*Maths*

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## February 03, 2006

### The Dollar Game

A friend related to me the following game: (Game as in Game Theory, I'm afraid)

A certain person is auctioning off a dollar. (Or a pound, or whatever) The deal is this – the highest bidder will win the prize, but every bidder – including losers – will be forced to pay whatever price they bid.

So what happens in the game? Notionally, two players can make a profit by agreeing to share the prize and refusing to escalate the situation. But such a deal is an inherently unstable one – each player has a lot to gain by screwing over the other. Furthermore, additional players can jump in and demand their share, blackmailing the cooperators by threatening to take all the prize for himself.

At each stage in the game, then, the player always profits by raising his offer to be above that of his rival. And unlike with the various pricing games we learn about in economics, this game doesn't just stop with zero profit. Even when the players are bidding above the prize, they still have an incentive to bid up a little more in order to try and claw back $1 worth of their losses. Extrapolating, we get to the conclusion that the players all end up paying infinite amounts of money for $1.

The only way to win the game is not to play.

Which is pretty cool. I wonder if this can be used to describe some real world situations…

## January 31, 2006

### 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.