# Cost of Conciseness in Sponsored Search Auctions

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5 on this graph, and assigns advertisers to slots according to this matching. We will use M to denote the weight of the maximum matching on G, and number advertisers so that bidder i is assigned to slot i in this matching. Let M i denote the weight of the maximum weight matching on G when all edges incident to bidder i are removed. Then p i, the VCG price for bidder i, is p i = M i + v i i M. 3 Inefficiency of oblivious strategies In this section, we explore the loss in efficiency when restricting a bidder to oblivious bidding strategies (i.e., her single bid is a function only of her full vector of values). We find that in terms of efficiency, no matter how bidders choose to encode their bids, performance is very poor amongst this class of strategies. This is in stark contrast with situations that allow bidders to express their full spectrum of private values in a single bid (e.g. see 2.3), where maximum efficiency is attainable. Let g : R k R be the function that a bidder uses to map his value vector into a single bid. A GSP mechanism will be applied to this bid, so it is not effective for a bidder to simply encode their full vector of values into a single number (since the auctioneer cannot decode the single value into a full spectrum of bids). We show that when values are monotone, under mild assumptions on the behavior of g, the worst case loss in efficiency is lower-bounded by a factor k, even when the seller knows the ordering of the slots; i.e., oblivious bidding strategies are not efficient. In fact, we show that this is tight; there are auctions for which this factor is obtained. For i = 1,..., k, define f i (x) = g(x,..., x, 0,..., 0), where x is repeated i times. We will assume in this section that the auctioneer breaks ties deterministically. However, we could alternatively assume that, for all x, there exists x > x such that f i (x ) > f(x), i.e., f is eventually increasing in x for fixed i (this is a very weak restriction on a bidding strategy, and simply says that the reported bid is eventually increasing for a particular form of the value vector). Under this assumption, the proof of Theorem 1 can easily be modified so that there are no ties in reported single-bids, and the lower bound results will apply. Call g block-continuous if the induced functions f i are continuous for all i = 1,.., k. (Note that block-continuity is a much weaker condition that requiring g to be continuous.) Then we have the following theorem. Theorem 1. For every oblivious bidding strategy g that is block-continuous, there are click-monotone [resp., monotone] value vectors such that the efficiency under GSP c [resp., GSP v ] is as small as 1/k of the maximum efficiency. Conversely, there is an oblivious block-continuous bidding strategy g such that the efficiency under GSP c [resp., GSP v ] is at least 1/k of the maximum efficiency, for all click-monotone [resp., monotone] private value vectors. Proof. For the first part of the proof, we consider GSP v. Note that the same argument shows that GSP c can be inefficient as well, by setting clickthrough rates θ 1 =... = θ k = 1. So, suppose we can find x 2,..., x k 1 such that f 2 (x 2 ) = f 1 (1), f 3 (x 3 ) = f 1 (1),..., f k 1 (x k 1 ) = f 1 (1). We will show that if we can find such x i, then the efficiency can be as bad as 1/k of the maximum efficiency; if we cannot find such x i, then we can make the efficiency be a vanishingly small fraction of the optimal efficiency, so that k is still a lower bound. First suppose we can find such x i, i = 2,..., k 1. Let j = arg min i=1,...,k 1 x i. Consider the set of value vectors v 1 = (1, 0..., 0), v 2 = (x 2, x 2, 0..., 0),.. v k = (x k,..., x k, 0,..., 0), v k+1 = v j, 5

6 where v j = (x j,..., x j, 0..., 0) with x j repeated j times. Since the single bids reported by all bidders are the same, a possible allocation is one which assigns bidder k + 1 the first slot (we assumed that vectors are monotone) and bidders 1 through k 1 to slots 2 through k, for a total efficiency x j (since all other bidders except bidder k derive no value from their assignments). So the efficiency from a single bid auction can be as small as x j = min i=1,...,k x i 1 k k x i, where k i=1 x i is the maximum efficiency, obtained by assigning bidder i to slot i. Next we show that if such values cannot be found, then the efficiency can be arbitrarily bad, so the lower bound of k is trivially true. By our assumption that f i (x) is continuous in x, it is sufficient to consider the following two cases: There is an i, 2 i