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KeywordsAuction Combinatorial auction Game theory
Auctions are procedures for selling one or more items to one or more bidders. Auctions induce games among the bidders, so notions of equilibrium from game theory can be applied to auctions. Auction theory aims to characterize and compare the equilibrium outcomes for different types of auctions. Combinatorial auctions arise when multiple-related items are sold simultaneously.
First price auction: Each bidder submits a bid one of the bidders submitting the maximum bid wins, and the payment for the item is the maximum bid. (In this context “wins” means receives the item, no matter what the payment.)
Second price auction or Vickrey auction: Each bidder submits a bid, one of the bidders submitting the maximum bid wins, and the payment for the item is the second highest bid.
English auction: The price for the item increases continuously or in some small increments, and bidders drop out at some points in time. Once all but one of the bidders has dropped out, the remaining bidder wins and the payment is the price at which the last of the other bidders dropped out.
A key goal of the theory of auctions is to predict how the bidders will bid, and predict the resulting outcomes of the auction: which bidder is the winner and what is the payment. For example, a seller may be interested in the expected payment (seller revenue). A seller may have the option to choose one auction format over another and be interested in revenue comparisons. Another item of interest is efficiency or social welfare. For sale of a single item, the outcome is efficient if the item is sold to the bidder with the highest value for the item. The book of V. Krishna (2002) provides an excellent introduction to the theory of auctions.
Auctions Versus Seller Mechanisms
An important class of mechanisms within the theory of mechanism design are seller mechanisms, which implement the sale of one or more items to one or more bidders. Some authors would consider all such mechanisms to be auctions, but the definition of auctions is often more narrowly interpreted, with auctions being the subclass of seller mechanisms which do not depend on the fine details of the set of bidders. The rules of the three types of auction mentioned above do not depend on fine details of the bidders, such as the number of bidders or statistical information about how valuable the item is to particular bidders. In contrast, designing a procedure to sell an item to a known set of bidders under specific statistical assumptions about the bidders’ preferences in order to maximize the expected revenue (as in Myerson (1981)) would be considered a problem of mechanism design, which is outside the more narrowly defined scope of auctions. The narrower definition of auctions was championed by R. Wilson (1987). An article on Mechanism Design appears in this encyclopedia.
Equilibrium Strategies in Auctions
An auction induces a noncooperative game among the bidders, and a commonly used predictor of the outcome of the auction is an equilibrium of the game, such as a Nash or Bayes-Nash equilibrium. For a risk neutral bidder i with value xi for the item, if the bidder wins and the payment is Mi, the payoff of the bidder is xi − Mi. If the bidder does not win, the payoff of the bidder is zero. If, instead, the bidder is risk averse with risk aversion measured by an increasing utility function ui, the payoff of the bidder would be ui(xi − Mi) if the bidder wins and ui(0) if the bidder does not win.
The second price auction format is characterized by simplicity of the bidding strategies. If bidder i knows the value xi of the item to himself, then for the second price auction format, a weakly dominant strategy for the bidder is to truthfully report xi as his bid for the item. Indeed, if yi is the highest bid of the other bidders, the payoff of bidder i is ui(xi − yi) if he wins and ui(0) if he does not win. Thus, bidder i would prefer to win whenever ui(xi − yi) > ui(0) and not win whenever ui(xi − yi) < ui(0). That is precisely what happens if bidder i bids xi, no matter what the bids of the other bidders are. That is, bidding xi is a weakly dominant strategy for bidder i.
Nash equilibrium can be found for the other types of auctions under a model with incomplete information, in which the type of each bidder i is equal to the value of the object to the bidder and is modeled as a random variable Xi with a density function fi supported by some interval [ai, bi]. A simple case is that the bidders are all risk neutral, the densities are all equal to some fixed density f, and the Xi’s are mutually independent. The English auction in this context is equivalent to the second price auction: in an English auction, dropping out when the price reaches his true value is a weakly dominant strategy for a bidder, and for the weakly dominant strategy equilibrium, the outcome of the auction is the same as for the second price auction. For the first price auction in this symmetric case, there exists a symmetric Bayesian equilibrium. It corresponds to all bidders using the bidding function β (so the bid of bidder i is β(Xi)), where β is given by β(x) = E[Y1|Y1 ≤ x]. The expected revenue to the seller in this case is E[Y1|Y1 < X1], which is the same as the expected revenue for the second price auction and English auction.
