Example. A Bradley--Terry comparison calculation [ftip-002T]
Example. A Bradley--Terry comparison calculation [ftip-002T]
A declared difference between two reward scores maps to one Bradley--Terry preference probability.
Under the Bradley--Terry link, \[ \Pr (y^+\succ y^-\mid x) =\frac {\exp r(y^+)}{\exp r(y^+)+\exp r(y^-)} =\sigma (2-0.5) =\sigma (1.5)\approx 0.8176. \] Reversing the pair gives probability \(1-0.8176=0.1824\).
The fixed scores instantiate Definition [ftip-002Q] and determine the displayed probability. Whether one scalar reward can represent every judge is a separate modeling question.