If I were designing the experiment, I would have the control group be to play a different game instead of having it be maths instructions.
You generally don’t want test subjects to know whether they are in the control condition or not. So if you’re going to make it be maths instructions, you probably shouldn’t tell them what the experiment is designed to test at all, until you’re debriefing at the end. If you tell people you are recruiting that you are testing the effects of playing computer games on statistical reasoning, then the people in the control condition won’t need to realize that what you’re really testing is whether your RPG in particular helps people think about statistics. They can just play HalfLife 2 or whatever you pick for them to play for a few minutes, and then take your tests afterwards.
I haven’t studied this in general, but I have read a decent amount about the history of a couple cities, and based on those examples, can say with confidence that no modern city comes remotely close to the density that people would choose absent regulations keeping density down.
Tokyo today is less densely populated per square meter ground than late medieval Edo was, and late medieval Edo had no plumbing and basically no buildings taller than three stories. (I don’t think there are historical examples of cities with no height restrictions and no density restrictions because until 1885, nobody knew how to build a skyscraper, so height restrictions existed indirectly through limitations of engineering—technically, they still do.)
All of the evidence I’m familiar with suggests that people would choose to be very densely concentrated if it wasn’t for regulations limiting their density.
The favelas of Brazil are generally considered a stepping stone towards urban living by their residents. Most of their residents don’t live there because they need to; they live there because they would prefer to leave the places they came from (generally the countryside). There’s pretty strong evidence globally and historically that, when given the option, people deliberately choose urban poverty over rural poverty. People migrate from villages to slums, and they don’t move back. This is happening in Brazil, Kenya, Tibet, and India today. It happened historically in the United States and the U.K. This exhausts my knowledge of the history of human migration patterns, but I assume that the cases I don’t know anything about are roughly consistent with the places I do know something about.
Air pollution from density of residency is unlikely to ever be self-limiting. 19th century London had way worse air pollution than any modern city, caused by coal-burning urban factories being everywhere, not to mention that everyone also burned coal for heat in the winter. (They lacked the technology to track air pollution back then, but it was bad enough that it effectively limited life expectancy to 30, so pretty bad. Incidentally, high polluting urban factories were priced out of existing in urban settings more than they were regulated out of existing in them.) Most cities also end up having a high percentage of their residents primarily travel by not-car, because traffic gets to be horrendous in everywhere but the nimbyest of cities. Outside the U.S., most cities are also designed around encouraging people to get around by not-car.
Asian countries generally permit much higher urban density than Western countries, and this seems to greatly increase the percentage of people who prefer to live in urban settings, and more or less prevent suburbs from developing. (I assume this happens because people are much less likely to be priced out of being able to live in a city, and that the preference for living outside of a city mainly comes from costs.)
Population density and price per square foot of livable space are highly correlated. I strongly suspect the density causes the increase in price; pretty sure the increase in price doesn’t cause the increase in density.
By the way, Bloomberg News has a section called “Citylab” that is primarily focused on urban planning. I highly recommend it to anyone interested in the subject.