Collusion-Resilient Revenue In Combinatorial Auctions

Transcription

1 Collusion-Resilient Revenue In Combinatorial Auctions Silvio Micali and Paul Valiant December 11, 2007 Abstract Little is known about generating revenue in combinatorial auctions in particular, the VCG algorithm has no revenue guarantees. We put forward a dominant-strategy, probabilistic mechanism that guarantees a very meaningful revenue benchmark even in a very adversarial setting: that is, in the absence of any Bayesian information and in the presence of omniscient and perfectly coordinated collusive players. We prove that our mechanism is optimal among all dominant-strategy ones.

2 1 Introduction When a single good is for sale, the second-price mechanism allocates the good to the highest bidder, for a price equal to the second highest bid. The revenue so generated cannot be much disrupted by collusive players, even if they are perfectly coordinated and well informed about the other players valuations. We trace such robustness to the fact that the secondprice mechanism is dominant-strategy truthful (DST) and that it achieves a player-monotone revenue benchmark. By such a benchmark, we mean one whose value cannot but increase when more players join the auction. This is indeed the case for the second-price benchmark, and it is thus quite clear that, no matter how collusive players might bid, the revenue fetched by the seller is at least the second highest bid of the independent players. Combinatorial auctions, however, are more complex: there are multiple goods for sale, and each player may value any subset of them for an arbitrary amount. For such auctions very little is known about generating revenue, never mind in a robust fashion. Thus a natural question arises in combinatorial auctions: namely, Can a DST mechanism meaningfully approximate a meaningful player-monotone benchmark? On the mechanism side, the famous VCG mechanism generalizes the second-price one to combinatorial auctions, but here its revenue is no longer a player-monotone benchmark. Indeed, the VCG revenue is extremely fragile: its revenue may be high for a specific profile of bids, and disappear altogether if a new player with a suitable valuation for the goods enters the arena. Moreover, the VCG mechanism does not offer any revenue guarantee in the presence of even two sufficiently informed and coordinated collusive players. On the benchmark side, MSW, the maximum social welfare (i.e., the sum of the values received by all players in the best possible allocation of the goods), certainly is an attractive and player-monotone benchmark, but we prove that it cannot be approximated in the slightest by any dominant-strategy mechanism. (This is quite obvious for deterministic mechanisms, but less so for probabilistic ones.) In sum, to robustly extract revenue in combinatorial auctions, we need new benchmarks and new mechanisms. Thus, We put forward MSW, a revenue benchmark that meaningfully generalizes the secondprice benchmark. Indeed, MSW is defined to be the maximum social welfare after disregarding the bid(s) of the star player, that is, the one with the highest bid (for some subset of the goods). Simultaneously, 1

3 We put forward a simple probabilistic mechanism M whose expected revenue, in any combinatorial auction with n player and m goods, provably exceeds (disregarding small constants) a fraction 1 log min{n,m} of MSW. Given the lack of general revenue results in combinatorial auctions, M s performance is theoretically meaningful. In addition, since (log min{n, m}) 1 is a very slowly decreasing function, it may also be practically meaningful when either n or m is sufficiently small. In particular, when there are fewer than 299 goods for sale, M always returns at least 10% of the value of MSW relative to the independent players alone. Remarkably, this is true no matter how many collusive players might participate in the auction (hundreds, thousands, or even millions), and no matter how perfectly they might coordinate their bids. Still: is significantly better performance possible? Surprisingly, we prove that the answer is no. In order to hope to outperform M, one would need to assume either that (1) the players valuations are of some restricted form; or (2) a weaker notion of equilibrium is provably sufficient for the context at hand. Finally, in seeking to understand what revenue can be guaranteed in combinatorial auctions, we prove that probabilism is essential in our mechanism. That is, we prove than any deterministic DST mechanism will perform exponentially worse than M in at least some auctions. 2 The Problem of Guaranteeing Revenue in Combinatorial Auctions 2.1 Combinatorial Auctions and Their Basic Revenue Problem A (non-bayesian, n m) combinatorial-auction context is described as follows. There is a set of players N = {1,..., n} and a set of m goods G. A valuation is a function from G s subsets to R +, and each player i has a private valuation T V i, which we refer to as i s true valuation. An outcome consists of (1) a profile (i.e., a vector indexed by the players) P = P 1,..., P n, where P i R + is the price to be paid by player i, and (2) an allocation A = A 0, A 1,..., A n, where A i is the subset of goods allocated to player i, and A 0 the set of unallocated goods. For each outcome Ω = (A, P ), the utility of player i is defined via his utility function u i as follows: u i (T V i, Ω) = T V i (A i ) P i, that is, i s true value of the goods allocated to him minus the price he pays. Note that such a context is fully described by just N, G, and the true-valuation profile T V, which in turn determine the outcome space and 2

