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I enjoyed reading Graham Cormode’s guide for the adversarial reviewer in computer science (pdf)

h/t Haris

My spouse told me about a talk show she has seen on TV, whose subject was “the difficulty faced by women whose pants size is 46″ (equivalent to 38 in the US). Apparently fashion stores sell women’s pants up to size 44, and stores who specialize in large sizes do not sell this size. The women that are in-between find themselves in a difficult position.

Several representatives of fashion chains provided few explanations:

1) Cost: larger pants require more fabric, which increase the pant’s cost. Pants are sold at the same price, regardless of size: pants of size 36 and pants of size 44 sell at the same price; in particular, thinner women subsidize the rest of the population (but this is irrelevant for this post). Pants of size 46 will require more fabric, but will sell at the regular price. The manufacturers therefore have no reason to make them.

There are (at least) two responses for this point: (a) one can charge more for size 46, and (b) one can have in general differential price for pants: the larger the pants, the higher the price. 200cl cup costs more than a 150cl cup. Why not the same price differentiation with pants?

2) Design: Apparently pants of size 42  are not identical to pants of size 36, after multiplying all dimensions by the correct ratio of 42/36. The designer has to make additional adaptations so that the pants will fit bodies of different sizes. Thus, if the manufacturer has to make pants of a new size, size 46, additional designing work should be invested.

This argument is not too strong either. First, the additional work adds to the production cost, which can affect the pants price. Second, similar adaptations have to be done for every model of pants and every size: 38, 40, 42. I am pretty sure that there are usual tricks that designers do to adapt pants of one size to another size. So it is not clear to me why one additional size makes the designer’s life so difficult.

3) Positioning: One (male)  representative of a fashion chain said that they position themselves as a store for young beautiful women. What will their customers think if they see a larger woman wearing the same cloths that they wear? Since we are talking about female customers, I asked my spouse whether she agrees with this argument. She said that contrary to what the representative said, she would love to see a larger woman wearing the same cloths that she wears. She will then be able to say “these pants look much better on me than on her”.

So why don’t we find pants of size 46 in stores?

As finals week approach, I scrutinize the papers more closely looking for inspiration for examination questions. My eye was recently caught by an article in the NYT about most favored nation clauses and a suit brought against Blue Cross & Blue Shield (BCBS) in Michigan. The gist of the matter is this. BCBS asked hospitals it contracted with to serve their policy holders to give it most favored nation (MFN) status. There are two kinds of MFN. Regular MFN where the supplier promises the buyer it will always get a price no higher than lowest price the supplier offers to any other buyer. And, MFN+, where the supplier promises the buyer their price will be some % lower than the lowest price offered to any other buyer. BCBS apparently had MFN promises from 45 hospitals (all with < 100 beds) and MFN+ promises from >25 others which required BCBS to receive at least a 23% discount compared to other insurance providers. A full list of the relevant hospitals can be found here.

The benefits to a buyer  from MFN status are well known. My colleague Sandeep Baliga has articulated them on his blog. It raises prices for BCBS’s competitors. Hospitals should also  benefit, because it reduces their incentives to cut prices to fill beds. Why? A price to cut to another insurance provider in order to gain their business, requires a price to cut to BCBS as well. If BCBS fills a lot of beds, that might be sizable. It is, since BCBS is the largest provider in the region covering 60% of Michigan’s commercially insured residents. Indeed, BCBS should be willing to increase payments to hospitals to get a MFN. Thus, the effect of the MFN is to raise prices to all insurance providers, but raises the prices of BCBS’s competitors even more!

If you buy this, then hospitals should jump at the chance to offer BCBS MFN or MFN+ status. However, in order to get the 45 small hospitals I noted above to sign on, BCBS threatened to slash payments to them by 16% if they did not offer MFN status. We come now to the interesting issue that the newspapers have missed.  When exactly does it make sense to demand an MFN and to offer one?

Let us look at the choice that faces an individual hospital. To simplify, suppose that 70%  of the hospitals capacity is filled by by BCBS policy holders and the other 30% by policy holders of another provider, say RCRS. Suppose BCBS comes to the hospital and offers to increase the payment it makes in return for the hospital raising the price to RCRS even more. Should the hospital accept?

