First price auction

The first price auction (also first price sealed bid auction) is an auction in which the bidders submit their bids once and concealed. The bidder with the highest bid wins the auction and has to pay his own highest bid.

In contrast to the first price auction, there is the second price auction , in which the bidders also submit their bids once and in secret and the bidder with the highest bid wins the auction; however, he only has to pay the second highest bid.

If the object to be auctioned is of purely private value and the bidders are risk-neutral , the first price auction is strategically equivalent to the Dutch auction , while the second price auction is strategically equivalent to the English auction.

Auction history

First auctions appear for the first time in Greek documents 500 BC. Chr. On. At that time women were auctioned off in a kind of Dutch auction . While very pretty women got relatively high bids, the seller had to give a dowry or other monetary offer to less attractive women in order to successfully close the auction. In fact, it was forbidden to sell women outside of an auction.

At the time of Jesus Christ , auctions were popular in the Roman Empire to sell parts of the family estate or the spoils of war . For example, the Roman emperor Mark Aurel auctioned furniture to pay off his debts .

Auctions in the United States can be traced back to the early 17th century when the first Pilgrim Fathers moved there. Plants , imports , shingles, animals , tools , tobacco , slaves and even entire farms were sold through auctions . Selling through auctions was the fastest, most efficient way to turn possessions into cash.

At the time of the civil war in the USA, the name " Colonel " , which is still partly used today, was created for an auctioneer: At that time, the colonels of the military usually sold spoils of war.

In Europe , records of auctions appear for the first time in the " Oxford English Dictionary " in 1595. In the late 17th century the London Gazette wrote about auctioning art in coffee houses and inns . The famous auction houses Sotheby’s and Christie’s were founded in 1744 and 1766 respectively.

The first auctions in the Netherlands were held in 1887 to sell fruit and vegetables . At the same time, fishermen in Germany were selling their catch through auctions.

At fish markets in Japan , dried fish used to be auctioned through the first price auction. The procedure was as follows: The bidders submitted their bids in a box on a slip of paper. After a predetermined time, the auctioneer opened the box and announced the winner.

Today, many central banks such as the German Bundesbank , the European Central Bank or the Treasury of the United States of America use the first price auction to award government bonds . The so-called multi-price auction procedure (English: discriminatory auction) is mostly used, in which there can be several surcharges at different interest rates .

In addition, the first price auction is usually used as the award procedure when awarding construction contracts. However, the role of buyer and seller is reversed here. Therefore, the bidder with the lowest bid wins.

A variant of the first price auction is the so-called " Swiss auction": This type of auction is also used when building contracts are awarded, but with the difference that the winner of the auction can also reject the auctioned item. The name comes from the fact that the Swiss construction industry partly uses this award procedure for construction contracts. Architects prefer this type of auction because construction contracts keep changing the actual contract and there is no reason to work with someone who doesn't want to do the work.

Optimal bidding strategy for continuously distributed reviews

The optimal bidding strategy of a bidder for any continuously distributed evaluations is to submit the expected highest evaluation of all other bidders, given this expected highest evaluation is lower than the bidder's own evaluation.

Assumptions

• There are bidders for a single object. Each bidder rates the item to be auctioned , i.e. the maximum amount that the bidder is willing to pay for the item. ${\displaystyle n}$ ${\displaystyle i}$ ${\displaystyle v_{i}}$

• Each evaluation is identical and independent on , with , distributed with the corresponding distribution function and the associated density function . ${\displaystyle v_{i}}$ ${\displaystyle [0,{\overline {v}}]}$ ${\displaystyle v_{i}\in [0,\infty )}$ ${\displaystyle F(x)=P(X\leq x)}$ ${\displaystyle f(x)={\frac {dF(x)}{dx}}=F^{\prime }(x)}$

• ${\displaystyle E[v_{i}]<\infty ,\quad \forall v_{i}\in [0,{\overline {v}}]}$.

• The bidders are risk-neutral and the item to be auctioned is of purely private value.

