Tài liệu Pricing communication networks P14 doc

14 Auctions An auction is a sale in which the price of an item is determined by bidding. Flowers, wine, antiques, US treasury bonds and land are sold in auctions. Takeover battles for companies can be viewed as auctions (and indeed, the Roman empire was auctioned by the Praetorian Guards in A.D. 193). Auctions are commonly used to sell natural resources, such as oil drilling rights, or even the rights to use certain geostationary satellite positions. Government contracts are often awarded through procurement auctions. There is the advantage that the sale can be performed openly, so that no one can claim that a government official awarded the contract to the supplier who offers him the greatest bribe. In Section 9.4.4 we saw how instantaneous bandwidth might be sold in a smart market in which the price is set by auction. In recent years, auctions have been used in the communications market to sell parts of the spectrum for mobile telephone licenses. Some of these have raised huge sums for the government, but others have raised less than expected. An auction can be viewed as a partial information game in which the valuations that each bidder places on the items for sale is hidden from the auctioneer and the other bidders. The game’s equilibrium is a function of the auction’s rules, which specify the way bidding oc- curs, the information bidders have about the state of bidding, how the winner is determined and how much he must pay. These rules can affect the revenue obtained by the seller, as well as how much this varies in successive instances of the auction. An auction is economically efficient, in terms of maximizing social welfare, if it allocates items to bidders who value them most. We emphasize that designing an auction for a particular situation is an art. There is no single auctioning mechanism that is provably efficient and can be applied in most situ- ations. For example, in spectrum auctions some combinations of spectrum licenses are more valuable to bidders than others, and so licenses must be sold in packages, using some sort of combinatorial bidding. As we explain in Section 14.2.2, this greatly complicates auction design. One can prove important theoretical results about some simple auction mechanisms, (such as the revenue equivalence theorem of Section 14.1.3). They are not easily applied in many real life situations, but they do provide insights into the problems involved. The purpose of this chapter is to provide the reader with an introduction to auction theory and some examples of how it can be used in pricing communications services. Auction theory is now a very well-developed area of research, and we can do no more than give an introduction and some interesting results. We have previously discussed how the mechanism of tatonnement can be used to maximize social welfare in resource allocation problems (Section 5.4.1). In tatonnement, price is varied in response to excess demand (positive or Pricing Communication Networks: Economics, Technology and Modelling. Costas Courcoubetis and Richard Weber Copyright  2003 John Wiley & Sons, Ltd. ISBN: 0-470-85130-9 310 AUCTIONS negative) until demand exactly matches supply. One crucial property of any tatonnement mechanism is that prices should be able to increase or decrease until that point is reached. Auction mechanisms do not usually allow prices to fluctuate in both directions. Tatonnement can take a large number of steps. Some auctions take place in just one step, with little information exchange between the buyers and seller. In general, auctions are more restricted than tatonnement, and do not necessarily maximize social welfare. However, they have the advantage that they can be faster and simpler to implement. A second requirement for the tatonnement mechanism to work is that customers should make truthful declarations of their resource needs for given posted prices. This will happen if the market has many customers, with no customer being so large that he can affect the price by the size of his own demand. That is, customers are price takers. Auctions, however, can be efficient even when there are a small number of bidders, although the optimal strategy for some may be not to tell the truth. There are two important and distinct models for the way bidders value items in an auction. In the private value model, each bidder knows the value that he places on a given item, but he does not know the valuations of other bidders. As bidding takes place, his valuation does not change, although he may gain information about other bidders’ valuations when he hears their bids. In the common value model , all bidders estimate their valuation of the item in the same way, but they have different prior information about that value. Suppose, for example, a jar of coins is to be auctioned. Each bidder estimates the value of the coins in the jar, and as bidding occurs he adjusts his estimate on the basis of what others say. For example, if most bidders make higher bids than his own, a bidder might feel that he