Case 1: RCRS can go elsewhere.

If the hospital accepts BCBS’s offer, RCRS sees a price increase. If it can find another hospital with capacity and low price, it leaves. In which case, the hospital is left with 30% unused capacity. Thus BCBS’s price must be large enough to compensate for the lost profits from RCRS’s departure. However, from BCBS’s point of view, RCRS’s costs are as they were before, and BCBS has just raised its costs!

Case 2: RCRS cannot go elsewhere.

Then, neither can BCBS. In which case the hospital is in the drivers seat when bargaining with BCBS and RCRS. It could raise prices anyway on both.

Lets try another tack. Could BCBS’s size and its threat to reduce payments (I presume by shrinking volume) persuade a hospital to offer it MFN status? Suppose, as one of the hospitals, I decline BCBS’s offer. It reduces the # of its policy holders that come to me. These policy holders have to go somewhere. Perhaps they go to a hospital that accepted BCBS’s offer. In which case RCRS will be looking to shift their policy holders from that hospital to mine perhaps, because I have a lower price. Again, if the numbers work out I have no reason to say yes to BCBS.

Perhaps, what is missing from the discussion are the policy holders. What if policy holders care about the hospital they will be sent to. So, if RCRS is turfed out of a hospital, its policy holders `stay behind’ as it were by switching to BCBS. This again puts bargaining power in the hands of the hospital, because insurance companies need  the right mix of hospitals to attract policy holders.

So, why did the hospitals accept the MFN clauses.

In a recent post I described stopping games in continuous time; I mentioned that in this class of games, an equilibrium exists if there are at most two players, and it does not necessarily exist if there are at least three players. David Rahman, who visited Tel Aviv, asked me about the existence of correlated equilibrium in stopping games in continuous time. My first guess was that the three-player game that I described in that post had no correlated equilibrium. Last night I did the necessary calculations, and I was surprised to learn that this game possesses a correlated equilibrium: a mediator chooses a pair of players, each pair is chosen with probability 1/3, and that pair should stop at time 0, while the third player continues. There are additional correlated equilibria, but this is the simplest one. Does there always exist a correlated equilibrium in stopping games in continuous time? I bet that the answer is negative, but I am ready to be surprised again. Can anyone come up with a proof that a correlated equilibrium always exists, or with an example of a game without a correlated equilibrium?

I was asked by J. Bowen, in my pricing class, why Progressive Insurance (i.e., Flo) informed potential buyers of the prices charged by the competition. Why tell them they can get the same policy for less elsewhere.? A nice question since it required me to put on my thinking cap. First some homework. A visit to the Progressive web site to see how they come up with a quote. Visitors are asked a series of questions about the car (if one were trolling for car insurance), driving habits etc. (does the competition ask the same questions?). Then a choice of polices and a quote. This is followed by a link to comparable offerings provided by competitors. However, in the absence of a credit score, they are only able to furnish price ranges for some of the  substitute offerings. Thus, in some cases comparisons may not be easy to make.

Now, lets turn to possible explanations.

1) They exaggerate the rivals price since it may be hard to verify. If true, it makes the whole thing uninteresting.

2) The opportunity to search for a low cost attracts potential buyers. Once on the site, inertia kicks in (laziness, warm glow etc) and as long as the price difference is small enough, they don’t depart. This may be true. If so, there is nothing about Progressive in this regard and therefore other companies should do the same.

3) For certain kinds of customers they tend to have offerings that a priced lower than the competition and others they may be roughly at par. Thus, on the special customers they don’t actually run the risk of losing them when offering a comparison.

I like (3). But, it raises a question. How is it that on these certain customers they can offer lower prices? They must have a cost advantage. Where does it come from.

From Gad Ben Zvi, another student in the class, comes an answer. Progressive believes that it can more accurately price the risk because of the specific questions it asks. Questions that others don’t. That is the cost advantage.

The comments on Eilon’s last post turned into a dialogue on whether it is of any use for MBAs to learn game theory. I will make some brief comments on this.

