RevenueEquivalence145

RevenueEquivalence145 - Revenue Equivalence Ichiro Obara...

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Revenue Equivalence Ichiro Obara UCLA February 15, 2010 Obara (UCLA) Revenue Equivalence February 15, 2010 1 / 27
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Revenue Equivalence Revenue Equivalence Obara (UCLA) Revenue Equivalence February 15, 2010 2 / 27
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Revenue Equivalence Revenue Equivalence? We have seen that the seller’s (expected) revenue is the same for the first price auction and the second price auction. Is this a coincidence? We show that the seller’s revenue is the same for “every” auction . Obara (UCLA) Revenue Equivalence February 15, 2010 3 / 27
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Revenue Equivalence General Auction Consider any auction that satisfies the following conditions. I A seller sets the reserve price b 0 . I Bidders submit a bid b 0 or opt out. I The bidder making the highest bid wins. If every bidder opts out, then the seller keeps the object. I Each bidder is treated equally (anonimity). I There exists a symmetric Bayesian Nash equilibrium ( b , .... , b ) where b ( v ) is strictly increasing in v when making a bid and the lowest type’s expected payment is 0. The basic FPA and SPA is a special case with 0 reserve price. Obara (UCLA) Revenue Equivalence February 15, 2010 4 / 27
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Revenue Equivalence SBNE If a bidder with v participates in equilibrium, then every bidder with v 0 > v participates in equilibrium. This is because v 0 can get at least as large an expected payoff as v . So there is critical value v * such that a bidder participates in auction if and only if v v * . Hence the symmetric BNE looks like the next graph. Obara (UCLA) Revenue Equivalence February 15, 2010 5 / 27
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Revenue Equivalence SBNE b(v) v* b 0 Obara (UCLA) Revenue Equivalence February 15, 2010 6 / 27
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Revenue Equivalence Theorem This is one of the most important theorems in the theory of auctions. Revenue Equivalence Theorem For any auction that satisfies the above conditions, the expected revenue for the seller is the same at the SBNE. Obara (UCLA)
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RevenueEquivalence145 - Revenue Equivalence Ichiro Obara...

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