Vickrey auction

As a (hidden ) second price auction (English second- price [ sealed bid ] auction) is referred to in the auction theory, a form of an auction, in which although the highest bidder is awarded the contract, but in the end only has to pay the second highest bid. The bids will be submitted once each so that they are not known to the other bidders ( " hidden ", as in the case of disposal in an envelope that will be opened only after the end of the bidding process ). After their theoretical founder, Nobel laureate William Vickrey, second- price auctions is referred to as Vickreyauktionen.

Second- price auctions are to be distinguished from the so-called Erstpreisauktionen, although they use the same format and an identical allocation mechanism ( the highest bidder wins), in which the winner but also need to pay the discharged his own bid.

• 4.1 Assumptions and related to IPV model
• 4.2 Properties 4.2.1 balance
• 4.2.2 Revenue and lack of revenue equivalence
• 4.2.3 Example

• 5.1 theory
• 5.2 Experimental results in the IPV case

• 7.1 Vickrey -Clarke - Groves mechanism
• 7.2 Generalized Second -Price Auction

Historical classification

The first formal analysis of a second- price auction provides Vickrey (1961 ), who constructed the second- price auction, on a theoretical level, because they (always leave under certain conditions identical results as the well-known English auction in which bidders successively higher bids until it finally offers no other bidders more ) produces. Vickrey was henceforth also repeatedly as the "inventor " of the auction format simply postulated overlooking the apparent lack of previous examples of this auction type.

With considerable time lag even earlier examples of the use of the format were led in practice in the literature, however. Lucking - Reiley (2000 ) shows to be around for the stamp market that are there came long before Vickreys work second price auctions are used. Having had approached already in the 1870s in the concentrated in New York collector market at least one retailer to the format of the second- price auction, by complementing the hitherto usual English Format Recalling high cost of getting foreign collectors to simple ways to submit one-off pre- bids had dated Lucking - Reiley, the first of it in this market identified full-fledged second- price auction to the year 1883. fact be on the ( U.S. ) market for stamps since the 1930s, even in a majority second- price auctions have been used.

Moldovanu and Rabin (1998 ) have anecdotally indicated that already in correspondence of Johann Wolfgang von Goethe is found from 1797 is an example of a special form of second- price auction. In a letter to the publisher Friedrich Vieweg Goethe describes the procedure to sell a manuscript as follows:

Analysis framework

More generally, it is at auctions mechanisms that make it possible to assign one or more objects of a certain number of bidders. The extent of " bidders " are mentioned, they do not necessarily represent the buyer's side - just think of an auction in which offer multiple sellers a works contract which falls to the lowest bidder -, however this serves in practice as theory as a reference case: a seller offers an object and the one bidder who submits the highest bid for that, it gets. This role distribution also follows the present article.

Second- price auctions can be distinguished according to the nature of the individual value estimates, raise the bidders regarding the object. In this sense, we can identify two distinct basic formats: In an auction with private valuations all bidders know their own appreciation for the object with certainty. The valuations of other bidders, however, they do not know with certainty and their own appreciation of the object would not change even if they knew it. In auctions with interdependent valuations is the bidders own appreciation, however, not known with certainty; rather, they will appreciate this even only by certain signals that are correlated with the true appreciation. Would they submit the information (signals) to know the other bidders, also their own appreciation would possibly change. Because the analysis under these two formats in some cases significantly different, these cases are discussed below in separate sections each.

The structured and clearly defined structure of auctions suggests to analyze them by means of game-theoretic methods. Due to the uncertainty about the value estimate (s ) is at auctions ( in particular: second price auctions) to games with incomplete information: There is always at least one player who does not know the payoff function of at least one other player. These kinds of games can be played in the tradition of Harsanyi (1967, 1968) modeled as games with imperfect information. This approach opens a structured way for the formal characterization of a ( second- price) auction based on a number of components:

Private value estimates

Assumptions

In the following, it is assumed that a single-stage second- price auction, which the assumptions of the so-called IPV model is sufficient ( to English independent private values ​​, eg " independent private values' ). The hypothetical sequence of such auctions can be described as follows:

In the basic model outlined here, the auction always relates it to an object. The bidders are risk- neutral, ie they maximize their expected payoff in their actions, and are not subject to budget constraint. Furthermore, they are able in the case of a profit to pay the costs incurred. Finally, each player, apart from the realizations of other players all the aforesaid known characteristics, and thus particularly the distribution function F. In the Extensions section, some of these assumptions are abandoned.

Properties

The Payoff function in a second- price auction, the shape is described by

Given. In words: If a bidder Highest Bidder, as corresponds to his payoff to the difference between the value that the object has for him, and the cost (ie the second highest bid ) incurred by him; he is not the highest bidder, resulting in a payoff of zero. The case is often resolved by randomization (so that each of the h highest bidder wins with probability 1 / h), or equivalently, for simplification, by the commandments initially numbered and are level on the one can win with the highest / lowest number.