k such that f i+1 (x) < f i (x i ) x: Consider a set of bidders with the following values: i=1 v 1 = v 2 =... = v k 1 = (x i,..., x i, 0,..., 0), v k = (x, x,..., x, 0,..., 0), where x is repeated i + 1 times (and x i is repeated i times for bidders 1 through k 1). Since the maximally efficient allocation assigns some slot between 1 and i + 1 to bidder k for total efficiency at least x, whereas the single-bid allocation does not assign any of the top i + 1 slots to bidder k, since there are at k 1 i + 1 bidders with a higher single bid. Thus the single-bid efficiency is arbitrarily small. There is an i, 2 i k such that f i+1 (x) > f i (x i ) x: Consider a set of bidders with the following values: v 1 =... = v k 1 = (x,..., x, 0,... 0), where x 0 is repeated i + 1 times, and v k = (1, 0,..., 0). The most efficient allocation has an efficiency greater than 1, obtained by assigning bidder k to slot 1, but the single-bid allocation does not assign the first slot to bidder k but rather to some bidder 1,...,, k 1 for a total efficiency of at most kx 0 as x 0. So the single-bid efficiency can be arbitrarily bad here as well. Now, to show that there is a block-continuous bidding strategy whose efficiency is guaranteed to be within factor k of maximum, observe that the following function meets this lower bound for GSP c : g(v 1 i, v 2 i,..., v k i ) = max j=1,...,k vj i /θ j = v 1 i /θ 1 (by monotonicity) This follows since the maximum efficiency for GSP c is bounded by max vi 1 /θ 1 k max g(v i), i=1,...,n S {1,...,n}, S =k i S and the efficiency obtained from using g is at least max i g(v i ). Likewise, we use g(v 1 i,..., vk i ) = v1 i for the GSP v auction. Again, we see that its efficiency is never smaller than 1/k times the maximum possible. 6

7 3.1 Inefficiency with non-monotone values The situation is even worse when the private values are not monotone; we show that it is not possible to make any guarantee on efficiency in this case. Theorem 2. For any oblivious strategy g, there are value vectors such that the single-bid efficiency under single-bid auctions is arbitrarily smaller than the maximum possible efficiency. Proof. We consider the GSP v auction. The GSP c auction is similar. Let v 1 = (0, x), v 2 = (1, 0), v 3 = (1, 0) and 1 x, and consider that there are two slots being auctioned off. The maximum efficiency is 1 + x, obtained by placing either bidder 2 or 3 in the top slot and bidder 1 in the second slot. Case I: g(0, x) > g(1, 0), then GSP v will place bidder 1 in the first slot and one of bidders 2 and 3 in the second slot, for a total efficiency of 0. Case II: g(0, x) < g(1, 0), then GSP v will place bidders 2 and 3 in the first two slots, for a total efficiency of 1. Case III: g(0, x) = g(1, 0), then if GSP v deterministically chooses based on bidder identity, there is an assignment of identities to vectors with efficiency 1. Even if the tie is broken by a random coin flip, we may add an arbitrarily large number of bidders with valuation v i = (1, 0), so that the probability v 1 is assigned to the second slot is arbitrarily small, and the expected efficiency approaches 1. 4 Equilibrium with single bids Having seen that oblivious strategies can be highly inefficient, we now investigate equilibria of single-bid auctions. In this section, we prove the following surprising result: the (full-spectrum) VCG outcome is also an envy-free equilibrium of GSP c [and of GSP v ] so long as the values are strictly click-monotone [respectively, strictly monotone], where an envy-free outcome means an outcome where for every bidder i (numbering bidders according to the slots they are assigned), v i i p i 0, and for every slot j, v i i p i v j i p j, where p j is the (current) price for slot j. That is, bidder i (weakly) prefers slot i at price p i to slot j at price p j, for all j i. Our proof is comprised of two main parts. In the first part, we show that when the values are strictly click-monotone [respectively, strictly monotone], any envy-free outcome is a realizable equilibrium in GSP c [resp., GSP v ]. The second part gives a direct proof that the VCG outcome is always envy-free in auctions that match bidders to at most one item each (this proof holds even when the spectrum of values are not monotone, although we do not need it here). Theorem 3. Any envy-free outcome on k slots and n > k bidders in which prices are nonnegative and values are strictly click-monotone [resp., strictly monotone] is a realizable equilibrium in GSP c [resp., GSP v ]. 1 Proof. We give the proof for the GSP c auction. The proof for the GSP v follows by setting θ j = 1 for all slots j. As above, label bidders so that the envy-free outcome we consider assigns bidder i to slot i, at price p i (note that p i is a price per-impression, or simply the cost of slot i). Further, recall that p i = b i+1 θ i. We first show that p i /θ i p j /θ j for all i < j (which shows that b i+1 b j+1 ). 1 For technical reasons, we say an envy-free outcome is a realizable equilibrium so long as there is a set of bids leading to an equilibrium that agrees with the outcome for every bidder having nonzero value for her slot. We allow bidders that have zero value for their slots to be moved to other slots for which they have zero value (so long as they continue to pay 0 for the new slot). 7

