How Rational Should You Be?

A problem that comes up when we discuss game theory is that it can sometimes be hard to logically arrive at the best solution. It took us a few minutes to solve the lions and Christians example as a class and it’s fair to wonder if individuals playing games will always reach the solution. Besides the difficulty of finding some solutions, there is the question of how hard your opponent will work to find that solution.

If you’ve played the two-thirds game, you’ve seen this. The goal of the game is to pick a number from 0 to 100 that will be two-thirds the average of the numbers everyone in the game picks. If 10 people played and they picked numbers such that their average was 60, the player who picked the closest to 40 would win (60*2/3  = 40). If you look at the game, you’ll probably realize that the average cannot be higher than 100, meaning you should never guess a number higher than 67. So now 67 becomes ceiling, meaning it makes no sense to guess anything less than 67*2/3 or about 45. If you continue this process you should get to 0 as the best guess.

Now that you’ve found the solution, would you actually play it? When I had 10 people play the game for a class last semester, the average was about 20 and the winning player guessed 13. The fact that the game didn’t reach 0 represents a combination of the fact that some people won’t realize the true solution and some people believe other people won’t reach it.

This article uses another game, the Traveler’s Dilemma, to illustrate these problems. In this game, you are and an opponent are trying to place a value on a broken vase between $2 and $100. If you agree on the price you will both receive that price to replace it. If you differ, you’ll both receive the lower value and the person who suggested the lower value will get a $2 bonus. The high value person will get a $2 penalty.

If you both think about the game completely logically, you will both end up saying $2. But no one actually does that. In fact, when researchers had professional game theorists play, two-thirds of the the guesses were above $95.

The fact that you can find a solution to a game doesn’t always mean you should play the solution. The article suggests that the best way to decide what you will do is to base it off of what you believe your opponent will do. Playing rationally when your opponent plays irrationally can actually reward your opponent. There could be a penalty for being too rational.