( Vickrey 1961 :) The placing a bid in the amount of self-worth is a weakly dominant strategy in a second price auction.

To see this, consider one that it can not be optimal to offer any amount.

• A bid is not optimal. I Provides instead with suitable, he gets the object still ( and at the same price ) if, but new addition, even if, which is also assumed to be a positive payoff can be realized.

• A bid is not optimal. I Provides instead with suitable, he gets the object still ( and at the same price ) if, but new no longer if, but which would have been assumed to be a negative payoff realized.

Consequently, should be given neither below nor above the value estimate precisely because they never higher, but sometimes realizes a lower payoff. The statement shall otherwise independent of the above distributional assumptions ( iid); it depends only on the assumption of private value estimates. Second- price auctions are therefore incentive compatible.

The following charts illustrate the optimality of appreciation according bidding. Shown is the payoff of i ( vertical axis) is a function of the highest not own bid (horizontal axis), where the three situations that one's own precept of self-worth equivalent (case 1), including (case 2 ) or above (case 3) lies. The payoff function is always there, drawn in red, where the player would be victorious with his bid. It can be seen easily that in the case of appreciation of the proper bidding maximum expected payoff is achieved.

However, the identified here weak dominant strategy induces only one of many Bayesian Nash equilibria. For example, let and be the appreciation of bidders drawn from a uniform distribution over. Then and characterize ( and a whole continuum of other strategies ) is also an equilibrium, even if not "perfect" in the sense of Selten ( 1975) (so-called Trembling -hand perfect equilibrium). Rather, such equilibria are unstable: If a player plays only a little probability mass that his opponent makes a mistake, breaking the equilibrium collapses. If bidder 2 believes that bidder 1 with high probability ( and not, for sure ) 3 and with low probability provides a different amount, then bidder 2 would have a (slight ) chance to win the auction with strictly positive payoff if he is more than 0 provides, and if the bid is not above his appreciation, he has to worry also no losses. A general characterization of all equilibria of a second- price auction - even for the case discussed in detail below asymmetric bidders - deliver flower and Heidhues ( 2004).

We call a one -object auction to be efficient when the object is allocated to the tenderer who it among all bidders ex post the highest appreciation is applied - regardless of the price that is paid. This is obviously the case with the outlined second price auction.

If one arranges the value estimates (more precisely, the random variables of the value estimates) of all bidders except i, in descending order, we obtain a ranking with. This is called a k-th order statistic estimates the value of all bidders except i The first order statistics, follow a distribution G with continuous density. This distribution can also be quantified: Because each individual appreciation is drawn independently from the same distribution function F, is the IPV framework.

The expected cost of bidder i be using this terminology

Where for the second equality we use the fact that is offered in balance in the amount of appreciation. After the revenue equivalence theorem, this term corresponds to the expected cost at a Erstpreisauktion as well as in a whole class of other auction formats.

From the perspective of the seller, it is important to maximize the expected proceeds. This is of course just around the n-fold expectation value of individual bidders cost. This expected value again is known, in fact, because of the distribution in accordance with the model assumptions is unknown. It is therefore

Reference being made for the derivation of the last equation to a footnote.

Extensions

Minimum prices

While maintaining the above described IPV framework can examine yourself, what changes have arisen at the basic model when minimum prices (also: reservation prices ) introduces in the bidding process. Let r about a minimum price, which applies to all bidders. If the own appreciation below the minimum price, it is a weakly dominant strategy, dont bid (or, equivalently here, to offer zero ), because never a positive payoff can be realized. If the appreciation about it, so it is analogous to the above considerations in IPV basic model again weakly dominant to bid in the amount of self-worth. The weakly dominant strategy is therefore

The total expected cost of a bidder with consist of the probability-weighted costs which arise when the second highest bid is below the reserve price, ( first summand ) and the expected value of the costs which arise when the second highest bid is above the reserve price ( second summand ):

With.

