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}

Specific Objectivity Define odds \(o\qty(j,j’) = \log \frac{P\qty(j > j’)}{P\qty(j’ > j)}\), we have:

\begin{equation} o\qty(j, j’’) - o\qty(j’,j’’) = o\qty(j,j’) \end{equation}