Second price auction

As a (concealed) Vickrey auction ( English second-price [sealed bid] auction ) is called in the auction theory an auction where the highest bidder is awarded the contract, it not his own price, but only the second-highest bidder must pay. The bids are submitted once in such a way that they are not known to the other bidders (“hidden”, as when submitting an envelope that is only opened after the bidding process has ended). After their theoretical founder, the Nobel Prize Winner (Economy) William Vickrey , second-price auctions are also known as Vickrey auctions .

First price auctions have the same hidden format and the highest bidder wins, but he has to pay the bid he made himself.

Historical classification

The first formal analysis of a second-price auction is provided by Vickrey (1961), who constructs the second-price auction on a theoretical level because, under certain conditions, results are identical to those of the well-known English auction (in which bidders submit higher and higher bids one after the other until finally no other bidder outbids ) produces. In view of the apparent lack of earlier examples of this type of auction, Vickrey was from then on repeatedly postulated as the “inventor” of the auction format par excellence.

With a considerable time lag, earlier examples of the use of the format in practice have also been cited in the literature. Lucking-Reiley (2000) shows for the postage stamp market, for example, that second price auctions were used there long before Vickrey's work. After at least one dealer in the collectors' market, which was concentrated in New York, had already approached the format of the second-price auction in the 1870s by adding simple options for submitting one-off pre-bids to the English format that had been customary up until then, with reference to the high costs for traveling to foreign collectors Lucking-Reiley dates the first full second-price auction he identified on this market to the year 1883. In fact, second-price auctions have actually been mostly used on the (American) market for postage since the 1930s.

Moldovanu and Tietzel (1998) anecdotally point out that an example of a special form of a second price auction can be found in Johann Wolfgang von Goethe's correspondence from 1797. In a letter to the publisher Friedrich Vieweg , Goethe describes the procedure for selling a manuscript as follows:

"I am inclined to leave Mr. Vieweg in Berlin an epic poem Herrmann and Dorothea, which will be about 2000 hexameters strong, to the publisher [...] As far as the fee is concerned, I will send Mr. Oberconsistorialrath Böttiger a sealed ticket, which contains my request and awaits what Mr. Vieweg thinks he can offer me for my work. If his offer is less than my demand, then I take back my sealed slip of paper unopened, and the negotiation fails, if it is higher, I ask no more than the slip to be opened by Mr. Oberconsistorialrath. "

Analysis framework

In general, auctions are mechanisms that make it possible to assign one or more objects to a specific number of bidders. As far as "bidders" are mentioned, they do not necessarily have to represent the buyers' side - think of an auction in which several sellers bid for a construction contract that goes to the lowest bidder - but this serves as a reference case in practice and theory: A seller offers an object and the bidder who makes the highest bid for it receives it. This article follows this distribution of roles.

Second price auctions can be differentiated according to the nature of the individual valuations that bidders bring up with regard to the property. In this sense, two distinct basic formats can be identified: In an auction with private valuations , all bidders know their own appreciation for the object with certainty. However, they are not sure about the valuations of other bidders and their own appreciation of the property would not change if they found out about it. In auctions with interdependent valuations, however , bidders are not certain of their own esteem; rather, they only value it themselves by means of certain signals that are correlated with the true appreciation . If they were to know the information (signals) of the other bidders, their own appreciation might also change. Since the analysis framework of these two formats differs significantly in some cases, these cases are dealt with in the following in separate sections.

The structured and clearly regulated structure of auctions suggests analyzing them using game theory methods. Due to the uncertainty about the appreciation (s), auctions (in particular: second price auctions) are games with incomplete information : there is always at least one player who does not know the payoff function of at least one other player. Such games can be modeled in the tradition of Harsanyi (1967, 1968) as games with imperfect information . This approach opens up a structured possibility for the formal characterization of a (second price) auction based on a number of components:

1. The set of all players (here: bidders): with a typical element . ${\displaystyle {\mathcal {I}}=\{1,\ldots ,n\}}$ ${\displaystyle i}$
2. The set of all types (here: valuations) that a player can (potentially) accept (type possibilities set), with a typical element . For the sake of simplicity, be for everyone ; is therefore the maximum possible appreciation. ${\displaystyle i}$ ${\displaystyle \Theta _{i}}$ ${\displaystyle v_{i}}$ ${\displaystyle \Theta _{i}=\Theta =[0,{\overline {v}}]}$ ${\displaystyle i}$ ${\displaystyle {\overline {v}}}$
3. A distribution over the type possibilities set. ${\displaystyle F}$
4. The set of all possible strategies (in this case possible bids) for players , . This amount is given uniformly for all players . ${\displaystyle i}$ ${\displaystyle {\mathcal {B}}_{i}}$ ${\displaystyle {\mathcal {B}}_{i}={\mathcal {B}}=\mathbb {R} ^{\geq 0}}$
5. The preferences , represented by individual payoff functions (with the bid of and his appreciation). ${\displaystyle \pi _{i}(b_{1},\ldots ,b_{n};v_{i},\ldots ,v_{n})\in \mathbb {R} }$ ${\displaystyle b_{i}}$ ${\displaystyle i}$ ${\displaystyle v_{i}}$

Private appreciations

Assumptions

In the following, a single-stage second-price auction is assumed, which meets the assumptions of the so-called IPV model (in English, independent private values , i.e. “independent private valuations”). The hypothetical course of such an auction can be described as follows:

1. Each bidder is valued at random according to the function ; is the realization of a random variable that is independently and identically (iid) distributed according to a monotonically increasing distribution function . The density function of , , is continuous; have full support . Note that the ratings are therefore symmetrical in the sense that each bidder's rating is taken from the same distribution. ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle F}$ ${\displaystyle v_{i}\in \Theta }$ ${\displaystyle v_{i}}$ ${\displaystyle V_{i}}$ ${\displaystyle F}$ ${\displaystyle F}$ ${\displaystyle f\equiv F'}$ ${\displaystyle F}$
2. ${\displaystyle i}$experiences the realization of , (but not the appreciation of other bidders ). (In reality, of course, he already knows it. The fictitious process described, in which the player only experiences the appreciation at a certain point in time, only ensures that the appreciation is a move from a distribution.) ${\displaystyle V_{i}}$ ${\displaystyle v_{i}}$ ${\displaystyle j\in {\mathcal {I}}\setminus \{i\}}$
3. The bidders decide independently of each other about their bid. The chosen bid depends on your own appreciation for the property and is denoted by, where . Because the valuations are balanced and informed the bidders as equal, fully rational payoff maximizers will choose the same strategies, the function differs not between bidders and is dispensed at times on the index, so therefore , . ${\displaystyle b_{i}}$ ${\displaystyle b_{i}=\beta _{i}(v_{i})}$ ${\displaystyle \beta _{i}}$ ${\displaystyle b_{i}=\beta (v_{i})}$ ${\displaystyle \beta :\Theta \rightarrow {\mathcal {B}}}$
4. The bidders realize their payoffs . ${\displaystyle \pi _{i}}$

In the basic model outlined here, the auction always relates to one object. The bidders are completely rational and, in particular, also risk-neutral , which means that they maximize their expected payoff in their actions. You are not subject to any budget restrictions and are therefore able to cover the costs incurred in the event of a profit. Finally, apart from the realizations of other players , each player is familiar with all of the above features, including in particular the distribution function . The Extensions section abandons some of these assumptions. ${\displaystyle V_{i}}$ ${\displaystyle F}$

properties

Payoff

The payoff function in a second price auction of the form described is through

${\displaystyle \pi _{i}(v_{i})={\begin{cases}v_{i}-\max _{j\neq i}b_{j}&\mathrm {falls} \;b_{i}>\max _{j\neq i}b_{j}\\0&\mathrm {falls} \;b_{i}<\max _{j\neq i}b_{j}\end{cases}}}$.

given. In words: If a bidder is the highest bidder, his payoff corresponds to the difference between the value that the property has for him and the costs (i.e. the second highest bid) that he incurs; if he is not the highest bidder, the result is a payoff of zero. The case is often resolved by randomization (so that each of the highest bidders has a probability of winning) or equivalent, for practical simplicity, by first numbering the bids and, in the event of a tie, letting the highest / lowest number win. ${\displaystyle b_{i}=\max _{j\neq i}b_{j}}$ ${\displaystyle h}$ ${\displaystyle 1/h}$

Optimal bid amount

(Vickrey 1961 :) Placing a bid in the amount of one's own appreciation is a weakly dominant strategy in a second price auction .

To see this, consider that bidding any amount may not be optimal . ${\displaystyle b_{i}\neq v_{i}}$

• A bid is not optimal. If he bids with a suitable option instead , he will still receive the property (and at the same price) if , but now also if , which, based on the assumption, can also achieve a positive payoff. ${\displaystyle b_{i}<v_{i}}$ ${\displaystyle i}$ ${\displaystyle b_{i}+\epsilon <v_{i}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \max _{j\neq i}b_{j}<b_{i}}$ ${\displaystyle b_{i}<\max _{j\neq i}b_{j}<b_{i}+\epsilon }$

• A bid is not optimal. If he bids with a suitable one instead , he will still receive the property (and at the same price) if , but no longer if if , which would have been assumed to have resulted in a negative payoff. ${\displaystyle b_{i}>v_{i}}$ ${\displaystyle i}$ ${\displaystyle b_{i}-\epsilon >v_{i}}$ ${\displaystyle \epsilon >0}$ ${\displaystyle \max _{j\neq i}b_{j}<b_{i}-\epsilon }$ ${\displaystyle b_{i}-\epsilon <\max _{j\neq i}b_{j}<b_{i}}$

Consequently, you should neither bid below nor above the estimate, because you never get a higher, but sometimes a lower payoff. The statement also applies regardless of the above distribution assumptions (iid); it only depends on the assumption of private valuations. Second-price auctions are therefore incentive-compatible .

The following graphics illustrate the optimality of valued bidding. The figure shows the payoff of (vertical axis) depending on the highest non-own bid (horizontal axis), given the three situations that your own bid corresponds to your own appreciation (case 1), is below (case 2) or above (case 3) . The payoff function is always drawn in red where the player would be victorious with his bid. It is easy to see that if the bid is valued, the greatest possible expected payoff is achieved. ${\displaystyle i}$

Case 1: Bid equal to your own appreciation

Case 2: bid below one's own appreciation

Case 3: Commandment above one's own appreciation

Balance (s)

The weakly dominant strategy identified here induces only one of many Bayesian Nash equilibria. For example, let and be from a appreciation of bidders equal distribution over drawn. Then and (as well as a whole continuum of other strategies) also characterize an equilibrium , albeit not a "perfect" one in the sense of Selten (1975) (so-called trembling-hand-perfect balance ). Rather, such equilibria are unstable: If a player puts even a little probability that his opponent makes a mistake, the equilibrium collapses. If bidder 2 believes that bidder 1 has a high probability (and not: with certainty) 3 and has a low probability of bidding an amount below the value of bidder 2, then bidder 2 would have a (low) chance of closing the auction with a strictly positive payoff win if he bids more than 0, and if the bid is not above his estimate, he does not have to fear any losses. A general characterization of all equilibria in a second price auction - also for the case of asymmetric bidders discussed in detail below - is provided by Blume and Heidhues (2004). ${\displaystyle {\mathcal {I}}={1,2}}$ ${\displaystyle [0,1]}$ ${\displaystyle b_{1}=3}$ ${\displaystyle b_{2}=0}$

Efficiency

A one-object auction is said to be efficient if the object is assigned to the bidder who has the highest appreciation of all bidders ex post - regardless of the price that is paid. This is obviously the case with the outlined second price auction.