Analogous to the above consideration at no minimum price closes at present the question of viewing the auction from the seller's perspective. The expected revenue of the seller is according to the n-fold expected value of the individual cost, and therefore so

Where the last equality follows by integration by parts. You Involves that the object typically also for the seller a certain - here designated - holds value, follows the expected revenue obtained the last term is based on the idea that the seller always not sold ( and therefore implemented ) if no bidder offers above the minimum price. Solving the resulting optimization problem, we get the following result, which is reproduced here without derivation:

( Riley and Samuelson 1980, Laffont and Maskin 1980 :) Be distributes the appreciation of all independent and identically and the bidders are risk neutral. Then the ex ante revenue-maximizing reserve price is independent of the number of bidders and we have:

Note that second- price auctions with minimum prices are no longer necessarily efficient. Is about and is a strictly positive minimum price charged, so there is no bidder with positive probability, which provides at least the minimum price, but at the same time a bidder with a strictly positive value estimate below the minimum price. The auction is therefore inefficient. In addition, the case described illustrates a commitment problem. While it is redemptive maximizing for the seller ex ante, to set the minimum price, it is ex post sometimes just no longer redeeming maximizing: If the seller states that all bidders have bid less than the reserve price, he could still realize a positive revenue by he nonetheless still sells the property to the highest bidder. There exists, in other words, a profitable renegotiation. This in turn results in a credibility problem that can incentivize bidders to submit lower bids.

In reality, the imposition of minimum prices is not observed regularly, though this strategy according to the above consideration, the waiver of a minimum price dominates weak. The reason for this may exist in unrecognized costs incurred by a bidder before he brings his experience in appreciation ( for example, travel expenses incurred in order to take the object in inspection ). Engelbrecht - Wiggans (1987 ) shows for the case of bidders, all of whom have prior to the acceptance of the said costs have identical information about the likely value of the object, an equilibrium in pure strategies (deterministic equilibrium) in which the revenue -optimizing minimum price of reference case is different and the imposition of a minimum price can reduce the expected proceeds. Levin and Smith ( 1994) model the decision-making process of potential bidders and explicitly construct an equilibrium in mixed strategies ( stochastic equilibrium), in the setting of a minimum price also has a negative effect on the proceeds.

Budget constraints

Game theory initially changed in the analysis of second- price auctions with budget constraints compared to the case without the possibility of type Quantity: uncertainty not only about the appreciation of other players, but also on their available budget. We denote with the budget of i It is now, that type option set of player i in the case of limited budgets with typical element, the set of possible budget is constructed analogously to the set of possible value estimates. The types are still independent and identically distributed according to a density function; the bidding function will be denoted by.

The payoff structure loud as follows:

It differs from the IPA's basic model by the fact that in the case of a victory not incurred costs in the amount of the second highest bid in each case. Provides a player that is more than is available to him, he finds a negative payoff.

Note first that each bid is a weakly dominated strategy. If i order wins and the second highest bid is above his budget, he realized a negative payoff; he wins and the second highest bid at or below his budget, he would have won with a bid in the amount of its budget. It is shown by further consideration that by

The weakly dominant strategy is given: If the budget constraint is not binding and is therefore a weakly dominant strategy to offer in the amount of appreciation (see the above reflections on the IPA base model). If it is analogous to the weakly dominant consideration in projecting sales to offer in the amount of the budget.

Looking at the payoff functions, is illustrated the result. Shows you the trivial case of a non -binding budget condition and is limited to the case, then shows below to Case 1 that a bid on the budget constraint always with positive probability implies a negative payoff. A bid below the budget constraint is again not optimal, because you thereby with a corresponding amount of the second highest bid waives the profit of the object ( with a positive payoff ).

Risk aversion

Maximize bidder no longer their expected payoff (risk neutrality ), but subject to a degree of risk aversion ( risk aversion ), the change in auction formats often their optimal bidding strategies. In a second- price auction in the basic model outlined IPV not there is a corresponding influence on the other hand; the arguments that have been given above to show the weak dominance appreciation proper bidding will not be affected by the presence of risk aversion. The expected proceeds thus does not change. The revenue equivalence for Erstpreisauktion is nevertheless no longer assured because changing the local strategy choice.

Asymmetry

One of the key assumptions in the standard IPA Framework ( and one of the conditions for the validity of the revenue equivalence ) is that the value estimates are trains from the same distribution. This must however not be the case in reality. An example of this is that an art dealer needs a work to complete his collection and thus a synergistic effect, which is for a "normal " Bidder without relevance. The question of the nature and the number of equilibria in the case of asymmetry, however, not trivial and generally depends on the respective distributions from. A special complication arises through renegotiations in the form of re-selling the property. In this case, the object of winning bidder after the auction has sold to another bidder.

Within the IPV framework, as mentioned, it is a weakly dominant strategy to offer in the amount of self-worth, and the auction has furthermore an efficient equilibrium. These two results derived above are also valid for the case that bidders are asymmetric. Hafalir and Krishna (2008) show for the case of two bidders with strictly monotonically increasing and continuous distribution functions and that appreciation contemporary bid even represents a perfect Bayesian Nash equilibrium.

Even less clear are so far the revenue implications of asymmetry. Cantillon (2008 ) shows for n bidders that the distribution of proceeds of an asymmetric second- price auction with distributions of the symmetric benchmark auction with will be stochastically dominated in the first order and that the aggregation of ex-ante payoffs from the asymmetric auctions are always higher than those in the symmetric benchmark auction. Chen and Xu (2012 ) indicate, however, be borne in mind that this result does not seem to be particularly robust. Constructed to other, equally plausible symmetric benchmarks, the result can be reversed, so that the seller proceeds from an asymmetric second- price auction may be higher under conditions other than in a symmetrical.

