This post describes and outlines the optimal strategies for three different auction types; ascending bid, first price, and second price auctions. A variety of auction types exist beyond the main three discussed in this post, each with its own optimal strategy. Conclusions are drawn from the similarities and differences between the models in order to arrive at a central theorem describing all auctions. Finally, applications for industry and government use are briefly reviewed, and a few methods in which auctions are modified are explored.
For the purposes of this post, an auction consists of a finite set of bidders and a single seller attempting to sell a single object. Each bidder has their own positive monetary value of the object. Note that these values will vary between bidders, and each bidder's value of the object remains constant throughout the auction. A bidder will only want to buy an object if the price is less than their value, thus providing the bidder with a net gain. An assumption is made that bidding ties will be resolved using a fair coin flip. With these assumptions we can model three types of auctions: ascending bid, second price, and first price. It is important to make the distinction that both first price and second price auctions are sealed auctions, where all bidders submit their bid in a hidden manner (i.e. an envelope), while an ascending bid auction is conducted out loud, synchronously.
An ascending bid auction is perhaps the most famous and commonly depicted auction method. It is performed over time in a sequence of periods. At each period the auctioneer will announce a price, and then each bidder chooses to remain in the auction or to leave. At each proceeding period the announced price increases. Bidders who do not intend to buy the object at the announced price will leave the auction. The object is awarded to the last remaining bidder at a price equivalent to the announced price during the period in which the last other bidder left the auction. In the event that all remaining bidders leave the auction during the same period, the object shall be randomly awarded to one of those bidders at a price equivalent to the announced price during the last period with remaining bidders.
A rational bidder will choose to stay in an ascending bid auction until the announced price is greater than or equal to their monetary value of the object. It would be suboptimal to leave the auction before such an announced price because the bidder may lose out on an opportunity to purchase the object for a price lower than their monetary value of the object. Such a bidder would miss an opportunity for a net gain. It is also suboptimal for a bidder to remain in an auction after the announced price is equal to or greater than their monetary value of the object. The bidder could win the object thus being forced to pay equal to or greater than their monetary value of the object creating a net loss (or no net change). The outcome in an ascending bid auction with all rational bidders is that the object will be awarded to the bidder with the highest monetary value at the price of the second highest bidder’s value.
In a second price auction each bidder submits a sealed bid. All bids are reviewed and the object is awarded to the highest bidder at the price of the second highest bid. In the event that highest bid is shared between multiple bidders, the object is randomly awarded to one of those bidders at that price. Note that in this special case the second highest bid is also the highest bid.
A rational bidder in a second price action will make a bid equal to their monetary value of the object (true value bid). By bidding in this manner, the rational bidder has guaranteed that the auction will result in one of the following outcomes: They do not win the auction, or they win the auction and pay a price that is equal to their monetary value of the object, or they win the auction and pay a price that is less than their monetary value of the object. In the first two possible outcomes the rational bidder creates no net change. The third possible outcome clearly results in a net gain. It would be suboptimal for a bidder to to submit a bid greater than their value of the object (overbid), because of the case where another bid exists between the bidder’s bid and the bidder’s value of the object. In the event that the bidder’s bid is the highest bid, the bidder would win the object and be forced to pay an amount greater than their value of the object, a net loss. Overbidding will never result in a net gain where true-value bidding would not have resulted in the same net gain. It would also be suboptimal for a bidder to bid below their value of the object (underbid). This may result in the bidder losing the auction to a bid that is above their bid, but below their value of the object, forfeiting a net gain. It is also true that underbidding will never result in a net gain where true-value bidding would not have resulted in the same net gain. Neither underbidding or true value bidding will ever result in a net loss. The outcome in a second price auction with all rational bidders is that the object will be awarded to the bidder with the highest value for the object at the price of the second highest bidder’s value.
In a first price auction each bidder submits a sealed bid. All bids are reviewed and the the object is awarded to the highest bidder at that price (highest bid). In the event that highest bid is shared between multiple bidders, the object is randomly awarded to one of those bidders at that price.
Unlike the other two auction types, in a first price auction a bidder’s optimal strategy depends on what they know (or don’t know) about the distribution of other bidders’ valuations. This is to say that a rational participant will base their own bid off of both their value of the object and how they perceive others to value the object. Suppose there is a first price auction between two people, each with a unique value for the object. Also suppose that both players know the other’s value. Note that this is not standard for a first price auction. In this special case, the lower value participant would never bid over (or equal to) his value. Knowing this, the higher value participant would optimally place a bid for the lower value. The higher value participant will be awarded the object and pay a price equal to the other participant’s value; the lower value.
Now suppose that the two participants do not know the other’s value, but instead assume that it is uniformly distributed from zero to a known maximum. While bidding one’s true value of the object yields the highest chance of winning the auction, it does so without ever providing net gain. The optimal strategy will provide the best tradeoff between maximizing the likelihood of winning and maximizing the net gain once won. In this case, hand-waving the mathematics, the optimal strategy is to bid half of one’s value of the object.
Extrapolating upon this simple two participant auction, (brace yourself for handwaving) it can be shown that for a first price auction where all bids are uniformly distributed the optimal strategy is to bid (1 - 1/N)*V, where N is the number of participants and V is one’s value for the object. The outcome of all participants abiding by this strategy is that the participant with the highest value wins the object at a price equal to the expected value of the second-highest. However, this is only the optimal strategy when all players are perfectly rational and the bids are uniformly distributed.
