The Golden Ratio in Economics
Rounding to three digits because it feels right
This originally appeared as an email to my parents. Congratulations on being privy to our communication.
For the duration of its existence, economics has been based on a theory of ‘rationality’, stating that people will act the ways which maximize their monetary take home, regardless of all else. (Robinson Crusoe gets mentioned frequently)
This has always been a rough approximation, but for the past few decades alternatives have been explored via research, to see if some other theory fits humans better than perfect rationality.
In particular, today I studied why we act differently in ‘game’ situations than rationality would predict. A game is anything with players and a defined strategy set. An example would be a prisoner’s dilemma, or even chess (which would be immensely hard to model).
One group of economists has studied this via a theory of ‘cognitive hierarchy.’ They are analyzing games under the assumption that we solve problems in steps. A popular game, often used in the first meeting of game theory 101, is a good example of this. (This game is called a ‘Keynesian Beauty Contest’, after an analogy that appears in Keynes’ famous book A General Theory of Employment, Interest, and Money) The game is this: everyone in a room is given a slip of paper, and the following instruction: ‘Write a number on the sheet of paper between 0 and 100. Whomever gets closest to 2/3 of the average of the room wins $20.
You can see the steps of the problem here: Even assuming that everyone else puts the highest possible number (100), then 2/3 of that is 66. So, the highest you should ever put is 66. If you assume everyone is that smart, you say that the highest average could be 66, so you’d put the two thirds of 66, which is 45. If you think everyone is at least that smart, you put… etc. Essentially, you go down steps to solve the problem. An economically rational person, for example, would iterate infinitely, and realize that the only rational choice is to play 0, and everyone else would play zero, and you’d split the money.
Of course, we are not rational, and the general result of the game ends with an average around twenty or thirty. The researchers I mentioned above have proposed the cognitive hierarchy theory for why something like this happens, and here is how it works.
People are made such that they will be smart enough or hard working enough to go through k steps of a process. Being human, however, we are obnoxiously self confident and assume that no one we are playing with is as smart or smarter than us. (We think no one else will play k steps or k+1 steps.) It is pretty clear from both evidence and intuition (thinking more takes energy, and we’re lazy) that the more steps you go, the fewer people go that far, so the number of people making each number of steps declines as you go to higher and higher k. As it turns out, this is approximated quite well with a Poisson distribution, which is quite nice because a Poisson distribution is defined by one parameter, let’s call it z.
Now, with our Poisson distribution, we want to figure out what that value of z might be. From a look at a decade’s worth of multi-step games, it would appear that the number of people who don’t take the time to think at all (and take k = 0 steps, and don’t go through the ‘two thirds of 100 is 66' step above), and add them to the number of people who only take k = 1 steps (who see that you can’t play above 66, but don’t think any further), you see that this sum is almost exactly double the number who take two steps (and play 45). This is enough information to find z in our Poisson distribution. Poisson is quite cool because the variable z not only defines the whole process, but it is also the mean. In our particular manipulation, the mean is the average number of steps a person takes when solving a game. And when we have (# 0 steps) + (# 1 steps) = 2 * (# 2 steps), that value of z becomes…. 1.618, aka the golden ratio.
So not only does the golden ratio give us pleasant rectangles and cool looking shells, it is perhaps also the average number of steps a human is willing to think in a given moment.
Important Note: Nothing mentioned here is statistically significant. In one test, when rounded to three decimal places, this result popped up. But wouldn't it be fun if it were true?
