How do we take individual rankings and produce a group ranking?

This is the central question in voting.

We might simply disregard all of the individual rankings except for one, maybe the president of the club, and make her ranking the group ranking. (In this voting method the president is called a dictator.) Although this is perhaps the simplest method, it is unlikely that the other club members will be happy about it.

There is no requirement that everyone's vote should count, or that every vote counts the same. (The U.N.  Security Council is an example of this.) However, in most political elections every person's vote carries the same weight, and we will assume this for all of our examples.

Suppose our only choices for toppings are pepperoni, sausage, mushrooms, or anchovies. Each person in the club could fill out a ballot, which might look like:

Joan  Richard  Suzanne 
Sausage  Anchovies  Pepperoni
Pepperoni Sausage Mushrooms
Mushrooms  Mushrooms  Sausage 
Anchovies  Pepperoni Anchovies

With four choices, there are 24 possible orderings. If we had five choices, there would be 120 possible orderings. 

If the club were large, actually listing all of the ballots in full would take a lot of space and would be hard to use. So, next we'll do several things to make the ballots easier to understand.