Optimal Combinatoric Auctions with Single-Minded Bidders

Transcription

1 Optimal Combinatoric Auctions with Single-Minded Bidders John O. Ledyard March 21, 2007 Abstract Combinatoric auctions sell K objects to N people who have preferences defined on subsets of the items. The direct version of the optimal auction satisfies incentive compatibility (it is a dominant strategy to report true values), voluntary participation (bidders are not worse off through participation) and maximizes the expected revenue of the auctioneer among such auctions. I characterize the optmal auction for the special case of single-minded bidders. The optimal auction is not a Vickrey-Clarke-Groves mechanism. Thus auctions that are efficient or in the core are not optimal. A detailed example suggests improvement over the VCG mechanism can be large. The performance of three well-known combinatoric auction designs (SMR, RAD, Sealed bid) in economic experiments are compared to the predicted performance of the optimal auction. The sealed bid auction achieves higher revenue than the optimal auction. RAD achieves higher revenues than SMR but less than the optimal auction. Keywords: combinatoric auctions, optimal auction, mechanism design, core, experiment JEL categories: C7, C9, D44 A shorter version of this paper, without the experimental results, was accepted for and presented at EC 07, the 8th ACM conference on Electronic Commerce, I would like to thank the significant help of several referees for that conference in catching typos and adding references. Division of Humanities and Social Sciences, California Institute of Technology, Mail Code , Pasadena, California, USA jledyard@caltech.edu 1

2 1 Introduction The combinatoric auction problem is to design an auction to sell M heterogeneous objects to N buyers in a way that is optimal and, because the seller does not know the valuations of the buyers, incentive compatible. By now much has been written about this problem, most of it focused on dominant strategy auctions. Vickrey [17], Clarke [5], and Groves [7] have been rediscovered, analyzed and characterized. But, there is a problem with VCG auctions. They are not revenue maximizing. It is true that, as long as voluntary participation constraints are ignored, the VCG auctions are efficient. However, efficiency does not imply maximum revenue. Characterizing the class of revenue maximizing, dominant strategy auctions remains an open question. Myerson [14] characterized the expected revenue maximizing auction for a single item. Although he took a Bayesian approach, the auction he found is a dominant strategy mechanism. In the symmetric case, the desired auction can be implemented by the seller posting a reservation price, buyers bidding their value, awarding the item to the highest bidder if their bid is higher than the reservation price, and charging the winning bidder the highest of the second highest bid or the reservation price. This is a VCG auction with a reservation price. In the symmetric case, the auction is efficient conditional on participation. Extending the Myerson analysis beyond one good has proven difficult. Ulku [16] has generalized Myerson s [14] to multiple goods when private information remains one-dimensional. Armstrong [1] has solved the problem for 2 goods and linear utility functions. Avery and Hendershott [2] and Manelli and Vincent [12] also carry out their analysis under the assumption of linear utility functions. But in many problems of interest there are synergies or complementarities across items. This is believed to be the case in the FCC bandwidth auctions and is certainly the case in most logistics acquisition auctions. The problem with complements has remained intractable. 1 I am able to make some limited progress by restricting attention to environments with single-minded buyers; that is, buyers who care only about one subset of the items. 2 I am able to characterize the expected revenue maxi- 1 In a recent paper, Malakhov and Vohra [11] have applied a shortest path approach to the problem under the assumption of that each agent may have only a finite number of types. They provide a solution for some special cases. 2 This is not so unrealistic. The initial FCC broadband auction was dominated by re- 2

3 mizing, dominant strategy auctions that satisfy voluntary participation. The optimal auctions are not VCG auctions; that is, they do not select expost efficient allocations. One implication is that direct auctions that choose allocations in the core are not optimal. Another is that if an indirect auction coupled with a particular behavior (such as straight-forward bidding) does produce allocations in the core, then either that auction does not maximize expected revenue or that behavior is not incentive compatible. I describe two particular implementations of the optimal auction. I show that the optimal auction can be implemented in a direct revelation form with the use of individualized bid preferences, which depend only on a bidder s own bid, and pivot prices. Further, if valuations are believed by the auctioneer to be uniformly distributed, then the optimal auction can be implemented using reserve prices. In the final section, an example is provided to illustrate that the increase in expected revenue can be significant. In some cases more than a 100% increase can be achieved. I also provide the results from some economic experiments in which we tested three familiar auction formats in environments identical to the example. This allows me to use the theoretical optimal auction as a benchmark in evaluations. 2 The Optimal Combinatoric Auction Problem 2.1 The Basic Framework A combinatoric auction is used to sell M objects to N buyers who have preferences for bundles of these discrete private goods. The buyers are denoted i = 1,..., N. The seller 3 is denoted i = 0. For each object, let x i m = 1 if object m is allocated to i and let x i m = 0 otherwise. Each buyer has a quasilinear utility function for objects and money U i (x i, a i, θ i ) = V i (x i, θ i ) a i gional firms. Each of them had holes iin their wireless networks and it was common knowledge what these were for each company. That is, everyone knew exactly what packages the big bidders were after. What was not known was how much they were willing to pay. 3 It is possible to accomodate multiple sellers in this model. [9].I do not do so in this paper. 3

