The Sure Thing

If we prefer action a to b both under an event and under its complement, then we should just prefer a to b. This is Savage's sure-thing principle. In spite of its intuitive- and simple-looking nature, for which it gets almost immediate acceptance, the sure thing is not a logical principle. So where does it get its support from? In fact, the sure thing may actually fail. This is related to a variety of deep and foundational concepts in causality, decision theory, and probability, as well as to Simpsons' paradox and Blyth's game. In this paper we try to systematically clarify such a network of relations. Then we propose a general desirability theory for nonlinear utility scales. We use that to show that the sure thing is primitive to many of the previous concepts: In non-causal settings, the sure thing follows from considerations of temporal coherence and coincides with conglomerability; it can be understood as a rationality axiom to enable well-behaved conditioning in logic. In causal settings, it can be derived using only coherence and a causal independence condition.