Continuous SSB representation of preferences

We propose a topological variant of skew-symmetric bilinear (SSB) representation of preferences. First, semi-Fishburn relations are defined by assuming convexity and coherence, a newly considered topological property. We show that lower and upper semi-Fishburn relations admit the existence of a minimal element and a maximal element, respectively. Then axiom of “balance” is stated and we prove that a binary relation has a continuous SSB representation if and only if it is a balanced (lower and upper semi-)Fishburn relation. The relationship between the above definitions and the original axioms of (algebraic) SSB representation is fully discussed. Finally, by applying this theory to probability measures, we show the existence of a maximal preferred measure for an infinite set of pure outcomes, thus generalizing all available existence theorems of (algebraic) SSB representation. Note that by using this framework to, e.g., finitely additive measures, one may develop a non-probabilistic variant of SSB representation as well.