbors to attack members of other religious groups regardless of their policy positions. My neighbors most exercised by such religious attacks did not attend the convention. The newcomer won.

Although the focus of the convention was the election of new leadership, the bulk of the time was devoted to changing the rules governing the party. There were no significant areas of contention, but a room full of people in the grip of Robert's Rules of Order can have trouble reaching a common goal. In addition, an occasional freelancer would offer an ill considered amendment that would send the group into strange byways that strongly testified against the reputed wisdom of crowds.

As the convention dragged into the afternoon, the assembled became hungry and restless. A GOP auxillary group offered water and snacks for sale. Unfortunately, election law has become so ridiculous that the purchase of a donut is regarded as a political contribution. In order to avoid corrupting our political process by hordes of conventioneers receiving lucrative government contracts in return for a cruller, the purchase of a donut or bottle of water required the buyer to fill out forms providing employer, address, phone number, and endless other nonsense.

The county convention was definitely messier than the state convention I attended last summer. There were fewer experts to guide us along. Nonetheless, despite the crudeness of the process, I admire those who are willing to give up a beautiful weekend morning in order to make political sausage, and maybe, if they are lucky, leverage a box of glazed donuts into ill gotten government largesse.

## Sunday, March 27, 2011

## Sunday, March 13, 2011

### Love of Linearity

A function F is said to be linear if it satisfies

(i) F(x+y) = F(x) + F(y), and

(ii) F(sx) = sF(x).

Heuristically, a linear function satisfies a generalized distributive property.

A function G(x) is called affine if G(x)=F(x)+c, where F is linear and c is a constant. Mathematicians and the general public often apply the term linear to any affine function. History is probably on the side of this broader usage, as the graph of an affine function (of one variable) is a line.

Mathematicians love linear functions because they are so easy to analyze. An entire semester college course is typically devoted to their study. Moreover, the object of a differential calculus course is to study the approximation of functions by linear functions. Crudely, a function is differentiable, if in sufficiently small regions, it is well approximated by affine functions (the requisite affine function depending on the small region in question).

We spend several years in elementary school teaching children that multiplication satisfies properties (i) and (ii). We then spend several years in high school (occasionally spilling over into college) convincing students that most functions are not linear. Our success in this latter effort is clearly limited. For example, we often see data analysis accompanied by a computation of the affine function which best approximates the data, even when there is no theoretical or experimental reason to suspect the underlying phenomenon is described by a linear function. In fact, in most natural systems, linearity seems highly unlikely. Remember, the graph of a one dimensional affine function is a line marching onward in perpetuity. If you do not think that a stock price or a population of herring is likely to increase (or decrease) at a constant rate forever, then a linear model is clearly ruled out. Nonetheless, our affinity for linearity is extremely strong, as demonstrated by our tendency to see straight rows in scatter planted corn fields.

I was drawn to think about our bias toward linearity, by a WSJ article,

Balanced Budget vs. the Brain. The focus of the article is the putative irrationality of the average economic actor, who fears risk more than he values gain. The author cites psychologists Daniel Kahneman and Amos Tversky who asked students to bet on coin flips.

"If the coin landed on heads, the students had to pay the professors $20. Messrs. Kahneman and Tversky wanted to know how big a potential payoff their students would demand for exposing themselves to this risk. Would they accept a $21 payoff for tails? What about $30?

If the students were rational agents, they would have accepted any payoff larger than the potential $20 cost. But that's not what happened. Instead, the psychologists found that, when people were asked to risk $20 on the toss of a coin, they demanded a possible payoff of nearly $40. "

Question: why is it rational to value a $20 dollar gain to be worth the risk of a $20 dollar loss? If you own one house (and do not have a large bank account), is it a rational gamble to risk it in an even bet in exchange for an additional house? Are we irrational if we do not view the prospect of owning two houses to be worth the risk of becoming homeless? It is easy to construct numerous examples of this kind. I think the correct interpretation of the study is the pair of observations :

(1) the value we place on assets is not a linear function of their dollar value, and

(2) Jonah Lehrer, the author of the WSJ piece, has an irrational expectation that linearity is ubiquitous in our highly nonlinear world.

I bet (but not $20) that if the students were risking a 20 cent loss, their acceptable payoff would be closer to 20 cents. This is an expression of my irrational expectation that differentiability can frequently be found in our highly nonsmooth world.

As the French are wont to say, "Vive la nonlinearity."

(i) F(x+y) = F(x) + F(y), and

(ii) F(sx) = sF(x).

Heuristically, a linear function satisfies a generalized distributive property.

A function G(x) is called affine if G(x)=F(x)+c, where F is linear and c is a constant. Mathematicians and the general public often apply the term linear to any affine function. History is probably on the side of this broader usage, as the graph of an affine function (of one variable) is a line.

Mathematicians love linear functions because they are so easy to analyze. An entire semester college course is typically devoted to their study. Moreover, the object of a differential calculus course is to study the approximation of functions by linear functions. Crudely, a function is differentiable, if in sufficiently small regions, it is well approximated by affine functions (the requisite affine function depending on the small region in question).

We spend several years in elementary school teaching children that multiplication satisfies properties (i) and (ii). We then spend several years in high school (occasionally spilling over into college) convincing students that most functions are not linear. Our success in this latter effort is clearly limited. For example, we often see data analysis accompanied by a computation of the affine function which best approximates the data, even when there is no theoretical or experimental reason to suspect the underlying phenomenon is described by a linear function. In fact, in most natural systems, linearity seems highly unlikely. Remember, the graph of a one dimensional affine function is a line marching onward in perpetuity. If you do not think that a stock price or a population of herring is likely to increase (or decrease) at a constant rate forever, then a linear model is clearly ruled out. Nonetheless, our affinity for linearity is extremely strong, as demonstrated by our tendency to see straight rows in scatter planted corn fields.