For comparison between first and second- price auction show Hafalir and Krishna (2008) for asymmetric distributions that satisfy Myersons regularity condition that the proceeds exceeding that of a Erstpreisauktion with repeat purchase of a second- price auction with resale. Kirkegaard (2012), in generalization of results in Maskin and Riley (2000 ) shows for the two- bidder case, that the proceeds from a Erstpreisauktion in an asymmetric environment without re- purchasing options is higher than the long as the distribution of out of a second -price auction, is "strong" bidder flat and a higher dispersion than the "weak " bidder.

Collusion

Like other auction formats are also second- price auctions for collusion -prone. In a Kollusionsumfeld a subset of the bidders, with cardinality R closes together to form a so-called bidding ring to keep the price of the object as small as possible. This is achieved by means that not all members of the ring offer individually, but only the ring as a whole makes a bid, so that a total instead of n bidders only n 1 -R offer bidders. ( In practice, this is realized so that all non- winning bidders submit bids in the ring below the floor price or invalid bid. ) In a second- price auction can be as one proceeds from the sale of reducing market distortion can be achieved. If the second highest bidder in the ring namely a greater appreciation for the object is applied as each bidder outside of the ring, so the victorious ring (or the member with the highest appreciation ) will pay a lower price than if there had been no collusion.

From the macro point of view changes in the described environment strategically initially not much. For the ring, it is still weak dominant, in the amount of his " appreciation " ( that is the appreciation of that with the highest such ) has to offer, and, as it is weakly dominant for all bidders outside the ring to provide the level of their appreciation. Moreover, the probability of winning each tenderer is untouched outside the ring of its existence. It can be, outgoing even show that the expected cost of nichtkolludierenden tenderers shall not be affected by the existence of a ring. The problem to study the allocation in the ring itself to identify ie the bidder with the highest esteem, the members need to be incentivized, their appreciation truthfully opposed to a " center " ( center) of the ring ( which coordinates the operations within the cartel ) disclose. For this purpose, there is the so-called (second -price ) pre -auction knockout ( PACT ) ( Graham and Marshall 1987), in fact, an incentive- compatible algorithm. The implementation by PACT is as follows:

The following results on the effects of a bidding ring in a second- price auction are reproduced here without derivation.

(Graham and Marshall 1987 :)

Assessment

The fact that should be offered exactly at the level of self-worth, simplifies the bidding process from the perspective of the bidder, because establishing its own tender offer, it does not require any distributional assumptions about the other bidders bid heights. Compared to that in the case of private valuations strategically equivalent ascending auction (also: English auction ) offers the second- price auction has the advantage that it is not necessary for tenderers to prolonged bidding process to attend ( " bid " ), since this stage in a sealed bid format is already completed by the unique bidding. For example, it was in the format of the English auction for a payoff -optimizing customers originally used on the Internet auction site Ebay necessary to ensure about on the appointment of an agent, special software or own presence that one out actually the end of the auction is actively involved in the bidding process; this effort for buyers presented here format.

Rothkopf, Teisberg and Kahn ( 1990) give two reasons why second- price auctions are rarely encountered in practice. On the one hand this is due to the fact that the bidders to worry about being cheated. As the winning bidder pays the second highest bid, it depends on the seller / auctioneer this also correctly estimated and not even inserts a unique bid to drive the price up. Second, it is precisely the fact that it is optimal to disclose its own appreciation, discouraging bidders. A participant in a second price auction has regularly not to inform a significant incentive, other bidder or a seller on the actual appreciation. To think is to situations where auction results are renegotiated; here would potentially about the bargaining power of a company suffering because previously in effect, the cost structure has been disclosed by the strategy compliant bid. This point can be traced in practice on the aforementioned collectors' market for stamps. Remedy may provide technological innovations that make the pricing process more transparent and less prone to manipulation.

Interdependent Valuations

And assumptions related to IPV model

The assumptions of the IPV model appear in practice often not very plausible, because the own appreciation is affected by the appreciation of other bidders. This may stem from personal indecision, however, resulted sometimes from the uncertainty of the resale price - white, a bidder about the low value estimates of its bidders, it will reduce its expected resale proceeds, which may affect his bidding; yet another possibility is that a portion of the consumption object is implemented as application consumption. Such a scenario leads to a model with interdependent valuations, and indeed interdependent in the sense that it would change the appreciation of a bidder, if he would know the value estimate ( some of) the other bidders. Formally, this is described so that each bidder is notified a signal ( in IPV basic model that was then directly his appreciation ); individual appreciation then results as a function of all signals,

This approach allows for two special cases: the IPV basic model, if one

Sets, and on the other hand, the common value model, if for all i, if so, in words, each bidder the object an identical (but unknown ) appreciation brings to.

