Consider the independent private values model with N possibly asymmetric bidders. Suppose we restrict attention to efficient individually rational, incentive-compatible direct selling mechanisms; i.e., those that always assign the object to the bidder who values it most.

(a) What are the probability assignment functions?

(b) What then are the cost functions?

(c) What cost functions among these maximise the seller’s revenue?

(d) Conclude that among efficient individually rational, incentive-compatible direct selling mechanisms, a second-price auction maximises the seller’s expected revenue. (What about the other three standard auction forms?)

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