Four agents are observing infinite streams of outcomes in ${\{S,F\}}$. None of them knows the future outcomes and as good Bayesianists they represent their beliefs about unknowns as probability distributions:

• Agent 1 believes that outcomes are i.i.d. with probability ${1/2}$ of success.
• Agent 2 believes that outcomes are i.i.d. with probability ${\theta}$ of success. She does not know ${\theta}$; She believes that ${\theta}$ is either ${2/3}$ or ${1/3}$, and attaches probability ${1/2}$ to each possibility.
• Agent 3 believes that outcomes follow a markov process: every day’s outcome equals yesterday’s outcome with probability ${3/4}$.
• Agent 4 believes that outcomes follow a markov process: every day’s outcome equals yesterday’s outcome with probability ${\theta}$. She does not know ${\theta}$; Her belief about ${\theta}$ is the uniform distribution over ${[0,1]}$.

I denote by ${\mu_1,\dots,\mu_4\in\Delta\left(\{S,F\}^\mathbb{N}\right)}$ the agents’ beliefs about future outcomes.

We have an intuition that Agents 2 and 4 are in a different situations from Agents 1 and 3, in the sense that are uncertain about some fundamental properties of the stochastic process they are facing. I will say that they have structural uncertainty’. The purpose of this post is to formalize this intuition. More explicitly, I am looking for a property of a belief ${\mu}$ over ${\Omega}$ that will distinguish between beliefs that reflect some structural uncertainty and beliefs that don’t. This property is ergodicity.

Abraham Neyman and Sergiu Hart are two of the prominent mathematical game theorists to date. Neyman contributed immensely to the study of the Shapley value, stochastic games, and repeated games and complexity. Hart contributed significantly to the study of correlated equilibrium and adaptive processes leading to it, value theory, and formation of coalitions.
Both Abraham and Sergiu will be 66 next year. To celebrate this rare occasion, the Center for the Study of Rationality at the Hebrew University of Jerusalem organizes two conferences, one in honor of each of them. The conference in honor of Abraham will be held on June 16–19, 2015, and the conference in honor of Sergiu will follow on June 21–24, 2015.
Mark the dates and reserve tickets.

You may have heard about ResearchGate, the so called facebook of scientists. Yes, another social network. Its structure is actually more similar to twitter: each user is a node and you can create directed edges from yourself to other users. Since I finally got rid of my facebook account (I am a Bellwether. In five years all the cool guys will not be on facebook), I decided to try ResearchGate. I wanted a stable platform to upload my preferable versions of my papers so that they will be the first to pop up on google. Also, I figured if I am returning to blogging then I need stuff to bitch about. ResearchGate only partially fulfill the first goal, but it does pretty well with the second.

Last week I wrote a post about two issues with Elsevier’s e-system, which is the system that all journals run by Elsevier, including Games and Economic Behavior and Journal of Mathematical Economics, use for handling submissions: the fact that sometimes reviewers can see the blinded comments that other reviewers wrote to the editor, and the user agreement that allows Elsevier to change its terms without notifying the users.

After I corresponded with the editors of Games and Economic Behavior and Journal of Mathematical Economics and with the Economics Editor of Elsevier, the reason for the privacy breach became clear: the e-system allows each editor to choose whether the blinded comments of one referee to the author and the blinded comments of one referee to the editor will be seen by other reviewers. For each type of blinded comments the editor can decide whether to show it to all reviewers or not. Each editor makes his or her own choice. I guess that often editors are not aware of this option, and they do not know what was the choice that the previous editor, or the one before him, made.

Apparently, the configuration of Games and Economic Behavior was to allow reviewers to see only the blinded comments to the author, while the configuration of Journal of Mathematical Economics was to allow reviewers to see both types of blinded comments. Once the source of the problem became clear, Atsushi Kajii, the editor of Journal of Mathematical Economics decided to change the configuration, so that the blinded comments of reviewers to the editor will not be seen by other reviewers. I guess that in few days this change will become effective. Elsevier also promised to notify all of its journals, in which the configuration was like that of JME, about this privacy issue, and let the editors decide whether they want to keep this configuration or change it. In case this configuration remains, they will add a warning that warns the referee that the blinded comments can be read by other reviewers.

I am happy that the privacy breach came to a good end, and that in the future the e-system will keep the privacy the referees.

