An Operations Research Tragedy

Hallo Señor Royal-Dominique Fennell.  Take a look at this film composed of several concepts in operations research and the management sciences, cross-posted from Palam to Idlewild:

The concepts we included were supply chain management, queuing theory, marketing, and risk analysis.  One other segment could have been on scheduling in which the hero is not able to have a job interview due to scheduling issues.  Also, we could have done health care analytics as a follow-on segment in which the hero recovers and gets back on top. 

YouTube is one of the tools of social business, especially for marketing.  Producing content specifically for YouTube and the web as opposed to just uploading existing physical-world content makes a difference in popularity:

Khan said that his videos and those on OCW have fundamentally different approaches, which may explain Khan Academy’s popularity. OCW lectures by MIT professors can be over an hour long and hit on many different topics. Khan’s videos are short — 10 to 15 minutes — and focus on very particular concepts.

“I think having that 10 minute video of someone walking through [a concept] conversationally, and they know you’re viewing this on the internet, I think those things could be pretty powerful,” said Khan.

You’ll notice that he highlighted the conversational aspect of his videos.  Although an old-school piece of web content, the Britney Spears Guide to Semiconductor Physics, wasn’t necessarily conversational, it did have something to distinguish it from regular non-web content.  Do you think clay animation + true accurate descriptions of technical concepts could be useful to people and become popular?

Conversations

Sat sri akal Señor Fauja Singh. Sorry about lacking on the return dialogue for the last few months.  A one-way dialogue is not a conversation by any stretch of the imagination and it doesn’t help with shaping our sciences.

“Science,” the physicist Werner Heisenberg once wrote, “is rooted in conversations.” As he saw it, scientists are rarely solitary thinkers but people who constantly talk: about ideas, findings, research techniques, and unresolved problems.  Some of these conversations last for a few minutes or hours.  But others continue for years or decades, shaping careers, disciplines, and even institutions.

The Google Scholar effect you have pointed out is very interesting and a real manifestation of how technology changes the way we as humans work.  There’s something else you’re well-aware of that hasn’t happened as much yet in universities and other academic institutions, but is reshaping the industrial/corporate world: the virtual office, where everyone works remotely and communicates via telephone, email, instant messaging, video chat, etc.  Some people think that this will increase productivity (however one wants to define productivity), but I am not sure.  While all the tools of social business are great, I don’t think they’re an adequate substitute for conversations of “random, serendipitous encounters that always seem to happen” (if I may borrow the phrase you wrote in describing the 2006 LIDS Student Conference). 

The part that is missing is the randomness and serendipity.  If you use Google Scholar, you won’t randomly read the paper printed before or after the one you’re looking for.  If you use social business tools exclusively, you won’t randomly run into a colleague in the hallway.  (Expertise recommendation only works when you are looking for specific expertise.)  I know you are interested in studying how new, creative science is done — I wonder if there is an analogy to evolution.  For evolution to be successful, you need both the random mutation and the natural selection; you can’t have just one or the other.

Let’s see how my view on social business evolves during Lotusphere next month after random, serendipitous encounters that always seem to happen at conferences.

Strategy and Design

As I had mentioned in my previous post, I had been on a Smarter Cities Challenge engagement in September, looking at combating the problem of vacant properties in Syracuse.  Besides spending time with a statistician, I also spent time with technology strategists.  This inspired me to read a Harvard doctoral dissertation by Charles Jason Woodward, Architectural Strategy and Design Evolution in Complex Engineered Systems.  He began his career at IBM before going to Harvard.  The beginning of the abstract is as follows.

Engineers have traditionally been trained to solve design problems without regard to the potentially competing interests of other designers. But just as technology strategists are increasingly drawn into the technical minutiae of product development decisions, engineers are increasingly exposed to the competitive forces that shape their requirements and the resources at their disposal. I propose that architectural strategy—the application of strategic thinking to system design problems—should therefore be integrated with the theory and practice of engineering design.

I found several parts of it to be rather interesting, and so I thought I’d share a thought or two from it with you.  First a working definition of strategic interdependence:

…artifacts are composed of many interacting parts, typically designed by teams of people spread across many organizations.  In turn, they function as parts of larger systems, such as the Internet and the global transportation network, whose designs evolve without centralized coordination or control.  Although design decisions are dispersed in such systems, the interests of the designers are often intertwined.  When one designer’s decisions affect the outcomes of another’s design process and vice versa, their designs are strategically interdependent.  Strategic designers act with an awareness of their influence on other designers, which often arises from technological interdependence among the artifacts being designed. 