Expected costs and expected income

If one assigns the value estimates (more precisely the random variables, the value estimates) of all bidders except , in descending order, the result is a ranking with . Here is referred to as th order statistic of the valuations of all bidders except . The first order statistic ,, follows a distribution with constant density . This distribution can also be quantified: Because each individual appreciation is drawn independently from one and the same distribution function , is in the IPV framework . ${\displaystyle i}$ ${\displaystyle V_{1},\ldots ,V_{i-1},V_{i+1},\ldots ,V_{n-1}}$ ${\displaystyle Y_{1:n-1},\ldots ,Y_{n-1:n-1}}$ ${\displaystyle Y_{1:n-1}\geq \ldots \geq Y_{n-1:n-1}}$ ${\displaystyle Y_{k:n-1}}$ ${\displaystyle k}$ ${\displaystyle i}$ ${\displaystyle Y_{1:n-1}}$ ${\displaystyle G}$ ${\displaystyle g\equiv G'}$ ${\displaystyle F}$ ${\displaystyle G=F^{n-1}}$

Using this terminology, the expected cost to bidders is ${\displaystyle i}$

${\displaystyle {\begin{aligned}m(v_{i})&=\mathrm {Prob} (\mathrm {Gewinn} )\cdot \mathbb {E} (\mathrm {zweith{\ddot {o}}chstes} \;\mathrm {Gebot} |v_{i}\;\mathrm {ist} \;\mathrm {h{\ddot {o}}chstes} \;\mathrm {Gebot} )\\&=G(v_{i})\cdot \mathbb {E} (Y_{1:n-1}|Y_{1:n-1}<v_{i})\end{aligned}}}$,

where for the second equation the fact is used that in equilibrium the amount of appreciation is offered. According to the revenue equivalence theorem , this expression also corresponds to the expected costs for a first price auction and for a whole class of other auction formats.

From the seller's perspective, it is important to maximize the expected revenue. Of course, this is precisely the expected value of the individual bidding costs. This expectation, in turn, is in fact known because the distribution of is known according to the model assumptions. So it is ${\displaystyle n}$ ${\displaystyle V_{i}}$

${\displaystyle n\cdot \mathbb {E} \left[m(V)\right]=n\int _{0}^{\overline {v}}m(v)f(v)\mathrm {d} v=\mathbb {E} \left(Y_{2:n}\right)}$,

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

Extensions

Minimum prices

While maintaining the IPV framework described above, it is possible to examine which changes to the basic model result if minimum prices (also: reservation prices) are introduced into the bidding process. For example, be a minimum price that applies to all bidders. If your own appreciation is below the minimum price, it is a weakly dominant strategy not to bid (or, equivalently, to bid zero), because a positive payoff can never be realized. If the appreciation is higher, then, analogous to the considerations above in the IPV basic model, it is again weakly dominant to bid in the amount of one's own appreciation. So the weakly dominant strategy is ${\displaystyle r}$

${\displaystyle \beta _{i}(v_{i})={\begin{cases}v_{i}&\mathrm {falls} \;v_{i}\geq r\\0&\mathrm {sonst} \end{cases}}}$

The full expected costs for a bidder with are composed of the probability-weighted costs that arise if the second highest bid is below the minimum price (1st summand) and the expected value of the costs that arise if the second highest bid is above the reserve price (2nd summand): ${\displaystyle v_{i}\geq r}$

${\displaystyle m(v_{i},r)=rG(r)+\int _{r}^{v_{i}}yg(y)\mathrm {d} y=v_{i}G(v_{i})-\int _{r}^{v_{i}}G(y)\mathrm {d} x}$

with . ${\displaystyle v_{i}\geq r}$

Analogous to the above consideration without a minimum price, this raises the question of considering the auction from the point of view of the seller. The expected revenue of the seller is correspondingly the -fold expected value of the individual costs, therefore ${\displaystyle n}$

${\displaystyle n\cdot \mathbb {E} \left[m(v,r)\right]=n\int _{r}^{\overline {v}}m(v,r)f(v)\mathrm {d} v=n\int _{r}^{\overline {v}}\left[vf(v)+F(v)-1\right]G(v)\mathrm {d} v}$,

where the last equation follows by partial integration . If one takes into account that the object typically also has a certain value - here denoted by - for the seller , this follows for the expected revenue , whereby the last term is based on the idea that the seller always does not sell (and therefore realizes) if no bidder bids above the reserve price. Solving the resulting optimization problem gives the following result, which is given here without derivation: ${\displaystyle v_{0}}$ ${\displaystyle n\cdot \mathbb {E} \left[m(v,r)\right]+F(r)^{n}v_{0}}$ ${\displaystyle v_{0}}$

(Riley and Samuelson 1980; Laffont and Maskin 1980 :) Let the appreciation for everyone be distributed independently and identically and the bidders be risk-neutral. Then the ex ante revenue-maximizing minimum price is independent of the number of bidders and the following applies: ${\displaystyle V_{i}}$ ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle r^{*}}$

${\displaystyle r^{*}=v_{0}+{\frac {1-F\left(r^{*}\right)}{f\left(r^{*}\right)}}}$

Example: is evenly distributed over ; (blue curve) or (red curve). At can be achieved with a reserve price of the maximum expected proceeds. The ex ante revenue-optimizing minimum price increases in ; with it is already about here . ( Source code / calculation ) ${\displaystyle V_{i}}$ ${\displaystyle [0,10]}$ ${\displaystyle v_{0}=0}$ ${\displaystyle v_{0}=2}$ ${\displaystyle v_{0}=0}$ ${\displaystyle r^{*}=5}$ ${\displaystyle r^{*}}$ ${\displaystyle v_{0}}$ ${\displaystyle v_{0}=2}$ ${\displaystyle r^{*}=6}$

Note that secondary price auctions with minimum prices are no longer necessarily efficient. If, for example , a strictly positive minimum price is charged, there is a positive probability that there is no bidder who offers at least the minimum price, but at the same time there is a bidder with a strictly positive assessment below the minimum price. The auction is consequently inefficient. In addition, the case described illustrates a commitment problem. While it is ex ante revenue-maximizing for the seller to set the minimum price, it is sometimes no longer revenue-maximizing ex post : If the seller finds that all bidders have offered less than the minimum price, he could still generate positive revenue by Nevertheless, he still sells the object to the highest bidder. In other words, there is a profitable renegotiation. This in turn results in a credibility problem that can incentivize bidders to submit lower bids. ${\displaystyle v_{0}=0}$

In reality, the setting of minimum prices cannot regularly be observed, although, according to the above considerations, this strategy weakly dominates the waiver of a minimum price. The reason for this may be disregarded costs that a bidder incurs before he finds out about his appreciation (for example, travel costs that are incurred to inspect the property). Engelbrecht-Wiggans (1987) shows for the case of bidders who all have identical information about the probable value of the property before accepting the said costs, an equilibrium in pure strategies (deterministic equilibrium) in which the revenue-optimizing minimum price from Reference case differs and the raising of a minimum price can reduce the expected revenue. Levin and Smith (1994) explicitly model the decision-making process of potential bidders and construct a balance in mixed strategies (stochastic equilibrium) in which setting a minimum price also has a negative effect on the proceeds.

Budget constraints

In game theory, the analysis of second-price auctions with budget restrictions changes compared to the case without initially the number of possible types: uncertainty no longer only prevails about the appreciation of other players, but also about their available budget. Denote by the budget of . It is now , that type of opportunity set of players in the event of limited budgets with typical element , the amount of possible budgets analogue designed the set of possible valuations. The types are still distributed independently and identically according to a density function ; the bidding function is designated with . ${\displaystyle w_{i}}$ ${\displaystyle i}$ ${\displaystyle \Theta _{i}^{c}\equiv [0,{\overline {v}}]\times [0,{\overline {w}}]}$ ${\displaystyle 0<{\overline {v}}\leq {\overline {w}}}$ ${\displaystyle i}$ ${\displaystyle (v_{i},w_{i})}$ ${\displaystyle f(v_{i},w_{i})}$ ${\displaystyle \beta _{i}^{c}}$

The payoff structure is as follows:

${\displaystyle \pi _{i}(v_{i},w_{i})={\begin{cases}v_{i}-\max _{j\neq i}b_{j}&\mathrm {falls} \;b_{i}>\max _{j\neq i}b_{j}\;\mathrm {und} \;b_{i}\leq w_{i}\\-\infty &\mathrm {falls} \;b_{i}>\max _{j\neq i}b_{j}\;\mathrm {und} \;b_{i}>w_{i}\\0&\mathrm {falls} \;b_{i}<\max _{j\neq i}b_{j}\end{cases}}}$

It differs from the basic IPV model in that, in the event of a win, costs in the amount of the second highest bid are not always incurred. If a player bids more than is available, he experiences a negative payoff.

First of all, note that any bid is a weakly dominated strategy. If this wins and the second highest bid is above his budget, he realizes a negative payoff; if he wins and the second highest bid is equal to or below his budget, he would have won with a bid equal to his budget. It is shown by further consideration that through ${\displaystyle b_{i}>w_{i}}$ ${\displaystyle i}$

${\displaystyle \beta _{i}^{c}(v_{i},w_{i})=\min\{v_{i},w_{i}\}}$

the weakly dominant strategy is given: If , the budget restriction is not binding and it is accordingly a weakly dominant strategy to bid in the amount of the appreciation (see the above considerations on the IPV basic model). If so, it is weakly dominant, analogous to the consideration in the paragraph above, to bid in the amount of the budget. ${\displaystyle w_{i}\geq v_{i}}$ ${\displaystyle w_{i}<v_{i}}$

Looking at the payoff functions, the result becomes clear. If one ignores the trivial case of a non-binding budget condition and limits oneself to the case with , then case 1 below shows that a bid above the budget limit always implies a negative payoff with a positive probability. A bid below the budget limit is again not optimal, because if the second highest bid is at the same level, you will forego the profit of the property (with a positive payoff!). ${\displaystyle w_{i}<v_{i}}$

Case 1: bid above budget constraint

Case 2: bid below budget limit

Risk aversion

If bidders no longer maximize their expected payoff (risk neutrality), but are subject to a certain risk aversion ( risk aversion ), their optimal bidding strategies often change in auction formats. In a second price auction in the outlined IPV basic model, however, there is no corresponding influence; the arguments given above to show the weak dominance of valued bidding are not affected by the existence of risk aversion. As a result, the expected proceeds will not change. The revenue equivalence to the first price auction is no longer guaranteed because the strategy chosen there is changing.

asymmetry

One of the central standard assumptions in the IPA framework (and one of the requirements for the validity of revenue equivalence) is that the estimates are moves from the same distribution. In reality, however, this does not have to be the case. An example of this is, for example, that an art dealer still needs a work to complete his collection and this creates a synergy effect that is irrelevant for a “normal” bidder. The question of the nature and the number of equilibria is not trivial in the case of asymmetry and basically depends on the respective distributions. A special complication arises from renegotiations in the form of reselling the property. The winning bidder sells the object to another bidder after the auction has ended.

