# Constructing Valuation As a Function of Condition

Suppose there is an item that deteriorates with use, its condition expressed as a number between $$0$$ (completely broken condition) and $$1$$ (perfect condition), for which I want to assign a monetary value. To make the units nicer, let's think of the condition also as a monetary value; i.e., someone has guaranteed to pay me $$T$$ dollars if I want to sell the item at any time, where $$T$$ is its condition. $$M(T)$$ is an implicit multiplicative markup on $$T$$ (i.e., $$M(T) = 2$$ would mean I value the item at twice the guaranteed resale value), and my overall value is $$V(T) = M(T) * T$$.

The first intuition (which I'm treating as an axiom) is that $$V(T)$$ should be monotonically increasing in $$T$$. The second is that $$M(T)$$ should be monotonically decreasing in $$T$$.

I can also estimate my $$V$$ at a handful of points in $$T$$. So far, I have tried Lagrange Polynomial Interpolation on such a handful of points to output a candidate $$V(T)$$, but when I divide this by $$T$$ to produce the corresponding $$M(T)$$, the result violates the second axiom; namely, polynomials produce non-decreasing regions in $$M(T)$$ around the LPI input points.

Are there ways to construct a $$V(T)$$, in elementary but perhaps non-polynomial functions, so that the two axioms are satisfied, and it also passes through a handful of input points (the points can be trusted to themselves satisfy the axioms)? Outside the regions corresponding to $$0 \leq T \leq 1$$, of course, the behavior of $$V(T)$$ and $$M(T)$$ do not matter. Is there a standard way to express valuations for these sorts of items?

• How many is "a handful" of points in $T$? Does your $V(T)$ have to exactly fit the estimated points or is an approximate fit enough? Mar 9 at 14:56
• @VARulle An approximate fit is enough; the points are just subjectively determined by asking "about how much would I be willing to pay for the item at this particular condition?" I was using 5 points with my Lagrange Polynomial Interpolation, but I guess up to around 20 points would be manageable. Mar 9 at 15:22
• O.k., but then instead of exact interpolation why not just use a simple regression model? Even linear regression could do, depending on how your data points look like. Mar 10 at 9:29
• @VARulle Linear regression would be incompatible with the two axioms, wouldn't it? To be clear, I'm not looking to express merely a trend in my data points. I want a function I could hand to someone and say, "this is what I'm willing to pay for any $T$." It's just that I wouldn't be very sensitive to, i.e., a $2-3\%$ error. Mar 10 at 19:24
• Linear regression for V. If you get a positive intercept, then dividing by T gives you a decreasing M. Mar 12 at 13:12