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This lecture derives bidding rules for some auctions where there is incomplete information, and discusses the the virtues and shortfalls of alternative auction mechanisms.
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This lecture derives bidding rules for some auctions where there is incomplete information, and discusses the the virtues and shortfalls of alternative auction mechanisms. First we explore the concept of revenue equivalence, which is weaker than congruence, and applies to private value auctions where the bidders are risk neutral. Then we relax the conditions for revenue equivalence to apply, seeking to show the effects on bidding behavior and auction revenue. Finally we discuss the role of collusion and entry in auctions. Lecture 2Auction Design
Review of Lecture 1: Congruence revisited Using the concept of congruence we derived two rules for bidding: Rule 1: Pick the same reservation price in a Dutch auction that you would submit in a first price sealed bid auction. Rule 2: In private value auctions, or if there are only two bidders, choose the reservation price for an English or a Japanese auction, that you would submit in a second price sealed bid auction.
Review of Lecture 1:Second price auctions • We also proved that the bidding strategy in sealed bid second price auctions (and ascending auctions) is very straightforward if you know your own valuation. • Rule 3 : In a second price sealed bid auction, bid your valuation if you know it.
Relaxing congruence • In congruent auctions, the revenue to the auctioneer and the payoffs to each bidder are identical for every game history generated by a solution strategy profile. • This is a very strong form of equivalence, and often not met. The bidders and the auctioneer may be indifferent between two auctions that are not congruent to each other. • For example, suppose the auctioneer and the bidders only care about their expected utility from respectively conducting and participating in an auction, and did not care about whether each individual game history has the same outcome. • Can we show that such players might be indifferent to certain non-congruent auctions (where there is incomplete information)?
Revenue Equivalence Defined • The concept of revenue equivalence provides a useful tool for exploring this question. • Two auction mechanisms are revenue equivalent if, given a set of players their valuations, and their information sets, the expected surplus to each bidder and the expected revenue to the auctioneer is the same. • Therefore revenue equivalence is a less stringent condition than congruence. • Thus two congruent auctions are invariably revenue equivalent, but not all revenue equivalent auctions are congruent.
Why study revenue equivalence ? • If the auctioneer and the bidders are risk neutral, studying revenue equivalence yields conditions under which the players are indifferent between auctions that are not necessarily congruent. • Exploiting the principle of revenue equivalence can sometimes give bidders a straightforward way of deriving their solution bid strategies.
Preferences and Expected Payoffs Let: U(vn) denote the expected value of the nth bidder with valuation vn bidding according to his equilibrium strategy when everyone else does too. P(vn) denote the probability the nth bidder will win the auction when all players bid according to their equilibrium strategy. C(vn) denote the expected costs (including any fees to enter the auction, and payments in the case of submitting a winning bid).
An Additivity Assumption • We suppose preferences are additive, symmetric and private, meaning: • U(v) = P(v) v - C(v) • So the expected value of participating in the auction is additive in the expected benefits of winning the auction and the expected costs incurred.
A revealed preference argument • Suppose the valuation of n is vn and the valuation of j is vj. • The surplus from n bidding as if his valuation is vj is U(vj), the value from participating if his valuation is vj, plus the difference in how he values the expected winnings compared to to a bidder with valuation vj, or (vn – vj)P(vj). • In equilibrium the value of n following his solution strategy is at least as profitable as deviating from it by pretending his valuation is vj. Therefore: • U(vn) > U(vj) + (vn – vj)P(vj)
Revealed preference continued • For convenience, we rewrite the last slide on the previous page as: • U(vn) - U(vj) > (vn – vj)P(vj) • Now viewing the problem from the jth bidder’s perspective we see that by symmetry: • U(vj) > U(vn) + (vj – vn)P(vn) • which can be expressed as: • (vn– vj)P(vn) > U(vn) - U(vj)
A fundamental equality • Sayvn> vj.which then impliesP(vn)> P(vj). • Putting the two inequalities together, we obtain: • (vn – vj) P(vn)> U(vn) - U(vj) > (vn – vj) P(vj) • Writing: • vn = vj + dv • yields • which, upon integration, yields
Revenue equivalence • This equality shows that in private value auctions, the expected surplus to each bidder does not depend on the auction mechanism itself providing two conditions are satisfied: • 1. In equilibrium the auction rules award the bid to the bidder with highest valuation. • 2. The expected value to the lowest possible valuation is the same (for example zero). • Note that if all the bidders obtain the same expected surplus, the auctioneer must obtain the same expected revenue.
A theorem Assume each bidder: - is a risk-neutral demander for the auctioned object; - draws a signal independently from a common, strictly increasing, cumulative continuous distribution function. Consider auction mechanisms where - the buyer with the highest signal always wins - the bidder with the lowest feasible signal expects zero surplus. Then the same expected revenue will be generated by the auctions, and each bidder will make the same expected payment as a function of her signal.
