Dominance-Based Decision Theory

Decision theory has been plagued with a variety of problems almost from the start. Some, like the Ellsberg paradox [Ellsberg, 1961] merely seem to point to the problems actual people have in calculating probabilities and utilities. Others, like the Newcomb paradox and its relatives, have motivated a re-foundation of decision theory on causal grounds rather than evidential grounds [Joyce, 1999]. Recent work of Harris Nover, Alan Hájek, and Mark Colyvan has revealed more problems with the way decision theory deals with the infinite, brought out in [Nover and Hájek, 2004] with the Pasadena Game a variant of the St. Petersburg game that puzzled the 18th century founders of decision theory, which has no expected value, rather than the infinite one of St. Petersburg. I believe that a different formalism may deal better with these infinities, and may suggest interesting new ways to think about the problems of causality and undefined probability that affect the others. The goal of decision theory has traditionally been to explicate the way that (perhaps idealized) rational agents choose (or ought to choose) a single action to perform from among the actions available to them at any moment. The traditional theory has done this by associating each action with a real number, known as its expected utility, and saying that a rational agent will perform the action whose expected utility is highest (or one among such, if there is more than one). This theory has been supported by representation theorems [Joyce, 1999, p. 96, p. 127] showing that any agent whose preferences among actions satisfy certain seemingly plausible assumptions can always be interpreted as having utility and credence functions according to which she is maximizing expected utility. It has also been bolstered by theorems of probability, like the Strong Law of Large Numbers, and the Central Limit Theorem, together showing that an agent who behaves otherwise will almost certainly have lower utility overall than an agent who maximizes expected utility. However, these results seem almost too good to be true. If all we wanted was a way to pick one action from among many at any moment, then assigning real numbers (as opposed to some other ordered objects) seems like overkill. And this is right the great power of these results demands very strong conditions. First of all, the agent must at least implicitly have a well-defined probability for each