"Auctions and Bidding"

Prepared by L. Tesfatsion

Prepared by: Leigh Tesfatsion Date: 10 October 1999 For: Electricity Project Meeting, October 11 NOTES ON FIRST PART OF R. Preston McAfee and J. McMillan, "Auctions and Bidding," Journal of Economic Literature XXV (June 1987), 699-738. The notes below summarize, and in places comment upon, Sections I-VII (pages 699--716) of the McAfee/McMillan article on auctions and bidding. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx I. WHY STUDY AUCTIONS (pages 699-701) and II. TYPES AND USES OF AUCTIONS (pp. 701--703) Some of the most exciting of the recent advances in microeconomic theory have been in the modeling of strategic behavior under asymmetric information. Comment: STRATEGIC BEHAVIOR occurs if one agent recognizes that the utility (net benefits) he will receive from undertaking various actions depend on the actions undertaken by other agents as well. Consequently, it is to the agent's possible benefit to either estimate what the actions of these other agents will be, or even to attempt to influence what the actions of these other agents will be. Consequently, strategic behavior can have both backward-looking and forward-looking (anticipatory) aspects. Backward: If the other agent does this, what should I do? Forward: If I do this, what will the other agent do? ASYMMETRIC INFORMATION means that different agents condition their expectations and actions on different information sets. For example, I may know my own true valuation for an object up for sale but not the true valuation that some other potential buyer places on the object. An AUCTION is a market institution with an explicit set of rules determining resource alloation and prices on the basis of bids to buy and/or offers to sell from the market participants. The study of auctions provides one way of approaching the question of price formation in markets in which market participants have asymmetric information and behave strategically on the basis of their information. A more practical reason for studying auctions is that auctions are of considerable empirical importance. REAL-WORLD EXAMPLES: (pp. 701--702) Artwork, books, antiques, agricultural produce, mineral rights, U.S. Treasury bills, corporations, and gold are examples of items sold by auction. Government procurement -- for many government contracts, firms submit sealed bids, and the contract is required by law to be submited to the highest bidder. Procurement by firms of inputs from other firms through sealed-bid tenders. Auctioning of rights to import (with an import quota providing an upper bound on these rights to import). Auctioning of rights to produce noxious wastes (with an overall limit on the total amount of wastes that can be emitted providing an upper bound on these rights). Auctioning of airport time slots to competing airlines. Auctioning of frequency bandwidths for airwave transmission. Why are auctions used rather than other selling devices such as posting a fixed price? Main reason given: Some products have no standard value, so that the value of the product has to be discovered through a bidding process. DOUBLE AUCTIONS are auctions in which several buyers and sellers submit bids and offers simulataneously. In contrast, for a ONE-SIDED AUCTION, either bids are solicited from buyers by a monopolist seller for items to be sold, or offers are solicited from sellers by a monopsonist buyer for items to be bought. THIS PAPER WILL FOCUS ON SINGLE-SIDED AUCTIONS ORGANIZED BY MONOPOLIST SELLERS. QUESTIONS ABOUT THESE TYPES OF AUCTIONS TO BE SURVEYED IN THIS PAPER INCLUDE: What is the best form of auction for a monopolist seller to use in any particular set of circumstances? Should the monopolist seller impose a reserve price? If so, at what level? Can the monopolist seller design the auction so as to achieve price discrimination among the bidders (thus enabling the seller to extract more surplus)? Is it ever in the monopolist seller's interest to require payment from unsuccessful bidders (as well as the successful bidder)? Is it both feasible and desirable to make payment depend on something other than buyer bids? In particular, on something correlated with the true value of the item as perceived by the successful bidder (e.g., use royalties)? Should the monopolist seller release any information he has about the item's "true" value? What can the monopolist seller do to counter collusion among the bidders? FOUR BASIC TYPES OF SINGLE-SIDED AUCTIONS THAT CAN BE ORGANIZED BY MONOPOLIST SELLERS: (pp. 702-703) The ENGLISH AUCTION is the auction form most commonly used for the selling of goods. In the English auction, the price is successively raised until only one bidder remains. This can be done