Auction theory

Auction theory is a branch of applied economics that 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 confluence of the price between the buyer and seller is an economic equilibrium. Auction theorists design rules for auctions to address issues that can lead to market failure. The design of these rulesets encourages optimal bidding strategies in 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 have two aspects: 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 valuations by other bidders.

History

A purportedly historical event related to auctions is a custom in Babylonia, namely when men make an offers 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.
Non-cooperative games have a long history, beginning with Cournot's duopoly model. A 1994 Nobel Laureate for Economic Sciences, John Nash, proved a general-existence theorem for non-cooperative games, which moves beyond simple zero-sum games. This theory was generalized by Vickrey 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.

Auction types

There are traditionally four types of auctions that are used for the sale of a single item:

First-price sealed-bid auction in which bidders place their bids in sealed envelopes and simultaneously hand them to the auctioneer. The envelopes are opened and the individual with the highest bid wins, paying the amount bid. This form of auction requires strategic considerations since bidders must not only consider their own valuations but other bidders' possible valuations. The first formal analysis of such an auction was by Vickrey. For the case of two buyers and uniformly distributed values, he showed that the symmetric-equilibrium strategy was to submit a bid equal to half of the buyer's valuation.
Second-price sealed-bid auctions which are the same as first-price sealed-bid auctions except that the winner pays a price equal to the second-highest bid. The logic of this auction type is that the dominant strategy for all bidders is to bid their true valuation. William Vickrey was the first scholar to study second-price valuation auctions, but their use goes back in history, with some evidence suggesting that Goethe sold his manuscripts to a publisher using the second-price auction format. Online auctions often use an equivalent version of Vickrey's second-price auction wherein bidders provide proxy bids for items. A proxy bid is an amount an individual values some item at. The online auction house will bid up the price of the item until the proxy bid for the winner is at the top. However, the individual only has to pay one increment higher than the second-highest price, despite their own proxy valuation.
Open ascending-bid auctions are the oldest, and possibly most common, type of auction in which participants make increasingly higher bids, each stopping bidding when they are not prepared to pay more than the current highest bid. This continues until no participant is prepared to make a higher bid; the highest bidder wins the auction at the final amount bid. Sometimes the lot is sold only if the bidding reaches a reserve price set by the seller.
Open descending-bid auctions are those in which the price is set by the auctioneer at a level sufficiently high to deter all bidders, and is progressively lowered until a bidder is prepared to buy at the current price, winning the auction.

Most auction theory revolves around these four "basic" auction types. However, other types have also received some academic study. Developments in the world and in technology have also influenced the current auction system. With the existence of the internet, online auctions have become an option.

Online auctions are efficient platforms for establishing precise prices based on supply and demand. Furthermore, they can overcome geographic boundaries. Online auction sites are used for a variety of purposes, such as online "garage sales" by companies liquidating unwanted inventory. A significant difference between online auctions and traditional auctions is that bidders on the internet are unable to inspect the actual item, leading to differences between initial perception and reality.
Auction process

There are six basic activities that complement the auction-based trading process:

Initial buyer and seller registration: authentication of trading parties, exchange of cryptography keys when the auction is online, and profile creation.
Setting up a particular auction event: describing items sold or acquired and establishing auction rules. Auction rules define the type of auction, starting date, closing rules, and other parameters.
Scheduling and advertising, as well as grouping of items of the same category to be auctioned together, is done to attract potential buyers. Popular auctions can be combined with less-popular auctions to persuade people to attend the less popular ones.
Bidding step: bids are collected and bid control rules of the auction are implemented.
Evaluation of bids and closing the auction: winners and losers are declared.
Trade settlement: payment to seller, transfer of goods, fees to agents.
Auction envelope theorem

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

Benchmark model

The benchmark model for auctions, as defined by McAfee and McMillan, is as follows:

All of the bidders are risk-neutral.
Each bidder has a private valuation for the item, which is almost always independently drawn from some probability distribution.
The bidders possess symmetric information.
The payment is represented only as a function of the bids.
Win probability

In an auction a buyer bidding wins if the opposing bidders make lower bids.
The mapping from valuations to bids is strictly increasing; the high-valuation bidder therefore wins.
In statistics the probability of having the "first" valuation is written as:
With independent valuations and N other bidders

The auction

A buyer's payoff is

Let be the bid that maximizes the buyer's payoff.
Therefore

The equilibrium payoff is therefore

Necessary condition for the maximum:
when
The final step is to take the total derivative of the equilibrium payoff

The second term is zero. Therefore

Then

Example uniform distribution with two buyers. For the uniform distribution the probability if having a higher value that one other buyer is.
Then
The equilibrium payoff is therefore.
The win probability is.

Then
.
Rearranging this expression,
With three buyers, , then
With buyers
Lebrun provides a general proof that there are no asymmetric equilibriums.

Optimal auctions

Auctions from a buyer's perspective

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. For example, there might be a rule that all buyers pay a nonrefundable bid. 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.

Symmetric auctions with correlated valuation distributions

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

Auctions from the seller's perspective (revenue maximization)

Quite independently and soon after, used the revelation principle to characterize revenue-maximizing sealed high-bid auctions. 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 sub-intervals.
Relaxing each of the four main assumptions of the benchmark model yields auction formats with unique characteristics.

Risk-averse bidders incur some kind of cost from participating in risky behaviours, which affects their valuation of a product. In sealed-bid first-price auctions, risk-averse bidders are more willing to bid more to increase their probability of winning, which, in turn, increases the bid's utility. This allows sealed-bid first-price auctions to produce higher expected revenue than English and sealed-bid second-price auctions.
In formats with correlated values—where the bidders' valuations of the item are not independent—one of the bidders, perceiving their valuation of the item to be high, makes it more likely that the other bidders will perceive their own valuations to be high. A notable example of this instance is the winner’s curse, where the results of the auction convey to the winner that everyone else estimated the value of the item to be less than they did. Additionally, the linkage principle allows revenue comparisons amongst a fairly general class of auctions with interdependence between bidders' values.
The asymmetric model assumes that bidders are separated into two classes that draw valuations from different distributions.
In formats with royalties or incentive payments, the seller incorporates additional factors, especially those that affect the true value of the item, into the price function.

The theory of efficient trading processes developed in a static framework relies heavily on the premise of non-repetition. For example, an auction-seller-optimal design involves the best lowest price that exceeds both the seller's valuation and the lowest possible buyer's valuation.