8 Suppose i < j. By definition of an envy-free outcome, ( ) v θ j j j θ j pj θ j v j j p j v i j p i ( v i ) j θ i θ i pi θ i ( v i ) j θ j θ i pi θ i θ j ( pi θ i pj θ j v j j θ j pi θ i ) by click-monotonicity by click-monotonicity Further, we see equality can occur only if v j j = 0. Now, consider the following set of bids: Bidder j bids p j 1 /θ j 1 for all j, where for convenience, we set p 0 > p 1, θ 0 < θ 1, and p j = 0, θ j = 1 for all j > k. From the above argument, we see that the jth largest bid is indeed p j 1 /θ j 1 ; however, there may be ties. Hence, according to GSP c, bidder j is assigned slot j at a price b j+1 θ j = p j, for j = 1,..., k, assuming that the auctioneer breaks ties in the right way. We now appeal to the fact that values are strictly click-monotone to remove this assumption. We now show that bidding ties occur only when both bids are 0, and the bidders have no value for the slots they are tied for. To see this, first note that if v j j = 0, then p j = 0, by the envy-free condition, hence p j /θ j = 0. Hence, if i < j and p i /θ i = p j /θ j, it must be the case that v j j = 0, implying p i/θ i = p j /θ j = 0. That is, bids are tied only when both bids are 0. Now, suppose bidder j bids 0. We will show that v j j = 0. To this end, notice that bidder j bids p j 1 /θ j 1 = 0, which implies p j /θ j = 0, from the above argument. So p j 1 = p j = 0. By the envyfree condition, v j 1 j p j 1 v j j p j, hence v j 1 j v j j. But this violates strict click-monotonicity, unless v j 1 j = v j j = 0. Hence, bidder j has value 0 for slot j. Putting this together, we see that the auctioneer only breaks ties between bidders that bid 0, and have no value for any slot they are tied for. (Hence, these ties may be broken arbitrarily and still satisfy our goal.) We finish by showing that this envy-free outcome is indeed an equilibrium. Suppose bidder i bids lower, so she gets slot j > i rather than i. She then pays p j, and by definition of envy-free, does not increase her utility. On the other hand, if bidder i bids higher, getting slot j < i, then she pays p j 1, with utility v j i p j 1 v j i p j vi i p i, since prices decrease with slot rankings. Therefore, this set of bids leads to an envy-free equilibrium in GSP c. Next we show that the VCG outcome is envy-free in auctions that match each bidder to at most one item, even when the values are not monotone. We do not need clickthrough rates here at all, since the VCG price is simply a price per slot, or per-impression. We first prove a general property of maximum weight matchings on bipartite graphs. While this lemma is superficially similar to Lemma 2 in [1], we note that the proof is quite different: the proof in [1] crucially uses the fact that an advertiser s value in a slot is the product of a value-per-click and a clickthrough rate which decreases with slot number, and simply does not work in our setting. Lemma 1. Let G, M, M i, and M j be as above. Then if advertiser j is assigned to slot j in a maximum matching,m j M i + v j i vj j. Proof. Fix a maximum matching on G, say M, and without loss of generality, relabel the advertisers so that advertiser i is assigned to slot i for each i = 1,..., k in M. The proof of the lemma proceeds by taking a maximum matching of G with i removed, and using it to produce a matching of G with j removed. This new matching will have weight at least M i + v j i vj j, showing that M j must also be at least this large. Fix a maximum matching of G with i removed, call it M i. If there is more than one such maximum matching, we will take M i to be one in which advertiser j is matched to slot j, if such a maximum matching exists. Either advertiser j is matched to slot j in M i or not. We consider each case in turn. 8

