Let's suppose we have a variable $x$ with a domain $X \in [0,1000]$ and two utility functions $uf_1(x)$ and $uf_2(x)$ that describe the utility of $x$ with respect to two different properties. We assume that both functions model a linear relationship but in the opposite direction, i.e. $uf_1(x) = x/\max(X)$ and $uf_2(x) = 1-(x/\max(X))$. An usual and simple way to compose both utility functions to a total utility value would be the weighted sum method: $ws(x) = (w_1 \cdot uf_1(x)) + (w_2 \cdot uf_2(x))$. Finally, we seek for the $x$ that maximizes $ws(x)$. For $w_1 = w_2 = 0.5$, the utility of all $x$ is equal (0.5). For a slight preference of property 1 ($w_1 = 0.51$, $w_2 = 0.49$), the highest utility value is reached for $x = 1000$. But this is a too sensitive behavior and is actually not what I want to have.
I would rather expect the following behavior of an appropriate composition function:
- For an equal weighting ($w_1 = w_2 = 0.5$), I expect an unambiguous maximum at $x = 500$.
- For a slight preference of property 1 ($w_1 = 0.51$, $w_2 = 0.49$) that benefits from higher $x$, I expect the maximum to be at a slightly higher value than 500.
- Let's assume there is a third property and a corresponding utility function $uf_3(x)$ that behaves exactly like $uf_2(x)$, i.e. $uf_3(x) = uf_2(x)$, I would expect the maximum to be at $x = 333$ for an equal weighting ($w_1 = w_2 = w_3 = 0.33$)
- and so on ...
That finally leads to two questions:
- Does it make sense to you to expect that described behavior from a consistent composition function based on the given use-case?
- If yes, is there a generic mathematical model that realizes this expected behavior?
For equal weighting the utility as a function of $X$ is constant. And for an linear function (which the linear combination of linear utility functions has to be), the maximum will always be at the boundary, so either $x=0$ or $x=1,000$. You need a non-linearity to have a maximum that is not on the boundary.