Oh Maths, Why Has Thou Forsaken Me?
Writing about web page http://www.channel4.com/4money/ontv/deal_or_no_deal/
A tv show, currently being show on Channel 4 on Saturdays called Deal or No Deal has caught my attention. The general gist of the show is this:
There are a number of boxes containing a known selection of cash amounts – what is not know is how the amounts are distributed. A player selects one box at random.
The game then proceeds round by round. With each round, one of the boxes is randomly selected, it's contents shown, and then eliminated. Every 2 or 3 rounds, the player is given an offer based on the average of the remaining box amounts. The player can take this offer and quit the game, or he can carry on, until eventually every box is eliminated except the one he initially selected, whose contents he wins. This is essentially the only choice the player has.
Question: How best does one play the game?
The obvious thing to do, as a statistician , is to stop playing when the expectation is decreasing. I.e. calculate the average win the next time you get an offer over all possible intervening outcomes, and stop once this is below what you currently have. The problem is that this doesn't work. It turns out that such expectations are exactly the same as the current average. (Proof is exercise for the reader)
A friend suggested an alternative – to stop playing the moment you are above the initial average – i.e. quit when you are ahead. The argument is that you remove the cases where you start off winning, but eventually lose. Unfortunately, you also lose the cases where you win and keep on winning. And guess what, these cancel out and you don't change your average winnings.
It seems that you can change the distribution of the winning probabilities only so much as you keep the average winnings the same. (Maybe this can be proven using some theorem from that of random walks?) So in the end, the best strategy is decided by the utility function of the player – how much is that extra 1000 pounds worth to him? Personally, I'd go for the reverse of my friend's strategy – quit when I fall below my initial expectation. This means that I guarantee a reasonable win, but have a long tail of probabilities going upwards, giving me a chance at the big one. Interestingly, this corresponds exactly to the much vilified idea of 'luck'. (Possibily an evolutionary explanation?)
Wikipedia has a good article on this, of course. It notes I've been a little bit naive with my analysis - the offer is a proportion of the expectation that is increasing with time. So essentially, the game lets players trade average winnings for reduced risk. It also references an unusually interesting social science article that I currently aren't arsed to read because I have two essays to write. The article (slogan = Tomorrow's Research Today! – really very creepy if you think about it) notes that the game is actually pretty good economics experiment into risk aversion, and there's a nice variety of different behaviours.
So, um, read it if you are interested.