Combinatorial Auctions CSCI 1951K/2951Z 2020-04-01

Combinatorial Auctions CSCI 1951k/2951z 2020-04-01

We introduce several alternative combinatorial auctions, other than the classic VCG mechanism. We argue that sequential and simultaneous Vickrey auctions suffer from immediate deﬁciencies, citing unsuccess- ful examples of real-world wireless spectrum auctions. Simultaneous ascending auctions, in contrast, offer more promise.

1 Multi-Parameter Ascending Auctions

Recall the VCG mechanism, which generalizes the second-price (Vickrey) auction to the multi-parameter setting. This mechanism suffers from several anomalies, which you explored in Homework 3. In addition, when valuations have exponential complexity, the communication complexity of VCG is likewise exponential, and the computation of a VCG outcome is potentially exponential as well, since solving for a welfare-maximizing allocation is NP-hard. A combinatorial auction is an auction for multiple goods, where bidders value goods in combination: i.e., bundles, or subsets, of goods. Moreover, these values do not generally decompose “nicely.” Let vi : A → R denote bidder i’s valuation function, deﬁned on the G set of possible allocations A = 2 . Thus, vi(X) denote i’s value for bundle X ⊆ G. Note that in the special case of a singleton bundle {j}, we simply write vi(j) to denote i’s value for good j. Two special (and relatively nice) cases of combinatorial valuations are additive and unit demand valuations. Given a bundle of goods X ⊆ G, bidder i’s valuations are additive iff

vi(X) = ∑ vi(j), j∈X and unit demand iff

vi(X) = max vi(j). j∈X

Other (less nice, but prevalent) combinatorial valuations of interest include complementary and substitutable valuations. Given two bundles of goods X, Y ⊆ G, bidder i’s valuations satisfy the former iff

vi(X ∪ Y) ≥ vi(X) + vi(Y),

and the latter

vi(X ∪ Y) ≤ vi(X) + vi(Y). combinatorial auctions 2

Examples of complements include left and right shoes. The value of the pair far exceeds the sum of the individual values. Examples of substitutes include an iPhone and and Android. Presumably you don’t have a real need for one, if you already have the other. Having noted the shortcomings of VCG, and the prevalence of combinatorial valuations, we would be remiss if we did not explore alternative combinatorial auction designs. Two obvious alternatives include sequential and simultaneous sealed-bid auctions. That is, it might make sense to auction off goods separately, either one after another or all at the same time, using, say, standard Vickrey auctions. These two n-dimensional designs eliminate the communication complexity of VCG, although it may be NP-hard for bidders to solve the requisite value or demand queries. Imagine, for example, a set of goods of size m, and a set of bidders with unit-demand valuations. Let’s consider running m sequential Vickey auctions in such an environment. What can possibly go wrong? (Think for a moment before reading on.) The problem with this arrangement is, as the auctions proceed, more and more bidders win goods, so competition in later auctions decreases, causing prices to likewise decrease. Consequently, the high-value bidders face a strategic decision—how to bid so that they win a good eventually, but not when prices are still high. Another possible auction design is simultaneous Vickrey auctions. Here, unit-demand bidders—and bidders with any non-additive combinatorial valuations, really—face the exposure problem. They are forced to bid in all auctions, as they have only one shot at the goods, but unit-demand bidders—who demand exactly one good— would not want to bid too high on too many goods, and risk winning more than one, or too low on too few, and risk winning none at all. In 1959, the (eventual) Nobel laureate Ronald Coase lamented the power of the FCC to control the operators of radio and television spectrum. He argued that welfare would be better served if the spectrum were instead allocated to those who valued it most.1 In 1994, a 1 variant of Coase’s proposal was ﬁnally adopted; in that year, the FCC sold spectrum licenses via auction, and raised over $600 million for the U.S. Treasury Department. However, the early spectrum auction designs were far from perfect. For example, in one of the earliest auctions in New Zealand in 1990, revenue was projected to be $250 million, but the auction real- ized only $36 million. The goods on offer were essentially identical spectrum licenses, and the mechanism was simultaneous Vickrey auctions. For one license, the highest bid was $100,000, while the second highest bid was $6; for another license, the highest bid was $7 million, while the second highest bid was $5,000. These bids were all combinatorial auctions 3

