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Auction Theory an introduction DAI Hards October 16 th.

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1 Auction Theory an introduction DAI Hards October 16 th

2 Introduction Auctions are the most widely-studied economic mechanism. Auctions refer to arbitrary resource allocation problems with self-motivated participants: Auctioneer and bidders Auction (selling item(s)): one buyer, multiple bidders) e.g. selling a cd on eBay Reverse Auction (buying item(s)): one buyer, multiple sellers e.g. procurement We’ll discuss auction, though the same theory holds for reverse auction

3 Historical note Reports that auctions was used in Babylon 500 B.C. Reports that auctions was used in Babylon 500 B.C. 193 A.D. After having killed Emperor Pertinax, Prætorian Guard sold the Roman Empire by means of an Auction 193 A.D. After having killed Emperor Pertinax, Prætorian Guard sold the Roman Empire by means of an Auction

4 Where auctions are used nowadays? Treasury auctions (bill, notes, Treasury bonds, securities) Treasury auctions (bill, notes, Treasury bonds, securities) Has been used to transfer assets from public to private sector Has been used to transfer assets from public to private sector Right to drill oil, off-shore oil lease Right to drill oil, off-shore oil lease Use the EM spectrum Use the EM spectrum Government and private corporations solicit delivery price offers of products Government and private corporations solicit delivery price offers of products Private firms sell products (flowers, fish, tobacco, livestock, diamonds) Private firms sell products (flowers, fish, tobacco, livestock, diamonds) Internet auctions Internet auctions

5 Questions Information problem: the seller has usually incomplete information about buyers’ valuations (else, he just need to set the price as the maximum valuation of the buyer)  what pricing scheme performs well even in incomplete information setting (is auction better suited for a given problem? Does a type of auction yield greater revenue?) Information problem: the seller has usually incomplete information about buyers’ valuations (else, he just need to set the price as the maximum valuation of the buyer)  what pricing scheme performs well even in incomplete information setting (is auction better suited for a given problem? Does a type of auction yield greater revenue?) For the buyer, what are good bidding strategies? For the buyer, what are good bidding strategies?

6 Terminology Criterion of comparison: Criterion of comparison: –Revenue: expected selling price –Efficiency: the object ends up in the hands of the person who values it the most (resale does not yield to efficiency) Private Value : no bidder knows with certainty the valuation of the other bidders, and knowledge of the other bidders’ valuation would not affect the value of the particular bidder Private Value : no bidder knows with certainty the valuation of the other bidders, and knowledge of the other bidders’ valuation would not affect the value of the particular bidder Pure common value : the actual value is the same for ever bidders but bidders have different private information about the what that value actually is (e.g. auction of an oil field and the amount of oil is unknown, different bidders have different geological signals, learning another signal would change the valuation of a bidder). Pure common value : the actual value is the same for ever bidders but bidders have different private information about the what that value actually is (e.g. auction of an oil field and the amount of oil is unknown, different bidders have different geological signals, learning another signal would change the valuation of a bidder). Correlated value: agent’s value of an item depends partly on its own preferences and partly on others’ values for it Correlated value: agent’s value of an item depends partly on its own preferences and partly on others’ values for it

7 Agents care about utility, not valuation Auctions are really lotteries, so you must compare expected utility rather than utility. Risk attitude speak about the shape of the utility function: – – linear utility function refers to risk-neutrality  optimize her/his expected payoff – –Concave utility function refers to risk-aversion (u’>0 and u’’<0) – –convex utility function refers to risk-seeking (u’>0 and u’’>0) The types of utility functions, and the associated risk attitudes of agents, are among the most important concepts in Bayesian games, and in particular in auctions. Most theoretical results about auction are sensitive to the risk attitude of the bidders.

8 Outline Single-item Auctions Single-item Auctions –Common auctions forms –Equivalence between auctions –Revenue equivalence Multi-unit Auction Multi-unit Auction Multi-item Auction Multi-item Auction