As mentioned above, within the IPA framework, bidding in the amount of one's own appreciation is a weakly dominant strategy, and the auction also has an efficient balance. These two results derived above also apply in the event that the bidders are asymmetrical. Hafalir and Krishna (2008) show for the case of two bidders with strictly monotonically increasing and steady distribution functions or that bidding according to appreciation even represents a perfect Bayesian Nash equilibrium . ${\displaystyle F_{1}}$ ${\displaystyle F_{2}}$

The revenue implications of asymmetry are even less clear. Cantillon (2008) shows for bidders that the revenue distribution of an asymmetrical second price auction with distributions from the symmetrical benchmark auction is dominated by stochastic in first order and that the aggregated ex-ante payoffs from the asymmetrical auction are always higher than those in the symmetrical auction Benchmark auction. However, Chen and Xu (2012) point out that this result does not appear to be particularly robust. If one constructs other, equally plausible, symmetrical benchmarks, the result can be reversed, so that the seller proceeds from an asymmetrical second price auction can be higher than in a symmetrical auction under different conditions. ${\displaystyle n}$ ${\displaystyle (F_{1},\ldots ,F_{n})}$ ${\displaystyle (F,\ldots ,F)}$ ${\displaystyle F=\left[F_{1}(x)\cdot \ldots \cdot F_{n}(x)\right]^{(1/n)}}$

To compare the first and second price auctions, Hafalir and Krishna (2008) show for asymmetrical distributions that meet Myerson's regularity condition that the proceeds from a first price auction with repurchase exceed those from a second price auction with resale. Kirkegaard (2012), generalizing the results in Maskin and Riley (2000), shows for the two-bidder case that the proceeds from a first-price auction in an asymmetrical environment with no repurchase opportunities are higher than that from a second-price auction, provided that the "Strong" bidder is flatter and has a higher spread than that of the "weak" bidder.

collusion

Like other auction formats, second price auctions are prone to collusion . In a collusive environment , a subset of bidders with cardinality form a bidding ring in order to keep the price for the property as low as possible. This is achieved in such a way that not all members of the ring bid individually, but only the ring as a whole submits a bid, so that overall, instead of bidders, only bidders bid. In practice, this is implemented in such a way that all non-highest bidders in the ring submit bids below the minimum price or invalid bids. In a second price auction, a market distortion that reduces the sales proceeds can be achieved. If the second highest bidder in the ring has a higher appreciation for the object than any bidder outside the ring, the winning ring (or its member with the highest appreciation) will pay a lower price than if there had been no collusion. ${\displaystyle {\mathcal {R}}\subseteq {\mathcal {I}}}$ ${\displaystyle R}$ ${\displaystyle n}$ ${\displaystyle n-R+1}$

From a macro perspective, not much will change strategically in the environment described. It is still weakly dominant for the ring to bid in the amount of its "appreciation" (that is the appreciation of the one with the highest such), and it is also weakly dominant for all bidders outside the ring to bid in the amount of their appreciation . In addition, the probability of winning any bidder outside the ring is unaffected by its existence. It can even be further shown that the expected costs of the non-colluding bidders are not influenced by the existence of a ring. The problem is the study of the allocation in the ring itself. Namely, in order to identify the bidder with the highest appreciation, members must get an incentive appreciation truthfully against a "center" (center) of the ring (the operations coordinated within the cartel) to disclose. The so-called (second-price) pre-auction knockout (PAKT) (Graham and Marshall 1987) is indeed an incentive-compatible algorithm for this purpose. The implementation with PAKT works as follows:

1. The (risk-neutral) center is charged a fixed amount to each member of the ring and selected with a random variable of the greater of and the difference between the second highest appreciation of all the ring members minus the highest appreciation of all bidders outside the ring. ${\displaystyle q}$ ${\displaystyle q=\mathbb {E} (\delta )/R}$ ${\displaystyle \delta }$ ${\displaystyle 0}$
2. Each of the ring members makes a bid to the center. ${\displaystyle R}$
3. The highest bidding member of the ring will be determined as the only bidder for the ring and all other members are recommended not to participate in the auction or with a bid below the reserve price in the main auction.
4. If the highest bidding member wins the item in the main auction, it not only has to pay the second highest bid of all bidders to the auctioneer of the main auction, but also has to pay the ring center the difference between the second highest bid within the ring and the second highest bid in the main auction (if the difference is positive). ${\displaystyle n-R+1}$

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

(Graham and Marshall 1987 :)

1. Be the ring size given . Then the expected payoff of a ring member is higher, the lower the minimum price that the seller sets in the main auction. ${\displaystyle R}$ ${\displaystyle 2\leq R\leq n}$ ${\displaystyle r}$
2. Be the reserve price in the main auction. Then the expected payoff of a ring member is higher the larger the ring size . ${\displaystyle r}$ ${\displaystyle R}$

rating

The fact that you should bid in the amount of your own appreciation simplifies the bidding process from the bidder's point of view, because no distribution assumptions about the bid amounts of other bidders are required to determine your own bid. Compared to the strategically equivalent ascending auction (also: English auction) in the case of private valuations, the second-price auction has the advantage that it is not necessary for the bidders to attend the bidding process for a longer period of time ("bidding"), since it is a one-step bid Sealed bid format has already been concluded by submitting a one-off bid. For example, in the format of the English auction originally used on the Internet auction platform Ebay , it was necessary for a payoff-optimizing customer to ensure, for example, by employing an agent, special software or their own presence, that one would actually still be able to do so at the end of the auction is actively involved in the bidding process; this effort does not apply to the format presented here.

Rothkopf, Teisberg and Kahn (1990) cite two reasons why second price auctions are rarely found in practice. On the one hand, this is due to the fact that the bidders fear being cheated. Since the highest bidder pays the second highest bid, he depends on the seller / auctioneer correctly numbering this and not adding his own bid to drive the price up. On the other hand, the fact that it is optimal to reveal one's own appreciation is a deterrent for bidders. A participant in a second price auction regularly has a considerable incentive not to inform other bidders or a seller about the actual appreciation. One should also think of situations in which auction results are renegotiated; Here, for example, a company's bargaining power would potentially suffer from the fact that the cost structure was actually disclosed beforehand through the strategy-compliant bid. This point can also be seen in practice on the collectors' market for postage stamps mentioned above. Technological innovations that make the pricing process more transparent and less susceptible to manipulation may help.

Interdependent valuations

Assumptions and connection to the IPV model

The assumptions of the IPV model often do not appear very plausible in practice because one's own appreciation is influenced by the appreciation of other bidders. This may result from personal indecision, but sometimes also results from the uncertainty about the resale price - if a bidder knows about the low esteem of his fellow bidders, he will reduce his expected resale proceeds, which can influence his bidding behavior; yet another possibility is that part of the object consumption is realized as validity consumption . Such a scenario leads to a model with interdependent valuations, namely interdependent in the sense that the valuation of a bidder would change if he knew the valuation of (certain) other bidders. Formally, this is described in such a way that every bidder is given a signal (in the IPV basic model this was then directly his appreciation); the individual appreciation then results as a function of all signals, ${\displaystyle S_{i}}$

${\displaystyle V_{i}=\nu _{i}(S_{1},\ldots ,S_{n})}$.

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

${\displaystyle V_{i}=\nu _{i}(S_{1},\ldots ,S_{n})=S_{i}}$

sets, and on the other hand the common value model, if for all , if so, in words, every bidder shows the object an identical (but not known) appreciation. ${\displaystyle V_{i}=V=\nu (S_{1},\ldots ,S_{n})}$ ${\displaystyle i}$

If another bidder receives a higher signal, your own appreciation should always be at least as high as it would be if the other bidder received a lower signal. In the case of one's own signal, it is even true that a higher signal always goes hand in hand with a strictly higher appreciation. So it is true

${\displaystyle {\frac {\partial \nu _{i}}{\partial S_{j}}}\geq 0}$for everyone as well as for everyone . ${\displaystyle i,j\in {\mathcal {I}}}$ ${\displaystyle {\frac {\partial \nu _{i}}{\partial S_{i}}}>0}$ ${\displaystyle i\in {\mathcal {I}}}$

${\displaystyle \nu _{i}}$is also twice continuously differentiable . The signals are constructed analogously to the appreciation in the IPV case; it is with the maximum signal value. In order to allow interdependencies, the signals are combined to form a -valent random variable ; this in turn follows a distribution with a density function . The assumption of symmetry finds its way into the model by assuming that it is symmetrical in its arguments, in the following sense: For every -digit permutation , that applies , and this for all . It is therefore irrelevant for the probability of a certain signal configuration which bidder receives which signal. Symmetry also applies to the valuation structure of the bidders. To do this, first define with . be symmetric in the last arguments (that is, they can be swapped arbitrarily without affecting the function value of ). Further be . ${\displaystyle S_{i}\in [0,{\overline {s}}]}$ ${\displaystyle {\overline {s}}}$ ${\displaystyle [0,{\overline {s}}]^{n}}$ ${\displaystyle \mathbf {S} =(S_{1},\ldots ,S_{n})}$ ${\displaystyle F}$ ${\displaystyle f}$ ${\displaystyle f}$ ${\displaystyle n}$ ${\displaystyle \tau :{\mathcal {I}}\rightarrow {\mathcal {I}}}$ ${\displaystyle f(s_{1},\ldots ,s_{n})=f(s_{\tau (1)},\ldots ,s_{\tau (n)})}$ ${\displaystyle (s_{1},\ldots ,s_{n})\in [0,{\overline {s}}]^{n}}$ ${\displaystyle \nu _{i}(\mathbf {S} )\equiv u(S_{i},\mathbf {S} _{-i})}$ ${\displaystyle \mathbf {S} _{-i}\equiv (S_{1},\ldots ,S_{i-1},S_{i+1},\ldots ,S_{n})}$ ${\displaystyle u}$ ${\displaystyle n-1}$ ${\displaystyle u}$ ${\displaystyle u(0,\mathbf {0} )=0}$

Finally, the signals "affiliated" are (affiliated) , that in this way a higher signal at another bidder increases the likelihood that the own signal is also higher (positive Affiliation). For the formal definition, reference is made to a footnote. Note that this in no way implies that a player knows all of the other signals.

properties

balance

Denote

${\displaystyle \eta (s_{i},y_{\setminus i})\equiv \mathbb {E} (V_{i}|S_{i}=s_{i},Y_{1:n-1}^{\setminus i}=y_{\setminus i})}$

- with the random variable of the highest signal of all other bidders except - the conditional expected value of the appreciation of the bidder when he experiences the signal and when the second highest signal of all the remaining bidders is given. Ultimately, that applies ${\displaystyle Y_{1:n-1}^{\setminus i}}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle s_{i}}$ ${\displaystyle y_{\setminus i}}$

${\displaystyle s_{i}>s_{i}'\Rightarrow \eta (s_{i},y_{\setminus i})>\eta (s_{i}',y_{\setminus i})}$

for everyone . ${\displaystyle y_{\setminus i}}$

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

${\displaystyle \beta (s_{i})=\eta (s_{i},s_{i})}$

given.

This means that in the (Bayesian) equilibrium all players bid as if they knew that the highest signal of the other bidder corresponds to their own signal.

