Game Theory Notes on Subgame Perfectness and the Centipede Game

I've just written up notes for my game theory class on a paradox of sequential rationality and on the Centipede Game.

Most-Used Bible Verses

TopVerses.com has a list of Bible verses that they somehow calculate are most-used. It's interesting to see what's on it, and a good list of important verses. They don't give actual counts, though, so it's just ordinal.

Ranges and Codomains

I just learned a useful math term: CODOMAIN. Consider the function f(x) = 3 +5x as defined over the intervals of x in [0, 10] and f(x) in [0, \infinity). The DOMAIN is [0,10]. The RANGE is [3, 50]. The CODOMAIN is [0, \infinity). This mapping is one-to-one, but not onto, so the range and codomain are not identical.

Phelps on Capitalism and Innovation

Nobel laureate Edmund Phelps has an interesting essay, "Economic Justice and the Spirit of Innovation," in the October 2009 First Things.

The issue of morality in economics is neither the fairness of income distribution nor the stability of financial systems. It is how human institutions can be shaped to correspond to human nature—to man’s nature as an innovator.... Prosperity and the development of the human spirit are linked in the dynamism of the economy. The dynamism of the American economy over the past two hundred years was strong, and that helps to explain why prosperity was high both in the sense of high employment and the sense of a high degree of personal satisfaction compared to that in other countries.... That is the positive moral content of economics—to realize an anthropology that starts with innovative human nature: homo innovaticus, not homo economicus. Existing economics has a negative moral content in that it treats economic factors as though they were pieces on a game board rather than human beings who learn, discover, and innovate. Politicians play the same game, channeling resources from one activity or social group to another without considering the effect on the creativity and judgment exercised within the economy and thus the deep rewards the economy imparts or fails to impart.... Even now, in the midst of an economic downturn, there are signs of vitality that weren’t present in the 1950s. There is exuberance, however irrational, in the banking system, and some originality here and there in hedge funds and private equity, and still some inventiveness in Silicon Valley. Although they may have caused more problems than they were worth, the exotic, new financial instruments showed that America is still the world’s leader in invention. They reflect America’s capacity to create. Unfortunately, the markets were unsophisticated and set mistaken asset prices.

Constructing a Risky Density Function

My colleague Haizhen Lin found a neat trick from someone in the math department. Suppose you have a density f(x) and you want to construct a pointwise less risky function, as in my paper cited below. You can use this: f(a, x) = (1/a) f( .5 - .5/a +x/a) If a=1, f(a,x) = f(x). If a is small, f(a,x) tends to get big because of the 1/a portion, and it gets very big for x=0, but for x far from 0, the f becomes small because the argument becomes very big, distant from 0. "When Does Extra Risk Strictly Increase the Value of Options?" The Review of Financial Studies, 20(5): 1647-1667 (September 2007). It is well known that risk increases the value of options. This paper makes that precise in a new way. The conventional theorem says that the value of an option does not fall if the underlying option becomes riskier in the conventional sense of the mean-preserving spread. This paper uses two new definitions of ``riskier'' to show that the value of an option strictly increases (a) if the underlying asset becomes ``pointwise riskier,'' and (b) only if the underlying asset becomes ``extremum riskier.'' Paper in tex or pdf ( http://www.rasmusen.org/published/Rasmusen07-RFS-options.pdf).

Leo Strauss Audio File of Him Teaching Meno

The University of Chicago has posted an audio file of Leo Strauss lecturing on Meno as the first session of his political philosophy class. I should listen through it sometime. Will I? There are so many wonderful opportunities that we let slide by. We professors ought to be sitting in on each others' classes, for example, but that is rare.

Occupations over the Decades

DD refers me to a good site like the baby name site I once blogged but showing over decades the most common occupations: http://flare.prefuse.org/apps/job_voyager . This site also tells you how to set up your own graphics of this kind.

A New Word: Jobligation

Jobligation: an obligation arising from your job--- something you must attend because of work. From Mr. Lileks.

Dynamic Pricing for Hockey Tickets

Via Marginal Revolution, a news story tells of daily-changing prices for hockey games:

Similar to airline pricing, the best prices are often found early. Dynamic pricing will provide fans with great prices starting from the initial on-sale on Sept. 12. The upper level single-game ticket prices can go up or down based on a variety of factors, including league standings, opposing team, star players, day of the week, and real time supply and demand. Dynamic pricing for upper level tickets will continue all season. Fans will be able to check out the current prices at any time at DallasStars.com.

Cromwell's Rule

AJ refers me to Wikipedia's article on Cromwell's Rule:

Cromwell's rule, named by statistician Dennis Lindley, states that one should avoid using prior probabilities of 0 or 1, except when applied to statements that are logically true or false. (For instance, Lindley would allow us to say that \Pr(2+2 = 4) = 1.)

The reference is to Oliver Cromwell, who famously wrote to the synod of the Church of Scotland on August 5, 1650 saying

I beseech you, in the bowels of Christ, think it possible that you may be mistaken.

As Lindley puts it, if a coherent Bayesian attaches a prior probability of zero to the hypothesis that the Moon is made of green cheese, then even whole armies of astronauts coming back bearing green cheese cannot convince him.

This seems reasonable. But is there a psychological problem if we are sure of nothing in the world? We might be haunted by having to always to do a substantive Bayes's Rule calculation. Maybe not, though. The substance of Lindley's idea is the story of the astronauts bringing green cheese-- we will throw away our heuristic solid belief if that happens.

The Blue-Eyed Islander Puzzle

Here is a well-known puzzle that I will probably be teaching next week. An island starts with 2 blue-eyed people and 48 green-eyed, but the people do not know these numbers. If a person ever decides his eyes are blue, he must leave the island at dawn the next day. There are no mirrors and people may not talk about eye color, but they see each others' faces.

What will happen? -- nobody leaves.

Now an outsider comes to the island and says, "At least one of you has blue eyes".

The next dawn, nobody leaves, but on the second dawn, both blue-eyed people leave.

The reason: Both blue-eyed people realize there are either 1 or 2 blue-eyed people. When nobody leaves on the first dawn, each realizes that there must be 2-- and he is one of them.