Regarding the second issue, Elsevier is not willing to change its user agreement. Reading the user agreements of other publishers, like Springer and INFORMS, shows that user agreements can be reasonable, and not all publishers keep the right to change the user agreement without notifying the users. The Economics Editor of Elsevier wrote: “This clause is not unreasonable as the user can choose to discontinue the services at any time.” As I already wrote in the previous post, I choose to discontinue the service.

When I give a presentation about expert testing there is often a moment in which it dawns for the first time on somebody in the audience that I am not assuming that the processes are stationary or i.i.d. This is understandable. In most modeling sciences and in statistics stationarity is a natural assumption about a stochastic process and is often made without stating. In fact most processes one comes around are stationary or some derivation of a stationary process (think the white noise, or i.i.d. sampling, or markov chains in their steady state). On the other hand, most game theorists and micro-economists who work with uncertainty don’t know what is a stationary process even if they have heard the word (This is a time for you to pause and ask yourself if you know what’s stationary process). So a couple of introductory words about stationary processes is a good starting point to promote my paper with Nabil

First, a definition: A stationary process is a sequence ${\zeta_0,\zeta_1,\dots}$ of random variables such that the joint distribution of ${(\zeta_n,\zeta_{n+1},\dots)}$ is the same for all ${n}$-s. More explicitly, suppose that the variables assume values in some finite set ${A}$ of outcomes. Stationarity means that for every ${a_0,\dots,a_k\in A}$, the probability ${\mathop{\mathbb P}(\zeta_n=a_0,\dots,\zeta_{n+k}=a_{n+k})}$ is independent in ${n}$. As usual, one can talk in the language of random variables or in the language of distributions, which we Bayesianists also call beliefs. A belief ${\mu\in\Delta(A^\mathbb{N})}$ about the infinite future is stationary if it is the distribution of a stationary process.

Stationarity means that Bob, who starts observing the process at day ${n=0}$, does not view this specific day as having any cosmic significance. When Alice arrives two weeks later at day ${n=14}$ and starts observing the process she has the same belief about her future as Bob had when he first arrives (Note that Bob’s view at day ${n=14}$ about what comes ahead might be different from Alice’s since he has learned something meanwhile, more on that later). In other words, each agent can denote by ${0}$ the first day in which they start observing the process, but there is nothing in the process itself that day ${0}$ corresponds to. In fact, when talking about stationary processes it will clear our thinking if we think of them as having infinite past and infinite future ${\dots,\zeta_{-2},\zeta_{-1},\zeta_0,\zeta_1,\zeta_2,\dots}$. We just happen to pop up at day ${0}$.

The last of the trio, Harold Kuhn, passed away on July 2nd, 2014. Upon hearing the news, I was moved to dig up some old lecture notes of Kuhn’s in which KTK is stated an proved. I’ve been carrying them around with me since 1981. From the condition they are in, this must have been the last time I looked at them. With good reason, for as I re-read them, it dawned upon me how much of them I had absorbed and taken to be my own thoughts. Kuhn motivates the KTK theorem by replacing the non-linear functions by their first order Taylor approximations. This turns the exercise into a linear program. The LP duality theorem suggests the theorem to be proved, and the separating hyperplane theorem does the rest. For details see the relevant chapter of my book. The notes go on to describe Kuhn and Tucker’s excitement and subsequent despair as they uncover a counterexample and the need for a constraint qualification.

William Karush, who passed in 1997, had arrived at the same theorem many years earlier in his 1939 University of Chicago Masters Thesis (Kuhn-Tucker is 1951). When Kuhn learned of Karush’s contribution through a reading of Takayama’s book on Mathematical Economics. Upon doing so he wrote Karush:

In March I am talking at an AMS Symposium on “Nonlinear Programming – A Historical View.” Last summer I learned through reading Takayama’s Mathematical Economics of your 1939 Master’s Thesis and have obtained a copy. First, let me say that you have clear priority on the results known as the Kuhn–Tucker conditions (including the constraint qualification). I intend to set the record as straight as I can in my talk.

The missive closes with this paragraph:

Dick Cottle, who organized the session, has been told of my plans to rewrite history and says you must be a saint’ not to complain about the absence of recognition. Al Tucker remembers you from RAND, wonders why you never called this to his attention and sends his best regards.

Karush’s reply, 6 days later, equally gracious:

Thank you for your most gracious letter. I appreciate your thoughtfulness in wanting to draw attention to my early work. If you ask why I did not bring up the matter of priority before, perhaps the answer lies in what is now happening – I am not only going to get credit for my work, but I am going to crowned a “saint”.