First price sealed bid private value action for wireless licenses
Second price sealed bid private value action for wireless licenses
The Revenue from Private Value Auctions Since any auction satisfying the conditions for the theorem can be used to calculate the expected revenue, we select the second price, or English auction, to accomplish this task.
Steps for deriving expected revenue • The expected revenue from any auction satisfying the conditions of the theorem, is the expected value of the second highest bidder. • To obtain this quantity, we proceed in two steps: • 1. derive and analyze the probability distribution of the highest valuation, • 2. and then derive the probability distribution of the second highest bidder.
The unconditional probability of winning • From the perspective of the auctioneer, or an outsider who does not know the valuation of any player, each player has an equal chance of winning the auction. • Given N bidders, each of whom has the same chance of winning, the probability of bidder n winning the auction is 1/N. • Each player knows his own valuation vn and consequently has more information than the auctioneer. • In the solution to a second price sealed bid auction, each bidder submits her own valuation to the auctioneer, and therefore the winning bidder is the player with the highest valuation.
The probability of winning an auction conditional on your own valuation • That is the probability of winning an auction in equilibrium is just the probability of being endowed with the highest valuation. • If valuations are identically and independently distributed with cumulative probability distribution function F(v), the probability that v1, for example, is the highest of the N valuations equals: Pr{(v2 v1) (v3v1). . . (vNv1)} = Pr{v2 v1}x Pr(v3v1) x . . . x Pr{vNv1} = F (v1) x F (v1) x . . . x F (v1) (N-1 times) = F (v1)N-1
The probability of the median valuation winning an auction • Conditioning on your own valuation, as N increases the probability of winning the auction declines at a much faster rate than 1/N does. • For example if v₁ is the median of the distribution, the probability of winning the auction when N=2 is 0.5. • But the probability of winning the auction when N=10 is • 0.59= 1. 9531×10-3 • which is orders of magnitude less than 1/10.
The probability of winning an auction with a median valuation as a function of the number of bidders
Expected revenue in a private value auction • The expected revenue to the auctioneer is the expected value of the second highest valuation. This can be calculated as: • Hence the expected revenue for a private values auction satisfying the conditions of the theorem is this formula.
Bidding Rules for Private Value Auctions Armed with the formula on the previous slide, we can also derive the solution bidding strategies for auctions that are revenue equivalent to the second price sealed bid auction.
Bidding function in a first price sealed bid auction • Consider, for example a first price sealed bid auctions with independent and identically distributed valuations. • In a symmetric equilibrium to first price sealed bid auction, we can show that a bidder with valuation vn bids:
An example: the uniform distribution • Valuations are uniformly distributed within the closed interval . In this case • which implies:
Bidding function with the uniform distribution • Thus in the case of the uniform distribution the equilibrium bid of the player with valuation v is to bid a weighted average of the lowest possible valuation and his own, where the weights are respectively 1/N and (N-1)/N:
Comparison of bidding strategies • The bidding strategies in the first and second price auctions markedly differ. • In a second price auction bidders should submit their valuation regardless of the number of players bidding on the object. • In the first price auction bidders should shave their valuations, by an amount depending on the number of bidders.
All pay sealed bid auction with private values • The revenue equivalence theorem implies that the amount bidders expect to pay in an all-pay auction as in all other auctions satisfying the conditions of the theorem. • In contrast to a first or second price sealed bid auctions where only the winner bidder pays his bid or the second highest bid in an all pay auction losers also pays their bids. • The amount paid by the nth bidder is certain, and not paid with the probability of winning the auction, that is F(vn)N-1. • By the revenue equivalence theorem the amount each bidder expect to pay in the first two auctions, upon seeing their valuation, equals the amount the bidder actually does pay in all pay auction.
Expected revenue in all pay auction • In an all pay auction the expected revenue from a bidder with vn bids F(vn)N-1 multiplied by the amount he would bid in a first price auction. • This is:
When does Revenue Equivalence Fail? • Bidders might be risk averse, not risk neutral. • The private valuations of bidders might be drawn from the probability distribution that are not identical. • The theorem does not apply when bidders receive signals about the value of the object to them that are correlated with each other. • Collusion and entry deterrence are also considerations that auctioneers should account for.
Attitudes towards risk in second price sealed bid auctions with private values • It remains a weakly dominant strategy for each player to bid his or her valuation. • The optimal bidding strategy for the second price sealed bid auction (and also the Japanese and English auctions) is independent of a bidder's attitude towards risk and uncertainty when private values are drawn from a common probability distribution.
Attitudes towards risk in first price sealed bid auctions with private values • A strategy of bidding your valuation guarantees exactly zero surplus. • If you place a lower bid than your valuation your expected surplus initially increases until it reaches the maximum for a risk neutral bidder, and then falls, but the variance of the surplus increases as well. • A risk averse gambler is willing to trade a lower expected value to reduce the amount of uncertainty, he accordingly bids higher than a risk neutral bidder.