10 5 Bidding strategies Now, we consider the following question: while the VCG outcome is indeed realizable as an efficient equilibrium of single-bid auctions, is there a natural bidding strategy for bidders, such that when all bidders bid according to this strategy in repeated auctions for the same keyword (so that the private values do not change), they converge to this efficient outcome? We define an history-independent bidding strategy, g, to be a set of functions g 1,..., g n, one for each bidder i, where g i takes the bids b from the previous round of the auction, together with the vector of private values v i, and outputs a nonnegative real number, which is the bid of player i 2 We call the history-independent strategy myopic if the value of g i does not depend on the bid of player i from last round. Finally, we say a vector of bids b is a fixed point for history-independent strategy g, if after bidding b, the bidding strategy continues to output that same vector of bids b, i.e., g i ( b; v i ) = b. Throughout this section, we will focus primarily on GSP v for clarity, although the results also hold for GSP c. First, we show that for any bidding strategy that always has an envy-free fixed point, there are value vectors where bidders must bid more their maximum value over all slots. This puts them at risk to lose money (i.e., have negative utility) if other bidders change their bids, or new bidders join the auction. We call a strategy safe if it never require bidders to bid beyond their maximum value. Although it would not be surprising to see bidders occasionally bid values that are not safe, it seems questionable whether they would continue to bid this way repeatedly. Theorem 6. There is no safe, history-independent bidding strategy that always has an envy-free fixed point whenever the private values are strictly monotonic. Proof. Consider the following auction (GSP v ), in which three bidders vie for two slots: Bidder 1 values slot 1 at 20 and slot 2 at 0; bidder 2 values slot 1 at 15 and slot 2 at 14; bidder 3 values slot 1 at 19 and slot 2 at 0. Let us examine an envy-free fixed point, showing that bidder 2 must bid above her maximum value (i.e., 15). If bidder 1 does not get slot 1, then clearly she will be envious (unless the cost is greater than 20, in which case the assignment cannot be envy-free). If bidder 1 does not pay at least 19 for slot 1, then bidder 3 will be envious. Further, slot 2 must be assigned to bidder 2, for otherwise she will be envious (or someone will have to pay more than their value for slot 2, violating the envy-free condition). Hence, bidder 2 must bid at least 19, showing that she must bid above her maximum value. We further show that no bidding strategy has the VCG outcome as a fixed point, under a few minor assumptions. These assumptions stem from our need to stop bidders from encoding information in their bids, essentially artificially allowing the players to calculate the fixed point and bidding accordingly. As such, recall that in the VCG outcome, when the values are strictly monotone, all non-zero bids are distinct, and determined. Further, the highest bidder is free to bid any value larger than some quantity (determined by the VCG prices). Here, we refer to the VCG bids as the set of bids supporting the VCG outcome in which the highest bidder bids her utility for the top slot. Then we have the following: Theorem 7. There is no myopic bidding strategy that always attains the VCG outcome as a fixed point, as defined above, even if that VCG outcome does not require bidders to bid above their maximum value over all slots. Proof. First consider the following auction (GSP v ), in which three bidders vie for two slots: Bidder 1 values slot 1 at 30 and slot 2 at 16; bidder 2 values slot 1 at 15 and slot 2 at 14; bidder 3 values slot 1 at 13 and slot 2 at 10. The VCG outcome of this auction has bidder 1 bid 30, bidder 2 bid 13, and bidder 3 bid 10. Now, suppose bidder 3 instead valued slot 1 at 12 and slot 2 at 10. Then the VCG outcome of this auction has bidder 1 bid 30, bidder 2 bid 12, and bidder 3 bid 10. So they cannot both be fixed points of the myopic strategy, since the input to g 2 is the same in both cases bidder 1 bids 30 and bidder 3 bids 10, and private values are slot 1 at 15 and slot 2 at 14 (the bid of bidder 2 is ignored). 2 Notice that our definition actually allows a kind of collaboration between bidders, despite the fact that in practice, we do not expect this to happen. 10