9 Single Item Auction

10 English (first-price open-cry = ascending) Protocol: Each bidder is free to raise his bid. When no bidder is willing to raise, the auction ends, and the highest bidder wins the item at the price of his bid Protocol: Each bidder is free to raise his bid. When no bidder is willing to raise, the auction ends, and the highest bidder wins the item at the price of his bid Strategy: Series of bids as a function of agent’s private value, his prior estimates of others’ valuations, and past bids Strategy: Series of bids as a function of agent’s private value, his prior estimates of others’ valuations, and past bids Best strategy: In private value auctions, bidder’s dominant strategy is to always bid a small amount more than current highest bid, and stop when his private value price is reached Best strategy: In private value auctions, bidder’s dominant strategy is to always bid a small amount more than current highest bid, and stop when his private value price is reached Variations: Variations: –In correlated value auctions, auctioneer often increases price at a constant rate or as he thinks is appropriate (japonese auction) –Open-exit: Bidder has to openly declare exit without re-entering possibility => More info to other bidders about the agent’s valuation

11 First-price sealed-bid Protocol: Each bidder submits one bid without knowing others’ bids. The highest bidder wins the item at the price of his bid Protocol: Each bidder submits one bid without knowing others’ bids. The highest bidder wins the item at the price of his bid Single round of bidding Single round of bidding Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations Best strategy: No dominant strategy in general Best strategy: No dominant strategy in general Strategic underbidding & counterspeculation Strategic underbidding & counterspeculation Can determine Nash equilibrium strategies via common knowledge assumptions about the probability distributions from which valuations are drawn Can determine Nash equilibrium strategies via common knowledge assumptions about the probability distributions from which valuations are drawn Variant: k th price Variant: k th price

12 Example Values are uniformly distributed on [0,1] Values are uniformly distributed on [0,1] The equilibrium bid is (N-1)*x/N Where –x is the valuation of the bidder –N is the number of bidders (proof)

13 Dutch (descending) Protocol: Auctioneer continuously lowers the price until a bidder takes the item at the current price Protocol: Auctioneer continuously lowers the price until a bidder takes the item at the current price Strategically equivalent to first-price sealed-bid protocol in all auction settings Strategically equivalent to first-price sealed-bid protocol in all auction settings Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations Best strategy: No dominant strategy in general Best strategy: No dominant strategy in general –Lying (down-biasing bids) & counterspeculation –Possible to determine Nash equilibrium strategies via common knowledge assumptions regarding the probability distributions of others’ values –Requires multiple rounds of posting current price Dutch flower market, Ontario tobacco auction, Filene’s basement, Waldenbooks Dutch flower market, Ontario tobacco auction, Filene’s basement, Waldenbooks

14 Vickrey (= second-price sealed bid) Protocol: Each bidder submits one bid without knowing (!) others’ bids. Highest bidder wins item at 2nd highest price Protocol: Each bidder submits one bid without knowing (!) others’ bids. Highest bidder wins item at 2nd highest price Strategy: Bid as a function of agent’s private value & his prior estimates of others’ valuations Strategy: Bid as a function of agent’s private value & his prior estimates of others’ valuations Best strategy: In a private value auction with risk neutral bidders, Vickrey is strategically equivalent to English. In such settings, dominant strategy is to bid one’s true valuation Best strategy: In a private value auction with risk neutral bidders, Vickrey is strategically equivalent to English. In such settings, dominant strategy is to bid one’s true valuation –No counterspeculation –Independent of others’ bidding plans, operating environments, capabilities... –Single round of bidding Widely advocated for computational multiagent systems Widely advocated for computational multiagent systems Old [Vickrey 1961], but not widely used among humans Old [Vickrey 1961], but not widely used among humans Revelation principle --- proxy bidder agents on www.ebay.com, www.webauction.com, www.onsale.com Revelation principle --- proxy bidder agents on www.ebay.com, www.webauction.com, www.onsale.com

15 All Pay (e.g.lobbying activity) Protocol: Each bidder is free to raise his bid. When no bidder is willing to raise, the auction ends, and the highest bidder wins the item. All bidders have to pay their last bid Protocol: Each bidder is free to raise his bid. When no bidder is willing to raise, the auction ends, and the highest bidder wins the item. All bidders have to pay their last bid Strategy: Series of bids as a function of agent’s private value, his prior estimates of others’ valuations, and past bids Strategy: Series of bids as a function of agent’s private value, his prior estimates of others’ valuations, and past bids Best strategy: ? Best strategy: ? –In private value settings it can be computed (low bids) Potentially long bidding process Potentially long bidding process Variations Variations –Each agent pays only part of his highest bid –Each agent’s payment is a function of the highest bid of all agents

16 In a Nutshell English Auction Second-Price Sealed Bid i.e Vickrey Second-Price Sealed Bid i.e Vickrey First-Price Sealed Bid Dutch Descending Price Strong Weak Private Value Sealed Bid FormatOpen Format

