Suggest edit — Preference Elicitation

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---
title: "Preference Elicitation"
type: concept
related: [Preference Elicitation]
source: https://www.jemoka.com/posts/kbhpreference_elicitation/
confidence: high
status: active
---

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 &ldquo;do you like system \(a\) or system \(b\)&rdquo;.
first, we collect a series of design variables \((a_1, a_2, a_3 &hellip;)\) and \((b_1, b_2, b_3&hellip;)\) and we ask &ldquo;which one do you like better&rdquo; say our user WLOG chose \(b\) over \(a\) so we want to design a \(w\) such that \(w^{\top} a &lt; w^{\top} b\) meaning, we solve for a \(w\) such that&hellip; \begin{align} \min_{w}&amp;\ \sum_{i=1}^{n} (a_{i}-b_{i})w^{\top} \\ \text{such that}&amp;\ \bold{1}^{\top} w = 1 \\ &amp;\ 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 \(w^{T} a &gt; w^{T} b\).
Let&rsquo;s say we had two bags, each with \(a = \qty(1,3,6)\) and \(b = \qty(7,1,2)\).
This means:
\begin{equation} 1 w_1 + 3 w_2 + 6 w_3 &gt; 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 &gt; 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 &lt; 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.