$\chi^2$ test on user preferences 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? 
 A: I suspect whuber's answer is (as usual) more replete than what I am about to type.  I admit, I may not fully understand whuber's answer... so what I am saying may not be unique or useful.  However, I did not notice where in whuber's answer the nesting of preferences under individuals as well as the nesting of preferences within test-cases was considered.  I think given the question asker's clarification that:

The cases are indeed a random sample of all possible cases. I think an
  analogy is the following: the election is determined by what happens
  at the polls, but I do have for each voter their party affiliation. So
  it would be almost expected that a candidate from one party appeals to
  the voters affiliated with that party, but this is not necessarily a
  given, a great candidate can win in his party and win over people form
  the other party.

... these are important considerations.  Therefore, perhaps what is most appropriate is not $\chi^2$ but a multi-level logistic model.  Specifically in R I might cast something like:
lmer(PreferenceForM1~1+(1|RaterID)+(1|TestCaseID),family=binomial)

PreferenceForM1 would be coded as 1 (yes) and 0 (no).  Here an intercept over 0 would indicate an average rater's preference for method 1 on an average test case.  With samples near the lower bounds of usefulness for these techniques, I'd probably also use pvals.fnc and influence.ME to investigate my assumptions and the effects of outliers.
The basic question about ties here seems well answered by whuber.  However, I'll (re-)state that it seems that ties reduce your ability to observe a statistically significant difference between the methods.  In addition, I'll claim that eliminating them may cause you to over-estimate the preference individuals have for one method versus the other.  For the later reason, I'd leave them in.
