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One reason I like thinking about probability and statistics is that my raw intuition does not fit well with the theory, so again and again I find myself falling into the same pits and enjoying the same revelations. As an impetus to start blogging again, I thought I should share some of these pits and revelations. So, for your amusement and instruction, here are three statistics questions with wrong answers which I came across during the last quarter. The mistakes are well known, and in fact I am sure I was at some level aware of them, but I still managed to believe the wrong answers for time spans ranging from a couple of seconds to a couple of weeks, and I still get confused when I try to explain what’s wrong. I will give it a shot some other post.

— Evaluating independent evidence —

The graduate students in a fictional department of economics are thrown to the sharks if it can be proved at significance level {\alpha=0.001} that they are guilty of spending less than eighty minutes a day on reading von Mises `Behavioral econometrics’. Before the penalty is delivered, every student is evaluated by three judges, who each monitors the student in a random sample of days and then conducts a statistical hypothesis testing about the true mean of daily minutes the student spend on von Mises:

\displaystyle  \begin{array}{rcl} &H_0: \mu \ge 80\\&H_A: \mu<80\end{array}

The three samples are independent. In the case of Adam Smith, a promising grad student, the three judges came up with p-values {0.09, 0.1, 0.08}. Does the department chair have sufficient evidence against Smith ?

Wrong answer: Yup. The p-value in every test is the probability of failing the test under the null. These are independent samples so the probability to end up the three tests with such p-values is {0.09\cdot 0.1\cdot 0.08<0.001}. Therefore, the chair can dispose of the student. Of course it is possible that the student is actually not guilty and was just extremely unlucky to get monitored exactly on the days in which he slacked, but hey, that’s life or more accurately that’s statistics, and the chair can rest assured that by following this procedure he only loses a fraction of {0.001} of the innocent students.

— The X vs. the Y —

Suppose that in a linear regression of {Y} over {X} we get that

\displaystyle Y=4 + X + \epsilon

where {\epsilon} is the idiosyncratic error. What would be the slope in a regression of {X} over {Y} ?

Wrong answer: If {Y= 4 + X + \epsilon} then {X = -4 + Y + \epsilon'}, where {\epsilon'=-\epsilon}. Therefore the slope will be {1} with {\epsilon'} being the new idiosyncratic error.

— Omitted variable bias in probit regression —

Consider a probit regression of a binary response variable {Y} over two explanatory variables {X_1,X_2}:

\displaystyle \text{Pr}(Y=1)=\Phi\left(\beta_0 + \beta_1X_1 + \beta_2X_2\right)

where {\Phi} is the commulative distribution of a standard normal variable. Suppose that {\beta_2>0} and that {X_1} and {X_2} are positively correlated, i.e. {\rho(X_1,X_2)>0}. What can one say about the coefficient {\beta_1'} of {X_1} in a probit regression

\displaystyle \text{Pr}(Y=1)=\Phi\left(\beta_0'+ \beta_1'X_1\right)

of {Y} over {X_1} ?

Wrong answer: This is a well known issue of omitted variable bias. {\beta_1'} will be larger than {\beta_1}. One way to understand this is to consider the different meaning of the coefficients: {\beta_1} reflects the the impact on {Y} when {X_1} increases and {X_2} stays fixed, while {\beta_1'} reflects the impact on {Y} when {X_1} increases without controlling on {X_2}. Since {X_1} and {X_2} are positively correlated, and since {X_2} has positive impact on {Y} (as {\beta_2>0}), it follows that {\beta_1'>\beta_1}.

Here, recounting the debate before the Bin Laden operation.

[The President] said, I have to make this decision what is your opinion. He started with the National Security adviser, the Secretary of State and he ended with me. Every single person in that room hedged their bet, except Leon Panetta. Leon said GO. Everyone else said 49, 51, this…

It got to me. Joe what do you think? I said “You know I didn’t know we have so many economists around the table. We owe the man a direct answer.

Me: First, is `economist’ a synonym to somebody who can’t give a direct answer ? I thought we have a better reputation. Second, the way I see it 49,51 is actually a direct answer and a bold prediction. It means that if they start making independent attempts to kill Bin Laden, and substantially less or substantially more than half of these attempts succeed then the president has all the reasons to give these guys the boot.

So, wikipedia is dark today in protest of an initiative in congress to block sites that link to sites that infringe on copyrighted intellectual property. Ever noticed before how many times a day you use wikipedia ?

