Auction theory vijay krishna pdf download

Then from 9. This implies that the losing bidder has no incentive to raise his because v22 bid, since if he were to do so and win the auction, it would be at a price that is too high. Thus, if there is an increasing solution to 9. We omit a proof of the existence of such 9. Note that when there are only two bidders, the English auction is strategically equivalent to the sealed-bid second-price auction.

This is because the only information available in the open format—that the other bidder dropped out—comes too late to be of any use, since it signals only that the auction is over.

The fact that the two auctions are equivalent means that Proposition 9. This equivalence does not hold, of course, once there are three or more bidders. Now information that one of the bidders has dropped out can be used by the remaining bidders to update their behavior in the remaining auction. Example 9.

All other comparisons have v22 1 3 12 33 13 are straightforward. If x2 and x3 are both greater than 12 , then for all x1 , v1 is greater than both v2 and v3. Thus, the average crossing condition implies the single crossing condition and is equivalent to it when there are only two bidders.

The average crossing condition is depicted in Figure 9. In that case, the average value is also p and the left-hand panel depicts the inequalities in 9. As in Figure 9. Suppose that the valuations v satisfy the average crossing condition. Before proceeding with the proof of Proposition 9. First, as noted earlier, when there are only two bidders, the average crossing condition is equivalent to the single crossing condition.

How strong is the average crossing condition relative to the single crossing condition when there are three or more bidders? Suppose further that these are rescaled so that they lie in the unit simplex the labels on the vertices, e1 ,e2 , and e3 , denote the three unit vectors. When there are only two bidders, this strategy is simple since the dropping out of either bidder ends the auction. In this case, a strategy of a bidder can depend only on his own signal—and that is why with only two bidders, the English and sealed-bid second-price auctions are equivalent.

If there are more than two bidders, however, the strategy of a bidder in an English auction is more complicated. We saw in Chapter 6 that when bidders are symmetric, the number of active bidders and the prices at which the bidders who dropped out did so were important, but their identities were not. With asymmetric bidders, the identities of the bidders are also important since the inference that bidder 1, say, would draw if bidder 2 dropped out at a price p may be very different from the inference he would draw if instead, bidder 3 were to drop out at the same price p.

Lemma 9. For purposes of exposition, the arguments that follow assume that it is never the case that two bidders drop out simultaneously at the same price.

The arguments can be easily extended to account for simultaneous exits. Bidder 1 cannot affect the price he pays for the object. The break-even conditions 9. This makes a difference only if he stays active until all other bidders have dropped out and he actually wins the object.

Then the break-even conditions 9. Differentiating 9. As an example, consider the subauction where one of the bidders—say, N—with signal xN has dropped out. Proceeding recursively in this way results in strategies satisfying 9. Lemmas 9. To complete the proof of Proposition 9.

Consider the case when all bidders are active. Suppose that the signals are x1 , x2 ,. In a completely symmetric model, however, a stronger property holds: At every stage, the bidder who drops out is the bidder with the lowest value among those remaining. The same is true when values are separable into private and common components.

Bidders need not drop out according to increasing values, however, when bidders are asymmetric and values are interdependent. The average crossing condition permits such interesting behavior to emerge in equilibrium. The valuations are the same as in Example 9. Fix x1 and cient equilibrium.

This is a contradiction. It should be noted that the results of this chapter are all of an ex post nature; the equilibrium strategies depend only on the valuations vi and not on the distribution of signals. Show that the equilibrium strategies so determined survive the iterated elimination of weakly dominated strategies and are the only strategies to do so. In that case, the distribution matters only via its effect on the valuations.

In step 1, discard all weakly dominated strategies for both bidders. In step 2, discard all weakly dominated strategies in the reduced game obtained after step 1. In step 3, discard all weakly dominated strategies in the reduced game obtained after step 2.

Continue in this fashion. The weak form of the single crossing condition was introduced by Dasgupta and Maskin The general case of three or more bidders was considered by Krishna The average crossing condition and Proposition 9. It is also close to being necessary.

