| Sumario: | Expected utility theory often falls silent, even in cases where the correct rankings of options seems obvious. For instance, it fails to compare the Pasadena game to the Altadena game, despite the latter turning out better in every state. Decision theorists have attempted to fill these silences by proposing various extensions to expected utility theory. As I show in this paper, such extensions often fall silent too, even in cases where the correct ranking is intuitively obvious. But we can extend the theory further than has been done before—I offer a new extension, Invariant Value Theory, which deals neatly with those problem cases and also satisfies various desirable conditions. But other prima facie desirable conditions, including Independence, the theory violates. Is this a problem for the proposal? It may not be—in a new impossibility result, I show that no theory can satisfy Independence in full generality without violating several other conditions that together seem just as plausible.
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