4 where θ i Θ i is i s type and a i is i s payment. 4 We normalize i s utility, without loss of generality, by letting V i (0, θ i ) = 0 for all θ i. The seller s revenue is a 0 = N i=1 ai. The seller s holdings of unsold items is x 0. For this paper I will assume that the seller has no utility for the items. So the seller s utility function is U 0 (x 0, a 0, θ 0 ) = a 0 and Θ 0 = {0}. We will say that θ = (θ 0, θ 1,..., θ N ) is an environment and that Θ = Θ 0 Θ 1... Θ N is the class of environments. Definition 1 An allocation is (x, a) {0, 1} M(N+1) R N+1. An allocation is feasible iff x X = {x {0, 1} M(N+1) N i=0 xi m = 1 for all m} and a Y = {a R N+1 N i=0 ai = 0}. We will let Z = X Y. It is also important to understand the infomation structure. I make the minimalist assumption that no buyer knows anything about the other buyers. But I will assume that the seller knows something. In particular, I depart from most of the computer science literature, but use the approach of most economists, and assume that the seller is a risk-neutral, Bayesian who has beliefs about the types of bidders. In order to make the problem tractable I will assume that the seller believes the bidders are single-minded; that is, each bidder cares about only one package. Let x i be the package that bidder i likes. Then V i (x i, θ i ) = V i (x i, θ i ) if x i x i and V i (x i, θ i ) = 0 otherwise. Let T i = {t i t i = V i (x i, θ i ) for some θ i Θ i }. We can rewrite buyer i s utility as t i g i (x i ) where g i (x i ) = 1 if x i x i and g i (x i ) = 0 otherwise. The seller is assumed to be a Bayesian and to know x i for all i and to believe that the types t i are independently distributed with distribution functions F i (t i ) on T i. We will assume that T i = [t i, t i ] for convenience. 2.2 Optimal Auctions In this paper we are looking for the auction design that maximizes the seller s revenue, a 0. Without further constraints, this is not very interesting since one could make a 0 as large as one wanted by setting a i, for i 0, as large as desired. Instead, it is standard to impose a Voluntary Participation constraint. 5 Each buyer is assumed to have an outside option whose value, 4 We could also represent utility for i as 2 M numbers, one for each subset of the M items, representing willingness-to-pay and let t i R 2M. 5 Voluntary Participation is sometimes called Individual Rationality. 4

5 for this paper, I will normalize at 0. 6 Definition 2 An allocation satisfies the Voluntary Participation constraint if and only if t i g i (x i ) a i 0 for all i (1) An allocation (x, a) maximizes revenue (in t) subject to voluntary participation if and only if it solves the following problem: max x X N a i (2) i=1 subject to (1). If the auctioneer knew the utility functions of all the bidders, or equivalenty knew each t i, then she could solve this by first solving the efficiency problem max x X N t i g i (x i ). (3) i=1 and then charging each agent an amount a i = t i g(x i ). But the auctioneer does not know t i and so must ask for that information. This is usually done in the form of bids, either sealed bids or iteratively. Buyers will of course respond strategically, usually by understating their true willingness to pay. This requires the seller to anticipate the buyers strategic behavior when processing the bids. In this paper I use the standard, minimalist assumption that buyers will use a dominant strategy if that is available to them and focus my attention on auctions which offer buyers a dominant strategy. 7 Analytically, strategic behavior is integrated into the optimal auction problem by applying the Revelation Principle (see e.g., [14]) which states that any outcome we can attain with any auction process in which bidders use 6 In general the value of the outside option can depend on an agent s type. That is, it can be V oi (t i ). 7 This approach does ignore the possibility that there are auctions without dominant strategies that might yield better solutions to the optimal auction problem. It is, of course, well known that no such possibility exists for the efficiency problem. Whether such possibilities exist for the revenue maximization problem depends on what assumption one is willing to make about the strategic behavior of the buyers. 5

