Dynamic Pricing and Auction - Department of Industrial Engineering ...

Revenue in second priceauctionn bidders independently draw their valuations from distribution function F(.) and density function f (.)Bidders' values: { v1, v2, ... , v n} (under IID assumption,i.e., i each is a independent and identicallydistributed random variable)Order statistics: { v(1) , v(2) , ... , v( n )} (arranged in ascending order)thn!Density of k lowest: f( v( k ))=f v F v F v( k −1)!( n−k)!k−1n−k().[ ()] .[1 − ()]Uniform distribution case, i.e., for U[0,1]th n!k−1Density of k lowest: f( v( k )) = . v .[1 −v]( k−1)!( n−k)!(This is a β distribution with paremeters k and ( n- k+ 1)thkMean value of k order statistic: E( v( k ))=( n + 1)nd( n −1)Mean value of 2 order statistic: E( v(2)) =( n +1)(expected revenue for the auctioneer)n−k