# Ranking estimation with partial data

Consider a problem where we ask a number of people to select and rank their top three choices out of a number of options. The set of options is the same for everyone, and they all have to rank their top three choices.

How can I go about the estimating the ranking of the options given data collected? How can I model the problem?

For example, say that we get the following data (only showing the first four options, and the rankings whenever someone included them in the ranking):

Option A: 1
Option B: 1,3,1,2,1
Option C: 3,3
Option D: 2,2,2


Where the way to read the above is:

• First row: Only one person chose A as their top choice
• Second row: Three people chose B as their top choice, one person chose it as their second best choice and one more person chose it as its third choice.

A trivial, but unhelpful answer would be to average the ranking for each option. That would be:

Option A: 1
Option B: (1+3+1+2+1)/5 = 1.6
Option C: (3+3)/2 = 3
Option D: (2+2+2)/3 = 2


The above averaging is unhelpful because it would not consider the number of people that chose a given option, and given the way the experiment is designed, the more often a given option is chosen in the top 3 list, the higher the ranking should be.

How can I go about modeling this problem?