6 dominant equilibrium strategies can also be attained using a direct revelation mechanism that is dominant strategy Incentive Compatible. 8 Definition 3 A direct revelation mechanism asks each bidder for a report, or bid, b i t i, that identifies her type and which is submitted without seeing the other s submissions. Given these reports, (b 1,..., b N ), the auctioneer chooses an allocation according to the rules (x, a)(b) = η(b). A feasible, direct revelation mechanism for the environments T is a mapping η : T 1... T N Z. Definition 4 A direct revelation mechanism η for T is dominant strategy incentive compatible if and only if for all t T, all i = 1,..., N, and all b i T i, t i g i (η i x(t/b i )) η i a(t/b i ) t i g i (η i x(t)) η i a(t) (4) where (t/b i ) = (t 1,..., t i 1, b i, t i+1,..., t N ). That is, a direct revelation mechanism is dominant strategy incentive compatible if and only if it is a dominant strategy to honestly report one s type. The goal is to find an optimal, feasible, direct revelation mechanism that satisfies dominant strategy incentive compatibility and voluntary participation. Since the auctioneer is a risk-neutral Bayesian, this means maximizing expected revenue subject to incentive compatibility and voluntary participation. Letting η(t) = (x(t), a(t)), this can be written as: max η N i=1 subject to x(t) X for all t T, (4) and (1). a i (t)df (t) (5) Below, I provide a solution to this problem. I do so by first finding the expected revenue maximizing Bayesian incentive compatible auction, an 8 Of course using the revelation principle sidesteps all issues of computational complexity or communication constraints. That is, it is assumed that the mechanism and the participants can make all optimizing calculations and can communicate anything requested. Thus, one needs to keep in mind that any optimal auction found using the revelation principle is only an upper bound on those that can be practically implemented. Nevertheless, such upper bounds can provide valuable guidance in practical mechanism design. 6

7 analytically easy problem. I then modify that auction slightly so that it is a dominant strategy auction with the same expected revenue. Since dominant strategy mechanisms are also Bayesian incentive compatible this gives me the optimal dominant strategy auction. The rest of the paper proceeds as follows. First, I supply a brief background on VCG and Myerson auctions. Next, I derive the optimal Bayesian auction. Then, I derive and characterize the optimal dominant strategy mechanism. Finally, an example and some experimental results are given. 3 A Bit of Background 3.1 VCG Auctions In order to solve (5) directly, we need a tractable way to characterize the class of mechanisms, η, that satisfy (4) and (1). Characterizations have been given, see for example Roberts [15] and Bikhchandani et. al. [3], but tractability remains an open problem. One well-known class of mechanisms that are dominant strategy incentive compatible is the class of Vickrey-Clarke-Groves mechanisms. For the combinatoric auction problem, these mechanisms are given by functions x V CG (t), a V CG (t) where x V CG (t) arg max x X a V GCi (t) = j i N t i g i (x i ) (6) i=1 t j x j (t) + max x F t j (x j ) + h j (t j ). The functions h j ( ) are arbitary but cannot depend on t j. Notice that if h i (t j ) 0 then i ai (t) 0 for all t T, but we are not guaranteed that voluntary participation will be satisfied or that revenue will be maximized. In fact it is easy to provide examples that show that revenue can be as low as 0. It is known that VCG mechanisms are the only mechanisms that produce an efficient allocation, i.e. ones that maximize i V i (x i, θ i ). But they are not the only dominant strategy, direct revelation mechanisms. Perhaps we can do better. j i 7

8 3.2 Myerson Auctions In Myerson [14] the optimal auction is characterized for the case of one commodity, M = 1, and where V i (1, t i ) = t i and V i (0, t i ) = 0. Let w i (t i ) = 1 F (ti ). (7) f(t i ) w i ( ) is called i s virtual valuation by Myerson. Let α i (t i ) = max{0, max j i {w j (t j )}}. Assume a regularity condition that w i (t i ) is non-decreasing in xt i. The Myerson optimal auction allocation is then given by the function x M (t) where And, letting x Mi (t) = 1 if w i (t i ) α i (t i ) x Mi (t) = 0 otherwise. t Mi (t i ) = inf{t i x Mi (t) = 1}, the payments are given by the function a M (t) where a Mi (t) = x Mi (t)t Mi (t i ) Remark 1 If F i (t i ) = F j (t j ) for all i, j, let r solve w i (r ) = 0. Let β i (t i ) = max{r, max j i {t j }}. Then x Mi (t) = 1 if t i β i (t i ) x Mi (t) = 0 otherwise and a Mi (t) = x i (t)β i (t i ). That is, if i has the largest value and that value is at least as large as the reserve price then i wins and pays the largest of the second highest value or the reserve price. All those who don t win don t pay anything. If we think of the seller as a bidder with true value equal to r, then this is just the VCG mechanism from (6). Remark 2 In the Myerson Optimal Auction, although his analysis imposes only a Bayesian Incentive Compatibility (BIC) constraint and not a Dominant Stratetegy Incentive Compatibility (DIC) constraint, it is a dominant strategy for bidders to bid their true type t i. Since the set of mechanisms satisfying DIC and Voluntary Participation (VP) is contained in the set of mechanisms satisfying BIC and VP, the Myerson Optimal Auction solves problem (5) when there is only one unit to sell. 8

