By Vijay Krishna
This can be a excellent ebook at the subject, yet we want a "Dummies" model. Auctions have gotten part of daily life -- been on EBay in recent years? performed the inventory industry? I want Dr. Krishna had a model of this fabric with out the rigorous math that i'll in an undergrad path.
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Extra info for Auction Theory
6) shows that the ex ante expected payment of a bidder is now Z ω ¤ £ mA (x, r) f (x) dx E mA (X, r) = r Z ω y (1 − F (y)) g(y) dy = r (1 − F (r)) G(r) + r What is the optimal, or revenue maximizing, reserve price from the perspective of the seller? Suppose that the seller attaches a value x0 ∈ [0, ω). This means that if the object is left unsold, the seller would derive a value x0 from its use. Clearly, the seller would not set a reserve price r that is lower than x0 . Then the overall expected payoﬀ of the seller from setting a reserve price r ≥ x0 is £ ¤ Π0 = N × E mA (X, r) + F (r)N x0 Diﬀerentiating this with respect to r, we obtain dΠ0 = N [1 − F (r) − rf (r)] G(r) + N G(r)f(r)x0 dr Now recall that the hazard rate function associated with the distribution F is defined as λ (x) = f(x)/ (1 − F (x)) .
In the absence of a reserve price, the object will always be sold to the highest bidder and in the symmetric model studied here, that is also the bidder with the highest value. Thus, both the first- and second-price auctions allocate eﬃciently in the sense that the object ends up in the hands of the person who values it the most. If the seller sets a reserve price r > 0, however, there is a positive probability that the object will remain in the hands of seller and this is ineﬃcient. This simple observation implies that there may be a trade-oﬀ between eﬃciency and revenue.
2. 2) in Appendix A again. This shows that the bid is, naturally, less than the value x. Since · ¸N−1 F (y) G(y) = G(x) F (x) the degree of “shading” (the amount by which the bid is less than the value) depends on the number of competing bidders and as N increases, approaches 0. Thus, for fixed F , as the number of bidders increases, the equilibrium bid β I (x) approaches x. It is instructive to derive the equilibrium strategies explicitly in a few examples. 1 Values are uniformly distributed on [0, 1].
Auction Theory by Vijay Krishna