Receives another bidder a higher signal, the own appreciation should always be at least as high as they would be at a lower signal of the other bidder. In the case of own signal even applies, that a higher such is always associated with a strictly higher appreciation. So Hence we have

Is also twice continuously differentiable. Signals are constructed analogously to the appreciation in the IPV case; it is with the maximum signal value. To allow interdependencies, sums put together the signals to a - valued random variable; which in turn follows a distribution F with density function f, the symmetry assumption is thereby incorporated into the model that is assumed for f that it is symmatrisch in his arguments, in the following sense: applies for each n-ary permutation that, and this for all. So it's irrelevant to the probability of a particular signal configuration which bidder receives that signal. Symmetry also applies to the appreciation structure of the bidders. To do this, define first with. u is symmetric in the arguments, the last (i.e., they can be reversed as desired without affecting the function value of u ). Furthermore, it is.

Finally, the signals " affiliated " ( affiliated ) such that a higher signal at another bidder increases the probability that the own signal is also higher (positive affiliation ). For the formal definition, please refer to a footnote. Note that this in no way implies that a player knows all the other signals.

Properties

Balance

Denote

- With the random variable of the highest signal of all other bidders except i - the conditional expectation of appreciation of bidder i when he hears the signal and when the second highest signal of all remaining bidders is given by. Finally, applies that

For everyone.

Milgrom and Weber ( 1982): The symmetric equilibrium of a second price auction with interdependent valuations is through

Given.

That is the ( Bayesian ) equilibrium offer all players as if they knew that the highest signal of Mitbietenden corresponds to its own signal.

Notably, if all bidders follow the strategy described and i would receive the signal, as was his expected payoff for a bid of b

By assumption and, consequently, the payoff is maximized when. Intuitive, one increases the upper limit of integration from 0 continues, the value of the integral will always first further (because the integrand is positive); as soon as it exceeds, the integral value but decreased again (because the integrand is negative then ).

Revenue and lack of revenue equivalence

The equilibrium bidding strategy is the expected revenue of the seller

With the conditional distribution given by and the corresponding density.

Using the Affiliations property can further be shown that the expected revenue from a second price auction, at least as high as that in a Erstpreisauktion. In addition, it is at most as high as in the English auction. There is thus no revenue equivalence. Intuitively, one can clearly make the difference between second- price and English auctions in particular. For bidders view one can only achieve a pension with affiliated signals described in a second- price auction, by taking advantage of the private information available. The stronger the costs now depend on the private information of the other players, the closer they are related to the information of the winner together ( because the signals are affiliated ). This is an English auction (where the cost depends on the information of all the other bidders ) is more the case than in a second- price auction ( in which the costs of the information depend another bidder ).

Example

The following Klemperer (1999) borrowed example illustrates the conflicting considerations.

Be the number of bidders. Each bidder i receives a signal that is drawn independently from a uniform distribution over, where v is the value of the object. Now assume that the bidders ex ante have no knowledge of v, so that all values ​​are equally likely classified as of v. This implies that a higher value of the probability of a higher value of V increases, and thus also requires a higher signal value of the other bidder likely affiliations which the property is satisfied. In a second- price auction in equilibrium provides each bidder equal to the conditional expectation of appreciation, given his own signal and given that the highest signal of the other bidders is the same as this. So you take out the other winning bidders from consideration, i assumes that he is the one with the highest signal is distributed under the same trains from the interval.

Now Consider, first, that is always present in a uniform distribution over the expected k- highest value of n independent trains through. This is nothing else than the k-th order statistic, ie for. The result for the expectation formation of i: i expected average and accordingly offers (Equation 1).

What is the expected revenue? Average, the bidder with the second highest signal, the signal ( equation 2). That is, on average, is the bid (Equation 2 into Equation 1) - the average income from the auction.

The winner's curse

Auctions with interdependent valuations are - unlike those with private value estimates - prone to perceptual distortions. Bidders at the time of bidding an estimate of the value of the object. Even if the respective estimates are unbiased, bidders must comply with the information value of their (potential) own victory involve: Who will win the auction, will have had one of the highest appreciation expectations regarding the object, which in turn should raise doubts about their own estimation arise. The non- inclusion of this information in its own bid choice can result in substantial Payoffeinbußen by itself. We call this form of adverse selection as the " winner's curse " ( winner's curse ).

The existence of the winner's curse is usually analyzed in experimental work based on the extreme case of a common- value auction, and there is also in second price auctions, a largely reliable result.

