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Sunday, August 09, 2009

Just Pick a Number at Random.

Just pick the only random number.

John Mauldin explains that to believe in the Efficient Markets Hypothesis one needs to believe 6 impossible things before breakfast.

He provides an illustration of a game in one does not win by playing the unique Nash equilibrium strategy. Rather the way to win is to say 17.

This game can be easily replicated by asking people to pick a number between 0 and 100, and telling them the winner will be the person who picks the number closest to two-thirds the average number picked. The chart below shows the results from the largest incidence of the game that I have played - in fact the third largest game ever played, and the only one played purely among professional investors.

The highest possible correct answer is 67. To go for 67 you have to believe that every other muppet in the known universe has just gone for 100. The fact we got a whole raft of responses above 67 is more than slightly alarming.

[click the same old link for the figure]

You can see spikes which represent various levels of thinking. The spike at fifty reflects what we (somewhat rudely) call level zero thinkers. They are the investment equivalent of Homer Simpson, 0, 100, duh 50! Not a vast amount of cognitive effort expended here!

There is a spike at 33 - of those who expect everyone else in the world to be Homer. There’s a spike at 22, again those who obviously think everyone else is at 33. As you can see there is also a spike at zero. Here we find all the economists, game theorists and mathematicians of the world. They are the only people trained to solve these problems backwards. And indeed the only stable Nash equilibrium is zero (two-thirds of zero is still zero). However, it is only the ‘correct’ answer when everyone chooses zero.

The final noticeable spike is at one. These are economists who have (mistakenly…) been invited to one dinner party (economists only ever get invited to one dinner party). They have gone out into the world and realised the rest of the world doesn’t think like them. So they try to estimate the scale of irrationality. However, they end up suffering the curse of knowledge (once you know the true answer, you tend to anchor to it). In this game, which is fairly typical, the average number picked was 26, giving a two-thirds average of 17. Just three people out of more than 1000 picked the number 17.

This fact made my spine tingle. It was a blast from the past. I attended the
Hampshire College Summer Studies in Mathematics (HCSSiM) in which a running joke was the claim that 17 is the only random number.

Although the entire student body changes every year, HCSSiM has a number of well-established traditions. Some of the oldest of these traditions deal with the repeated use, in mathematical problems, jokes, and lectures, of the number 17 (and its multiples) and the Yellow Pig. These two traditions were started by Professor Kelly, who collects objects with yellow pigs on them and examples of the number 17 in print.

1 comment:

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