11 In [2], the authors show that under the model of a single value-per-click for all slots as in [4, 7], there is a myopic strategy, based on envy-free bidding, that does indeed have the VCG outcome as a fixed point. The result above shows that this does not work in our more general setting. We might wonder whether we can prove even stronger results on bidding strategy. Unfortunately, there is an unnatural bidding strategy that converges to the VCG outcome. Specifically, suppose that the private vector values in a GSP c [resp., GSP v ] auction are strictly click-monotone [resp., strictly monotone]. Then there is an history-independent bidding strategy that converges to the VCG outcome. The proof of this, which relies heavily on the ability to encode private values in bids, can be found in the full version of this paper. The strategy allows bidders to encode their private values in their bids, which are then used by the other players to calculate the VCG outcome and bid accordingly. This is unnatural for (at least) two reasons: bidders should not be able to encode much information in a bid, and bidders will not cooperate with each other consistently. The two negative results we give above use the fact that bidders are restricted in their ability to artificially encode information. We leave open the question of whether we can prove similar negative results by restricting the cooperation between players. 6 The monotonicity assumption Throughout this paper, we show a number of results that rely on the private values to be monotone, or clickmonotone. Although previous work has (implicitly) assumed click-monotonicity [4, 7, 1], this assumption may not always be true. Here, we give several examples demonstrating the problems that can occur with non-monotone values (in addition to 3.1, which shows the inefficiency that can occur with oblivious bidding). Instability in GSP v. We first give a simple, but compelling, reason for assuming monotone private values. Put simply, the auction simply does not work in general without this assumption: the auctioneer must pick some ordering of the slots to assign bidders in decreasing order of bid, but this is not meaningful when all bidders do not agree on the ordering. For example, suppose there are just three slots, and bidders 1,2,3 value slot 1 at 100,90,80, respectively, and slots 2,3 at 0. Further, suppose bidders 4,5,6 value slot 3 at 100,90,80, respectively, and slots 1,2 at 0. Let us assume that the auctioneer orders slots so that the highest bidder gets slot 1, the second highest gets slot 2, and the third gets slot 3. Bidder 1 can bid high in order to obtain slot 1. Everyone else has no incentive but to bid 0: bidders 2,3 do not want slots 2 or 3, so will not pay; bidders 4,5,6 want slot 3, but bidding above 0 puts them into slot 2, which they do not want. Instability in GSP c. Unfortunately, when private values for slots are monotone, but not click-monotone, there are situations in which GSP c has no equilibrium outcome at all. This is not surprising given that bidder s values are not really per-click values: the intuition behind our counterexample is that when the value for a click is higher in slots with lower clickthrough rate, then the most desirable slot is not the slot for which the most money is charged. Therefore, everyone will vie for this slot, and we cannot reach an equilibrium because the auction mechanism does not allow us to charge more for the more desirable item. Theorem 8. GSP c does not always have an equilibrium, even when the private values are strictly monotone. Proof. We prove the theorem by giving an example for which there is no equilibrium outcome under GSP c. The bidder valuations are given in the table below. We assume clickthrough rates of θ 1 = 1 and θ 2 = 0.1, which are bidder independent. Slot 1 Slot 2 Clickthrough rate Bidder A 5 1 Bidder B 5 1 Bidder C

13 References [1] G. Aggarwal, J. Feldman, and S. Muthukrishnan. Bidding to the top: VCG and equilibria of positionbased auctions. Workshop on Approximation and Online Algorithms, [2] M. Carry, A. Das, B. Edelman, I. Giotis, K. Heimerl, A. Karlin, C. Mathieu, and M. Schwarz. Greedy bidding strategies for keyword auctions. ACM Conference on Electronic Commerce, [3] E. H. Clarke. Multipart pricing of public goods. Public Choice, 11:17-33, [4] B. Edelman, M. Ostrovsky, and M. Schwarz. Internet advertising and the generalizaed second price auction: Selling billions of dollars worth of keywords. Second Workshop on Sponsored Search Auctions, Ann Arbor, MI. June, [5] T. Groves. Incentives in teams. Econometrica, 41: , [6] H. Leonard. Elicitation of honest preferences for the assignment of individuals to positions. the Journal of Political Economy, 91(3): , [7] H. R. Varian. Position auctions. To appear in International Journal of Industrial Organization, [8] W. Vickrey. Counterspeculation, auctions, and competitive sealed tenders. Journal of Finance, 16:8-37,

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