The game of life

A couple of weeks ago, I was reading some lecture notes on game theory and I came across a really neat game.

After discussing the very basics of game theory and decision making theory, the author of the lectures gives an exercise which I found really interesting and enjoyable, at the point that I went ahead and gave it as a quiz to my business calculus class.

To my surprise, most of my class got the right answer, which was truly a grateful feeling. The game is really simple so anybody can understand it, but in my opinion, it represents many aspects of real life.

It is as follows:

Every student is to write down a real $x_i$ number in between 0 and 10 inclusively. After doing so, one computes the mean $\bar{x}$ of all of the students' bets and each student's grade is given by

$10-\left|x_i-\frac{2}{3}\bar{x}\right|$

This might look a really simple task, and by no means a game at all, but it is a game of strategy and common sense.

Our desire as a students is to maximize our grade, but that depends on the average choice of the class, which might complicate a bit the analysis of a best strategy to pick our $x_i$.

It is not hard to see that a global best strategy is to pick $x_i=0$, as if everybody is a good and logical player, having all bets equal to $0$ would give each student's grade to be $10$, which is the best possible.

So, our personal best strategy should be to pick $0$, but in real life, not all players are good thinkers or really logical, so at the end of the day, our best strategy won't give us the best out come possible.

In a sense, we can think of this game as rewarding you if you somehow think average, and most of the times, the average thinking is not precisely the most wise and logical.

By the way the game was set up, we can see that it neither rewards the average thinking as much as someone that was 2/3 away from it. If you are 2/3 away from the average, you'll get full credit, and this is somehow what happens in real life. Usually not the average people get the best outcome nor the people that plays the best, but people that are in between.

This gives a really good example that in most occasions, your outcome does not depend only on your own strategy, but also in someone else's strategy, and that making the best decisions and taking the best choices does not guarantee your success.