Independence and indifferent points imply continuity

Abstract

I propose the new axiom of Indifferent Points (IP) that can replace continuity axioms in classical expected utility representations under the Independence Axiom over a finite set of prices. IP asserts the existence of a set of indifferent points that span a hyperplane. In the case of three prices, often used for illustrations, the decision maker only needs to show indifference between two distinct lotteries. IP does not imply any of the established continuity axioms and is even strictly weaker than mixture continuity and solvability.