The main theoretical construct introduced by Woodward is a generalization of a design structure matrix (DSM) (which finds all kinds of uses) and is called a design structure network (DSN).  It is rather expressive and is used to develop the theoretical construct of a system design game (SDG), where dynamics come into play.  As he says:

If architectural strategy were a game, then a design structure network would be an image of the playing field at a snapshot in time.  While a still picture may display evidence
of motion, one can neither predict the players’ trajectories nor explain their history without knowing something about the forces that propel them.  In the context of strategic system design, these forces arise from interactions among agents (such as designers, consumers, regulators, and financiers) that are mediated by artifacts and institutions.  

I know you have had some recent interest in dynamics of systems, so I wonder if this idea would be of interest.  In any case, Woodward goes on to perform Monte Carlo experiments to play out large SDGs and develops insight into system design with architectural constraints.  Let me not give away the punchlines.

In my earliest research work, I used Monte Carlo simulations to derive results, but have not used that methodology since.  Perhaps it is time to return to it: I just need a good problem.  Any suggestions?  Simulating a (smarter) city?   

The Drug called Google Scholar

I was just reading a little bit about dose-response functions in medicine and how they may have different shapes.  Somehow I had assumed that most treatments would have monotonic effects, but in fact they may not.  Consequently, there is often a need to perform statistical tests to see whether there is a monotonic trend. 

Continuing on with my scientometrics meme (this is getting worse than the birds, eh?), I went ahead and collected data for all years from 2004 to 2011 (the 2011 set does not yet contain all theses).  I had previously demonstrated that there were indeed more references in 2010 than in 2004, but was that a coincidence, or is there an actual monotonic trend?  If you plot out the mean and median, it does seem like there might be a noisy upward trend, but a formal test would be nice.  Of course, I am thinking of “technological progress” as the dose and the number of references as the response.

Looking around the internet for the appropriate statistical test for monotonicity, I found that this area of order-restricted inference is actually not at all well-settled.   This is particularly the case for unbalanced designs and non-parametric settings, as here.  Often this is due to computational difficulties. 

Notwithstanding, I decided to follow the regression-style method of Tukey et al. that “combines all the allowed principles of witchcraft.”  As Tukey et al. argue, using a unified regression is better than pairwise KS-tests or their equivalents, as one might have considered trying. 

In contrast to their setting  where doses have actual measures, e.g. in milligrams, in my setting it is very unclear what the “dose of technological progress” is.  Hence, rather than considering arithmetic, ordinal, and arithmetic-logarithmic candidate dose scalings and using the one with minimal p-value, I restricted myself to only ordinal scaling. As Capizzi et al. say, “the use of regression on a single scaling may generate controversy and doubt about one’s motives, especially in a regulatory environment,” but oh well.

Note that unlike traditional uses of regression for parameter estimation, here the goal is detection: to detect whether or not there is a monotonic trend.

So we have a sequence of ordinal doses, sample sizes, and mean responses as follows.

	2004	2005	2006	2007	2008	2009	2010	2010
Ordinal Dose	0	1	2	3	4	5	6	7
Sample Size	62	68	109	95	108	100	92	47
Mean Response	98.87	104.01	98.11	102.21	101.57	107.23	111.83	108.15

With this data, I went ahead and used SPSS to perform linear regression, getting a positive slope of 1.75 and a p-value for the hypothesis test against the null hypothesis of slope 0 of 0.09.  Hence, there is evidence in favor of a positive trend (at the 90% confidence level).  If one were using matlab, then the functions regstats and linhyptest would be useful.

Although I had learned a little statistics when doing some connectomics work in the past (e.g. not to make this mistake), I am certainly learning much more these days.  For example, when I was doing a Smarter Cities Challenge project for three weeks last month, I spent a good chunk of time with a statistician, who was big on what the data shows and picked up some tricks and tips. 

Incidentally, you might have some interest in these two new neuroscience papers, from Allerton and NIPS, though maybe you have already found them by browsing, searching, or being alerted.

Maheshwar Prasad Varshney

Our interview subject is M. P. Varshney, a vānaprasti who can often be found at the Celestial Knowledge Foundation.

Kush: You were involved in the establishment phases of many engineering colleges, including the Indian Institute of Technology (Kharagpur), Motilal Nehru Regional Engineering College, King Mongkut’s University of Technology Thonburi, and the University of Technology (Baghdad).  What were some of the common challenges that you faced?