public information, revealing how much money was left on the table. In a later rendition, New Zealand instead ran simultaneous first-price auctions. The auction still did not meet its revenue projections, but the outcome was less embarrassing. In March 2000, Switzerland auctioned off licenses for three blocks of spectrum sequentially, again using Vickrey auctions. The first two licenses, which were identical (28 MHz blocks), sold for rather dif- ferent prices: 121 million Swiss francs and 134 million Swiss francs, respectively. But third auction, for a larger block of spectrum (58 MHz), was the most problematic—it sold for only 55 milion! Having observed the failure of sequential and simultaneous sealed-bid auctions in practice, economists turned to simultaneous ascending auctions as a possible spectrum auction design. (Sequen- tial ascending auctions would suffer from the same shortcomings as sequential sealed-bid auctions.) Indeed, ascending auctions afford the opportunity for price discovery, which, in the simultaneous context, can help bidders hone in on their preferred bundles. They also afford the opportunity for value discovery. Bidders need not evaluate all bundles, only those which can be procured at reasonable prices. Eventually, the FCC (and many other nations, most notably the UK) adopted the simultaneous multi-round auction (SMRA)— simultaneous ascending auctions—as the standard format in which to sell spectrum licenses. In this design, participants bid on all goods simultaneously, essentially responding to value queries (though jump bidding as in eBay may be limited), with all auctions likewise closing simultaneously. The exposure problem remains an issue in SMRA, however. To mitigate this problem, another auction design was proposed, namely the combinatorial clock auction (CCA).2 This auction also comprises 2 simultaneous ascending auctions, but with demand queries, so that bidders report a set of goods—specifically, one of their preferred bundles—rather than a vector of bids. Prices then increase on all goods that are overdemanded. Both of these auction designs are enhanced with activity rules to encourage all bidders to participate in the price (and value) discovery process. Among other things, these rules are designed to preclude exiting and then re-entering the auction. While the CCA has been in use by the FCC since 2008, laboratory experiments have established that SMRA, which is much simpler than CCA, yields both greater efficiency and greater revenue than CCA. Consequently, we adopt a variant of SMRA (essentially, simultaneous eBay auctions) for our course final project. In the next section, we show that simultaneous ascending auctions (SAAs) yield strong incentive guarantees, assuming additive valua- combinatorial auctions 4

tions. They also yield strong performance guarantees, as each individual auction is welfare maximizing up to e.3 In later lectures, we 3 Problem 3, Part 2 will attempt to generalize these results beyond additive valuations, with some, albeit limited, success.

2 Simultaneous Ascending Auctions: Incentive Guarantees

Our goal in this section is to establish that SAAs yields strong incentive guarantees. To get started, we must articulate these incentive goals, in the context of indirect mechansisms. We will thus begin by deﬁning what it means for an indirect mechanism to be DSIC. But before we can do that, we put forth an analog of truthful bidding in indirect mechanisms.

Deﬁnition 2.1. A bidding strategy in an indirect auction is called sincere if the bidder responds to all queries, and does so truthfully, meaning in a way that honestly reﬂects their values.

For example, a sincere bidder in an ascending auction for a single good would answer “yes” to the demand query “Would you like the good for $10?” iff their value for the good were at least $10. N.B. In what follows, when we refer to indirect mechanisms, we have in mind ascending auctions with demand queries. The results also apply to ascending auctions with value queries (e.g., eBay auctions), but are far less interesting in this case.

Deﬁnition 2.2. An indirect mechanism is DSIC if sincere bidding is a dominant strategy. Likewise, an indirect mechanism is EPIC or BIC, if sincere bidding is an EPNE or BNE, respectively.

Recall that DSIC is a particularly strong property of an auction. Indeed, the (vanilla) English auction is not DSIC. Here is a counterexample that establishes this claim.

Example 2.3. We show by counterexample that sincere bidding is not a dominant-strategy equilibrium in the English auction. Indeed sincere bidding is not even an EPNE or BNE. In this example, a bidder can obtain slightly more (less than e) expected utility by deviating from sincere bidding, even when the others bid sincerely. Suppose there are two bidders i and j, with (deterministic) values vi = 3.5 and vj = 2.5 for the good. Suppose further, e = 1. If bidder i bids sincerely, they drop out at round 3, winning the good at price 3, and earning utility 0.5. If, on the other hand, bidder i drops out at round 2 (i.e., at the same time as bidder j), they earn the good with probability 1/2, in which case their utility is 3.5 − 2, and their expected utility is 0.75. Hence, dropping out early yields greater combinatorial auctions 5

expected utility for bidder i than bidding sincerely in this example, even when bidder j bids sincerely.

