Wednesday, March 24, 2010

Rationality Equilibrium

As a way to make my Economics class more interesting, each homework assignment has an 'Extra Credit Game' that illustrates a principle of economics or game theory.  A game that I have used three times now, with a slight variation, is:

Extra Credit Game: Choose a number from zero to 100. You will get one point if you choose the number that is closest to one-half of the average of all numbers chosen by the class.

Before reading on, pick the number that you think will win.

This game has a lot of strategic depth, if you think about it a bit.  It tests your rationality and your ability to predict human behavior.  There are several possible levels of thinking:

Level 0: Some people will just not think about the problem and pick a number at random.  Anyone who picks a number above 25 is clearly a Level Zero thinker.

Level 1: If you assume that everyone else guesses at random, then you will figure that the average will be 50, so you will choose 25.  That is, if you chose 25 it marks you as a Level One thinker who assumes that everyone else is a Level Zero thinker.

Level 2:  If you think that everyone else is a Level 1 thinker, then you will think that the average will be 25, and you will pick 12.

There are also in-between levels.  If you picked a number between 12 and 25, it means you think that the rest of the population is divided between Level 0 and Level 1 thinkers.  (Or you could be a Level Zero person who just happened to guess low.)

Level 3:  If you think that everyone else is a Level 2 thinker, then you will think that the average will be 12, and you will pick 6.

This process continues, up until the level of 'Hyperrationality'  A hyperrational thinker is one who is capable of thinking through all consequences of everything and assumes that everyone else does the same thing.  A hyperrational person will pick zero for the problem.

Game theory assumes that everyone is hyperrational, so the only equilibrium for the problem is one where everyone picks zero.

There is a difference between being hyperational and being wise.  Wise people understand that the world is full of people who, to put it kindly, do not really think through things, and adjust their estimates accordingly.  In order to win, you have to accurately judge the rationality of the other people playing the game.

This game shows how hard it is to make economic predictions about things like consumer behavior or housing prices or the stock market, because the 'correct' prediction will be based on the actions of other people, many of whom are also trying to figure out what you are doing and who will change their actions based on the signals you generate.

The first time I ran the game, the class average was 17.2 so the winning number was 9.  The second time I ran it, the average was 8.5 and the winning number was 4.

The last time, after the homework had been assigned but before it was due, one of the students asked, in class, "If we all pick zero, will we all get the point for winning?"  I said yes.

This student is the 'community organizer' of the class.  Whenever I run any kind of coordination game, she sends emails out to the class telling them what to do so everyone gets the maximum number of points.  She told everyone, in class, to pick zero, and later sent an email to everyone with the same instructions.

This is a great example of why economics models that assume rational, or even hyperrational actors, make good predictions about the world.  Casual observation, plus lots of lab experiments, have shown that people act irrationally the first time they are confronted with a problem.  In my class, five people picked numbers greater than 25 in the first game.  But economic theory is about the real world, where people have a chance to see what happens as a result of their choices, and learn from their mistakes.  People may be bad at thinking things through, but they are good at responding to feedback and learning through experience.  It does not take many feedback cycles for behavior to converge to the hyperrational.

Unfortunately for our community organizer, and most of the class, chaos happens no matter how much you try to plan.  There were three people who did not pick zero.  I have no idea why.  They were actually in class when she asked the question, and they must have gotten the email.  Maybe they just did the homework at the last minute and forgot about the plans.  Two of the people picked higher numbers, and one picked the number '1' which turned out to be the winning number.  I do not think that this was a conspiracy, based on what I know of the social patterns of the class.  In any case, there is zero motive to arrange such a thing, because the winner would have still gotten a point if she picked 0, and there is no grade curve.

This illustrates another thing that economists know.  Even if everyone has an incentive to cooperate, and if someone is actively helping the cooperation, there will always be random shocks to the system.  No matter how much you try, it is really hard to get people to all do the 'right' thing.  Any system you design has to be robust to these shocks.  A small shock, or a small act of stupidity, can make everything fall apart if you are not careful.  Never assume that people will always listen to you or do the right thing or even avoid doing utterly stupid things.

So in conclusion, 'self-interested rationality, plus a random error term' is the best model we have of predicting human behavior.

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