4 the utility functions. For such a context, a combinatorial-auction mechanism is a (possibly probabilistic) function M mapping a profile of valuations V to an outcome (A, P ) such that A i is empty and P i is 0 whenever V i is the null valuation. 1 An n m context C = (N, G, T V ) and an n m mechanism M define a (n m) combinatorial auction: namely, the game G = (C, M) envisaged to be played as follows. First, each player i (independently of the others) chooses a valuation BID i on inputs T V i, N, and G. Then, an outcome (A, P ) is obtained by evaluating M on BID, the profile of all such valuations. We refer to the so chosen valuations as bids, to emphasize that they need not coincide with the players true valuations. In such a game, a strategy is a (possibly probabilistic) way for a player to choose his bid. As for any game, in a rational play of an auction the players end up in equilibrium a profile of strategies σ = (σ 1,..., σ n ) such that no player i has an incentive to deviate from his strategy σ i if the other players stick to theirs. Accordingly, the basic revenue problem can be informally stated as follows: Find a mechanism M such that, for every context C, the resulting combinatorial auction (C, M) yields high revenue for a reasonable equilibrium σ. 2.2 The Worst Natural Setting and Natural Solution-Pairs The basic revenue problem for combinatorial auctions is already difficult. Yet, we wish to investigate and provide solutions to a more adversarial version of it: namely, the problem of generating revenue in the worst natural setting. We characterize such a setting by the following three assumptions. Equilibrium Uncertainty. An auction having an equilibrium (or several equilibria) σ yielding high revenue may not guarantee high revenue at all. Indeed, as for any other game, an auction typically has plenty of equilibria, and the one ultimately played may not be σ. In the worst (rational) setting, we assume that the equilibrium ultimately played will be the one generating least revenue among those equilibria τ having a modicum of rationality, that is, those composed by weakly non-dominated strategies. 2 Bayesian Ignorance. In a Bayesian setting, the players true valuations are assumed to be drawn from a distribution D. Clearly, knowledge of such a distribution can be 1 This guarantees that any player can opt out (i.e., win no goods and pay nothing) by bidding the null valuation. 2 Informally, a strategy τ i is weakly dominated for player i if i has another strategy ψ i that is always at least as good as τ i and sometimes strictly better. 3

5 very helpful for designing high-revenue mechanisms, particularly when D is of a suitable form. 3 In the worst setting, however, no Bayesian information is available. Devilish Collusion. Bidder collusion is a very well recognized problem both in theory and in real auctions [19]. This problem is particularly severe in combinatorial auctions (as illustrated later on). The severity of this problem decreases when we can assume some restrictions on the ability of collusive players to coordinate themselves. Aiming at guaranteeing revenue, we envisage no special restrictions in the worst setting. In particular, collusive players can make side payments to one another, and enter any secret and binding agreement of their choosing. More ominously, the worst natural setting envisages a malicious external entity (the Devil) which, with the sole purpose of lowering the auction s revenue and with full knowledge of the players true valuations, is free to introduce any number of additional players whose bids he freely chooses. 4 We consider any revenue achievable in such an adversarial setting as guaranteed revenue, because it is a fortiori achievable in any more realistic one. Accordingly, the problem addressed this paper can be rephrased as follows: How much revenue can be guaranteed in combinatorial auctions? Let us now argue that a natural solution to this problem consists of a revenue benchmark and a dominant-strategy truthful (DST) auction mechanism satisfying some simple properties. Here, by a revenue benchmark we mean a function from sequences of valuations for the same set of goods to non-negative numbers; and by a DST auction mechanism we mean an auction mechanism for which for any player i, (1) bidding his true valuation is at least as good as any other strategy, no matter what bids the other players might choose; and (2) i cannot be charged more than he bids. Definition 1. Let B be a revenue benchmark, M a DST auction mechanism, and g(n, m) a function with range [0, 1]. Then, we say that (B, M) is a natural solution with revenue guarantee g(n, m) if the following two properties hold: 1. B is player monotone: For any valuation profile S and any sub-profile T of S, B(S) B(T ); and 3 For single-good auctions, Myerson [18] has put forward mechanisms that generate optimal revenue essentially whenever the players have independent valuations of the good, a quite general condition. For such a case, he is able to optimally set minimum winning bids without undue risk of pricing out players. By contrast, in combinatorial auctions no general distributions for the players true valuations are known for which optimal revenue mechanisms exist. 4 I.e., the Devil has a totally different type of utility, and can force the collusive players to bid against their own interests in the auction, by offering external compensation/punishment. 4

6 2. M g(n, m)-achieves B: For any valuation profile V in a n m auction, the revenue of M(V 1,..., V n ) is g(n, m) B(V 1,..., V n ). If Condition 2 holds only for all n, m k, we say that (B, M) has revenue guarantee g(n, m) for n, m k. If (B, M) is a natural solution pair, then mechanism M is well suited to generate revenue in the worst natural setting. In fact, M s dominant-strategy truthfulness successfully handles equilibrium uncertainty: when the auction mechanism is DST, no rational player player has reason to prefer any other bid to his true valuation. 5 Further, M does not rely on any Bayesian information: the only inputs of M (and of benchmark B) solely consist of specific valuations, even if these were indeed drawn from a suitable distribution D. Finally, the player monotonicity of the benchmark supports the fundamental economic principle that increased competition among buyers is good for the seller. And a mechanism M g(n, m)-achieving a player-monotone benchmark B implies that the revenue guaranteed to the seller increases with the arrival of additional players. 6 This is formally stated by the following, trivially proven lemma, where the function D models the Devil and, being universally quantified, all possible ways to lower revenue. Lemma 1. For any set of goods G of cardinality m, any sequence V 1,..., V s of G s valuations, and any function D whose range consists of tuples of G s valuations, letting (V s+1,..., V n ) = D(V 1,..., V s ), the revenue of M(V 1,..., V n ) is g(n, m) B(V 1,..., V s ). In other words, A player-monotone revenue benchmark guarantees that, if a set I of players bids independently, then the benchmark relative to the entire set of players N (possibly including collusive or irrational players) will be at least as high as the benchmark relative to the set I. Of course, the attractiveness of a natural solution pair ultimately depends on two things: 5 From the perspective of an individual player i, DST mechanisms also remove any strategy uncertainty as well. In a general equilibrium σ, σ i is i s best course of action only assuming that he believes that every other player j will chose to stick to his prescribed strategy σ j. Such beliefs, however, may be hard to justify and constitute an additional condition that weakens the strength of the equilibrium. Quite differently, when σ is a dominant-strategy equilibrium, no reliance on such beliefs is necessary to support σ: σ i is i s best response to any possible strategies of the other players. 6 Note that guaranteed revenue is different from actual revenue, and the latter may go down when additional players join the auction. For instance, if (B, M) is a natural solution pair with revenue guarantee 1/2, then it is conceivable that (i) for a given valuation profile, the value of B is 10 and the actual revenue returned by M is 15, while (ii) adding one more player increases B to 12 and thus increases the revenue guarantee to 6 but the actual revenue of M decreases to 12. 5