by having an auctioneer announce prices, or by having biddersf call the bids themselves, or by having bids submitted electronically with the current best bid posted. The essential feature of the English auction is that, at any point in time, each bidder knows the level of the current best bid. The final remaining bidder receives the item in return for paying the current best bid for the item. The DUTCH auction is the converse of the English auction. The auctioneer calls an initial high price and then successively lowers the price until a bidder accepts the current price. With the FIRST-PRICE SEALED-BID AUCTION, each potential buyer simultaneously submits a single sealed bid for the item, and the highest bidder is awarded the item for the price he bid. With the SECOND-PRICE SEALED BID AUCTION (OR VICKREY AUCTION), each potential buyer simultaneously submits a single sealed bid for the item, and the highest bidder is awarded the item but pays a price for the item equal not to his own bid but rather to the second-highest bid received. [In general, this gives each bidder an incentive to bid his true valuation for the item; for his bid only determines whether or not he wins the bid, not what he pays to the seller.] In practice, many variations of these four basic auction forms are in use. For example, the seller sometimes imposes a RESERVE PRICE, discarding all bids if they are too low. Bidders may be allowed only a LIMITED TIME for submitting bids. The auctioneer may charge bidders an ENTRY FEE for the right to bid. Payment may be made to depend not only on bids but also on something correlated with the true value of the item, e.g., ROYALTIES. In an English auction, the auctioneer may set a MINIMUM ACCEPTABLE INCREMENT to the highest existing bid. The seller, instead of selling the item as a unit, may offer for sale SHARES in the item. III. THE ABILITY TO MAKE COMMITMENTS (pp. 703--704) In theoretical studies of single-sided auctions organized by monopolist sellers, bargaining problems are typically side-stepped by presuming that the monopolist seller has all of the bargaining power. More precisely, it is assumed that the organizer of the auction has the ability to commit himself in advance to a set of policies. The advantage of commitment is that procedures can be adopted that induce the bidders to bid in ways deemed desirable by the auction organizer (the monopolist seller). There are several ways in which an auction organizer can achieve commitment: publicly available book of rules; pledge of his own reputation -- the cost of reneging on a current commitment might be the inability to commit oneself credibly in future transactions, and hence the loss of future bargaining power. Nevertheless, it does not follow from the fact that one party has the ability to make commitments that he can extract all of the gains from trade. What limits his bargaining power is the asymmetry of information. The seller can exploit competition among the bidders to drive up the price; but usually the seller will not be able to drive the price up so far as to equal the valuation of the bidder who values the item the most, because the seller does not know what this valuation is. IV. THE NATURE OF UNCERTAINTY (pp. 704--707) Asymmetry of information is the crucial element of an auction problem. Asymmetry of information means that both the seller and the bidders are uncertain about how other agents in the auction process will behave. How the seller and bidders respond to uncertainty depends on their attitude towards risk. IT IS ASSUMED THROUGHOUT THE REMAINDER OF THIS PAPER THAT THE MONOPOLIST SELLER IS RISK NEUTRAL. COMMENT: An agent is RISK NEUTRAL if he is indifferent between participating or not participating in "fair lotteries," i.e., in lotteries which have zero expected return. [In this case the utility function of the agent is linear.] An agent is RISK AVERSE if the utility he attaches to the expected return of a lottery, received as a lump sum payment, is STRICTLY GREATER than the utility he attaches to actual participation in the lottery. [In this case the agent would be willing to pay a small "premium" to avoid participation in the lottery, i.e., to avoid risk, and his utility function is strictly concave.] An agent is RISK LOVING if the utility he attached to expected return of a lottery, received as a lump sum payment, is STRICTLY LESS than the utility he attaches to actual participation in the lottery. [In this case the agent would be willing to pay a small "premium" in order to participate in the lottery, i.e., to take on risk, and his utility function is strictly convex.] Differences among the bidders' valuations of an item can arise from uncertainty for a variety of reasons. Consider the following two polar cases (p.705): 1. INDEPENDENT-PRIVATE VALUES MODEL. At one extreme, suppose each bidder knows precisely how highly he values the item. However, he does not know anyone else's valuation, and he perceives any other bidder i's valuation as a draw from some probability distribution F_i (and "knows" this is how other bidders perceive his own valuation). 