Preference Elicitation — Jemoka Knowledge Base

For weighted sum method for instance, we need to figure a w such that:

\begin{equation} f = w^{\top}\mqty[f_1 \\ \dots\\f_{N}] \end{equation}

where weight w \in \triangle_{N}. To do this, we essentially infer the weighting scheme by asking “do you like system a or system b”. first, we collect a series of design variables (a_1, a_2, a_3 …) and (b_1, b_2, b_3…) and we ask “which one do you like better” say our user WLOG chose b over a so we want to design a w such that w^{\top} a < w^{\top} b meaning, we solve for a w such that… \begin{align} \min_{w}&\ \sum_{i=1}^{n} (a_{i}-b_{i})w^{\top} \ \text{such that}&\ \bold{1}^{\top} w = 1 \ &\ w \geq 0 \end{align} unlike the rest of everything, we are MAXIMIZING here idk why example assume: if we prefer a to b, then w^{T} a > w^{T} b. Let’s say we had two bags, each with a = \left(1,3,6\right) and b = \left(7,1,2\right). This means:

\begin{equation} 1 w_1 + 3 w_2 + 6 w_3 > 7 w_1 + w_2 + 2 w_3 \end{equation}

Doing algebra, this gives:

\begin{equation} -6 w_1 + 2 w_2 + 4 w_3 > 0 \end{equation}

Recall this is also a probability, so we have:

\begin{equation} 1 - w_1 - w_2 = w_3 \end{equation}

Finally, solving this gives:

\begin{equation} 5w_1 + w_2 < 2 \end{equation}

This is what we call a halfspace, which further bounds weights that are possible. Its a line which bounds the space of weights down. Combining each of the halfspaces together gets a piecewise linear graph. Taking say the centroid of the remaining space will give you the desired result.