M. P. Varshney: The challenges faced in the 4 projects relating to establishment of new engineering colleges specified by you had little in common.  This will be clear as I talk of each one of these in succession.

1. Indian Institute of Technology, Kharagpur.  It was conceived on the lines of MIT in the USA to be the only institute of excellence in India for advanced technological studies and research.  Things went on in this direction in the beginning, but soon four more IITs were established for political considerations and subsequently the number increased to seven.  Now, under orders from the government another eight IITs are being set up with no buildings, equipment, faculty, etc. and students for undergraduate studeies admitted therein are studying in the existing seven IITs that were and are already short in the number of faculty members for their normal operation.  Thus the lofty ideals of setting up one IIT (at Kharagpur) have been thrown to the winds.  Personally I learnt a lot about the problems faced in setting up the electrical engineering department at IIT Kharagpur that stood in good stead in my career.
2. Motilal Nehru Regional Engineering College, Allahabad.  This was one of ten engineering colleges set up for providing facilities primarily for undergraduate engineering education in different regions in India and did not aim at being colleges of excellence like the five IITs existing at the time.  The idea, no doubt, was good but execution faulty.  At the college mentioned, all the powers (administrative, academic, etc.) were vested in the principal, an engineer returned from the state irrigration department who had no experience or knowledge at all about engineering education.  This led to many problems that persisted for years even after that individual was finally removed.  Being head of electrical engineering department there, I was thus faced with problems and challenges in setting up the department, laboratories, etc. in the new engineering college.
3. University of Technology, Baghdad, Iraq.  This was a project aided by UNESCO where some necessary equipment was supplied by UNESCO and international experts were deputed to advise the local authorities in setting up the new university.  There then existed a college of engineering as part of the Baghdad University and they were opposed to the setting up of this new center for higher engineering education.  Unlike India, the number of Iraqis qualified to teach were inadequate and many foreigners were recruited for teaching jobs.  Language was another barrier as teaching was done in English of which the students had poor knowledge.  Also the administration was in the hands of poorly qualified personnel who would manage not to abide by the advise of the experts.  Te funds provided by UNESCO were in accordance with an agreement with the Iraqi government at a higher level politically and that resulted in a number of challenges and problems where the experts had no say.
4. King Mongkut’s University of Technology, Thonburi, Thailand.  This too was a UNESCO assisted project.  There was similarity in the challenges faced here as those at the University of Technology, Baghdad.  Here the project consisted of upgrading an existing Thonburi Technical Institute to a higher level where students would get university degrees instead of a lowe rlevel technical diploma.  There was opposition from the nearby existing college of engineering at Bangkok University, similar to what the position was in Baghdad.  The faculty at the institute was poorly qualified for upgradation and no foreign teachers were recruited.  The authorities waited that the faculty members who had been sent abroad for higher studies under UNESCO fellowships come back.  In fact much more could have been achieved with greater cooperation and coordination with Thai authorities.

K: So in one word, politics. 

You are an electrical engineer by formal education, but for two decades now you have been primarily been studying the Bhagavad Gītā.  There exist electrical engineers who have attempted to understand the Gītā using the math and science of today, for example Robert W. Newcomb using field theory.  Should one attempt to understand the Gītā through the lens of modern science and engineering, or does it require a completely different thought process?

MPV: I am of the opinion that it may not be possible to understand the Bhagavad Gītā, or for that matter any text related to spiritual studies, which is what Bhagavad Gītā primarily is, using the so-called scientific process.  The Bhagavad Gītā makes a distinction between the spirit and matter, and primarily deals with spirit soul, while the other deals with material phenomena exclusively.  Even though the spirit and matter coexist, the twain never meet.  Spirit and matter, being diametrically opposite, it seems unlikely that any reconciliation between the two approaches may be possible.

Lav: Interesting thoughts on the distinction between spiritual and material phenomena.  As you know, the name of this blog is information ashvins and that both of us have invested some time thinking about the nature of  information.  In some of my writing, I have tried to make a distinction between matter and information, or energy and information.  To me it, seems that information is intertwined with, but somewhat distinct from material phenomena.  What do you think about informational phenomena?

MPV: It’s my view that information, energy, and matter come under one and the same category.  Spiritual (soul) consciousness is distinct from material consciousness and as I mentioned earlier, the twain don’t meet.  To quote Albert Einstein: “When I read the Bhagavad Gita and reflect about how God created this universe everything else seems so superfluous.”

