I have a maths question. Suppose that we are scoring n individuals on their performance in an area where there is significant uncertainty. We are categorizing them into a low number of categories, say 4. Effectively we're thereby saying that for the purposes of our scoring, everyone with the same score performs equally well. Suppose that we say that this means that all individuals with that score get assigned the mean actual performance of the individuals with that that score. For instance, if there were three people who got the highest score, and their perfomance equals 8, 12 and 13 units, the assigned performance is 11 units.

Now suppose that we want our scoring system to minimise information loss, so that the assigned performance is on average as close as possible to the actual performance. The question is: how do we achieve this? Specifically, how large a proportion of all individuals should fall into each category, and how does that depend on the performance distribution?

It would seem that if performance is linearly increasing as we go from low to high performers, then all categories should have the same number of individuals, whereas if the increase is exponential, then the higher categories should have a smaller number of individuals. Is there a theorem that proves this, and which exacty specifies how large the categories should be for a given shape of the curve? Thanks.

"Spreading quicker" may not be the best question to ask. The question I'm more interested in is, What is the relationship between speed of communication, and the curve that describes innovation over time?

A good model for this is the degree of genetic isolation in a genetic algorithm. Compare two settings for a GA. One allows mating between any two organisms in the population. Another has many subpopulations, and allows genetic exchange between subpopulations less frequently.

Plot the fitness of the most-fit organism in each population by generation. The first GA, which has fast genetic communication, will initially outstrip the second, but it will plateau at a lower level of fitness, and all the organisms in the population will be identical, and evolution will stop. This is called premature convergence.

The second GA, with restricted genetic communication, will catch up and pass the fitness of the first GA, usually continuing on to a much higher optimum, because it maintains homogenous subpopulations (which allows adaptation) but a diverse global population (which prevents premature convergence).

Think about the development of pop music. As communication technology improved, pop stars like Elvis could be heard, seen, and their records marketed and moved across the entire country more efficiently than marketing local musicians, and replaced live performers with recorded music. On one hand, you could live in Peoria and listen to the most-popular musicians in the country. On the other, by 1990, American pop music had nearly stopped evolving. Rebecca Black could become popular across the nation in a single week, but the amount of innovation or quality she produced was negligible.

Basically, rapid communication gives people too much choice. They choose things comfortably similar to what they know. Isolation is needed to allow new things to gain an audience before they're stomped out by the dominant things.

You need to state your preferences as a function of the long-term trajectory of the entropy of ideas, rather than as any instantaneous quantity.

What about non-elite groups? (...) they are likely to be heavily influenced by the cognitive elite, especially in the longer run.

I think they are likely to be influenced by whom they consider high-status. If you succeed to make a conspiracy theory website seem like an authoritative source (it must seem as professional as the mainstream media, except that it brings "news you will not hear elsewhere"), that's all you need.

Why would anyone make a professionally looking conspiracy theory website? Aren't "professionality" and "crackpot thinking" kinda opposed in real life? Yeah, the genuine crackpots usually also have low regards for mainstream design or marketing. But a professionally looking conspiracy theory website is a great vehicle for political propaganda. So a foreign government may spend a lot of money and professional work to make a high-status conspiracy website, where the conspiracies are filtered, and only those that are neutral or convenient for the owner are published.

This is not a hypothetical scenario. Russians already have such website with online radio in my country. I see advertisements for it almost everywhere. The content is more or less: "Things your government wants to keep secret from you: Vaccination causes autism. West is the source of all evil. Russia is a paradise. Being a member of European Union is bad for you, because they will kidnap and abuse your children!" And of course, the website is "independent and alternative".

I think I will want to see further the further debates in the US presidential race in this format. At the moment I don't see a clear link where I can express that preferance and get an email when the next debates get released with annotations.

In order to be fact-checked, a statement has to be truth-apt in the first place. That is, it has to be the sort of statement that is capable of being true or false.

A lot of political arguments aren't truth-apt; they amount to cheering ("Georgism, boo! Synarchism, yay!") as opposed to historical claims ("Countries that adopt goat control have seen their arson rate double") or even theoretical claims ("The erotic calculation problem predicts that college-educated adults will move out of states that ban vibrators").

But besides new software applications and better debating formats, we also need something else, namely a raised awareness among the public what a great problem politicians' careless attitude to the truth is. They should ask themselves: are people inclined to mislead the voters really suited to shape the future of the world?

I watched the beginning of your annotation of the Republican debate. I think you did a good job at annotating it. The annotations add to the experience of watching the debate, which is likely the most important thing for making it impactful.

There were a few technical issues where the annotation freezed and didn't update (I'm using Firefox).

Overall fantastic article. One small request:

If they don’t have any special reason not to, people without special knowledge defer to the scientific consensus on technical issues.

Citation needed

Great post.

Another great resource is the ABC's fact checker, for those political news watchers to see whether the politicians they agree with or disagree with are factually correct or incorrect.

And, to actively compare one's ideological beliefs with theoretical truisms, such as those in game theory

Instead most people seem to believe themselves to be politically rational, and hold that as a very important value (or so I believe).

That depends a bit on the enviroment. There are cases where people care very much of signaling group alliance. On the left you find people competing on being more ideological pure than the next person.

I had one experience with an extremly smart person, which politically influential parents and maybe a future political career on her own who once quite explicetly said, that's what she was doing after she read a room wrongly and excused for that (it wasn't a even a public event but party internal).

I'm uncomfortable with the question distinguishing between "richer than most" and "richer than nearly all". Logically if I answer "richer than most" is true.

There justification of the answer makes little sense as you have to count the number of countries which are richer to find out and not take the EU average.