The Envelope Paradox


The confusion arises because, in the real world, there must exist a finite upper boumd on how much money an envelope can contain, a fact which our player tends to ignore. Thus the double or something philosophy, valid for most situations that could occur, becomes invalid should the player open an envelope that contains an amount whose double exceeds that upper bound. And, when considering the average winnings that a player can expect in this game, a huge price is paid when he deliberately switches after having selected the richest envelope.

Player strategies can be quantified by considering the average winnings that result from their playing this game many, many times. To keep things simple, consider the analagous game in which an urn contains N pairs of envelopes, each pair stapled together, and that each envelope contains an amount of dollars that is a finite power of 2. Thus it contains the pairs (21, 22), (22, 23), (23, 24),....(2N, 2N+1). The player now selects a random envelope pair from out of the urn, and then separates the pair, selecting one envelope as his initial choice. The player opens this envelope, notes its content, and, consistent with the double or something philosophy, pockets, instead, the contents of the other envelope. It can be shown that his average winnings are identical to that had he not switched, and is given by

Wav = 3(2N - 1)/N

So it simply doesn't matter which envelope was selected, or whether the player switched or not, assuming average return as the criterion.

Now consider the advantage held by a player who is privy to to the value of N, and thus to the contents of the richest envelope, namely 2N+1. This player, upon observing the contents of his initial envelope choice, adopts the optimum policy of always switching to the other envelope unless his envelope holds 2N+1 dollars. Now it turns out that his average winnings are

Wav = 3.5(2N - 0.857)/N

If N is reasonably large, this represents a 17% improvement over the previous cases where the value of N was unknown to the player.

So what should someone do if fortunate enough to be offered this game, but without any knowledge of N? One is faced with the trade-off between the switching policy just marginally better almost all of the time, and the secure holding policy which guards against disaster in a single unlikely event. The above examples make it clear that a reasonable recipe demands some commitment, at least some lower bound, to the value of N, which can, at best, be a guess. Whether to switch or hold would then follow.