Two envelopes, each containing a check, areplaced in front

Chapter 7, Problem 7.78

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Two envelopes, each containing a check, areplaced in front of you. You are to choose one ofthe envelopes, open it, and see the amount of thecheck. At this point, either you can accept thatamount or you can exchange it for the check inthe unopened envelope. What should you do? Is itpossible to devise a strategy that does better thanjust accepting the first envelope?Let A and B,A < B, denote the (unknown)amounts of the checks, and note that the strategythat randomly selects an envelope and alwaysaccepts its check has an expected return of(A + B)/2. Consider the following strategy: LetF() be any strictly increasing (that is, continuous)distribution function. Choose an enveloperandomly and open it. If the discovered check hasthe value x, then accept it with probability F(x) andexchange it with probability 1 F(x).(a) Show that if you employ the latter strategy,then your expected return is greater than(A + B)/2.Hint: Condition on whether the first envelopehas the value A or B.Now consider the strategy that fixes a value xand then accepts the first check if its value isgreater than x and exchanges it otherwise.(b) Show that, for any x, the expected returnunder the x-strategy is always at least(A + B)/2 and that it is strictly larger than(A + B)/2 if x lies between A and B.(c) Let X be a continuous random variable on thewhole line, and consider the following strategy:Generate the value of X, and if X = x,then employ the x-strategy of part (b). Showthat the expected return under this strategy isgreater than (A + B)/2.