1. Condorcet Method Another group-ranking method

2. Development of Condorcet Method  As we have seen, different methods of determining a group ranking often give different results.  For this reason, the marquis de Condorcet, a French mathematician, philospher, and good friend of Jean-Charles de Borda, proposed that any choice that could obtain a majority over every choice should win.

3. Condorcet Method  The Condorcet method requires that a choice be able to defeat each of the other choices in a one-on-one contest.  Again consider the preference schedules from the last section.

4. Condorcet Example (CORRECTED)!  We must compare each choice with every other choice: A A A A BBB B C C C C D D D D 8567

5. Condorcet Example (cont’d)  Begin by comparing A with each of B, C and D.  A is ranked higher than B on 8 schedules and lower on 18.  Because A can not obtain a majority against B, it is impossible for A to be the Condorcet winner.

6. Choice B  Next consider B.  We have already seen that B is ranked higher than A, so let’s now compare B with C.  B s ranked higher on 8+5+7 = 20 schedules and lower on 6.  Now compare B with D. B is ranked higher on 8 + 5 + 6 = 19 schedules and lower on 7.  B could be the Condorcet winner since it has a majority over the other choices.

7. Tabular results  The table on the left shows all of the possible one-on- one contests in this example.  To see how a choice does in one-on-one contests, red across the row associated with that choice. ABCD ALLL BWWW CWLW DWLL

8. Disadvantages  Although the Condorcet method seems ideal, it sometimes fails to produce a winner.  Try this example!

9. You Try: Condorcet Example A A A B B BC C C 20

10. Condorcet Example Explained  Notice that A is preferred to B on 40 of the 60 schedules but the A is preferred to C on only 20.  For this reason, there is no Condorcet winner.

11. Introduction of Paradoxes  You may expect that if A is preferred to B by a majority of voters and if B is preferred to C by a majority of voters, then a majority of voters would prefer A to C.  However, we have just seen that this is not necessarily the case.

12. Transitive Property  Do you remember the transitive property?  If we consider the relation “greater than (>)”, we see that if a > b and b>c, then a>c.  However, according to the Condorcet method the transitive property does not always hold true.  When this happens using the Condorcet method, it is known as a Condorcet Paradox.

13. Extra Practice 1.The choices in the set of preferences shown represent three bills to be considered by a legislative body. The members will debate two of the bills and choose one of them. The chosen bill and the third bill will then be debated and another vote taken. Suppose you are responsible for deciding which two will appear on the agenda first.

14. Practice (cont’d)  If you strongly prefer Bill C, which two bills would you place first on the agenda? Why? A A A B B B C C C 4030

15. Practice (cont’d) a.Use a runoff to determine the winner in the set of preferences. A AA A B B BB C C C C 8 14 322