should increase his estimate of the value of the coins. In this case, the winner generally over-estimates the value (since he has the highest estimate), and so it is likely that he pays more than the jar of coins is worth. This is known as the winner’s curse (about which we say more in Section 14.1.7). Sometimes a bidder’s valuation is a function of both private information and of information revealed during the auction. For example, suppose an oil-lease is to be auctioned. The value of the lease depends both upon the amount of oil that is in the ground and the efficiency with which it can be extracted. Bidders may have different geological information about the likely amount of oil, and have different extraction efficiencies, and so make different estimates of the value of the lease. During bidding, bidders reveal information about their estimates and this may be helpful to other bidders. There are many other considerations that come into play when designing auctions. The seller may impose a participation fee, or a minimum reserve price. An auction can be oral (bidders hear each other’s bids and make counter-offers) or written (bidders submit closed sealed-bids in writing). In an oral auction, the number of bidders may be known, but in a sealed-bid auction the number is often unknown. Oral auctions proceed in a progressive manner, taking many rounds to complete, while sealed-bid auctions may take only a single round. All these things can influence the way bidders compete; by making them compete more fiercely, the seller’s revenue is increased. In Section 14.1 we describe some types of auction and summarize some important theoretical results. These concern auctions of a single item. However, one may wish to sell more than a single item. In a multi-object auction, multiple units of the same or of different items are to be sold. Such auctions can be homogeneous or heterogeneous, depending on the items to be sold are identical or not; discriminatory or uniform price, depending on whether identical items are sold at different or equal prices (this distinction only applies to homogeneous auctions); individual or combinatorial, depending on whether bids are allowed only for individual items or for combinations of items; sequential or simultaneous, depending on the whether items are auctioned one at a time or all at once. We take up SINGLE ITEM AUCTIONS 311 these issues in Section 14.2. Note, however, that we opt for an informal presentation of the multi-object auction, as there are few rigorous results. In summary, auctions are mechanisms for allocating resources in situations in which there is incomplete information and traditional market mechanisms do not provide incentives for participants truthfully to declare the missing information. Auction design takes account of this lack of information and can improve the equilibrium properties of the underlying games. We conclude the chapter in Section 14.3 by summarizing its ideas in the context of a highspeed link whose bandwidth is put up for sale by auction. 14.1 Single item auctions 14.1.1 Take it or leave it Pricing In this section, we consider the sale of a single item by auction. For the purposes of comparison, we begin with analysis of a selling mechanism that is not an auction, but which could be used under the same conditions of incomplete information that pertain when auctions are used. Suppose a seller wishes to sell a single item. He does this simply by making a take- it-or-leave-it offer, at price p. If any customers wants to buy the item at that price, then it is sold; otherwise it is not sold, and the seller obtains zero revenue. If more than one customer wants the item at the stated price, then there must be a procedure for deciding who gets it. However, the seller still receives revenue of p. Suppose customers are identical and their private valuations are independent and identically distributed as a random variable X, with distribution function F.x/ D P.X Ä x/. Given knowledge of this distribution, the seller wants to choose p to maximize his expected revenue. Let x. p/ denote the probability the item is sold. Then x. p/ D 1  F. p/ n (14.1) Let f . p/ D F 0 . p/ be the probability density function of X. By maximizing the expected revenue, of px.p/, we find that the optimal price p Ł should satisfy p  1  F. p/ n nF.p/ n1 f . p/ D 0 (14.2) For example, if valuations are uniformly distributed on [0; 1], then F.x/ D x,and we find that the optimal price is p Ł D .n C 1/ 1=n . The resulting expected revenue is n.n C 1/ .nC1/=n .Forn D 2, the optimal price is p Ł D p 1=3. The seller’s expected revenue is .2=3/ p 1=3(D 0:3849). Note that, because there is a positive probability that the item is not sold, this method of selling is not economically efficient. We have seen this before in Chapter 6; if a monopolist seeks only to maximize his own revenue then there is often a social welfare loss. For the example above, the maximum valuation is the maximum of n uniform random variables distributed on [0; 1]; it is a standard result that this has expected value n=.n C 