Auction theory

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

research object

An auction is generally understood here to be a mechanism for allocating one or more goods. The preferences of the participants are their private willingness to pay or reservation prices for goods or bundles of goods. Both direct mechanisms are considered, in which the players' options for action represent bids for goods or bundles of goods, as well as indirect mechanisms in which the players, for example, indicate indifference quantities with regard to given goods prices. The auction result, i.e. the allocation and the cash payments, depends solely on the bids. This results in two essential properties of auctions:

Anonymity : The auction result does not depend on the identity of the bidders, but is symmetrical with regard to the participants, and

Universality : The auction rules abstract from the specific properties of the good, i. H. the same type of auction can be used in a variety of markets.

In contrast to the general mechanism design, auction theory assumes that the participants have quasi-linear utility functions , i.e. that differences in preferences can be compensated for by cash payments. Furthermore, the auction rules are well known and the auction participants behave strategically , thus maximizing their private benefit. Participant preferences are represented as random quantities, so the mechanism design problem is applied to a Bayesian game . Auction theory typically analyzes the outcome of equilibrium auctions . In contrast, the behavior of human participants in auctions is examined in experimental economics . Assuming incomplete information results in results other than those shown here, e.g. B. the winner's curse .

Auction processes are mainly examined with regard to two possible properties:

Efficiency : The allocation should (in equilibrium) maximize the sum of the individual benefits. Since payment flows are only allowed between auction participants, they add up to zero and therefore do not play a role in efficiency. To optimize an auction design in terms of efficiency, payments are only defined as an incentive for the participants. Auction design with the aim of maximizing efficiency primarily plays a role if the public sector wants to allocate resources such as radio spectrum to participants in the private sector. One example of this was the auction of UMTS licenses in Germany in 2000. The classic economic theory is indifferent to payment flows within an economy and does not examine the long-term industrial economic effects of the auction result. ( Spectrum auction )

Revenue maximization : Here an excellent auction participant, the seller , is defined and the auction design is chosen so that its benefit is maximized. In addition to applications in the private sector, tenders are usually designed as revenue-maximizing auctions.

Modeling the goods

If a single, indivisible good is to be auctioned, one speaks of an auction .
Several similar (homogeneous) goods are auctioned in a multiple- goods auction.
An auction for different (heterogeneous) goods is called a combinatorial auction .
Finally, consider auctions for divisible goods .

Modeling the utility functions

A distinction is made between two approaches to model the benefit of the participants in the auction item.

In the model of private values , the benefit for each participant is an individual preference . This preference is usually modeled as a random variable . If the random variables of the participants are independent of one another, the model of private independent benefit results . In the standard model it is also assumed that the bidders are symmetrical, i.e. that is, their preferences are all equally distributed.

If the benefit of participants from a common variable depend Speak of the model of shared values ( common values ). Examples of applications for this model are the auctioning of a wallet with unknown content or a license for the extraction of raw materials in a certain territory. The theory here models information asymmetries between the participants as private signals that are correlated with the underlying common variable.

The theory also examines mixed cases from these two categories. In the example of the license to mine raw materials, the value of the raw material to be extracted is the same for all participants, but there may be companies that have an advantage in terms of being equipped with specific technology or suitable personnel.

Auctions

Standard auction types

There are a number of traditional auction types especially for auction auctions.

The easiest to analyze in the private values model is the second price auction with hidden bids or Vickrey auction . Here, the bidders submit a bid for the auction item independently of one another, of which the highest wins. The winner pays the price of the second highest bid . For the bidders, truthful bidding is the weak dominant strategy and the auction is efficient.

The classic first price auction with hidden bids 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 first price auction is also efficient.

Other pricing options are conceivable, such as the third-party price auction or the so-called all-pay auction, in which the bidders pay their bid value regardless of whether they win the bid.

Other forms of auction provide for open bids. The best-known basic form is the English auction , in which increasing bids are submitted sequentially and openly. The bids of competing bidders can be understood here as signals about their type. An analysis of this requires a model with common value elements. At the Dutch auction , a clock running backwards shows the price; it stops as soon as a bid is placed and the winner pays the price shown.

Revenue equivalence

An important result of the auction theory is the theorem on revenue equivalence. In the case of the auction of a single good in the model with private values, it says the following:

Suppose the types of bidders are independent and identically distributed, and the bidders are risk neutral. Also, assume that two auction designs meet the following requirements:

Bidders with a private value of 0 have an expected benefit of 0 from participating in the auction.
The allocation in equilibrium does not differ between the two auctions.

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

This applies in particular to first, second, third and all-pay auctions, which all lead to the same expected sales revenue.

Reservation prices

A reservation price defines a minimum price for the surcharge. If this is not achieved, the goods remain with the seller. A reservation price influences the allocation in equilibrium. Auctions with different reservation prices lead to different revenues.