Comparing first and second price sealed bid auctions • Revenue generated by a second price auction is independent of the bidders' preferences over uncertainty, since bidding is unaffected. • The revenue generated by the first price auction is the same as the revenue generated by a second price auction when bidders are risk neutral. • Therefore risk averse bidders generate more revenue in a first price auction than they would in a second price auction, and they generate more revenue in a first price auction than do risk neutral bidders.
Asymmetric valuations • In many auctions where there are private valuations, the bidders have different uses for the auctioned object, and this may be common knowledge to all the bidders. • Bidder knows the probability distributions that each of the other valuations are drawn from, he will typically use that information when making his own bid. • This affects the revenue equivalence theorem, and also the auctioneer's preferences towards different types of auctions.
An example of asymmetry • Instead of assuming that all bidders appear the same to the seller and to each other, suppose that bidders fall into two recognizably different classes. • Instead of there being a single distribution F(v) from which the bidders draw their valuations, there are two cumulative distributions, F₁(v) and F₂(v). • Bidders of type i∈{1,2} draw their valuations independently from the distribution Fi(v). • Let fi(x) denote the probability density function of Fi(x).
Asymmetry in a first price auction with only two bidders • The private valuation of the first bidder is drawn from a probability distribution F₁(v) that stochastically dominates the probability distribution for the other probability distribution F₂(v). • In fact we make a stronger assumption, that for all v • Then b₁(v)< b₂(v). The intuition is to bid aggressively from weakness and vice versa.
Incomplete information about the type of the bidder • Suppose each bidder sees his valuation, but does not immediately learn whether he comes from the high or low probability distribution. • At that point the bidding strategy cannot depend on which probability distribution the valuation comes from. • Then each bidder is told which probability distribution his bid is drawn from. • How should he revise his bid? The second (first) bidder learns that the first (second) bidder is more likely to draw a higher (lower) valuation than himself, realizes the probability of winning falls (rises), so adjusts his upwards (downwards).
Auction Design In this part of the lecture we relax the independence assumption for revenue equivalence to apply, discuss the winner’s curse, and show the effects on bidding behavior and auction revenue. Finally we discuss the role of collusion and entry in auctions.
Relaxing independence • The revenue equivalence theorem applies to situations in which the valuation of each is bidder is independently distributed, and this is is what we have been focusing on in the first part of this lecture. • This is not always a useful assumption, because in many situations a bidder would be informed if he had information about the object on the auction block that another bidder had, and would use the information in a similar way. • What happens when the signals that bidders get about the value of the auctioned item are positively correlated?
Symmetric Valuations • What happens when the signals that bidders get about the value of the auctioned item are positively correlated? • We relax independence and consider the class of symmetric valuations, which have two defining features: • 1. All bidders have the same utility function. • 2. Each bidder only cares about the collection of signals received by the other bidders, not who received them. • Thus we may write the valuation of bidder n as:
An example: Value of the object not known to bidders • Suppose the value of the object to each bidder is the same, but this value is unknown to each bidder. The nth bidder receives a signal sn which is distributed about the common value v, and write • sn = v + n where n E[v|Information of n] – v • where n is independently distributed. • More generally, each bidder might place more significance on their own draw, but still attach some value to the assessments of others.
Summary so far • When comparing two (or more) auctions, we should consider the following questions: • Are the auctions congruent, or if not, revenue equivalent? • Are the bidders risk neutral? • In private valuation auctions, are bidders drawing from the same probability distribution? • In common valuation auctions, are bidders drawing from the same distribution of signals?
The winner's curse • When other bidders have information that you lack about the value of the object for sale, winning the auction may cause you to decrease your conditionally expected value of the object. • Failure to take into account the bad news about others' signals that comes with any victory is called the winner's curse. • The winner's curse describes the fact that winning an auction may convey new and unfavorable information about the item. • Because all other bids are less than the winning bid, the expected value of the item to the winning bidder might fall when the outcome of the auction is announced.
The expected value of the item upon winning the auction • If the nth bidder wins the auction, he will realize his signal exceeded the signals of everybody else, that is • sn ≡max{s₁,…,sN} • so he will condition the expected value of the item on this new information. • His expected value is now the expected value of vn conditional upon observing the maximum signal: • E[vn| sn ≡ max{s₁,…,sN}] • This is the value that the bidder should use in the auction, not E[vn|sn], because he should recognize that unless his signal is the maximum he will receive a payoff of zero.
Defining the Winner’s Curse • The winner's curse can be defined as: • W(sn) ≡ E[vn|sn] - E[vn| sn ≡ max{s₁,…,sN}] • Since the max operator is a convex increasing function of its arguments • it follows that W(sn) is a negative function. • Although bidders should take the winner's curse into account, there is widespread evidence that novice bidders do not take this extra information into account when placing a bid.
