[peripetylabs]

In the case of three people sharing one pizza:

Slice the pizza in four, and eat three of the quarters (the two people who aren't slicing picking first of course). Repeat with the remaining quarter until what's left is not worth being disappointed about.

I would never actually do this, but it's a good illustration of limits of geometric series.

[xentronium]

You assume that any pizza can be reliably sliced into four more or less even parts. Also you assume that cutter is interested in getting the best share. That is not always the case.

Imagine that slicer makes 1 huge and 3 tiny pieces. He obviously won't get the biggest one, but the second chooser will be mistreated too. Thus, evil divider can starve the second chooser to death.