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4820–11 Actions Geir B. Asheim Auctions Introduction Analysis 4820–11 Revenue equalization Reservation price Geir B. Asheim Relaxing assumptions Department of Economics, University of Oslo ECON4820 Spring 2010 Last modiﬁed: 2010.04.27 Why are auctions important? 4820–11 Actions Geir B. Asheim Introduction Types Outline Analysis Revenue equalization Practical: Much used In ﬁnancial markets Procurement auctions by governments and ﬁrms Houses, agricultural products, art, etc. Radio spectrum licences, electricity, transport Reservation price Relaxing assumptions Theoretical: Basis for important theoretical work Empirical: Testing-ground for economic theory What is an auction? 4820–11 Actions Geir B. Asheim Introduction Types Outline Analysis One seller Small number of potential buyers Bidding by buyers determines allocation and price Revenue equalization Reservation price Relaxing assumptions Important variant for procurements and contracts: Reverse auction One buyer Small number of potential sellers Bidding by sellers determines allocation and price Application, alternatives, and concerns 4820–11 Actions Geir B. Asheim Introduction Types Outline Analysis Revenue equalization Reservation price Relaxing assumptions When are auctions used? A unique object Uncertainty about who should get the object Uncertainty about the object’s value Seller can commit to the procedure Alternatives to auctions Market — decides who gets the object, but how to determine the price? Bargaining — determines the price, but how to decide who is the counterpart? Handing out for free Two concerns for an auction For society: Is the object bought by the bidder with the highest willingness to pay? (Eﬃciency ) For the seller: Is the price the highest possible? Types of auctions 4820–11 Actions Geir B. Asheim Introduction Types Outline Analysis Revenue equalization Reservation price Relaxing assumptions The descending-bid (Dutch) auction The price is successively lowered until the ﬁrst bidder enters Used for ﬂowers in the Netherlands The ascending-bid (English) auction The price is successively raised until only one bidder remains Used for art and collectibles First-price sealed-bid auction Simultaneous bids, highest bid win, pays highest bid Used for real estate, government procurement Second-price sealed-bid (Vickrey) auction Simultaneous bids, highest bid win, pays second-highest bid Used for stamps etc. Outline 4820–11 Actions Geir B. Asheim Introduction Types Outline Analysis of bidding behavior under standard assumptions Bidders are risk-neutral Private values Symmetric bidders Analysis Revenue equalization Reservation price Relaxing assumptions Important result: Revenue equalization theorem Auction form doesn’t matter for eﬃciency & exp. price Does the seller want to impose a reservation price? Relaxing the standard assumptions Bidders are risk-averse Common values Asymmetric bidders Standard assumptions 4820–11 Actions Geir B. Asheim Introduction Analysis English Vickrey 1st price Bidders are risk-neutral Each bidder’s valuation is i.i.d. Revenue equalization Reservation price Relaxing assumptions Each bidder knows only her own valuation Seller knows only the c.d.f. from which the values are drawn Bidding behavior in English auctions 4820–11 Actions Geir B. Asheim Introduction Analysis English Vickrey 1st price Revenue equalization Reservation price Relaxing assumptions Continued bidding is proﬁtable as long as own valuation > current price This strategy is a best response ind. of others’ behavior (weakly dominates any other strategy) The winner is the one with the highest valuation (⇒ eﬃciency) Price is (just above) second highest valuation Bidding behavior in sealed-bid 2nd price auctions 4820–11 Actions Geir B. Asheim Introduction Analysis English Vickrey 1st price Revenue equalization Reservation price Relaxing assumptions v Bidder’s valuation b Bidder’s bid a Largest bid from others If v < a, the bidder does not want to win, and bidding b ≤ v ensures this. If v > a, the bidder wants to win, and bidding b ≥ v ensures this. Conclusion: Bidding b = v is a best response, weakly dominating any other strategy. Everyone bidding their true value is a Bayesian Nash equilibrium. This equilibrium is characterized by: The winner is the one with the highest valuation (⇒ eﬃciency) The price equals the second-highest valuation 2nd price auction corresponds to English auction (w.r.t. bidding strategy, winner and price) Bidding behavior in sealed-bid 1st price auctions 4820–11 Actions Geir B. Asheim Bidder trades oﬀ two concerns: Bidding b < v reduces her chances to win; not good reduces the paid price if she wins; good Introduction Analysis English Vickrey 1st price Revenue equalization Reservation price Relaxing assumptions This trade-oﬀ makes the optimal bid smaller than v . Others behaving likewise, makes the opt. bid even smaller. Conclusion: A symmetric equilibrium is characterized by: The winner is the one with the highest valuation (⇒ eﬃciency) The price equals the highest bid, which is less than the highest valuation 1st price auction corresponds to Dutch auction (w.r.t. bidding strategy, winner and price) Calculating the equilibrium bid strategies is diﬃcult Formal analysis of sealed-bid 1st price auctions 4820–11 Actions Geir B. Asheim Introduction Analysis English Vickrey 1st price Revenue equalization Reservation price Relaxing