Example: is evenly distributed , there are 3 bidders and . Be it . The difference between and is shown . ( Source code ) ${\displaystyle S_{i}}$ ${\displaystyle V=\sum S_{i}}$ ${\displaystyle s_{i}=6}$ ${\displaystyle \eta (s_{i},y_{\setminus i})}$ ${\displaystyle \eta (y_{\setminus i},y_{\setminus i})}$

If all bidders followed the strategy described and received the signal , his expected payoff would be for a bid of ${\displaystyle j\neq i}$ ${\displaystyle i}$ ${\displaystyle s_{i}}$ ${\displaystyle b}$

${\displaystyle {\begin{aligned}\pi (s_{i})&=\int _{0}^{\beta ^{-1}(b)}\left[\eta (s_{i},y_{\setminus i})-\beta (y_{\setminus i})\right]g(y_{\setminus i}|s_{i})\mathrm {d} y_{\setminus i}\\&=\int _{0}^{\beta ^{-1}(b)}\left[\eta (s_{i},y_{\setminus i})-\eta (y_{\setminus i},y_{\setminus i})\right]g(y_{\setminus i}|s_{i})\mathrm {d} y_{\setminus i}\end{aligned}}}$.

According to premise and , consequently, the payout is maximized if . Intuitive: If the upper integration limit is increased further and further starting from 0, the value of the integral initially increases further and further (because the integrand is positive); as soon as it exceeds, the integral value decreases again (because the integrand is then negative). ${\displaystyle s_{i}>y_{\setminus i}\Rightarrow \eta (s_{i},y_{\setminus i})-\eta (y_{\setminus i},y_{\setminus i})>0}$ ${\displaystyle s_{i}<y_{\setminus i}\Rightarrow \eta (s_{i},y_{\setminus i})-\eta (y_{\setminus i},y_{\setminus i})<0}$ ${\displaystyle \beta ^{-1}(b)=s_{i}}$ ${\displaystyle y_{\setminus i}}$

Revenue and lack of revenue equivalence

With the balanced bidding strategy, the seller's expected revenue is

${\displaystyle {\begin{aligned}\mathbb {E} \left[\beta \left(Y_{1:n-1}^{\setminus i}\right)\left|S_{i}=s_{i},s_{i}>Y_{1:n-1}^{\setminus i}\right.\right]&=\mathbb {E} \left[\left.\eta \left(Y_{1:n-1}^{\setminus i},Y_{1:n-1}^{\setminus i}\right)\right|S_{i}=s_{i},s_{i}>Y_{1:n-1}^{\setminus i}\right]\\&=\int _{0}^{s_{i}}{\frac {\eta (y_{\setminus i},y_{\setminus i})\cdot g(y_{\setminus i}|s_{i})}{G(s_{i}|s_{i})}}\mathrm {d} y_{\setminus i}\end{aligned}}}$

with the conditional distribution of given and the associated density. ${\displaystyle G(\cdot |s_{i})}$ ${\displaystyle Y_{1:n-1}^{\setminus i}}$ ${\displaystyle S_{i}=s_{i}}$ ${\displaystyle g(\cdot |s_{i})}$

Using the affiliations property, it can also be shown that the expected proceeds from a second-price auction are at least as high as those from a first-price auction. In addition, it is at most as high as in the English auction. There is therefore no longer any revenue equivalence. In particular, one can intuitively understand the difference between a second price auction and an English auction. From the bidder's point of view, one can only achieve a rent in a described second price auction with affiliated signals by using the available private information. The more the costs depend on the private information of the other players, the more closely they are related to the information of the winner (because the signals are affiliated). This is more the case in an English auction (in which the costs depend on the information of all other bidders) than in a second-price auction (in which the costs depend on the information of another bidder).

example

The following example, borrowed from Klemperer (1999), illustrates the above considerations.

Be the set of bidders. Each bidder experiences a signal that is drawn independently from a uniform distribution , whereby the value of the object is. Let us now assume that the bidders have no knowledge of ex ante , so that all values ​​of are classified as equally likely. This implies that a higher value of increases the probability of a higher value of and at the same time makes a higher signal value of the other bidders more probable, whereby the affiliation characteristic is fulfilled. In a second price auction, each bidder bids in equilibrium in the amount of the conditional expected value of the appreciation, given his own signal and given that the highest signal from the other bidders is just as high as this. If one takes the other highest bidder out of consideration, it is assumed that he is the one with the highest signal among evenly distributed moves from the interval . ${\displaystyle {\mathcal {I}}=\{1,\ldots ,n\}}$ ${\displaystyle i}$ ${\displaystyle s_{i}}$ ${\displaystyle [v-1/2,v+1/2]}$ ${\displaystyle v}$ ${\displaystyle v}$ ${\displaystyle v}$ ${\displaystyle s_{i}}$ ${\displaystyle v}$ ${\displaystyle i}$ ${\displaystyle n-1}$ ${\displaystyle [v-1/2,v+1/2]}$

First consider that in a uniform distribution over the expected -highest value from independent moves is always given by . This is nothing else than the -th order statistic , so for . This results in the following for the formation of expectations : expects that on average and accordingly offers (equation 1). ${\displaystyle [{\underline {v}},{\overline {v}}]}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle {\underline {v}}+[(n+1-k)/(n+1)]({\overline {v}}-{\underline {v}})}$ ${\displaystyle k}$ ${\displaystyle Y_{k:n-1}}$ ${\displaystyle k=1}$ ${\displaystyle Y_{1:n-1}}$ ${\displaystyle i}$ ${\displaystyle i}$ ${\displaystyle s_{i}={\underline {v}}+[(n-1)/n](v+1/2-v+1/2)=v-1/2+(n-1)/n}$ ${\displaystyle s_{i}+1/2-(n-1)/n}$

What is the expected proceeds? On average, the bidder with the second highest signal has the signal (equation 2). That is, the average bid (equation 2 inserted into equation 1) is the average proceeds from the auction. ${\displaystyle s_{j}={\underline {v}}+[(n-1)/(n+1)](v+1/2-v+1/2)=v-1/2+(n-1)/(n+1)}$ ${\displaystyle \left[v-1/2+(n-1)/(n+1)\right]+\left[1/2-(n-1)/n\right]}$

Curse of the winner

theory

Auctions with interdependent valuations are - unlike those with private valuations - susceptible to distortion of perception. At the time of submitting a bid, bidders have an estimate of the value of the property. Even if the respective estimates are undistorted , bidders must include the informational value of their (potential) own victory: Whoever wins the auction will also have had one of the highest expectations of appreciation for the property, which in turn should raise doubts about one's own estimate. Failure to include this information in your own bid selection can result in significant payoff losses. One calls this form of adverse selection as " the winner's curse " (winner's curse) .

Experimental results

The passing of the winner's course is usually analyzed in experimental work on the basis of the extreme case of a common value auction and is a largely reliable result there even in the case of second-price auctions.

Kagel, Levin and Harstad (1995) conduct a number of second price auctions in which the study participants are not informed of the actual value of the object. ("Object" is not to be understood materially in this context; what is meant is rather the claim of the winner to receive the claimed (monetary) value in the end.) The bidders are given private signals, each from an even distribution over an interval randomly drawn around the actual value. In later rounds they were also informed of the lowest signal from their opponents, whereupon in an environment with four or five bidders an insignificant increase in the average profit of the bidders results, but the average profit with six or seven bidders falls sharply (and consistently is negative). Because there is also evidence that the players do not change their bidding function with different numbers of bidders, the authors conclude that this observation is probably due to the influence of the curse of the winner (which is naturally more pronounced with several players).

Avery and Kagel (1997) analyze the behavior of bidders in a second-price auction in which the value of the object is the sum of two independent random variables that are equally distributed over and over . Each bidder is informed of the realization of one of the signals ( or ). The balanced strategy in this setting would be by or given, but the actual bid amount significantly similar to the naive estimator in the form of unconditional expected value of the property value . In addition, with low signals, almost all victorious bidders lose money. The phenomenon weakens as the experience of the bidders increases. ${\displaystyle [{\underline {s}},{\overline {s}}]}$ ${\displaystyle S_{i}}$ ${\displaystyle S_{j}}$ ${\displaystyle s_{i}}$ ${\displaystyle s_{j}}$ ${\displaystyle b_{i}=2s_{i}}$ ${\displaystyle b_{j}=2s_{j}}$ ${\displaystyle s_{i}+({\overline {s}}+{\underline {s}})/2}$

Rationalizability?

The implicit explanation of the winner's curse usually consists in the inability of the bidders to adjust their assumptions about the property value based on the informational value of their win. On the other hand, proposals have been put forward in the more recent literature that rationalize the phenomenon of the winner's curse . Crawford and Iriberri (2007), for example, design a model that explains the reactions in an auction using the so-called level approach (Stahl and Wilson 1995, Nagel 1995). The behavior of each bidder is drawn from a common distribution of all possible "decision-maker types", which differ from one another in terms of their level of ability (L0 types behave naively and non-strategically). L1 types give the best answer to L0 types with their bid , L2 types the best answer to L1 types and so on. The authors come to the conclusion that a special level structure in a second price auction can produce a winner's course . ${\displaystyle k}$ ${\displaystyle k}$

Eyster and Rabin (2005) explain to explain the concept of a "cursed equilibrium" (cursed equilibrium) . In such a system, bidders correctly predict the bid distribution of the other bidders and react optimally to it. However, their perception error is that they misjudge the connection between the bids of other bidders and their signals. The higher the “ cursedness” , the higher bidders estimate the probability that other bidders will bid across all signals in the amount of the average of the other bids (and not as specified by their own strategy). In the case of maximum cursedness , bidders assume that there is no connection between the bids and the signals of other players. This can lead to incorrect valuations in the second price auction.

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

Strategic relationship

theory

Be through

${\displaystyle A=\left\langle {\mathcal {I}};v_{i},\ldots ,v_{n};F_{1},\ldots ,F_{n};b_{1},\ldots ,b_{n};\pi _{1}',\ldots ,\pi _{n}'\right\rangle }$

and

${\displaystyle B=\left\langle {\mathcal {I}};v_{i},\ldots ,v_{n};F_{1},\ldots ,F_{n};b_{1},\ldots ,b_{n};\pi _{1}'',\ldots ,\pi _{n}''\right\rangle }$

given two auctions in the sense described above. The two auctions are said to be strategically equivalent if there is a real and a tuple such that ${\displaystyle k>0}$ ${\displaystyle (z_{1},\ldots ,z_{n})\in \mathbb {R} ^{n}}$

${\displaystyle \pi _{i}''(b_{1},\ldots ,b_{n};v_{1},\ldots ,v_{n})=k\cdot \pi _{i}'(b_{1},\ldots ,b_{n};v_{1},\ldots ,v_{n})+z_{i}}$

for all tuples . Strategic equivalence means that the two games assign identical players to identical strategy rooms; they may only differ in the individual initial configuration and the relative unit in which the payoffs are paid out. Obviously, for strategically equivalent games, their Nash equilibria are identical. ${\displaystyle (b_{1},\ldots ,b_{n};v_{1},\ldots ,v_{n})}$

The second price auction is strategically related to the English auction. You can visualize this with the help of the following consideration: In a second price auction, as shown above, a player optimally bids in the amount of his own appreciation and, if he is the highest bidder, pays the next higher bid. In an ascending auction, the bidders will outbid each other until there are no more two bidders whose value is higher than the current bid amount; The last bid is accordingly just marginally higher than the penultimate bid (and therefore approximately the same) if the bidding behavior is optimal. This closes the circle to the second-price auction: In both auction types, the one with the highest valuation wins in (unambiguous) equilibrium, and in both cases he incurs expenses in the amount of the second highest valuation among all bidders.