• The distribution function and the number of bidders are common knowledge , i.e. known to every bidder. ${\displaystyle F(x)}$ ${\displaystyle n,n\in \mathbb {N} }$

• The strategy of a bidder is a function that determines each rating's own bid. ${\displaystyle i}$ ${\displaystyle \beta _{i}:[0,{\overline {v}}]\rightarrow \mathbb {R} _{+}}$

• The balance is a symmetrical balance, so each bidder follows the same strategy . ${\displaystyle b_{i}=\beta (v),\quad \forall i\in \{1,...,n\}}$

• The payment of the bidder with evaluation of the object to be auctioned and bid is ${\displaystyle i}$ ${\displaystyle v_{i}}$ ${\displaystyle b_{i}}$

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

Derivation of the optimal bid strategy

Be the gamer's command . It is never optimal to choose a bid , because in this case the bidder will definitely get the object, but he can present himself better by reducing his bid, because he will still get the object but have to pay less. It follows that one only has to consider the case . In addition, a bidder with a rating would never submit a positive bid, as he would then make a loss if he won the auction . So it applies . ${\displaystyle b_{i}}$ ${\displaystyle i}$ ${\displaystyle b_{i}>\beta ({\overline {v}})}$ ${\displaystyle b_{i}\leq \beta ({\overline {v}})}$ ${\displaystyle 0}$ ${\displaystyle \beta (0)=0}$

The bidder gets the object if he makes the highest bid, so . Since is monotonically increasing , the following applies: ${\displaystyle i}$ ${\displaystyle \max _{j\neq i}\beta _{j}(v_{j})<b_{i}}$ ${\displaystyle \beta }$

${\displaystyle \max _{j\neq i}\beta _{j}(v_{j})=\beta _{j}(\max _{j\neq i}v_{j})=\beta (Y_{1})}$

with the highest rating of the other players. ${\displaystyle Y_{1}}$ ${\displaystyle n-1}$

The bidder will always be awarded the contract if . ${\displaystyle i}$ ${\displaystyle \beta (Y_{1})<b_{i}\Longleftrightarrow Y_{1}<\beta ^{-1}(b_{i})}$

His expected payout is now

${\displaystyle E[b_{i},b_{-i}^{\ast },v_{i}]=G\left(\beta ^{-1}(b_{i})\right)\cdot (v_{i}-b_{i})}$

with distribution function of . ${\displaystyle G}$ ${\displaystyle Y_{1}}$

Maximizing the expected payout over leads to ${\displaystyle b_{i}}$

${\displaystyle {\frac {\partial E[b_{i},b_{-i}^{\ast },v_{i}]}{\partial b_{i}}}={\frac {g\left(\beta ^{-1}(b_{i})\right)}{\beta '\left(\beta ^{-1}(b_{i})\right)}}\cdot (v_{i}-b_{i})-G\left(\beta ^{-1}(b_{i})\right){\overset {!}{=}}0}$

with density function of . ${\displaystyle g}$ ${\displaystyle Y_{1}}$

Since the equilibrium is symmetrical ( ), the following differential equation now follows : ${\displaystyle b_{i}=\beta (v)}$

${\displaystyle G(v)\beta '(v)+g(v)\beta (v)=vg(v)}$ or

${\displaystyle {\frac {d}{dv}}\left(G(v)\beta (v)\right)=vg(v).}$

With the initial value condition you get the optimal bid strategy: ${\displaystyle \beta (0)=0}$

${\displaystyle \beta (v)={\frac {1}{G(v)}}\int _{0}^{v}yg(y)dy=E[Y_{1}|Y_{1}<v].}$

Or with the help of partial integration :

${\displaystyle \beta (v)=E[Y_{1}|Y_{1}<v]=v-\int _{0}^{v}{\frac {G(y)}{G(v)}}dy.}$

Thus, the optimal bidding strategy of a bidder is to submit the expected highest rating of all other bidders, given this highest rating is lower than his own rating.

Expected proceeds of the seller

The seller's expected revenue is the expected second highest rating of all bidders.

