Auction theory

Branch of applied economics regarding the behavior of bidders in auctions

Auction theory is an applied branch of economics which deals with how bidders act in auctions and researches how the features of auctions incentivise predictable outcomes. Auction theory is a tool used to inform the design of real world auctions. Sellers use auction theory to raise higher revenues while allowing buyers to procure at a lower cost. The conference of the price between the buyer and seller is an economic equilibrium. Auction theorists design rules for auctions to address issues which can lead to market failure. The design of these rulesets encourages optimal bidding strategies among a variety of informational settings. The 2020 Nobel Prize for Economics was awarded to Paul R. Milgrom and Robert B. Wilson “for improvements to auction theory and inventions of new auction formats.”

Auctions facilitate transactions by enforcing a specific set of rules regarding the resource allocations of a group of bidders. Theorists consider auctions to be economic games that differ in two respects: format and information. The format defines the rules for the announcement of prices, the placement of bids, the updating of prices, when the auction closes, and the way a winner is picked. The way auctions differ with respect to information regards the asymmetries of information that exist between bidders. In most auctions, bidders have some private information that they choose to withhold from their competitors. For example, bidders usually know their personal valuation of the item, which is unknown to the other bidders and the seller; however, the behaviour of bidders can influence the personal valuation of other bidders.

One of the historical events related to auctions that has been reported is a custom in Babylonia, namely when men try to make an offer to women in order to marry them. The more familiar the auction system is, the more situations where auctions are conducted. There are auctions for various things, such as livestock, rare and unusual items, and financial assets. No cooperative games have a long history beginning with Cournot's duopoly model. The 1994 Nobel Laureate for Economic Sciences, John Nash, proved a general existence theorem for non cooperative games which move beyond simple zero sum games. This theory was generalized by Vickrey (1961) to deal with the unobservable value of each buyer. By the early 1970s, auction theorists had begun defining equilibrium bidding conditions for single object auctions under most realistic auction formats and information settings. Recent developments in auction theory consider how multiple object auctions can be performed efficiently.

The auction envelope theorem defines certain probabilities expected to arise in an auction.

The revelation principle is a simple but powerful insight. In 1979 proved a general revenue equivalence theorem that applies to all buyers and hence to the seller. Their primary interest was finding out which auction rule would be better for the buyers. A rule might be, for example, that all buyers pay a nonrefundable bid. (Such auctions are conducted on line.). The equivalence theorem shows that any allocation mechanism or auction that satisfies the four main assumptions of the benchmark model will lead to the same expected revenue for the seller. (Buyer i of with value v also has the same "payoff" or "buyer surplus" across all auctions.)

The first model for a broad class of models was Milgrom and Weber's (1983) paper on auctions with affiliated values. In a recent working paper general asymmetric auctions, Riley (2022) provided a characterization of equilibrium bids for all value distributions. Each buyer's value can be positively or negatively correlated. The revelation principle as applied to auctions is that the marginal buyer payoff or "buyer surplus" is P(v), the probability of being the winner. In every participant efficient auction, the probability of winning is 1 for a high value buyer. The marginal payoff to a buyer is therefore the same in every such auction. The payoff must therefore be the same as well.

Quite independently and soon after, ) used the revelation principle to provide a general characterization of a revenue maximizing sealed high bid auction. In the "regular" case this is a participation efficient auction. Setting a reserve price is therefore optimal for the seller. In the "irregular" case it has since been shown that the outcome can be implemented by prohibiting bids in certain subintervals. Relaxing each of the four main assumptions of the benchmark model yields auction formats with unique characteristics. Usually, but not always, a private values model assumes that the values are independent across bidders, whereas a common value model usually assumes that the values are independent up to the common parameters of the probability distribution. A more general category for strategic bidding is the affiliated values model, in which the bidder's total utility depends on both their individual private signal and some unknown common value. Both the private value and common value models can be perceived as extensions of the general affiliated values model. When it is necessary to make explicit assumptions about bidders' value distributions, most of the published research assumes symmetric bidders. This means that the probability distribution from which the bidders obtain their values (or signals) is identical across bidders. In a private values model which assumes independence, symmetry implies that the bidders' values are "i. i. d." – independently and identically distributed. An important example (which does not assume independence) is Milgrom and Weber's "general symmetric model" (1982).

One of the major findings of auction theory is the revenue equivalence theorem. Early equivalence results focused on a comparison of revenue in the most common auctions. The first such proof, for the case of two buyers and uniformly distributed values was by In 1979 proved a much more general result. (Quite independently and soon after, this was also derived by ). The revenue equivalence theorem states that any allocation mechanism or auction that satisfies the four main assumptions of the benchmark model will lead to the same expected revenue for the seller (and player i of type v can expect the same surplus across auction types).