Reasonable Doubt

Just to build on my previous post, I thought I’d go ahead and actually do the scientometrics experiment I had described.  I had data on the number of references in MIT EECS doctoral theses from the year 2010.  I also collected the same data for the year 2004.

My first step was to exclude theses that had “chapter-style” references rather than all at the end as in “monograph-style”.  Luckily (at least from my viewpoint), this “chapter-style” trend is not at all popular in EECS: for 2004 data, I excluded 4/66 theses and for 2010 data, 6/98 theses.  On the contrary, as noted by Nils T. Hagen in a paper “Deconstructing doctoral dissertations: how many papers does it take to make a PhD?“:

The traditional single-authored monograph-style doctoral dissertation has declined into relative obscurity in the natural and biomedical sciences (cf. Lariviere et al. 2008), and been largely superseded by a modern PhD thesis consisting of a collection of multi-authored manuscripts and publications (e.g. Powell 2004).

After expurgating the data, I wanted to see whether there was a huge difference in the distributions.  In terms of first order moments, in 2004 the average number of references was 99 whereas in 2010 the average number of references was 112.  This definitely suggests that there has been an increase in number of references, as I had conjectured.  To do a full distributional characterization, I plotted the empirical cumulative distribution functions, as seen below.

Again, it looks like there are generally more references in 2010 than in 2004.  But is there a way to quantify it?  In their paper, “Critical Values for the One-Sided Two-Sample Kolmogorov-Smirnov Statistic,” Mitchell H. Gail and Sylvan B. Green say:

The Kolmogorov-Smirnov one-sided two-sample statistic is used to test the null hypothesis F = G against the alternate F > G where F and G are distribution functions.  If the random variables X and Y correspond to F and G, respectively, then the one-sided alternative is that Y is stochastically greater than X.  For example, one is often interested in the one-sided alternative that survival times (Y) with a new medical treatment are longer than survival times (X) with conventional therapy.  The two-sided alternative is of less interest in this case.

This seems perfect, so I ran the one-sided two-sample Kolmogorov-Smirnov test on the expurgated data, using matlab.  In particular with the command:

 [h,p] = kstest2(d_2010,d_2004,0.05,'smaller')  

I got the result h = 1 and p = 0.0380.  This implies that there is a statistically significant difference between the two distributions, and in particular theses from 2010 have more references than theses from 2004.

You might be worried that the Kolmogorov-Smirnov test is designed for continuous-valued data rather than discrete-valued data as here, but don’t be alarmed.  As noted by W. J. Conover in a paper “A Kolmogorov Goodness-of-Fit Test for Discontinuous Distributions“:

Studies of the Kolmogorov test with discontinuous distributions appear to be quite limited.  The Kolmogorov test is known to be conservative if F(x) is discrete.

and by G. E. Noether in a paper “Note on the Kolmogorov Statistic in the Discrete Case“:

A simple demonstration of the conservative character of the Kolmogorov test in the case of discrete distributions is given.

But even with this assurance, one might wonder why the 95% confidence value is the right one to choose or even whether the p-value approach to statistics is the right one overall. 

On the 0.95 value, I was reading a paper by Simon that tried to quantify what regular people use as their standard of proof, e.g. in jury trials.  She concluded that the standard of reasonable doubt was between 0.70 and 0.74, so maybe 0.95 is not the most appropriate choice.  What do you think?

On the null hypothesis approach to statistics, I’ve recently been learning more about Bayesian data analysis methods, and am starting to feel that they might be a better way to go, but I am still not sure.  What is your take?

Notwithstanding, standard statistics do have really neat results, e.g. the Dvoretzky–Kiefer–Wolfowitz inequality, which somehow doesn’t seem to be used as much as it could be.  Are there any standard machine learning applications that you are aware of?

Da Bears

That was some good stuff in your previous post.  Yeah, it is amazing how quickly scientific papers are generated in the worldwide system.  I believe this causes a certain sense of information overload that many people feel.  Moreover, not only are there an increasing number of papers, but there has also been an emergence of putatively new fields of study, like synthetic biology, connectomics, and service science.  Of course it remains to be seen whether these fields remain viable or whether they collapse.

I was reading this paper “Google Effects on Memory: Cognitive Consequences of Having Information at Our Fingertips” by Sparrow, Liu, and Wegner that recently appeared in Science.  As the authors say, the internet has caused a huge technological shift in how information can be used:

In a development that would have seemed extraordinary just over a decade ago, many of us have constant access to information. If we need to find out the score of a ball game, learn how to perform a complicated statistical test, or simply remember the name of the actress in the classic movie we are viewing, we need only turn to our laptops, tablets, or smartphones and we can find the answers immediately. It has become so commonplace to look up the answer to any question the moment it occurs that it can feel like going through withdrawal when we can’t find out something immediately.