The expected payment from the buyer with the highest bid is

${\displaystyle m(v)=G(v)\cdot E[Y_{1}|Y_{1}<v]=\int _{0}^{v}yg(y)dy.}$

The ex ante expected payment from the buyer is

${\displaystyle E[m(v)]=\int _{0}^{\overline {v}}m(x)f(x)dx=\int _{0}^{\overline {v}}\left(\int _{0}^{v}yg(y)dy\right)f(x)dx=\int _{0}^{\overline {v}}\left(\int _{y}^{\overline {v}}f(x)dx\right)yg(y)dy=\int _{0}^{\overline {v}}y(1-F(y))g(y)dy.}$

The expected revenue of the seller is now

${\displaystyle E[R]=n\cdot E[m(v)]=n\cdot \int _{0}^{\overline {v}}y(1-F(y))g(y)dy.}$

With the help of the order statistics , the expected revenue of the seller now results:

${\displaystyle E[R]=E[Y_{2}]}$

with the second highest rating of all bidders. The expected revenue of the seller is currently the expected second highest rating of all bidders. ${\displaystyle Y_{2}}$ ${\displaystyle n}$ ${\displaystyle n}$

Revenue equivalent to the second price auction

→ Main article : Revenue equivalence theorem

The revenue equivalence theorem states that for goods with a purely private value and risk-neutral bidders, the expected revenue of the seller in the first and second price auction is the same.

Example: Optimal bidding strategy for evenly distributed evaluations of the object to be auctioned

If the evaluations are evenly distributed , then the following applies to the associated density function : ${\displaystyle [0,{\overline {v}}]}$

${\displaystyle f(x)={\begin{cases}{\frac {1}{\overline {v}}}&x\in [0,{\overline {v}}]\\0&x\notin [0,{\overline {v}}]\end{cases}}.}$

From this it follows for the distribution function :

${\displaystyle P(X\leq x)=F(x)=\int _{0}^{x}f(t)dt=\int _{0}^{x}{\frac {1}{\overline {v}}}dt=\left[{\frac {1}{\overline {v}}}t\right]_{0}^{x}={\begin{cases}{\frac {1}{\overline {v}}}x&x\in [0,{\overline {v}}]\\0&x\notin [0,{\overline {v}}]\end{cases}}.}$

The following applies for the distribution of the highest order statistics of the other bidders: ${\displaystyle G(v)}$ ${\displaystyle n-1}$

${\displaystyle G(v)=\left[F(v)\right]^{n-1}=\left({\frac {1}{\overline {v}}}v\right)^{n-1}}$

and this results in the following optimal bid strategy:

${\displaystyle \beta (v)=v-\int _{0}^{v}{\frac {G(y)}{G(v)}}dy=v-\int _{0}^{v}\left({\frac {F(y)}{F(v)}}\right)^{n-1}dy={\frac {n-1}{n}}\cdot v.}$

In particular:

${\displaystyle {\frac {\partial \beta (v)}{\partial n}}={\frac {1}{n^{2}}}\cdot v>0,\quad \forall v>0,}$

${\displaystyle \lim \limits _{n\to \infty }\beta (v)=\lim \limits _{n\to \infty }{\frac {n-1}{n}}\cdot v=v.}$

The bid is strictly increasing in the number of bidders and with a large number of bidders the bid goes against the own evaluation of the property and thus against the payment . ${\displaystyle v}$ ${\displaystyle 0}$

Extensions

Risk-averse bidders

Risk-averse bidders have higher equilibrium bids than risk-neutral bidders.

Each bidder now has a Von Neumann Morgenstern utility function with , and as a disbursement function . Instead of maximizing the expected payout as in the case of risk neutrality , the expected benefit is now maximized. The equilibrium strategies are characterized by a growing and differentiable function with given. The optimization problem of a bidder with evaluation is therefore through ${\displaystyle u\colon \mathbb {R} \rightarrow \mathbb {R} _{+}}$ ${\displaystyle u(0)=0}$ ${\displaystyle u'>0}$ ${\displaystyle u''<0}$ ${\displaystyle \gamma :[0,{\overline {v}}]\rightarrow \mathbb {R} }$ ${\displaystyle \gamma (0)=0}$ ${\displaystyle i}$ ${\displaystyle v_{i}}$

${\displaystyle \max _{b_{i}\in [0,{\overline {v}}]}G(b_{i})\cdot u(v_{i}-\gamma (b_{i})).}$

given. The first order condition is now

${\displaystyle g(b_{i})\cdot u(v_{i}-\gamma (b_{i}))-G(b_{i})\cdot \gamma '(b_{i})\cdot u'(v_{i}-\gamma (b_{i})){\overset {!}{=}}0.}$