Interestingly, the English auction is also not DSIC up to e,4 where 4 Problem 1, Part 1 e is the price increment, fixed for the duration of the auction. The modified English auction, however, modified with an activity rule that precludes bidders from exiting the auction and re-entering at some later round, is DSIC up to e5 (though it is also not DSIC6). 5 Problem 2, Part 2 Given that the modified English auction is DSIC up to e, we might 6 Problem 2, Part 1 hope that k parallel modified English auctions (assuming, in general, heterogeneous goods) might likewise be DSIC up to O(e) (e.g., ke), assuming the bidders valuations are additive, so that their interests across auctions are decoupled. But as it happens, k parallel modified English auctions are not DSIC up to O(e), as the following counterexample shows. (This counterexample actually shows that k parallel vanilla English auctions are not DSIC up to O(e), but imposing the activity rule of the modified English auction would not change the outcome, as it is not violated.)

Example 2.4. Suppose goods a and b are being auctioned off in two parallel English auctions, and assume two bidders, i and j. Let vi(a) = vi(b) = 3, and vj(a) = vj(b) = 1. Suppose bidder j declares that unless bidder i forfeits a on the very ﬁrst round, they will bid on both goods forever, but if i does forfeit a, then j will forfeit b (which wasn’t rightly theirs to win, anyway). Given this opposing strategy, it is in bidder i’s interest to (insincerely) make no attempt to procure a so that they will at least win b, obtaining a utility of 2, rather than 0.

At this point, we might decide that it is time to throw in the towel. But wait; there are other fish in the sea! Yes, there are other equilibrium concepts besides DSIC that are worthy of our attention. Specif- ically, we now set our sights on designing EPIC auctions, another worthy goal, which it turns out we can (at least sometimes) achieve. Remark 2.5. Though weaker than DSIC, EPIC is still a worthwhile goal. Recall, for example, that we have derived only a BNE for first- price auctions, not an EPNE. In particular, while we know that the equilibrium of first-price auctions assuming uniformly distributed valuations involves shading bids by a known constant (n−1/n), this equilibrium is not an EPNE. In expectation over all other-bidder values, this strategy is optimal (i.e., symmetrically, a best response). But it is clearly not optimal for an agent to shade in this way ex-post; given a particular choice of other-bidder values, the optimal bid is the second-highest value plus a small increment. Another problem in the homework exercises is to prove that vanilla English auctions are EPIC up to e.7. In other words, assuming 7 Problem 1, Part 2 combinatorial auctions 6

others are bidding sincerely, sincere bidding is an e-best response in vanilla English auctions. Once we assume that others are bidding sincerely, we eliminate the possibility of wacky (and irrational) bidding strategies of the form used in Example 2.4. This strategy created de- pendencies across the parallel auctions, which can be eliminated by assuming sincere bidding on the part of all other bidders. Indeed, as a single (vanilla) English auctions is EPIC up to e, k parallel (vanilla) English auctions are EPIC up to ke, assuming additive valuations. We provide a more rigorous proof of this claim in the next lecture, after presenting a recipe for the design of EPIC ascending auctions. In summary:

• The English auction is not DSIC.8 The English auction is not even 8 Example 2.3 DSIC up to e.9 9 Problem 1, Part 1

• The modiﬁed English auction, so that re-entry is forbidden (once a bidder stops bidding, they can never bid again), is also not DSIC.10 10 Problem 2, Part 1 It is however, DSIC up to e.11 11 Problem 2, Part 2

• k parallel English auctions (modiﬁed or not), assuming additive valuations and, in general, heterogeneous goods, are not DSIC, even up to O(e).12 12 Example 2.4

• The (vanilla; unmodiﬁed) English auction is not EPIC (or even BIC).13 It is, however, EPIC up to e.14 13 Example 2.3 14 Problem 1, Part 2 • k parallel English auctions (vanilla or modiﬁed), assuming additive valuations and, in general, heterogeneous goods, are also EPIC up to ke.