Right; or if you don't decide exactly, at least you have to do (believe or not believe) one or the other. I would say that the model breaks down. Mathematics (or at least the particular mathematical model being used) is not capable of describing this situation, but that doesn't make the situation itself meaningless. (That would be a version of the map/territory fallacy.) [...] Here I disagree with you. I would say that you have not given enough information. It is as if you gave the same problem statement but with the word ‘infinite’ removed (so that we only know whether the utilities are positive or negative). It may seem as if you have given all of the information: the probabilities and the utilities. But the mathematics which we use to calculate everything else out of those values breaks down, so in fact you have not given all of the information. One important missing piece of information is the ratio of the first positive utility to the second. That and two other independent ratios would be enough information, if they're all finite. (If not, then we might need more information.) And don't tell me that these ratios are undefined; the mathematical model that calculates the ratios from the information given breaks down, that's all. In fact, there is an atlernative mathematical model of decision which deals only in ratios between utilities; if you'd followed that model from the beginning, then you would never have tried to state the actual utilities themselves at all. (For mathematicians: instead of trying to plot these 4 utilities in a 4-dimensional affine space, plot them in a 3-dimensional projective space.) [...] Right; the proper conclusion of the argument is not to believe, but to try to believe. And if you buy the argument, then you should try very hard!