9 4 Bayesian Analysis of the Combinatorial Auction In this section, we go over the basics of Bayesian auction design, because it is a theoretical short-cut to finding the expected revenue maximizing, dominant strategy, combinatorial auction. The Bayesian auction design problem replaces the requirement of dominant strategy incentive compatibility (4) with the weaker concept of Bayesian incentive compatibility. 9 Given a mechanism η, let R i (b i, t i ) = t i g i (ηx(t/b i i )df (t t i ) ηa(t/b i i )df (t t i ) (8) be the expected payoff to i if they have type t i and bid b i. Definition 5 A direct revelation mechanism η is Bayesian Incentive Compatible if and only if for all t T,all i = 1,..., N, and all b i T i, R i (t i, t i ) R i (b i, t i ) (9) That is, a direct revelation mechanism is Bayesian incentive compatible if and only if it is a Bayes equilibrium for all agents to honestly report their type. Definition 6 A direct revelation mechanism η satisfies Bayesian Voluntary Participation if and only if for all t T,all i = 1,..., N, and all b i T i, R i (t i, t i ) 0 (10) Remark 3 Note that if a mechanism is dominant strategy incentive compatible then it is Bayesian incentive compatible. And, if a mechanism satisfies voluntary participation then it satisfies Bayesian voluntary participation. The converses need not be true. 9 This implicitly assumes that bidders have beliefs about other bidders types and about other bidders beliefs and so on. It also implicitly assumes that these beliefs are common knowledge. Since we are merely using these results as a theoretical bridge to dominant strategy incentive compatibility, we do not need to worry about the practical implications of these knowledge assumptions here. 9

10 The optimal Bayes auction problem is to find a mechansim η(t) = (x(t), a(t)) that solves max η N i=1 subject to x(t) X for all t T, (9) and (10). a i (t)df (t) (11) There is a standard approach to solving the optimal Bayes auction problem which we sketch here for those unfamiliar with this literature. 10 Rewrite R i (b i, t i ) as t i Q i (b i ) A i (b i ). Incentive compatibility requires that b i be chosen to maximize R i. The first and second order conditions for this are, when evaluated at b i = t i, t i dq i (t i )/dt i da i (t i )/dt i = 0 (12) dq i (t i )/dt i 0 (13) The first order condition (12) implies that A i (t i ) = A i (t i )+ t i sdq i (s). To get t i the second order condition (13) first notice that the second order condition for maximization is t i d 2 Q i (t i )/dt i2 d 2 A i (t i )/dt i2 0. Then differentiate (12) with respect to t i to get t i d 2 Q i (t i )/dt i2 d 2 A i (t i )/dt i2 + dq i (t i )/dt i = 0. (13) then follows. The second order condition implies that dr i (t i, t i )/dt i 0 and so voluntary participation requires that t i Q i (t i ) A i (t i ) 0. Since the auctioneer wants to maximize revenue, A i (t i ) will equal zero. Integrating the second term of A i (t i ) by parts and using all of the facts, we see that A i (t) = t i Q i (t i ) t i Q i (s)ds. So the optimal Bayesian auction t i problem reduces to choosing x(t) to solve max N a i (t)df (t) = max N i=1 i=1 A i (t)df i (t) 10 See Ledyard and Palfrey [9] for a detailed application of this approach to a more general class of problems. 10

11 [ ] t i = max t i Q i (t i ) Q i (s)ds df i (t i ). x X t i Integrating the second term by parts and expanding Q i allows us to rewrite this as [ (t i 1 F ) ] i x i df. max x X To solve this we need only solve, for each t T, max x X f i ( t i 1 F i (t i ) f i (t i ) ) x i. (14) The maximization problem (14) determines the allocation rule. The payment rule is then derived from this and the First Order Condition for Incentive Compatibility (12). Letting x (t) be the solution to (14), and Q i (t i ) = x i (t)df (t t i ), the (expected) payment rule is: A i (t i ) = t i t i sdq i (s) (15) To ensure that (14) and (15) are indeed the optimal auction, we must assume a regularity condition 11 on F i that t i 1 F i (t i ) is non-decreasing in t i for f i (t i ) all i and t i. This guarantees that the x (t) which solves (14) has the property that dq i /dt i 0, the second order condition for incentive compatibility. 5 The optimal dominant strategy auction 5.1 The optimal auction In a very nice paper on the relationship between Bayesian mechanism design and dominant strategy mechanism design, Mookherjee and Reichelstein [13] 11 This condition is standard in the literature on auction design and requires that the virtual valuation w i, as defined in (7), is non-decreasing in t i. This in turn insures that the second order conditions for incentive compatibility are not binding at the solution to (14). 11

