Balancing acts

Grant Kajii and Polak (2000, Journal of Economic Theory) have identified weak decomposability as the key axiom to characterise the betweenness class for acts without the assumption of probabilistic sophistication, leaving, however, a small gap between necessary and sufficient conditions. We show how the balancing perspective naturally leads to a full characterisation, for monetary acts on a finite state space.

In our first theorem, we avoid some technical issues by working with the implicit notion of soundness, reflecting the proper working of a sliding balance. We sharpen the result using ratio monotonicity as the `missing' axiom, already known for lotteries as a consequence of first order stochastic dominance, but not necessarily satisfied under weak decomposability for acts.
It guarantees soundness, and in fact induces a semi-strict form of monotonicity of utility in balance setpoints.

Finally, we show how the balancing perspective offers an intuitive way to incorporate partial acts in the framework without reference to conditioning, replacing the notion of a null state by a nil outcome. This leads to a very straightforward betweenness axiom: the balance point of an act should be in between those of its parts. Our final theorem characterises the betweenness class for acts using this axiom of balance point betweenness, combined with a perhaps unexpected requirement of embeddability.