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lecture-auctions

# lecture-auctions - Lecture XVI Auctions Markus M Mbius o...

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Lecture XVI: Auctions Markus M. M¨obius April 14, 2002 Readings for this class: P. Klemperer - Auction Theory: A Guide to the Literature (especially parts of the appendix - the main text provides an excellent introduction to auction theory but is optional) 1 Introduction Auctions are extremely common. Natural resources such as wireless spec- trum, oil and minerals etc. are auctioned of. There are several important types oF auctions: ascending price auction second-price auction sealed bid (±rst price) auction all-pay auction (good model For legislative lobbying, war oF attrition) There are many possible scenarios For agents’ types. The two most im- portant ones are: Private Value environment: each agent’s valuation oF the good is drawn i.i.d. From some distribution F on [ v , v ]. Common value auction: agents’ valuations are correlated (oil leases). Easy Formulation: v i = V ( t 1 , ..., t n )where t i are agents’ signals. (1) 1

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2 Some Simple Solved Examples We frequently work with values (signals) drawn from the uniform distribu- tion. In that case it is useful to know that the expected value of the ith order statistic is v + n +1 k n ( v v )( 2 ) 2.1 Sealed Bid Auction (Private Value) In the sealed bid auction where the players’ valuations are independently uniformly distributed on [0 , 1] the unique BNE is: f 1 ( v 1 )= v 1 2 f 2 ( v 2 v 2 2 Proof: To verify that this is a BNE is easy. We just show that each type of each player is using a BR: E v 2 ( u 1 ,f 2 ; v 1 ,v 2 )=( v 1 b 1 ) Prob ( f 2 ( v 2 ) <b 1 )+ 1 2 ( v 1 b 1 ) ( f 2 ( v 2 b 1 ) We assume b 1 £ 0 , 1 2 ¤ . No larger bid makes sense given f 2 . Hence: E v 2 ( u 1 2 ; v 1 2 v 1 b 1 )2 b 1 This is a quadratic equation which we maximize by the FOC: 0=2 v 1 4 b 1 (3) Hence b 1 = v 1 2 .
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lecture-auctions - Lecture XVI Auctions Markus M Mbius o...

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