Kagel, Levin and Harstad ( 1995) through a series of second- price auctions in which the study participants are not informed of the actual value of the object. ( " Object" is not material to understand in this context;. Rather meant is the claim of the winner to get paid also the alleged ( monetary) value at the end ) Tenderers shall be notified private signals, each consisting of a uniform distribution over an interval be drawn about the actual value of around randomly. In later rounds, they were also given the lowest signal of the opponent known beyond, then in an environment with four or five bidders a non-significant increase in the average earnings of bidders results, however, strongly decreases the average gain in six or seven bidders ( and consistently is negative ). Because at the same time, there is evidence that players do not change their bidding function at different bidders figures, the authors conclude that this observation is probably due to the influence of the ( two or more players naturally more pronounced ) Curse of the winner.

Avery and Kagel (1997) analyze the bidding behavior in a second- price auction, in which the value of the object is measured as the sum of two independent, uniformly distributed random variables and over. Each bidder is notified of the realization of one of the signals (or). The equilibrium strategy in this setting would be given by or, but the actual amount of the bid clearly resembles the naive estimator in the form of the unconditional expected value of the object value. In addition, at low signals realize almost all winning bidder losses. The phenomenon diminishes with increasing experience of the bidder.

The implicit declaration of the winner's curse is usually the inability of bidders to adjust their assumptions about the object value from the information value of their victory. On the other hand, proposals have been put forward, rationalize the phenomenon of the winner's curse in the recent literature. Crawford and Iriberri (2007 ) is about designing a model that explains the reactions in an auction with the level -k approach so-called (Steel and Wilson 1995, Nagel 1995). The behavior of the tenderer is drawn from a common distribution of any " decision types " that differ from each other in their degree of proficiency (L0 - types behave as naive and not strategic). L1 types giving it with their bid the best answer to L0 types, L2 types, the best response to L1 - types and so on. The authors come to the conclusion that a specific level -k structure in a second- price auction can produce a winner's curse.

Eyster and Rabin (2005) reasons to explain the concept of a " cursed equilibrium " ( cursed equilibrium ). In such bidders say the bid distribution of other bidders ahead correctly and optimally respond to it. Your perception error, however, lies in the fact that they assess the relationship between the bids submitted by other bidders and their signals wrong. The higher the " Verfluchungsgrad " ( cursedness ), the higher the probability of a bidder appreciate that offer other bidders in the average of the other commandments of all signals away (and not so as by their own strategy specified). In the case of maximum cursedness bidders go about assuming that there is no connection between the Commandments and the signals of other players. This can result in the second- price auction to mispricing.

Ivanov, Levin and Niederle (2010) test the validity of these explanations experimentally using a common- value second- price auction. The authors compare the bidding behavior in an environment where excessive bidding by the theories outlined above is rationalized, with the behavior of bidders in situations where such explanations are unlikely. In the latter, no decrease excessive bids can be determined, which raises as an explanation of the winner's curse doubts about the plausibility of these approaches.

Strategic relationship

Theory

Be by

And

Two auctions given in the sense described above. It describes the two auctions as strategically equivalent if there is a real and a tuple, so

For all tuples. Strategic equivalence means, therefore, that the two identical players Games assign identical strategy spaces; they are allowed only in the individual's initial endowment and the relative unit in which the payoffs are paid differ. Obviously, applies to strategically equivalent games that their Nash equilibria are identical.

The second- price auction has a close strategic relationship to the English auction. Let us recall this about using the following consideration: In a second- price auction, a player above has shown, optimally, in the amount of one's appreciation and, if he is the highest bidder pays the next highest bid. In an ascending auction, the bidders are as long outdo each other until there are no more two bidders whose value estimates are based on the current bid price; the final bid accordingly at optimal bidding behavior just marginally higher than the second last bid ( and so far so approximatively equal). This brings us full circle to the second price auction: In both auction types in the ( unique) equilibrium wins one with the highest esteem, and in both cases, expenses incurred in the amount of the second highest esteem among all bidders.

Unlike the Erstpreisauktion and the Dutch (descending ) auction, however, second- price auction and English auction apparently are not strategically equivalent. The optimal strategies are not generally identical. Illustrative you can imagine roughly corresponding to the above, that one of the bidders does not know his own appreciation. One example is that the license to operate a Koltanmine by auction to be auctioned; while the assessment of the expected return (and thus also the appreciation of the license) is fraught with uncertainty because the quality and quantity of the mined coltan are ex ante the immediate insight hidden. In such a situation with uncertainty about the value estimates an English auction as shown on standing yields higher (expected ) prices as a second- price auction.

