Pricing Algorithms

A quite amusing post I stumbled upon describes pricing algorithms used by book retailers at Amazon.com, and what happens when two retailers use such algorithms. I am pretty sure that economists can say much about the use of these algorithms, optimal pricing algorithms, whether the customer gains or loses when retailers use them, and other similar questions. Alas, I am a mathemtician, so all I can do is read the post and smile.

Symbolab: Customer Review

Symbolab is a new search engine that allows one to search for equations. A great idea for those of us who, for example, forgot how the model of “stochastic game” is called, but remember that in stochastic games one is interested in the discounted sum of payoffs. The search engine presumably searches through articles to look for the equation that you look for.

Excited about the prospect that forgetting words is no longer devastating, I immediately searched for the formula of the discounted sum of payoffs:

$\sum _{t=1}^{\infty }\left(\lambda \left(1-\lambda \right)^{t-1}r\left(s_t,a_t\right)\right)$

Unfortunately 0 results were found. I did not give up and change the payoff function from “r” to “c” and then to “u”. Maybe the engine prefers costs or utilities. When I searched for the equation with costs, I got one result; a sum related to convex sets which is not related whatsoever to discounted sum. When I searched for the equation with utilities the engine gave up. I got 174 results! but they were of various sums, none of them (at least, not the 30 top ones which I looked at) involved discounted sums. There were references to statistics, ergodic theory, and various other topics, but no economics or game theory.

Maybe stochastic games are not that common on the net. Let’s look for something simpler:

$u_i\left(\sigma \right)\ge \:u_i\left(\sigma _{-i},\sigma ‘_i\right)$

which is the condition that defines Nash equilibrium. 0 results again. I gave up.

Unfortunately, it seems that we will have to continue remembering names of concepts, and will have to consult with colleagues when we need results in topics we are not familiar with. Magic has not been found yet.

Prize for Stability and Market Design

It was a pleasure to awaken to the news that this years Nobel (memorial) in Economics went to Shapley and Roth (Schwartz lecture series hits bullseye again). Even more had it gone to me…..but, have yet to see pigs in the sky.

Pleasure turned to amusement as I heard and read the attempts of journalists to summarize the contributions honored. NPR suggested that it was about applying statistics. Forbes had a piece that among Indians would be described as putting shit in milk. This always makes me wonder about the other things they get wrong in the subjects I have no knowledge of. Nevertheless, I will plug one outlet for reasons that will become obvious upon reading it.

In this post, I set myself the task of seeing if I can do a better job than the fourth estate of conveying the nature of the contribution that was honored, Oct 15th, 2012. Here goes.

The fictional decentralized markets of the textbook, like the frictionless plane in a vacuum used in physics, are a useful device for establishing a benchmark. Real markets, however, must deal with frictions and the imperfections of their participants. One such market is for College Admissions in the US that is largely decentralized This decentralization increases uncertainty and raises costs. Students, for example, must forecast which colleges are likely to accept them. The greater the uncertainty in these forecasts the more `insurance’ is purchased either by applying to a large set of colleges or aiming `low’. On the college side, this insurance makes yields difficult to forecast. Increasing acceptance rates to increase yields has the effect of driving application numbers in the future down, so, waiting lists grow. These problems could be eliminated were one to switch to a centralized admissions market. How would one design such a centralized process? This is the question that the work of Shapley (and the late David Gale) and Alvin Roth addresses.

A major hurdle that a centralized market for college admissions must overcome is that it must match students with colleges in a way that respects the preferences and incentives of both parties. A centralized market cannot force a student to attend a college she does not want to, or require a college to accept a particular student. There is always the threat that the participants can pick up their marbles and walk away. If enough do, the incentives for the students and colleges that participate in the centralized market decline. Would a student participate in an centralized market if certain brand name colleges opted out? The work of (Gale &) Shapley was the first to formalize this concern with designing centralized markets that would be immune defections on the part of participants, i.e., stable. Their seminal paper articulated a model and a mechanism for matching students to colleges that would be stable in this sense. Alvin Roth’s own work builds on this in a number of ways. The first is to use the notion of stability to explain why some centralized markets fail. Second, to highlight the importance of other sources of instability associated with, say timing. Participants may wish to `jump the clock’. Colleges, for example, offering admission to high school students in their junior year when there might be less competition for that student. The third, is to use the ideas developed in other contexts to allocate scarce resources where money as a medium of exchange is ruled out, most notably kidneys.

