Extensions To The Problem
Since the two envelopes problem became popular, many authors have studied the problem in depth in the situation in which the player has a prior probability distribution of the values in the two envelopes, and does look in Envelope A. One of the most recent such publications is by McDonnell and Douglas (2009), who also consider some further generalizations.
If a priori we know that the amount in the smaller envelope is a whole number of some currency units, then the problem is determined, as far as probability theory is concerned, by the probability mass function describing our prior beliefs that the smaller amount is any number x = 1,2, ... ; the summation over all values of x being equal to 1. It follows that given the amount a in Envelope A, the amount in Envelope B is certainly 2a if a is an odd number. However, if a is even, then the amount in Envelope B is 2a with probability, and a/2 with probability . If one would like to switch envelopes if the expectation value of what is in the other is larger than what we have in ours, then a simple calculation shows that one should switch if, keep to Envelope A if .
If on the other hand the smaller amount of money can vary continuously, and we represent our prior beliefs about it with a probability density, thus a function which integrates to one when we integrate over x running from zero to infinity, then given the amount a in Envelope A, the other envelope contains 2a with probability, and a/2 with probability . If again we decide to switch or not according to the expectation value of what's in the other envelope, the criterion for switching now becomes .
The difference between the results for discrete and continuous variables may surprise many readers. Speaking intuitively, this is explained as follows. Let h be a small quantity and imagine that the amount of money we see when we look in Envelope A is rounded off in such a way that differences smaller than h are not noticeable, even though actually it varies continuously. The probability that the smaller amount of money is in an interval around a of length h, and Envelope A contains the smaller amount is approximately . The probability that the larger amount of money is in an interval around a of length h corresponds to the smaller amount being in an interval of length h/2 around a/2. Hence the probability that the larger amount of money is in a small interval around a of length h and Envelope A contains the larger amount is approximately . Thus, given Envelope A contains an amount about equal to a, the probability it is the smaller of the two is roughly .
If the player only wants to end up with the larger amount of money, and does not care about expected amounts, then in the discrete case he should switch if a is an odd number, or if a is even and . In the continuous case he should switch if .
Some authors prefer to think of probability in a frequentist sense. If the player knows the probability distribution used by the organizer to determine the smaller of the two values, then the analysis would proceed just as in the case when p or f represents subjective prior beliefs. However, what if we take a frequentist point of view, but the player does not know what probability distribution is used by the organiser to fix the amounts of money in any one instance? Thinking of the arranger of the game and the player as two parties in a two person game, puts the problem into the range of game theory. The arranger's strategy consists of a choice of a probability distribution of x, the smaller of the two amounts. Allowing the player also to use randomness in making his decision, his strategy is determined by his choosing a probability of switching for each possible amount of money a he might see in Envelope A. In this section we so far only discussed fixed strategies, that is strategies for which q only takes the values 0 and 1, and we saw that the player is fine with a fixed strategy, if he knows the strategy of the organizer. In the next section we will see that randomized strategies can be useful when the organizer's strategy is not known.
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