7 1. The quality of the benchmark, and 2. The quality of the revenue guarantee. Indeed, the identically-0 benchmark can be trivially achieved with revenue guarantee 1, and the highest imaginable benchmark can be trivially achieved with revenue guarantee 0. Of course too, natural solution pairs are sufficient conditions to guarantee revenue in the worst natural setting. As such, they might very well be unnecessarily demanding, but are certainly appealingly simple. 2.3 Natural Solution Pairs: Single-Good vs. Combinatorial Auctions When there is a single good for sale, things become very simple: the true valuation of a player i is a non-negative real number T V i, representing the value to i of the item for sale; only a single player wins the good, and only he pays a positive price. For such auctions, denote by 2P the famous second-price mechanism (i.e., the mechanism returning the player whose bid is highest as the winner, the second-highest bid as the winner s price, and 0 as the price of every other player); and by 2V the Second-Valuation benchmark (i.e., the function returning the second highest valuation). 7 Now, the benchmark 2V is clearly player-monotone, and the mechanism 2P is well known to be DST. Therefore, (2V, 2P ) is a very attractive natural solution pair: indeed, its benchmark is reasonably high and its revenue guarantee is as high as possible. The situation ceases to be so rosy in combinatorial auctions. To be sure, the 2P mechanism is a special case of the famous VCG mechanism which indeed applies to combinatorial auctions. Let us quickly recall the definition of this more general mechanism, establishing our notation along the way. In a combinatorial auction, the social welfare relative to a profile of valuations V and an allocation A denoted by SW (V, A) is the sum of the values actually accrued by the players. That is, SW (V, A) = V 1 (A 1 )+ +V n (A n ). The best allocation relative to a valuation profile V, denoted by BestA(V ), is the allocation whose social welfare is maximum. The actual social welfare of this allocation is called the maximum social welfare, and is denoted by MSW (V ). With this preamble, given a valuation profile V for the goods for sale, the VCG mechanism returns an allocation A and a price profile P as follows: A = BestA(V ) and P i = MSW (V i ) SW (V i, BestA(V )), that is the maximum social welfare relative to 7 In either case, ties are broken arbitrarily. 6

8 all valuations in V except i s one, minus the social welfare of all players except i in the best allocation relative to V. The primary objective (and claim to fame) of the VCG mechanism is efficiency in the economic sense. 8 However, although revenue and efficiency are not unrelated, and although it too is dominant-strategy truthful, the revenue generated by the VCG mechanism suffers from three (interrelated) deficiencies, illustrated below assuming that the good for sale are always a and b and that V i is the true valuation of player i. (For each subset S, we assume V i (S) to be 0 unless explicitly defined to be positive. The number of players is deducible form the number of valuations.) Revenue Fragility. Let V 1 ({a}) = V 1 ({a, b}) = 1000 and V 2 ({a, b}) = In this context the VCG mechanism returns revenue However, if another player 3 with a valuation V 3 ({b}) = V s ({a, b}) = 1000 appears, then the VCG revenue drops to 0. That is, the VCG revenue is not player-monotone in combinatorial auctions. Revenue Unattainability. Let V k 1 ({a}) = V k 1 ({a, b}) = V k 2 ({a, b}) = V k 3 ({b}) = V 3 ({a, b}) = k. Then, for every k, the VCG mechanism returns revenue 0 on input the triple of valuations V k = (V k 1, V k 2, V k 3 ). This performance is in strident contrast with the fact that other DST mechanisms (in particular, ours) generate revenue going to infinity for this sequence. 9 Collusion Vulnerability. Let V 1 ({a}) = V 1 ({a, b}) = 500; V 2 ({a, b}) = 1000; and V 3 ({b}) = V s ({a, b}) = Given such valuations, the VCG mechanism would return revenue of Such revenue, however, is very vulnerable to collusion, despite the fact that the VCG is DST. Indeed, the DST property solely guarantees that an independent player has nothing better to do than bidding his true valuation. But it guarantees nothing of the sort about two or more players! Indeed, if players 1 and 3 are collusive and know that player 2 values only the subset {a, b}, and for no more than one million. Then, while player 2 bids 1000 for {a, b}, player 1 bids one million for {a} and player 3 one million for {b}. Thus each of 1 and 2 obtains his desired set by the VCG mechanism paying 0. In sum, the VCG mechanism is wonderful for efficiency, but inadequate for revenue. The 8 Indeed, best allocations appear to be computationally hard, but the amount of computation is orthogonal to the traditional goals of game theory, and is not a concern of this paper either. 9 For instance, consider the following bundling mechanism M: for each player i, compute the personal welfare to i of all goods (i.e., the maximum social welfare of just i s valuation, that is MSW (V i )); allocate all goods to the player, w, with the highest personal welfare; let w pay the second highest personal welfare, and let every other player pay 0. It is easy to show that M is dominant-strategy truthful, and that the revenue it generates for the profile V k grows to infinity with k. 7

