Call a direct selling mechanism pi(·), ci(·), i = 1,…, N deterministic if the pi take on only the values 0 or 1.

(a) Assuming independent private values, show that for every incentive-compatible deterministic direct selling mechanism whose probability assignment functions, pi(vi, v−i), are nondecreasing in vi for every v−i, there is another incentive-compatible direct selling mechanism with the same probability assignment functions (and, hence, deterministic as well) whose cost functions have the property that a bidder pays only when he receives the object and when he does win, the amount that he pays is independent of his reported value. Moreover, show that the new mechanism can be chosen so that the seller’s expected revenue is the same as that in the old.

(b) How does this result apply to a first-price auction with symmetric bidders, wherein a bidder’s payment depends on his bid?

(c) How does this result apply to an all-pay, first-price auction with symmetric bidders wherein bidders pay whether or not they win the auction?

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