Experimental results in the IPV case

In the experimental literature has repeatedly found that the strategic equivalence between second- price and English auctions for private value estimates can not be fully replicated. Kagel, Harstad and Levin (1987 ) divide to study the strategic equivalence with their study participants that initially a number 'll drawn from a uniform distribution over a given interval; but was not disclosed. Instead, the private value estimates of the participants were now successively drawn from a uniform distribution over (where all was known ) and the respective participants informed (and only these ). For this reason, we also speak of a so-called induced value experiment, because the ' values' of the outside set ( " induced "). It has now been investigated in several rounds of betting, how to provide the participants with different auction formats. The authors ( compared to the actual dominant strategy ) excessively high bids in second- price auctions permanently ( an average of 11 percent too high), while the result in an English auction was compatible with the dominant strategy. You suspect may be causing that this stems from the false perception that winning percentage would rise by higher bid without incurring real (because of the second- price format ) costs. Harstad (2000) confirmed this result. He also observed as already Kagel and Levin (1993 ) that the excessive bidding does not change significantly with repeated performing a second- price auction and explains the so arising difference between English auction and second- price auction with repeated second- price auctions so that bidders a negative feedback mechanism for over-bidding missing because they can still realize a positive profit even at high bids to what they mistakenly interpreted as confirmation of their strategic choice.

However, individual work come to results that is consistent with the prediction of strategic equivalence in line or even suggest the levy to lower bids. Lucking -Riley (1999) walks through a field experiment and examined the bidding in an auction of identical Magic: The Gathering trading card on the internet through various auction mechanisms of time. The study does not produce evidence of income differences between second- price and English auction. Shogren et al. (2001) divide the participants in a laboratory setting in addition to their appreciation in addition to that it would be in their best interest to provide the level of their esteem; the authors come to the conclusion that in this environment, about 33 percent of the bids below and nearly 11 percent exceed the esteem this.

Morgan, Steiglitz and Reis ( 2003) rationalize excessive bids on a theoretical level by the assumption that the bidder a negative value through profits of their comrades realize ( hence so mischievous act ). More recently, experimental studies have been conducted to verify such explanatory hypotheses. Andreaonia, Che and KimC (2007) bidders share from round to round randomized into groups and investigate differences in bidding behavior of bidders with the highest esteem and those who are convinced to lose. The authors find evidence that, in principle, hold bidders to their weak dominant strategy, but then it ( upwards) differ when they become convinced to lose anyway and are able to influence the selling price. The result is the extent consistent with the Boshaftigkeitsmotiv Morgan, Steiglitz and Reis ( 2003).

Cooper and Fang ( 2008) experimentally investigate the empirical consistency of a number of potential Declaration, namely the Boshaftigkeitsmotiv Morgan, Steiglitz and Reis ( 2003) ( Hypothesis 1 ), which at the top briefly lectured Declaration Kagel, Harstad and Levin (1987 ) ajar existence of bounded rationality ( hypothesis 2 ) as well as a new motif, the joy of winning hypothesis ( " joy of victory" ) ( hypothesis 3). Hypothesis 2 is constructed by the authors so that bidders underestimated in determining their bid, the importance of a Bid for the expected payoff in case of victory, while taking into account their importance for winning percentage complete; this would explain the difference to the English auction, because there is readily apparent to everyone how a Bid affects the payoff in case of victory, so in the English auction ( transported large bids ) the perception of distortion should be less pronounced. Hypothesis 3 was that bidders learn the content on monetary factors also largely a positive benefit from the profit, resulting in their appreciation (and thus the optimal bid ) increased accordingly. The authors find evidence for the hypotheses 2 and 3, and - contrary to Morgan, Steiglitz and Reis ( 2003) - against hypothesis 1

Garratt, Walker and Wooders (2004, 2012) criticize the fact that the vast majority of experimental work on the issue of strategic equivalence with inexperienced bidders (students) is carried out and recruit their test subjects instead of experienced sellers on the Internet auction site eBay. In addition, they enable bidders to think about their bids longer than usual. In contrast to the results of other studies, the authors ( 67 38 to 41 percent vs. 6 percent in Kagel, Harstad and Levin ) make their participation certificate holders an approximately equal share of bids above and below the appreciation resistant and also no greater tendency to wertschätzungsgemäßem bidding. Roth and Levin (2008) conjecture with reference to psychological studies, the result is so far little surprising, as the holders of participation certificates were experienced seller and not the buyer, and are used for this reason in mind to buy at a low price to connect to a high Price to sell what extent performing an activity and why it is no reason to assume that they are more prone to emit the theoretical optimal bid.