Unlike the first price auction and the Dutch (descending) auction, however, the second price auction and the English auction are obviously not strategically equivalent. Even the optimal strategies are not generally identical. Illustratively, one can imagine, according to the above statements, that one of the bidders does not know his own appreciation. An example would be that the license to operate a coltan mine is to be auctioned by way of an auction; The evaluation of the expected yields (and thus also the appreciation of the license) is fraught with uncertainty because the quality and quantity of the coltan extracted are hidden from direct insight ex ante . In such a situation with uncertainty about the valuations, an English auction yields higher (expected) prices than a second price auction, as shown above.

Experimental results in the IPV case

Finding

In the experimental literature it has been stated time and again that the strategic equivalence between second price and English auction cannot be fully replicated in private valuations. To investigate strategic equivalence, Kagel, Harstad and Levin (1987) inform their study participants that a number is first drawn from a uniform distribution over a given interval; but was not communicated. Instead, the private valuations of the participants were drawn one after the other from an equal distribution (which was known to all) and communicated to the respective participants (and only these!). For this reason, one speaks of what is known as an induced value experiment , because the “valuations” are specified (“induced”) from outside. In several rounds of bidding, it was examined how the participants bid in different auction formats. The authors found excessively high bids in second price auctions (compared to the actual dominant strategy) (on average 11 percent too high), while the result in an English auction was compatible with the dominant strategy. They suspect the cause is that this stems from the wrong perception that the probability of winning would increase through higher bidding without incurring any real costs (because of the second price format). Harstad (2000) confirms this result. Like Kagel and Levin (1993), he observes that excessive bidding does not change significantly when a second-price auction is carried out repeatedly and explains the resulting difference between an English auction and a second-price auction in repeated second-price auctions with the fact that bidders have a negative feedback mechanism in the case of excessive bids is lacking because they can still realize a positive profit even if the bids are too high, which they mistakenly regard as confirmation of their strategic choice. ${\displaystyle x_{0}}$ ${\displaystyle x_{0}}$ ${\displaystyle x_{i}}$ ${\displaystyle [x_{0}-\epsilon ,x_{0}+\epsilon ]}$ ${\displaystyle \epsilon }$

However, individual studies also come to results that are consistent with the forecast of strategic equivalence or that even suggest that bids should be too low. Lucking-Riley (1999) leads 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 any evidence of differences in revenue between second price and English auctions. Shogren et al. (2001) inform the participants in a laboratory setting, in addition to their appreciation, that it would be in their best interest to bid in the amount of their appreciation; The authors come to the conclusion that in this environment around 33 percent of the bids fall short of the appreciation and just under 11 percent exceed it.

causes

Morgan, Steiglitz and Reis (2003) rationalize inflated bids on a theoretical level by assuming that the bidders realize a negative benefit through the profits of their fellow campaigners (thus acting maliciously). Only recently have experimental studies been carried out to test such explanatory hypotheses. Andreaonia, Che and Kimc (2007) randomly assign bidders to groups from round to round and examine the differences in the bidding behavior of bidders who are most valued and those who are convinced that they will lose. The authors find evidence that bidders generally adhere to their weakly dominant strategy, but deviate from it (upwards) if they are convinced that they are losing anyway and are able to influence the selling price. The result is so far consistent with the evil motive of Morgan, Steiglitz and Reis (2003).

Cooper and Fang (2008) experimentally investigate the empirical consistency of a number of potential explanations, namely the wickedness motive of Morgan, Steiglitz and Reis (2003) (Hypothesis 1), which was based on the explanation by Kagel, Harstad and Levin (1987) briefly referred to above Presence of limited rationality (hypothesis 2) and a new motive, the joy of winning hypothesis (“Freude am Sieg”) (hypothesis 3). Hypothesis 2 is constructed by the authors in such a way that, when determining their bid, bidders underestimate the importance of a bid increase for the expected payoff in the event of a win, while fully considering its importance for the probability of winning; This would explain the difference to the English auction, because everyone can easily see how a bid increase affects the payoff in the event of a win, so that in the English auction the (high bids) distortion of perception should be less pronounced. Hypothesis 3 has as its content that bidders experience a positive benefit from the profit beyond monetary factors, which increases their appreciation (and thus the optimal bid) accordingly. The authors found evidence for hypotheses 2 and 3 and - in contradiction to Morgan, Steiglitz and Reis (2003) - against hypothesis 1.

Garratt, Walker and Wooders (2004, 2012) criticize that the vast majority of experimental work on the question of strategic equivalence is carried out with inexperienced bidders (students) and instead recruit their test subjects from experienced sellers on the Internet auction platform Ebay . They also allow bidders to think about their bids longer than usual. In contrast to the results of other studies, the authors found an approximately equal proportion of bids above and below the appreciation among their participants (38 to 41 percent vs. 67 to 6 percent for Kagel, Harstad and Levin [1987]) and none either greater tendency to bid in accordance with appreciation. Roth and Levin (2008), with reference to psychological studies, speculate that the result is not surprising insofar as the participants were experienced sellers and not buyers, and for this reason were used to buying at a low price and then at a high one To sell price, which in this respect represents a different activity and why there is also no reason to assume that they tend more towards submitting the theoretical optimal bid.

Practical implementations

The format of the second price auction is generally considered to be rare in practice. On the Internet, purely second-price auctions are used, especially on some platforms on which collector's items are auctioned. Some providers combine different bidding channels. For example, the stamp dealer Sandafayre offers weekly second price auctions with extensive options for submitting bids, which can be done not only via the website, but also by post or fax. This would only be possible with difficulty using what is probably the most common auction format, the English auction, and in this respect points to a distinct advantage of sealed bid formats. Other Internet auction platforms such as Ebay do not use a pure second price auction; however, their implementation of the English auction de facto comes very close to the second price format. The price entered by bidders only acts as a so-called proxy bid; The actual bid is, in Ebay's own description , "compared by our system with the bids of other bidders and only increased by the smallest possible amount that is necessary for you to continue to be the highest bidder". This happens at most until the proxy bid is reached. As a result, the victorious highest bidder incurs costs equal to the second highest bid plus a small amount, which corresponds approximately to the result from a second-price auction. Lucking-Riley (2000) examines the auction formats on 142 auction sites on the Internet and finds only five examples of pure second price auctions, but 65 examples of modified English auctions with the possibility of proxy bids.

Second price auctions have also been used in the past for auctioning frequency block licenses. Such licenses mostly relate to the right to use the respective frequency block for a specific purpose - such as the transmission of television signals ; Sometimes they also allow you to decide for yourself how to use the licensed frequency block. When the New Zealand government auctioned such frequency block licenses for the first time in 1990, they conducted several simultaneous second price auctions with no reserve prices (one for each block). As a result, instead of the hoped-for 240 million New Zealand dollars (NZ $), the government achieved only 36 million. In one case, a company that had bid NZ $ 100,000 won the bid for only NZ $ 6; in another, a bid of NZ $ 7,000,000 resulted in a cost of just NZ $ 5,000. The reasons for this are generally the absence of minimum prices and, on a more fundamental level, the failure to take into account interdependencies (certain licenses are in a substitutive or complementary relationship to one another, which cannot be taken into account in the case of second price auctions that are carried out simultaneously and thus leads to unnecessarily "random" results).

Generalizations

Vickrey-Clarke-Groves Mechanism

→ Main article : 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). Based on the work of William Vickrey (1961), Edward H. Clarke (1971) and the generalization of Theodore Groves (1973), the VCG mechanism describes a general method for implementing efficient and incentive-compatible allocations in multi-object auctions. In fact, it can even be shown that the VCG mechanism is the one with the highest revenue of all efficient, incentive-compatible (strategy proof) and individually rational mechanisms. In special cases, the VCG mechanism is reduced to the second price auction. Finally, an additional generalization enables interdependent valuations to be taken into account within the multi-object setting of the VCG mechanism. In this case, however, it is no longer an auction mechanism in the narrower sense because it requires the auctioneer to be familiar with the valuation functions . ${\displaystyle \nu _{i}}$

See the Vickrey-Clarke-Groves Mechanism article for details .

Generalized Second-Price Auction

A further generalization is provided by Edelman, Ostrovsky and Schwarz (2007), who recognize the use of a second price method in the auctioning of advertising space on Internet search engines; Used for the first time in 2002 at the latest, this type of auction is said to have generated revenues of around ten billion dollars as early as 2006. Search engine operators proceed, for example, in such a way that advertising spaces on a page are filled from top to bottom with ads in descending order of the bids submitted for them. If a user then clicks on an ad, the relevant advertising customer will be charged in the amount of the next highest bid as part of a pay-per-click procedure .

In general, the Generalized Second Price Auction (GSP auction) is characterized by the following features: It is a multi-item auction with objects (in the example: advertising spaces) and risk-neutral bidders (advertisers). The number of clicks on an ad on position , in a specified period of time to call with and advertisers , a click on their ad value. (It is irrelevant for the appreciation of such a click on which position an ad clicked was displayed.) For the sake of simplicity and without loss of generality, determine that the positions are numbered in descending order, i.e. for the ad with the highest number of clicks ( ), and so on. If a user of the search engine enters a corresponding keyword, this sets the allocation mechanism in motion: the operator determines the last bid submitted for each bidder and fills the advertising spaces from top to bottom until all are occupied or, alternatively, until the last one participating bidders is allocated, whereby each bidder can in no case occupy more than one place. Be the -highest bidder and be his bid. Then the payoff of amounts to ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle i}$ ${\displaystyle 1\leq i\leq m}$ ${\displaystyle \alpha _{i}}$ ${\displaystyle k}$ ${\displaystyle 1\leq k\leq n}$ ${\displaystyle s_{k}}$ ${\displaystyle i=1}$ ${\displaystyle \alpha _{i}}$ ${\displaystyle k}$ ${\displaystyle b_{k}}$ ${\displaystyle g(i)}$ ${\displaystyle i}$ ${\displaystyle b(i)}$ ${\displaystyle i}$

${\displaystyle \alpha _{i}\cdot \left[s_{g(i)}-b(i+1)\right]}$.

The authors show that in a GSP auction, unlike in the context of the VCG mechanism, it is not a dominant strategy to bid according to one's own appreciation; Correspondingly, a large amount of strategic bidding behavior has already been empirically proven in this market on various occasions. In contrast to the VCG mechanism in general, there is no balance in dominant strategies.

In some more recent applications of the model, the bidder side of the auction is explicitly modeled, which conceptually sets it apart from the classic GSP setting. In this context, it is also possible to model externalities that arise under the assumption that the number of clicks is greater for advertisements placed higher than those placed further down; This then has the consequence that successful bidders with a higher ranking negatively influence the average return of the other advertisers.

