Why does the original Game Boy support four shades of grey? [closed]

While Nolonar's answer is true from a technical perspective, it's not the actual reason why the developers chose to go with a monochrome system. Gunpei Yokoi, the creator of the Game Boy, intentionally designed the system monochrome to force games to be more abstract, thus preventing players from getting too caught up in details. As he put in in a 1997 interview:

The technology was there to do color. But I wanted us to do black and white anyway. If you draw two circles on a blackboard, and say “that’s a snowman”, everyone who sees it will sense the white color of the snow, and everyone will intuitively recognize it’s a snowman. That’s because we live in a world of information, and when you see that drawing of the snowman, the mind knows this color has to be white. I became confident of this after I tried playing some Famicom games on a black and white TV. Once you start playing the game, the colors aren’t important. You get drawn, mentally, into the world of the game.

The full interview can be read here: http://www.techspot.com/news/61318-console-gaming-now-fascinating-1997-interview-nintendo-legendary.html

While I haven't studied the internal workings of the Game Boy, I have designed and built a controller for a four-gray-level LCD, for use with display panels that were designed for on/off control only. To get four-level grayscale from a panel that's designed for on-off control, one must be able to do the following:

1. Switch instantly between two display buffers.

2. Time the display switching so that the first buffer will be shown for one scan, the second for two, the first for one, the second for two, etc.

3. Set the refresh rate about 3x as fast as would otherwise be needed to avoid flicker [in practice, it can be a bit less than 3x as fast; I used 100Hz].

Adding the ability to switch between two display buffers at the proper times and boosting the refresh rate are both things that can be done quite cheaply. While the same approach might in theory be used to achieve an eight-level or sixteen-level grayscale, doing those things would require significantly increasing display refresh rates. While pushing things to eight gray levels or even sixteen might be feasible, the comparative benefit obtained by doing so would be slight compared with the benefits of going from on-off control to four-level control.

[BTW, some common controllers have the ability to automatically flip between two buffers, but show each one for two scans rather than doing a 1-2-1-2 pattern; I don't know why they don't have 1-2-1-2 since it provides four gray levels rather than three, doesn't require quite as high a refresh rate, and is at least as effective if not moreso at preventing display polarization].