# Raynaud

**Raynaud**, **Arrow-Raynaud's method**^{[1]} or **Pairwise-Elimination** is a Condorcet method in which the loser of the strongest pairwise defeat is repeatedly eliminated until only one candidate remains. Raynaud can also be described as the sequential loser-elimination method that uses Minmax as its base method. Defeat strength is usually measured as either the absolute number of votes cast for the winning side (*winning votes*), or the number of votes for the winning side minus those for the losing side (*margins*). It was devised by Hervé Raynaud.^{[1]}

## Criterion compliances

Raynaud satisfies Condorcet. Since it is a sequential loser-elimination method, it thus automatically satisfies the Smith criterion. Furthermore, it satisfies ISDA.

Raynaud fails the Monotonicity criterion. Even when winning votes are used as the measure of defeat strength, Raynaud fails the Plurality criterion and the Strong Defensive Strategy criterion. A variant called *Raynaud(Gross Loser)* does satisfy the Plurality criterion.

### ISDA

Raynaud passes ISDA. This is because Raynaud can be thought of as hinging a candidate's victory chances on the successive evaluation of the ordering of all defeats from strongest to weakest; since all candidates in the Smith set have no pairwise defeats to any candidates not in the Smith set, there is no way for Smith set candidates' elimination chances to be impacted by the existence or non-existence of candidates not in the Smith set.

### Plurality criterion

*Raynaud(Gross Loser)* satisfies the Plurality criterion. It successively eliminates the candidate with the fewest votes for him in any pairwise contest. In this way, it is not possible to eliminate candidate *A* before candidate *B* when *A* has more first preferences than *B* has any preferences, since this situation means that the minimum number of votes for *A* in any contest is greater than the maximum number of votes for *B* in any contest. This variant was devised by Chris Benham.

Example:

- 25: A>B>C
- 40: B>C>A
- 35: C>A>B

A beats B (60 to 40) beats C (65 to 35) beats A (75 to 25), so there is no Condorcet winner. If using winning votes as the defeat strength measure, then the C>A defeat is strongest, and so A is eliminated (Raynaud(Gross Loser) does the same). Then, since B beats C, C is eliminated for being the only candidate remaining with a defeat, and B wins.

### Monotonicity criterion

This example is due to Kevin Venzke.^{[2]}
In the election

- 36: A>B>C
- 34: B>C>A
- 30: C>A>B

C is eliminated and A wins. Then raise A on five of the B>C>A ballots to get

- 41: A>B>C
- 29: B>C>A
- 30: C>A>B

where B is eliminated and C wins.

## Notes

Raynaud's calculation can be simplified by only eliminating candidates until there is a Condorcet winner.

## References

- ↑
^{a}^{b}Arrow, Kenneth J.; Raynaud, Hervé (1986).*Social Choice and Multicriterion Decision-Making*. The MIT Press. ISBN 0-262-01087-9. - ↑ Venzke, K. (2005-12-07). ""Talk: Monotonicity criterion: Problem in the example?"".
*Wikipedia*.