Conditions under which thurstone case III representations for binary choice probabilities are also Fechnerian

Abstract

By a Thurstone Case III representation for binary symmetric choice probabilities P x, y we mean that there exist functions F, μ, σ > 0 such that P x,y = F[ (μ(x) − μ(y)) (σ 2(x) + σ 2(y)) 1 2 ] . We show that the constraint σ = constant, or μ = ασ + β, α ≠ 0, is both necessary and sufficient for a Thurstone Case III representation to be Fechnerian, i.e., to be reexpressable as as P x, y = G( u( x) − u( y)) for some suitably chosen functions G, u.