Independence and indifferent points imply continuity

TL;DR

This paper introduces the Indifferent Points (IP) axiom as a weaker alternative to continuity axioms in expected utility theory, requiring only indifference between two lotteries for three prices.

Contribution

The paper proposes the IP axiom, replacing traditional continuity assumptions with a weaker condition based on indifferent points spanning a hyperplane.

Findings

IP does not imply traditional continuity axioms.

IP is strictly weaker than mixture continuity and solvability.

Indifference between two lotteries suffices for three prices.

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.