A functional equation arising in multi-agent statistical decision theory
The authors obtain the measurable solution of the functional equation \[ (1)\quad f(x_ 1y_ 1,...,x_ ny_ n)f((1-x_ 1)(1-y_ 1),...,(1- x_ n)(1-y_ n))= \] \[ f(x_ 1(1-y_ 1),...,x_ n(1-y_ n))f((1- x_ 1)y_ 1,...,(1-x_ n)y_ n)\quad as \] \[ (2)\quad f(x_ 1,...,x_ n)=c \exp \sum^{n}_{i=1}a_ i(x_ i-x^ 2_ i)\cdot \prod_{i}x_ i^{b_ i}, \] using results of Jarai. The same result has been obtained by \textit{Abou-Zaid} using induction [M. Phil. thesis, University of Waterloo (1984)]. The reviewer and \textit{C. T. Ng} [J. Appl. Probab. 17, 271-277 (1980; Zbl 0428.94006)] had obtained the measurable solution of \(f_ 1(pq)+f_ 2(p(1-q))+f_ 3((1-p)q)+f_ 4((1-p)(1- q))=f_ 5(p)+f_ 6(q),\) a more general equation than (1) when \(n=2\).
scientific article; zbMATH DE number 4004821