Moreover, they go on to describe experiments that demonstrate how technology has changed the nature of human cognition itself.  They essentially demonstrate that “our internal encoding is increased for where the information is to be found rather than for the information itself.”  They discuss their results as follows:

These results suggest that processes of human memory are adapting to the advent of new computing and communication technology. Just as we learn through transactive memory who knows what in our families and offices, we are learning what the computer “knows” and when we should attend to where we have stored information in our computer-based memories. We are becoming symbiotic with our computer tools, growing into interconnected systems that remember less by knowing information than by knowing where the information can be found. This gives us the advantage of access to a vast range of information, although the disadvantages of being constantly “wired” are still being debated.

Maybe the Google Scholar generation is fine with the growing volume of information due to these cognitive changes.  From introspection, I certainly feel that I don’t know all that much in the scientific literature, but rather I either know where to find it or feel confident that I could search for it if I needed to.  Recently I’ve slowly been learning a more empirical approach to life, so perhaps I should offer more evidence than simply introspection.

On the plane ride back from St. Petersburg, I was telling someone that I probably have a much more “referency” writing style than others.  (Although one might cynically feel that references are a way to show off erudition, in my approach to writing I include references because I know things “by reference” and also to give proper attribution.)  To test this hypothesis using scientometrics, I went through all 98 doctoral theses in EECS at MIT from the year 2010.  Although I am definitely in the top three, with 371 references, I do not hold the top spot.  That distinction is held by Umit Demirbas with his thesis on Low-cost, highly efficient, and tunable ultrafast laser technology based on directly diode-pumped Cr:Colquiriites, which has 416 references.  Your thesis comes in at #5 with 230 references, so it seems that you too have something of a referency writing style.  As a point of comparison on the other side, Mike Rinehart’s thesis on The value of information in shortest path optimization has 19 references.

To make an actual generational argument though, perhaps I should get the numbers from another year, like 2004.  Anyone up for crowdsourcing the data collection?  Also what would be an appropriate statistical test for me to look up to make such an argument?

Shifting gears to the start of the football season, in a certain sense I am more intrigued by UConn than by Syracuse itself.  Of course this is primarily due to Paul Pasqualoni and George DeLeone, a chief architect of the freeze option offense.  People often used to say that the Syracuse football playbook was too big and confusing; certainly larger than at other schools.  Coming back to the question of too much or too little, I wonder if there is a way to make an argument about the pluses and minuses of strategic complexity in a competition like football with bounded agents; perhaps following the lines of Daskalakis?

Anyway, let me leave it there and not mention anything about cognitive history or the difficulty of ranking multivariates, or my new found fear of non-human hominids.

Goldilocks

Welcome home Señor Rasputin the mad monk. How was St. Petersburg?

In the Klosterman article, I liked how Ato Boldon said, “sprinters believe that — someday — somebody will run the 100 meters and the clock will read 0.00.”  Related to scientific progress, modeling records, and crowding, I recently came across two passages.  First, from the Financial Times:

Da Vinci was able to achieve so much, so broadly, because so little was known. It was possible to make leaps forward in scientific understanding armed with little more than a keen eye and a vivid imagination. Those times are long gone. Approximately 3,000 scientific articles are published per day – roughly one every 10 seconds of a working day. We can now expect that these papers will, each year, cite around five million previous publications. And the rate of production of scientific papers is quadrupling every generation. The percentage of human knowledge that one scientist can absorb is rapidly heading towards zero.

Second, from the IEEE Spectrum:

Given any prospective problem, a search may reveal a plethora of previous work, but much of it will be hard to retrieve. On the other hand, if there is little or no previous work, maybe there’s a reason no one is interested in this problem. You need something in between. Moreover, even in defining the problem you need to see a way in, the germ of some solution, and a possible escape path to a lesser result, like the runaway truck ramps on steep downhill highways.

Timing is critical. If a good problem area is opened up, everyone rushes in, and soon there are diminishing returns. On unimportant problems, this same herd behavior leads to a self-approving circle of papers on a subject of little practical significance. Real progress usually comes from a succession of incremental and progressive results, as opposed to those that feature only variations on a problem’s theme.