In symmetrical equilibrium, the following applies to all bidders and thus: ${\displaystyle b_{i}=v}$ ${\displaystyle i}$

${\displaystyle \gamma '(v)={\frac {u(v-\gamma (v))}{u'(v-\gamma (v))}}\cdot {\frac {g(v)}{G(v)}}.}$

If the bidders are risk-neutral, and thus applies ${\displaystyle u(v)=v}$

${\displaystyle \beta '(v)=(v-\beta (v))\cdot {\frac {g(v)}{G(v)}}.}$

Here refers to the equilibrium strategy for risk-neutral bidders. Since is strictly concave and , applies and thus ${\displaystyle \beta (\cdot )}$ ${\displaystyle u(\cdot )}$ ${\displaystyle u(0)=0}$ ${\displaystyle {\frac {u(v)}{u'(v)}}>v,\quad \forall v>0}$

${\displaystyle \gamma '(v)={\frac {u(v-\gamma (v))}{u'(v-\gamma (v))}}\cdot {\frac {g(v)}{G(v)}}>(v-\gamma (v))\cdot {\frac {g(v)}{G(v)}}.}$

If also applies . According to the assumption , it now follows : ${\displaystyle \beta (v)>\gamma (v)}$ ${\displaystyle \beta '(v)<\gamma '(v)}$ ${\displaystyle \beta (0)=\gamma (0)=0}$ ${\displaystyle \forall v>0}$

${\displaystyle \gamma (v)>\beta (v).}$

In this way, risk-averse bidders have higher equilibrium bids than risk-neutral bidders . The risk-averse bidder wants to insure himself against the probability of losing the auction by placing a higher bid .

Example: Bidders with constant relative risk aversion and equally distributed ratings

The bidder's payment with evaluation of the item to be auctioned and the bid is now ${\displaystyle i}$ ${\displaystyle v_{i}}$ ${\displaystyle b_{i}}$

${\displaystyle u(b_{i},b_{-i},v_{i})={\begin{cases}(v_{i}-b_{i})^{r}&b_{i}>\max _{j\neq i}b_{j}\\0&b_{i}<\max _{j\neq i}b_{j}\end{cases}}}$

with . ${\displaystyle r\in (0,1)}$

Furthermore applies and and thus also ${\displaystyle f(x)={\begin{cases}{\frac {1}{\overline {v}}}&x\in [0,{\overline {v}}]\\0&x\notin [0,{\overline {v}}]\end{cases}}}$ ${\displaystyle F(x)={\begin{cases}{\frac {1}{\overline {v}}}x&x\in [0,{\overline {v}}]\\0&x\notin [0,{\overline {v}}]\end{cases}}}$

${\displaystyle G(v)=\left[F(v)\right]^{n-1}=\left({\frac {1}{\overline {v}}}v\right)^{n-1}.}$

Maximizing the expected payout leads to the optimal bidding strategy

${\displaystyle \gamma (v)={\frac {n-1}{n+r-1}}\cdot v.}$

Comparing the two cases of risk neutrality and risk aversion at equally distributed valuations, applies to : . ${\displaystyle r\in (0,1)}$ ${\displaystyle \gamma (v)>\beta (v)}$

Seller with reservation price

If the seller has a reservation price, i.e. a price below which he is not ready to sell the object to be auctioned, the optimal bidding strategy of a bidder is to offer the expected maximum of the reservation price and the highest rating of all other bidders, given this highest rating is smaller than the bidder's own rating.

If the seller has a reservation price, the price achieved is at least , since no bidder with an evaluation can achieve a positive profit. In addition, in the symmetrical equilibrium for the bidding strategy , a bidder with a valuation only wins the auction if all other bidders have submitted lower bids and then wins the auction with a bid equal to The following applies to the optimal bid strategy in the case : ${\displaystyle r>0}$ ${\displaystyle r}$ ${\displaystyle i}$ ${\displaystyle v_{i}<r}$ ${\displaystyle \beta (r)=r}$ ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle r}$ ${\displaystyle v\geq r}$

${\displaystyle \beta (v)=E[\max {(Y_{1},r)}|Y_{1}<v]=r\cdot {\frac {G(r)}{G(v)}}+{\frac {1}{G(v)}}\int _{r}^{v}yg(y)dy.}$

Asymmetrical bidders

With 2 asymmetrical bidders whose ratings are not equally distributed, the bidder bids higher in equilibrium, whose ratings are stochastically lower.