9 problem, however, is quite more general: indeed For combinatorial auctions, not only are no results known about guaranteeing revenue in the worst setting, but also no general results about generating revenue are known even in the Bayesian setting. 3 Our Results We contribute four results about guaranteeing revenue in combinatorial auctions. We present their statements in this section, and their proofs in the appendix. Our positive result consists of putting forward a natural solution pair with a reasonable revenue guarantee. We find this solution attractive because of the conceptual simplicity of its probabilistic mechanism, M, and the fact that our benchmark is a very natural generalization of the second-valuation benchmark of single-good auctions. Personally, however, we find more interesting our negative results. In particular, we prove that no DST mechanism can significantly outperform M on our benchmark, and that an exponential performance gap exists between M and any deterministic DST mechanism. Therefore, while our positive result establishes reasonably attractive properties of a specific DST mechanism, our negative ones establish intrinsic limitations of all possible DST auction mechanisms, and advance our understanding of what revenue can be guaranteed in combinatorial auctions. (The usefulness of such understanding is perhaps evidenced by the fact that we have first proved our negative results, and that these have then guided us to discover our positive one!) 3.1 Our Benchmark What revenue benchmarks should we choose for natural solution pairs in combinatorial auctions? An obvious temptation is to consider M SW. In fact, M SW upper-bounds the revenue achievable by any auction mechanism. Further, M SW is a player-monotone benchmark. However, as the next theorem shows, it is not achievable with any positive revenue guarantee. Theorem 1. For any DST mechanism M, positive function g(, ), and integers n, m, there exists a valuation profile V for an n m auction context such that the revenue of M(V 1,..., V n ) < g(n, m) MSW (V 1,..., V n ). In other words, M SW is too perfect: aiming to approximate it in combinatorial auctions in a sufficiently adversarial setting is tantamount to aiming for eternal life in a national 8

10 health-care system. But in looking for alternatives, we must resist another natural temptation: namely, choosing a next-to-perfect benchmark that can only be achieved in specific contexts. Instead, our goal is to understand what revenue can be guaranteed in all combinatorial contexts. And it is with this goal in mind that we put forward the following benchmark MSW. Definition 2. Let V = (V 1,..., V n ) be a sequence of valuations of a finite set of goods G, and let i and S be such that V i (S) V j (T ) for all j [1, n] and all T G. Then, we define MSW (V ) = MSW (V i ), where V i = (V 1,..., V i 1, V i+1,..., V n ). We refer to such a player i as the star player, and denote him by the symbol, so as to capitalize on standard game theoretic notation in our symbolic manipulations. (If two or more such players exist, then player is the lexicographically first among them.) In words, MSW removes the star player and then computes the maximum social welfare for the valuations of the remaining players. It should then be obvious that MSW is player monotone, and that it is a generalization of the 2V benchmark: that is, MSW = 2V in single-good auctions. Indeed, in single-good auctions, (1) removing the highest bid is tantamount to removing the star player, and (2) the maximum social welfare of the remaining valuations is the highest remaining bid, and thus the second highest of the original bids. A Surprising Example. values the ith good. Consider the following context, where each player i only {a} {b} {c} {d} {e} {f} {g} Example A (Every subset not explicitly and positively valued by a player i is assumed to be valued 0 and is not shown.) Without resorting to bundling (i.e., without deciding a priori that certains subsets of the goods must be sold together), generating revenue for the valuations of Example A is not 9

11 straightforward, because the players are not competing for the sets they value. In particular, the VCG mechanism returns no revenue for such valuations. By contrast, the value of MSW is 6000 for these valuations, and thus we have the right to expect reasonably high revenue from a mechanism that, like our M, is always guaranteed to reasonably approximate MSW. Indeed, the expected revenue of M for the valuations of Example A is greater than Notably, as we shall see, M achieves this performance without bundling, because it is a general-purpose mechanism designed to perform reasonably in all cases. A proper bundling mechanism could return even higher revenue in the case of Example A, but would perform very poorly in other cases. 3.2 Our Mechanism Our mechanism M is simple and probabilistic. In constructing it, some of our choices are dictated by our desire for M to be DST; others by our desire for M to generate revenue approximating MSW. At the highest level, the idea is that of trading efficiency for revenue. We obtain M by starting with an underlying, deterministic, DST, and high-efficiency mechanism M. We then modify M so as convert some of its efficiency to revenue. The first approach to implement such a plan consists of three stages. In the first stage, we run M on the profile of bids provided by the players and obtain an allocation A and a profile of prices P. In the second stage, we raise all prices in P by a fixed amount ρ. In the third stage we decide decide the final allocation and prices as follows. If player i wins a set of goods S in A, and if P i + ρ is less than i s bid for S, then we finally allocate S to i for a price of P i + ρ. Else, S will go unallocated, and i pays 0. This modification of M may cause a loss of revenue from some players, but such loss may be compensated by additional revenue from other players. Thus: How should prices be raised, and by how much, to get better revuenue? As we shall argue later on, the revenue of any deterministic mechanism can only poorly approximate our benchmark. Thus, we shall raise prices probabilistically. Further, in light of our benchmark, one natural choice for ρ is a fraction α of MSW (BID), where the scaling factor α is probabilistically chosen between 0 and 1. This approach, however, needs to be refined. To begin with, to ensure that M is DST, we do not want player i s price to depend on his bid, and MSW (BID) may indeed depend on BID i. This problem is traversed by continuing to choose α probabilistically between 0 and 1, but then raising price P i by αmsw (BID i ) instead. Our analysis will support that this small change does not alter the ability to achieve the chosen benchmark. At the same time, such a modification of M 10

