A functional equation arising in multi-agent statistical decision theory

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.