12 provide a theorem which shows that any Bayes incentive compatible mechanism with the appropriate monotonicity can be converted to a dominant strategy mechanism with the same expected utility for all i and all t i. Under the regularity condition, the mechanism in the previous section has the required monotonicity. It is really easy to see how this works in our case. First, we rewrite the allocation rule for the optimal Bayesian auction (14) in a more intuitive way. Let b i (t i ) = inf{t i g i (x (t)) = 1 in (14)}. (16) We refer to b i (t i ) as the Pivot Price since it is the minimum amount that i must bid to receive her package given the others types, t i. Because of the assumption that t i 1 F i is non-decreasing we can write the allocation rule f i as g i (x (t)) = 1 iff t i b i (t i ) (17) Next, integrate (15) by parts and reorganize to get the expected payment function, ] A i (t i ) = [ t i g i (x (t)) t i t i g i (x (t))ds df (t t i ) (18) It is straight-forward to compute that the expression inside the large brackets is simply b i (t i )g i (x (t)) To get the optimal dominant strategy auction, we use the allocation rule (17). For the payment rule, we simply remove the expectations operation from the Bayesian rule (18). The result is g i (x (t)) = 1 iff t i b i (t i ) (19) a i (t) = b i (t i )g i (x (t)) (20) It is easy to see that (19) and (20) satisfy the constraint that truth is a dominant strategy and yield the same expected revenue as the optimal Bayesian mechanism. They also satisfy voluntary participation since i pays nothing unless t i b i (t i ) and then pays b i (t i ) which is less than or equal to t i. Because dominant strategy incentive compatible mechanisms are also Bayesian incentive compatible mechanisms, (19) and (20 ) are a solution to our optimal auction problem (5). 12

13 5.2 Bid Preferences One way to implement the optimal auction is to use individualized bid preferences. Each i submits a sealed bid, b i. After each i submits their bid, the auctioneer adjusts each bid by subtracting the amount p i (b i ) = (1 F i (b i ))/f i (b i ). The amount p i (b i ) can be thought of as a bidder specific preference - those with lower p i have a higher preference. Let c i = b i p(b i ) be this adjusted bid. The auctioneer solves the winner determination problem by choosing x (b) to solve max c i g i (x) x X If g i (x (c)) = 1 then i is a winner, gets his preferred bundle, and pays b i (c i ). Since b i = t i is a dominant strategy, this is the optimal auction. One observation worth making 12 is that bidder i s preference, p i (b i ), depends only on i s own bid and not on the bids of others. Even though combinatoric auctions involve complex tradeoffs across multiple commodities and individuals, the optimal auction, with single-minded bidders, is implemented without requiring complicated preferences. The preference compensates for the incentive costs of the private information known by the buyer but not the seller. 5.3 Reserve Prices For the special case when F i (t i ) is the uniform distribution over [l i, L i ], the optimal auction can be implemented by using individualized reservation prices, instead of preferences. The auctioneer sets a reserve price for each i equal to r i = L i /2 and, upon receiving bids b i, solves max x X (b i r i )g i (x) Since (1 F i (t i ))/f i (t i ) = (L i t i ), this is equivalent to using a preference of p i = L i b i. As usual, i pays the pivot price b i (b i ). Several obvious observations can be made here. First, each i must bid higher than r i if they are to participate. Second, the reservation prices do not depend on the lower bound of the distribution. Third, if the prior of the seller is that the upper bounds of the distributions depend only on the number of items in a bidder s preferred package, then the seller can use reserve prices on items, instead of individualized reserve prices. 12 I thank David Parkes for pointing this out to me. 13

14 5.4 Comparison to VCG It is interesting to compare the optimal mechanism to the VCG mechanism. The primary difference is in the allocation rules. For the VCG mechanism one chooses x V CG (t) to solve max x X t i g i (x(t).) For the optimal auction one chooses x OA (t) to solve max x X ( t i 1 F i f i ) g i (x(t).) The payment schemes for each auction, given the allocation rules, are the same. For k = V CG, OA, let b k i (t i ) = min{t i g i (x k (t)) = 1} be the minimum value of t i such that i wins their pachage in auction k. Then for both k = V CG and k = OA, i pays a ki (t) = b k i (t i )g i (x i (t)) The VCG auction maximizes the sum of bids (or reported values) while the optimal auction maximizes the sum of virtual bids (or virtual reported values). It is rare that these different objective functions yield the same allocation. Thus, in most environments, the optimal auction is not ex-post efficient. There are other allocations that would make the seller and buyers all better off after observing the valuations. One implication of this is that auctions that choose allocations in the core are not optimal. 6 An Example and Experimental Results Although the designs of the optimal auction and the VCG auction look very similar in spirit and form, they can yield remarkably different levels of expected revenue. In this section we provide an example which illustrates the extent of the difference. This example also serves as basis for economic experiments that compare the performance of three familiar auctions. 6.1 An Example There are two items to allocate, K = 2, and three bidders, N = 3. Let the preferred packages of each bidder be x 1 = (1, 0), x 2 = (0, 1), x 3 = (1, 1) The 14