Web links

• Benedikt Fehr : Second price auctions: Conceived by Goethe, used by Ebay. FAZ.net , December 22, 2007.
• Takashi Kunimoto: Lecture Notes on Auctions (PDF file, 0.7 MB). Lecture notes, McGill University, 2008.

literature

• Lawrence M. Ausubel: Auctions (Theory). In: Steven N. Durlauf, Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, doi : 10.1057 / 9780230226203.0073 (online edition).
• Paul Klemperer (Ed.): The Economic Theory of Auctions. 2 volumes. Edward Elgar, Cheltenham and Northampton 2000, ISBN 1-85898-870-5 (both volumes). [Collection of important articles and contributions to auction theory.]
• Vijay Krishna: Auction Theory. 2nd Edition. Academic Press, San Diego et al. a. 2010, ISBN 978-0-12-374507-1 .
• Jayson L. Lusk and Jason F. Shogren: Experimental Auctions. Methods and Applications in Economic and Marketing Research. Cambridge University Press, Cambridge u. a. 2007, ISBN 978-0-521-85516-7 .
• Andreu Mas-Colell, Michael Whinston, and Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-19-507340-1 .
• Flavio M. Menezes and Paulo K. Monteiro: An Introduction to Auction Theory. Oxford University Press, Oxford and New York 2005, ISBN 978-0-19-927598-4 .
• Paul Milgrom: Putting Auction Theory to Work. Cambridge University Press, Cambridge u. a. 2004, ISBN 0-521-53672-3 .
• Eric Rasmusen: Games and Information. An Introduction to Game Theory. 4th ed. Wiley-Blackwell, Malden 2007, ISBN 978-1-4051-3666-2 . [Chapter 13: Auctions; as a draft version also online: http://www.rasmusen.org/GI/chapters/chap13_auctions.pdf (PDF file, 0.6 MB)]
• Nikolaĭ N. Vorob'ev: Game Theory. Lectures for Economists and Systems Scientists. Translated by Samuel Kotz. Springer, New York a. a. 1977, ISBN 0-387-90238-4 .

Remarks

1. ^ William Vickrey: Counterspeculation, Auctions, and Competitive Sealed Tenders. In: The Journal of Finance. 16, No. 1, 1961, pp. 8-37 ( JSTOR 2977633 ).
2. ↑ This is the subject of the #Strategic Relationship section .
3. ↑ For example Michael H. Rothkopf, Thomas J. Teisberg and Edward P. Kahn: Why Are Vickrey Auctions Rare? In: Journal of Political Economy. 98, No. 1, 1990, pp. 94-109 ( JSTOR 2937643 ), here p. 95.
4. ^ David Lucking-Reiley: Vickrey Auctions in Practice: From Nineteenth-Century Philately to Twenty-First-Century E-Commerce. In: The Journal of Economic Perspectives. 14, No. 3, 2000, pp. 183-192 ( JSTOR 2646925 ).
5. ^ Benny Moldovanu and Manfred Tietzel: Goethe's Second-Price Auction. In: Journal of Political Economy. 106, No. 4, 1998, pp. 854-859 ( JSTOR ).
6. ↑ Quotation from Inge Jensen (ed.): Sources and testimonials on the printing history of Goethe's works. Part 4: The individual prints. Akademie-Verlag Berlin, Berlin 1984, p. 650. Allegedly the content of this ticket read as follows: “I am sending you a manuscript in the sealed connection. If Mr. Vieweg does not want to pay 200 Friedrichsd'or for this, he can return the pack without unsealing it. ”Jensen doubts the authenticity of the same.
   From an auction theoretical point of view, the result of this auction was ultimately not very productive. Goethe's adviser Karl August Böttiger wrote to Vieweg on January 16: “Now it came down to the main point, the fee. I don't want to compromise myself, he says, but I don't want to hurt the publisher either. Now he communicated the thought to me, which can also be read on the attached slip of paper signed by himself. The sealed ticket […] is really in my office. So you say what you can and will give? // I place myself in your position, dearest viewer, and feel what a bystander who is your friend can feel. // After what I already know about Göschen, Bertuch, Cotta and Unger's fees, allow me to add only one thing: you cannot bid below 200 Fr [iedsrichs] d'or. ”(Quoted from Jensen op. cit., p. 651) In fact, this was of course exactly the amount that Goethe had written on the slip of paper and Vieweg finally offered exactly 200 Friedrichsd'or; It is obvious that Böttiger gave Vieweg a corresponding hint with his estimate. Accordingly, Jensen speculates, “[m] ortiger knew the amount of conversations demanded by Goethe” (Jensen ibid.). For job certificates cf. Benny Moldovanu and Manfred Tietzel: Goethe's Second-Price Auction. In: Journal of Political Economy. 106, No. 4, 1998, pp. 854-859 ( JSTOR ).
7. ↑ See, for example, R. Preston McAfee and John McMillan: Auctions and Bidding. In: Journal of Economic Literature. 25, No. 2, 1987, pp. 699-738 ( JSTOR 2726107 ), here pp. 705 f .; the terminology follows Krishna 2010, p. 2 f.
8. ^ John C. Harsanyi: Games with Incomplete Information Played by “Bayesian” Players, I – III. Part I. The Basic Model. In: Management Science. 14, No. 3, 1967, pp. 159-182 ( JSTOR 2628393 ).
9. ^ John C. Harsanyi: Games with Incomplete Information Played by "Bayesian" Players, I-III. Part II. Bayesian Equilibrium Points. In: Management Science. 14, No. 5, 1968, pp. 320-334 ( JSTOR 2628673 ); John C. Harsanyi: Games with Incomplete Information Played by "Bayesian" Players, I-III. Part III. The Basic Probability Distribution of the Game. In: Management Science. 14, No. 7, 1968, pp. 486-502 ( JSTOR 2628894 ).
10. ↑ See the article Bayes game for details .
11. ↑ Instead of the utility function - which expresses preferences in general - often only a disbursement / payoff function is considered. Conceptually, however, it is by no means necessary that actors only focus on monetary gains or losses. In auctions in the traditional sense, however, the monetary interpretation is usually obvious, which is why this article also uses the corresponding terminology.
12. ↑ See Krishna 2010, p. 13; Mas-Colell / Whinston / Green 1995, p. 865, footnote 8.
13. ^ William Vickrey: Counterspeculation, Auctions, and Competitive Sealed Tenders. In: The Journal of Finance. 16, No. 1, 1961, pp. 8-37 ( JSTOR 2977633 ).
14. ↑ See Krishna 2010, p. 13.
15. ↑ See Ausubel 2008. For more on the term, see John O. Ledyard: Incentive Compatibility. In: Steven N. Durlauf, Lawrence E. Blume (Eds.): The New Palgrave Dictionary of Economics. 2nd Edition. Palgrave Macmillan 2008, doi : 10.1057 / 9780230226203.0769 (online edition).
16. ^ Based on Daron Acemoglu and Asu Ozdaglar: Incomplete Information: Bayesian Nash Equilibria, Auctions and Introduction to Social Learning. Lecture Notes, MIT. 2009, accessed on July 28, 2013, p. 28.
17. ↑ See Paul R. Milgrom: Rational Expectations, Information Acquisition, and Competitive Bidding. In: Econometrica. 49, No. 4, 1981, pp. 921-943 ( JSTOR 1912511 ); for the following example ibid, p. 939.
18. ↑ Reinhard Selten : Reexamination of the perfectness concept for equilibrium points in extensive games. In: International Journal of Game Theory. 4, No. 1, 1975, pp. 25-55, doi : 10.1007 / BF01766400 .
19. ^ Andreas Blume and Paul Heidhues: All equilibria of the Vickrey auction. In: Journal of Economic Theory. 114, No. 1, 2004, pp. 170-177, doi : 10.1016 / S0022-0531 (03) 00104-2 .
20. ↑ See for example Kunimoto 2008, p. 105.
21. ↑ This terminology differs from that used in statistical literature. There the -th order statistic is not a random variable of the -highest value, but of the -lowest. ${\displaystyle k}$ ${\displaystyle k}$ ${\displaystyle k}$
22. ↑ See Krishna 2010, pp. 13 f., 17 f.
23. ↑ See Krishna 2010, pp. 17 ff.
24. ↑ It is

    ${\displaystyle n\int _{0}^{\overline {v}}m(v)f(v)\mathrm {d} v=n\int _{0}^{\overline {v}}\left(\int _{0}^{v}zg(z)\mathrm {d} z\right)f(v)\mathrm {d} v}$

    according to the definition of (see above). After swapping the integration variables, this is equivalent to ${\displaystyle m(v_{i})}$

    ${\displaystyle n\int _{0}^{\overline {v}}\left(\int _{z}^{\overline {v}}f(v)\mathrm {d} v\right)zg(z)\mathrm {d} z=n\int _{0}^{\overline {v}}\left(1-F(z)\right)zg(z)\mathrm {d} z}$.

    The density function of the second order statistics regarding all value estimates , but is contaminated thereto, and therefore the expected revenue equals with the above definition of the density of the first-order statistics of the estimated value of the remaining straight bidder ${\displaystyle Y_{2:n}}$ ${\displaystyle f_{2:n}}$ ${\displaystyle n(1-F(Y))f_{1:n-1}(Y)}$ ${\displaystyle g}$ ${\displaystyle n-1}$

    ${\displaystyle \int _{0}^{\overline {v}}zf_{2:n}(z)\mathrm {d} z=\mathbb {E} (Y_{2:n})}$,

