he best-looking frog ... but he doesn't fancy me .. and I don't know why not!
http://plus.maths.org/issue48/featur...gham/kiss2.jpg A frog's perspective.
There are lots of other things that we could add to our mathematical model to make it more realistic. For example, in real life, if you kiss the second best frog, you don't have to stay in the enchanted forest. Unless you're an incurable romantic who thinks that there's just one perfect person out there for you, you can be very happy with frog number 2. Maybe you're more interested in avoiding a very bad frog. What's more important, making sure you bag frog number 1 or avoiding frogs 51 to 100? The strategy you should choose depends upon what you're trying to achieve.
Now let's imagine that the princess, who I have so far avoided describing, has a face like a bag of spanners. When the handsome prince appears, he may take one look at her and run for his life. Partner choice is a two-way business. You must choose and be chosen. If you kiss a frog whose number is too far from yours, disaster is probably just around the corner. And how do you know what your number is anyway? You may think you know, but it's not what you think that matters. It's only by finding out who will agree to date you that you start to get some idea of what your number is. All of these features can be, and have been, added to mathematical models.
There's more to life than snogging
One of the great things about maths is that it has so many different uses. In fact, the same bit of maths can have lots of different applications, and the fairytale of the frog prince is an example of this. Let me retell the fairytale one last time. In this version, the princess's three brothers have been turned into frogs and thrown into the pond. She still has to find them by kissing them, the frogs are still numbered between 0 and 1 and her brothers have the three largest numbers. The witch is very generous and allows her to kiss 20 of the frogs. As long as her brothers get a kiss, all four of them can go home. If you think this sounds easy, you'd be right. In fact, with the right strategy, the chances of a brother missing out on his kiss are tiny. Again, there are decision numbers, but these are very low to start with, so that the princess's lips will be kept busy at the start of the parade of frogs. However, after each of the first two kisses, the decision numbers increase and, after she has kissed three frogs, she kisses any frog numbered in the top three so far.
http://plus.maths.org/issue48/featur...gham/pylon.jpg Don't marry this.
So what's that got to do with partner choice? I don't know of anyone who arranges their love life in such a bizarre way. In fact, this is a mathematical model for a game played between the regulators and the suppliers of electricity in the UK. Part of the money paid by industrial customers to their electricity supplier is based on their consumption of electricity during the three half hour periods during each year when the total consumption of energy across the whole country is highest. From the regulator's point of view, this is meant to encourage industry to cut back its consumption during times when the national grid gets close to its operating capacity. From the point of view of a large industrial consumer of electricity, they want to know when these half hour periods are, so that they can shut down their factories and save themselves lots of money. The problem is that nobody knows when these three periods (called the
triads) are until the year ends and the electricity consumption figures are added up. Industry is therefore prepared to pay a lot of money to anyone who can accurately predict these triad periods for them (
call the triads). They know that they'll have to turn off their factories more than just three times to hit the triads reliably, but aren't prepared to do this more than about 20 times per year.
So, here's the fairytale again. Predictions of energy consumption appear in sequence (frogs jump out of the pond), we want to make sure we advise our customers to turn off their factories when we think this will be a triad (the princess must kiss her three brothers), but we mustn't do this more than 20 times (the princess can kiss no more than 20 frogs). If energy consumption figures were just random numbers between 0 and 1, this would be pretty easy, and no one would make any money from predicting triads.
http://plus.maths.org/issue48/featur...gham/frog2.jpg 'Kiss me!'
However, and here we're back at step 6b of my description of mathematical modelling, there are lots of complications. I'll tell you about just two of them. Firstly, energy consumption figures are basically random numbers, but each number depends to some extent on the number before, in the same way that the weather today is a reasonable predictor of the weather tomorrow. This means that we need another mathematical model for how these numbers behave, and, back in the real world, we have to predict them. Secondly, calling a triad affects the numbers that we're trying to predict. Say we predict that electricity consumption will be high enough to be a triad period, we call it, and our customers close down their factories. Electricity consumption is then lower than it would have been (which is, of course, why the regulators set up the game in this way). There is
negative feedback built into the system. But we may still need to call the triad, because otherwise the factories stay on, electricity consumption is higher, and the princess's brother stays in the pond forever!