With very few exceptions, professors at leading business schools are chosen for their research prowess and hence their skill at analytic thinking and modeling, and have spent little time if any in the business world. Many will do some consulting, but this comes later in their career and is not the basis on which they are hired. How do future managers benefit from professors who have never managed?

Experience may be everything, but it is not the only thing. Or, as Ricky puts it, “Experience is inevitable. Learning is not.” Just as important a cornerstone as experience are models by which to organize experience. The research professor will teach in part via war stories (cases), just as a business veteran would. These are not his war stories, of course, so what it the advantage of his expertise? The advantage comes via in-depth understanding of models. Why is that so important? Models organize the information from the case into well-defined causes and effects. Then you can learn not only what the successful decision was in a given case, but how to adjust this decision for related cases, and, just as important, when the concepts from the case do not apply. A good professor will be able to make clear the assumptions that go into the model, and when they do or do not apply in real life. You always may come across situations where none of the models seems to help, and have to improvise. But the same would certainly be true if you learned from non-theoretical professors; their experience might not be closely enough related to yours. Worse, it might be close enough that you think you should make similar decisions, but you miss a key difference, one which a good model would have drawn your attention to.

A source of real-life war stories that can be better understood with the help of game theory is Co-optition by Brandenburger and Nalebuff. The contents of this book are a great response to anyone who is not sure why theorists might be at a business school.

I stumbled upon a paper by Anna Dreber, David Rand, Drew Fudenberg and Martin Nowak, whose title is the title of this post. This paper reports on the following experiment: players play the repeated prisoner’s dilemma, where there is an additional action: punish the opponent. At every period each player has to decide whether to cooperate (I lose 1, you get 2), to defect (I get 1, you lose 1), or to punish (I lose 1, you lose 4).  The game is discounted: after each period an unfair coin is tossed: with probability 25% the game ends, with probability 75% it continues. In particular, the expected length of the game is 4 periods.

Results: the players who had the highest gains did not punish each other. The players who came next are those who punished but were not punished, and the players who fared worse are those who were punished. When you look at the expected length of the game and at the payoffs, you realize why this happens. The authors run more correlation tests to derive additional conclusions; you can press this link (or the one above) and read it for yourselves.

Why do I tell you about this paper? Because I do not like experiments. I do not see what we gain from them. The title of this paper, for example, is “Winners Don’t Punish”. Is this the right conclusion? I do not think so. Because the game is short, there is not much time for learning, for using a punishment. If you were punished, you lose 4, and there is hardly any time to regain your losses. And even if the game were longer, and players could teach each other that defection leads to punishment and is therefore not profitable, will an experiment on \$10 teach us anything about decisions in the real world? The Roman Empire used to punish heavily. And it worked, as long as it was the strongest. Most dictators punish, and some last decades. So in real life, if you are strong, you can punish as much as you wish. I still wait to see the experiment that will convince me that it reveals some insight on real life.

NYU, with outposts in Buenos Aires, Shanghai, Singapore, Tel Aviv, and Abu Dhabi is on it’s way to becoming the Starbucks of higher education. Cornell, Georgetown, Northwestern and CMU each have a franchise in the desert sands. Beyond the dunes, over the seas, in the utter east, one will find Yale, Johns Hopkins, MIT and Georgia Tech. This interest in setting up shop anywhere east of Pirate Alley is not confined to US universities. UK and  Australian universities have been trying as well.

Many follow in the footsteps of B-schools. INSEAD, Chicago’s GSB (I know, I should say Booth, but the acronym BSB seems insulting), Duke’s Fuqua (say it like a Canadian and it is insulting) and Kellogg had already established outposts and alliances on distant shores at least a decade earlier.

Why the lust to go global? Particularly in locations that espouse values inimical to free inquiry. Please, no answers that juxtaposes `global’ and `interconnected’. Lets try a couple of others.

1) Its important to expose students and faculty to the wider world.

OK. One could achieve this in ways that do not require building campuses elsewhere. Choosing to admit students from around the world. Encouraging students to spend an extended period abroad at a foreign University or College. Selecting faculty from around the world. Supporting and encouraging research with an international dimension.

2) Maintain or increase share of able students, faculty and attractive employment opportunities.