17 Setting for Private Value Auctions N potential bidders. Bidder i is assigned a value of X i to the object –Each X i is i.i.d. on some interval [0, ω ] according to the cumulative distribution F –Bidders I knows her/his x i and also that other bidders values are i.i.d. according to F –Bidders are risk neutral (seek to maximize their expected payoffs) –The number of bidders and the distribution F are common knowledge. Symmetry Symmetry The distribution of values is the same for all bidders. WE can consider that all bidders are alike, hence an optimal bidding strategy for one should also be an optimal strategy for the others  symmetric equilibrium

18 Results for private value auctions Dutch strategically equivalent to first-price sealed-bid Dutch strategically equivalent to first-price sealed-bid Risk neutral agents => Vickrey strategically equivalent to English Risk neutral agents => Vickrey strategically equivalent to English All four protocols allocate item efficiently (assuming no reservation price for the auctioneer) All four protocols allocate item efficiently (assuming no reservation price for the auctioneer) English & Vickrey have dominant strategies  no effort wasted in counterspeculation English & Vickrey have dominant strategies  no effort wasted in counterspeculation Which of the four auction mechanisms gives highest expected revenue to the seller? Which of the four auction mechanisms gives highest expected revenue to the seller? Assuming valuations are drawn independently & agents are risk neutral: The four mechanisms have equal expected revenue!

19 Reserve Price in Private Values A seller can reserve the right to not sell the object if the price is below a reserved price r A seller can reserve the right to not sell the object if the price is below a reserved price r  Bidders with value x { "@context": "http://schema.org", "@type": "ImageObject", "contentUrl": "http://images.slideplayer.com/13/3598027/slides/slide_19.jpg", "name": "Reserve Price in Private Values A seller can reserve the right to not sell the object if the price is below a reserved price r A seller can reserve the right to not sell the object if the price is below a reserved price r  Bidders with value x

20 Revenue Equivalence Theorem In all auctions for k units with the following properties – –Buyers are risk neutral – –Private Value, with values independently and identically distributed over [a,b] (technicality – distribution must be atomless) – –Each bidder demands at most 1 unit – –Auction allocates the units to the k highest bids (efficiency) – –The bidder with the lowest valuation has a surplus of 0 (i.e. a bidder with a value of 0 has an expected payment of 0)  a buyer with a given valuation will make the same expected payment, and therefore all such auctions have the same expected revenue

21 Application of the Revenue Equivalence Theorem Helps to find some equilibrium strategy Ex: compute the equilibrium bid in an all pay auction or in a third price auction Ex: compute the equilibrium bid in an all pay auction or in a third price auction In the case where the number of bidders is uncertain, we can compute the equilibrium bid strategy for a first price auction In the case where the number of bidders is uncertain, we can compute the equilibrium bid strategy for a first price auction

22 Revenue equivalence ceases to hold if agents are not risk-neutral Risk averse Agents Risk averse Agents – for bidders: Dutch, first-price sealed-bid ≥ Vickrey, English Compared to a risk neutral bidder, a risk averse bidder will bid higher (“buy” insurance against the possibility of loosing) (utility of winning with a lower bid < utility consequence loosing the object) –For auctioneer auctioneer: Dutch, first-price sealed-bid ≤ Vickrey, English Risk-Seeking Agents – –The expected revenue in third-price is greater than the expected revenue in second-price (English) – –Under constant risk-attitude: (k+1)-price is preferable to k-price

23 Revenue equivalence ceases to hold if it is not Private Value Results for non-private value auctions Dutch strategically equivalent to first-price sealed-bid Dutch strategically equivalent to first-price sealed-bid Vickrey not strategically equivalent to English Vickrey not strategically equivalent to English All four protocols allocate item efficiently All four protocols allocate item efficiently Winner’s curse: each bidder must recognize that she/he wins the objects only if she/he has the highest signal, failure to take into account the bad news about others’ signal can lead the bidder to pay more than the prize it is worth. Winner’s curse: each bidder must recognize that she/he wins the objects only if she/he has the highest signal, failure to take into account the bad news about others’ signal can lead the bidder to pay more than the prize it is worth. –Common value auctions: –Agent should lie (bid low) even in Vickrey & English Revelation to proxy bidders? Thrm (revenue non-equivalence ). With more than 2 bidders, the expected revenues are not the same: Thrm (revenue non-equivalence ). With more than 2 bidders, the expected revenues are not the same: English ≥ Vickrey ≥ Dutch = first-price sealed bid ˜ v 1  E[v| ˆ v 1,b( ˆ v 2 )  b( ˆ v 1 ),...,b( ˆ v N )  b( ˆ v 1 )]

