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
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
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.