Auctions Introduction Definition: An Auction is a selling institution that elic- its information from potential buyers in the form of bids and where the outcome (i.e. who obtains the objects and who pays how much) is determined solely by this information. This implies that auctions are universal (i.e. any object can be sold by means of an auction) and anonymous (i.e. the identity of the bidders does not matter, hence if the bids of two bidders are ex- changed, the allocation and payments are exchanged accordingly and no other bidder is affected).
Procurement auctions used to buy a good from po- tential sellers work exactly correspondingly. Hence we can restrict the discussion to auctions employed to sell goods. Auctions are useful when the seller is unsure about the valuations (i.e. the maximal willingness to pay) of the buyers. Otherwise he could just offer the good to the buyer with the highest valuation at a price just below this valuation. Auctions have been used for a long time, e.g. govern- ment bonds, drilling rights. Recent important applica- tions: privatization, spectrum auctions, internet auc- tion platforms.
Different auction formats are evaluated on the basis of revenue (expected selling price) and efficiency (allo- cation to the bidder with the highest valuation). For practical purposes simplicity and the susceptibility to collusion are further (and possibly more) important cri- teria.
Common (Single Unit) Auction Forms 1. First-Price auction: All bidders submit a sealed bid, the highest bidder wins and pays his bid 2. Second-Price auction: All bidders submit a sealed bid, the highest bidder wins and pays the second highest bid 3. English Ascending Price auction (Japanese auction): The auctioneer continuously raises the price until only one bidder remains active, who obtains the object at the price where the auction ended (i.e. where the second to last bidder dropped out).
4. Dutch Descending Price auction: The auctioneer starts with a high price (presumed to be higher than the maximal valuation) and continuously lowers the price until one bidder signals to buy the object at the current price. A number of other auctions formats is possible. Some might appear as unusual when thought of as a classi- cal auction, but less so when seen in another context (e.g. an all-pay auction, where all bidders pay their bids seems unusual for selling a painting, but a patent-race, or lobbying are essentially all-pay auctions). Some auc- tions may at a first glance not conform to a straight- forward idea of an auction.
Single Unit Private Value Auctions Definition: Bidders are said to have private values if each bidder knows the value of the object to himself (and only to himself) for sure at the time of bidding. Otherwise, if the value of the object to a bidder may depend on information that other bidders have, values are said to be interdependent. An extreme case is that of a pure common value, where the value of the object is the same for all bidders, but unknown by the time of bidding. Private values do not have to be statistically indepen- dent and in the case of interdependent values the sig- nals of the bidders can still be statistically independent. Independent private values are, however, the standard case.
Equivalences For the single unit case, the First-Price and the Dutch auction are equivalent in a strong sense, they are strate- gically equivalent (the games have the same normal form). The available strategies to a bidder consist of plainly choosing one number, the bid in the FPA or the price where the bidder would agree to buy the object in the DA in case it has not been sold yet. Also the outcomes are derived from the strategies in the same way: the bidder choosing the highest number wins and pays this number. A bidder does not learn anything in DA, because when he does learn something the auction is over (and the fact that no other bidder has bought the object yet is not informative, because the strategies condition on this event in the first place).
Between the Second-Price and the English auction there is a weaker form of equivalence, because in the Eng- lish auction (with more then two bidders) a bidder can learn something by the drop-out prices of other bid- ders and could in principle condition his strategy on his observations (hence the games are not strategically equivalent). But in the private value case, the informa- tion gathered from the drop-out prices of other bidders is not informative and hence the equilibria in both auc- tions are identical.
Equilibria in the Symmetric Model There is a single object for sale to N bidders. Bidder i assigns a value X i (this is a random variable) and the X i are independently and identically distributed in some interval [0 , ω ] with distribution function F which has a continuous density with full support f = F ′ . While ω = ∞ is allowed, E [ X i ] < ∞ is assumed.
Bidder i knows the realization x i of X i but only that the other bidders’ values are distributed according to F. Except for the realizations of the values, all aspects of the model, in particular F and N, are common knowl- edge. Bidders are risk neutral, they try to maximize expected profits. Bidders do not face liquidity constraints, i.e. bidder i is willing and able to pay up to x i . An auction determines a game with the strategies being bid functions β i : [0 , ω ] → R + . The focus will be on symmetric equilibria, because bidders are symmetric.
