Auction theory

The auction theory is a specialized field of game theory and is part of the mechanism design theory. It deals with auctions as market instruments with explicit rules that determine the way how such allocation of resources and the resulting price based on bids by market participants.

• 5.1 Classical Mehrgutauktionen 5.1.1 Demand Reduction

• 5.2.1 Calculation of the optimal allocation
• 5.2.2 VCG mechanism
• 5.2.3 Round Auctions 5.2.3.1 Ausubel - Milgrom proxy auction

Object of investigation

As auction a mechanism for allocation of one or more goods is understood here in general. The preferences of the participants are their private willingness to pay or reservation prices for goods or goods bundle. One considers both direct mechanisms in which options for action of the players representing bids for goods or bundle of goods, as well as indirect, in which state the player, for example, indifference amounts based on specified goods prices. The auction result, ie the allocation and payments of money, depends solely on the Commandments. This has two important properties of auctions:

In contrast to the general mechanism design, it is assumed in the auction theory that the participants have quasi-linear utility functions, so that differences in preferences can be compensated by money payments. Furthermore, the auction rules are well known and the auction participants behave strategically, ie maximize their private benefit. The user preferences are represented as random variables, therefore, applied the mechanism design problem in a Bayesian game. The auction theory typically analyzes the results of auctions in balance. The behavior of human participants in auctions on the other hand studied in experimental economics. Under assumption he Incomplete information other result, as shown here, for example, results the winner's curse.

Auction procedures are thereby studied mainly in terms of two possible properties:

Modeling of the goods

• If a single indivisible good to be auctioned, one speaks of a Eingutauktion.
• Several similar (homogeneous ) goods are sold in a Mehrgutauktion.
• An auction for various (heterogeneous ) goods is called combinatorial auction.
• Finally we considered auctions for divisible goods.

Modeling of the utility functions

There are two different approaches to model the benefit of participants in the auction item.

In the model of private values ​​, the benefits for each participant is measured as the individual preference. This preference is modeled usually as a random variable. Are the random variables of the participants independently, there is the model of independent private benefit. In the standard model one assumes additionally that the bidders are symmetric, ie that their preferences are all subject to the same distribution.

If the benefit of participants depend on a common variable, it is called the model of shared values ​​(common values). Examples of applications for this model include the auctioning of a purse with unknown content or a license for the exploitation of resources in a certain territory. The theory modeled here information asymmetries between the parties as private signals that are correlated with the underlying common variable.

The theory also examines Mixed cases of these two categories. In the example, the license for resource extraction, although the value of the raw material to be lifted is same for all participants, but there may be companies that have an advantage in the specific technology or equipment with appropriate personnel.

Eingutauktionen

Standard auction types

Especially for Eingutauktionen there are a number of traditional auction types.

In the private values ​​model the easiest to analyze is the second- price sealed-bid auction with or Vickreyauktion. Here give the bidder are each independently a bid for the auction item from, of which the highest wins. The winner pays the price of the second highest bid. Here is to tenderers truthful bidding weakly dominant strategy and the auction is efficient.

The classic Erstpreisauktion with sealed-bid allows the explicit calculation of the equilibrium strategies. In the Nash equilibrium, the participants offer less than their private value (bid shading ), there are no dominant strategies. The Erstpreisauktion is efficient.

Other variants of pricing are conceivable, such as the third- price auction, or the so-called all -pay auction in which the bidder regardless of whether the contract is awarded, pay their bid value.

Other types of auctions provide an open bidding. Best-known basic form, the English auction, are sequentially and open at the rising bids. The bids of competing bidders can be understood as signals about their type here. An analysis of which requires a model with common- value elements. In the Dutch auction shows a count-down clock to the price; it stops as soon as a bid is placed, and the winner pays the price shown.

Revenue equivalence

An important result of auction theory is the revenue equivalence theorem on the ( revenue equivalence theorem ). In the event the auction of a single good in the model with private values ​​, it states the following:

Suppose the bidder types are independent and identically distributed, and the bidders are risk neutral. Suppose further, two auction designs meet the following requirements:

• Bidder with private value 0 have an expected utility of 0 from the auction participation.
• The allocation in equilibrium does not differ in the two auctions.