Here is what I don’t get about this whole idea of “copyrighted intellectual property”. Is it something advocated on moral grounds or on economic grounds ? I mean, when Bob sneaks into Alice’s vineyard and eats the grapes without permission, we view it as a moral atrocity; It’s just a wicked thing to do; It invokes the wrath of the gods; Moses explicitly forbade it. To be sure, it’s hard to pin down what exactly makes the vineyard belong to Alice without getting into a recursive definition of ownership, and if we try tracing back the vineyard from one legitimate owner to another we arrive to the first man who just fenced a piece of land and said “This is mine”. But here the economic argument kicks in — Most of us don’t begrudge this initial act of illegitimate fencing because the bastard who committed it was the founder of civil society. We like the idea of civil society. We like prosperity and growth. Without protection of private property we will have none of these.

But what about protection of “intellectual property” ? Clearly this is not a necessary condition for a civil society. It’s also not a necessary condition for production of knowledge and culture. We had Plato and Archimedes and Cicero and Shakespeare and Newton before it occurred to anybody that Bob has to gets Alice’s permission to reproduce a code that Alice wrote. Coming to think about it, when did the concept of intellectual property pop up anyway ? Waitaminute let me just check it up on wikipedia. Oops.. What did we ever do before wikipedia ?

The White House thinks that intellectual property is justified on economic grounds

Online piracy is a real problem that harms the American economy, and threatens jobs for significant numbers of middle class workers and hurts some of our nation’s most creative and innovative companies and entrepreneurs. It harms everyone from struggling artists to production crews, and from startup social media companies to large movie studios.

I wonder if this assertion backed by some empirical research ? I realize some people lose their job because of online piracy. Also, Some people lost their jobs following the introduction of ATMs. But we view ATMs as positive development since it made a certain service way cheaper. My guess is that the same is true about intellectual piracy — it makes distribution of culture and knowledge cheaper and therefore makes also the production of culture and knowledge cheaper. True, some companies, particularly the established ones, are damaged by intellectual theft. Other companies, particularly startups, benefit. One may argue that intellectual piracy destroys incentive to produce and therefore no new culture or knowledge will be produced absent some protection for intellectual property. But this is a claim that can be empirically checked no ? We live in a world of file sharing and user generated (often stolen) content sites. Are there less books written ?

Embassy Suite hotel Saturday morning. Photo by Jacob Leshno.

They say that when Alfred Tarski came up with his theorem that the axiom of choice is equivalent to the statement that, for every set {A}, {A} and {A\times A} have the same cardinality, he first tried to publish it in the French PNAS. Both venerable referees rejected the paper: Frechet argued there is no novelty in equivalence between two well known theorems; Lebesgue argued that there is no interest in equivalence between two false statments. I don’t know if this ever happened but it’s a cool story. I like to think about it everytime a paper of mine is rejected and the referees contradict each other.

Back to game theory, one often hears that the existence of Nash Equilibrium is equivalent to Brouwer’s fixed point theorem. Of course we all know that Brouwer implies Nash but the other direction is more tricky less known. I heard a satisfying argument for the first time a couple of months ago from Rida. I don’t know whether this is a folk theorem or somebody’s theorem but it is pretty awesome and should appear in every game theory textbook.

So, assume Nash’s Theorem and let {X} be a compact convex set in {\mathbf{R}^n} and {f:X\rightarrow X} be a continuous function. We claim that {f} has a fixed point. Indeed, consider the two-player normal-form game in which the set of pure strategies of every player is {X}, and the payoffs under strategy profile {(x,y)\in X^2} is {-\|x-y\|^2} for player I and {-\|f(x)-y\|^2} for player II. Since strategy sets are compact and the payoff function is continuous, the game has an equilibrium in mixed strategies. In fact, the equilibrium strategies must be pure. (Indeed, for every mixed strategy {\mu} of player II, player 1 has a unique best response, the one concentrated on the barycenter of {\mu}). But if {(x,y)} is a pure equilibrium then it is immediate that {x=y=f(x)}.

Update I should add that I believe that the transition from existence of mixed Nash Equilibrium in games with finite strategy sets to existence of mixed Nash Equilibrium in games with compact strategy sets and continuous payoffs is not hard. In the case of the game that I defined above, if {\{x_0,x_1,\dots\}} is a dense subset of {X} and {(\mu_n,\nu_n)\in \Delta(X)\times\Delta(X)} is a mixed equilibrium profile in the finite game with the same payoff functions and in which both players are restricted to the pure strategy set {\{x_1,\dots,x_n\}}, then an accumulation point of the sequence {\{(\mu_n,\nu_n)\}_{n\geq 1}} in the weak{^\ast} topology is a mixed strategy equilibrium in the original game.

