(the Exchange Paradox) You're playing the following monetary game against an opponent, with a referee also taking part; imagine playing this game with two of your friends, which will provide context for the amounts of money that would reasonably be involved in playing it. The referee has two envelopes (numbered 1 and 2 for the sake of this problem, but when the game is played the envelopes have no markings on them), and (without you or your opponent seeing what she does) she (the referee) puts 3m in envelope 1 and $2 at in envelope 2 for some m > 0 (let's treat m as continuous in this problem, even though in practice it would be rounded, let's say to the nearest dollar). You and your opponent each get one of the envelopes at random. You open your envelope secretly and nd $3: (your opponent also looks secretly in his envelope), and the referee then asks you if you want to trade envelopes with your opponent. You reason that if you trade, you will get either $ or $2 3:, each with probability %. This makes the expected value of the amount of money you'll get if you trade equal to (g) ($33) + (g) ($233) = 3'75\"\(b) [40 total points for this part of this problem] (Bayesian decision theory) (i) Suppose that for you in this game, money and utility coincide (or at least suppose that utility is linear in money for you with a positive slope). Use Bayesian decision theory, through the principle of maximizing expected utility, to show that you should offer to trade envelopes if and only if pg) m1). Make a sketch of what condition (3) implies for a decreasing p. [10 points] (iii) One possible example of a continuous decreasing family of priors on M is the exponential distribution indexed by the parameter /\\ > 0, which represents the reciprocal of the mean of the distribution. Identify the set of conditions in this family of priors, as a function of a: and A, under which it's optimal for you to trade. Does the inequality you obtain in this way make good intuitive sense (in terms of both a: and A)? Explain briey. [20 points] (c) [10 points] Looking carefully at the correct argument in paragraph 2 of this problem, identify the precise point at which the argument in the rst paragraph breaks down, and specify what someone who believes the argument in paragraph 1 is implicitly assuming about the prior distribution p(m)