Equilibrium for Auctions with Interdependent Valuations
Seminal work of Milgrom and Weber (1982) addresses the performance of the above three auction formats in case the bidders do not know the value of the item, but each bidder i has a private signal Xi about the value Vi of the item to bidder i. The values and signals (X1, …Xn, V1, …, Vn) can be interdependent. Under the assumption of invariance of the joint distribution of (X1, …Xn, V1, …, Vn) under permutation of the bidders and a strong form of positive correlation of the random variables (X1, …Xn, V1, …, Vn) (see Milgrom and Weber 1982 or Krishna 2002 for details), a symmetric Bayes-Nash equilibrium is identified for each of the three auction formats mentioned above, and the expected revenues for the three auction formats are shown to satisfy the ordering R(firstprice) ≤ R(secondprice) ≤ R(English). A significant extension of the theory of Milgrom and Weber due to DeMarzo et al. (2005) is the theory of security-bid auctions in which bidders compete to buy an asset and the final payment is determined by a contract involving the value of the asset as revealed after the auction.
Combinatorial auctions implement the simultaneous sale of multiple items. A simple version is the simultaneous ascending price auction with activity constraints (Cramton 2006; Milgrom 2004). Such an auction procedure was originally proposed by Preston, McAfee, Paul Milgrom, and Robert Wilson for the US FCC wireless spectrum auction in 1994 and was used for the vast majority of spectrum auctions worldwide since then Cramton (2013). The auction proceeds in rounds. In each round a minimum price is set for each item, with the minimum prices for the initial round being reserve prices set by the seller. A given bidder may place a bid on an item in a given round such that the bid is greater than or equal to the minimum price for the item. If one or more bidders bid on an item in a round, a provisional winner of the item is selected from among the bidders with the highest bid for the item in the round, with the new provisional price being the highest bid. The minimum price for the item is increased 10 % (or some other small percentage) above the new provisional price. Once there is a round with no bids, the set of provisional winners is identified. Often constraints are placed on the bidders in the form of activity rules. An activity rule requires a bidder to maintain a history of bidding in order to continue bidding, so as to prevent bidders from not bidding in early rounds and bidding aggressively in later rounds. The motivation for activity rules is to promote price discovery to help bidders select the packages (or bundles) of items most suitable for them to buy. A key is that complementarities may exist among the items for a given bidder. Complementarity means that a bidder may place a significantly higher value on a bundle of items than the sum of values the bidder would place on the items individually. Complementarities lead to the exposure problem, which occurs when a bidder wins only a subset of items of a desired bundle at a price which is significantly higher than the value of the subset to the bidder. For example, a customer might place a high value on a particular pair of shoes, but little value on a single shoe alone.
A variation of simultaneous ascending price auctions for combinatorial auctions is auctions with package bidding (see, e.g., Ausubel and Milgrom 2002; Cramton 2013). A bidder will either win a package of items he bid for or no items, thereby eliminating the exposure problem. For example, in simultaneous clock auctions with package bidding, the price for each item increases according to a fixed schedule (the clock), and bidders report the packages of items they would prefer to purchase for the given prices. The price for a given item stops increasing when the number of bidders for that item drops to zero or one, and the clock phase of the auction is complete when the number of bidders for every item is zero or one. Following the clock phase, bidders can submit additional bids for packages of items. With the inputs from bidders acquired during the clock phase and supplemental bid phase, the auctioneer then runs a winner determination algorithm to select a set of bids for non-overlapping packages that maximizes the sum of the bids. This winner determination problem is NP hard, but is computationally feasible using integer programming or dynamic programming methods for moderate numbers of items (perhaps up to 30). In addition, the vector of payments charged to the winners is determined by a two-step process. First, the (generalized) Vickrey price for each bidder is determined, which is defined to be the minimum the bidder would have had to bid in order to be a winner. Secondly, the vector of Vickrey prices is projected onto the core of the reported prices. The second step insures that no coalition consisting of a set of bidders and the seller can achieve a higher sum of payoffs (calculated using the bids received) for some different selection of winners than the coalition received under the outcome of the auction. While this is a promising family of auctions, the projection to the core introduces some incentive for bidders to deviate from truthful reporting, and much remains to be understood about such auctions.
Summary and Future Directions
Auction theory provides a good understanding of the outcomes of the standard auctions for the sale of a single item. Recently emerging auctions, such as for the generation and consumption of electrical power, and for selection of online advertisements, are challenging to analyze and comprise a direction for future research. Much remains to be understood in the theory of combinatorial auctions, such as the degree of incentive compatibility offered by core projecting auctions.
- Ausubel LM, Milgrom PR (2002) Ascending auctions with package bidding. BE J Theor Econ 1(1): Article 1Google Scholar
- Krishna V (2002) Auction theory. Academic, San DiegoGoogle Scholar
- Wilson R (1987) Game theoretic analysis of trading processes. In: Bewley T (ed) Advances in economic theory. Cambridge University Press, Cambridge/New YorkGoogle Scholar