You asked if there is some distribution that would model scientific progress.  I don’t think that the modeling would be much different, even including crowding, from other types of models in the theory of records described in the book by Barry Arnold et al. you linked to, with one qualification.  How do you quantify scientific progress?  It is not simple to measure like sprints or floods.  (The Eurekometrics plots come with the qualification that they are of “areas where discovery – not simply scientific output – is well-defined and may be easily quantified.”)

By the way, Barry Arnold was Bill Hanley‘s advisor.

Overcrowding

Very interesting take on overcrowding.  I must admit I hadn’t thought too much about the physical volumetric notion of overcrowding until you brought it up, but it is an interesting idea.  As you know, things in high dimensions do get concentrated; as Han wrote on p. 7 of his book Information-Spectrum Methods in Information Theory, “That is, almost all probability is concentrated on the dust Tn if n is sufficiently large.”  But with people, I’m not sure what exactly is the notion of dimension, or of surface area, or of volume.  

Notwithstanding, a recent paper by Boudreau, Lacetera, and Lakhani on crowdsourcing contests makes the following statement:

Research in economics suggests that increasing the number of competitors who are admitted to a contest will reduce the likelihood of any one competitor winning, thereby reducing incentives to invest or exert effort and lowering overall innovation outcomes (Che and Gale 2003, Fullerton and McAfee 1999, Taylor 1995).  Similar predictions and findings on negative incentive effects have been found in research in sociology and psychology (Bothner et al. 2007, Garcia and Tor 2009). Overall, the literature has generally recommended against free entry into contests, with some models specifically determining the ideal number of competitors to be just two (Che and Gale 2003, Fullerton and McAfee 1999).

Relatedly, I assume you’ve heard of Metcalfe’s Law, right?  This asserts that the value of a network is the square of the number of members.  Such a law doesn’t take overcrowding into effect at all.  Indeed, McAfee and Oliveau argued against Metcalfe’s Law by citing several effects: saturation, cacophony, contamination, clustering, and search costs.  Cacophony is clearly an effect of overcrowding: “When too many users of a network make interaction difficult—in a crowded Internet chat room, for example—cacophony is the result.”  Disappointingly, the solution to cacophony that they propose is to limit new members or to kick out existing members.

A totally different kind of overcrowding is in scientific fields due to herding behavior.  It seems like many people jump on bandwagons when new tools emerge that allow previously unknowable things to become knowable: this could be a newish mathematical insight as in compressed sensing or a newish sensing technology like electron microscopy.  By the way, what happened to your electron micrographs?  Anyway, I recently followed a link from Solomon Hsiang‘s blog, Fight Entropy, about Eurekometrics.  One of the points is that the ease of scientific discovery decays with time, so maybe that explains some of these bandwagons.

Speaking of ease of accomplishment as a function of time, what do you think of Klosterman’s view on the 100m world record?  For those who like visuals, here is a graphical depiction from Wikipedia:

and for good measure, here is the 200m world record:

I wonder if there is some record value distribution method to study scientific progress, as there is for flood levels or world records in athletics.  Moreover, how does overcrowding play into it?

Nobody Goes There Anymore. It’s Too Crowded.

Señor Henri de Toulouse-Lautrec, that number of connections doesn’t include people from the past that you might encounter during Midnights in Paris, does it?

At the SSP Workshop last week, in which I presented a bound on reject option risk for ensemble classification and our work on minimax Bayes risk error quantization, Vikram Krishnamurthy brought up the paper A Global Game with Strategic Substitutes and Complements in his plenary talk.  One key notion in that work is summarized by the following quote by Yogi Berra:

Nobody goes there anymore.  It’s too crowded.

The idea is that when frequenting an establishment, your utility depends on the number of other patrons also frequenting that establishment.  When there is no one else there, you receive no utility.  As it fills up, you get more utility.  When it starts becoming crowded, however, your utility goes down.  This utility function is quasiconcave and applies equally well to sensor networks and cognitive radio as pointed out by Krishnamurthy.  An example of such a function is below, where α is the filling fraction of the establishment.

Does this sort of utility function arise in the crowdsourcing you have been studying?  Do you think that this utility model is at all related to Figure 2(A) of your neuroscience paper or to the surface area of hyperspheres that you pointed out here previously?  (In your neuroscience work, there is a volume constraint with neurons occupying some volume.  Similarly, in an establishment, there is a volume constraint with patrons occupying some volume.  In both cases, more is better until the space starts getting crowded.  For the hypersphere, again there is a volume constraint.)