2. COMMON-VALUE MODEL At the other extreme, suppose the item being bid for has a single objective value V -- namely, the amount the item is worth on the market. However, no one knows this true value V. Rather, each bidder i has a perceived value v_i for the item modelled as an independent draw from some conditional distribution H(v_i|V). All agents know the general form H(.|.) of the conditional distribution function, but they do not observe V. A further choice to be made by the modeller of an auction regarding sources of uncertainty for market participants is the extent to which bidders are recognizably different from each other. The following two cases will be considered below (p. 706): 1. ASYMMETRIC BIDDERS In the independent-private values model, each agent i is modelled as drawing his valuation v_i from a DIFFERENT probability distribution F_i, known to all agents. 2. SYMMETRIC BIDDERS In the independent-private values model, each agent i is modelled as drawing his valuation v_i from the SAME probability distribution F, known to all agents. Finally, another modelling consideration arising from uncertainty is that the amount of payment can only be made contingent upon variables that are observable to both buyer and seller. For example, in mineral-rights auctions, royalties make the payment depend upon the amount of oil extracted as well as the winning bid. Consequently, an auction organizer needs to consider the feasibility of such contingent payments (whether commonly observed variables exist) as well as the desirability of such contingent payments. BENCHMARK MONOPOLISTIC SELLER AUCTION MODEL (Page 706): A1. There are a finite number n of bidders, and each bidder is risk neutral (together with the monopolistic seller). A2. The independent-private values assumption applies. That is, the item valuation v_i of bidder i is an independent drawing from a distribution F_i, i = 1,...,n. A3. The bidders are symmetric. That is, in A2, F_i = F for all i. A4. Payments by bidders to the seller are a function only of bids. The analysis below starts with the benchmark model and then examines various weakenings of the benchmark model assumptions. The following assumptions are maintained throughout this analysis. MAINTAINED ASSUMPTIONS: (pp. 706--707) (a) Each of the n bidders knows the rules of the auction that the seller has chosen and committed himself to. (b) Bidder i knows his own valuation v_i, i = 1,...,n . (c) Each bidder i is assumed to know the number of bidders n, their risk attitudes, and the probability distributions of valuations. (d) Each bidder i knows that every other bidder j knows that he knows the auction aspects listed in (a)-(c), and that everyone else knows that bidder i knows that everyone else knows this, and so on (the so-called "common knowledge" assumption). COMMENT: Assumption (d) implies each agent knows the form of each other agent's information set, but not the precise values of the elements of this information set -- e.g., each bidder j knows that bidder i knows his own valuation v_i, but bidder j may not know what precise value v_i takes on. (e) The notion of "equilibrium" that all agents are assumed to use is "Bayes-Nash equilibrium." COMMENT: Roughly defined, a STRATEGY for a particular player i participating in a game is a rule (contingency plan) that specifies what action player i should take in each situation he might feasibly encounter. A configuration (s_1,...,s_n) of strategies for players 1,...,n in an n-player game is called a NASH EQUILIBRIUM if for each given i, and given s_j for all j not equal to i, player i has no incentive to deviate from the strategy s_i. Consider an auction in which n bidders are bidding for a single item from a monopolist seller under the maintained assumptions (a) through (d), above. A BIDDING STRATEGY for this game is a rule that specifies a feasible bid b_i for bidder i for each possible valuation v_i that bidder i might have, conditional on his beliefs regarding the valuation of each other bidder j. A BAYES-NASH EQUILIBRIUM for this game is then a configuration (B_1,...,B_n) of bidding strategies for bidders 1,...,n that satisfies the following two properties: (i) (B_1,...,B_n) is a Nash equilibrium; and (ii) all bidder beliefs about the valuations of other bidders are correct. See, for example, Robert Gibbons, "An Introduction to