Calculation in the standard model

For the standard model, the average payment in the second-price auction ${\ displaystyle r}$ can be calculated with the reservation price , a bidder with the type and the average proceeds. Let the distribution function be of type . We write . Note that the random size is the highest ranking statistic of the independently identically distributed random sizes . Let be the distribution and density functions of . It arises and for the seller proceeds ${\ displaystyle m ^ {II} (x, r)}$ ${\ displaystyle i}$ ${\ displaystyle t> r}$ ${\ displaystyle F}$ ${\ displaystyle t_ {j}}$ ${\ displaystyle t ^ {(1), n-1} = \ max _ {j \ neq i} t_ {j}}$ ${\ displaystyle t ^ {(1), n-1}}$ ${\ displaystyle n-1}$ ${\ displaystyle t_ {j} (j \ neq i)}$ ${\ displaystyle G}$ ${\ displaystyle g}$ ${\ displaystyle t ^ {(1), n-1}}$ ${\ displaystyle m ^ {II} (t, r) = rG (r) + \ int _ {r} ^ {t} yg (y) \, dy}$

{\ displaystyle {\ begin {aligned} R ^ {II} (r) & = nE (m ^ {II} (t, r)) \\ & = n \ int _ {r} ^ {\ infty} m ^ {II} (t, r) f (t) \, dt \\ & = nr (1-F (r)) G (r) \, + \, n \ int _ {r} ^ {\ infty} s (1-F (s)) g (s) \, ds \ end {aligned}}}

To determine the revenue-maximizing reservation price , you determine the first-order condition , adjust to and obtain the condition for the revenue-optimal reservation price ${\ displaystyle r}$ ${\ displaystyle {r ^ {*}} ^ {II} = {\ frac {1-F ({r ^ {*}} ^ {II})} {f ({r ^ {*}} ^ {II} )}}}$

It follows that the expected payment of Erstpreisauktion from the proceeds of equivalence with matches. On the other hand, it obviously applies to the equilibrium strategy that ${\ displaystyle m ^ {I} (t, r)}$ ${\ displaystyle m ^ {II} (t, r)}$ ${\ displaystyle \ beta ^ {I}}$

${\ displaystyle m ^ {I} (t, r) = G (t) \ beta ^ {I} (t)}$ and one receives ${\ displaystyle \ beta ^ {I} (t, r) = r {\ frac {G (r)} {G (t)}} + {\ frac {1} {G (t)}} \ int _ { r} ^ {t} yg (y) \, dy}$

Multi-lot auctions

In multi-lot auctions, a distinction is made between models in which the auction of several indistinguishable copies of a good are considered from those with heterogeneous goods.

Classic multi-lot auctions

Let be the number of goods. The utility functions are written as demand vectors. The demand for bidders is where the incremental utility is for a kth additional good. As a rule, one considers the case of decreasing incremental utility , so it is assumed that holds. ${\ displaystyle n}$ ${\ displaystyle i}$ ${\ displaystyle b ^ {i} = (b_ {1} ^ {i}, \ ldots, b_ {n} ^ {i})}$ ${\ displaystyle b_ {k} ^ {i}}$ ${\ displaystyle b_ {1} ^ {i} \ geq b_ {2} ^ {i} \ geq \ ldots \ geq b_ {k} ^ {i}}$

A standard multi-item auction is an auction in which the allocation is efficient on the basis of the bids submitted, i.e. H. in which the k highest bids (chosen from all for all j and i) are awarded the contract. ${\ displaystyle b_ {j} ^ {i}}$

As a generalization of the first price auction, the auction with discriminatory prices is ideal . Here a bidder pays the sum of his winning bids.

The second price auction offers two conceivable generalizations: on the one hand, the uniform price auction , in which the highest rejected bid is chosen as the uniform price and each bidder pays the uniform price multiplied by the number of goods allocated to him.

Finally, the Vickrey-Clarke-Groves mechanism can be applied to the case of multiple lots. This has the characteristic of implementing truthful bidding in dominant strategies and of being efficient.

Demand reduction

The equilibrium strategy for the uniform price auction has the property that a truthful bid is made for the first good , but for all further goods the bid is reduced compared to the real benefit. For the auction with discriminatory prices, the bids for all goods are reduced. From this it can be deduced that the unit auction and auction with discriminatory prices are inefficient if bidders ask for several goods.

Multi-lot auctions with heterogeneous goods

Calculation of the optimal allocation

Be given a multitude of goods. The utility function of the bidders assesses bundles of goods here and has the form . ${\ displaystyle G}$ ${\ displaystyle (v ^ {i} (B): B \ subseteq G)}$

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

${\ displaystyle {\ begin {aligned} \ max & \ sum _ {i \ in I} \ sum _ {B \ subseteq G} v ^ {i} (B) x_ {i} (B) \\ {\ text {so that}} \\\ sum _ {B \ subseteq G} x_ {i} (B) & \ leq 1 & {\ text {for all}} i \ in I \\\ sum _ {i \ in I} \ sum _ {B \ subseteq G: g \ in B} x_ {i} (B) & \ leq 1 & {\ text {for all}} g \ in G \\ x_ {i} (B) & \ in \ {0.1 \} \ end {aligned}}}$

${\ displaystyle x_ {i} (B)}$ is the allocation function. expresses that bundle B is allocated to bidder i. The first constraint is that each bidder only gets one bundle. The second secondary condition ensures that each good is allocated at most once. The objective function maximizes the total utility. ${\ displaystyle x_ {i} (B) = 1}$ ${\ displaystyle g \ in G}$

The problem is NP-complete .