    what was to be shown. See Krishna 2010, p. 17 ff.
25. ↑ See also Krishna 2010, p. 21.
26. ^ John G. Riley and William F. Samuelson: Optimal Auctions. In: American Economic Review. 71, No. 3, 1981, pp. 381-392 ( JSTOR 1802786 ), here p. 384; see. also Krishna 2010, p. 21.
27. ^ John G. Riley and William F. Samuelson: Optimal Auctions. In: American Economic Review. 71, No. 3, 1981, pp. 381-392 ( JSTOR 1802786 ), here p. 383.
28. ^ John G. Riley and William F. Samuelson: Optimal Auctions. In: American Economic Review. 71, No. 3, 1981, pp. 381-392 ( JSTOR 1802786 ), here pp. 385 f.
29. ^ Jean-Jacques Laffont and Eric Maskin: Optimal Reservation Price in the Vickrey Auction. In: Economics Letters. 6, 1980, pp. 309-313, doi : 10.1016 / 0165-1765 (80) 90002-6 .
30. ↑ See Krishna 2010, p. 22 f. The result is actually valid even for a large class of auction formats. In addition, John G. Riley and William F. Samuelson: Optimal Auctions. In: American Economic Review. 71, No. 3, 1981, pp. 381-392 ( JSTOR 1802786 ).
31. ↑ See Krishna 2010, p. 24.
32. ↑ See Krishna 2010, p. 24 f. An implementation of the commitment problem in the seller's decision problem about the amount of the minimum price is for example in Ela Glowicka and Jonathan Beck: A note on reserve price commitments in the Vickrey auction. Internet http://mpra.ub.uni-muenchen.de/6669/1/MPRA_paper_6669.pdf (PDF file, 0.2 MB), accessed on October 18, 2013 outlined.
33. ↑ Richard Engelbrecht-Wiggans: On Optimal Reservation Prices in Auctions. In: Management Science. 33, No. 6, 1987, pp. 763-770 ( JSTOR 2632261 ).
34. ↑ ^{a b} The terminology follows John H. Kagel and Dan Levin: Auctions. A Survey of Experimental Research, 1995 - 2008. Mimeo (for reprint in John H. Kagel, Alvin E. Roth (Eds.): The Handbook of Experimental Economics. 2nd edition), 2008, Internet http: //www.econ .ohio-state.edu / kagel / Auctions_Handbook_vol2.pdf (PDF file, 2.2 MB), accessed on July 28, 2013.
35. ^ Dan Levin and James L. Smith: Equilibrium in Auctions with Entry. In: The American Economic Review. 84, No. 3, 1994, pp. 585-599 ( JSTOR 2118069 ).
36. ↑ The presentation follows (simplifying) Yeon-Koo Che and Ian Gale: Standard Auctions with Financially Contrained Bidders. In: Review of Economic Studies. 65, No. 1, 1998, doi : 10.1111 / 1467-937X.00033 . See also Krishna 2010, p. 42 ff.
37. ↑ See Krishna 2010, p. 38.
38. ↑ See Isa Hafalir and Vijay Krishna: Asymmetric Auctions with Resale. In: American Economic Review. 98, No. 1, 2008, pp. 87–112, doi : 10.1257 / aer.98.1.87 , here p. 97.
39. ↑ Isa Hafalir and Vijay Krishna: Asymmetric Auctions with Resale. In: American Economic Review. 98, No. 1, 2008, pp. 87–112, doi : 10.1257 / aer.98.1.87 , here p. 98.
40. ^ Estelle Cantillon: The effect of bidders' asymmetries on expected revenue in auctions. In: Games and Economic Behavior. 62, No. 1, 2008, pp. 1–25, doi : 10.1016 / j.geb.2006.11.005 .
41. ↑ Definition: Be and random variables with support and distribution functions respectively . Then stochastically dominates in the first order if the following applies to all : Cf. Marc S. Paolella: Fundamental Probability. A computational approach. Wiley, Sussex 2006, ISBN 978-0-470-02594-9 , p. 144. ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle S\subset \mathbb {R} }$ ${\displaystyle F_{A}}$ ${\displaystyle F_{B}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle x\in S}$ ${\displaystyle F_{A}(x)\geq F_{B}(x)}$
42. ↑ Jihui Chen and Xu Maochao: Asymmetry and Revenue in Second-Price Auctions: A Majorization Approach. Illinois State University Working Paper, 2012, Internet Archive Link ( December 2, 2013 memento on the Internet Archive ), accessed November 22, 2013.
43. ↑ Isa Hafalir and Vijay Krishna: Asymmetric Auctions with Resale. In: American Economic Review. 98, No. 1, 2008, pp. 87–112, doi : 10.1257 / aer.98.1.87 , here p. 99.
44. ↑ Definition: The distributions , are regular in the sense of Myerson (1981), if: ${\displaystyle F_{i}}$ ${\displaystyle i\in {\mathcal {I}}}$

    ${\displaystyle v_{i}-{\frac {1-F_{i}(v_{i})}{f_{i}(v_{i})}}}$

    is strictly increasing in . See Roger B. Myerson: Optimal Auction Design. In: Mathematics of Operations Research. 6, No. 1, 1981, pp. 58-73 ( JSTOR 3689266 ). ${\displaystyle v_{i}}$
45. ↑ René Kirkegaard: A Mechanism Design Approach to Ranking Asymmetric Auctions. In: Econometrica. 80, No. 5, pp. 2349-2364, 2012, doi : 10.3982 / ECTA9859 .
46. ↑ Eric Maskin and John Riley: Asymmetric Auctions. In: The Review of Economic Studies. 67, No. 3, 2000, pp. 413-438 ( JSTOR 2566960 ).
47. ↑ For the following cf. Krishna 2010, p. 158 f.
48. ^ Daniel A. Graham, Robert C. Marshall: Collusive Bidder Behavior at Single-Object Second-Price and English Auctions. In: Journal of Political Economy. 95, No. 6, 1987 ( JSTOR 1831119 ).
49. ^ Daniel A. Graham and Robert C. Marshall: Collusive Bidder Behavior at Single-Object Second-Price and English Auctions. In: Journal of Political Economy. 95, No. 6, 1987 ( JSTOR 1831119 ), Theorem 2 and 3.
50. ↑ See Krishna 2010, p. 163 ff.
51. ↑ Michael H. Rothkopf, Thomas J. Teisberg and Edward P. Kahn: Why Are Vickrey Auctions Rare? In: Journal of Political Economy. 98, No. 1, 1990, pp. 94-109 ( JSTOR 2937643 ).
52. ↑ See David Lucking-Reiley: Vickrey Auctions in Practice: From Nineteenth-Century Philately to Twenty-First-Century E-Commerce. In: The Journal of Economic Perspectives. 14, No. 3, 2000, pp. 183-192 ( JSTOR 2646925 ), here pp. 189-190.
53. ^ See, for example, David Lucking-Reiley: Auctions on the Internet: What's Being Auctioned, and How? In: The Journal of Industrial Economics. 48, No. 3, pp. 227-252, doi : 10.1111 / 1467-6451.00122 .
54. ^ With Robert B. Wilson: Competitive Bidding with Disparate Information. In: Management Science. 15, No. 7, 1969, pp. 446–448 (the article was already presented as a working paper in 1966) and Armando Ortega-Reichert: Models for Competitive Bidding under Uncertainty. PhD Thesis, Stanford University, 1968 (partly reprinted in Klemperer 2000), common value models were opposed to the IPV model early on; Paul R. Milgrom: Rational Expectations, Information Acquisition, and Competitive Bidding. In: Econometrica. 49, No. 4, 1981, pp. 921-943 ( JSTOR 1912511 ) analyzes the symmetrical equilibrium in a common-value second-price auction. The generalist approach of auction models with interdependent valuations described and also used below follows - for all common auction formats - Paul R. Milgrom and Robert J. Weber: A Theory of Auctions and Competitive Bidding. In: Econometrica. 50, No. 5, 1982, pp. 1089-1122 ( JSTOR 1911865 ).
55. ↑ The presentation follows Kunimoto 2008, chapter 7; similar to Krishna 2010, chapter 6.
56. ↑ See for details Kunimoto 2008, p. 91 ff .; Milgrom 2004, p. 181 ff .; Krishna 2010, Appendix D.
57. ↑
    

    Component-wise maximum and minimum.
    Formally, let the random variables with a common strictly positive density be distributed over a product of intervals . is called (positively) affiliated if and only if that applies to all ${\displaystyle \mathbf {S} =S_{1},\ldots ,S_{n}}$ ${\displaystyle f(\mathbf {S} )}$ ${\displaystyle {\mathcal {S}}=\times _{i\in {\mathcal {I}}}[0,{\overline {s}}]\subset \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {S} }$ ${\displaystyle \mathbf {s} ',\mathbf {s} ''\in {\mathcal {S}}}$

    ${\displaystyle f(\mathbf {s} '\downarrow \mathbf {s} '')f(\mathbf {s} '\uparrow \mathbf {s} '')\geq f(\mathbf {s} ')f(\mathbf {s} '')}$.

    Here referred to the component-wise minimum of and , therefore , and the component-wise maximum . ${\displaystyle \mathbf {s} '\downarrow \mathbf {s} ''}$ ${\displaystyle \mathbf {s} '}$ ${\displaystyle \mathbf {s} ''}$ ${\displaystyle \left(\min(s_{1}',s_{1}''),\ldots ,\min(s_{n}',s_{n}'')\right)}$ ${\displaystyle \mathbf {s} '\uparrow \mathbf {s} '}$ ${\displaystyle \left(\max(s_{1}',s_{1}''),\ldots ,\max(s_{n}',s_{n}'')\right)}$

    By taking the logarithm, it is easy to show that the random variables (in this case the signals) are affiliated if and only if the density function of the random variables is log-supermodular, where log-supermodular is if and only if ${\displaystyle f}$

    ${\displaystyle \ln f(\mathbf {s} '\downarrow \mathbf {s} '')+\ln(\mathbf {s} '\uparrow \mathbf {s} '')\geq \ln f(\mathbf {s} ')+\ln f(\mathbf {s} '')}$.

    The latter condition is, as Topkis (1978) already pointed out, with two continuous differentiations of equivalent to that ${\displaystyle f}$

    ${\displaystyle {\frac {\partial ^{2}\ln f}{\partial s_{i}\partial s_{j}}}\geq 0}$

    for all , . See Donald M. Topkis: Minimizing a Submodular Function on a Lattice. In: Operations Research. 26, No. 2, 1978, pp. 305-321 ( JSTOR 169636 ), here p. 310; Kunimoto 2008, p. 93 (Claim 7.2); Milgrom 2004, p. 182. ${\displaystyle i,j\in {\mathcal {I}}}$ ${\displaystyle i\neq q}$