Games and Economic Behavior, Journal of Economic Theory, Journal of Mathematical Economics, and Economics Letters are four journals that publish game theoretic papers and are published by Elsevier. They all use Elsevier e-system to handle submissions. I already talked in the past about the difficulty of operating these e-systems. Rakesh discussed the boycott against Elsevier. Recently I had some experience that made me stop using the Elsevier’s system altogether, even though I serve on the editorial board of Games and Economic Behavior. I will not use Émile Zola’s everlasting words for such an earthly matter; I will simply tell my experience.

1) The e-system seems to be sometimes insecure.

I was surprised when a referee with whom I consulted on the evaluation a paper (for GEB) told me that the system showed to him the private message that the other referee wrote to me, and that the same thing happened to him with JME. To prove his point, he sent to me screenshots with the private letter of the other referee for JME.

2) The user agreement of Elsevier is a contract that one should never agree to sign.

I guess no one bothered to read the user agreement of Elsevier. I did. The first paragraph binds us to the agreement:

This Registered User Agreement (“Agreement”) sets forth the terms and conditions governing the use of the Elsevier websites, online services and interactive applications (each, a “Service”) by registered users. By becoming a registered user, completing the online registration process and checking the box “I have read and understand the Registered User Agreement and agree to be bound by all of its terms” on the registration page, and using the Service, you agree to be bound by all of the terms and conditions of this Agreement.

The fourth paragraph, titled “changes” says that any change made to the contract is effective immediately, and so it binds you. If you want to make sure they did not add some paragraph to which you disagree, you must read the whole user agreement every time you use the system.

Elsevier reserves the right to update, revise, supplement and otherwise modify this Agreement from time to time. Any such changes will be effective immediately and incorporated into this Agreement. Registered users are encouraged to review the most current version of the Agreement on a periodic basis for changes. Your continued use of a Service following the posting of any changes constitutes your acceptance of those changes.

I contacted Elsevier about the user agreement and got the following response:

The Elsevier website terms and conditions (see http://www.elsevier.com/legal/elsevier-website-terms-and-conditions) cannot be customized upon request; however, these terms and conditions do not often change and notification would be provided via the “Last revised” date at the bottom of this page.  The current terms and conditions were Last revised: 26 August 2010.

Well, it is comforting that they did not make any change in the past four years, but will Elsevier’s CEO agree to open an account in a bank that has the “change” paragraph in the contract?

I stopped using the e-system of Elsevier, both as a referee and as an editor.

One of the most engaging books of academic politics, is C. P. Snow’s The Masters’. In it,  a scene describing the debate between factions (Science and Arts) over who should be elected to a college fellowship. The master is in the Arts camp. One of the Science camp urges his candidate upon the college with these words (from memory, as my copy is beyond my reach): “He has written the absolute last word on the subject.” To which the master, responds: “Why can’t you chaps ever have the first word on the subject?” As the narrator, Eliot notes, it was an impolitic response, but recognizes that the master could not resist because it felt good on the tongue.

From Swansea comes another example  of the inability to resist something that felt good on the tongue. A note from the head of Swansea University’s school of management to his colleagues (do they still have those at UK universities?):

Some wags call for the removal of some or all of the school’s top management team. Yes, well don’t hold your breath. Or actually, do.

Roscoff is a village at the north-west corner of France, located on a small piece of land that protrudes into the English canal. Right here, in 1548, the six-year-old Mary, Queen of Scots, having been betrothed to the Dauphin François, disembarks.

As far as I understood, the most common sights in the area are tourists and sea food. As far as I can tell, the main advantage of Roscoff is the Laboratoire Biologique, which is used to host conferences. Every now and then the French game theory group makes use of this facility and organizes a conference in this secluded place. The first week of July was one of these nows and thens. This is my third time to attend the Roscoff conference, and I enjoyed meeting colleagues, the talks, and the vegetarian food that all non-sea-food eaters got.

Here I will tell you about one of the talks by Roberto Cominetti.

