Transitivity and Simple Scalability
Last edited: September 9, 2026Simple Scability: \(u : X \to \mathbb{R}\) and \(F\) strictly increasing in the first argument, strictly decreasing in the second argument, with \(P\qty(j \succ j’) = F\qty(u_{j}, u_{j’})\)
Independence: \(P \qty(j \succ j’’) \geq P\qty(j’ \succ j’’) \Leftrightarrow P\qty(j \succ j’’’) \geq P\qty(j’ \succ j’’’)\)
Stochastic transitivity: \(P\qty( j \succ j’) \geq \frac{1}{2}\), \(P\qty(j’ \succ j’’) \geq \frac{1}{2}\), then \(P\qty(j \succ j’’) \geq \max \qty[P\qty(j \succ j’), P\qty(j’ \succ j’’)]\)
Agents Index
Last edited: September 9, 2026Lectures
cool thing
https://trafilatura.readthedocs.io/en/latest/
project thoughts
mixture of depth + speculative decoding
Bradley-Terry Preference Model
Last edited: September 9, 2026Suppose you are only observing binary cases, how do we induce Choice Data that are ordered preferences?
The following three statements are equivalent.
Brady Terry For \(u: X \to \qty(0, \infty)\) with
\begin{equation} P\qty( j \succ j’) = \sigma \qty(u_{j} - u_{j’}) \end{equation}
Triangle property for all \(a,b,c\)
\begin{equation} P\qty(j \succ j’) P\qty(j’ \succ j’’) P\qty(j ’’ \succ j) = P\qty(j \succ j’’) P \qty(j’’ \succ j’) P\qty(j’ \succ j) \end{equation}