VCG mechanism

Here, too, the Vickrey-Clarke-Groves mechanism implements efficiency in dominant strategies in the private values model. The theorem of Ausubel and Milgrom characterizes the class of utility functions in which it is guaranteed that the result of the VCG mechanism lies in the core , i.e. is stable under the formation of a coalition.

Round auctions

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

Bidders do not have to calculate their utility functions for all goods bundles, but can limit themselves to the bundles that are most attractive to them, taking into account the given feedback.

Bidders do not have to disclose their complete utility functions , but only disclose their preferences piece by piece, similar to the openly rising English auction for a single good.

On the other hand, the bidders can learn from each other with their valuations , which may play a special role if the value model contains common value elements.

Ausubel-Milgrom proxy auction

In this round auction, the bidder as feedback received in each round bundle of prices , ranging as bid a list of the most attractive for them bundle based on the given prices (Indifferenzmenge). A provisional allocation is determined on the basis of the round bids. If every bidder receives an element of the indifference quantity, the auction is over and the bid price is paid. Otherwise, the price for the bundles in the indifference quantities of the bidders not considered will be increased by one increment.

If the bidders bid truthfully (i.e. they state the correct amount of indifference in each round), the auction ends with a result that is on the one hand in the core and on the other hand gives the bidders maximum profit overall among all core elements.

The following also applies:

If all bidders have substitutive value functions, truthful play forms a Nash equilibrium.

The reverse applies: Assuming there are at least 4 bidders and the set of possible utility functions includes the set of additive utility functions . If bidder 1 has a non-substitutive utility function , additive utility functions can be constructed for bidders 2, 3 and 4 so that truthful play is not a Nash equilibrium.

{\ displaystyle {\ mathcal {V}}}

{\ displaystyle {\ mathcal {V}} _ {\ text {add}}}

{\ displaystyle v_ {1}}

{\ displaystyle v_ {2}, v_ {3}, v_ {4}}

literature

Lawrence M. Ausubel: Auctions (theory). In: Steven N. Durlauf and Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, Internet http://www.dictionaryofeconomics.com/article?id=pde2008_A000217 (online edition), doi : 10.1057 / 9780230226203.0073 .
Siegfried Berninghaus, Karl-Martin Ehrhart and Werner Güth: Strategic games. 2nd edition Springer, Berlin 2006, ISBN 978-3-540-28414-7 (online version: doi : 10.1007 / 3-540-28488-5 ).
Peter Cramton, Yoav Shoham and Richard Steinberg (Eds.): Combinatorial Auctions. MIT Press, Cambridge 2006, ISBN 0-262-03342-9 .
Paul Klemperer (Ed.): The Economic Theory of Auctions. E. Elgar, Cheltenham 2000, ISBN 1-85898-870-5 .
Paul Klemperer: Auctions. Theory and Practice. Princeton University Press, Princeton 2004, ISBN 0-691-11925-2 .
Vijay Krishna: Auction Theory. 2nd ed. Academic Press, Burlington u. a. 2010, ISBN 978-0-12-374507-1 .
Paul Milgrom : Putting Auction Theory to Work. Cambridge University Press, Cambridge 2004, ISBN 0-521-53672-3 .

Remarks

^ See R. Preston McAfee and John McMillan: Auctions and Bidding. In: Journal of Economic Literature. 25, No. 2, 1987, pp. 699-738 ( JSTOR 2726107 ), here p. 701.
↑ See Krishna 2010, p. 6.
↑ See Michael H. Rothkopf, Aleksandar Pekeč and Ronald M. Harstad: Computationally Manageable Combinational Auctions. In: Management Science. 44, No. 8, 1998, pp. 1131-1147 ( JSTOR , EBSCOhost ).
↑ Going back to Lawrence M. Ausubel and Paul R. Milgrom: Ascending Auctions with Package Bidding. In: Frontiers of Theoretical Economics. 1, No. 1, 2002, Internet http://www.ausubel.com/auction-papers/package-bidding-bepress.pdf , accessed on October 28, 2012; see. also Milgrom 2004, pp. 324-333.

[1] See R. Preston McAfee and John McMillan: Auctions and Bidding. In: Journal of Economic Literature. 25, No. 2, 1987, pp. 699-738 ( JSTOR 2726107 ), here p. 701.

[2] See Krishna 2010, p. 6.

[3] See Michael H. Rothkopf, Aleksandar Pekeč and Ronald M. Harstad: Computationally Manageable Combinational Auctions. In: Management Science. 44, No. 8, 1998, pp. 1131-1147 ( JSTOR , EBSCOhost ).

[4] Going back to Lawrence M. Ausubel and Paul R. Milgrom: Ascending Auctions with Package Bidding. In: Frontiers of Theoretical Economics. 1, No. 1, 2002, Internet http://www.ausubel.com/auction-papers/package-bidding-bepress.pdf , accessed on October 28, 2012; see. also Milgrom 2004, pp. 324-333.