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is Pr [your draw x is k -th highest] = μ n 1 k 1 ·{ 1 F ( x ) } k 1 1 F ( x ) } n k See footnote 2 . 2. Consider a second price auction with n bidders. (a) Compute the seller’s expected revenue (in symmetric Bayesian Nash equilibrium) when each bidder’s value is independently and uniformly distributed on [0 , 2] . Note: CDF of Order Statistic There are n bidders. Assume valuation v is drawn independently from the Cumulative Distribution Function (henceforce CDF) F ( x ). (1) G 1 ,n ( x ): CDF of highest valuation among n draws G 1 ,n ( v )= { F ( x ) } n = F ( x ) | {z } prob that a draw is below x n The interpretation is (very) easy. To make the highest draw be less than or equal to x ,a l l n inde- pendent draws have to be smaller than x . (2) G 2 ,n ( x ): CDF of highest valuation among n draws G 2 ,n ( x { F ( x ) } n + n (1 F ( x )) { F ( x ) } n 1 = { F ( x ) } n | {z } (i) + n (1 F ( x )) { F ( x ) } n 1 | {z } (ii) = { F ( x ) } n + n |{z} n possibilities , · (1 F ( x )) | {z } one draw is above x, F ( x ) } n 1 | {z } ( n 1) draws are below x The interpretation of this CDF is little bit tricky.
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