You are currently browsing the category archive for the ‘game theory’ category.

With the end of the Mayan Long Count Calendar today, it seems only fair to join the melee and talk a little about the end of the world, oh – excuse me: the End of the World. It at least deserves to be capitalised. First of all let me say that if you genuinely believe the world is going to end I am not out to mock you (although why you would spend your last hours on the internet reading my blog is not entirely clear). As ever, I’m just going to talk about some thoughts I’ve had on the subject and explain what I believe is a rational response to prophecies of impending doom.

But more of that later. Prophecies of impending doom seem to come around every so often, my first encounter with the idea was in about 1984 when I was 9. It was round about the end of September, a documentary on BBC2 announced that one particular prophecy had it that the end of the world was nigh. “in fact,” claimed the presenter, “very nigh: next Thursday”. That, it is safe to say, scared the 9 year old me very much indeed – so much so that I had to have the day off school the following Thursday. As I recall, we had a class outing planned to Saddlers-Wells theatre that day which I missed as a result. I was kind of looking forward to it at the time.

Needless to say, the world did not end but I learned a lesson that day about uncertainty and not believing everything you see on TV and the fact that adults don’t always necessarily know what they’re talking about. Since then predictions of the end of the world have come and gone, and I’ve developed what I believe is a sane a sensible response to them. It is as follows: You think the world is going to end on Saturday? I’ll bet you £50 it won’t. Or an equivalent quantity in your local currency.

This is based on the fact that, in the absence of evidence that a doomsday scenario is on the cards, I don’t believe that it will. If the world is still there, I’m £50 up. If the world ends, well I’m not likely to have to make good on the debt and this fact may actually be of some small comfort as the meteor slams into the Earth’s crust / the tail of the comet seeds a deadly virus / the antichrist, harbinger of the unrighteous, punishes the unrighteous.

This is the essence of Game theory. I play a certain strategy, I get a certain payoff in a certain circumstance. In this case I choose to bet. If I win the bet I am richer, if I lose the bet everyone is dead. On the other had I could choose not to bet, in which case if the world survives I am no richer but if the world ends everyone still dies. A rational agent should therefore choose to make the bet. Of course, this implies that someone is willing to bet with me, but seeing as this hypothetical person must presumably believe that the world is going to end they have nothing to lose and are probably thinking about rewarding ways to spend their last hours, in which case a bit of a wager might be fun, and the prospect of me having to admit that all my fancy book learnin’ hasn’t helped me one bit might also soften the blow a little.

I could look at this another way: way do I believe that the world isn’t going to end? You could argue that none of the previous predictions have proved true so why should this one. This, however, is very faulty reasoning. The end of the world is something that that, by definition, will only happen once. So the fact that it is still here after all those false predictions is just confirmation bias – if any of them had proved true we would not be here to ask the question.

This is where we get into some interesting questions about statistics and how we interpret probability. Classically, probability was interpreted as a frequency of events: given a large number of trials you would expect one result x percent of the time, and other y percent of the time and so on. This is called a frequentist approach, and it works well when you are dealing with lots of repeatable events, like coin tosses. If I flip a coin, I expect heads 50% of the time and tails 50% of the time (in a vanishingly small number of cases it might land on its side, but we’ll ignore that), so the probability of heads is 0.5.

This idea works well for coins tosses, but less well for the end of the world. The end of the world won’t repeat, but I would still like to attach a probability to it happening: surely on a day when there is no prophecy or threat of nuclear destruction the probability is lower than a day when there is? Here we get into the realm of Bayesean statistics. Bayesean statistics is built on the idea of condition probabilities: “given that this has happened, the probability that that will happen is…” and allows you to include prior information, such as the amount of evidence for a particular outcome. This means that I can say things like

“given the recent observation of a huge fiery comet of doom headed directly for central London, the probability of the end of the world is…”

or

“given that major changes in various numbering systems have come and gone without incident and the evidence of any effect of overflow in numbering time based on arbitrary start dates and dynamical range is thin at best, the probability of the end of the world on 21st December 2012 is not statistically different than in was on the 20th or 22nd Decamber 2012”.

So a combination of Baysean statistics, with its ability to incorporate prior knowledge into estimates of probability, and Game Theory, with its notion of strategy and expectation, I will therefore bet you £50 that the world does not end today. In fact, I’m not the first one to come to this conclusion. See you tomorrow in the 13th Long Count period!