Auction theory
Auction theory is an applied branch of economics which deals with how bidders act in auction markets and researches how the features of auction markets 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.” Introduction. 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, the auction close, 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. 1994 Nobel Laureate for Economic Sciences, John Nash, designed a generalized theory of auctions as a non-cooperative game which moves beyond simple zero-sum games. This theory was vital for the theorisation of auctions since the goal  of auctions is to assign an object to the buyer who will make the most use of it for the highest price, thus maximising value for both buyer and seller. Nash developed a way for auctions to facilitate absolute gains for society. Vickrey (1996 Nobel Laureate) and Harsanyi (1994 Laureate) extended on Nash's equilibrium specifying ways in which equilibrium can be reached under informational settings. By the 1990s, auction theorists had defined equilibrium bidding conditions for single-object auctions under most realistic auction formats and information settings. The state-of-the-art considers how multiple-object auctions can be  performed efficiently; Robert B. Wilson and Paul Milgrom won The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2020 for their work in defining these auctions. Some other state-of-the-art auction designs are Product-Mix Auctions which allows 'package bidding' which was implemented by Paul Klemperer in response to the 2007 Northern Rock Bank Run to sell troubled debt, and Position Auctions which implement a generalized Second-price auction which Google has used to sell ads on Internet search keywords efficiently. Auction types. There are traditionally four types of auction that are used for the  allocation of a single item: Most auction theory revolves around these four "basic" auction types. However, others have also received some academic study (see ). Benchmark model. The "benchmark model" for auctions, as defined by McAfee and McMillan (1987), offers a generalization of auction formats, and is based on four assumptions: The benchmark model is often used in tandem with the "Revelation Principle", which states that each of the basic auction types is structured such that each bidder has the incentive to report their valuation honestly. The two are primarily used by sellers to determine the auction type that maximizes the  expected price. This optimal auction format is defined such that the item will be offered to the bidder with the highest valuation at a price equal to their valuation, but the seller will refuse to sell the item if they expect that all of the bidders' valuations of the item are less than their own. Relaxing each of the four main assumptions of the benchmark model yields auction formats with unique characteristics: Game-theoretic models. A game-theoretic auction model is a mathematical game represented by a set of players, a set of actions (strategies) available to each player, and  a payoff vector corresponding to each combination of strategies. Generally, the players are the buyer(s) and the seller(s). The action set of each player is a set of bid functions or reservation prices (reserves). Each bid function maps the player's value (in the case of a buyer) or cost (in the case of a seller) to a bid price. The payoff of each player under a combination of strategies is the expected utility (or expected profit) of that player under that combination of strategies. Game-theoretic models of auctions and strategic bidding generally fall into either of the following  two categories. In a private values model, each participant (bidder) assumes that each of the competing bidders obtains a random "private value" from a probability distribution. In a common value model, the participants have equal valuations of the item, but they do not have perfectly accurate information about this valuation. In lieu of knowing the exact valuation of the item, each participant can assume that any other participant obtains a random signal, which can be used to estimate the true valuation, from a probability distribution common to all bidders. 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 earlier published theoretical research addressing properties of auctions among asymmetric bidders is Keith Waehrer's 1999 article. Later published research include Susan Athey's 2001 Econometrica article, as well as Reny and Zamir (2004). The first formal analysis of auctions was by William Vickrey (1961 ). Vickrey considers two buyers bidding for a single item. Each buyer's value, v, is an independent draw from a uniform distribution with support [0,1]. Vickrey showed that in the sealed first-price auction it is an equilibrium bidding strategy for each bidder to bid half his valuation. With more bidders, all drawing a value from the same uniform distribution it is easy to show that the symmetric equilibrium bidding strategy is To check that this is an equilibrium bidding strategy we must show that if it is the strategy adopted by the other n-1 buyers, then it is a  best response for buyer 1 to adopt it also. Note that buyer 1 wins with probability 1 with a bid of (n-1)/n so we need only consider bids on the interval [0,(n-1)/n]. Suppose buyer 1 has value v and bids b. If buyer 2's value is x he bids B(x). Therefore, buyer 1 beats buyer 2 if Since x is uniformly distributed, buyer 1 bids higher than buyer 2 with probability nb/(n-1). To be the winning bidder, buyer 1 must bid higher than all the other bidders (which are bidding independently). Then his win probability is Buyer 1's  expected payoff is his win probability times his gain if he wins. That is, It is readily confirmed by differentiation that U(b) takes on its maximum at It is not difficult to show that B(v) is the unique symmetric equilibrium. Lebrun (1996) provides a general proof that there are no asymmetric equilibria. Revenue equivalence. 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). Relaxing these assumptions can provide valuable insights for auction design. Decision biases can also lead to predictable non-equivalencies. Additionally, if some bidders are known to have a higher valuation for the lot, techniques such as price-discriminating against such  bidders will yield higher returns. In other words, if a bidder is known to value the lot at $X more than the next highest bidder, the seller can increase their profits by charging that bidder $X – Δ (a sum just slightly inferior to the sum is willing to pay) more than any other bidder (or equivalently a special bidding fee of $X – Δ). This bidder will still win the lot, but will pay more than would otherwise be the case. Winner's curse. 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". In an equilibrium of such a game, the winner's curse does not occur because the bidders account for the bias in their bidding strategies. Behaviorally and empirically , however, winner's curse is a common phenomenon, described in detail by Richard Thaler. Optimal auctions. With identically and independent distributed private values, Riley and Samuelson (1981) 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). JEL classification. In the Journal of Economic  Literature Classification System C7 is the classification for game theory and D44 is the classification for auctions. Applications to business strategy. 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. Auction Theory won a Nobel Prize. Two professors from Stanford University Paul Milgrom and Robert Wilson have successfully improved the auction theory and inventions of new auction formats. They have announced Simultaneous multiple round auctions (SMRA), a new format of action theory. Which combined the benefit of English auction and sealed bid auction. Simultaneous multiple round auctions are deemed to solve the problem by Federal Communications Commission (FCC). Due to FCC wanting to sell all of its telecommunication frequency slots by using  traditional auction, however, it will eventually turn out to give away the licenses for free or end up with a telecom monopoly in the United States. Eventually, simultaneous multiple round auctions have generated $20 billion in US dollars profit for the US government. The process of Simulations multiple round auctions is there had three to four round auctions. Every bidder has to seal their auction price; the auctioneer has to announce the highest price auction to every bidder every round. All the bidders can adjust and change their auction price and strategy after they listen to the highest auction  price this round. The auction will maintain until the highest price of bidding price this round is lower than the last round’s highest bidding price. SMRA is the first distinguishing feature is that this auction is going place simultaneously for different frequencies, therefore, it seriously increases the cost of speculators, by the reason sealed bidding can assure all bidding is reflecting all bidder’s valuation of price to the products. The second difference is that bidding takes place in numerous rounds and announced the highest price of bidding, hence, 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 being active in the auction. It has substantially increased information for the bidder to understand the highest bidder’s price because every round after the bidding the host will announce the highest bidder's price.