1/.This is the expected social welfare gain if the item is allocated to the bidder with the highest valuation. For n D 2, this is 2=3(D 0:6666). However, under take it or leave it pricing, the expected social welfare gain can shown to be 1  p Ł  .1  p Ł / .nC1/ =.n C 1/.Forn D 2, this is only 0:6094. In all the above, we have assumed that the seller knows the distribution of the bidders’ valuations. If he does not have this information, then he cannot determine the optimal ‘take it or leave it’ price. He also has a problem if his prior beliefs are mistaken. Suppose, for 312 AUCTIONS the example with n D 2, he believes that valuations are uniformly distributed on [0; 1] and sets the price optimally at p Ł D p 1=3. Say, however, he is mistaken: bidders valuations are actually uniformly distributed on [0; 0:5]. Then, as p Ł > 0:5, he never sells. It would have been better if he had auctioned the item, thus ultimately selling it to the highest bidder. Even if the seller does know the distribution of bidders’ valuations, he can do better by auctioning. As we see below, one can design auction rules that increase the expected revenue and make auctioning the most profitable selling method. One way to do this is to introduce a minimum price that must be paid by the auction’s winner. This reserve price has the effect of increasing the average price paid by the winner. In our example, he could set a reserve price of 1=2 and would obtain expected revenue of 0:4167 (see Section 14.1.4). 14.1.2 Types of Auction We now describe some of the most popular types of auction. In the ascending price auction (or English auction), the auctioneer asks for increasing bids by raising the price of the item by small increments, until only one bidder remains. Or perhaps bidders place increasing bids by shouting. The item is awarded to the last remaining bidder, at the price of the last bid at which all other bidders had withdrawn. It is clear that in this type of auction the winner is the bidder with the highest valuation, and he pays a price equal to the second highest valuation. Unique items, such as artworks, tend to be sold in English auction, in order to find an unknown price. Another version of this auction is used in Japan; the price is displayed on a screen and raised continuously. Any bidder who wishes to remain active keeps his finger on a button. When he releases the button he quits the auction and cannot bid again. In a reverse procedure to the English auction above, the Dutch auction starts by setting the price at some initial high value. A so-called ‘Dutch clock’ displays the price and continuously decreases it until some bidder decides to claim the item at the price displayed. Multiple items (such as fish or flowers) tend to be sold in Dutch auctions; this speeds up the time the sale takes. The price is lowered until demand matches supply. In the next two types of auction, bidders submit sealed-bids and the one with the greatest bid wins. The auctions differ in the price charged to the winner. Under the first-price sealed- bid auction, the winner pays his bid. In this auction, the bidder has to decide off-line how much he should bid. This is equivalent to deciding off-line at what price he would claim the item in a Dutch auction, since in that auction no information is revealed until the first bid, at which point the auction also ends. Thus, we see that the Dutch auction and first-price sealed-bid auction are completely equivalent. In the second-price sealed-bid auction, the winner pays the second highest bid. This is also known as a Vickrey auction, after its inventor. An important property of the Vickrey auction is that it is optimal for each bidder to bid his true valuation. To understand why this is so, note that a bidder would never wish to bid more than his valuation, since his expected net benefit would then be negative. However, if he reduces his bid below his valuation, he reduces the probability that he wins the auction, but he does not affect the price that he pays if he does win (which is determined by the second highest bidder). Thus, he does best by bidding his true valuation. The winner is the bidder with the greatest valuation and he pays the second greatest valuation. But this is exactly what happens in the English auction, in which a player drops out when the price exceeds by a small margin his valuation, and so the winner pays the valuation of the second-highest bidder. Thus, we see that the English and Vickrey auctions are equivalent. Other auctions include the all-pay auction, in which all bidders pay their bid but the highest bidder wins the object, and the k-price auction, in which the winner pays the kth SINGLE ITEM AUCTIONS 313 largest bid. Some of these auctions can be easily extended to multiple units. For example, in the two-unit first-price sealed bid auction the participants with the two greatest bids are winners and pay the third largest bid. Multi-unit auctions require bidders to follow much more complex strategies. We return to multi-unit auctions in Section 14.2. 