In an ascending bid auction, the winner is the last participant remaining at a price equivalent to the second highest value and pays that value. In a second price auction, the winner bids their true value and pays the amount of the second highest value. In a first price auction, the winner bids the expected value of the second highest value and pays that amount. In all three types of auction, the outcome remains the same in a game of rational players; the object will be awarded to the participant with the highest value for the object at the price of the second highest participant’s value. This brings about the Revenue Equivalence Theorem (RET). The RET states that all auction types result in the same revenue to the seller and the same net payoffs to the bidders, assuming the basic auction rules, all bidders are rational, and that the highest bidder always wins.
The discussion of the RET and optimal strategies all assume that all participants are rational. Depending on the type of auction, the optimal bidding strategy may be affected due to the presence of suboptimal strategies carried out by irrational participants. In an ascending bid auction and second price auction, the presence of irrational participants does not change the optimal bidding strategy. Bidding one’s true value (or in an ascending bid auction, up to the true value) is still the optimal strategy. No matter the other participants’ bidding strategies, the optimal strategy holds. This is known as a dominant strategy. In a first price auction, the presence of an irrational participant breaks down the assumption of a known bidding distribution. Thus the optimal first price auction strategy is not robust to irrational participants and therefore is not dominant.
The seller, who is separate from the auctioneer to prevent tampering, may attempt to manipulate the auction by introducing fake bidder accomplices. In an ascending price auction the fake bidders will try to arrive at a price between to two highest values. This will have the effect of shaving margin between the highest and second highest values by making a new inflated second highest value. Given that the highest bidder pays the second highest value, this results in greater revenue for the seller. However, this may lead to a fake bidder winning the auction. The same logic follows for a second price auction. The only difference lies in that the fake bidders must know the value of the two highest values. Thus, second price auctions are mostly robust to seller manipulation, while ascending bid auctions are less so. First price auctions are only partially robust to seller manipulation. By simply adding more bidders in the form of fake bidders, the seller has caused all rational bids to increase. Introducing irrational fake bidders, as shown above, also affects previously optimal bidding strategies, increasing the selling price.
Auctions are a common method for a seller to increase their revenue on certain objects compared with traditional retail. Auctions are common in the antiques, advertising, collectables, commodities, real estate, and second-hand goods businesses. While the ascending bid auction is the most used, many markets tend to use other auction types due to more flexible timing options. Auctions are often utilized by the government to sell decommissioned vehicles and large commodities. Famously, the government auctioned off the right to use airwaves for electromagnetic broadcasts. This allows companies to purchase the right to a specific frequencies for radio, television, internet, and cellular data.
The online advertising industry is well known for its use of auctions. Search engines along with online advertising services act as sellers of advertisements, while other companies act as the bidders. Commonly, online advertising works on a per-click or per-impression basis. The auction is needed to determine which companies will pay the most per click/impression of their ad. Many sites use a generalized second price auction. This is simply a second price auction with multiple winners. The highest bid wins first and pays the value of the second highest bid, the second highest bid wins second and pays the value of the third highest bid, and so on.
A variety of types of auctions exist outside of the three auctions highlighted in this report. Each auction type will have its own optimal strategy. Dutch auctions function as the opposite ascending bid auctions; announced prices start high and lower with each period. The winner is the first bidder willing to accept an announced price. The outcome and strategy of Dutch auctions are identical to first price auctions. Dutch auctions are often used to facilitate the quick turnover of goods. All-Pay auctions are auctions in which each bidder pays for their bid, not just the winning bids. These auctions are most often seen at charity events or for the study political lobbying. Bidding fee auctions, also known as penny auctions, function like an ascending bid auction that charge a fixed price for every bid placed. Sites that operate penny auctions have recently become popular on the internet. As a result of both the winners and losers paying, both penny auctions and all pay auctions generally have a lower winning price than other auctions for the same object.
A reserve price is common addition to many auction types. In an auction with a reserve, the object may not be sold if the winning bid is not greater than the reserve price. This allows the seller to guarantee a price to part with their object. Note that setting a reserve violates the assumption that the highest bid wins in the RET. It imposes another set of constraints which may affect a bidders optimal strategy.
This report describes and outlines the optimal strategies for three different auction types; ascending bid, first price, and second price. In an ascending bid auction, bidders choose whether to remain in or exit the auction as the auctioneer continues to announce higher prices. The object is awarded to the last remaining bidder at a price equivalent to the announced price during the period in which the last other bidder left the auction. First price and second price auctions are sealed auctions, with each bidder placing a single hidden bid. In a second price auction, the bids are reviewed and the the object is awarded to the highest bidder at the price of the second highest bid. In a first price auction, all bids are reviewed and the the object is awarded to the highest bidder at that price.
For ascending bid and second price auctions the dominant strategy is to bid one’s monetary value of the object. There is no dominant strategy for a first price auction. The optimal strategy for a first price auction varies depending on what a participant believes the distribution of bids will be. The optimal strategy will provide the best tradeoff between maximizing the likelihood of winning and maximizing the net gain once won. In all three types of auction, the outcome remains the same in a game of rational players; the object will be awarded to the participant with the highest value for the object at the price of the second highest participant’s value. This is described by the Revenue Equivalence Theorem.
A variety of auction types exist beyond the main three discussed in this report. Each auction type will have its own optimal strategy, robustness to irrational bidders, and robustness to seller manipulation. The ascending bid auction is the most common and most well-known, but many auction types are used depending on the situation. Auctions are commonly used in industry, most notably in online advertising. Online advertising services typically use a generalized second price auction to assess bids of dollars per click.