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The price bidder 1 pays is the random variable β III (Y2 ), where, Y2 is the secondhighest of the N − 1 other values. The density of Y2 , conditional on the event that Y1 < x, can be written as (N−1) f2 (y | Y1 < x) = 1 (N−1) F1 (x) (N−2) (N − 1) (F (x) − F (y)) f1 (y) where (N − 1) (F (x) − F (y)) is the probability that Y1 exceeds Y2 = y but (N−2) (y) is the density of the highest of N − 2 values. 4), we obtain that Z x Z (N−2) β III (y) (N − 1) (F (x) − F (y)) f1 (y) dy = 0 x yg(y) dy 0 and differentiating with respect to x, this implies that Z x (N−2) (N − 1) f(x) β III (y)f1 (y) dy = xg(x) 0 = x × (N − 1) f (x)F (x)N−2 since G (x) ≡ F (x)N−1 .

The type (x00 , 1) is, as it were, the richest member of the family with types (x, w) such that min{x, w} = x00 . 1 depicts the set of types who bid the same in a second-price auction as does type (x, w) . This consists of all types on the thin-lined right angle “Leontief iso-bid” curve whose corner lies on the diagonal. As before, let m II (x, w) denote the expected payment of a bidder of type (x, w) in a second-price auction. 9) to be the set of types who bid less than type (x00 , 1) in a second-price auction.

We have thus argued that if all other bidders are following the strategy β, a bidder with a value of x cannot benefit by bidding anything other than β(x); and this implies that β is a symmetric equilibrium strategy. 3 First-Price Auctions 1 2 .................................................. © ©© © β ©© © © © © © ©© 0 .................................................. 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.