14.1.3 Revenue Equivalence A simple auction model for which we can give a full analysis is the Symmetric Independent Private Values (SIPV) model. It concerns the auction of a single item, in which both seller and bidders are risk neutral. To understand the idea of being risk neutral, imagine that a seller has a utility function that measures how he values the payment he receives. If his utility function is linear he is said to be risk-neutral. His average utility (after repeating the auction many times) is the same as his utility for the average payment, and hence the variability of the payment around its mean does not reduce the average utility of the seller. If the utility function is concave then the seller is risk-averse; now the average utility is less than the utility of the average payment, and this discrepancy increases with the variability of the payment. Suppose each bidder knows his own valuation of the item, which he keeps secret, and valuations of the bidders can be modelled as independent and identically distributed random variables. Some important questions are as follows. 1. Which of the four standard auctions of the previous section generates the greatest expected revenue for the seller? 2. If the seller or the bidders are risk-averse, which auction would they prefer? 3. Which auctions make it harder for the bidders to collude? 4. Can we compare auctions with respect to strategic simplicity? Let us begin with an intuitive, but important, result. Lemma 1 In any SIPV auction in which (a) the bidders bid optimally, and (b) the item is awarded to the highest bidder, the order of the bids is the same as the order of the valuations. Proof Suppose that under an optimal bidding strategy a bidder whose valuation is v bids so as to win with probability p.v/.Lete. p/ be the minimal expected payment that such a bidder can make if he wants to win the item with probability p. Assume v 1 and v 2 are such that v 1 >v 2 , but p.v 1 /<p.v 2 /. If this is true, then it is simple algebra to show that, with p i D p.v i /, [ p 1 v 2  e. p 1 /] C [ p 2 v 1  e. p 2 /] > [ p 1 v 1  e. p 1 /] C [ p 2 v 2  e. p 2 /] Thus, either p 1 v 2  e. p 1 /> p 2 v 2  e. p 2 /,or p 2 v 1  e. p 2 />p 1 v 1  e. p 1 /.Inother words, either it is better to win with probability p 1 when the valuation v 2 ,oritisbetterto win with probability p 2 when the valuation is v 1 , in contradiction to our assumptions. We are forced to conclude that p.v/ is nondecreasing in v. By assumption (b) in the lemma statement, this means that the optimal bid must be nondecreasing in v.  We say that two auctions have the same bidder participation if any bidder who finds it profitable to participate in one auction also finds it profitable to participate in the other. The following is a remarkable result. 314 AUCTIONS Theorem 4 (revenue equivalence theorem) The expected revenue obtained by the seller is the same for any two SIPV auctions that (a) award the item to the highest bidder, and (b) have the same bidder participation. We say this is a remarkable result because different auctions can have completely different sets of rules and strategies. We might expect them to produce different revenues for the seller. Note that revenue equivalence is for the expectation of the revenue and not for its variance. Indeed, as we see in Section 14.1.5, auctions can have quite different properties so far as risk is concerned. Proof of the revenue equivalence theorem Suppose there are n participating bidders. As above, let e. p/ denote the minimal expected payment that a bidder can make if he wants to win with probability p. The bidder’s expected profit is ³.v/D pv e. p/,where p D p.v/ is chosen optimally and so, since ³ must be stationary with respect to any change in p, we must have v  e 0 . p/ D 0. Hence, d dv e. p.v// D e 0 . p/ dp dv D v dp dv Integrating this directly and then by parts gives e. p.v// D e. p.0// C Z v 0 w dp.w/ dw dw D vp.v/  Z v 0 p.w/ dw (14.3) where clearly e. p.0// D e.0/ D 0, since there is no point in bidding for an item of value 0. Thus, e. p.v//, which is the expected amount paid by a bidder who values the item at v, depends only upon the function p.Ð/. We know from Lemma 1 that if bidders bid optimally then bids will be in the same order as the valuations. It follows that if F is the distribution function of the valuations, then p.w/ D F.w/ n1 , independently of the precise auction mechanism. The expected revenue can therefore be computed from (14.3) as P n iD1 E v i e. p.v i // D nE v e. p.v//.  