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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.

### Lecture 2Auction Design

### Review of Lecture 1: Congruence revisited

### Auction Design

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.

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.

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).

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.

Revenue Comparisons forSymmetric Auctions

- We can rank the expected revenue generated in symmetric equilibrium for auctions where valuations are also symmetric.
- There are two basic results. In a symmetric auction:
- 1. The expected revenue from a Japanese auction is higher than what an English auction yields.
- 2. The expected revenue from an English auction exceeds a first price sealed bid auction.

Differential information

- We have discussed several types of information structures in auctions:
- - perfect foresight
- - independently distributed private valuations
- - symmetric valuations.
- But one important case we have not touched yet, when some bidders know more about the common value of the object than other bidders do.

Bidding with differential information

- An extreme form of dependent signals occurs when one bidder know the signal and the others do not.
- How should an informed player bid?
- What about an uninformed player?

An English common value auction experiment on asymmetrically informed bidders

Perspective of the less informed bidder

Suppose the uninformed bidder always makes the same positive bid, denoted b. This is an example of a pure strategy.

Is this pure strategy part of a Nash equilibrium?

The best response of the informed bidder is to bid a little more than b when the value of the object vi is worth more than b, and less than b otherwise.

Therefore the uninformed bidder makes an expected loss by playing a pure strategy in this auction. A better strategy would be to bid nothing.

A theorem

The argument in the previous slide shows that the uninformed bidder plays a mixed strategy in this game.

One can show that in equilibrium the informed bidder bids according to the strategy

(x) = E[V|V x]

and that the uninformed bidder chooses a bid at random from the interval [0, E[V]] according to the probability distribution H defined by

H(b) = Prob[(V) b]

The field of bidders

- The last part of this lecture discusses two issues outside the bidding process itself that nevertheless may affect the outcome of the auction.
- They are:
- Collusion amongst bidders
- Determining the number of bidders
- The degree of collusion and entry deterrence may affect how the auctioneer and the bidders rank different types of auctions that are revenue equivalent, or even strategically equivalent.

Collusion

- First we discuss the scope for collusive behavior under different auction mechanisms.
- Collusion between bidders is only possible if there is a mechanism for determining within a designated bidding ring which bidder has the highest valuation, and then ensuring members of the ring do not break the collusive agreement in the bidding.
- We focus on the second point. Sometimes determining which member has the highest valuation is trivial (for example in common value auctions), and sometimes it can be resolved through an auction within the ring for the right to present the only serious bid to the auctioneer.

Evidence about collusion in auctions

- Section 1 of the Sherman Act pertains to trusts and illegal constraints to trade.
- Over three quarters of the criminal cases filed in the 1980s under its provisions were in auctions markets.
- Given the difficulty in finding sufficient grounds to prosecute this illegal activity, one can safely conclude that collusive behavior in auctions is often a serious concern for the seller.

What are the gains from collusive behavior?

- Consider a bidding ring of R members out of a total of N bidders.
- The goal of the ring is to internally solve which of the players have the highest valuation, and then make one (serious) bid instead of R bids.
- With fewer effective bidders in the auction, the expected price of the object is lower.
- If all the bidders are able to collude, meaning N = R, the auction reduces to a game of bilateral bargaining between the auctioneer and the single bidder.

Bidding rings in second price sealed bid auctions

- For example, suppose the player with the highest valuation in the ring bids it, the other bidders in the ring submit the auctioneer’s reservation price, and the bidders outside the ring respond optimally but individually by bidding their true valuations.
- Then the ring benefits from colluding if:
- 1. The high bid from the ring wins the auction
- 2. The second highest valuation of bidders in the ring exceeds the highest valuation of all the bidders outside the ring.

Enforcing collusion in first and second price sealed bid auctions

- One reason why second price auctions are rarely used in practice is because they are more susceptible to collusion.
- In a first price sealed bid auction, a member of the bidding ring must submit a bid over the ring’s low price to win the auction. Providing his valuation exceeds the ring’s bid, there is an opportunity to make profits by deviating from the ring’s decision. This makes collusive bidding harder to enforce.
- Contrast this with a second price auction. In order to win the auction a bidder in the ring who breaks the collusive agreement must pay the reported value of the ring. The agreement by the ring is self enforcing!

Ascending auctions encourage communication amongst the bidders

A key issue for firms attempting to collude is agreeing how to share the spoils.

In a multiunit ascending auction, bidders can use the early stages when prices are still low to signal their views about who should win which object, and then when consensus has been reached, tacitly agree to stop pushing prices up.

By contrast, bidders cannot easily achieve the same coordination in simultaneous sealed-bid or descending auctions, in which each player simultaneously makes at most a single “best and final” offer to each object.

Low reservation prices encourage collusion

Reducing the reserve price increase the potential gains from joint-bidding or colluding, because the gains from colluding are greater.

Therefore the auctioneer should set a reservation price that is higher than the opportunity cost of failing to sell the auctioned item if it reduces the probability that bidders will collude.

Entry and the provision of information about the auctioned object

- Should the auctioneer encourage more players to enter the auction?
- Potential bidders could, for example, be encouraged to bid in an auction, by providing them with services that help them to value the object for sale.
- Note that simply paying people to participate in the auction would not achieve any useful purpose, because anyone could accept the payment and then make a very low bid.

Encouraging entry in private valuation auctions

- If another bidder enters a private valuation auction, the the level of the highest or the second highest valuation might increase.
- In either case, the revenue from the auction would increase.
- Therefore the expected revenue from holding a private valuation auction increases with the number of bidders.

Encouraging entry in common value auctions

- The argument for encouraging participation is less definitive.
- In an example discussed earlier we found that the auctioneer gains from having a (second) uninformed bidder enter the auction even though the uninformed bidder is indifferent between bidding or not.
- However one can construct examples of common value auctions in which the winning bid falls as the number of bidders increases.

Encouraging entry to avoid collusion

The lower the number of bidders, the easier it is for them to reach a collusive agreement.

Thus another reason to encourage entry is to reduce the probability for facing a bidding cartel.

Summary and Synthesis

- 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?
- Are the auctions symmetric?
- Is collusion between bidders a possibility?
- Should information about the auction be given to bidders?
- Not all these questions have easy answers, but that is where the value of experiments is especially evident.

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