I am trying to rank some related/unrelated items and asking for your help in the identification of a solution.

Let assume that we ask a person whether he likes fruits and vegetables, and he replied that he likes fruits more than vegetables.

Among the fruits he says that he likes oranges, bananas, and apples in the same order; and among the vegetables, he says that he likes potato and peas respectively.

So the problem is to find overall ranking of these five items, i.e. oranges, bananas, apples, potato, and peas based on the person's ranking, e.g. does he like potato more than apple, or banana?

I was thinking converting the rankings to probabilities (which currently I don't know how to) such that $P(X=fruit)+P(X=vegetable)=1$ and $\sum P(Y=fruit_{i}|X=fruit)=1$ and $\sum P(Y=vegetable_{i}|X=vegetable)=1$

Then ranked the items based on the result of conditional probability, i.e. $P(Y=orange)=P(Y=orange|X=fruit)P(X=fruit)$. Thus for each item I can calculate the probability and hence rank the items finally.

There are two main problems in this approach:

-How to calculate probabilities from rankings? and

-Since the number of items is different in each group, e.g. 3 fruits and 2 vegetables, the assignment of the probabilities may not be aligned between the groups.How to assign the probabilities to each item in different groups?

Can you please provide any direction to approach the problem?



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