Balancing Acts

When exactly does a preference ordering on acts have an additive state-dependent utility representation per indifference curve? Following the analogy of a sliding balance, a perfect reflection of the well-known betweenness models for lotteries, we find three answers for monetary acts on a finite state space. In the first two, we exploit the axiom of weak decomposability, identified by Grant, Kajii, and Polak (2000) as the key axiom to extend betweenness to acts, but leaving the continuity and monotonicity of state utility in setpoints an open issue. Our first theorem confirms the necessity of continuity, while the second shows that monotonicty in setpoints requires the additional axiom of ratio monotonicity . It formalises the idea that sliding a setpoint towards a gain decreases its momentum compared to losses. This observation, in turn, suggests to extend the primitive of a preference ordering with a benchmark for discerning gains from losses within an act. Thus we arrive at the Additive Sliding Balance Utility model for orderings anchored to such a benchmark. In the third theorem, we characterise this class with an even more straightforward betweenness axiom: the balance point of an act should be in between those of its parts.