Applicable Game Theory," _Journal of Economic Perspectives_ 22 (Winter 1997), pages 127--149. V. THE BENCHMARK MODEL: COMPARING AUCTIONS USING THE REVENUE EQUIVALENCE THEOREM (pp. 707--711) Which of the four simple auction types (English, Dutch, first-price sealed bid, second-price sealed bid) should the monopolistic seller choose when organizing his auction? One result can be mentioned immediately: THE DUTCH AUCTION YIELDS THE SAME OUTCOME AS THE FIRST-PRICE SEALED BID AUCTION. This is because the situation facing a bidder is exactly the same in each auction: The bidder must choose how high to bid without knowing the other bidders' decisions; if he wins, he pays a price equal to his own bid. Because of this, the Dutch auction does not need to be separately considered from the first-price sealed bid auction. More interesting is the following famous revenue-equivalence theorem. For a proof of this theorem, see pages 707--710. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx REVENUE EQUIVALENCE THEOREM (p. 707): Under the maintained assumptions (a)-(e) and the benchmark model assumptions A1-A4, the **expected** revenue of the monopolistic seller under the English auction, the Dutch auction, the first-price sealed bid auction, and the second-price sealed bid auction are precisely the same. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx COMMENT 1 (page 710): It is important to note that the revenue equivalence theorem does not imply that the actual outcomes of the four auction forms will always be exactly the same. Rather, it simply says that these four auction forms have the same **expected** revenue prior to implementation under the stated assumptions. Roughly speaking then, each of these four auction forms would generate the same revenue **on average** for the seller if they were implemented repeatedly with different random valuation draws for the n bidders each time. COMMENT 2 (page 710): Note, also, another practical distinction between these four auction forms. In either an English auction or a second-price sealed bid auction, it is easy for the bidder to decide how high to bid. In the first case, the bidder should remain in the bidding until the current best price reaches his valuation. In the second case, the bidder should submit a bid equal to his true valuation. In contrast, in a Dutch auction or first-price sealed bid auction, the bidder must submit a sealed bid, and typically it is optimal to submit a bid that is some amount less than one's true valuation. Exactly how much less depends upon the probability distribution F perceived to govern other bidders' valuations and the exact number of competing bidders. See, e.g., the analytic representation for the optimal bidding rule for the first-price sealed bid auction (and Dutch auction) derived in equation (5), page 709. COMMENT 3 (footnote 13, page 710): Given the assumption that the monopolistic seller is risk neutral, the revenue-equivalence theorem implies that the seller will be indifferent among the four auction forms under the stated assumptions: they each have the same expected revenue, and the seller does not care about variance (risk). On the other hand, it can be shown that the **variance** of revenue is lower in an English auction or second-price sealed bid auction than in a first-price sealed bid auction or Dutch auction under the assumptions of the revenue-equivalence theorem. Consequently, if the seller were instead assumed to be risk averse, he would prefer the English or second-price sealed bid auction. COROLLARIES TO THE REVENUE EQUIVALENCE THEOREM (page 711) Under the assumptions of the revenue equivalence theorem: 1. Increasing the number n of bidders increases the expected revenue of the seller. This is because, as the number of bidders increases, the bids submitted by these bidders increase. [Note that it is assumed here that bidders do not incur a cost in preparing their bids, and the seller does not incur a cost in checking the credentials of bidders -- i.e., there are no organizational costs to consider as n increases.] 2. As the number n of bidders approaches infinity, the price paid to the seller tends to the highest possible valuation, implying that all gains from trade go to the seller. COMMENT: The authors claim this is true because, as n approaches infinity, the second-highest valuation approaches the highest possible valuation. But this need not be the case, even if the distribution function has compact support. Consider a highest valuation v(n) = 1+e for some positive e and a sequence of other valuations v(i) = 1 - 1/i, i = 1,...,n-1. As n goes to infinity, the second-highest valuation v(n-1) approaches 1 from below, hence remains strictly bounded below v(n). 