12 is guaranteed to be DST. Two choices now remain to fully specify M: that of the underlying mechanism M and that of scaling factor α. For M, as we plan to turn efficiency into revenue, it is natural to choose the VCG mechanism, since it has optimal efficiency. (However, any M whose efficiency is a sufficiently high fraction of M SW would work too, leaving room for computationally more tractable auction mechanisms and other desiderata.) To choose α, we are actually guided by one of our impossibility results discussed in the next subsection, and indeed discovered before M. Informally, this result states that, in a n m auction, no DST mechanism can guarantee revenue greater than a logarithmic (in µ = min{n, m}) fraction of MSW (BID). With this limitation in mind, we choose α by means of a continuous exponential distribution. (Discrete exponential distributions were used in simpler settings by [13] and [3]. 10 ) Our specific selection of constants is solely justified by our desire to optimize our revenue guarantee. Definition 3. We denote by M the auction mechanism that on input BID, a profile of n bids for a set of m goods, computes an outcome (A, P ) as follows: 1. Pick a scaling factor α [0, 1] as follows: Let µ = min{n, m} and c n,m be the constant > 2 that solves the equation e x 2 = xµ. 1 Flip a coin whose probability of Heads is c n,m 1. If Heads, choose α = 0. If Tails, draw r uniformly from [ (c n,m 2), 0] and choose α = e r. 2. Compute the provisional allocation A and the profile of provisional prices P = V CG p (BID) respectively the allocation and the prices of the VCG mechanism for the bid profile BID and then the set of provisional winners W consisting of all players that obtain a non-empty subset of goods in A. 3. For each i W compute i s offer price P i + αmsw (BID i ). If i s bid BID i (A i) exceeds i s offer price, set A i = A i and P i = P i + αmsw (BID i ); otherwise set A i = and P i = For instance, the goal in [3] was to obtain high-revenue in the unlimited supply model. Recall that in this model there are arbitrarily many copies of each good for sale, and thus competition among players for the same good is not an issue. Consequently, in this simpler setting their mechanism is content with producing a single global price, paying which any player can obtain as many copies of every good he wants. They call their pricing structure the amusement-park model, in analogy to what happens in amusement parks where payment of the fixed entrance fee enables one to enjoy as many rides as he wants. 11

13 Remarks We note that c n,m is uniquely defined since for µ 1 the function f n,m (x) = e x 2 xµ is negative at x = 2, goes to infinity as x increases, and has positive second derivative everywhere. Notice that although each price P i is personalized, it is obtained via the same choice of scaling factor α. Were we in a Bayesian setting, where different players have different distributions for their valuations, then we would optimally choose a separate scaling factor α i for each player i. For all combinatorial auctions, we prove the following about our mechanism and our benchmark. Theorem 2. (MSW, M) is a natural solution pair with revenue guarantee 1/c n,m. Concrete Revenue Consequences. As implied by c n,m s very definition, 1/c n,m is a very slowly decreasing function of n and m; namely: 1/c n,m = Θ(1/ log min{n, m}). That is, disregarding constants, 1/c n,m equals 1/ log min{n, m}. Therefore, when either n or m is reasonably small, M s revenue is expected to be a reasonable portion of the potential value expressed by our MSW benchmark. For instance, In the case of Example A, n = 7 and 1/c n,m > 17%. Accordingly, because here the value of our benchmark is 6000, Theorem 2 implies that M s expected revenue in Example A must be greater than This performance is indeed guaranteed for any n m auction with a MSW benchmark of 6000 and min{n, m} = 7. But Example D specifies a very special case of such auctions, and M s expected revenue with Example A s valuations is even better. Indeed, it can be separately shown to be greater than If there are fewer than 299 goods for sale (perhaps a plausible scenario), then it can be shown (see Corollary 1) that, regardless of whether the number of players is in the tens, the thousands, or the millions, that M always returns as revenue 10% of MSW relative to all independent players, no matter how combinatorially complex the valuations may be, and no matter how collusively, irrationally, or maliciously the other players may bid. 3.3 Optimality of Our Mechanism We prove two results about M s revenue performance, relative to our benchmark and any other possible DST mechanism. Namely, M is (1) precisely optimal asymptotically and (2) 12

14 optimal within a constant factor in every auction. Crudely put, we prove there is no way, in dominant strategies, to extract significantly more revenue than M for all combinatorial auctions. Let us now to express these properties more precisely. Definition 4. Let opt(n, m) denote the smallest x R + for which there exists a DST mechanism M n,m whose (expected) revenue is at least 1 MSW x (BID) for any n m bid profile BID. Theorem 3. lim n,m c n,m opt(n, m) = 1 and n, m c n,m < 5 opt(n, m). 3.4 The Necessity of Probabilism Our mechanism demonstrates once again the power of randomization. Indeed, we prove the following. Theorem 4. For any deterministic auction mechanism M, (MSW, M) has revenue guarantee 1 min{n,m} 1 for n, m 2. Thus an exponential gap exists between the revenue guaranteed by our M and that guaranteed by any deterministic DST mechanism. 4 Conclusions Collusive players affect the practical meaningfulness of many mechanisms. However, we believe and hope that the revenue robustness achieved by our natural solution pair (MSW, M) is exportable to other mechanism-design settings. In addition, the traditional game-theoretic assumption of the perfect rationality of all players strikes us as often being too risky in practice. 11 We thus look forward to a more thorough investigation of the degree of protection achievable against collusive and/or somewhat irrational players. Acknowledgements We are grateful to Sergei Izmalkov for his many comments and suggestions. 11 For instance, consider the well known game of choosing an integer between 0 and 100 and rewarding the player whose integer is closest to half of the average of chosen integers. Would you really choose 0 when playing it among strangers on the street? 13

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16 [13] Guruswami, V., Hartline, J.D., Karlin, A.R., Kempe, D., Kenyon, C., and McSherry, F. On Profit-Maximizing Envy-free Pricing. Symposium on Discrete Algorithms, pp , [14] Jain, K. and Vazirani, V. Applications of Approximation Algorithms to Cooperative Games. Symposium on Theory of Computing [15] Lehmann, D., O Callaghan, L.I., and Shoham, Y. Truth Revelation in Approximately Efficient Combinatorial Auctions. ACM Conf. on E-Commerce, pp , [16] Likhodedov, A. and Sandholm, T. Approximating Revenue-Maximizing Combinatorial Auctions. The Twentieth National Conference on Artificial Intelligence, pp , [17] Moulin, H. and Shenker, S. Strategyproof Sharing of Submodular Costs: Budget Balance Versus Efficiency. Economic Theory, 18: , [18] Myerson, R. Optimal Auction Design. Mathematics of Operations Research, 6: 58-73, [19] Porter, R. Detecting Collusion. Review of Industrial Organization, 26:2: , [20] Vickrey, W. Counterspeculation, Auctions, and Competitive Sealed Tenders. J. of Finance, 16:8-37,