Practical implementations

The format of the second- price auction is considered, in general practice as rare. On the Internet, find purely second price auctions in particular on some platforms use where Collectibles will be auctioned. Some vendors combine doing various Bietkanäle. So, for example, offers the stamp dealer Sandafayre weekly second price auctions with extensive possibilities of bidding that can be done not only through the website, but also by mail or via fax. This would be the use of the most common auction format, the auction English, very difficult and, it refers to a distinct advantage of sealed bid formats. Although other Internet auction sites such as Ebay do not use pure second- price auction; their implementation of the auction English is de facto, however, the second- price format very close. The input received from tenderers price merely acts as a so -called proxy bid; the actual bid will be doing, " compared our system with the commandments of other bidders and only increased to the smallest possible amount that is needed for you to continue to be the highest bidder " in the self- description of Ebay. This happens only as long until the proxy bid has been reached. In the result, ie fall for the victorious highest bidder costs in the amount of the second highest bid plus a small amount of what approximatively corresponds to the result of a second- price auction. Lucking -Riley (2000) examined the auction formats to 142 auction sites on the internet and finds only five examples of purely second price auctions, but 65 examples of modified English auctions with the possibility of proxy bids.

Second- price auctions have been used in the past and in the Versteigung of spectrum block licenses. Such licenses relate mostly to the right, the respective frequency block for a specific purpose - such as the transmission of television signals - to use; sometimes they also provide the enhanced ability to decide on the use of the licensed frequency block. The New Zealand government introduced at their first auction such spectrum block licenses in 1990 simultaneously several second- price auctions with no minimum prices through (one for each block ). In the result, the government achieved instead of the hoped-for 240 million New Zealand dollars ( NZ $ ) only 36 million. In one case, a company that had offered NZ $ 100,000, $ the contract at a price of only 6 NZ received; in another resulted in a bid of NZ $ 7,000,000 in costs of only NZ $ 5,000. The causes of this are generally the absence of minimum prices and, on a more fundamental level, the failure to take account of interdependencies ( certain licenses to each other in substitutive or complementary relationship, which in simultaneously performed second- price auctions can not be considered and in this respect brings unnecessary " random " results ).

Generalizations

Vickrey -Clarke - Groves mechanism

It can be shown that second- price auctions represent a special case of a much more general allocation mechanism, the Vickrey -Clarke - Groves mechanism ( VCG ) mechanism. Dating back to work of William Vickrey (1961 ), Edward H. Clarke ( 1971) and the generalization of Theodore Groves (1973 ) describes the VCG mechanism, a general method to implement efficient and incentive compatible allocations in multi- object auctions. In fact, can even show that the VCG mechanism among all efficient, incentive compatible ( strategy proof ) and individually rational mechanisms is the one with the highest revenue. In the special case, the VCG mechanism reduces to the second- price auction. An additional generalization it finally allows to take into account interdependent valuations within the multi -Settings objects of the VCG mechanism. In this case, however, it is no longer an auction mechanism in the strict sense, because it requires that the auctioneer is aware of the appreciation functions.

For details, refer to the article Vickrey -Clarke - Groves mechanism.

Generalized Second -Price Auction

Another generalization deliver Edelman, Ostrovsky and Schwarz ( 2007) who recognize the use of a second- price method in the auction advertising space on Internet search engines; no later used in 2002 for the first time, should this type of auction have spawned in 2006 revenues in the order of ten billion dollars. Search engines go about it, for example, so that advertising space is filled on one side from top to bottom with ads in descending order of votes for them commandments. When a user clicks then so is the advertising customer concerned is suffering on a display as part of a pay- per-click method, the amount of the next highest bid.

In general, the Generalized Second -Price Auction distinguished ( GSP auction ) by the following features: It is a multi- object auction with m objects (in the example: Advertising squares ) and n risk neutral bidders ( advertisers ). The number of clicks on an ad on position i, in a specified period of time you call with an advertiser and k, , clicking on its display is worth it. ( It does not matter for the appreciation of such clicks on the position where a clicked ad was displayed.) We put for simplicity and without loss of generality further notes that the positions are numbered in descending order, that is for the display with the highest number the clicks (), and so on. Is now a user of the search engine a corresponding keyword, so this sets the allocation mechanism in response: The operator shall determine for each bidder k whose last bid submitted and fills the ad space from top to bottom until all are occupied or, alternatively, to the last participating bidder is allocated, each bidder can occupy more than one space in any case. Be the i- highest bidder, bidder and was his given bid. Then the payoff of i amounts to

The authors show that there is no dominant strategy in a GSP auction unlike under the VCG mechanism, according to the self-worth to offer; a great deal of strategic bidding behavior was according to this market now also many various empirically proven. An equilibrium in dominant strategies exist otherwise than under the VCG mechanism is not in general.

In some recent applications of the model, the bidders of the auction is modeled explicitly, what sets them apart conceptually from the classical GSP setting. In this context, can be also about externalities model that arise under the assumption that the click rates are greater at higher positioned than advertisements placed further down; this has a negative influence that successful bidder then result in higher placement the average yield of the remaining advertiser.