# How to rank a product with 90% score by more number of people better than a product with 100% score by less number of people?

First of all, I really couldn't think of a right way to phrase the question title. Below, I have reduced the problem that I am trying to solve to a similar one that could be easily understood. I searched for similar questions on this forum but really couldn't find one that's close enough/similar to mine. Please help.

Let's say, for example, that I purchase camera on Amazon. I would rather purchase a camera with a 4.9 star (out of 5 star max.) rating by say, 100 people than purchase a 5 star rated camera by, say, 10 people.

In other words, I want the product with 4.9 star rating out of 100 people to be the winner than the one with 5 star by 10 people.

Could you please help me how can I achieve this mathematically? Is there subjective opinion involved here that would make it difficult to tackle the problem mathematically? While my mind understands what it wants, translating it into a problem which I could program a computer to do has been challenging me for quite some time now.

There are a variety of related problems.

Rank aggregation (combining rankings of preferences for various users into one overall ranking) in the literature, and appears in books on voting theory, game theory, economics, operations research, etc.

You may want to read these slides for some details.

Recommender systems - Predict the preferences a user would have towards an item. This is the basic problem that underlies the Netflix Prize. Other terms to look at are collaborative filtering. The case for Amazon is presented in this patent.

When designing these systems, one has to deal with robustness (such as how stable the predictions are), scalability and security against adversaries (for example, if Google's PageRank algorithm was sensitive to certain types of inputs which gave higher placing in the search results, people would exploit it for financial gain), privacy as well as trading off exploring new items someone may like versus items you know they'll already like (trading exploration with exploitation).

Its a difficult problem to formulate in order to balance these traits (such as how to associate a score with the reliability of a review or a user to rank their relative importance) and is an active area of research (with full conferences dedicated to related problems like ACM RecSys).

• Thank you for the pointers. I will have a look at them and will accept your answer afterwards. Commented May 11, 2014 at 6:01