There are 2 bidders with ratings and , who are or are distributed independently on or with distribution functions . The strategies in the balance are and . These strategies are monotonically growing , differentiable and have an inverse function and . The same applies as in the symmetrical case and furthermore that if, for example , bidder 1 would win the auction with probability 1, if his rating is, he still wins if he would reduce his bid by an infinitesimally small amount . ${\displaystyle v_{1}}$ ${\displaystyle v_{2}}$ ${\displaystyle [0,{\overline {v_{1}}}]}$ ${\displaystyle [0,{\overline {v_{2}}}]}$ ${\displaystyle F_{1}(x)}$ ${\displaystyle F_{2}(x)}$ ${\displaystyle \beta _{1}}$ ${\displaystyle \beta _{2}}$ ${\displaystyle \phi _{1}=\beta _{1}^{-1}}$ ${\displaystyle \phi _{2}=\beta _{2}^{-1}}$ ${\displaystyle \beta _{1}(0)=\beta _{2}(0)=0}$ ${\displaystyle \beta _{1}({\overline {v_{1}}})=\beta _{2}({\overline {v_{2}}})={\overline {b}}}$ ${\displaystyle \beta _{1}({\overline {v_{1}}})>\beta _{2}({\overline {v_{2}}})}$ ${\displaystyle {\overline {v_{1}}}}$ ${\displaystyle \epsilon >0}$

Given player plays his strategy , the expected payoff of player with valuation and bidding is ${\displaystyle j=1,2}$ ${\displaystyle \beta _{j}}$ ${\displaystyle i\neq j}$ ${\displaystyle v_{i}}$ ${\displaystyle b<{\overline {b}}}$

${\displaystyle E[b,b_{-i}^{\ast },v_{i}]=F_{j}(\phi _{j}(b))\cdot (v_{i}-b).}$

Derive after leads to ${\displaystyle b}$

${\displaystyle {\frac {\partial E[b,b_{-i}^{\ast },v_{i}]}{\partial b}}=\phi _{j}'(b)\cdot F_{j}'(\phi _{j}(b))(v_{i}-b)-F_{j}(\phi _{j}(b)){\overset {!}{=}}0.}$

In equilibrium, and with , follows: ${\displaystyle v_{i}=\phi _{i}(b)}$ ${\displaystyle F_{j}'(x)=f_{j}(x),\quad \forall j=1,2}$

${\displaystyle \phi _{j}'(b)={\frac {F_{j}(\phi _{j}(b))}{f_{j}(\phi _{j}(b))\cdot (\phi _{i}(b)-b)}}.}$

For this system of differential equations one can only give an explicit solution for a few special cases. But if, for example, the evaluations of bidder 1 are stochastically higher than those of bidder 2, i. H. for and applies ${\displaystyle {\overline {v_{1}}}\geq {\overline {v_{2}}}}$ ${\displaystyle \forall x\in (0,{\overline {v_{2}}})}$

${\displaystyle {\frac {f_{1}(x)}{F_{1}(x)}}>{\frac {f_{2}(x)}{F_{2}(x)}},}$

so follows

${\displaystyle \beta _{1}(x)<\beta _{2}(x),\quad \forall x\in (0,{\overline {v_{2}}}).}$

The "weak" bidder 2 bids more aggressively than the "strong" bidder 1 due to its stochastically lower ratings.

Dependent valuation or auctioning of objects with general value

When auctioning objects of general value, the highest bidder is subject to the winner's curse: he systematically bids higher than he would have to to win the auction.

