Stochastic Differential Equation for Approval Ratings and Scoring Systems

Abstract

Feedback and rating/scoring systems are used almost everywhere nowadays. Generally customers are being asked to rate a product on a scale of 1 to 10, or 1 to 5 or on any other positive real number scale. The average score then would provide for the qualitative assessment of the product. For example Movies would be rated by their viewers and critics alike. The mean or some king of weighted mean of there feedback scores would then become their "APPROVAL RATING" or "AR" for short. Approval ratings thus are measures of central tendency for approval/disapproval for a product. But these approval ratings keep changing as new feedback data is punched into the system. Hence they are dynamic (stochastic in nature) with their value changing as and when more feedback data keep flowing in. The most simple feedback system is where a customer is just asked whether they liked/disliked a product. The sequence of responses then would look like the output of a coin toss experiment (with heads/approval would be scored as 1 and tails/disapproval would be 0). At any point of time the overall mean would then be reported as the Approval Rating Score. The sequence of all running means, updated with the latest feedback data from customers, would form the "APPROVAL RATING VECTOR" or "ARV" for short. If we take a look at a sequence of evolution of the approval rating score with newer feedbacks, we would realise that even though we are using the last available data as the latest rating the whole path and the data hidden behind that path is never actually used. In Finance for example a measure called IRR (Internal Rate of Return) would describe the whole yield curve. While IRR is a single number, it is derived by using the information from the whole path followed by the yield curve. Similar mechanism is also present in the field of quantitative finance where the zero coupon bond prices would be calculated from the expected value of the continuous compounding of the rates which could be described by any popular short rate model. The short rate model parameters are calculated by taking in information of whole path of evolution of the interest rate. While the future is predicted by calculating the expectation over all possible future paths that the interest rate could take. The point is that the expectation is taken over the whole path. The prevailing interest rate today is not the only component used for pricing the zero coupon bond. Approval ratings however are just described by the present mean/weighted mean, and do not use the information provided by the path it took to reach to the present value, devoiding us the power to predict its future.