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lecture6_revenue

lecture6_revenue - Expected Revenue Here we calculate the...

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Expected Revenue Here we calculate the expected revenue under the efficient equilibrium bidding strategies for the first- and second-price auction formats. In a first-price auction with F ( · ) uniform on [0,100], the symmetric equilibrium bidding strategy has each bidder bid ( n - 1) / n times their value. Hence, expected revenue is simply ( n - 1) / n times the expected value of the highest value. I.e., Expected revenue is n - 1 n E v (1) ] = n - 1 n Z 100 0 vf (1) ( v ) dv = n - 1 n Z 100 0 v nv n - 1 100 n dv = ( n - 1) Z 100 0 v n 100 n dv = ( n - 1) 100 n v n +1 n + 1 100 0 = n - 1 * 100 . Rod Garratt ECON 177: Auction Theory With Experiments
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In a second-price auction it is a weakly dominant strategy to bid your value, and hence expected revenue is simply the expected value of the second-highest value. In case where F ( · ) is uniform on [0,100], E v (2) ] = Z 100 0 vf (2) ( v ) dv = Z 100 0 vn ( n - 1)( v n - 2 100 n - 1 - v n - 1 100 n ) dv = n ( n - 1) Z 100 0 ( v n - 1 100 n - 1 - v n 100 n ) dv = n ( n - 1) ˆ v n n 100 n - 1 100 0 - v n +1 ( n + 1)100 n 100 0 !
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