assumptions n bidders, vi ∈ [v , vh ], i ∈ {1, . . . , n}; F (vi ): c.d.f. Consider a symmetric equilibrium (bidders are ex ante identical): Bi (vi ) = B(vi ) for all i Each player believes that Bj (vi ) = B(vi ) for all j = i Assume that B > 0 ⇒ Bid = bi implies vi = B −1 (bi ) The probability that bi is the winning bid: [F (B −1 (bi ))]n−1 Bidder i’s expected proﬁt: πi = (vi − bi )[F (B −1 (bi ))]n−1 i FOC: 0 = ∂π ∂bi , implying that ∂πi dbi ∂πi −1 (b ))]n−1 = [F (v )]n−1 i = ∂π i i ∂vi + ∂bi dvi = ∂vi = [F (B Integrating, setting πi = 0 if vi = v (since zero prob. of v winning then): π(vi ) = vi [F (x)]n−1 dx Two expressions for bidder i’s proﬁt must be equal: v i (vi − B(vi ))[F (vi )]n−1 = πi (vi ) = v [F (x)]n−1 dx dπi dvi ⇒ B(vi ) = vi − v n−1 dx i v [F (x)] [F (vi )]n−1 Common for all four kinds of auctions 4820–11 Actions Eﬃciency: Object to the bidder with highest valuation Geir B. Asheim Revenue equivalence: All four kinds give the seller the same expected revenue under the standard assumptions Introduction Analysis Revenue equalization Reservation price Relaxing assumptions Result (Revenue equalization theorem) Any auction mechanism in which (i) the object always goes to the buyer with the highest signal, and (ii) any bidder with the lowest-feasible valuation expects zero surplus, yields the same expected revenue (and results in each bidder making the same expected payment as a function of her valuation) An increase in # of bidders increases the expected price More bidders increases the expected 2nd-highest valuation Diﬀerence: Bid more diﬃcult to calculate in sealed-bid 1st-price (Dutch) auctions than in sealed-bid 2nd-price (English) auctions Does the seller want to impose a reservation price? 4820–11 Actions Geir B. Asheim Introduction Analysis Revenue equalization Reservation price Relaxing assumptions A parallel situation: The monopolist’s problem A monopolist trades oﬀ two concerns: Wants to sell large quantities ⇒ Low price Wants to earn a proﬁt per unit sold ⇒ High price Optimal trade-oﬀ: Price above marginal cost Auction: Seller trades oﬀ the same two concerns: Wants to sell the object ⇒ Low reservation price Wants to earn a proﬁt if the object is sold ⇒ High reservation price Optimal trade-oﬀ: Reservation price above own valuation Three cases (v1 , v2 two highest valuations; r reserv. price): (i) v1 > v2 > r : Increasing r has no eﬀect (ii) v1 > r > v2 : Increasing r increases the price (iii) r > v1 > v2 : r prevents sale Optimal reservation price with 1 bidder 4820–11 Actions Geir B. Asheim Introduction Analysis Bid r or nothing Seller’s own valuation: v0 Seller’s expected proﬁt: π(r ) = r [1 − F (r )] + v0 F (r ) Reservation price FOC: 0 = [1 − F (r )] − rf (r ) + v0 f (r ) (r ) ⇒ v0 = r − 1−F f (r ) ≡ J(r ) i.e., Marginal cost = Marginal revenue Relaxing assumptions Equivalently: r = J −1 (v0 ) Revenue equalization Eﬃciency with a reservation price: With a reservation price, the object may not be sold, even if there exists a bidder with v > v0 Ex ante eﬃciency vs. ex post eﬃciency Relaxing the standard assumptions 4820–11 Actions Geir B. Asheim Introduction Analysis Revenue equalization Reservation price Relaxing assumptions Risk-averse bidders Common values Asymmetric bidders Bidders are risk averse Common values Asymmetric bidders Risk-averse bidders 4820–11 Actions Geir B. Asheim Introduction Analysis Revenue equalization Reservation price Relaxing assumptions Risk-averse bidders Common values Asymmetric bidders In a sealed-bid 1st price auction, risk-averse bidders bid higher than risk-neutral ones. An increase in the bid (1) increases the chance of winning (2) reduces earning in case of winning Risk-aversion makes (1) more important relative to (2) No eﬀect of risk-aversion in a sealed-bid 2nd price auction The seller gains more in a sealed-bid 1st-price auction than in a sealed-bid 2nd price auction Common value auctions 4820–11 Actions Geir B. Asheim Introduction Analysis Revenue equalization Reservation price Relaxing assumptions Risk-averse bidders Common values Asymmetric bidders All bidders have identical valuations Each bidder does not know the value, but receives a signal Each bidder does not know the other bidders’ signals Examples: A jar of coins Buying for resale Buying a project with common uncertainty for all bidders (an unexplored petroleum ﬁeld w/equally capable bidders) Winner’s curse Bidder receiving the most optimistic signal wins Bidders must take into account that winning is “bad news” (since it means that the true value is less than the received signal indicates) by scaling down their bids In an English auction, bidders learn from each other during the bidding process. Reduces the winner’s-curse problem. Asymmetric bidders 4820–11 Actions Geir B. Asheim Introduction Example: Public procurement – domestic vs. foreign ﬁrms. Suppose foreign ﬁrms are more cost eﬀective than domestic ones Analysis Revenue equalization English and sealed-bid 2nd-price auctions are still eﬃcient Reservation price Sealed-bid 1st-price auction is no longer eﬃcient Relaxing assumptions Risk-averse bidders Common values Asymmetric bidders It is optimal to discriminate between bidder groups; ⇒ possible that winner is not the lowest-cost bidder In the example: It is optimal to discriminate in favor of the domestic ﬁrm. This favoring increases the change of getting an ineﬃcient supplier but also lowers the bid from the eﬃcient ﬁrms