Brouwer’s fixed point theorem states that every continuous function $f$ that is defined on a compact and convex subset $X$ of a Euclidean space has a fixed point. When the function $f$ is a contraction, that is, when there is $ρ ∈ [0,1)$ such that $d(f(x),f(y)) ≤ ρ d(x,y)$ for every $x,y \in X$, then Banach’s fixed point theorem tell us that there is a unique fixed point $x*$ and there is an algorithm to approximate it: choose an arbitrary point $x_0 ∈ X$ and define inductively $x_{k+1} = f(x_k)$. The sequence $(x_k)$ converges to $x*$ at an exponential rate.

When the function $f$ is non-expansive, that is, $d(f(x),f(y)) \leq d(x,y)$ for every $x,y \in X$, there may be more than a single fixed point (e.g., $f$ is the identity) and the sequence defined above need not converge to a fixed point (e.g., a rotation in the unit circle).

In his talk, Roberto talked about a procedure that does converge to a fixed point when $f$ is non-expansive. Let $(α_k)$ be a sequence of numbers in $(0,1)$. Choose $x_0 ∈ X$ in an arbitrary way and define inductively $x_{k+1} = α_{k+1} f(x_k) + (1-α_{k+1}) x_k$. Surprisingly enough, under this definition the distance $d(x_k,f(x_k))$ is bounded by

d(x_k,f(x_k)) ≤ C diameter(X) / \sqrt( α_1 (1-α_1) + α_2 (1-α_2)  + … + α_n (1-α_n) ),

where C = 1/\sqrt(π).

In particular, if the denominator goes to infinity, which happens, for example, if the sequence $(α_k)$ is constant, then the sequence $(x_k)$ converges to a fixed point. Since the function that assigns to each two-player zero-sum strategic-form game its value is non-expansive, this result can become handy in various situations.

This is a good opportunity to thank the organizers of the conference, mainly Marc Quincampoix and Catherine Rainer, who made a great job in organizing the week.

Its not: why haven’t I one won? I have. A sixth form science prize. In my salad days I would day dream about winning the big one (to the sound of Freddy Mercury crooning We Are The Champions’). But, one ages and comes to term with one’s mediocrity.

In the good old days, when men were men and sheep were nervous, prizes were awarded for accomplishing particular tasks. The French academy of sciences, for example, established in 1781, I think, a system of prizes or contests. A committee would set a goal (in 1766 it was to solve the 6 body problem, in 1818 it was explain the properties of light) and submissions judged after a deadline and a prize, if merited, awarded. One sees that today with the X -prize and the Clay prize. The puzzle with these prizes is would the challenge they highlight not be undertaken in their absence? For example, resolving P = NP has been around well before the Clay prize and many a bright young thing had already given it serious thought.

Many prizes are achievement’ awards, given out in recognition of a great accomplishment after the fact. Some are awarded by learned societies and named in honor of an ancient worthy (Leibniz, Lagrange, Laplace etc.) Others are funded by private individuals (Nobel, Nemmers, Simons etc.).

Some learned societies have a surfeit of prizes (Mathematics) that are concentrated in the hands of a few. Indeed, one might be able to construct a partial order of the prizes and come to the conclusion that some prize X can only be awarded provided prize Y has already been secured. Once again, there is the incentive question. It is hard to imagine the prize winner strives and continues to do so in the anticipation of winning further prizes. If the purpose of the prize is to honor the work (rather than the individual) why give the $$’s to the individual? Perhaps better to take the$$’s, divide them up and hand it to junior researchers in the same area telling them they have received it in honor of X, a pioneer of the field.

Other learned societies have very few prizes (American Economic Association). There is the Clark medal (famous), Walker medal (discontinued after Nobel), Ely Lecture and the Distinguished Fellow (who?). No doubt, this is a great comfort for the members’ status anxiety. Although I have it heard it said that a paucity of awards can adversely affect a discipline in that it lessens its members chances of securing grants. No doubt this is why some learned societies have prizes for every age group and speciality one can imagine: best under 40 in applied nobble nozing theory.

Why do private individuals fund prizes? Nobel is the archetype. Is it a way to purchase reflected glory? The founders of Facebook and Google are famous in their own right, so it is hard to see how a prize will burnish their images. Perhaps they genuinely wish to support research into topic X. One can easily imagine more effective ways to do this via grants, fellowships and conferences. Indeed, both the Kavli and Simons do just this (in addition to handing out prizes). Perhaps its advertising. If one wishes to publicize the importance of some field, does awarding a generous prize buy more publicity than a simple advertisement or cultivating journalists? Unclear. How many have heard of the recent breakthrough’ prizes?