There are bidders with evaluation . The true value of the object to be auctioned is with evenly distributed on . Each bidder has an estimate of the true value . The value is the accuracy of the estimation of the bidder of the true value , said regardless of on uniformly distributed with density function are. The estimates of the bidders are faithful to expectations , because the following applies: ${\displaystyle n}$ ${\displaystyle v_{i}}$ ${\displaystyle V}$ ${\displaystyle V}$ ${\displaystyle [{\underline {V}},{\overline {V}}]}$ ${\displaystyle i}$ ${\displaystyle v_{i}=V+\varepsilon _{i}}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle i}$ ${\displaystyle V}$ ${\displaystyle \varepsilon _{i}}$ ${\displaystyle V}$ ${\displaystyle [-\varepsilon ,\varepsilon ]}$ ${\displaystyle f(\varepsilon )={\frac {1}{2\varepsilon }}}$

${\displaystyle E[v_{i}]={\frac {1}{2\varepsilon }}\int _{-\varepsilon }^{\varepsilon }V+\varepsilon _{i}d\varepsilon _{i}=V.}$

Thus, all estimates of the bidders are in the interval or the bidder knows that the true value lies in the interval . ${\displaystyle [V-\varepsilon ,V+\varepsilon ]}$ ${\displaystyle i}$ ${\displaystyle [v_{i}-\varepsilon ,v_{i}+\varepsilon ]}$

Maximizing the expected payout leads to the optimal bid strategy:

${\displaystyle \beta (v_{i})=(v_{i}-\varepsilon )+{\frac {2\varepsilon }{n+1}}\cdot \exp \left(-{\frac {n}{2\varepsilon }}\cdot ({\underline {V}}+\varepsilon -v_{i})\right).}$

In particular:

${\displaystyle {\frac {\partial \beta (v_{i})}{\partial n}}<0,}$

${\displaystyle \lim \limits _{n\to \infty }\beta (v_{i})=v_{i}-\varepsilon }$.

The optimal bid is the lowest value of the object based on the estimate plus a surcharge, which is lower, the more bidders participate in the auction. ${\displaystyle v_{i}-\varepsilon }$ ${\displaystyle v_{i}}$

The winner of the auction is subject to the curse of the winner : If the bidder would bid only on the basis of his own estimate of the true value, the optimal bid is the same as in the case of objects with a purely private valuation. However, this estimate neglects the information that the winner of the auction had the highest estimate and thus the bid submitted is higher than the optimal bid. ${\displaystyle v_{i}}$

Comparison of theory and empiricism

Although the Dutch auction and the first price auction are strategically equivalent for auctions of objects with purely private valuations and risk-neutral bidders, there are some differences in experiments . The prices achieved in a first-price auction are significantly higher than in a Dutch auction. One possible explanation for this is that in a Dutch auction the price goes down in 50-cent steps, while in a first price auction, bids do not have to be submitted in 50-cent steps.

If the number of bidders increases, experiments have shown that the amount of bids submitted also increases.

If you compare the English auction, Dutch auction, first and second price auction with regard to their efficiency in the sense of Pareto optimality , the English auction is the most efficient, followed by the second price auction, first price auction and finally the Dutch auction.

From the auctioneer's or seller's point of view, the first price auction is most desirable because it achieves the highest prices of all four auction types.

