Simple 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 \(P\qty(j \succ j’’) \geq \max \qty[P\qty(j \succ j’), P\qty(j’ \succ j’’)]\)