Game Theory and Speed Dating
As I’ve mentioned in a previous blog, each year – in collaboration with our Economics and Business teachers – I run a special GAME THEORY maths challenge (as part of our on-going weekly maths challenge series). This year was no exception and at a special talk to our Economics & Business Club, I introduced this year’s challenge – the mathematics of SPEED DATING!
SPEED DATING is a relatively new phenomen where single men and women meet and each person gets to spend a few minutes talking, one at a time, to all the people of the opposite sex; for example, if 10 men and 10 women took part, each man would get perhaps 5 minutes to talk to one of the women and then after his 5 minutes was done he would move over to start talking to the next women for 5 minutes, and so on until he had spoken to all 10 women. The idea is that, hopefully, one of the people you speak to is someone you really like – and if they like you too, then you might go on a proper date!
But the problem is this; if, whilst speed dating, you are talking to someone you really like – and let’s say it is the 4th person you have talked to that night – should you ask them out? You see, there are 6 more people to see and you might like one of them even more! And what if you ask them out and they say no? Well then you could move on and see the next 6 people and maybe you’ll really like one of them and want to ask them out! But if they have seen that you’ve been declined they might think that they were only your second choice and decline you too! A useful and very similar example appears in the famous Russell Crowe film ‘A Beautiful Mind’ about the life of the game theorist John Nash. In the film, Russell Crowe’s character (Nash) recounts an example where a group of women walk into a bar in which there is already a group of men. All the women are pretty but one in particular is especially attractive; and, in turn, each of the men approaches the most attractive women but is rebuffed. They then turn their attention to the other, less attractive but nonetheless very pretty, women only to be rebuffed by them too – because they think the men are only seeing them as their second choice. The result? All the men and all the women stay single. But, as John Nash subsequently argues, if the men had approached the other women first they would have accepted their invitation and all the men and women would have coupled up, with the exception of the most attractive women!
Now this is clearly a gross simplification of what would happen in real-life; and I have no idea whether the real John Nash ever used this example – after all, Hollywood films are allowed to apply a little poetic licence in order to create a more vivid image! But the central idea is valid; in certain circumstances it is better to settle for something less than the best, just to ensure that you do achieve something you’re happy with – after all, I would love for my football team to win 10-0 in their next game (incidentally, I support Spurs and our next game is against Blackpool – so please join me in wishing Spurs good luck!) but, if I’m honest, I would be very happy if we just won our game; in this situation it is the winning which matters to me, not how comprehensively we win. And the same idea is often used in FINANCE where a large number of financial transactions that are carried out are designed to be ones which produce a very small but very safe profit – rather than risky but very profitable deals. Now obviously not all financial transactions are safe ones that make a small profit – otherwise we might not be in the current dismal financial environment we find ourselves – but a lot of them are, and a very large number of small but safe profit-making deals, when added together, can a produce a very large income for the company or investor in question. On the other hand, when it comes to matters of love surely you don’t want to settle for a safe option, someone you merely like? Don’t you want to go for your one true love; the man or woman who is that special someone for you?
In life we sometimes settle for the safe option – like in some forms of financial trading – and sometimes we go all out for what we want – like in love. And it is these ideas which lie behind this year’s challenge. Rather than students having to decide which man or woman they wanted to ask on a date, to simplify things students just had to pick the highest number (as opposed to the most attractive man or woman) from 6 numbers selected at random – the trick was that they would be shown one number at a time and had to decide, before seeing the next number, if they wanted to pick the number they’ve just seen (just as when speed dating you see each person one at at time). Let me illustrate this with an example. Suppose I produce a number at random – say 57 – you immediately have to decide whether you want to pick that number; if you say yes then you cannot change your mind and if a number greater than 57 is subsequently selected, you lose. If you don’t select 57 then when I show you another number – say 138 – you have to decide whether you want to pick that number or not. And so on until we reach the sixth, and final, number. If the number you picked was the highest of the 6 numbers, you win; if not then you lose! Students could play as many rounds of this game as they like, with the two students from each class that finished with the highest average number of wins, making it through to the grand final – which was won by one of my A-level students, Sephy from China!
Clearly, because you don’t know what numbers are going to be drawn next, you cannot guarantee to win at this game. However, the idea of this game – and other GAME THEORY games – is to decide whether or not there is a strategy you can play that gives you a good chance of success. And there is. But I will save that for another blog!
Get Ahead in Maths 