I was drawn to think about our bias toward linearity, by a WSJ article,

Balanced Budget vs. the Brain. The focus of the article is the putative irrationality of the average economic actor, who fears risk more than he values gain. The author cites psychologists Daniel Kahneman and Amos Tversky who asked students to bet on coin flips.

"If the coin landed on heads, the students had to pay the professors $20. Messrs. Kahneman and Tversky wanted to know how big a potential payoff their students would demand for exposing themselves to this risk. Would they accept a $21 payoff for tails? What about $30?

If the students were rational agents, they would have accepted any payoff larger than the potential $20 cost. But that's not what happened. Instead, the psychologists found that, when people were asked to risk $20 on the toss of a coin, they demanded a possible payoff of nearly $40. "

Question: why is it rational to value a $20 dollar gain to be worth the risk of a $20 dollar loss? If you own one house (and do not have a large bank account), is it a rational gamble to risk it in an even bet in exchange for an additional house? Are we irrational if we do not view the prospect of owning two houses to be worth the risk of becoming homeless? It is easy to construct numerous examples of this kind. I think the correct interpretation of the study is the pair of observations :

(1) the value we place on assets is not a linear function of their dollar value, and

(2) Jonah Lehrer, the author of the WSJ piece, has an irrational expectation that linearity is ubiquitous in our highly nonlinear world.

I bet (but not $20) that if the students were risking a 20 cent loss, their acceptable payoff would be closer to 20 cents. This is an expression of my irrational expectation that differentiability can frequently be found in our highly nonsmooth world.

As the French are wont to say, "Vive la nonlinearity."

## Sunday, March 6, 2011

### Study Abroad: A Free Lunch for American Universities?

I have often wondered why universities promote their study abroad programs so heavily. Brochures proclaim:

"Come to Wonder University, where we allow you to study somewhere else!"

At first I thought this was simply a marketing ploy. Prospective students learn that if they matriculate at Wonder University, the University will help them to convince their parents that funding a semester long vacation in Europe will further their education. Wonder University becomes more attractive to students desiring a long European vacation, and applications rise. Wonder's competitors then have to promote study abroad so that they don't lose the best tourists to Wonder.

I have, however, heard numerous stories of colleagues fighting with students and the administration over overseas mathematics courses. My colleagues assert that most mathematics courses available to our students in study abroad programs are so much weaker than our own courses that they cannot be allowed to count towards our major. Again and again students are devastated to hear that their "grand tour" will actually slow their progress toward a degree. The more such stories I hear, the less I believe my marketing explanation. Although there often seems to be a chasm between the concerns of the faculty and the attitudes of the student life administration, perhaps something other than marketing is at play here. When trying to understand strange behavior, a common dictum is to follow the money trail. What financial incentives do colleges have to push study abroad programs?

The Wall Street Journal recently published an article on the growing numbers of Americans eschewing American universities, enrolling in European universities instead. The driver: money. Tuition and fees at Oxford are around $5500 for Brits and around $20,500 for Americans. St. Andrews charges approximately $3000 for Brits and $20,500 for Americans. The University of Chicago and Stanford University, for example, both charge around $40,000. The difference is significant to most of us. Given this large financial gap, how much money do students save when they study abroad? At Stanford, the tuition for a year's study in California is $40,500. Their study abroad program charges $40,500. Chicago also charges its students the same tuition to study in balmy Chicago or in Scotland. Who pockets the $20,000 difference? Perhaps the American universities and their overseas partners split the difference. It would be amusing to know the details. Clearly there is enough money to be made to make everyone happy. Even the students are not losing financially, unless their foreign adventure forces them to pay for an extra semester of tuition in the U.S. Perhaps there is an occasional free lunch after all.

"Come to Wonder University, where we allow you to study somewhere else!"

At first I thought this was simply a marketing ploy. Prospective students learn that if they matriculate at Wonder University, the University will help them to convince their parents that funding a semester long vacation in Europe will further their education. Wonder University becomes more attractive to students desiring a long European vacation, and applications rise. Wonder's competitors then have to promote study abroad so that they don't lose the best tourists to Wonder.

I have, however, heard numerous stories of colleagues fighting with students and the administration over overseas mathematics courses. My colleagues assert that most mathematics courses available to our students in study abroad programs are so much weaker than our own courses that they cannot be allowed to count towards our major. Again and again students are devastated to hear that their "grand tour" will actually slow their progress toward a degree. The more such stories I hear, the less I believe my marketing explanation. Although there often seems to be a chasm between the concerns of the faculty and the attitudes of the student life administration, perhaps something other than marketing is at play here. When trying to understand strange behavior, a common dictum is to follow the money trail. What financial incentives do colleges have to push study abroad programs?

The Wall Street Journal recently published an article on the growing numbers of Americans eschewing American universities, enrolling in European universities instead. The driver: money. Tuition and fees at Oxford are around $5500 for Brits and around $20,500 for Americans. St. Andrews charges approximately $3000 for Brits and $20,500 for Americans. The University of Chicago and Stanford University, for example, both charge around $40,000. The difference is significant to most of us. Given this large financial gap, how much money do students save when they study abroad? At Stanford, the tuition for a year's study in California is $40,500. Their study abroad program charges $40,500. Chicago also charges its students the same tuition to study in balmy Chicago or in Scotland. Who pockets the $20,000 difference? Perhaps the American universities and their overseas partners split the difference. It would be amusing to know the details. Clearly there is enough money to be made to make everyone happy. Even the students are not losing financially, unless their foreign adventure forces them to pay for an extra semester of tuition in the U.S. Perhaps there is an occasional free lunch after all.

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