These conditions are quite weak; in particular, they are weaker than the average crossing condition. We then look for optimal, or revenue maximizing, mechanisms. A direct mechanism asks buyers to report their private information—in this case, their signals—and replicates the equilibrium outcomes of the original mechanism.

A simple example illustrates that the single crossing condition cannot be dispensed with even if we consider general, abstract mechanisms. Example We will now argue that the valuation functions must satisfy the single crossing condition. Thus, ex post incentive compatibility implies that the mechanism must be monotonic in values.

Each buyer is asked to report his or her signal. A buyer who does not obtain the object does not pay anything. The workings of the mechanism are illustrated in Figure In particular, it exceeds that of buyer j. Proposition This means that xi xi. In the case of private values, the generalized VCG mechanism reduces to the ordinary second-price auction, and in that case, of course, truth-telling is a dominant strategy.

The mechanism designer is assumed to have knowledge of the valuation functions vi , and the mechanism is then able to elicit information regarding the signals xi that is privately held by the buyers. Moreover, buyers with different valuation functions are treated differently so that the mechanism is not anonymous. Put another way, the incomplete nature of the information available to the seller—his knowledge consisted only of the underlying distributions and not the actual realized values—meant that he was unable to extract all the surplus from the buyers.

Actually, as we will see in this section, the key feature is not that the values are private but rather that they are independently distributed. This means that each buyer has private information that is exclusive in a strong sense. Not only does no one else know his value, but no one else knows anything that could provide even statistical information about it.

The informational rents accruing to the buyers come solely from this strong exclusivity of information. The surprising conclusion is that the slightest degree of correlation in information among the buyers allows the seller to extract all the surplus.

This constitutes a departure from All other features of the model remain unaltered, in essence. Although we derived this result in a context where signals were continuously distributed, the same is true when they are discrete variables. Thus, as long as the discrete single crossing condition Unlike the generalized VCG mechanism, however, the optimal mechanism depends critically on the distribution of signals.

This last feature allows it to extract all the surplus from buyers so that their expected payoffs are exactly zero. From the perspective of the seller, this is clearly the best possible outcome as he is able to, in effect, act as a perfectly price-discriminating monopolist.

This is just the discrete analog of the joint density function, f , in the case of continuously distributed signals. The main result of this section is as follows: Proposition Suppose that signals are discrete and the valuations v satisfy the single crossing condition. The previous result is quite remarkable in that it shows that with the slightest degree of correlation among the signals, the seller can prevent the buyers from sharing any of the surplus resulting from the sale.

Some remarks are in order. In that case, truth-telling is a dominant strategy in the optimal auction as well. How buyer i evaluates this lottery depends on his own signal since, given the statistical dependence among signals, for different realizations of Xi the expected payment implicit in the lottery is different. The lottery is, as it were, an entry fee that allows buyer i to participate in the workings of the generalized VCG mechanism and in expectation the buyer is just indifferent between entering and not.

He may end up paying something to the seller even if he does not get the object—and so, unlike the common auction formats, the mechanism is not ex post individually rational. In this case, it becomes increasingly untenable to maintain the assumption that buyers remain risk-neutral over the range of payoffs they may encounter while participating in a mechanism. Suppose that there are two potential buyers for one indivisible object.

Determine the expected revenue from this mechanism. Consider the following mechanism. The other buyer j pays nothing. Show that the mechanism described above is incentive compatible and individually rational. What is the expected revenue in the truthful equilibrium of this mechanism? Does the mechanism have other nontruthful equilibria? The generalized VCG mechanism and the single crossing condition were introduced there.

More recent work on the generalized VCG mechanism is contained in Ausubel They also show that this is the best possible result in general—in an example, no mechanism can leave the buyers with a surplus of exactly zero. McAfee and Reny extend this result to other mechanism design settings. This page intentionally left blank Chapter Eleven Bidding Rings We have looked at an assortment of models with varying features: the auction format, the valuation structure, the informational structure, and so on.