58. ^ Paul R. Milgrom and Robert J. Weber: A Theory of Auctions and Competitive Bidding. In: Econometrica. 50, No. 5, 1982, pp. 1089-1122 ( JSTOR 1911865 ), here pp. 1100 f.
59. ↑ See Krishna 2010, p. 98.
60. ↑ For the proof cf. Kunimoto 2008, p. 99 f .; Krishna 2010, p. 98 f .; Menezes and Monteiro 2005, p. 68 ff.
61. ↑ For the proof cf. Kunimoto 2008, p. 98 f .; Krishna 2010, p. 97 f .; Menezes and Monteiro 2005, p. 68 ff.
62. ↑ The intuition follows Paul Klemperer: Auction Theory. A Guide to the Literature. In: Journal of Economic Surveys. 13, No. 3, pp. 227-286, 1999, doi : 10.1111 / 1467-6419.00083 , here p. 235.
63. ^ Paul Klemperer: Auction Theory. A Guide to the Literature. In: Journal of Economic Surveys. 13, No. 3, pp. 227-286, 1999, doi : 10.1111 / 1467-6419.00083 , here pp. 259 f.
64. ↑ The setting is popular in both theoretical and experimental literature. Originally it should probably refer to John H. Kagel and Dan Levin: The Winner's Curse and Public Information in Common Value Auctions. In: The American Economic Review. 76, No. 5, 1986, pp. 894-920 ( JSTOR 1816459 ).
65. ↑ In detail on this subject John H. Kagel and Dan Levin: Common Value Auctions and the Winner's Curse. Princeton University Press, Princeton et al. a. 2002, ISBN 9780691016672 .
66. ↑ See John H. Kagel: Auctions. A Survey of Experimental Research. In: John H. Kagel, Alvin E. Roth (Eds.): The Handbook of Experimental Economics. Princeton University Press, Princeton and New Jersey 1995, pp. 501-585, here pp. 536 ff.
67. ^ John H. Kagel, Dan Levin and Ronald M. Harstad: Comparative static effects of number of bidders and public information on behavior in second-price common value auctions. In: International Journal of Game Theory. 24, No. 3, 1995, pp. 293-319, doi : 10.1007 / BF01243157 .
68. ↑ Christopher Avery and John H. Kagel, Second-Price Auctions with Asymmetric Payoffs: An Experimental Investigation. In: Journal of Economics & Management Strategy. 6, No. 3, 1997, pp. 573-603, doi : 10.1111 / j.1430-9134.1997.00573.x .
69. ↑ Christopher Avery and John H. Kagel, Second-Price Auctions with Asymmetric Payoffs: An Experimental Investigation. In: Journal of Economics & Management Strategy. 6, No. 3, 1997, pp. 573-603, doi : 10.1111 / j.1430-9134.1997.00573.x , here pp. 587 f.
70. ↑ Vincent P. Crawford and Nagore Iriberri: Level-k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner's Curse and Overbidding in Private-Value Auctions? In: Econometrica. 75, No. 6, 2007, pp. 1721-1770, doi : 10.1111 / j.1468-0262.2007.00810.x .
71. ^ Dale O. Stahl and Paul W. Wilson: On Players ′ Models of Other Players: Theory and Experimental Evidence. In: Games and Economic Behavior. 10, No. 1, 1995, pp. 218-254, doi : 10.1006 / game.1995.1031 .
72. ^ Rosemarie Nagel: Unraveling in Guessing Games: An Experimental Study. In: The American Economic Review. 85, No. 5, 1995, pp. 1313-1326 ( JSTOR 2950991 ).
73. ↑ Erik Eyster and Matthew Rabin: Cursed Equilibrium. In: Econometrica. 73, No. 5, 2005, pp. 1623-1672, doi : 10.1111 / j.1468-0262.2005.00631.x .
74. ↑ Asen Ivanov, Dan Levin and Muriel Niederle: Can Relaxation of Beliefs Rationalize the Winner's Curse? An experimental study. In: Econometrica. 78, No. 4, 2010, pp. 1435-1452, doi : 10.3982 / ECTA8112 .
75. ↑ Analog Vorob'ev 1977, p. 3 f .; Rodica Branzei, Dinko Dimitrov and Stef Tijs: Models in Cooperative Game Theory. 2nd Edition. Springer, Heidelberg a. a. 2008, ISBN 978-3-540-77953-7 (also doi : 10.1007 / 978-3-540-77954-4 ), p. 8; Robert J. Weber: Games in coalitional form. In: Robert Aumann and Sergiu Hart (Eds.): Handbook of Game Theory with Economic Applications. Vol. 2. Elsevier, Amsterdam a. a. 1994, ISBN 0-444-89427-6 , pp. 1285-1303 (also doi : 10.1016 / S1574-0005 (05) 80068-2 ), here pp. 1288 ff.
76. ↑ See Vorob'ev 1977, p. 4.
77. ↑ See for example Paul R. Milgrom and Robert J. Weber: A Theory of Auctions and Competitive Bidding. In: Econometrica. 50, No. 5, 1982, pp. 1089-1122 ( JSTOR 1911865 ), here pp. 1091 f.
78. ↑ On this for example generally Krishna 2010, p. 4 f.
79. ↑ See Krishna 2010, p. 5.
80. ↑ An overview of the literature up to 1995 can be found in John H. Kagel: Auctions. A Survey of Experimental Research. In: John H. Kagel, Alvin E. Roth (Eds.): The Handbook of Experimental Economics. Princeton University Press, Princeton and New Jersey 1995, pp. 501-585, here pp. 508-514 and on the literature between 1995 and 2008 in Dems. and Dan Levin: Auctions. A Survey of Experimental Research, 1995 - 2008. Mimeo (for reprint in John H. Kagel, Alvin E. Roth (Eds.): The Handbook of Experimental Economics. 2nd edition), 2008, Internet http: //www.econ .ohio-state.edu / kagel / Auctions_Handbook_vol2.pdf (PDF file, 2.2 MB), accessed on July 28, 2013. Cf. also Lusk and Shogren 2007, pp. 27–33.
    Regarding the result of non-replicability, it should be noted that in the earlier literature, Vicki M. Coppinger, Vernon L. Smith and Jon A. Titus: Incentives and Behavior in English, Dutch and Sealed-Bid Auctions. In: Economic Inquiry. 18, No. 1, 1980, pp. 1-22, doi : 10.1111 / j.1465-7295.1980.tb00556.x . Find evidence for a match between appreciation and bid size. From the outset, however, the authors did not allow any bids above the estimate.
81. ^ John H. Kagel, Ronald M. Harstad, and Dan Levin: Information Impact and Allocation Rules in Auctions with Affiliated Private Values: A Laboratory Study. In: Econometrica. 55, No. 6, 1987, pp. 1275-1304 ( JSTOR 1913557 ).
82. ↑ Ronald M. Harstad: Dominant Strategy Adoption and Bidders' Experience with Pricing Rules. In: Experimental Economics. 3, No. 3, 2000, pp. 261-280, doi : 10.1007 / BF01669775 .
83. ^ John H. Kagel and Dan Levin: Independent Private Value Auctions: Bidder Behavior in First-, Second- and Third-Price Auctions with Varying Numbers of Bidders. In: The Economic Journal. 103, No. 419, 1993, pp. 868-879 ( JSTOR 2234706 ).
84. ^ David Lucking-Reiley: Using Field Experiments to Test Equivalence between Auction Formats: Magic on the Internet. In: The American Economic Review. 89, No. 5, 1999, pp. 1063-1080 ( JSTOR 117047 ).
85. ↑ Jason F. Shogren et al. a .: A random nth-price auction. In: Journal of Economic Behavior & Organization. 46, No. 4, 2001, pp. 409-421, doi : 10.1016 / S0167-2681 (01) 00165-2 .
86. ^ John Morgan, Ken Steiglitz and George Reis: The Spite Motive and Equilibrium Behavior in Auctions. In: Contributions in Economic Analysis & Policy. 2, No. 1, Article 5, doi : 10.2202 / 1538-0645.1102 .
87. ↑ James Andreonia, Yeon-Koo Che and Jinwoo Kimc: Asymmetric information about rivals' types in standard auctions: An experiment. In: Games and Economic Behavior. 59, No. 2, 2007, pp. 240-259, doi : 10.1016 / j.geb.2006.09.003 .
88. ^ David J. Cooper and Hanming Fang: Understanding Overbidding in Second Price Auctions: An Experimental Study. In: The Economic Journal. 118, No. 532, pp. 1572-1595, doi : 10.1111 / j.1468-0297.2008.02181.x .
89. ^ Rodney J. Garratt, Mark Walker and John Wooders: Behavior in second-price auctions by highly experienced eBay buyers and sellers. UCSB Working Paper, 2004, Internet http://econ.ucsb.edu/~garratt/faculty/gww.pdf (PDF file, 0.3 MB), accessed on July 28, 2013.
90. ^ Rodney J. Garratt, Mark Walker and John Wooders: Behavior in second-price auctions by highly experienced eBay buyers and sellers. In: Experimental Economics. 15, No. 1, 2012, pp. 44-57, doi : 10.1007 / s10683-011-9287-3 .
91. ^ Alvin E. Roth and Dan Levin: Auctions. A Survey of Experimental Research, 1995 - 2008. Mimeo (for reprint in John H. Kagel and Alvin E. Roth (Eds.): The Handbook of Experimental Economics. 2nd edition), 2008, Internet http: //www.econ .ohio-state.edu / kagel / Auctions_Handbook_vol2.pdf (PDF file, 2.2 MB), accessed on July 28, 2013, here p. 13.
92. ↑ In this sense, instead of many Michael H. Rothkopf, Thomas J. Teisberg and Edward P. Kahn: Why Are Vickrey Auctions Rare? In: Journal of Political Economy. 98, No. 1, 1990, pp. 94-109 ( JSTOR 2937643 ); Lawrence M. Ausubel and Paul Milgrom: The Lovely but Lonely Vickrey Auction. SIEPR Discussion Paper No. 03-36, 2004, Internet http://www-siepr.stanford.edu/Papers/pdf/03-36.pdf , accessed on November 26, 2013, p. 1 f.
93. ^ David Lucking-Reiley: Auctions on the Internet: What's Being Auctioned, and How? In: The Journal of Industrial Economics. 48, No. 3, pp. 227-252, doi : 10.1111 / 1467-6451.00122 .
94. ↑ How do your sales work? ( Memento of November 25, 2013 in the Internet Archive ) In: sandafayre.com.
95. ↑ See Axel Ockenfels, David H. Reiley and Abdolkarim Sadrieh: Online Auctions. In: Terrence Hendershott (Ed.): Handbook of Information Systems. Vol. 1 (Economics and Information Systems). Elsevier, 2006, ISBN 978-0444517715 , pp. 571-628, here pp. 578 f.
96. ↑ Everything about bidding. ( Memento from November 26, 2013 in the Internet Archive ) In: ebay.de.
97. ↑ In purely formal terms, the Ebay format lacks a dominant strategy. However, it can be shown that strategies with commandments above one's own appreciation are dominant, cf. Axel Ockenfels and Alvin E. Roth: Late and multiple bidding in second price Internet auctions: Theory and evidence concerning different rules for ending an auction. In: Games and Economic Behavior. 55, No. 2, 2006, pp. 297-320, doi : 10.1016 / j.geb.2005.02.010 , here p. 301.
98. ^ David Lucking-Reiley: Auctions on the Internet: What's Being Auctioned, and How? In: The Journal of Industrial Economics. 48, No. 3, pp. 227-252, doi : 10.1111 / 1467-6451.00122 .
99. ↑ On this, Milgrom 2004, pp. 9-13.
100. ↑ See John McMillan: Selling Spectrum Rights. In: The Journal of Economic Perspectives. 8, No. 3, 1994, pp. 145-162 ( JSTOR 2138224 ), here p. 148.
101. ↑ See Milgrom 2004, p. 13.
102. ↑ On this, for example Krishna 2010, pp. 75 ff.
103. ^ William Vickrey: Counterspeculation, Auctions, and Competitive Sealed Tenders. In: Journal of Finance. 16, No. 1, 1961, pp. 8-37, doi : 10.1111 / j.1540-6261.1961.tb02789.x .
104. ^ Edward H. Clarke: Multipart Pricing of Public Goods. In: Public Choice. 11, 1971, pp. 17-33 ( JSTOR 30022651 ).
105. ^ Theodore Groves: Incentives in Teams. In: Econometrica. 41, No. 4, 1973, pp. 617-631 ( JSTOR 1914085 ).
106. ↑ In this sense, the VCG mechanism is not a specific auction rule; although often used there, it can be used for social choice problems in general.
107. ↑ See Krishna 2010, p. 75.
108. ↑ On this, for example Krishna 2010, p. 148 ff.
109. ^ Benjamin Edelman, Michael Ostrovsky and Michael Schwarz: Internet Advertising and the Generalized Second-Price Auction. Selling Billions of Dollars Worth of Keywords. In: American Economic Review. 97, No. 1, 2007, pp. 242-259, doi : 10.1257 / aer.97.1.242 .
110. ↑ In this sense also Hal R. Varian: Position Auctions. In: International Journal of Industrial Organization. 25, No. 6, 2007, pp. 1163–1178, doi : 10.1016 / j.ijindorg.2006.10.002 , here p. 1163 f. Varian and Edelman, Ostrovsky and Schwarz (2007) formalize the same auction structure independently of one another in the market for advertisements in search engines.
111. ↑ See for example Benjamin Edelman and Michael Ostrovsky: Strategic bidder behavior in sponsored search auctions. In: Decision Support Systems. 43, No. 1, 2007, pp. 192-198, doi : 10.1016 / j.dss.2006.08.008 ; Matthew Cary et al. a .: Greedy Bidding Strategies for Keyword Auctions. In: Proceedings of the 8th ACM conference on Electronic commerce. 2007, pp. 262-271, doi : 10.1145 / 1250910.1250949 .
112. ↑ Renato Gomes, Nicole Immorlica, Evangelos Markakis: Externalities in Keyword Auctions: An Empirical and Theoretical Assessment. In: Lecture Notes in Computer Science. 5929, 2009, pp. 172-183, doi : 10.1007 / 978-3-642-10841-9_17 .

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