17 Appendix A Preliminaries A.1 Additional Notation We consistently denote the set of players by N = {1,..., n} and the set of goods for sale by G = {g 1,..., g m }. Sub-profiles. Recall that a profile is a vector V indexed by N. If C N, the sub-vector of V indexed by C is denoted by V C, and is referred to as a sub-profile. If i N, we more simply write V i rather than V {i} ; V i rather than V N {i} ; and, if C N, V C rather then V N C. If S and T are disjoint subsets of N, then by V S V T be denote the sub-profile mapping each player i S T to (V S ) i if i S, and to (V T ) i otherwise. We extend to sub-profiles the functions SW, BestA and MSW as follows. For each valuation sub-profile V C and allocation A: SW (V C, A) = i C V i(a i ), BestA(V C ) = argmax A A(G) SW (V C, A), where A(G) denotes the set of all possible allocations of G, and MSW (V C ) = SW (V C, BestA(V C )). By convention, argmax s ties are broken lexicographically. Also by convention, for all X G, C N, and i C, if V i (X) = 0, then BestA(V C ) i X. Single-Minded Valuations. A valuation v of a finite set of goods G is single-minded if there exists a single set of goods S and x R + such that v(t ) = x whenever S T and 0 otherwise. We compactly represent such a single-minded valuation v by the pair (S, x). Mechanisms, Revenue, and Winners. Recall that an auction mechanism is a probabilistic function mapping a bid profile BID to pair (A, P ), where A is an allocation and P a price profile, satisfying the opt-out condition: P i = 0 whenever BID i is the null valuation. We thus view each mechanism M as two separate functions: an allocation function M a and a price function M p. That is, for all bid profiles BID: M(BID) = (M a (BID), M p (BID)). The expected revenue of mechanism M on bid profile BID is E[ i N M p(bid) i ]. 16

18 We define the set of winners in an allocation A as follows: W ins(a) = {i N : A i }. DST Equilibria. Let C = (N, G, T V ) be an auction context, and G = (C, M) an auction. Then, we say that a profile of bids BID is a dominant-strategy truthful (DST) equilibrium of G if (1) BID = T V and for all bid profiles BID and (2) players i N: and E[u i (T V i, M(T V i BID i))] E[u i (T V i, M(BID ))]. We say that an n m mechanism M is dominant-strategy truthful if for all n m auction contexts C, the auction G = (C, M) has a DST equilibrium. Two Main Properties of DST Mechanisms. Let us explicitly highlight two properties of DST mechanisms which we are going to use extensively. (The first is an immediate consequence of the opt-out condition that is, that by submitting the null valuation a player can guarantee that he wins nothing and pays nothing.) DST-1: For all (probabilistic or not) DST mechanisms M, players i, and bid profile BID, we have: 0 E[M p (BID) i ] E[BID i (M a (BID) i )]. DST-2: For all deterministic DST mechanisms M, players i, and bid profiles BID and BID such that BID i = BID i, we have: M a (BID) i = M a (BID ) a implies M p (BID) i = M p (BID ) i. B Proof of Theorem 1 We actually prove a slightly stronger result: namely, Theorem 1 : For any probabilistic DST mechanism M, any integers n and m, any player i, any bid sub-profile BID i for an n m auction context, any subset S of the goods, and any constant d > 0, there exists a single-minded bid (S, v) for player i such that, letting BID = BID i (S, v), we have [ ] E M p (BID) j < j N MSW (BID). d Proof. Let (S, x) be player i s true valuation for the goods; and consider the bid profile BID x = BID i (S, x). To prove the theorem, the idea is to have x grow to infinity and show that M cannot generate sufficient revenue for large enough x. This is obvious if 17

19 M were a deterministic DST mechanism because Property DST-2 implies that the price imposed to player i is a fixed function of BID i, and is thus fixed too but needs a bit more analysis due to M s probabilism, which is provided below. For the bid profile BID x, define Q x to be the probability that i wins set S, and p x to be i s expected payment (both taken over M s random choices). That is, Q x = P r[m a (BID x ) i = S] and p x = E[M p (BID x ) i ]. Define Q = sup x Q x, fix positive ɛ < 1 2d, and let (S, x ɛ) be such that Q xɛ > Q ɛ. Since (S, x) is i s true valuation, the dominant-strategy truthfulness of M implies that i s expected utility is higher when he bids (S, x) than when he bids any other valuation, and in particular (S, x ɛ ). That is, Q x x p x Q xɛ x p xɛ, which we rewrite as (Q x Q xɛ )x p x p xɛ. (1) Since Q Q x and Q xɛ > Q ɛ, we can rewrite Inequality (1) as p x ɛx + p xɛ. (2) Thus M s expected revenue relative to the bid profile BID x can be upper-bounded as follows for any choice of x: [ ] E M p (BID x ) j j [ (a) = E j i M p (BID x ) j ] +p x (b) MSW (BID i )+p x (c) MSW (BID i )+ɛx+p xɛ Indeed, (a) follows from the linearity of expectation; (b) from Property DST-1; and (c) from Inequality 2. Now, because MSW (BID i ) and p xɛ so that ɛx > MSW (BID i ) + p xɛ, Inequality 3 and our choice of ɛ imply [ ] E M p (BID x ) j j 2ɛx < x d MSW (BIDx ), d (3) are fixed, when x is large enough where the last inequality is true because MSW (BID x ) BID x i (S) = x. Thus for large enough x, the bid (S, x) satisfies our thesis. Q.E.D. 18

