Making Do Without Expectations

The Pasadena game invented by Nover and Hájek (2004) raises a number of challenges for decision theory. The basic problem is how the game should be evaluated: it has no expectation and hence no well-defined value. Easwaran (2008) has shown that the Pasadena game does have a weak expectation, raising the possibility that we can eliminate the value gap by requiring agents to value gambles at their weak expectations (where they exist). In this paper, I first prove a negative result: there are gambles like the Pasadena game that do not even have a weak expectation. Hence, problematic value gaps remain even if decision theory is extended to take in weak expectations. There is a further challenge: the existence of a 'Value gap' in the Pasadena game seems to make decision theory inapplicable in a number of cases where the right choice is obvious. The positive contribution of the paper is a theory of 'relative utilities', an extension of decision theory that lets us make comparative judgements even among gambles that have no well-defined value.