 Joyce’s point seems to be that an agent who pays the fair price of the wager will know for sure that she will actually be worse off after she has paid the fee. However, this seems to presuppose that actual infinities do exist. If only potential infinities exist, then the player cannot “pay” an infinite fee for playing the game. If so, we could perhaps interpret Joyce as reminding us that no matter what finite amount the player actually wins, the expected utility will always be higher, meaning that it would have been rational to pay even more. Decisions theorists analyze a means-ends notion of rationality, according to which it is rational to do whatever is the best means to one’s end. The player thus knows that paying more than what one actually wins cannot be the best means to the end of maximizing utility. This observation enables us to strengthen the original “paradox” (in which no formal contradiction is derived) into a stronger version consisting of three incompatible claims: