A functional equation arising in multi-agent statistical decision theory

Abstract

Using recent results of Járai we show that the measurable solutions of the functional equationf(x ₁ y ₁,...,x _n y _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(y ₁(1−x ₁),...,y _n(1 −x _n )), wheref: (0, 1)ⁿ → (0, ∞) and 0<x _i ,y _i <1,i=1,...,n, are of the form

$$f(x_1 ,...,x_n ) = c \exp \left( {\sum\limits_{i = 1}^n {a_i (x_1 - x_1^2 ))} \prod\limits_{i = 1}^n {x_i^{b_1 } ,} } \right.$$

wherec>0,a ₁,...,a _n andb ₁,..., _b are arbitrary real constants. This result enables one to characterize certain independence-preserving methods of aggregating probability distributions over four alternatives.