15 seller believes that t 1, t 2 are uniformly distributed on [0, 1] and t 3 is uniformly distributed on [0, a]. So w i (t i ) = 2t i 1 for i = 1, 2 and w 3 (t 3 ) = 2t 3 a. The optimal auction allocation γ (t) = g(x (t)) solves subject to max [ (2t 1 1)γ 1 + (2t 2 1)γ 2 + (2t 3 a)γ 3] γ i {0, 1}, γ 1 + γ 3 1, γ 2 + γ 3 1 For this example, if a < 2 then the optimal auction is not efficient when t 1 > 1/2, t 2 > 1/2 and t 3 > a/2 because 3 wins when t 1 + t 2 > t 3 > t 1 + t 2 + a 2 2. If a > 1, then the optimal auction is not efficient when t 3 > a/2 and either {t 1 > 1/2, t 2 < 1/2} or {t 1 < 1/2, t 2 > 1/2} because, for example in the first case, 3 loses when t 1 + a 1 2 > t 3. Continuing the example, we can compute expected revenue as a % of the maximum extractable. In doing so, we can see what percentage of the possible surplus a seller is able to extract using either the VCG mechanisms and the optimal auction. As a benchmark I indicate the expected revenue from the mechanism in which an allocation is randomly determined (there are 5 possible allocations) and individuals pay exactly their value for what they get. This is, of course, not incentive compatible. Table 1: Auction Revenue as a % of the Maximum Extractable if a=1 if a=2 if a=3 OA VCG Random Table 1 contains these percentages as a function of the upper bound, a, of the bidder who wants both items. Clearly the optimal auction is doing significantly better at generating expected revenue for all values of a. The optimal auction achieves between 40% and 140% improvement over the VCG auction. It is interesting that the most symmetric case, a = 2, which is also the most competitive situation, yields the highest rate of extraction while the least competitive case, a = 1, yields the highest percentage increase, 144%, of the optimal over the VCG. 15

16 6.2 Experimental Results Using exactly the same environment as in the previous example, I ran a series of economics experiments testing the performance of three familiar auctions: SMR, RAD, and SB. 13 SMR is the simultaneous, multi-round format devised for the FCC in which the items are auctioned off simultaneously in a series of rounds. SMR does not allow package bidding. RAD is a design proposed in [8] which adds package bidding and an individual item price computation to SMR. SB is the sealed bid auction with package bidding in which bidders pay what they bid. 14 We ran 3 sessions of experiments, one for each auction. In each session we had 9 subjects randomly assigned to 3 groups. Each group stayed together during the session but did not know the identity of the members. Each group ran through 10 auctions for 2 items with the same auction design. The first 2 were practice auctions and did not pay anything. The last 8 were real and the winners were paid. So there were a total of 24 auctions run for each auction design. Before each of the ten auctions, the three subjects in a group were randomly assigned to be one of Bidder 1, 2, or 3 and then given a randomly drawn value. Bidder 1 desired item A and had a value drawn from the uniform distribution on [0, 100]. Bidder 2 desired item B and had a value drawn from the uniform distribution on [0, 100]. Bidder 3 desired the package A,B and had a value drawn from the uniform distribution on [0, 200]. These parameters imply an expected revenue of 80.7 for the optimal auction. The expected value of the maximum extractable revenue is yielding a value of for the ratio of the expected optimal auction revenue to the expected maximum extractable. In the versions of SMR and RAD that I used, there were no activity or eligibility rules. Rounds were a maximum of 30 seconds long. The minimum bid increment, the amount per item that a bid needed to exceed the previous 13 A brief description of each auction is provided in the Appendix. 14 The experiments were run using jauctions, which has been developed at Caltech by Jacob Goeree. The jauctions software consists of a flexible suite of Java-based auction programs designed to handle a wide range of auction formats and bidding environments, including combinatorial auctions with bid-driven or clock-driven prices, private and common valuations, etc. Instructions, which are available on request, were structured around relevant screen shots of the jauctions program. I am grateful to Walter Yuan for explaining the intricacies of the softwar and to Dustin Beckett for assistance in managing the experiments. 16