The one common feature across all these models has been the assumption that bidders make their decisions independently; that they do not act in a concerted way. In other words, the bidders were assumed to be engaged in a noncooperative game. This chapter explores some issues that arise when a subset, or possibly all, of the bidders act collusively and engage in bid rigging with a view to obtaining lower prices.

The resulting arrangement—a bidding ring—resembles an industrial cartel and many of the issues surrounding cartels resurface in this context. How can the cartel enforce the agreed upon mode of behavior? How are the gains from collusion to be shared? How should economic agents on the other side of the market—in this case, the seller—respond to the operation of the cartel? While bidding rings are illegal, they appear to be widely prevalent.

Theoretical models of collusion among bidders involve a mix of cooperative and noncooperative game theory. The assumption of a common interval is made only for notational convenience and is easily relaxed. The equilibrium behavior of bidders in asymmetric second-price auctions, with or without a ring, is, however, quite transparent and allows us to focus our attention on a new set of issues surrounding collusion among bidders. It is in the interests of the bidding ring to make sure that the object goes in the hands of the ring member who values it the most, provided, of course, that it manages to win the object.

But information regarding their values is privately held by the members and in order to function effectively, a bidding ring needs to gather this information and then to divide the gains from collusion among its members. How, and whether, both tasks can be accomplished is a key question. In what follows, however, we temporarily put aside the question of the internal functioning of a ring—returning to it later—and, assuming that it functions effectively, seek to identify the resulting gains and losses to the various parties.

It is also weakly dominant for the ring to submit a bid equal to the highest value among its members—that is, Y1I. Equivalently, we may think of the ring as being represented at the auction by the member with the highest value in the ring.

The other members submit bids of 0, or if there is a reserve price, they bid at or below this price. The rest submit nonserious bids by bidding at or below the reserve price. In the present context, however, the bidding ring exerts no externality whatsoever on bidders who are not part of the ring. First, the probability that a bidder who is not a member of the ring will win the object is the same whether or not the ring is functioning; in both cases it is just the probability that she has the highest value among all bidders.

Once again, this increase comes solely at the expense of the seller. For future reference it is useful to summarize the main conclusions reached so far.

To do this, however, the ring has to induce its members to truthfully reveal private information regarding their values. In other words, the bidding ring faces a mechanism design problem akin to those considered in Chapter 5. Indeed, the mechanism design perspective offers many insights into the workings of bidding rings.

Consider a bidding ring I and a ring center, which coordinates the activities of the ring. These payments may be negative, since it may be necessary for the center to make transfers to members of the ring other than the winner. Temporarily, suppose that the internal mechanism is incentive compatible so that the ring member with the But this probability is the same as it would be if there were no bidding ring and all bidders behaved noncooperatively. But since the two mechanisms have the same allocation rule, the revenue equivalence principle Proposition 5.

This means that the expected payments of any bidder in the two mechanisms—with the operation of the ring and without—differ by at most a constant. Furthermore, the operation of the ring does not affect any bidder outside the ring.

In a second-price auction the cartel agreement—only the member with the highest value submits a serious bid; all others bid at or below the reserve price— is self-enforcing in the following sense. If such a member were to win the object, it would be at a price that exceeds his own value. This is because there is going to be at least one bid in the auction that exceeds his value: that of the cartel representative. The winner of this auction, called a preauction knockout PAKT , wins the right to represent the bidding ring at the main auction.

The PAKT is also conducted under second-price rules, and its workings are as follows. First, each member of the ring is asked to reveal his or her private value to the center.

This amount is easily determined if all ring members report their values truthfully. Figure In particular realizations, however, the sum of the transfers made by the center may exceed or fall short of what it receives. Certainly, there are circumstances in which the ring is unable to obtain the object, so there are no receipts while the lump-sum payments to the ring members have still to be made.

The PAKT is incentive compatible—no ring member can do better than to report his true value to the center. Indeed, it is a weakly dominant strategy for every ring member to report truthfully.

