Question on Condorcet (Schulze) Ranking Algorithms

I've been doing some research on ranking algorithms, and I've read your research with interest. The aspects of the Schulze algorithm that appeals to me is that respondents do not have to rank all options, the rank just has to be ordinal (rather than in order), and ties can be resolved. However, upon implementation, I'm having trouble showing that ties do not occur when using the algorithm.

I've put together a below real-life example, in which the result looks to be a tie. I can't figure out what I'm doing wrong. Is the solution simply to randomly select a winner when there is a tie? Could somone please have a look and offer any advice?

Thanks very much,
David

Who do you think are the best basketball players?

_ Kevin Garnett
_ Lebron James
_ Josh Smith
_ David Lee
_ Tyson Chandler

5 users (let’s just call them User A, User B, User C, User D and User E) answer the question in the following way:

User A :
1.Kevin Garnett
2.Tyson Chandler
3.Josh Smith
4.David Lee
5.Lebron James

User B:
1.Kevin Garnett
2.David Lee
3.Lebron James
4.Tyson Chandler
5.Josh Smith

User C:
1.David Lee
2.Josh Smith
3.Kevin Garnett
4.Lebron James
5.Tyson Chandler

User D:
1.Lebron James
2.Josh Smith
3.Tyson Chandler
4.David Lee
5.Kevin Garnett

User E:
1.Tyson Chandler
2.David Lee
3.Kevin Garnett
4.Lebron James
5.Josh Smith

If you put together the matrix of pairwise preferences, it would look like:


To find the strongest paths, you can then construct the grid to look like:



Now, the strongest paths are (weakest links in red):



So the new strongest paths grid is:



Kevin Garnett and David Lee would tie in this case.

dstevens said:

The aspects of the Schulze algorithm that appeals to me is that respondents do not have to rank all options, the rank just has to be ordinal (rather than in order), and ties can be resolved.

Click to expand...

I find the following reference in Wikipedia, "Schulze Method":

Although ties in the Schulze ranking are unlikely,^[4] they are possible. Schulze's original paper^[1] proposed breaking ties in accordance with a voter selected at random, and iterating as needed.

[1] Markus Schulze, A new monotonic, clone-independent, reversal symmetric, and condorcet-consistent single-winner election method, Social Choice and Welfare, volume 36, number 2, page 267–303, 2011. Preliminary version in Voting Matters, 17:9-19, 2003.
[4] Under reasonable probabilistic assumptions when the number of voters is much larger than the number of candidates

In your example the number of voters is equal to the number of candidates, so probabilistic arguments do not hold.