The winner's curse is a phenomenon which can occur in common value settings—when the actual values to the different bidders are unknown but correlated, and the bidders make bidding decisions based on estimated values. In such cases, the winner will tend to be the bidder with the highest estimate, but the results of the auction will show that the remaining bidders' estimates of the item's value are less than that of the winner, giving the winner the impression that they "bid too much". showed that in any auction or auction like action (such as the "War of Attrition") the allocation is "participant efficient", i. e. the item is allocated to a buyer submitting the highest bid with probability 1. They then showed that allocation equivalence implied payoff equivalence for all reserve prices. They then showed that discriminating against low value buyers by setting a minimum price or (reserve price) would increase expected revenue. Along with Myerson, they showed that the most profitable reserve price is independent of the number of bidders. There is a simple intuition. The reserve price only comes into play if there is a single bid. Thus it is equivalent to ask what reserve price would maximize the revenue from a single buyer. If values are uniformly distributed over the interval [0, 100], then the probability p(r) that this buyer's value is less than r is p(r) = (100 r)/100. Therefore the expected revenue is

p(r)*r = (100 r)*r/100 =(r 50)*(r 50) + 25
Thus the expected revenue maximizing reserve price is 50. Myerson (1981). also examined the question of whether it might ever be more profitable to design a mechanism that awards the item to a bidder other than one with the highest value. Surprisingly, this is the case. As Maskin and Riley then showed, this is equivalent to excluding bids over certain intervals above the optimal reserve price. Bulow and Klemperer (1996) have shown that an auction with n bidders and an optimally chosen reserve price generates a smaller expected profit for the seller than a standard auction with n+1 bidders (and no reserve price).

In the Journal of Economic Literature Classification System C7 is the classification for game theory and D44 is the classification for auctions.

Scholars of managerial economics have noted some applications of auction theory in business strategy. Namely, auction theory can be applied to preemption games and attrition games. Preemption games are a game where entrepreneurs will preempt other firms in entering a market with new technology before it's ready for commercial deployment. The value generated from waiting for the technology to become commercially viable also increases the risk that a competitor will enter the market preemptively. Preemptive games can be modeled as a first priced sealed auction. Both companies would prefer to enter the market when the technology is ready for commercial deployment; this can be considered the valuation of both companies. However, one firm might hold information stating that technology is viable earlier than the other firm believes. The company with better information would, then, enter the market and bid to enter the market earlier, even as the risk of failure is higher. Games of attrition are games of preempting other firms to leave the market. This often occurs in the airline industry as these markets are considered highly contestable. As a new airline enters the market, they will decrease prices to gain market share. This forces the incumbent airline to also decrease prices to avoid losing market share. This creates an auction game. Usually, market entrants will use a strategy of attempting to bankrupt the incumbent. Thus, the auction is measured in how much each firm is willing to lose as they stay in the game of attrition. The firm that lasts the longest in the game wins the market share. This strategy has been used more contemporaneously by entertainment streaming services like Netflix, Hulu, Disney+ and HBOMax who are all loss making firms attempting to gain market share by bidding on more expansive entertainment content.

Two professors from Stanford University, Paul Milgrom and Robert Wilson, won the 2020 Nobel Prize in Economics for advancing auction theory by inventing several new auction formats, including simultaneous multiple round auctions (SMRA). This format combines the benefit of both English auctions (open outcry) and sealed bid auctions. SMRAs are deemed to solve the problem of the Federal Communications Commission (FCC). Due to the FCC wanting to sell all of its telecommunication frequency slots by using traditional auction, it will eventually turn out to give away the licenses for free or end up with a telecom monopoly in the United States. The process of simultaneous multiple round auctions is that there are three to four round auctions. Every bidder seals their bid; the auctioneer announces the highest bid to all bidders each round. All the bidders can adjust and change their auction price and strategy after they listen to the highest bid this round. The auction will continue until the highest bid this round is lower than the last round’s highest bid. SMRA's first distinguishing feature is that the auction is taking place simultaneously for different items, therefore, it seriously increases the cost of speculators, for the same reason sealed bidding can ensure that all bidding is reflecting the bidder’s valuation of the products. The second difference is that the bidding takes place in numerous rounds and the highest price of bidding is announced each round, allowing bidders to learn more about their competitors' preferences and information and adjust their strategy accordingly, thus decreasing the effect of asymmetric information inside the auction. In addition, multiple round bidding can maintain the bidder's activity in the auction. It has substantially increased the information the bidder has about the highest bid because every round after the bidding the host will announce the highest bid.