This is because the second-price PAKT requires the winning member, if and when he obtains the object, to pay the second-highest of all N values. It is not an ex post balanced budget mechanism. We will refer to this right as the ticket, thereby avoiding confusion with the object being offered for sale at the main auction. The ticket has a positive imputed value for each ring member—the expected gain from participating in the main auction. As a starting point, suppose that the center sells the ticket by means of a genuine second-price auction in which the winning member actually pays the center the second-highest imputed value for the ticket.

In this case, the ring center would end up with a surplus for sure. But the second-price auction is the same as a Vickrey-Clarke-Groves VCG mechanism, introduced in Chapter 5, specialized to an auction context.

Thus, if the ticket is allocated using the VCG mechanism, the center would run a surplus. Now Proposition 5. The proof of Proposition 5. The balanced budget mechanism from Proposition 5. In contrast, a PAKT balances the budget only in expected terms but has the property that truth-telling is a dominant strategy. But what recourse does the seller have? Assuming that the collusion cannot be detected by the antitrust authorities and that the seller cannot make a credible case for the presence of a bidding ring, the only instrument left in her hands with which to counter the actions of the bidding ring is to set a reserve price.

Here we explore the issue of the optimal reserve price when the seller is aware that a bidding ring consisting of members in the set I is in operation and likely to act in concert in an upcoming auction. The object is sold if Bidding Rings and only if the highest of these, which is the same as Y1N , is greater than the reserve price r. The second term comes from the event that the secondhighest value, Z I , itself exceeds r, so the object is sold at a price equal to Z I.

Analogous to The addition of a bidder to a bidding ring causes the optimal reserve price for the seller to increase. In particular, the optimal reserve price with a bidding ring is always greater than the optimal reserve price with no ring. The increase in the optimal reserve price resulting from a cartel is illustrated in the following simple example.

Suppose that there are two bidders with values that are uniformly and independently distributed on [0, 1]. Now suppose that both bidders are members of a cartel and suppose that the seller sets a reserve price of r. To see this simply, consider an all-inclusive cartel.

Assuming that the highest value exceeds the reserve price set by the seller, such a cartel will try to obtain the object at the reserve price by submitting only one bid at this level and ensuring that no other bid exceeds this amount. But now consider a bidder whose value is greater than the reserve price but is not the highest.

Such a bidder has the incentive to cheat on the cartel agreement and, by submitting a bid that just exceeds the reserve price, win the object. Apart from physical coercion—always a possibility, especially given the criminal nature of the activity—the agreement may be enforced by repeated play.

Second, even if the bidders are ex ante symmetric, the operation of a cartel naturally introduces asymmetries among bidders. In particular, bidders not in the cartel face a different decision problem if there is a cartel in operation than if there is not. In what follows, we suppose that bidders are ex ante symmetric—that is, their values are drawn independently from the same distribution F.

We also suppose that the cartel is all-inclusive. These two assumptions together ensure symmetry among bidders. It should submit only one serious bid—at the reserve price. The winner of the PAKT then represents the cartel at the main auction, obtaining the good at the reserve price. First, suppose bidders bid individually—that is, there is no bidding ring.

Now suppose that the N bidders form a perfectly functioning bidding ring. Note: Since values are discrete, this will be in mixed strategies.

Now suppose that bidders 1 and 2 form a cartel. While the cartel cannot control the bids submitted by its members, it can arrange transfers and recommend bids. Further, suppose that the values of its members become commonly known among the cartel once it is formed. Find an equilibrium with the cartel, assuming that bidder 3 acts independently. Is it possible for the cartel to ensure that only one member submits a bid?

PAKT A single object is to be sold via a second-price auction to two bidders whose private values Xi are drawn independently from the uniform distribution on [0, 1].

Suppose that the bidders form a cartel. Find the equilibrium bidding strategies in the preauction knockout PAKT. Collusion-proof mechanism A single object is to be sold to two bidders with private values drawn independently from the uniform distribution on [0, 1].

The following mechanism is used to sell the object. Each bidder i submits a bid bi. The winner, bidder i, is awarded the object and asked to pay bi to the losing bidder j. If there is a tie, either bidder is assigned the role of a winner. Find a symmetric equilibrium of this mechanism assuming that the bidders act noncooperatively. Can the two bidders gain by forming a cartel and colluding against the seller? Their model and results were later extended to accommodate asymmetric bidders by Mailath and Zemsky Most of the material in this chapter is based on these two papers.

