Guessing 2/3 of Average
Let’s play a simple game.
Say, In a room of 20 ladies and gentlemen, each of them is required to write down a number between 0 to 100.
The one who got the number closet to the average of all players’ choices wins the game.
So it’s natural that the result would be around 50.
If everybody is rational enough, it is likely that the answer of everyone would be 50.
Now let’s make a small adjustment to the rule.
This time, the one who got the number closet to two-third of the average of all players’ choices wins the game.
So, is there a Dominant Strategy (i.e. the “best” strategy) to play this game?
Iterated elimination of dominated strategies
The Answer is YES, with the assumption of perfect Rationality.
The key is to think deeper and deeper.
Firstly, since the Winning Number is 2/3 of Average, even if everybody chooses 100, 2/3 of the Average is just 66.6667. Hence, it is senseless to guess above 66.6667.
Then we move on to think deeper.
If everybody knows the only possible range of guess lies within [0, 66.6667], with a 2/3 of this range, 2/3 of the Average should be below 66.6667 * 2/3 = 44.4444.
So the rational range of guesses now lies within [0, 44.4444].
As every player thinks deeper and deeper, by elimination, the answers of all rational players will eventually go to 0.
So given that all players think rationally, the best strategy should be writing down 0.
But is it true in reality?
What happened in reality?
Nagel has published a a paper in The American Economic Review Vol. 85 in 1995, describing the result of a real experiment of Guessing 2/3 of Average Game.
Here is the result:
As we can see, the averages were decreasing over periods as more and more players began to realize the trick. However, even after 4 periods of Games, the average was still above 0.
The experiment shows that guessing zero is unlikely to be the best strategy in reality. Instead, the best strategy should be guessing at “a moving target” towards zero.
Steps of Thinking
Nagel introduced a concept of “Steps of Thinking” to describe the behavior of players in the 2/3 of Average Game.
Firstly, We set r as the mean of guesses in the previous period, p as the eliminating factor on average (p = 2/3 here)
Nagel found in the experiment that the majority would choose 100p as their initial guesses.
So we assume the numbers raised in Period 1 were randomly distributed with a mean of 100p.
From Period 2 and on, players should begin to guess with the information of numbers raised in previous Periods.
Rationally thinking, the average of guesses should be decreasing with time due to p < 1.
Hence we can classify players with “Steps of Thinking” as the following:
- Those who guess much above mean r are “Random Guessers“
- Those who guess around mean r are “Trend Followers“
- Thinking one more step further, if most players will choose the Average in last period, r, the final result this time should be around
(2/3 of r). We call these people “Step 1 Thinkers“
- Thinking even one more step further, if others see things in the same way and propose
. They are “Step 2 Thinkers“
So we define “Steps of Thinking” as the following:
| Step | Upper Boundary | Lower Boundary | |
| Random Guessers | -1 | 100 |  |
| Trend Followers | 0 | |
 |
| Step 1 Thinkers | 1 |  |
 |
| Step 2 Thinkers | 2 |  |
 |
… An so on …
The Table below shows the result summarized by Nagel on Period 2 to 4 of the Game:
It seems that players tended to be “Step 2 Thinkers” and more players joined this group as time went on.
So the result may suggest us to think for 2-3 Steps more as other players would probably do so! 🙂
At last, here is a online game for Guessing 2/3 of Average, in which you will be guessing the 2/3 of average of numbers raised by previous 100 visitors.
See if you can guess it correctly!
Project Delta 