Guess 2/3 of the average

The game is…

• you compete against some other people
• each of you guess one number from [0, 100]
• compute 2/3rd’s of the average of the guessed numbers
• the winner is whoever is closest

For example, if the numbers guessed are 7, 28, 53, 77, then the 2/3rd’s the average is:

Therefore, whoever guessed 28 wins!!
So……….. what number should you pick?

Of course the answer depends on your thought process. Let’s take a look to see what numbers people picked:

So what number would you pick?

One should note that guessing any number that lies above 66.66 can NEVER be equal to 2/3rd’s the average. Why is this true? Take the example that everyone picks 100. Then the average is 100. So 2/3rd’s of this is 66.66. This is the highest that “2/3rd’s the average” can be.

Thus, any rational player, would pick a number between [0, 66.67].

Now, any rational player would realize that everyone else is going to pick a number in [0, 66.67]. And thus the highest that 2/3rd’s the average can be using numbers in [0, 66.67] is 44.444. So the rational person will not pick a number above 44.444, since there is no way it can be 2/3rd’s the average.

Thus, any rational player, would pick a number between [0, 44.44].

Now you can see you can repeat this process and eventually get yourself down to [0, epsilon]. Thus, if everyone was rational, they would all pick the number 0, and hence, it would be a tie game.

However, strangely enough, not everyone can work the argument all the way down to 0. Hence, you have people picking 22 and 33 (see the spikes in the graph). The amazing thing is that people pick numbers like 100! There is no way 100 can win, so clearly these people don’t understand the game at all, or never even thought about their answer. There is also a spike at 0, most likely by mathematicians and people from similar disciplines who are familiar with the problem. However, if they were really smart, they’d know that they can’t win picking 0, since it’s highly likely that not everyone will pick 0.