Auctions Introduction Definition: An Auction is a selling - - PowerPoint PPT Presentation

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

• f 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

• nly one bidder remains active, who obtains the
• bject 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,

• r 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

• f 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

• utcomes 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 Xi (this is a random variable) and the Xi 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[Xi] < ∞ is assumed.

Bidder i knows the realization xi of Xi but only that the

• ther 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 xi. An auction determines a game with the strategies being bid functions βi : [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: βII(x) = x. Proof: Assume to the contrary that bidder i bids zi >

• xi. This changes the outcome only if for p, the highest
• f the other bidders’ bids, xi < p < zi.

In this case bidder i now wins the object and makes a loss. Hence

• verbidding is dominated. Bidding zi < xi only changes

the outcome if for p, the highest of the other bidders’ bids, zi < p < xi. In this case bidder i now misses a profitable deal that he could have made by bidding xi. 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 xi and it cannot pay to stay in once the price exceeds xi, so it is (weakly) dominant to drop out at p = xi. 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 Y1 ≡ Y (N−1)

1

denote the highest value of the re- maining N − 1 bidders, i.e. Y1 is the first order statistic

• f X2, . . . , XN. The distribution function of Y1 is given

by G(y) = F(y)N−1, with density g. First-Price auction: In FPA, each bidder submits a sealed bid bi and payoffs are Πi =

• xi − bi if bi > maxj=i bj

0 if bi < maxj=i bj 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

• bject 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(Xi) = β(max i=1 Xi) = β(Y1),

hence if Y1 < 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 βI(x) = 1 G(x)

x

0 yg(y)dy = E[Y1|Y1 < x]

The equilibrium bidding strategy can be rewritten as βI(x) = E[Y1|Y1 < x] = x −

x

G(y) G(x)dy = x −

x

• F(y)

F(x)

N−1

dy 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) = xN−1 and hence βI(x) = N − 1 N x.

Revenue Comparison The expected payment of a bidder with value x is in SPA mII(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[Y1|Y1 < x] In FPA it is mI(x) = Pr(Win) · Amount Bid = G(x) · E[Y1|Y1 < x] = mII(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 seller (it is E

• Y (N)

2

• ).

Note, however, that for given 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).

The Revenue Equivalence Principle The observed identity of expected revenues between SPA and FPA holds in much more generality. Definition: An auction is standard if the highest bidder

• btains the object.

Proposition: Suppose values are iid and bidders are risk-neutral. Then any symmetric and increasing equi- librium of any standard auction, such that the expected payment of a bidder with value 0 is 0, yields the same expected revenue to the seller.

Proof: Let mA(x) be the expected payment in the symmetric equilibrium β of a standard auction A and let mA(0) = 0. Assume all bidders except for bidder 1 follow β and that bidder 1 with value x bids β(z) instead

• f β(x). Bidder 1 wins when β(z) > β(Y1) or z > Y1. His

expected payoff is ΠA(z, x) = G(z)x − mA(z) Note that mA(z) depends on β and z but not on x. Maximization yields ∂ ∂zΠA(z, x) = g(z)x − d dzmA(z) = 0 In equilibrium z = x is optimal, hence for all y d dymA(y) = g(y)y

Thus mA(x) = mA(0) +

x

0 yg(y)dy

=

x

0 yg(y)dy = G(x) · E[Y1|Y1 < x]

since by assumption mA(0) = 0. The right hand side does not depend on A and hence the expected payment

• f each bidder for a particular value does not depend
• n the particular auction, and therefore, the ex-ante

expected payment and thus the expected revenue of the seller do not either. QED

Example: Values are uniformly distributed on [0, 1] F(x) = x ⇒ G(x) = xN−1, thus for any standard auction with mA(0) = 0 mA(x) = N − 1 N xN E[mA(x)] = N − 1 N(N + 1) E[RA] = N · E[mA(x)] = N − 1 N + 1

Application: All-Pay Auction The revenue equivalence principle can be used to de- rive equilibria in other auctions, e.g. the all-pay auction where all bidders pay their bid. Consider an all-pay auction with symmetric, independent private values. Suppose there is a symmetric, increasing equilibrium such that mAP(0) = 0. The expected payment in an all-pay auction equals the bid and hence due to the revenue equivalence principle the equilibrium bid is βAP(x) = mA(x) =

x

0 yg(y)dy

Qualifications

• 1. Risk Aversion

Suppose bidders are risk-averse, all with the same utility function. In the case of iid private values, FPA yields a higher expected revenue than SPA. Intuition: In SPA risk-aversion does not change that bidding one’s value is a dominant strategy (by rais- ing the bid, one can increase the probability of win- ning, but only in cases where one does not want to win). In FPA at the equilibrium bid there is a perfect trade-off between the probability of winning and the amount won.

A risk-averse bidder will prefer, compared to a risk- neutral bidder, a smaller gain with a higher proba- bility and will hence choose a higher bid. By bidding higher he insures against ending up with 0.