20 C Proof of Theorem 2 We break Theorem 2 into two separate theorems. Namely, Theorem 2a: (MSW, M) is a natural solution pair. and Theorem 2b: (MSW, M) has revenue guarantee 1/c n,m. C.1 Proof of Theorem 2a Theorem 2a is quite straightforward. The key observation is that M is DST because it has been obtained from the VCG mechanism by modifications that trivially preserve the dominant-strategy truthfulness. The proof below is mostly given for completeness and selfcontainment sake. Definition 5. A function f : N V(G) N R + is called stable if, for each i N, there exists a function g i such that, for each BID V(G) N, f(i, BID) = g i (BID i ) i.e., f is independent of BID i. Definition 6. Given a deterministic mechanism M and a stable function f define M +f to be the mechanism defined as follows: on input BID V(G) N, 1. Compute the provisional allocation A = M a (BID), the profile of provisional prices P = M p (BID), and the set of provisional winners W = W ins(a ). 2. For each i W, if BID i (A i) P i + f(i, BID) then let P i = P i + f(i, BID) and A i = A i; otherwise let P i = 0 and A i =. Lemma 2. If M is DST, so is M +f for any stable f. Proof. Given an auction context C = (N, G, T V ), a bid profile BID V(G) N, and a player i N we have u i (T V i, M +f (T V i BID i )) (1) = T V i (M +f a (T V i BID i ) i ) M +f p (T V i BID i ) i (2) = max{0, T V i (M a (T V i BID i ) i ) M p (T V i BID i ) i f(i, BID i )} (3) = max{0, u i (T V i, M(T V i BID i )) f(i, BID i )} (4) max{0, u i (T V i, M(BID)) f(i, BID i )} (5) u i (T V i, M +f (BID)), 19

21 where: Equality (1) holds by the definition of the utility function u; equality (2) holds by the definition of M +f, and can be easily checked in each of the two cases M +f a (T V i BID i ) i = M a (T V i BID i ) i and M +f a (T V i BID i ) i = ; equality (3) holds by the definition of the utility function u; inequality (4) holds because M is dominant-strategy truthful; and inequality (5) holds because M +f (BID) either yields i utility 0 or utility u i (T V i, M(BID)) f(i, BID i ), and thus at most the maximum of these two quantities. Q.E.D. We now define a special class of probabilistic mechanisms. Definition 7. Let D a distribution over a set of deterministic n m auction mechanisms. Then we denote by S D the the probabilistic n m auction mechanism that, on input a profile of bids BID, first selects a mechanism M according to D and then returns the outcome M(BID). We refer to such a mechanism S D as above as a (n m) sampler. Lemma 3. If S D is a sampler and all mechanisms in D s support are DST, then S D is DST. Proof. Let our S D actually be an n m sampler, and let C = (N, G, T V ) be an n m auction context, BID V(G) n, and i N. Then we have E[u i (T V i, S D (T V i BID i ))] (1) = E [u i (T V i, M(T V i BID i ))] (2) M D E [u i (T V i, M(BID))] (3) = M D E[u i (T V i, S D (BID))], where equality (1) holds because the definition of S D ; inequality (2) because M is DST; and equality (3) because of the definition of S D. This chain of inequality proves that S D is DST. Q.E.D. Theorem 2a: (MSW, M) is a natural solution pair. Proof. The benchmark MSW is player-monotone, while mechanism M is a sampler satisfying the hypotheses of Lemma 3. Q.E.D. 20

22 C.2 Proof of Theorem 2b Theorem 2b: (MSW, M) has revenue guarantee 1/c n,m. Proof. By virtue of Definition 2, we need to prove that, whenever BID is a valuation profile for a n m auction, we have E[ i N M p (BID) i ] MSW (BID) c n,m. (4) For each player i, let S i be the (possibly empty) set player i provisionally wins, and let P i be the provisional price V CG p (BID) i. We divide our proof into two cases: in the first case the star player bids large enough that we derive the revenue bound solely based on the revenue M extracts from the star player. In the second case we must sum up the revenue that M extracts from each set-winning player. Case 1: BID (S ) > P + MSW (BID). Note that the right-hand side of the inequality of this case is always 0, thus BID (S ) > 0 always. This implies that S ; namely that is a provisional winner. As such, when mechanism M makes the offer P + α MSW (BID) where α 1, the offer price will always be at most player s bid for S, and hence player will always pay his offer price. Thus the expected revenue from player is just the expected offer price, namely 1 E[M p (BID) ] = c n,m 1 P + (1 ( = = P c n,m 1 ) ) 1 c n,m 1 + (1 1 c n,m 1 ) 1 c n,m 1 MSW (BID) (c n,m 2) P + (1 0 = P + MSW (BID) 1 e (cn,m 2) c n,m 1 MSW (BID) 1 µe (cn,m 2) c n,m 1 (c n,m 2) 1 c n,m 2 (P + e r MSW (BID)) dr 1 c n,m 1 ) 1 c n,m 2 MSW (BID) e r dr 0 = MSW (BID) 1 1 c n,m c n,m 1 = MSW (BID) c n,m, where the inequality follows because P 0 and µ 1, and the second to last equality is by the definition of c n,m, namely that c n,m µe (cn,m 2) = 1. Thus we have the desired result in this case. (c n,m 2) e r dr 21