3. For particular valuation distributions F, such as normal and uniform, an increase in the variance of the valuations v_i increases both the expected revenue of the seller and the rents of successful bidders. COMMENT: The RENT OF THE SUCCESSFUL BIDDER is the difference between his valuation and what he pays to the seller. VI. OPTIMAL AUCTIONS AND THE REVELATION PRINCIPLE (pp. 711--714) An AUCTION MECHANISM for a single-sided auction organized by a monopolistic seller is any process that takes as inputs the bids of the bidders and produces as its output the decision as to which bidder receives the item and how much any of the bidders will be required to pay. An auction mechanism is DIRECT if each bidder is simply asked to report his valuation of the item. An auction mechanism is INCENTIVE COMPATIBLE if the mechanism is structured such that each bidder finds it in his interest to report his valuation honestly. Given that the monopolist seller has the power to choose any auction mechanism, and that bidders cannot collude among themselves, a more fundamental question than revenue equivalence for the four basic auction mechanisms is as follows: WHAT IS THE OPTIMAL CHOICE OF AN AUCTION MECHANISM FOR THE MONOPOLISTIC SELLER, GIVEN THAT HE CAN CHOOSE AMONG **ALL** FEASIBLE AUCTION MECHANISMS? The tool used to address this question is the following principle. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx REVELATION PRINCIPLE (p. 712): For any auction mechanism for an auction organized by a monopolistic seller with n participating bidders, there is a direct, incentive-compatible auction mechanism with the same outcome. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx The revelation principle achieves honest revelation in the direct mechanism by designing the payoff structure in such a way that it is in the bidders' interests to be honest. The revelation principle shows that the modeller can limit his search for an optimal auction mechanism to the class of direct, incentive-compatible mechanisms. The optimal direct mechanism is founds as the solution to a mathematical programming problem involving two kinds of constraints (p. 712): (i) incentive-compatibility constraints, which state that a bidder cannot gain by misrepresenting their valuations; (ii) individual rationality or "free exit" constraints, which state that the bidders would not be bettr off if they refused to participate in the auction. The following important implication has been obtained as a corollary to the revelation principle. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx THE FORM OF THE OPTIMAL AUCTION MECHANISM (p. 713): Suppose the maintained assumptions (a)-(e) and the benchmark model assumptions A1-A4 hold. Define a function J(v) by J(v) = v - [1 - F(v))]/f(v), where f(v) is the density function corresponding to the valuation distribution function F(v). Suppose J(v) is an increasing function of v. [Note: This requires F(v) to be sufficiently concave.] Also, let v_L denote the lowest feasible valuation, and define a function B(v) by /v n-1 | [F(w)] dw /v_L B(v) = v - ---------------- n-1 F(v) Finally, let v_0 denote the valuation that the seller, himself, attaches to the item for sale. Then the seller's expected revenue is maximized by the following direct, incentive-compatible auction mechanism: (a) The seller asks each bidder i to report his valuation v_i, i = 1,...,n. (b) If J(v_i) < v_0 for i = 1,...,n, then the seller refuses to sell the item; (c) Otherwise, the item goes to the bidder i whose valuation v_i is highest in return for a payment to the seller equal to B(v_i). xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx COMMENTS: Let v_1 denotes the highest valuation of all n bidders. As shown on page 708, given the maintained and benchmark model assumptions, J(v_1) is the amount paid to the seller by the successful bidder under an English auction, equal to the successful bidder's valuation, v_1, minus the successful bidder's rent, [1 - F(v_1)]/f(v_1) Also, given the maintained and benchmark model assumptions, it is shown on page 709 that B(v) is the Bayes-Nash equilibrium bidding strategy for each bidder in a first-price sealed bid auction. Since (as can be shown) the function J(v) satisfies J(v) < v for all v, the "optimal" mechanism described in the Corollary may not be efficient. That is, it is possible by (b) that the seller keeps the item despite the existence of a bidder i whose valuation v_i is greater than the seller's valuation v_0. Condition (a) in effect establishes a "reserve price" for the seller. If condition (a) is not binding, then the optimal auction is equivalent to an English auction. