Satterlee.com - Crack the Code

Crack the Code

Arnold Kling Tuesday, December 03, 2002

From an article on the causes of prosperity. I like the game more than I like his article.

Below is a game that illustrates the difference between linear and nonlinear feedback. It is a standard game where the object is to "crack the code." There are three columns of colors. The computer will randomly pick one color in each column to form a secret code. For example, it might pick red-silver-red.

To win the game, you must guess the code. The computer will give you feedback, based on your guess. Having a correct color in the correct column is worth four (4) points. But having a right color in a wrong column is worth only one (1) point. For example, if the code is red-silver-red and you guess red-blue-silver, you get 4 points for putting red in the first column and 1 point for putting silver in the third column, for a total of five (5) points. You use this feedback to make your next guess. You keep refining your guesses until you solve for the code.

That is how it works with linear feedback, where we add the points to get a total. However, what if we use nonlinear feedback? For example, suppose that the point total you get is equal to the product of the points that you get in each of the three columns. So, if you guess red-red-silver, you get 4 points times 1 point times 1 point = 4 points. Note that if you guess red-blue-silver you will get nonlinear feedback of zero (0) points.

Got it? Now try the game. First play with linear feedback. After the computer says "You Won," press start to play a new game. Try playing several games. Using logical deduction, you should be able to win most games in five or six rounds.

Once you get the hang of the game, try switching to nonlinear feedback, and try playing a few games. Do you see how frustrating it can be?

Downloaded: December 5, 2002
From: www.techcentralstation.com

My back-of-the envelope analysis comes up with the following:

Total number of options = 64
Possible scores / number of positions

Score: 64 (4 * 4 * 4) / No. of positions = 1 (the Winner)
Score: 16 (4 * 4 * 1) / No. of positions = 0
Score: 4 (4 * 1 * 1) / No. of positions = 3
Score: 0 (X * X * 0) / No. of positions = 60
So, by the rules of this game, if you're almost there with 2 correct positions, the third position will be 0.
(Column 1 position is correct = 4 points.
Column 2 position is correct = 4 points.
Column 3 can not be worth 1 point.
{right color, wrong column} is not an option.
The only two options are:
{right color, right column} = 4 or
{wrong color, right column} = 0.)
Therefore, you end up with feedback that is the same as if you weren't even close.

Like they say, "close" doesn't count except in horse shoes.