I’ve received several e-mails and some comments about the “what’s 2/3 of the average” game, which I analyzed earlier this week. (If you haven’t read that analysis you might find this post hard to follow.) Some of them suggested that players are not rational if they do not select zero. That’s not quite right. I explained why in my analysis. Below I explain again and show how this game is a stylized model of an asset bubble.

What makes the game interesting is precisely that all players can be rational and still not select zero. Why? Because it takes more than rationality on the part of players for zero to be the winning answer. Rationality is doing what is in one’s best interest. It would not be in one’s best interest to select zero if one has reason to believe that a number larger than zero will be the winning answer. In that case one will regret one’s choice of zero.

Indeed, zero *will not be *the winning answer unless something much stronger than rationality holds, namely common knowledge of rationality. As explained in the analysis of the game, common knowledge of rationality occurs when everyone knows everyone knows everyone knows…(an infinite number of these)…everyone knows everyone is rational. Rationality by itself is not enough for the winning answer to be zero.

Something like this game, and the difference between rationality and common knowledge of rationality, arises in asset markets in the form of bubbles. It can be rational to participate in a market bubble. One can make a lot of money. One can also lose one’s shirt. I’m not suggesting it is wise to speculate in a market bubble, but it can be rational to do so.

To make the connection between asset bubbles and the “what’s 2/3 of the average” game explicit, imagine your guess of a number between 0 and 100 is the maximum price at which you’d be willing to buy a share of stock. Call your guess G (for “guess”). That’s the most you think the stock is worth for a purely speculative purchase. The actual market price is 2/3 of the average of all guesses. Call this value P (for “price”). What’s the value of the stock? Will you participate in this market?

For example, your guess could be G=$10 and 2/3 of the average of all guesses could be P=$20. In that case you could make a profit if you could buy at any value below $20. Your guess G=$10 is the most you’d be willing to pay. If you could buy at $10 and sell at the market price of $20, you’d make a profit of $10 per share. It would be rational for you to do so.

What if no one will sell to you at $10 but someone will sell to you at $15. Should you buy? Yes you should. But your guess of $10 was the maximum you’d pay. You won’t pay $15 or you’d have guessed $15 (or more). So you’re out of the market. You lost an opportunity. Your guess was too low! In fact, it’d be rational for you to pay up to $19.99 since you can turn around and sell at $20.

You can see there is an incentive to bid (guess G) higher than is suggested by appeal to common knowledge of rationality. You want to participate in this market (this game) to make money. You can’t make money by sitting on the sidelines. But that is exactly where you’ll be for sure if you guess G=$0.

This is still the “what’s 2/3 of the average” game. We’ve just interpreted the guesses and the winning value in a specific way. So we know the “right” price of the stock based on an argument of common knowledge of rationality is $0. But we also know that P (2/3 of the average of guesses) will not be $0. In fact it is likely to be close to $20 (*). Everyone who is able to buy the stock at a price below P can make a profit and they’re rational to do so. It is not irrational to set G above zero. In fact it can be a very smart thing to do (in this game).

Therefore, a speculative bubble for a worthless stock can develop for which the price is far above the “right” price. Many market participants are behaving rationally. The bubble exists because an assumption of common knowledge of rationality does not hold. But an asset bubble is not a one-shot game. Players buy and sell multiple times. Eventually additional iterations of assumptions of rationality emerge. The price begins to fall. The bubble bursts, the price goes to zero, and everybody knows it. Moreover, it is common knowledge.

(*) One has to be a bit careful. The version of the “what’s 2/3 of the average” game presented here is different than the one presented previously. In particular the payoffs are different. In the version presented before there was a winner: the player(s) that guessed closest to 2/3 of the average of the guesses. In the version in this post many players can profit (by differing amounts). Though someone may make the most money (per share) I haven’t really defined a “winner.” Payoffs (incentives) can change strategy. It is therefore possible that 2/3 of the average will be different in the two versions of the game. Strictly speaking it is not correct to assume that a number near 20 will be 2/3 of the average in this version even if it is for the previous version. Nevertheless, I’d bet a fortune that 2/3 of the average will not be zero in either game played by a population not familiar with it. After that population plays many times the answer will likely tend toward zero. How many iterations will it take to converge? I don’t know.