Abstract: Consider three stockholders in a company: A has 42 shares of stock, B has 38 shares, and C has 20. If a simple majority of shares is required in order to pass a motion, then any two out of the three voters voting together would win. So, despite the unequal stock holdings, all three stockholders possess the same amount of power. What if we require a three-fifths majority in order to pass a motion? Well, B and C no longer win by voting together, so their power decreases. If a two-thirds majority is required, then C's power has been further reduced (to 0 in fact!)

These stockholders form an example of a "weighted voting system". We will classify all of the possible weighted voting systems for n=3 and n=4 voters, and discuss how to generalize for an arbitrary number of voters. Our tools are polyhedral geometry and combinatorics.