Identifying high-order rationality is a forthcoming paper in Econometrica from my favourite behavioural game theorist, Terri Kneeland. For those of you who are not familiar with the inside baseball, Econometrica is one of the top two journals in economics and good economists can go their entire careers without publishing a paper in Econometrica. Getting your first paper published in Econometrica, as Kneeland has, is a pretty awesome achievement.
So what makes Kneeland’s paper so good? Well, it is a simple (yet novel) idea that was implemented in a very clean fashion.[1] Economic models of strategic interactions rely on the notion of rationality, where an agent is defined to be rational if they make optimal decisions given their available information.[3] This definition is actually rather weak and, as Kneeland demonstrates, most people satisfy rationality in simple environments. In some cases, however, economic models require a much stricter notion known as common knowledge of rationality. An agent is called first-order rational if they are rational. They are second-order rational if they are first-order rational and they believe that others are first-order rational. They are third-order rational if they are second-order rational and believe that others are also second-order rational. Common knowledge of rationality is the state where all agents are infinite-order rational.
The innovation in Kneeland’s paper is that she implemented a new experimental design that, with minimal assumptions, can identify a subject’s order of rationality. The process that Kneeland developed is as follows. Suppose that you are standing in a circle with a large number of other people. Everyone in the circle must choose a between A, B and C. You will receive an amount of money that depends on your choice and the choice of the person standing to your left. You are given a table outlining how much money you will receive in each situation. Consider the following table as an example. In this example, you will choose a row, the person to your left will choose a column, and you will receive the amount shown in the corresponding box. Which option would you choose?
| A | B | C | |
| A | 12 | 16 | 14 |
| B | 8 | 12 | 10 |
| C | 6 | 10 | 8 |
You might have noticed that the option A will give you your best outcome, irrespective of what the person to your left chooses. An economist would call option A a dominant strategy. A first-order rational person will always choose their dominant strategy, if they have one.
Next, consider the following table. (Again, you choose a row, and the person to your left chooses a column).
| A | B | C | |
| A | 20 | 14 | 8 |
| B | 16 | 2 | 18 |
| C | 0 | 16 | 16 |
In this case your choice is harder, because there are no dominant strategies for you. If you think that the person to your left will choose A then you should also choose A. But if you think the person to your left will choose C then you should choose B. How should you decide what to do?
In Kneeland’s experiment, you could also see the payoff tables for everyone else in the circle. So to decide what you should do, you should probably look at the payoff table for the person standing to your left. If they have a dominant strategy, then perhaps you think they will choose their dominant strategy and you may then respond accordingly. If you were to behave in this fashion then you would be displaying second-order rationality.
In other, more complicated, situations perhaps the person to your left doesn’t have a dominant strategy either but the person to their left does (that is, they person two spots to your left has a dominant strategy). Perhaps you think that the person to your left will notice that the person to their left has a dominant strategy, and that you could use this to predict the behaviour of the person directly to your left. Then, you can make your choice taking all of this information into account. If you do this then you are third-order rational.
This logical reasoning process can be traced back further to identify people of even higher orders of rationality[2]. This special circular structure (called a ring game) is exactly what Kneeland used to identify the order of rationality of her experimental subjects.
So what were the results? Kneeland finds that 93 of her subjects are first-order rational, 71 percent are second-order rational, 44 percent are third-order rational and only 22 percent are fourth-order rational. Kneeland is very cautious about over-interpreting her results, but in a blog post I need not be so careful. The results indicate, to me, that the standard assumption of rationality is not too far off being correct in simple environments. However, the results also indicate that at most 22 percent of subjects could possibly satisfy the common knowledge of rationality assumption even in a relatively simple environment, indicating that common knowledge of rationality is not the most realistic of assumptions in many environments.
[1] Econometrica tends to publish two types of papers. The first is simple and clean papers that are good fun to read. The second is technically complicated, but brilliant, papers that can take days or weeks for experts to understand, and are pretty much incomprehensible to everyone else.
[2] An interesting question is to ask how you should reason when no one in the circle has a dominant strategy. The standard (but not entirely uncontroversial) recommendation would be that you should choose a Nash equilibrium strategy, named after John Nash who was played by Russel Crowe in the movie A Beautiful Mind.
[3] Note that this definition of rationality doesn’t mention anything about selfishness, nor make any assumptions about preferences, nor include any other of the crazy stuff that the anti-economics crowd seem to think economists mean by the word “rational”. At the most basic level you are rational if you make “good” decisions, given your information at the time of the decision, for whatever your personal evaluation of “good” is.