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Tuesday, October 11, 2005

More things you can do with a math degree

Of course, the top prize for mathematics is the Field's Medal. But, you could also win a Nobel prize in another field, as John Nash (the subject of "A Beautiful Mind") did.

Or more recently:
Game theory economists share Nobel prize Israeli-American and American win for work on political, economic conflict

The Associated PressUpdated: 2:49 p.m. ET Oct. 10, 2005
STOCKHOLM, Sweden - A pair of game theorists who defined chess-like strategies in politics and business that can be applied to arms races, price wars and actual warfare won the Nobel Prize in Economic Sciences on Monday.
Israeli-American Robert J. Aumann and U.S. citizen Thomas C. Schelling won the award for research on game theory, a branch of applied mathematics that uses models to study interactions between countries, businesses or people.
The theory, devised in 1944 by John von Neumann and Oskar Morgenstern, is often used in a political or military context to explain conflicts between countries. More recently it has been used to map trends in the business world, ranging from how cartels set prices to how companies can better sell their goods and services in new markets.
“The understanding of game theory helps explain economic conflicts like price competition and trade wars,” said Jorgen Weibull, chairman of the prize committee. “I think the main impact is on economics, but it also applies to other social sciences.”
It said the pair’s work, which built on research by the 1994 winners of the same prize, could be applied to understand how merchant guilds, international trade treaties and even organized crime groups are formed and operate.[...]
Schelling, who teaches at the University of Maryland, used game theory in his 1960 book “The Strategy of Conflict” to focus on how the U.S. and the former Soviet Union maintained credible threats that were not likely to be used, given the threat of nuclear annihilation.
“If you have second-strike capacity, then it makes your opponent think twice,” said Carl-Gustaf Lofgren, a member of the prize committee.
In an interview with The Associated Press, Schelling said: “I use game theory to help myself understand conflict situations and opportunities.”
He said the prize committee linked the two laureates on the virtue of their respective research.
“They linked us together because he is a producer of game theory and I am a user of game theory,” said Schelling, who worked with the U.S. Marshall plan to rebuild Europe after World War II.
In the economic world, Aumann (pronounced OW-man) and Schelling’s work is used to prevent illegal cartels between rival companies, Lofgren said.
Two competitors can use game theory to agree on joint price structures that benefit both parties, thereby eliminating fair competition, he explained. But the theory also lets regulators pick up on signs of collusion and, ultimately, break up illegal cartels.
Monday’s award also highlighted developments in game theory that were lauded with the 1994 economics prize to Americans John Harsanyi and John Nash and German Reinhard Selten. Nash was portrayed in the 2001 Academy Award-winning film “A Beautiful Mind,” starring Russell Crowe.
Lofgren said Aumann’s theories differed from Nash’s by introducing an infinite repetition of the same game so as to find the best solution in long-term relationships instead of in a single encounter.
The difference is illustrated in the so-called “prisoner’s dilemma,” one of game theory’s best-known situations in which two partners in crime are put in separate cells and given an ultimatum: If one implicates the other, he may go free while his partner faces a firing squad.
Facing that situation once, both prisoners are likely to talk, meaning both would be executed, Lofgren said. However, if they could repeat the situation an infinite number of times and add the results of each action, they would realize that the best option is for both to keep mum, he said.
That supposition formed the basis for a TV game show, “Friend or Foe?” that aired on the Game Show Network in the U.S.
Aumann is a professor at the Center for Rationality at the Hebrew University of Jerusalem."
For the full article: URL:

Tuesday, October 04, 2005

One job you can get with a math major

One can get a nomination to be on the Supreme Court of the United States!

Hat tip to Harriet Miers, a mathematics major at SMU (prior to getting her law degree at SMU).

Yeah, I am a Democrat, but it is good to see a math major make good.

I haven't posted much here as math blog posts take time, and most of my math time has been devoted to teaching a "numerical methods" course.

This course is time consuming as
  1. My last numerical analysis course was a long time ago.
  2. I wasn't that good at it then.
  3. I am learning the software packages as we go along.

I am not sure as to how I feel about the software packages; in my day we wrote our own code which helped us learn how the algorithms worked. On the other hand, we can cover more topics since coding isn't slowing us down.

What I can say is that I finally learned where the error formulas in the Simpson's rule approximation formula comes from.

What is Simpson's rule? It is a way of approximating a definite integral by approximating the function being integrated in a piecewise fashion by parabollas. Basically, you chop up the interval that you are integrating over into "n" equal intervals, subdivide each interval into two pieces, and over each piece you fit a parabola (three points determine a unique parabola (a line is considered a degenerate parabola). And it is easy to integrate a quadratic function.

So, over each interval, there is a maximum error; if the function being integrated has four continuous derivatives (can differentiate the function 4 times and still get a continuous function) then the error over each piece is bounded by the absolute value of

(M/2880)(b-a)^5 where the interval runs from "a" to "b" and M is the absolute value of the maximum value of the 4'th derivative of the function being integrated, taken over the interval from a to b.

Where in the world does the 2880 come from?

Well, if one uses the 3'rd degree Lagrange polynomial to approximate f over the interval from a to b, the maximum error is (M/4!)(where M is as before and the points x0, x1, x2, x3 run from a to b. But if you integrate the Lagrange polynomial over the interval, the cubic term vanishes (integrates to zero) and what you have left is the integral of the Lagrange quadratic polynomial through x0, x1, x2, which can be set so x0 =a, x1 = (1/2)(a+b) and x2 = b. that is, this is the unique quadratic that one integrates in Simpson's rule!

So, to find the error, we integrate (M/4!)(x-x0)(x-x1)(x-x2)(x-x3) from a to b, setting x0, x1 and x2 as before. we set x3=x2 which you really can't do if you are doing an approximation, but remember we are obtaining an upper bound for the error.

Set h = b-a and then make the substitution x = a + th, then dx = hdt and

x-x0 becomes th, x-x1 becomes h(t-1/2), (x-x2)(x-x3) becomes (h(t-1))^2

and the limits of the integral become 0 to 1. Pull all of the "h's" out of the integral sign to get the h^5 term, and integrate (t)(t-1/2)(t-1)^2 from 0 to 1 to obtain 1/120

and then 1/120 * (1/24) =2400 + 480 = 2880.