A short presentation of a new paper by Ehud Lehrer and Dov Samet revisiting the question of characterizing situations in which no agreeable bet exist between players.  The paper answers the question for a countable state space. The case of arbitrary state space is still open.

For more GT papers clips of a different style look at posts of Rakesh and Noam.

Merry christmas !

There is not a single paper I published that I wouldn’t have changed in retrospect. Usually it’s because I regret giving in too quickly to bad “suggestions” from referees. But even when I was happy with the final version of the paper when I submitted it, I see things differently after a couple of months. Luckily, the journal system, with all its faults, at least don’t let us keep rewriting our papers. Otherwise, we might have got this


Here is Sam Harris, in an addendum to his post How Rich is Too Rich?

I would be interested to know if any economist has an economic argument against the following ideas:

Future breakthroughs in technology (e.g. robotics, nanotech) could eliminate millions of jobs very quickly, creating a serious problem of unemployment.

I am not suggesting that this is likely in the near term…. I am suggesting, however, that there is nothing that rules out the possibility of vastly more powerful technologies creating a net loss of available jobs and concentrating wealth to an unprecedented degree.

Read the rest of this entry »

Kudos to Eilon for organizing this great workshop. It was exciting to see the lecture hall in the English Literature building full every time I entered, and completely packed for Aumann’s talk. I had to sit on the stairs, as I used to do during my grad years when I was auditing rock-star professor Jerome Mandel’s Medieval English class and found out that the entire university were coalescing there to watch him performing the green knight’s entrance to King Arthur’s court. I only came to the afternoon sessions and still managed to attend talks about stochastic games, repeated games, implementations, auctions, coding, matching, decision theory, bounded rationality. I expected to meet mostly the math game theory community, but I recognized only a small fraction of the participants, who came from econ, math, cs, social sciences, philosophy. Is there any other discipline where a workshop can interest such a wide audience ? and is there any other discipline where a workshop in Israel can attract so many people from out of the country, including some of the big shots ?

I hear I missed an entertaining “ game theory whither” roundtable discussion. I am usually disappointed with these events, since all the seniors agree with each other about everything, and are often wrong about it to boot. But rumor has it that this time was different. The issue seems to have been the importance of cooperative game theory. One discussant suggested that a way to measure the impact of a topic is its coverage in textbooks, and a certain book was mentioned in which the core, Shapley value and bargaining are not even mentioned. I like this suggestion because of its circular nature: game theorists evaluate game theoretic ideas according to the impact they make on the same theorists who also produce these ideas. Wouldn’t life be easier if game theory was applicable for building bridges or curing cancer or making money in the stock market ? Easier perhaps, but not as interesting in my view. Anyway, the suggestion has caused some intellectual controversy. No food fight, but sufficiently close to be interesting.

And speaking of food, the humus you get in the cafeteria near the law school is an offense to all taste and decency, though non-Israelis still enjoyed it (no surprise, it’s still better than what you get in the states under `humus’). If you go to Jerusalem, Lina (just near the via dolorosa, where Jesus of Nazareth has walked twenty centuries ago) was pretty good. The best of the best is Ali Karawan in Jaffa, but I didn’t get to go there this time. And speaking of Jaffa, Rann Smorodinsky got our everlasting admiration for suggesting Haj Kahil for dinner. And btw, Rann didn’t invent Kalai-Smorodinsky bargaining solution when he was six, as somebody suggested to me. That Smorodinsky is the father.

A word about Aumann’s talk: He presented his argument in favor of correlated equilibrium, following his seminal 1987 paper and a more recent paper (pdf) in which he considers the interim stage, after the players receive some information, and asks what payoff  they can rationally expect to receive in the game. (The paper is quite known for his previous ambitious title “when all is said and done, how should you play and what should you expect”). Aumann starts by claiming that game theory should not tell you what to do (as in “play the Nash Equilibrium”) but should only tell you which procedure to follow, and the procedure is to maximize utility with respect to your beliefs. Presumably many readers of this blog are already aware of Aumann’s argument, so i’ll cut directly to the chase. The dubious part is the common prior assumption. To justify it Aumann appeals to a rational expectation argument. John Muth defined “rational expectations” as expectations that are the same as the predictions of the relevant economic theory. This is a seminal concept which in a way captures the essence of game theory and perhaps economics in general: The economic agent, the subject matter of economic theories, takes into account to the predictions of the theories. Whatever theory an economist uses to make predictions, the economic agents already respond to the same theory; Economists are no smarter than their subject matter (contrast with physicists who presumably are smarter than electrons.) But back to Aumann’s point: The prediction of the theory is, says Aumann, the common prior. But I have two reservations here: First, didn’t Aumann say that game theory doesn’t tell you what to do but only which procedure to follow ? So how come the common prior is a prediction about what players believe and what they do ? Second, there are many correlated equilibria in the game, so many possible predictions of the theory. How do the players know which one of them to respond to ?

