Taking the Two Envelope Paradox to the Limit
Southwest Philosophy Review, Vol. 25, No. 2, 2009
26 Pages Posted: 7 May 2010
Date Written: August 17, 2009
The original version of the two envelope paradox is not all that paradoxical. The fact that (a) one of two sealed envelopes contains twice as much money as the other does not imply that (b) the other envelope is equally likely to contain twice or half as much money as your envelope. And (b) is what is behind the familiar reasoning that the other envelope has greater expected utility than your envelope. However, there are strengthened versions of the two envelope paradox where the familiar reasoning seems to hold, but it still does not make any sense to prefer the other envelope. David Chalmers (1994) and John Norton (1998) have attempted to resolve the paradox by pointing out that the familiar reasoning is flawed in such cases because the expected utility of each envelope is infinite. However, simply noting this fact provides only a partial analysis of the paradox. We should not simply throw up our hands when we are confronted with infinite expectations. In particular, there are two envelope scenarios where the expected utility of each envelope is infinite, but where it does seem rational to prefer one envelope over the other. Thus, we need a way to reason about such scenarios that does not lead us back into a paradox. Peter Vallentyne (2000) has proposed that we reason about infinite lotteries by looking at the behavior of arbitrarily large finite lotteries in the limit. In this paper, I show how the same sort of technique can be used to complete the analysis of the two envelope paradox.
Keywords: Two Envelope, Paradox, Infinity, Expected Utility, Limit
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