A quantitative theory of preferences: Some results on transition functions

We investigate a general theory of combining individual preferences into collective choice. The preferences are treated quantitatively, by means of preference functions ϱ(a,b), where 0≦ϱ(a,b)≦∞ expresses the degree of preference of a to b. A transition function is a function Ω(x,y) which computes ϱ(a,c) from ϱ(a,b) and ϱ(b,c), namely ϱ(a,c)=Ω(ϱ(a,b),ϱ(b,c)). We prove that given certain (reasonable) conditions on how individual preferences are aggregated, there is only one transition function that satisfies these conditions, namely the function Ω(x,y)=x·y (“multiplication of odds”). We also formulate a property of transition functions called invariance, and prove that there is no invariant transition function; this “impossibility theorem” shows limitations of the quantitative method.