Let me mention another presentation, by Michael Rabin, whose 1957 paper about computability in game theory I will take with me to a desert island . Rabin presented a simple (“even financiers understood it”) scheme for conducting seal bids auctions which preserves the secrecy of the bids but enables the auctioneers to prove to all participants the truthfulness of the outcome (pdf ). So for example in a second price auction, the auctioneer will verifiably announce the identity of the winner and inform the winner about his price without revealing any information to anybody else. What I found unsatisfactory in this paper — and this is not a criticism of the paper since it is probably completely unrelated to it goal — is the fact that it has nothing to do with game theory: the strategic aspects of auctions are irrelevant here. To my understanding, this is a paper about verifiable computation of a function when the input is distributed between several agents. Auctions are just an example. This is cryptology at its best, but I wish we would see some papers where cryptology is directly relevant for game theory.

Oh, and I need to register a caveat to my previous insinuation that you cannot make omelets out of game theory. Matching theory is an exception. Nobody will suggest to look at game theory textbooks to evaluate the importance of Gale Shapley, since this algorithm and its variations are actually used in real-life situations like assigning med grads to hospitals, kidney pair donation, matching students to public schools. Matching theory lives outside the GT-community circulation of ideas. There were two great matching talks in the workshop, one of Itai about two body problem in the residency program and one by Federico about testable implication of stable matching with and without monetary transfer from aggregate data.

Photos ? Unfortunately I didn’t take any during the workshop. But we have some from the trip. Here is one taken by Itai Ashlagi.
See if you can identify the location and the participants

That’s it folks, thanks to everybody who encouraged me to blog more. Will try.

In those happy occassions when a paper of mine is accepted for publication and I get the proofs from the journal, I usually just sign the consent for publication form without taking more than a glimpse at the proofs. The way I see it, my job is to get papers into journals, after that the papers should fend to themselves. Besides, the journal’s changes are usually either harmless or for the better.

Last, week, however, was an exception, not in my action because I still just signed the form as usual, but in my response to the proofs. Even after the quick glimpse I noticed that something is wrong, and after reading further I was horrified to see that all the references I made to myself (and, as readers of this blog know, I talk about myself a lot) were replaced by references to myself and my imaginary friends. In short, the word `I’ was eliminated from the paper and replaced with `we’. I thought I will just be furious about it for a couple of days and then forget the whole thing, but after a week of fuming with no forgeting in the horizon, I have no other option but to move to plan B, which is venting my indignation on you, dear TLTC readers.

Now look, I am cool with “we” that means “one”, to celebrate the fact that the validity of mathematical statements is independent of the person who happens to claim them, as in “Dividing by \pi, we get that the game admits an equilibrium”. But sometimes the automatic replacement of `I’ with `we’ garbles the meaning of the sentence. When I write “I call such a sequence of variables a random play”, the singular pronoun implies that this is not a universally recognized definition, but one that I have invented for the current paper. Change this “I” to “we”, as the journal did, and the implication is lost. And sometimes `we’ for `one’ is just ridiculous, as in “We review Martin’s Theorem in the appendix”. It is one thing to say that every intelligent creature recognizes that the game admits an equilibrium, and another thing to say that every intelligent creature reviews Martin’s Theorem in the appendix.

In some cases the editors were gracious enough to grant my singular identity but then they changed the person. “I thank, I proved, I don’t know” became “The author thanks, the author proved, the author doesn’t know”, which is even worse since it reminds me the way Israeli politicians speak about themselves.

What annoys me of here is the complete arbitrariness of the rule. The word “I” is perfectly legitimate in everyday language but banned from scientifc writing. I can’t think of any other word that is discriminated in this way. I guess next time if I don’t want to be called `we’ or `the author’ in my own papers then I will use all the puculiarities of academic writing like `it is conjuctured’, as if conjectures grow up on trees without somebody taking responsibility for them. I never understood why writers use these constructions so often. Now we know.

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