Here’s a cute data analysis puzzle, which I’m amazed I didn’t encounter sooner in my line of work.
You run a website that sells guns and banjos, and one day you notice from your web analytics data that the conversion rate of your site (orders divided by visits) is steadily declining over time.
Realising that you essentially cater to two quite different needs, you look at the performance of your two main site sections: the gun section and the banjo section. There is no significant overlap between the people visiting these sections.
Here’s the problem. The conversion rates in both the gun and banjo sections of the site are going up over the same period that overall conversion is going down. How is this possible?
I’ll let the name of the idea do the talking: Rorschmap.
Puzzle Answer – Cyborgs beat Robots
In the last Things I invited you to guess who would win in a chess match in which humans and computers could team up in any combination.
I recently read of an empirical answer here, which makes the excellent point that there are actually three criteria at work in any team: the chess skill of the computer(s), the chess skill of the human(s), and the friction in the way they work together as a team.
Some may be surprised to learn the most basic observation from the event: that a team of human + computer is much stronger than even an extremely powerful chess-playing computer. As Kasparov puts it: “Human strategic guidance combined with the tactical acuity of a computer was overwhelming.” Humans are useful!
More impressively, the winner of the tournament was a team of two amateur players working with three computers. The lack of friction in their system of working together beat the raw power of chess-playing supercomputers and the strategic brilliance of grandmasters.
This has some serious implications, too. Most simply, since mediocre computers and mediocre humans are more common than highly skilled ones, and since systems can be invented once and then used by all, there is in some general sense much more potential to solve hard problems than we might otherwise have expected in the world.
More extremely, anyone worried about a technological singularity in which we invent AI that is smarter than us (leading to runaway self-improvement of the AI and a very dangerous 4 hours for humanity) can rest assured that human-AI combinations will probably be smarter than pure AI.
Tim Link – Competitive Sandwich Making
Last week Clare and I ran a game based on tessellating pieces of cheese to make the best sandwich for the Hide&Seek Sandpit event. You can read about it and see the photos on my project blog, Tower of the Octopus.
In a chess tournament in which anyone can use any means available to them to come up with their moves, who would win? Some possible answers to give an idea of what kind of thing I’m talking about here:
A high-ranking chess Grandmaster
A really good chess-playing supercomputer
A huge team of moderately skilled players with some method of combining their ideas
A moderately skilled player with access to a moderately good computer that can run some basic chess calculations
(I had wondered about this in an abstract way before, but recently found out that has actually been done. I’ll relate what happened in that event next week, but you could of course try to Google as well as guess the answer if you wanted).
(Via Phil): Art out of Science:
(Two views of the same thing. If your browser is up to it, you could try watching both videos simultaneously – start the bottom one 20s after the top):
Links Kickstarter is one of my favourite things on the internet: people with an idea for something get a platform from which to shout about it, and to collect pre-orders or donations from people that like the idea. If there’s enough interest, the project can go ahead, and everybody wins.
You can follow Kickstarter on Twitter, or go to their home page and scroll to the bottom to sign up for the weekly newsletter which highlights the most interesting projects.
IndieGoGo is similar but for reasons I can’t really pin down doesn’t work as well for me.
Crowdfunder is a UK version which I don’t tend to find as inspiring, but would probably be the best one for someone in the UK to create a project with (since Kickstarter requires a US bank account).
Overheard in the maths common room when I was studying for my PhD at Royal Holloway:
But nobody knows what probability is! Probability is defined in terms of randomness, and randomness is defined in terms of probability!
Answers to Monty Hall and the Two Envelopes Last week I asked about the Monty Hall problem, which I should have introduced before the Two Envelopes problem I set two weeks ago.
The Monty Hall problem has a nice Wikipedia page, the most helpful part of which is probably the decision tree showing all possible outcomes.
In brief, the answer is that you should switch after Monty shows you an incorrect door, but certain misguided instincts steer most people away from that choice. The Endowment Effect and Loss Aversion mean that regardless of probability, people fear they would regret “giving up” their first choice more than sticking with it if they end up losing.
The more subtle effect is an instinctive (or partially trained?) feeling that the choices of others have no effect on the probabilities of our own choices in these kind of contexts. This is true when the other person has just as much information as you, but that is not the case here – Monty knows where the car is, and uses that information to ensure that he always opens a door with a goat behind it. So he has more information than you, and when you see his choice you gain some information.
Or to give an answer that might go with the grain of instinct for some people, consider this: there is a 2/3 chance that the car is behind one of the doors you don’t pick. Monty shows you that it definitely isn’t behind one of them. So there’s still a 2/3 chance the car is behind the other one, and a 1/3 chance it’s behind the one you first chose.
As for the Two Envelopes, it turns out this is more difficult than I originally remembered. Again, there’s a great Wikipedia page on the subject, which has quite a lot of detail.
As Thomas noted, a key phrase missing from the subtly specious argument for swapping is “Without Loss of Generality” (WLOG), which one must always be careful to check whenever substituting a variable (in this case, the amount in the envelope) with a specific figure (£10 in the example I gave).
Is it true that the reasoning I gave based on having £10 in the envelope truly retains the generality of the problem – would the reasoning also hold for any other amount? In short, no. For example, there could be 1p in the envelope, or any odd number of pence, in which case we would have to conclude that we had the lesser envelope (although this still means you should swap). More dramatically (here we imagine the envelopes contain cheques, and that these cheques are totally reliable), your envelope could theoretically contain over half of all the money in the world, in which case you can be sure the other envelope contains less. More realistically, it could contain more than 1/3 the amount of money you expect the person filling the envelopes to be willing to give away, in which case you would strongly suspect the other envelope to contain the lesser amount.