The Pizza Ballot Revisited

In Lesson 1 we saw this example:
 A
S P
P
S
P
S
M
M
M
M
S
P
A
A
A

Recall that when using the plurality method the jumbo pizza ended up with anchovies. Also note that anchovies were the last choice of 11 voters. Since many people who don't like anchovies really hate anchovies, it could well be the case that these 11 people - a clear majority - might not want any of the biggest pizza. Although this means more for the 7 people who like anchovies, overall it doesn't seem like the fairest way to choose.

How can we overcome these difficulties? One way is to begin by eliminating candidates until only two are left and then choose head to head between these two. Now the winner will have a majority.

So how do we eliminate candidates?