If Universities are platforms for matching students, faculty and employers, then it helps to be close to where there are the deepest pool of able students, attractive employment opportunities etc. If one believes these pools will be located in India and the PRC, say, then it would make sense for a University to buy an option in the form of a satellite campus. If the bet comes out right, one should expect the University to pick up and move entirely to the satellite campus. If the intellectual center of the World is to shift east, why not move with it? Perhaps, in a score of years, Yale will call Madras (oops Chennai) home rather than New Haven. A homecoming of sorts!

3) \$\$\$\$\$

Galbraith once described the crusaders this way:

Beneath the mantled cross beat hearts firmly attuned to the value of real estate.

Something similar can probably be said of those who lead our Universities. If we have reached the upper limit of fees that can be charged (any more and we switch from becoming teachers to concierge’s…….are we there yet?), then, the only way to grow revenue to feed the search for knowledge is to increase the volume of students. How is one to do that without sacrificing quality? Go east, where the bodies are. In some cases, autarchs, plutocrats and apparatchiks will defray the costs of doing so.

Any other reasons?

Stopping games are simple multi-player sequential games; each player has two actions: to continue or to stop. The game terminates once at least one player decides to stop. The terminal payoff depends on the moment at which the game stopped, on the subset of players who decided to stop at the terminal time, and on a state variable whose evolution is controlled by nature. In other words, the terminal payoff is some stochastic process. If no player ever stops, the payoff is 0 (this is without loss of generality).

Stopping games arise in various contexts: wars of attrition, duels, exiting from a shrinking market to name but a few. Does an equilibrium exist in such games? If the game is played in discrete time and payoffs are discounted, then payoffs are continuous over the strategy space, and an equilibrium exists. What happens if the game is played in continuous time?

Surprisingly, if there are at least three players, an equilibrium may fail to exist, even if the payoffs are constant (that is, they depend only on the subset of players who decide to stop, and not on the moment at which the game is stopped (and there is no state variable). Consider the game in the following figure:

A three player stopping game

In this game, in every time instant t, player 1 chooses a row, player 2 chooses a column, and player 3 chooses a matrix. Each entry except the (continue,continue,continue) entry corresponds to a situation in which at least one player stops, so that the three-dimensional vector in the entry is the terminal payoff in that situation. The sum of payoffs in every entry is 0, and therefore whatever the players play the sum of their payoffs is 0. Each player who stops alone receives 1, and therefore each one would like to be the first to stop. It is an exercise to verify that the game does not terminate at time t=0: termination at time 0 can happen only if there is a player who stops with probability 1 at time 0, but if, say, player 1 stops at time 0 then it is dominant for player 2 to continue at time t=0, and then it is dominant for player 3 to stop at time t=0, but then it is dominant for player 1 to continue at time t=0.

Thus, in this game, the players stop with some positive probability (that is smaller than 1) at time t=0, and, if the game has not terminated at time 0, each one tries to stop before the other. If the game is played in continuous time, there can be no equilibrium. Don’t worry because the payoff is not discounted; adding a discount factor will not affect the conclusion.

It is interesting to note that in discrete time this game has an equilibrium: in discrete time the players cannot fight about who stops first after time t=0, because the first time in which they can stop after time t=0 is time t=1, so they all stop with some positive probability at time t=1, and, in fact, they stop with positive probability at every discrete time t.

So stopping games are one example in which a game in continuous time does not have an equilibrium, while the corresponding game in discrete time does have an equilibrium.

I voted today because I enjoy voting (and because I happen to have a fairly strong preference.) The tiny chance of being pivotal surely doesn’t compensate me for the time, but I enjoy seeing the vote totals and knowing that I influenced the final digit. I have a feeling that many people are like me, or equivalently, vote out of a sense of duty. This is a bit of a problem for the game-theory models of voting where rational people weigh the probability of being pivotal and intensity of preference against the “cost” (presumably time) of voting. These models tend to predict a tiny turnout. Now, I don’t doubt that more people vote in close elections, so the likelihood of being pivotal has *some* impact on turnout. Intensity of preference probably matters more, though. I know political scientists debate the determining factors in turnout ad infinitum; I’m not trying to break new ground here, just sharing some thoughts and perhaps starting a discussion.