Notice that there is also ‘expected net benefit equivalence’ for the bidders. To see this, observe that the bidders obtain an expected net benefit that is equal to the expected value of the item to the winner of the auction, minus the expected total payment made to the seller. Since the expected value of both these quantities are independent of the auction rules, it follows that the expected net benefit of the bidders is also independent of the auction rules. Since bidders are symmetric they share this surplus equally. It should be clear that all four auctions described in Section 14.1.2 satisfy the conditions of the revenue equivalence theorem. Let us work through an example in which the valuations, say v 1 ;:::;v n , are random variables, independent and uniformly distributed on [0; 1]. Let v .k/ denote the kth largest of v 1 ;:::;v n (the k-order statistic). A standard result is that E[v .k/ ] D k=.n C 1/. Hence in the Vickrey and English auctions the expected revenue is E[v .n1/ ] D .n  1/=.n C 1/. Using this, we can find the optimal bid in the first-price sealed-bid auction. By the theorem the expected revenue in this auction is the same as in the English auction, i.e. .n  1/=.n C 1/. Also, recall that p.v/ D F.v/ n1 D v n1 . Using (14.3), we easily find e. p.v// D .n  1/v n =n. This must be p.v/ times the optimal bid. So a bidder who values the item at v has an optimal bid of .n  1/v=n. This is a shaded bid, equal to the expected value of the second-highest valuation, given that v is the highest valuation. SINGLE ITEM AUCTIONS 315 14.1.4 Optimal Auctions An important issue for the seller is to design the auction to maximize his revenue. We give revenue-maximizing auctions the name optimal auctions. It turns out that a seller who wants to run an optimal auction can increase his revenue by imposing a reserve price or a participation fee. This reduces the number of participants, but leads to fiercer competition and higher bids on the average, which may compensate for the probability that no sale takes place. Let us illustrate this with an example. Example 14.1 (Revenue maximization) Consider a seller who wishes to maximize his revenue from the sale of an object. There are two potential buyers, with unknown valuations, v 1 , v 2 , that are independent and uniformly distributed on [0; 1]. He considers four ways of selling the object: 1. A take it or leave it offer. 2. A standard English auction. 3. An English auction with a participation fee c (which must be paid if a player chooses to submit a bid). Each bidder must choose whether or not to participate before knowing whether the other participates. 4. An English auction with a reserve price, p. The bidding starts with a minimum bid of p. Case 1 was analysed in Section 14.1.1. The best ‘take it or leave it’ price is p D p 1=3 and this gives an expected revenue of .2=3/ p 1=3(D 0:3849). Case 2 was analysed above. The expected revenue in the English auction was 1=3 (D 0:3333). Case 3. To analyse the auction with participation fee, note that a bidder will not wish to participate if his valuation is less than some amount, say v 0 . A bidder whose valuation is exactly v 0 will be indifferent between participating or not. Hence P.winning j v D v 0 /v 0 D c. Since a bidder with valuation v 0 wins only if the other bidder has a valuation less than v 0 ,wemusthaveP.winning j v D v 0 / D v 0 , and hence v 2 0 D c. Thus, v 0 D p c. To compute the expected revenue of the seller, we note that there are two ways that revenue can accrue to the seller. Either only one bidder participates and the sale price is zero, but the revenue is c. Or both bidders have valuation above v 0 , in which case the revenue is 2c plus the sale price of minfv 1 ;v 2 g. The expected revenue is 2v 0 .1  v 0 /c C .1  v 0 / 2 [2c C v 0 C .1  v 0 /=3] Straightforward calculations show that this is maximized for c D 1=4, and takes the value 5=12 (D 0:4167). Case 4. In the English auction with a reserve price p, there is no sale with probability p 2 . The revenue is p with probability 2 p.1 p/.Ifminfv 1 ;v 2 g > p, then the sale price is minfv 1 ;v 2 g. The expected revenue is 2 p 2 .1  p/ C  1 3 C 2 3 p Á .1  p/ 2 This is maximized by p D 1=2 and the expected revenue is again 5=12, exactly the same as in case 3. 316 AUCTIONS That Cases 3 and 4 in the above example give the same expected revenue is not a coincidence. These are similar auctions, in that a bidder participates if and only if his valuation exceeds 1=2. Let us consider more generally an auction in which a bidder participates only if his valuation exceeds some v 0 . Suppose that with valuation v it is optimal to bid so as to win with probability p.v/, and the expected payment is then e. p.v//. By a simple generalization of (14.3), we have e. p.v// D e. p.v 0 // C Z v v 0 w dp.w/ dw dw D vp.v/  Z v v 0 p.w/ dw Assuming the SIPV model, this shows that a bidder’s expected payment depends on the auction mechanism only through the value of v 0 that it implies. The seller’s expected revenue is nE v [e. p.v//] D n Z 1 vDv 0 Ä vp.v/  Z v wDv 0 p.w/ dw ½ f .v/ dv D n Z 1 vDv 0 vp.v/ f .v/ dv  n Z 1 wDv 0 Z 1 vDw p.w/ f .v/ dw dv D n Z 1 vDv 0 ý v f .v/  [1  F.v/]  F.v/ n1 dv Now differentiating with respect to v 0 , to find the stationary point, we see that the above is maximized where v 0 f .v 0 /  [1  F.v 0 /] D 0 We call v 0 the optimal reservation price. Note that it does not depend upon the number of bidders. For example, if valuations are uniformly distributed on [0; 1], then v 0 D 1=2. This is consistent with the answers found for Cases 3 and 4 of Example 14.1. If bidders’ valuations are independent, but