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slide32-nov18 - ECON 401 Auctions Revenue Siyang Xiong Rice...

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ECON 401: Auctions: Revenue Siyang Xiong Rice University November 18, 2011 Xiong (Rice University) ECON 401 November 18, 2011 1 / 17

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1st-price sealed-bid auction: revenue for simplicity, suppose there are two bidders, 1 and 2; v i . i D 1 or 2 / is i.i.d. distributed uniformly on [ 0 , 1 ] ; i.i.d. ± independently and identically distributed; uniform distribution on [ 0 , 1 ] ; cumulative distribution function (cdf): F i . k / D Pr [ v i : v i k ] D k ; probability distribution function (pdf): f i . k / D @ F i . k / @ k D 1; we know the bidding strategy is b . v i / D v i 2 . Xiong (Rice University) ECON 401 November 18, 2011 2 / 17
1st-price sealed-bid auction: revenue what is the distribution of the winning bid, max v 1 2 , v 2 2 ± ? max v 1 2 , v 2 2 ± is distributed on h 0 , 1 2 i according to the cdf G . k / : G . k / D Pr ² max n v 1 2 , v 2 2 o k ³ D Pr ² v 1 2 k , v 2 2 k ³ D Pr . v 1 2 k / ± Pr . v 2 2 k / D F 1 . 2 k / ± F 2 . 2 k / D 4 k 2 what is the expected revenue, i.e., the expected value of the winning bid, max v 1 2 , v 2 2 ± ? Xiong (Rice University) ECON 401 November 18, 2011 3 / 17

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1st-price sealed-bid auction: revenue the expected revenue is k D 1 2 Z k D 0 kdG . k / D k D 1 2 Z k D 0 kd 4 k 2 ± D k D 1 2 Z k D 0 8 k 2 dk D ² 8 3 k 3 ³ k D 1 2 k D 0 D 1 3 Xiong (Rice University) ECON 401 November 18, 2011 4 / 17
for simplicity, suppose there are two bidders, 1 and 2; v i . i D 1 or 2 / is i.i.d. distributed uniformly on [ 0 , 1 ] ; i.i.d. ± independently and identically distributed; uniform distribution on [ 0 , 1 ] ; cumulative distribution function (cdf): F i . k / D Pr [ v i : v i k ] D k ; probability distribution function (pdf): f i . k / D @ F i . k / @ k D 1; we know the bidding strategy is b . v i / D v i . Xiong (Rice University)

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This note was uploaded on 02/09/2012 for the course ECON 401 taught by Professor Siyang during the Spring '11 term at Rice.

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slide32-nov18 - ECON 401 Auctions Revenue Siyang Xiong Rice...

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