23 Case 2: BID (S ) P + MSW (BID ). Consider a provisional winner i, and consider his offer price P i + α MSW (BID i ). We note that when α = 0 the offer price for player i is just P i, which is less than or equal to BID i (S i ) since the V CG mechanism never charges players more than their bid; thus when α = 0 player i will accept the offer and pay P i. Since player i will pay the offer price whenever it is less than BID i (S i ), we have that i will pay whenever α < BID i(s i ) P i. MSW (BID i ) Recall that, by the definition of M, when α 0 we have α = e r. Thus this condition becomes r < log e BID i (S i ) P i MSW (BID i ). We note that r is also bounded to be at most 0, but the other condition BID i (S i ) P i MSW (BID i ) takes precedence since log e 0, as we show by an analysis of two cases: when i = the claim is equivalent to the condition of Case 2, BID (S ) P + MSW (BID ); otherwise, when i we have BID i (S i ) P i BID i (S i ) MSW (BID ) (BID i ) (where MSW (BID ) denotes the highest bid of the star player, which is higher than any other bid, including BID i (S i ) by assumption) which implies the 0 bound we wanted to prove. Thus the expected price paid by player i is exactly expressed as the following integral: P i BID c n,m 1 +(1 1 max { (c i (S i ) P } n,m 2),log i c n,m 1 ) e MSW (BID i ) 1 (c n,m 2) c n,m 2 (P i + e r MSW (BID i )) dr. We lower-bound this expression using the following two observations: first, since P i 0 we may remove this term from inside the integral; second, since the integrand is always BID i (S i ) P i MSW (BID i ) positive, if we decrease the upper limit of the integral to log e the integral can only decrease (where we use the standard convention that an integral with upper limit less than its lower limit is evaluated with limits reversed and negated). Thus we have E[M p (BID) i ] P i c n,m 1 + (1 1 loge c n,m 1 ) (c n,m 2) BID i (S i ) P i MSW (BID i ) ( P i = c n,m 1 + MSW (BID 1 i) e log e c n,m 1 1 ( = BIDi (S i ) e (cn,m 2) MSW (BID i ) ) c n,m 1 Summing up this inequality over all provisional winners i, we get 1 c n,m 2 (er MSW (BID i )) dr ) BID i (S i ) P i MSW (BID i) e (cn,m 2) E[ i W M p (BID) i ] ( 1 BID i (S i ) e (cn,m 2) c n,m 1 i W 22 i W MSW (BID i ) ).

24 Now notice that i W BID i (S i ) = MSW (BID). Further since W µ and MSW (BID i ) MSW (BID) we have i W MSW (BID i ) µ MSW (BID). Thus we have E[ M p (BID) i ] MSW (BID) 1 µe (cn,m 2) c i W n,m 1 = MSW (BID) 1 1 c n,m c n,m 1 = MSW (BID) c n,m MSW (BID) c n,m, where we invoke the definition of c n,m to derive the first equality. Thus we have the desired conclusion. Q.E.D. Remarks. Notice that our mechanism M requires that its underlying DST mechanism be reasonably efficient. Indeed, in the analysis of Case 2, we rely on the fact that the VCG algorithm is 100% efficient: namely, we rely i W BID i (S i ) = MSW (BID). If another DST mechanism is used, one should make sure that, for its provisional allocation A, BID i (A i ) is a sufficient fraction of MSW (BID). Notice that, when lower-bounding the revenue generated by M, the profile of prices returned by the underlying DST mechanism are essentially ignored. However, were we to simplify the definition of M by replacing the provisional prices with zeros, the resulting mechanism would not be DST. Corollary 1. Given a constant c > 2, for every auction context where min{ N, G } ec 2 c, E[ i N M p (BID) i ] MSW (BID). c Proof. This follows immediately from Theorem 2 after solving for µ in the definition of c n,m. Q.E.D. We note that for the value c = 10 we have ec 2 > 298 and thus this corollary says that c for any auction with either at most 298 players and unlimited goods, or at most 298 goods and unlimited players, we can guarantee expected revenue of 10% of our benchmark. Corollary 2. Given a constant c > 2, for every auction context (N, G, T V ), where min{ N, G } e c 2 c, E[ i N M p (BID) i ] MSW (T V ). c 23

25 Proof. From Theorem 2a, mechanism M has a dominant-strategy equilibrium where BID = T V, so the corollary follows immediately from the previous corollary. Q.E.D. D Proof of Theorem 3 D.1 A First Revenue Upper-Bound It is obvious from Property DST-1 that, for any bid profile BID, the revenue generated by any DST mechanism probabilistic or not cannot exceed M SW (BID). Let us now minimally extend this upper-bound. Namely, Lemma 4. For any n > 1 and any n m DST mechanism M, there exists a bid profile BID such that [ ] E M p (BID) i MSW (BID). i N Proof. Let BID be such that all players bid single-mindedly for the first good, specifically BID i = ({1}, 1) for all i. Then, for any possible allocation A of the goods, SW (BID, A) 1. Thus, no matter what the DST mechanism M might be, in expectation SW (BID, M a (BID)) 1. By Property DST-1, this implies that the expected revenue generated by M also is 1. Since obviously MSW (BID) = 1, profile BID satisfies our thesis. Q.E.D. D.2 Harmonic Distributions, Harmonic Pricing, and the Harmonic Revenue Bound Let us now introduce and prove a more interesting revenue upper-bound. Recalling that the harmonic series is j=1 1/j and that the ith harmonic number, H i, is i j=1 1/j, we prove that no n m DST mechanism can guarantee more than a 1/(H min{n,m} 1) fraction of MSW (BID). To this end, let us first introduce a simple distribution that we have found crucial for proving other impossibility results in general auctions. Definition 8. (Bounded-Harmonic Distributions) For any subset of goods S and positive integer k, we denote by h k S the distribution assigning, for each integer i [1, k], probability 1 to the single-minded valuation (S, 1). k i We now prove a property of DST mechanisms that may be of independent interest. 24