Then both auction designs lead to the same expected seller proceeds.

This is especially true for first-, second-, third -, and all- pay auctions, which all lead to the same expected sales proceeds.

Reservation Prices

A reservation price defined a minimum price for the contract. If this is not achieved, the estate remains with the seller. A reservation price affects the allocation in equilibrium. Auctions with differently chosen reservation prices lead to different revenue.

Calculation in the standard model

For the standard model, mean payment in the second- price auction with reservation price, a bidder with type and medium income can be calculated. Is the distribution function of the types. We write. Note that the random variable is the highest rank statistics of independent identically distributed random variables. Is the distribution and function of the density. It arises and proceeds for the seller

To determine the revenue -maximizing reservation price is determined the first order condition, is to enable and receives as a condition of redeeming optimal reservation price

From the revenue equivalence follows that the expected payment of Erstpreisauktion with matches. On the other hand, is obviously true for the equilibrium strategy that

And one obtains

Mehrgutauktionen

In Mehrgutauktionen one distinguishes models in which the auction of several indistinguishable copies of a good to be considered from those with heterogeneous goods.

Classic Mehrgutauktionen

Is the number of goods. The utility functions are written as demand vectors. The demand for bidders wherein the Incremental benefit for a k-th additional asset. One considers usually the case, falling for incremental benefits, ie assumes that applies.

As standard Mehrgutauktion refers auctions in which the allocation is efficient based on the bids, ie where the k highest bids ( chosen among all for all j and i) awarded the contract.

The second- price auction offers two possible generalizations: first, the uniform-price auction in which the highest rejected bid is chosen as the unit price and each bidder the unit price multiplied by the number of allocated to him freight paid.

Finally, one can apply the Vickrey -Clarke Groves mechanism to the case of Mehrgutauktion. This has to implement the property, truthful bidding in dominant strategies and be more efficient.

Demand reduction

The equilibrium strategy for the uniform price auction has the property that is offered truthfully for the first Good, for all other but when the commandment against the true benefit is reduced. For the auction with discriminatory prices, bids are reduced for all goods. This suggests that the unit auctions are inefficient with discriminatory prices, provided that bidders demand more goods.

Mehrgutauktionen with heterogeneous goods

Calculation of the optimal allocation

Be given a lot of goods. The utility function of bidder rated this bundle of goods and has the form.

To determine the efficient allocation here is the solution of an integer linear optimization problem is necessary:

Is the allocation function. expresses that the bundle B to bidder i is allocated. The first constraint states that each bidder gets only a bundle. The second constraint ensures that each good is allocated more than once. The objective function maximizes the total utility.

The problem is NP -complete.

VCG mechanism

Again implemented in the private values ​​model of Vickrey -Clarke - Groves mechanism efficiency in dominant strategies. The set of Ausubel and Milgrom characterizes the class of utility functions in which it is guaranteed that the outcome of the VCG mechanism is located in the core, ie is stable under coalition formation.

Around Auctions

For combinatorial auctions, there are a number of designs in which the final result is found in a series of rounds. In each round, the bidders will receive information in the form of a provisional allocation or prices, and may adjust their bids accordingly. A number of advantages are for round auctions called ( Cramton, Ascending Auctions, 2003):

Ausubel - Milgrom proxy auction

In this round auction, the bidders are given as feedback in each round bundle prices and rich as bid a list of the most attractive bundle for them on the basis of the given prices ( Indifferenzmenge ). Based on the round bids a provisional allocation is determined. Each bidder receives an element of indifference amount, the auction is over and it is the price offered paid. Otherwise, the price of the bundle in the indifference amounts of the unsuccessful tenderer will be increased by one increment.

Provide the bidder truthfully (that is, they give in each round, the correct amount of indifference ), the auction ends with a result which is partly in the core and on the other hand are the bidders in the sum of maximum profit among all the core elements.

Furthermore: