Weakening the Sure-Thing Principle: Decomposable Choice under Uncertainty

Savage motivated his Sure-Thing Principle b y arguing that, whenever an act would be preferred if an event obtains and preferred if an event did not obtain, then it should be preferred overall. The idea that it should be possible to decompose and recompose decision problems in this way has normative appeal. We show, however, does it does not require the full separability across events implicit in Savage's axiom. We formulate a weaker axiom that suffices for decomposability, and show that this implies an implicit additive representation. Our decomposability property makes local necessary conditions for for optimality, globally sufficient. Thus, it is useful in computing optimal acts. None of these results rely on probabilistic sophistication; indeed, our axiom is consistent with the Ellsberg paradox. If we assume probabilistic sophistication, however, then the axiom holds if and only if the agent's induced preferences over lotteries satisfy betweenness. Thus, the weak sure-thing principle forms part of an axiomatization of subjective betweenness theory, just as its stronger ancestor did for subjective expected utility