Problem Che and Kim have shown, quite remarkably, that there exist mechanisms which are immune to collusion by the bidders. A general survey of the area, emphasizing many open questions, has been written by Hendricks and Porter The statistics on the level of antitrust activity that relates to bid rigging reported on page are taken from a U.

Department of Justice study that is quoted by Hendricks and Porter Alternatively, the objects may be complements—that is, the value derived from a particular object may be greater if another has already been obtained. For instance, a philatelist may value a collection of stamps more than the sum of the values of the individual stamps. Similarly, how much an airline values an airport landing slot may increase with the number of slots it has already acquired.

Not surprisingly, when multiple objects are to be sold, many options are open to the seller. First, the seller must decide whether to sell the objects separately in multiple auctions or jointly in a single auction. Second, the seller must choose among a variety of auction formats, and there is a wide range of possibilities to choose from. We begin by outlining the workings of a few auction forms for the sale of multiple units of the same good at one go, returning to study multiple one at a time, sequential, or simultaneous auctions later.

All three are intended to be used in situations in which the marginal values are declining—that is, the value of an additional unit decreases with the number of units already obtained.

The uniform-price auction. The Vickrey auction. We will thus use these interchangeably. The allocation rule implicit in all three auctions may be framed in conventional supply and demand terms. For example, the demand function depicted in Figure We will refer to an auction in which the K highest bids are deemed winning and awarded objects as a standard auction. The three auctions introduced next are all standard but differ in terms of their pricing rules—how much each bidder is asked to pay for the units he is awarded.

The discriminatory pricing rule can also be framed in terms of the residual supply function facing each bidder. The discriminatory auction asks each bidder to pay an amount equal to the area under his own demand function up to the point where it intersects the residual supply curve. At the risk of stating the obvious, we caution the reader not to infer from Figure As we will see, bidding behavior in the three auctions differs substantially and reaching any conclusions regarding revenue is a delicate matter.

We adopt the rule that the market-clearing price is the same as the highest losing bid. The number of units that bidder i wins is just the number of competing bids he defeats. The basic principle underlying the Vickrey auction is the same as the one underlying the Vickrey-Clarke-Groves mechanism discussed in Chapter 5: Each bidder is asked to pay an amount equal to the externality he exerts on other competing bidders.

In the example, had bidder 1 been absent, the three units allocated to him would have gone to the other bidders: two to bidder 2 and one to bidder 3. According to the demand function submitted by him, bidder 2 is willing to pay b22 and b32 , respectively, for two additional units.

Similarly, bidder 3 is willing to pay b33 for one additional unit. Bidder 1 is asked to pay the sum of these amounts. The amounts that bidders 2 and 3 are asked to pay are determined in similar fashion. Unlike the uniform-price auction, however, it shares many important properties with the second-price auction and is, as we will argue, the appropriate extension of the second-price auction to the case of multiple units.

Other possible pricing rules exist; the range of available options is virtually unlimited. For example, in one variant of the uniform pricing rule, all units are sold at a price equal to the average of all the winning bids.

The price is then gradually lowered until a bidder indicates that he is willing to buy a unit at the current price. This bidder is then sold an object at that price and the auction continues— the price is lowered further until another unit is sold, and so on.

This continues until all K units have been sold. In a sealed-bid discriminatory auction, no such information is available. The two auctions are, however, weakly equivalent in the sense of Chapter 1—with private values the information acquired from the fact that one bidder is willing to buy at some price, while available, is irrelevant.

Each bidder indicates— by using hand signals, by holding up numbered cards, or electronically—how many units he is willing to buy at that price—in other words, his demand at that price. Novels, tale publication, and also other amusing e-books become so popular now. Besides, the clinical e-books will certainly likewise be the very best factor to choose, particularly for the pupils, instructors, medical professionals, entrepreneur, and various other careers that love reading.

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