heterogenous in their distributions, then one can proceed similarly. Let p i .v/ be the probability that bidder i wins when his valuation is v.Lete i . p/ be the minimum expected amount he can pay if he wants to win with probability p. Suppose that bidder i does not participate if his valuation is less than v 0i . Just as above, one can show that the seller’s expected revenue is n X iD1 E v i e i . p i .v i // D n X iD1 Z 1 vDv 0i Ä v  1  F i .v/ f i .v/ ½ f i .v/ p i .v/ dv (14.4) The term in square brackets can be interpreted as ‘marginal revenue’, in the sense that if a price p is offered to bidder i, he will accept it with probability x i . p/ D 1  F i . p/,andso the expected revenue obtained by this offer is px i . p/. Differentiating this with respect to x i ,wedefine MR i . p/ D d dx i  px i . p/ Ð D d dp  px i . p/ Ð  dx i dp D p  1  F i . p/ f i . p/ Note that the right-hand side of (14.4) is simply E[MR i Ł .v i Ł /], where i Ł is the winner of the auction. This can be maximized simply by ensuring that the object is always awarded to the bidder with the greatest marginal revenue, provided that marginal revenue is positive. We can do this provided bidders reveal their true valuations. Let us assume that SINGLE ITEM AUCTIONS 317 MR i . p/ is increasing in p,foralli. Clearly, v 0i should be the least v such that MR i .v/ is nonnegative. Consider the auction rule that always awards the item to the bidder with the greatest marginal revenue, and then asks him to pay the maximum of v 0i and the smallest v for which he would still remain the bidder with greatest marginal revenue. This has the character of a second-price auction in which the bidder’s bid does not affect his payment, given that he wins. So bidders will bid their true valuations and (14.4) will be maximized. Example 14.2 (Optimal auctions) An interesting property of optimal auctions with heterogeneous bidders is that the winner is not always the highest bidder. Consider first the case of homogeneous bidders with valuations uniformly distributed on [0; 1]. In this case, MR i .v i / D v i  .1  v i /=1 D 2v i  1. Hence the object is sold to the highest bidder, but only if 2v i  1 > 0, i.e. if his valuation exceeds 1=2. The winner pays either 1=2 or the second greatest bid, whichever is greatest. In the case of two bidders, with the seller’s expected revenue is 5=8. This agrees with what we have found previously. Now consider the case of two heterogeneous bidders, say A and B, whose valuations are uniformly distributed on [0; 1] and [0; 2], respectively. So MR A .v A / D 2v A  1, and MR B .v B / D 2v B  2. Under the bidding rules described above, bidder B wins only if 2v B  2 > 2v A  1and2v B  2 > 0, i.e. if and only if v B  v A > 1=2andv B > 1; so the lower bidder can sometimes win. For example, if v A D 0:8andv B D 1:2, then A wins and pays 0:7 (which is the smallest v such that MR A .v/ D 2v  1 ½ 2v B  2 D 0:4). 14.1.5 Risk Aversion As we have already mentioned, the participants in an auction can have different attitudes to risk. If a participant’s utility function is linear then he is said to be risk-neutral .Ifhis utility function is concave then he is risk-averse; now a seller’s average utility is less than the utility of his average revenue, and this discrepancy increases with the variability of the revenue. Hence a risk-averse seller, depending on his degree of risk-aversion, might choose an auction that substantially reduces the variance of his revenue, even though this might reduce his average revenue. The revenue equivalence theorem holds under the assumption that bidders are risk- neutral. One can easily see that if bidders are risk-averse, then first-price sealed-bid and Dutch auctions give different results from second-price sealed-bid and English auctions. For example, in a first-price auction, a risk-averse bidder prefers to win more frequently even if his average net benefit is less. Hence, he will make higher bids than if he were risk-neutral. This reduces his expected net benefit and increases the expected revenue of the seller. If the same bidder participates in a second-price auction, then his bids do not affect what he pays when he wins, and so his strategy must be to bid his true valuation. Hence, a first-price auction amongst risk-averse bidders produces a greater expected revenue for the seller than does a second-price auction. However, it is not clear which type of auction the risk-averse bidders would prefer. In general, this type of question is very difficult. The seller may also be risk-averse. In such a case, he prefers amongst auctions with the same expected revenue those with a smaller variance in the sale price. Let us compare a first and second-price auction with respect to this variance. Suppose bidders are risk-neutral. Let v .n/ and v .n1/ be the greatest and second-greatest valuations. In a second-price auction, the winner pays the value of the runner-up’s bid, i.e. v .n1/ . In a first-price auction he pays his bid, which is the conditional expectation of the valuation of the runner-up, conditioned on his winning