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx OPTIMAL AUCTION DESIGN COROLLARY (pp. 713--714): Each of the basic four auction forms -- English, first-price sealed bid auction, Dutch auction, or second-price sealed auction -- becomes an optimal auction mechanism in the sense of the preceding revelation principle corollary if it is supplemented by a reserve price r satisfying J(r) = v_0. xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx This optimal auction mechanism design finding described above is a powerful result. No restriction has been placed on the types of auction mechanisms that the seller could use. The seller could have several rounds of bidding, charge entry fees, etc. But the optimal auction mechanism design finding states that none of these more complicated auction mechanisms would increase his expected revenue above the level of the optimal auction mechanism described above. This completes the analysis of the benchmark model. In the next sections, the effects of relaxing the benchmark assumptions one at a time is examined. VII. ASYMMETRIC BIDDERS: PRICE DISCRIMINATION (RELAXING A3) Instead of assuming that all bidders appear the same to the seller and to each other (assumption A3), suppose that the bidders can be partitioned into two recognizably different classes: n_1 bidders with valuation drawn from a distribution F_1 with corresponding density function f_1 n_2 bidders with valuation drawn from a distribution F_2 with corresponding density function f_2 When bidders are asymmetric in this fashion, the first-price sealed bid auction can yield a different expected revenue for the seller than the English auction, hence the Revenue Equivalence Theorem breaks down. For the English auction, the bids still rise until the price reaches the second-highest valuation. Since the highest-valuation bidder is still the successful bidder, the outcome is still efficient. However, the first-price sealed bid auction is no longer guaranteed to be efficient. While it remains the case that, within a bidding class, the higher-valuation agents bid higher, this is in general not the case across classes since bidders from different classes perceive themselves to be facing different degrees of bidding competition. This results in differences in their optimally calculated bidding functions. The expected revenue for the seller in an English auction can be either higher or lower than for a first-price sealed bid auction, depending on exact parameter values. Moreover, neither the English auction nor the first-price sealed bid auction (augmented with a reserve price) is optimal for the seller when the bidders are asymmetric. Rather, let v_i_k denote the valuation of bidder i of type k, k = 1,2, and define [1 - F_k(v_i_k)] J_k(v_i_k) = v_i_k - ----------------- f_k(v_i_k) r_k = J_k(v_0) = reservation price for class k. The following can then be shown (p. 715) For the auction mechanism that maximizes the seller's expected revenue, the seller awards the item to the bidder i in class k with the highest value of J_k(v_i_k), assuming that v_i_k is such that J_k(v_i_k) exceeds r_k. Consequently, the optimal auction mechanism for the seller is now discriminatory, in the sense that there is a possibility that one bidder succeeds despite another bidder's having a higher valuation. This is so because asymmetry means that F_1 differs from F_2 so that J_1(v) differs from J_2(v). Suppose, for example, that classes 1 and 2 each contain precisely one bidder, and that v_1_2 is greater than v_1_1 ; J_1(v_1_1) is greater than J_2(v_1_2) , Then the bidder in class 1 with valuation v_1_1 is the successful bidder despite the fact his valuation is not as high as the valuation v_1_2 of the bidder in class 2. Which type of bidder receives preferential treatment depends on the relative shapes of the distribution functions F_1 and F_2. In particular, one has the following surprising finding (p. 715): If F_1 and F_2 are identical apart from their means (one is a translation of the other), then the class of bidders with the lower mean valuation is favored by the optimal auction. How can this be? By favoring the low-valuation bidders, the seller raises the probability of awarding the item to someone other than the bidder who values the item the most, which raises the probability the seller will receive a lower payment. On the other hand, the benefit of this favoritism to the seller is that the favoritism forces the bidders from the higher valuation class to bid higher than they otherwise would have, thus driving up the price on average. BOTTOM LINE CAUTION: An ostensibly non-discriminatory sealed bid auction results in ad-hoc discrimination when the bidders are asymmetric. One cannot always judge the degree of discrimination inherent in an auction mechanism by its superficial aspects.