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slide31-nov16

# slide31-nov16 - ECON 401 1st-price sealed-bid Auctions...

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ECON 401: 1st-price sealed-bid Auctions Siyang Xiong Rice University November 16, 2011 Xiong (Rice University) ECON 401 November 16, 2011 1 / 16

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1st-price sealed-bid auction: how should we bid? one object is for sale; for simplicity, suppose there are two bidders, 1 and 2; the bidder°s value, 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; Xiong (Rice University) ECON 401 November 16, 2011 2 / 16
1st-price sealed-bid auction: how should we bid? solution concept: Bayesian Nash Equilibrium (BNE); generalized notion of Nash Equilibrium, accomodating conditional belief; what is the strategy for each bidder? if bidder 1 observes v 1 D 0 . 1, he should bid some price ± call it b 1 . 0 . 1 / ; if bidder 1 observes v 1 D 0 . 4, he should bid some price ± call it b 1 . 0 . 4 / ; if bidder 1 observes v 1 D 0 . 8, he should bid some price ± call it b 1 . 0 . 8 / ; ... i.e., bidder 1°s strategy is a function b 1 : [ 0 , 1 ] ! R . similarly, bidder 2°s strategy is a function b 2 : [ 0 , 1 ] ! R . these strategies are formed before bidders observe their true value. Xiong (Rice University) ECON 401 November 16, 2011 3 / 16

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1st-price sealed-bid auction: how should we bid? since the two bidders are symmetric: v i are i.i.d. distributed; it is reasonable to consider symmetric bidding functions, i.e., b 1 . k / D b 2 . k / D b . k / I further, it is reasonable to assume b . k / is increasing in k : the higher the value, the more he bids. also, assume b . k / is di/erentiable. Xiong (Rice University) ECON 401 November 16, 2011 4 / 16
1st-price sealed-bid auction: how should we bid? suppose [ b . ± / , b . ± / ] is a symmetric BNE, i.e., given bidder 1 taking the strategy b . ± / , it is the best for bidder 2 to take the strategy b . ± / I given bidder 2 taking the strategy b . ± / , it is the best for bidder 1 to take the strategy b . ± / I consider bidder 1 who observes v 1 2 [ 0 , 1 ] . Given bidder 2 taking the strategy b . k / , how should he bid?

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