The work honored this year has its roots in a specialty of game theory, long considered unfashionable but one Shapley made deep contributions to: co-operative game theory. One can trace an unbroken line between the concern for stability in the design for markets and the abstract notions of stability discussed, for example, in the first book on game theory by von-Neuman and Morgenstern. It proves, once again, Keynes’ dictum:

“The power of vested interests is vastly exaggerated compared with the gradual encroachment of ideas.’’

Turtles, Fossils, and Conditional Probability

Le Monde tells us (in French) that researchers found 47-million-year-old fossils of nine mating pairs of turtles of species Allaeochelys crassesculpta in a lake in Germany. These fossils, which, according to the researchers, are the only fossils of mating pairs of animals to be found, taught the researchers a lot on this extinct species. But what caught my eye is a probabilistic statement made by one of the researchers:

“Des millions d’animaux vivent et meurent chaque année, et nombre d’entre eux se fossilisent par hasard, mais il n’y a vraiment aucune raison que ça arrive lorsque vous êtes en train de vous reproduire. Il est hautement improbable que les deux partenaires meurent en même temps, et les chances que les deux soient fossilisés à la fois sont encore plus maigres”, a expliqué Walter Joyce, de l’université allemande de Tübingen. Avec plusieurs couples, les probabilités s’amoindrissent encore.

Since the name Walter Joyce sounds English speaking, I searched for a version in English. I found this:

“No other vertebrates have ever been found like these, so these are truly exceptional fossils,” Joyce said. “The chances of both partners dying while mating are extremely low, and the chances of both partners being preserved as fossils afterward even lower. These fossils show that the fossil record has the potential to document even the most unlikely event if the conditions are right.”

The difference between the English version and the French version are slight yet important. The French reader learns that it is improbable that a pair of animals will die together while mating, and that it is even less probable that they will be fossilized together. The chances that this will happen to several mating pairs is even smaller. This is true if the death-while-mating+being-fossilized events of the different couples were independent. However, the article itself explains to the readers that here the events are dependent. During the mating process the turtles freeze, and sometimes sink to the bottom of the lake, where the water is poisonous. Thus, in this lake the probability of finding a fossil of a single turtle is less probable than finding the fossil of a mating pair of turtles, and the probability of finding several fossils of mating pairs is not necessarily lower than the probability of finding a single fossil of a mating pair… The English version does not correct this inaccuracy, but it does end with a note that can be interpreted as a correction of previous mistakes: “These fossils show that the fossil record has the potential to document even the most unlikely event if the conditions are right.”

Next week the semester starts, and I will get a new bunch of first-year undergrad students who are eager to study Introduction to Probability. I will ask them to read this article and spot the mistakes. I wonder how many will find what is wrong in this article.

Are Alumni Networks Valuable?

One reason given for the value of an MBA degree is the relationships that one develops with other students as well as the connection to the larger alumni network. Such relationships can eventually be used to open doors, secure a place at `the table’ and traded with others. While I’ve long since replaced the belief in `res ipsa loquitur‘ for `who you know matters’, I’m still not convinced by the relationship story.

Suppose introductions to gatekeepers and decision makers are scarce resources. When handing them out, why should I favor someone just because we attended the same institution? Presumably what matters is what good turn the other might do for me. Surely, this will depend on the position held rather than the school attended. Furthermore, why should the other’s academic pedigree suggest anything about the likelihood of the other returning the favor in the future? I know of no B-school that claims that it is selecting a class of future Cato’s.

True, favors are not requested or granted until a bond is established between the parties. Having something in common assists the formation of such bonds. But, why should having attended the same school be any more useful in this regard than a common interest in wine, golf or stamps?

Finally, if the alumni network is valuable, then merging two small networks should increase value for members of either network. Thus, in much the same way that some airlines share their frequent flyer programs (eg star alliance), we should see certain schools merging their networks. Haas and Anderson? Johnson School and Olin? One sees something like this at the executive masters level but not at the full time MBA level.