I've generated a user test to compare two methods: M1 and M2. I generate 40 test cases and show the result of each method on test case to 20 individuals, side by side, the individuals don't know what result came from which method. For each test case each person has to say if the result computed by M1 is better or M2 is better or they are equally good.

I want to know if M1 is better than M2. I add up all the results and generate 3-D histogram, votes for M1, votes for tie, and votes for M2. 

If I only looked at M1 and M2 as 2-D histogram. I know that if M1 and M2 were equally good this histogram would be uniform. Then I'll just perform $\chi^2$ test.

What I don't know how to model are the votes for tie. Here are two options I've thought of:

* The basis of chi-squared test is that histograms are mutually
exclusive and add up to one. It seems like the votes for tie can be
split in two and added to each M1 and M2 (and ties removed), but this
does not seem very principled.
* Another option is that I could just ignore the ties, that seems
flawed because it breaks the "add up to one" property. For example if
I had (M1:2, ties:98 M2:0) the difference between both methods would
be not statistically significant.

What else can I do? Am I looking at this incorrectly? This seems like a common problem people would face when modeling user votes. What is correct way to model the ties?