the auction, i.e. E.v .n1/ jv .n/ /.LetY D .v .n1/ jv .n/ / and apply the standard 318 AUCTIONS fact that .EY/ 2 Ä EY 2 .Thisgives E v .n/ h E v .n1/ .v .n1/ jv .n/ / 2 i Ä E v .n/ h E v .n1/ .v 2 .n1/ jv .n/ / i D Ev 2 .n1/ Subtracting from both sides the square of the expected value of the winner’s bid, i.e. E.v .n1/ / 2 , we see that the winner’s bid has a smaller variance in the first-price auction, and so a risk-averse seller would prefer a first-price auction. Let us verify this for two bidders whose valuations are uniformly distributed on [0; 1]. In the first-price auction, each bidder bids half his valuation, so the revenue is .1=2/ maxfv 1 ;v 2 g. In the second-price auction each bids his valuation and the revenue is minfv 1 ;v 2 g. Both have expectation 1=3, but the variances are 1=72 and 1=18, respectively. Thus, a risk-averse seller prefers the first-price auction. 14.1.6 Collusion It is important when running an auction to take steps to prevent bidders from colluding. Collusion occurs when two or more bidders make arrangements not to bid as high as their valuations suggest, and so reduce the seller’s revenue. Antique auctions are notorious for this. A number of bidders form a ‘ring’ and agree not to bid against one another and on whom the winner will be. This lowers the winning bid. Later, the winner distributes his gain amongst all the bidders, in proportion to their market power, so that all do better than they would have done by not colluding. In some spectrum auctions in the US, there have been instances of bidders using the final four digits of their multimillion dollar bids to signal to one another the licenses they want to buy. Thus, a critical characteristic of an auction is how susceptible it is to collusion. This depends upon what incentives there are for players to stand by the promises they make to one another when agreeing to collude. We can see that an ascending English auction is susceptible to collusion. Suppose the bidders meet and determine that bidder 1 has the greatest valuation. They agree that bidder 1 should make a low bid and win the object for a payment close to zero. No other bidder has an incentive to bid against bidder 1, since he cannot win without ultimately outbidding bidder 1; yet if he does so he would incur a loss. Thus, the agreement between the bidders is ‘self-enforcing’ and the auction is susceptible to collusion. In contrast, collusion is difficult in a Dutch auction, or in a first-price sealed-bid auction. There is nothing to stop a ring member bidding higher than was agreed. His defecting action becomes obvious, but the auction is over before anyone can react. This is one reason why first-price sealed-bid auctions are often preferred when auctioning large government contracts. There is also a matter of trusting the seller. He might want to manipulate the auction to raise prices. One way he can do this is by soliciting fake bids. In a first-price sealed-bid auction, such bids do not make any sense, since they could prevent the sale of the object (and the seller could anyway use a reserve price). In a second-price auction, fake bids could benefit the seller. If the seller has approximate knowledge of the highest bidder’s valuation, he could solicit a ‘phantom’ bid with a slightly smaller value, and hence obtain almost all the surplus of the bidder. 14.1.7 The Winner’s Curse Thus far we have discussed the private values model. In the common values model, i.e. where the item that is auctioned has a common unknown value, the winner is the bidder [...]... transfer, satellite link bandwidth, and TV broadcast licenses, the law may explicitly require that all market players be treated similarly, and so that uniform pricing be used Also, national or international law may prohibit differentiated pricing, as being ‘politically incorrect’ Liquid versus less-liquid designs Thus far, we have assumed that the pipeline’s capacity is auctioned in small equal units... but the seller does better by selling the two licenses as a bundle 14.3 Auctioning a bandwidth pipeline We conclude this chapter by summarizing its ideas in the context of auctioning a high bandwidth communication link, or pipeline The pipeline is to be sold for a period of time, such as a year Its bandwidth is to be divided in discrete units, and these units sold in a multi-unit auction We suppose... identical units of a good are to be auctioned, and we speak of a multi-unit auction Multi-unit auctions are of great practical importance, and have been applied to selling units of bandwidth in computer networks and satellite links, MWs of electric power, capacity of natural gas and oil pipelines In the simplest multi-unit auction, each buyer wants only one unit The auction mechanisms we have already . tatonnement, price is varied in response to excess demand (positive or Pricing Communication Networks: Economics, Technology and Modelling. Costas Courcoubetis. introduction to auction theory and some examples of how it can be used in pricing communications services. Auction theory is now a very well-developed area