6 10 A Bidding Model Let U,, U,, be independent random variables, each uniformly distributed over the interval (0, 1 These random variables represent successive bids on an asset that you are trying to sell, and that you must sell by time t 1, when the asset becomes worthless As a strategy, you adopt a secret number 0, and you will accept the first offer that is greater than 0 For example, you accept the second offer if U, 0 Suppose that the offers arrive according to a unit rate Poisson process (A 1) (a) What is the probability that you sell the asset by time t 1 (b) What is the value for 0 that maximizes your expected return (You get nothing if you don 't sell the asset by time t 1 ) (c) To improve your return, you adopt a new strategy, which is to accept an offer at time t if it exceeds 0(t) (1 t) (3 t) What are your new chances of selling the asset, and what is your new expected return

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Question: 6.10. A Bidding Model Let U,, U,, . . . be independent random variables, each uniformly distributed over the interval (0, 1]. These random variables

6.10. A Bidding Model Let U,, U,, . . . be independent random variables, each uniformly distributed over the interval (0, 1]. These random variables represent successive bids on an asset that you are trying to sell, and that you must sell by time t = 1, when the asset becomes worthless.

As a strategy, you adopt a secret number 0, and you will accept the first offer that is greater than 0. For example, you accept the second offer if U, <_ 0 while U, > 0. Suppose that the offers arrive according to a unit rate Poisson process (A = 1).

(a) What is the probability that you sell the asset by time t = 1?

(b) What is the value for 0 that maximizes your expected return? (You get nothing if you don't sell the asset by time t = 1.)

(c) To improve your return, you adopt a new strategy, which is to accept an offer at time t if it exceeds 0(t) = (1 - t)/(3 - t). What are your new chances of selling the asset, and what is your new expected return?

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