You're playing the following game against an opponent with a referee also taking part. 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 puts Sm in envelope 1 and $2 m in envelope 2 for some m>0 (let's treat m as contimuous in this problem, even though in practice it would have to be rounded to the nearest dollar or penny). You and your opponent each get one of the envelopes at random. You open your envelope secretly and find Sr (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 S or $2 x, each with probability j. This makes the expected value of the amount of money you'll get if you trade equal to () )+() ($2 x) = 5#, which is greater than the Sz you currently have, so you offer to trade. The paradox is that your opponent is capable of making exactly the same calculation. How can the trade be advantageous for both of you? The point of this problem is to demonstrate that the above reasoning is flawed from a Bayesian point of view; the conclusion that trading envelopes is always optimal is based on the assumption that there's no information obtained by observing the contents of the envelope you get, and this assumption can be seen to be false when you reason in a Bayesian way. At a moment in time before the game begins, let p(m) be your prior distribution for the amount of money M the referee will put in envelope 1, and let X be the amount of money you'll find in your envelope when you open it (when the game is actually played, the observed r, of course, will be data that can be used to decrease your uncertainty about M) A) Explain why the setup of this problem implies that P(X mM m), and use this to show that P(M = x/ X = x) = p(x)/ p(x) + px/2 and P(M = x/2/ X = x = p(x/2)/ p(x) + p(x/2)Demonstrate from this that the expected value of the amount Y of money in your opponent's envelope, given than you've found $x in the envelope you've opened, is p(x} p()p p{x) + P{3 | E(Y|X ) (2) B) Suppose that for you in this game, money and utility coincide (or at least suppose that utility is lnear 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 only if <2 p(x) (3) If you and two friends (one of whom would serve as the referee) real money in the envelopes, it would probably be the case that small amounts of money are more likely to be chosen by the referee than big amounts, which makes it interesting to explore condition (3) for prior distributions that are decreasing (that is, for which p(m2) < p(m) for m2 > m). Make a sketch of what condition (3) implies for a decreasing p. One possible example of a continuous decreasing family of priors on M is the erponential distribution indexed by the parameter A, which represents the reciprocal of the mean of the distribution. Identify the set of conditions in this family of priors, as a function of z and A, under which it's optimal for you to trade. Does the inequality you obtain in this way make good intuitive sense points were to actually play this game with (in terms of both r and A)? Explain briefly. C) Looking carefully at the correct argument in paragraph 2 of this problem, identify precisely the point at which the argument in the first paragraph breaks down, and specify what someone who believes the argument in paragraph is implicitly assuming about the prior distribution p(m).