• 2. Budget constraints

Suppose that bidder i has an absolute budget Wi. In SPA a bidder is more likely to be constrained than in FPA and hence FPA leads to higher expected revenues.

• 3. Symmetry

Consider the case where bidders are ex-ante asym- metric, i.e. their valuations are drawn from different distributions. In SPA, bidding one’s own value again remains a dominant strategy and hence SPA allocates effi- ciently even in case of asymmetries. In FPA, the weak bidder will bid more aggressively, because he faces a stochastically higher distribution

• f bids.

This more aggressive bidding of the weak bidder can lead to inefficiencies (if the weak bidder’s value is slightly smaller than the strong bidder’s), hence the revenue equivalence principle fails. Furthermore, there is no definite ranking of revenues between FPA and SPA.

• 4. Resale and Efficiency

One might argue that, since the auction reveals the valuations of the bidders, the auctioneer does not have to worry about efficient allocation because post-auction transaction will lead to an efficient al- location. This, however, is not correct even ab- sent transaction costs. If there is the opportunity for post-auction trade, bidders have an incentive not to completely reveal their valuation and hence post-auction bargaining takes place under incom- plete information which can lead to profitable deals being missed.

Single Unit Auctions with Interdependent Values As opposed to the case of private values, with interde- pendent values, the value of the object to bidder i, Vi is assumed to depend also on the information that other bidders have. Each bidder i has some private informa- tion Xi ∈ [0, ωi], called i’s signal. Vi is assumed to be given by a function Vi = vi(X1,X2, . . . , XN) that is non- decreasing in all the arguments, strictly increasing in Xi and twice continuously differentiable. Hence the value is completely determined by the signals. More general cases where there is some remaining uncertainty can be accommodated by considering the expected value given all signals (bidders are assumed to be risk neutral) vi(x1,x2, . . . , xN) = E[Vi|X1 = x1, . . . XN = xN]

Further assumption are vi(0, . . . , 0) = 0 and E[Vi < ∞]. Extreme cases: private values: vi(X1, . . . , XN) = Xi pure common value: Vi = V = v(X1, . . . , XN) for all i. Special case (mineral rights model): conditional on V = v, the signals Xi of the bidders are iid with E[Xi|V = v] = v.

The Winner’s Curse If values are not private, the estimate of the value to

• ne-self has to take information into account that is ob-

tained during or even after the auctions. In particular, it brings bad news if one wins the auction: bidder 1’s initial estimate of the value upon receiving the signal x is E[V |X1 = x]. Now if bidders are symmetric and fol- low the same strategy then winning the auction means that bidder 1 has the highest signal. The estimate of the value is then E[V |X1 = x, Y1 < x] < E[V |X1 = x]. Failure to take this into account, i.e. bidding plainly ac- cording to the initial signal instead of shading the bid below the initial estimate, leads to the winner’s curse, paying more than the value.

The Winner’s Curse magnitude increases with the num- ber of bidders. Note that the winner’s curse results from the failure to take the interdependence into account, it does not

• ccur in equilibrium.

Experimental Evidence: Kagel, Levin: Common Value Auctions and the Winner’s Curse, 2002, Princeton Uni- versity Press)

Equivalences Dutch and FPA are still strategically equivalent, be- cause that their normal forms are identical does not depend at all on the distribution of values and signals. But English and SPA are not equivalent with interde- pendent values, since information gathered in the Eng- lish auction (the drop-out prices of the other bidders) conveys valuable information about their signals and hence about one’s own value. In the case of only two bidders, however, the auction is over as soon as one ob- tains information and hence the two auctions are equiv- alent.

Consider again symmetric bidders, i.e. signals are drawn from the same distribution, and for bidder i the valua- tion does not change if we swap the signals of bidders j and k. Let v(x, y) = E[V1|X1 = x, Y1 = y] denote the expected value to bidder 1 if his signal is x and the highest of the other signals is y. Second-Price Auctions Proposition: Symmetric equilibrium strategies in the second price auction are given by βII(x) = v(x, x)

English Auctions In an English auction the bidders learn the prices where

• ther bidders drop out (in the symmetric model, the

identities of the bidders who dropped out are irrelevant) and as this information becomes available, this can (and should) influence when the remaining bidders plan to drop out. Symmetric equilibrium strategies are as follows:

• As long as no bidder has dropped out, drop out if

the price reaches the expected value assuming that all bidders have the same signal as yourself.

• If a bidder (say bidder k) drops out before you do,

infer his signal xk from the price where he dropped

• ut.
• Then drop out if the price reaches the expected

value given xk and assuming that all remaining bidders have the same signal as you.

• Continue in this fashion, i.e. infer the signals of the

bidders who drop out and recalculate the expected value.

An interesting implication is that the price at which you intend to drop out decreases over the course of the auction. Experimental evidence: CERGE-EI prep semester stu- dents learn this (while they do not learn to play the equilibrium, they learn to lower their drop-out prices when other bidders drop out).