Auction Theory an introduction DAI Hards October 16

Auction Theory an introduction DAI Hards October 16 th

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 e. Bay • Reverse Auction (buying item(s)): one buyer, multiple sellers e. g. procurement • We’ll discuss auction, though the same theory holds for reverse auction

Historical note • 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

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

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? ) For the buyer, what are good bidding strategies?

Terminology • 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 • 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

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.

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

Single Item Auction

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 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 • 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

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 Single round of bidding Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations Best strategy: No dominant strategy in general Strategic underbidding & counterspeculation Can determine Nash equilibrium strategies via common knowledge assumptions about the probability distributions from which valuations are drawn Variant: kth price

Example • 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)

Dutch (descending) • 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 Strategy: Bid as a function of agent’s private value and his prior estimates of others’ valuations 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

Vickrey (= second-price sealed bid) • Protocol: Each bidder submits one bid without knowing (!) others’ • • bids. Highest bidder wins item at 2 nd highest price 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 – No counterspeculation – Independent of others’ bidding plans, operating environments, capabilities. . . – Single round of bidding • Widely advocated for computational multiagent systems • Old [Vickrey 1961], but not widely used among humans • Revelation principle --- proxy bidder agents on www. ebay. com, www. webauction. com, www. onsale. com

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 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 settings it can be computed (low bids) • Potentially long bidding process • Variations – – Each agent pays only part of his highest bid Each agent’s payment is a function of the highest bid of all agents

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

Setting for Private Value Auctions N potential bidders. Bidder i is assigned a value of Xi to the object – Each Xi is i. i. d. on some interval [0, ω] according to the cumulative distribution F – Bidders I knows her/his xi 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 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

Results for private value auctions • Dutch strategically equivalent to first-price sealed-bid • Risk neutral agents => Vickrey strategically equivalent to • • • English All four protocols allocate item efficiently (assuming no reservation price for the auctioneer) English & Vickrey have dominant strategies no effort wasted in counterspeculation 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!

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 Bidders with value x<r are excluded from the auction Bidders change their strategy (can be computed) • A revenue maximizing seller should always set a reserve price r that exceeds her or his valuation x 0 Proof: compute the expected payoff of the seller, differentiate it with respect to the reserve price and observe it the derivative is positive at x 0 • Entry Fees: the auctioneer can use an entry fee: a nonrefundable amount that the bidder has to pay to participate to the auction. Note : there is a way to fix the entry fee such it is equivalent to using a reserved price as far as the agents that are excluded are concerned. • Trade-off: may improve revenue at the expense of efficiency (if seller set a reservation price which is too high)

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

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 • In the case where the number of bidders is uncertain, we can compute the equilibrium bid strategy for a first price auction

Revenue equivalence ceases to hold if agents are not risk-neutral • 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: 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

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 Vickrey not strategically equivalent to English 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. – Common value auctions: ˜ ˆ ˆ v = E [ v | v , b ( v ) < b ( v ), . . . , b ( v 1 1 2 1 N ˆ ) < b ( v )] – Agent should lie (bid low) even in Vickrey & English proxy bidders? 1 Revelation to • Thrm (revenue non-equivalence ). With more than 2 bidders, the expected revenues are not the same: English ≥ Vickrey ≥ Dutch = first-price sealed bid

Results for non-private value auctions • 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” • Mitigates the winner’s curse

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.

Multi-unit Auction

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

Setting for sealed bid auctions • 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 Value of the bid Number of units

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

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.

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

Vickrey Auction • 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 • For bidder i to win the kth unit, i’s kth highest bid must defeat the kth lowest competing bid

Some Open Auctions • Dutch Auctions • English Auctions • Ausubel Auctions

Multi-item auctions multiple distinguishable items for sale

Bundle bidding scenario

Bundle bidding scenario (console, television, cd player $1000)

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

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

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

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

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

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