I am trying to make the better user experience for my poll application and one of the ideas I have for this is breaking multi choices polls into binary polls.
Is the statistical result of a question with these 4 options (a or b or c or d) equal with sum of statistical results of these 6 binary questions?:
(a or b), (a or c), (a or d), (b or c), (b or d), (c or d)
assume I ask these questions from separate people and I ask these questions equally
for example: assume we have a society with 1000 members if we ask all of them the first question, people will answer it with this distribution: a:60%, d:20%, b:15%, c:5%
now assume if we ask those six binary questions from all members of that society and then we sum each vote of winners of each of those questions.
we ask 1000 times a or b and the answer is a:600,b:400
we ask 1000 times a or c and the answer is a:500,c:500
we ask 1000 times a or d and the answer is a:100,d:900 ....)
we sum number of votes on a and it is 1300 =600+500+100
we do a similar thing with b,c, and d, is this result similar to the result of the first question? (a:60%, d:20%, b:15%, c:5% ) and is the meaning of this result similar to the first one?
@Bruno mention to the inconsistency of the result of this approach and I think it can be solved if we do not show all combinations to all voters. If voter1 chooses "a" between "a" and "b" we do not show questions with any combination of "b".
In fact, what I wish to know from this community is: "Does anybody do statistical research on how results of these two types of questionnaires are different?" The problem with this research is that when you ask any of these question types from someone it affects the result of the second one (people want to show themselves rationally. and if we do not ask questions with the same items for example (a or b), (a or c) (a or b or c) from the same person we do not certain about the complete answer of this particular person and the data is insufficient for comparing) On the second chance if there is not any already done research on this I want to know what is the proper (psychologically proper) way to do this research?