A choice probability characterization of generalized extreme value models (Q800204)

A choice probability structure over a (finite) set of alternatives \(I_ n=\{1,...,i,...,n\}\) with value profiles in \({\mathbb{R}}^ n\) is a map \(P:I_ n\times {\mathbb{R}}^ n\to {\mathbb{R}}_+\) such that \(\sum_{i}P_ i(v)=1\) for all \(v\in {\mathbb{R}}^ n\). One says that \(P_ i(v)=Prob\{v_ i+x_ i>_{j\neq i}(v_ j+x_ j)\}\) is the probability that \(i\in I_ n\) is chosen when the decision maker behaves according to a random utility \(u_ i=v_ i+x_ i\) such that \(v_ i\) is observable and \(x_ i\) is unobservable and associated to a random variable \(x_ i.\) Let \(F_ n\) be the class of cumultative distribution functions F on \({\mathbb{R}}^ n\) generated by continuous probability densities. A choice probability structure P is representable by an additive random utility (ARU) iff there exists a random n-vector \((x_ 1,..,x_ n)\) with cumulative \(F\in F_ n\) such that for all \(v\in {\mathbb{R}}^ n\) and all \(i\in I, P_ i(v)=\int^{+\infty}_{-\infty}F^ i(v_ i+x-v_ 1,..,x,...,v_ i+x-v_ n)dx.\)The author considers of these: a choice probability structure which is ARU is called a generalized extreme value model (GEV) iff the cumulative distribution F is such that, for all \(x\in {\mathbb{R}}^ n\), \(F(x)=\exp [-G(e^{-x})]\), where \(G:{\mathbb{R}}^ n_+\to {\mathbb{R}}_+\) also called a generator is a linearly homogeneous function. The author proves that a choice probability structure P is \(G\in V\) iff (1) \(P_ i\neq 0\), \(i\in I_ n\); (2) \(P^ j_ i=P^ i_ j\), \(i\in I_ n\), \(j\in I_ n\), with \(P^ j_ i=\frac{\partial}{\partial v_ j}P_ i\) and \(P^ i_ j=\frac{\partial}{\partial v_ i}P_ j\); and (3) \(\bar P_ S\geq 0\), \(S\in {\mathfrak S}\), where \({\mathfrak S}\) denotes the class of ordered sequences from \(I_ n\), and \(\bar P_ S:{\mathbb{R}}^ n\to {\mathbb{R}}\), \(S\in {\mathfrak S}\) is defind recursively by letting for all \(S=\{s_ 1,s_ 2,...,s_ k\}\) and all \(v\in {\mathbb{R}}^ n:\) \[ P_{\{s_ 1...s_ k\}}(v)= \begin{cases} P_{s_ 1}(v)&\;if\;k=1,\\ \bar P^{s_ k}_{\{s_ 1,...,s_ k\}}(v)-P_{s_ k}(v)\bar P_{\{s_ 1,...,s_ k\}}(v)&\;if\;k>1. \end{cases} \] Some tests for GEV representability, properties of the ''test'' functions \(\bar P_ S\) and some behavioral interpretations of the model are also studied.