Analysis of “What’s 2/3 of the Average”

The 2/3 of the average problem posed on Friday is a well known puzzle in game theory, and it illustrates some fundamental game theoretic concepts. To recap, here’s the problem statement:

Suppose everyone in your town selects a real number between 0 and 100, inclusive (i.e. 0 and 100 are both possible choices, as is any other number between). The winner is the individual (or individuals) who selects the number closest to 2/3 of the average of numbers chosen. What number do you choose? Why?

A nice way to view the problem is to start by identifying the set of numbers that no rational player should select. Intuitively, most sense that the winning answer will not be 100. Why? Because the average of numbers no greater than 100 cannot be greater than 100. Thus, 2/3 of the average cannot be 100. One can make a better choice. In fact, 2/3 of the average of numbers no greater than 100 cannot be greater than 66.666… Therefore, it is irrational to select a number higher than 66.666… All numbers above 66.666… are weakly dominated strategies (game theory jargon) meaning that one cannot do worse and may do better by selecting a number outside this range.

If you are rational you will not select a number greater than 66.666… this suggests the next stage of analysis. Can you assume everyone else is rational? If so then all your opponents also eliminate numbers above 66.666… from consideration. This reduces the puzzle to selecting a number between 1 and 66.666… trying to get closest to 2/3 of the average.

Well, that’s the same game, only over a compressed range [0, 66.666…], and one for which we’ve already assumed that everyone assumes everyone is rational. We can make a similar argument as above. If within this game, all players assume everyone is rational then it is clear that nobody would select a number greater than 44.444… because it is impossible for 2/3 of the average of numbers between 1 and 66.666… to be any larger than 44.444… Note that we’re embedding the assumption that everyone assumes everyone is rational within the similar assumption we’ve already made at the previous stage (the one that got us from 100 down to 66.666…). So, what we’re really assuming to get down to 44.444… is that everyone assumes everyone assumes everyone is rational.

We can continue this nesting of assumptions of rationality and continue to compress the range of numbers over which the game would be played. After doing so an infinite number of times (exercise left to reader), we will find that that every player ought to select zero. The process of recursively eliminating sets of numbers that would be irrational to select is known as iterative elimination of weakly dominated strategies (jargon). The case that everyone chooses zero is the Nash equilibrium (jargon), which means nobody would regret their choice. Arrival at a Nash equilibrium requires more than just rationality on the part of players. It requires that everyone assumes that everyone assumes that everyone assumes … that everyone assumes that everyone is rational (an infinite nesting of those).

Do you think the winning choice would really be zero? I’ll bet not. Why? Because it is unlikely that everyone in your town would follow the chain of logic described above. Everyone may be rational. Everyone may assume everyone is rational. Everyone may assume that everyone assumes that everyone is rational. But at some point, some people implicitly are going to stop the sequence. In fact, on average, people only nest these assumptions to four levels. Evidence: 21.6 was the winning number in such a game with 19,196 participants organized by a Danish newspaper (histogram).

This illustrates the difference between perfect rationality and common knowledge. The latter is a special form of knowledge where everyone knows (or assumes) X and everyone knows everyone knows X and everyone knows that everyone knows that everyone knows X, and so on, an infinite number of times. This is precisely what is required to reason in the 2/3 of the average game that the winning answer would be zero.

Is there really any such thing as common knowledge of rationality? Empirical results of the 2/3 of the average game suggest not. At some point people begin to assume, at least implicitly, a lack of assumption of assumption of assumption… of rationality by others. That is, it is not the case that everyone knows that everyone knows that everyone knows that everyone is rational (or something like that). That makes the game a lot harder in reality than it would otherwise be. That’s also what makes such a simple game so interesting.

A subsequent post applies this game to speculative bubbles in financial markets.

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