17 rounds prices in order to be submitted, was 5 per item. Bidding rounds continued until no new bids were received. In SB there is only a single round of 90 seconds in length. In all formats, bids could be submitted only once per round and no withdrawals were allowed. The results of the auction experiments are summarized in Table 2. I also include numbers which indicate what the optimal auction (OA) would have achieved had subjects followed the dominant strategy of bidding their true value for the draws that did occur in the experiments. 15 Table 2: Experiment Results Mean (Std. Dev.) Revenue Efficiency Rev/Max Possible OA (38.52) 0.86 (.29) 0.59 (.23) SMR (43.16) 0.90 (0.20) 0.46 ((0.33) RAD (46.99) 0.97 ((0.09) 0.53 (0.30) SB (36.99) 0.96 (0.19) 0.74 (0.19), These findings confirm previous experimental results achieved in more complex environments in [10] and [4]. The RAD auction generates higher efficiencies and revenue than SMR does. 16 In fact in 2 of the 3 groups in the RAD auctions efficiencies were 100%. The other group attained 100% efficiency in 6 of their 8 auctions. In the SMR auctions, no group had 100% efficiency in all auctions. In one group there were 5 at 100%, in another there were 6 and in the other there were 7. In the SMR auctions, one can see the exposure problem faced by the large bidder at work. The average number of rounds for the SMR auctions was 7.46 compared with 5.65 for RAD. This is consistent with the findings in [10] that the package bidding in RAD shortens the length of the auction over SMR. The opposite finding in [4] that SMR was quicker than RAD came when RAD was run using the eligibility and activity rules of the FCC. Without those rules, RAD moves much faster. The surprising result to me from the economic experiments is that the Sealed Bid auction (with package bidding) produces significantly more revenue that either SMR or RAD and does so with efficiencies as high as RAD I did not run an economic experiment with the direct optimal auction design. 16 One of the reasons that the RAD efficiencies are as high as 97% is that for the simple environment we are using there is always a competitive equilibrium. 17 As in RAD, two of the SB auction groups produced 100% efficiency in all of their 8 auctions. One group missed 100% twice. 17

18 Indeed, the average revenue in the SB auctions was even higher than what would have been obtained by the optimal auction with SB extracting 74% of the maximum possible while the optimal auction promised only 59%. This is similar to findings in one-item auctions in which sealed bid auctions yielded higher revenues than ascending bid auctions. 18 Many attributed that to riskaverse bidders which could also be a contributing factor here. 7 Conclusions A characterization of expected revenue maximizing, dominant strategy, combinatoric auctions has been obtained by imposing the assumptions that buyers are single-minded and that the seller believes the types to be independently distributed. In one implementation, the optimal auction uses personalized bid preferences. These auctions are not VCG. and they rarely allocate resources efficiently, even if one conditions on participation. This means, in particular, that auctions which choose core allocations are not optimal. The experimental results confirm previous findings that RAD is more efficient, generates more revenue, and is faster than SMR. But two new findings emerge. First, both RAD and SMR are more efficient and generate less revenue than is optimal. Second, and perhaps most surprising, SB generates more revenue than even the optimal mechanism while achieving efficiencies as high as RAD. On average SB extracts 50% more of the maximum possible than the optimal auction. There is some reason to believe that this may because bidders are risk-averse, contrary to the assumptions underlying the optimal auction model. There are many open problems left. The obvious one is the characterization of expected revenue maximizing auctions when the bidders are not single-minded. Another is an examination of how well one could do with carefully selected reservation prices coupled with a VCG mechanism or some other sub-optimal auction design. Another involves solving for the optimal auction design in the presence of risk-aversion. 18 See, e.h. [6]. 18

19 References [1] M. Armstrong. Optimal multi-object auctions. Review of Economic Studies, 67(3): , July [2] C. Avery and T. Hendershott. Bundling and optimal auctions of multiple products. Review of Economic Studies, 67(3): , July [3] S. Bikhchandani, S. Chatterji, R. Lavi, A. Mu alem, N. Nisan, and A. Sen. Weak monotonicity characterizes deterministic dominant strategy implementation. Econometrica, 74(4): , [4] C. Brunner, J. Goeree, C. Holt, and J. Ledyard. Combinatorial engineering. unpublished manuscript, Caltech, [5] E. Clarke. Multi-part pricing of public goods. Public Choice, 11:17 33, [6] J. Cox, B. Roberson, and V. Smith. Theory and behavior of single object auctions. In V. L. Smith, editor, Research in Experimental Economics. JAI Press, [7] T. Groves. Incentives in teams. Econometrica, 41: , [8] A. Kwasnica, J. Ledyard, D. Porter, and C. DeMartini. A new and improved design for multi-object iterative auctions. Management Science, 51(3): , March [9] J. Ledyard and T. Palfrey. A general characterization of interim efficient mechanisms for independent linear environments. Journal of Economic Theory, in press, [10] J. Ledyard, C. Porter, and A. Rangel. experiments testing multiobject allocation mechanisms. Journal of Economics and Management Strategy, 6(3): , [11] A. Malakhov and R. V. Vohra. Single and multi-dimensional optimal auctions - a network approach. KSM, Northwestern University, [12] A. Manelli and D. Vincent. Multidimensional mechanism design: Revenue maximization and the multiple-good monopoly. unpublished manuscript, Arizona State, U of Maryland,