Second-Price and English auction: Proposition: In SPA and EA it is a weakly dominant strategy to bid one’s own valuation: β II ( x ) = x. Proof: Assume to the contrary that bidder i bids z i > x i . This changes the outcome only if for p, the highest of the other bidders’ bids, x i < p < z i . In this case bidder i now wins the object and makes a loss. Hence overbidding is dominated. Bidding z i < x i only changes the outcome if for p, the highest of the other bidders’ bids, z i < p < x i . In this case bidder i now misses a profitable deal that he could have made by bidding x i . Hence also underbidding is dominated. QED
The argument is even more obvious in EA: it cannot pay to drop out before the price p reaches x i and it cannot pay to stay in once the price exceeds x i , so it is (weakly) dominant to drop out at p = x i . Note that this result depends neither on risk neutrality nor on the symmetry of the bidders (not even on the independence of the distributions, only on values being private).
Some Notes on Order Statistics: Consider bidder 1. Let Y 1 ≡ Y ( N − 1) denote the highest value of the re- 1 maining N − 1 bidders, i.e. Y 1 is the first order statistic of X 2 , . . . , X N . The distribution function of Y 1 is given by G ( y ) = F ( y ) N − 1 , with density g. First-Price auction: In FPA, each bidder submits a sealed bid b i and payoffs are � x i − b i if b i > max j � = i b j Π i = 0 if b i < max j � = i b j A bidder will clearly not submit a bid equal to his val- uation, because this guarantees a profit of 0. Raising the bid implies a trade-off. The chance of winning the object are increased, but so is the expected price.
Assume that there is a symmetric equilibrium with in- creasing, differentiable strategy β. Obviously, bidding b > β ( ω ) is dominated, and β (0) = 0 . Bidder 1 wins if his bid b > max i � =1 β ( X i ) = β (max i � =1 X i ) = β ( Y 1 ) , hence if Y 1 < β − 1 ( b ) . His expected payoff is G ( β − 1 ( b )) · ( x − b ) . Maximizing w.r.t b yields g ( β − 1 ( b )) β ′ ( β − 1 ( b ))( x − b ) − G ( β − 1 ( b )) = 0 In a symmetric equilibrium b = β ( x ) implying G ( x ) β ′ ( x ) + g ( x ) β ( x ) = xg ( x ) d dx ( G ( x ) β ( x )) = xg ( x )
Since β (0) = 0 � x 1 β I ( x ) = 0 yg ( y ) dy = E [ Y 1 | Y 1 < x ] G ( x ) The equilibrium bidding strategy can be rewritten as � x G ( y ) β I ( x ) = E [ Y 1 | Y 1 < x ] = x − G ( x ) dy 0 � N − 1 � x � F ( y ) = x − dy F ( x ) 0 hence the degree of bid shading decreases in the num- ber of bidders. Example: Values are uniformly distributed over [0 , 1] . Then F ( x ) = x, G ( x ) = x N − 1 and hence β I ( x ) = N − 1 x. N
Revenue Comparison The expected payment of a bidder with value x is in SPA m II ( x ) = Pr(Win) · E [2nd highest bid | x is the highest bid] = Pr(Win) · E [2nd highest value | x is the highest value] = G ( x ) · E [ Y 1 | Y 1 < x ] In FPA it is m I ( x ) = Pr(Win) · Amount Bid G ( x ) · E [ Y 1 | Y 1 < x ] = m II ( x ) = Hence the expected payment for a bidder with value x is identical in FPA and SPA.
Thus the ex-ante expected payment of each bidder is the same and therefore so is the expected revenue of the � � Y ( N ) seller (it is E ). Note, however, that for given 2 values, the revenue is usually different. Furthermore, the revenues in SPA vary more than in FPA (e.g. in the case of 2 bidders with uniformly distributed values in [0 , 1] the maximal revenue in FPA is 1 2 , but in SPA it is 1.) Precisely, the distribution of equilibrium prices in SPA is a mean preserving spread of the distribution of prices in FPA. Hence a risk-averse seller would prefer FPA (given that the bidders are risk-neutral).
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