Voting Rules Are Different

Show that all voting rules introduced in the first lecture on preference aggregation are different (of course, apart from those rules mentioned to be equal) when the number of votes is odd.

1.1 Ideally, use as few elections (example instances) as possible to exclude that any pair of voting rules is equivalent. Explain patterns you have used.

Borda vs. Condorcet

Please inform yourself about the famous dispute between the two mathematicians Marie Jean Antoine Nicolas de Caritat, Marquis of Condorcet and Jean-Charles, chevalier de Borda.

2.1 Construct an election where the Borda winner is not a Condorcet winner.

2.2 Construct an election where two candidates have a better Borda score than the Condorcet winner has.

2.3 How many candidates can have a better Borda score than the Condorcet winner has depending on (i) the number of candidates, and (ii) the number of voters?

2.4 Show a bound that is as good as possible.

2.5 Explain the dispute of Borda and Condorcet it in 1-2 sentences.