20 [13] D. Mookherjee and S. Reichelstein. Dominant strategy implementation of bayesian incentive compatible allocation rules. Journal of Economic Theory, 56(2): , [14] R. Myerson. Optimal auction design. Mathematics of Operations Research, 6:58 73, [15] K. Roberts. The characterization of implementable choice rules. In J. J. Laffont, editor, Aggregation and Revelation of Preferences, pages North-Holland, [16] L. Ulku. Optimal combinatorial mechanism design. unpublished manuscript, Rutgers University, [17] W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16:8 37,

21 Appendix A: Rules for the SMR Laboratory Auctions Rounds and Bid Structure: All licenses are put up for bid simultaneously, and participants may only submit bids on individual licenses. The auction consists of successive rounds in which participants may place bids. Following each round, the high bid for each license is posted. These high bids then become the standing bids for the subsequent round. Acceptable Bids: In the first round, an acceptable bid must be equal to or exceed the initial price of 0 by 5 points (each point equaled 40 cents in the experiment). Subsequently, in order to be acceptable, a bid must exceed the provisionally winning bid for the license by at least 5 points. Bid Withdrawal: No withdrawal is allowed. Bidding Eligibility and Activity: used. No eligibility or activity rules were End of Round Feedback: At the end of each round, bidders receive information on all bids including provisionally winning bids. They do not see bidder ID numbers. Bidders also see the sum of their own values for the licenses that they are provisionally winning and prices that would be paid for the licenses if the auction had ended. Closing Rule: The auction closes after any round in which no new bids were placed. In this case provisionally winning bids become winning bids that are used to calculate auction earnings. Winning bidders pay what they bid. The experiment did not allow for defaults on payments, so gains were added to cumulative earnings and losses were subtracted. Appendix B: Rules for RAD Rounds and Bid Structure: This is a simultaneous, multi-round auction in which participants may submit bids on individual licenses or on combi- 21

22 nations of licenses (packages). Provisionally winning bids are calculated by maximizing seller revenue for the round. Acceptable Bids: In the first round, an acceptable bid must be equal to or exceed the minimum opening bid of 0 by 5 points for each license, or by 5 points times the number of licenses in a package. After each subsequent round, prices are calculated for each license on the basis of bids received in the previous round. The pricing rule calculates prices that reflect (as closely as possible) the marginal sales revenue of each license based on bids received. Prices for packages are the given by the sum of the prices for each license in the package. In order to be acceptable, a bid must exceed the price of a license or package at least 5 points times the number of licenses covered by the bid. Bid Withdrawal: No withdrawal is allowed. Bidding Eligibility and Activity: used. No eligibility or activity rules were End of Round Feedback: At the end of each round, bidders receive information on all bids including provisionally winning bids. They do not see bidder ID numbers. Bidders also see the prices for all licenses, the sum of their own values for the licenses and packages that they are provisionally winning, and the sum of prices that would be paid for those licenses and packages if the auction had ended. Closing Rule: The auction closes after any round in which no new bids were placed. In this case provisionally winning bids become winning bids that are used to calculate auction earnings. Winning bidders pay what they bid. The experiment did not allow for defaults on payments, so gains were added to cumulative earnings and losses were subtracted. Appendix C: Rules for the Sealed Bid Auction Rounds and Bid Structure: This is a one round auction in which participants may submit bids on individual licenses or on combinations of licenses (packages). 22

23 Acceptable Bids: In the first and only round, an acceptable bid must be equal to or exceed the minimum opening bid of 0 by 5 points for each license, or by 5 points times the number of licenses in a package. Bid Withdrawal: No withdrawal is allowed. Bidding Eligibility and Activity: used. No eligibility or activity rules were Closing Rule: The auction closes after the first round. Winning bids are computed and winning bidders pay what they bid. The experiment did not allow for defaults on payments, so gains were added to cumulative earnings and losses were subtracted. 23