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Exploring Ordinal Ballots: Ranking Candidates and Calculating Preferences

This text discusses the concept of ordinal ballots, where voters list their preferences with the most favored candidate at the top and the least favored at the bottom. Voters must rank all candidates in a complete, linear, and transitive manner. The article explores how many different ordinal ballots can be formed for varying numbers of candidates, highlighting the pattern found in factorials. It uses examples with two, three, and four candidates, introducing the significance of preferences in the context of voting for TV shows like The Office and Family Guy.

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Exploring Ordinal Ballots: Ranking Candidates and Calculating Preferences

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  1. Ordinal Ballots Preference Schedule Preference List

  2. Ordinal Ballots • List your choices in order with the favorite on top and ‘least favorite’ on bottom Ballots must be • Complete (you must rank all candidates) • Linear (no ties) • Transitive (If you prefer A to B and B to C, then you must prefer A to C.)

  3. How many different ordinal ballots are possible? • Try this first for two candidates A and B. • Next try it for three candidates A, B and C. • What about for four candidates A,B,C and D? • Can you see a pattern to the numbers? What is it?

  4. Factorials • 1! = 1 • 2! = 1·2 = 2 • 3! = 1·2·3 = 6 • 4! = 1·2·3·4 = 24 • Etc. • Note: 0! is defined to be 1.

  5. TV Program Vote Again • Vote again for The Office, Family Guy and The Mentalist, this time making a preference list with your top choice on top. • How many possible lists will there be?

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