24 Results for non-private value auctions Common knowledge that auctioneer has private info Common knowledge that auctioneer has private info Q: What info should the auctioneer release ? A: auctioneer is best off releasing all of it “No news is worst news” “No news is worst news” Mitigates the winner’s curse Mitigates the winner’s curse

25 The revelation principle (mechanism Design) In a revelation mechanism agents are asked to report their types (e.g.valuations for the good), and an action (e.g. decision on the winner and his/her payment) will be based the agents’ announcement. In general, agents may cheat about their types, but: Any mechanism that implements certain behavior (e.g. a good is allocated to the agent with the highest valuation,v, and he pays (1-1/n)v) can be replaced by another mechanism that implements the same behavior and where truth-revealing is in equilibrium.

26 Multi-unit Auction

27 Auctions with multiple indistinguishable units for sale Examples Examples –IBM stocks –Barrels of oil –Pork bellies –Trans-Atlantic backbone bandwidth from NYC to Paris –…

28 Setting for sealed bid auctions Each bidder sends a “bid vector” indicating how much she/he is willing to pay for each additional unit Each bidder sends a “bid vector” indicating how much she/he is willing to pay for each additional unit  Can be understood as a demand function Number of units Value of the bid

29 Pricing rules Auctioning multiple indistinguishable units of an item The discriminatory (or “pay your bid”) auction The discriminatory (or “pay your bid”) auction The uniform price auction The uniform price auction The Vickrey auction The Vickrey auction

30 Discriminatory auction Each bidder pays an amount equal to the sum of his bids that are among the K highest of the N*K bids submitted. Each bidder pays an amount equal to the sum of his bids that are among the K highest of the N*K bids submitted.

31 Uniform-price Auction Any price between the highest loosing bid and the lowest winning bid is possible Any price between the highest loosing bid and the lowest winning bid is possible  can choose the highest losing bid

32 Vickrey Auction Basic principle is the same as the Vickrey- Clarke-Groves mechanism (see Mechanism Design) Basic principle is the same as the Vickrey- Clarke-Groves mechanism (see Mechanism Design) A bidder who wins k units pays the k highest losing bids of the other bidders A bidder who wins k units pays the k highest losing bids of the other bidders For bidder i to win the k th unit, i’s k th highest bid must defeat the k th lowest competing bid For bidder i to win the k th unit, i’s k th highest bid must defeat the k th lowest competing bid

33 Some Open Auctions Dutch Auctions Dutch Auctions English Auctions English Auctions Ausubel Auctions Ausubel Auctions

34 Multi-item auctions multiple distinguishable items for sale multiple distinguishable items for sale

35 Bundle bidding scenario

36

37 (console, television, cd player $1000)

38 Bundle bidding scenario (television, music system, computer, $1600)

39 Bundle bidding scenario (cd player, console, music system $400)

40 Bundle bidding scenario ((console, television, cd player $1000), (television, music system, computer, $1600), (cd player, console, music system $ 400))

41 Bundle bidding scenario ((Computer, television, cd player $1000), (television, music system, console, $600), (cd player, console, music system $400))

42 Bundle bidding scenario ((Computer, television, cd player $1000), (television, music system, console, $600), (cd player, console, music system $400))

43 Bundle bidding scenario ((Computer, television, cd player $1000), (television, music system, console, $600), (cd player, console, music system $400))

44 Multiple-item auctions Auction of multiple, distinguishable items Auction of multiple, distinguishable items Bidders have preferences over item combinations Bidders have preferences over item combinations Combinatorial auctions Combinatorial auctions –Bids can be submitted over item bundles –Winner selection: combinatorial optimization NP-complete NP-complete

45 Source Vijay Krishna: Auction Theory (Academic Press) Vijay Krishna: Auction Theory (Academic Press) Paul Klemperer: Auction Theory: A guide to the literature (Journal of Economics Survey) Paul Klemperer: Auction Theory: A guide to the literature (Journal of Economics Survey) Elmar Wolfstetter: Auctions An Introduction Elmar Wolfstetter: Auctions An Introduction Tuomas Sandholm COURSE: CS 15-892 Foundations of Electronic Marketplaces (CMU) Tuomas Sandholm COURSE: CS 15-892 Foundations of Electronic Marketplaces (CMU)


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