See also

• Auction theory

Individual evidence

1. ↑ ^{a b} Eichberger, Jürgen: Grundzüge der Mikroökonomik. 2nd edition, Mohr Siebeck, Tübingen, 2004: p. 300
2. ↑ ^{a b c d} https://mikebrandlyauctioneer.wordpress.com/auction-publications/history-of-auctions/
3. ↑ ^{a b} http://www.econport.org/econport/request?page=man_auctions_briefhistory
4. ↑ ^{a b c} http://www.econport.org/econport/request?page=man_auctions_firstpricesealed
5. ↑ http://www.newyorkfed.org/research/current_issues/ci3-9.pdf pp. 1-2
6. ↑ http://www.deutsche-finanzagentur.de/de/institutionelle-investoren/primaermarkt/tenderverfahren/
7. ^ Krishna, Vijay: Auction Theory. 2nd edition, Academic Press, Amsterdam, Heidelberg a. a., 2010: p. 14
8. ^ Krishna, Vijay: Auction Theory. 1st edition, Academic Press, Amsterdam, Heidelberg a. a., 2010: pp. 18-19
9. ↑ Eichberger, Jürgen: Grundzüge der Mikroökonomik. 2nd edition, Mohr Siebeck, Tübingen, 2004: p. 299
10. ^ Krishna, Vijay: Auction Theory. 2nd edition, Academic Press, Amsterdam, Heidelberg a. a., 2010: p. 40
11. ↑ ^{a b} Krishna, Vijay: Auction Theory. 2nd edition, Academic Press, Amsterdam, Heidelberg a. a., 2010: p. 21
12. ^ Krishna, Vijay: Auction Theory. 2nd edition, Academic Press, Amsterdam, Heidelberg a. a., 2010: p. 46
13. ^ Krishna, Vijay: Auction Theory. 2nd edition, Academic Press, Amsterdam, Heidelberg a. a., 2010: p. 47
14. ↑ ^{a b c} Eichberger, Jürgen: Grundzüge der Mikroökonomik. 2nd edition, Mohr Siebeck, Tübingen, 2004: p. 302
15. ↑ for a detailed derivation, see Eichberger, Jürgen: Grundzüge der Mikroökonomik. 2nd edition, Mohr Siebeck, Tübingen, 2004: pp. 305-311
16. ↑ ^{a b} Eichberger, Jürgen: Grundzüge der Mikroökonomik. 2nd edition, Mohr Siebeck, Tübingen, 2004: pp. 307-308
17. ↑ Cox, James C., Bruce Roberson, and Vernon L. Smith. Theory and behavior of single object auctions. Research in experimental economics 2.1 (1982): pp. 26-27
18. ^ Coppinger, Vicki M., Vernon L. Smith, and Jon A. Titus. INCENTIVES AND BEHAVIOR IN ENGLISH, DUTCH AND SEALED ‐ BID AUCTIONS. Economic Inquiry 18.1 (1980): pp. 16-17
19. ↑ Kagel, John H., and Dan Levin. Independent private value auctions: Bidder behavior in first-, second-and third-price auctions with varying numbers of bidders. The Economic Journal (1993): p. 874
20. ↑ ^{a b} Coppinger, Vicki M., Vernon L. Smith, and Jon A. Titus. INCENTIVES AND BEHAVIOR IN ENGLISH, DUTCH AND SEALED ‐ BID AUCTIONS. Economic Inquiry 18.1 (1980): p. 22
21. ↑ Cox, James C., Bruce Roberson, and Vernon L. Smith. Theory and behavior of single object auctions. Research in experimental economics 2.1 (1982): p. 28

literature

• Vijay Krishna: Auction Theory . 2nd Edition. Academic Press, Amsterdam, Heidelberg a. a. 2010, ISBN 978-0-12-374507-1 .
• Paul Milgrom : Putting Auction Theory to Work . 1st edition. Cambridge Univ. Press, Cambridge 2004, ISBN 0-521-53672-3 .
• Jürgen Eichberger: Basics of microeconomics . 1st edition. Mohr Siebeck, Tübingen 2004, ISBN 978-3-16-148167-3 .
• Paul Klemperer: Auctions: theory and practice . 1st edition. Princeton Univ. Press, Princeton et al. a. 2004, ISBN 978-0-691-11426-2 .
• John H. Kagel, Alvin E. Roth: The handbook of experimental economics . 1st edition. Princeton Univ. Press, Princeton et al. a. 1995, ISBN 978-0-691-05897-9 .
• Cox, James C., Bruce Roberson, and Vernon L. Smith. Theory and behavior of single object auctions. Research in experimental economics 2.1 (1982)
• Coppinger, Vicki M., Vernon L. Smith, and Jon A. Titus. INCENTIVES AND BEHAVIOR IN ENGLISH, DUTCH AND SEALED ‐ BID AUCTIONS. Economic Inquiry 18.1 (1980)
• Kagel, John H., and Dan Levin. Independent private value auctions: Bidder behavior in first-, second-and third-price auctions with varying numbers of bidders. The Economic Journal (1993): pp. 868-879.
• Kagel, John H., and Dan Levin. The winner's curse and public information in common value auctions. The American economic review (1986): pp. 894-920.

Web links

• Paul Klemperer's website - u. a. online version of his book "Auctions: Theory and Practice"
• Vijay Krishna - Lecture materials available online for Vijay Krishna's auctions