AUCTIONS An Introduction y Elmar Wolfstetter April Humb oldtUniversitat zu Berlin Institut f Wirtschaftstheorie I Wirtschaftswissenschaftliche Fakultat Spandauerstr Berlin Germany email wolfwiwihub erlinde Abstract This is a fairly detailed review of auction theory It b egins with basic results on private value auctions with particular emphasis on the generality and limitations of the revenue equivalence of a large class of distinct auction rules The basic framework is then gradually mo died to admit for example risk aversion a minimum price entry fees and other xed costs of bidding multiunit auctions and bidder collusion There follows an intro duction to the theory of optimal auctions and to common value auctions and the asso ciated winners curse problem It closes with a sample of applications of auction theory in economics such as the regulation of natural monop olies the theory of oligop oly and the government securities market Diese Arb eit ist im Sonderforschungsb ereich Quantikation und Simulation Okonomischer Prozesse Humb oldtUniversitat zu Berlin entstanden und wurde auf seine Veranlassung unter Verwendung der ihm von der Deutschen Forschungsgemeinschaft zur Verfugung gestellten Mittel gedruckt y Comments byFriedel Bolle UweDulleck Peter Kuhbier Michael Landsb erger Wolfgang Leininger Georg Merdian and in particular by Dieter Nautz are gratefully acknowledged Contents Intro duction Private value auctions Some basic results on Dutch and English auctions Revenue equivalence theorem The case of uniformly distributed valuations Generalization Robustness Removing risk neutrality Removing symmetry Intro ducing a minimum price or participation fee Intro ducing numb ers uncertainty Endogenous quantity Multiunit auctions Removing indep endence correlated b eliefs Auction rings Optimal auctions Application of the revelation principle Illustration two bidders and twovaluations Discussion endogenizing bidder participation Common value auctions and the winners curse Further applications Auctions and oligop oly Natural gas and electric p ower auctions Treasury bill auctions Men prize the thing ungainedmore than it is Shakesp eare Troilus and Cressida Intro duction Supp ose you inherit a unique and valuable go o d say a painting by the late van Gogh or a copy of the edition of Adam Smiths Wealth of Nations For some reason you are in desp erate need for money and decide to sell You have a clear idea of your reservation price Of course you hop e to earn more than that You wonder isnt there some way to reach the highest p ossible price Howshouldyou go ab out it If you knew the p otential buyers and their valuations of the ob ject for sale your pricing problem would have a simple solution You would only need to call a nonnegotiable price equal to the highest valuation and wait for the right customer to come and claim the item Its as simple as that Information problem The trouble is that you the seller usually haveonly incomplete information ab out buyers valuations Therefore you have to gure out a pricing scheme that p erforms well even under incomplete information Your problem b egins to b e interesting Basic assumptions Supp ose there are N p otential buyers eachof whom knows the ob ject for sale well enough to decide howmuch he is willing V V Since you the seller to pay Denote buyers valuations by V N do not know V you must think of it as a random variable distributed by some joint probability distribution function CDF F v v v v N N F v v PrV v V v Similarlypotential buyers N N N know their own valuation but not that of others Therefore buyersalsoview valuations as a random variable except their own Cournot monop oly approach Of course you could stick to the takeit orleaveit pricing rule even as you are sub ject to incomplete information ab out buyers valuations You would then call a nonnegotiable price p at or ab oveyour reservation price r and hop e that some customer has the right valuation and is willing to buyYour pricing problem would then b e reduced toavariation of the standard Cournot monop oly problem The only unusual part would b e that the tradeo b etween price and quantity is replaced byone between price and the probabilityofsale Alluding to the usual characterization of Cournot monop olyyou can even draw a demand curve with the nonnegotiable price p on the price axis and the probability that at least one bidders valuation exceeds the listed price p p Gp GpF pp on the quantity axisas in Figurebelow And you would then pickthat price p that maximizes the exp ected value of your gain from trade pp r Equivalentlyyou can state the decision problem in quantity co ordinates as is customary in the analysis of Cournot monop oly where quantity is here represented by the probabilityofsale q Gp maxP q r q P q G q q Computing the rstorder condition the optimal price p can then b e char een marginal revenue acterized in the familiar form as a relationship b etw MR and marginal cost MC Gp r MC MR p G p p v Gp marginal revenue p G p p r q Gp demand v q Figure Optimal takeitorleaveit price As always this condition applies only if the maximization problem is wellb ehaved Wellb ehavedness requires that revenue is continuous and marginal revenue strict monotone increasing in p Example Suppose valuations V are independent and uniformly distributed i over the interval and the reservation priceis r Then for al l N N p GpF p p Hence the optimal takeitorleaveit priceis s N p N N The resulting expectedpricepaid price asked times probability of success is N pN p N Gp N p N N Both p N and pN are strict monotone increasing in N starting from p p and approaching as N Takeitorleaveit pricing is one waytogo But generally you can do b etter by setting up an auction This brings us to the analysis of auctions and the design of optimal auction rules Interestingly the Cournot monop oly price p will continue to play a role as optimal minimum price in the optimal auction setting as you will learn in the section on optimal auctions Auctions what where and why An auction is a bidding mechanism describ ed by a set of auction rules that sp ecify how the winner is determined and howmuch he has to pay In addition auction rules may restrict partici pation and feasible bids and imp ose certain rules of b ehavior Auctions are widely used in many transactions not just in the sale of art and wine Every week the Treasury auctionso billions of dollars of bills notes and Treasury b onds Governments and private corp orations solicit deliveryprice oers of pro ducts ranging from oce supplies to tires or con struction jobs The Department of the Interior sells the rights to drill oil and other natural resources on federally owned prop erties And private rms sell pro ducts ranging from fresh owers sh and tobacco to diamonds and real estate Altogether auctions account for an enormous volume of transactions Essentially auctions are used for three reasons sp eed of sale to reveal information ab out buyers valuations to prevent dishonest dealing b etween the sellers agent and the buyer For a simple pro of of monotonicitywork with lnp whichisobviously a monotone transformation ofp The latter agency role of auctions is particularly imp ortantifgovernment agen cies are involved in buying or selling As a particularly extreme example just think of the current privatization programs in Eastern Europ e Clearly if the agencies involved such as the Treuhandanstalt that is in charge of privatizing the state run corp orations of former communist East Germanywere free to negotiate the terms of sale the lucky winner probably would b e the one who made the largest brib e or p olitical contribution However if assets are put up for auction cheating the taxpayer is much more dicult and costly and hence less likely to succeed Popular auctions There are many dierent auction rules Actually the word itself is something of a misnomer Auctio means increase but not all auctions involve calling out higher and higher bids One distinguishes b etween oral and written auctions In oral auctions bidders hear each others bids and can makecounteroers each bidder knows his rivals In a written or closedseal bid bidders submit their bids simultane ously without revealing them to others often bidders do not even knowhow many rival bidders participate The