[jyl]

What was all that about? The idea of the game is to guess how many steps ahead the other players are operating, and then to think one step ahead of that.

So, to operate “zero steps ahead” is to simply randomly pick a number, without trying to “win”. There is no preferred “pick” since there’s no strategy involved.

“One step ahead” is to pick a number closest to the average of all numbers picked by people operating zero steps ahead. This “pick” would be around 50.

“Two steps ahead” is to pick a number that would be 1/2 of the average of all numbers picked by people operating one step ahead. This pick would be around 25.

“Three steps ahead” is to pick a number that would be 1/2 of the average of all numbers picked by people operating two steps ahead. This pick would be around 12.5.

And so on down, converging on a pick of 0 as one operates an increasing number of steps ahead. It is not necessarily better to operate more steps ahead, as we’ll see.

Based on your estimate as to how the playing population is distributed among zero-step-ahead-ers, one-step-ahead-ers, two-step-ahead-ers, etc, you could formulate a strategy. For example, if you think 40% of OT Board members are operating one step ahead and 60% are two steps ahead, you might pick 17.5 = 1/2 * (0.4 * 50 + 0.6 * 25). As it turns out, that is very close to the actual answer.

The winner is kjb who picked 17. The runner-up is Island 911 who picked 17.45847. Third place goes to Paco Anton who picked 15. These are close enough than I’ll send each of you a print - I will get working on them ASAP.

In total I got 33 entries. The arithmetic average of all the picks was 33.26, implying that the average player was operating about 1.5 steps ahead. The “right” answer was thus 16.63 = 1/2 * 33.26.

Picks were distributed as follows: from 0-10 there were 4, 10-20 were 8, 20-30 were 4, 30-40 were 5, 40-50 were 3, 50-60 were 5, 60-70 were 2, 70-80 were 1, 80-90 were 0, 90-100 were 1. Picks tended to cluster around 50, 33, 25, 17, 12, and 1.

I read about this experiment in an article on behavioural finance. Variants of this game have been played many times by various groups in different professions. The typical numbers of “steps ahead” was about 2. For example, when 20 CEOs played the game, they on average operated 1 step ahead. 32 US high school students who played were operating 1.6 steps ahead. Appx 30 CalTech students were operating appx 2.6 steps ahead. 124 Harvard economics students were operating appx 4.4 steps ahead. 136 game theorists were operating 4.3 steps ahead. Professional investors, in several games of up to 1000 players, were operating 2.3 to 3.2 steps ahead (although in some German games, they were operating <1 step ahead).

Perhaps one reason this game gets played mostly in financial circles may be that this is not a very common sort of problem. In many cases, we don’t need to “get ahead” of anyone – e.g. the proverbial “rocket scientist” has a difficult job, but at least the rocket isn’t alive and trying to outwit him. I speculate – this is purely speculation – that for an engineer, operating too many steps ahead might be a bad thing, as it might distract from solving the immediate critical problem.

In the situations where we are trying to get the drop on someone, we usually have a specific opponent – e.g. Kim Jong Il, the opposing chess player, the racing driver ahead of us. So here we have to think ahead of a human opponent, but at least we have a particular opponent to analyze.

But this sort of problem does come up in the financial markets, where everyone is trying to outwit everyone else but nobody has a specific individual opponent. For example, suppose you invest in the sort of things that might be affected if China floats its currency. Do you wait until China actually does float the yuan to see what happened and then make a decision? Do you try to anticipate the float and take a position in the yuan? Do you invest in things that will be affected by a change in the yuan’s value, e.g. Chinese real estate and wages? Do you invest in things that will be affected by those things, e.g. go short Chinese manufacturers and long Vietnamese and Indian manufacturers? Etc etc.

Anyway, I’ve always wanted to play this game, and so thanks to everyone for helping. Hope this is interesting! If anyone plays this game again, I'd be interested to hear the results.