(from Division of Labor)

My good friend, Neil Emerick in South Africa who runs Global Economic Software, sent me the following question. I pass it along to you here. If you'd like to add to his database of answers, e-mail him at
neilemerick[at]wol[dot]co[dot]za.

Imagine a company is running a competition for a prize of 10,000 dollars.
They challenge people to send in a number between 1 and 100. The prize will
go to the person(s) who picked the number that was 2/3rds of the average
number sent in.

What number would you send in?

With splitting the pot on ties, the only Nash equilibrium is for everyone to send in 0. Any other result is unstable, as everyone has incentive to move to the 2/3 point, which will tend to decrease the average towards 0. And once it is there, no one profits by moving. It also matters whether the metric is "closest to 2/3", or "closest to 2/3 from above". With the latter metric, all 0 is not stable, because if one person moves to 1, they win. It is unclear to me what the answer is in this case, its certainly much harder.

Another interesting point is that this game is ripe for collusion. Suppose everyone knows to pick 0. If my partners and I have a rough idea of how many people there are, some of us pick 100, and one of us picks the 2/3 point.