# Expressive Auctions for Externalities in Online Advertising

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4 whether the outcome will be S or M, as well as the pricing for S. The GSP 2D auction is defined below. Definition 3.1 (The GSP 2D auction). The mechanism GSP 2D takes as input bids (b i, b i) and compares b max to k+1 θi 1b i to decide whether the outcome should be S or M. If b max k+1 θi 1b i, the outcome is S with winning bidder max, whose payment is k+1 θ i 1b i per click. If b max k+1 θ i 1b i, assign the page to bidders 1,..., k and charge them according to GSP M pricing, i.e. bidder i (for i k) has to pay b i+1 per click. Note that the allocation rule compares against k+1 θ i 1b i, rather than against k θ ib i, i.e., the highest M-bid is completely ignored when deciding the outcome. This is because the natural allocation rule, which would be to compare b max with θ 1b 1 + θ 2b θ k b k, does not quite work: if the bidder with the highest M-value is different from the bidder max with the highest S-value, that bidder will always set b 1 = b max ɛ which changes the outcome to M at no cost to her (as long as there is some other non-zero bid b i), since the pricing when the outcome is M according to GSP remains b 2. That is, using the natural allocation rule would imply that the only possible equilibria are those with outcome M, defeating the purpose of designing a more expressive auction. In the remainder of this section, we will investigate the efficiency and revenue of the equilibria of this mechanism. The restriction to using GSP M when the outcome is M does cause a potential loss in efficiency and revenue with respect to V CG 2D, unlike the case with one-dimensional valuations where all envy free equilibria of GSP M are efficient and dominate V CG M in terms of revenue. However, as we show below, GSP 2D has fairly nice properties nonetheless: both the efficiency and revenue of all reasonable equilibria of GSP 2D are guaranteed to be at least within a factor 1/3 and 1/2 respectively of the optimal efficiency and revenue. (By reasonable equilibria, we mean equilibria of the mechanism where losers bid at least their true value; we show such equilibria always exist. Further, when the outcome is M, we will restrict ourselves, as in [20] and [8], to envy free equilibria, since the efficiency and revenue guarantees for GSP M relative to V CG 1D themselves hold only for envy-free equilibria of GSP M.) The easy lemma below, which follows immediately from individual rationality, will be used repeatedly in the following two subsections. Lemma 3.1. In any equilibrium of GSP 2D with outcome M, b i+1 v i for every i k. 3.1 Efficiency We consider two cases, one where the efficient outcome is S, and the other where the efficient outcome is M, and analyze the efficiency of the equilibria of GSP 2D. Note that we prove our efficiency results for all reasonable equilibria, rather than only showing that there exists one equilibrium with these properties. Due to want of space, the proofs of the following results are omitted, and can be found in the full version of the paper [10]. Theorem 3.1. If the efficient outcome is S (v max > k θ iv [i]), there is no equilibrium of GSP 2D with outcome M. Proposition 3.1. Suppose the efficient outcome is M. Every equilibrium of GSP 2D with outcome S where losers bid at least their true values has efficiency at least 1/3 of the optimal. Any envy-free equilibrium with outcome M is efficient. 3.2 Revenue In this section, we compare the revenues of equilibria in GSP 2D with the revenue of V CG 2D. Theorem 3.2. Suppose the efficient outcome is S. The revenue in any equilibrium of GSP 2D where losers bid at least their true values is at least half of the revenue of V CG 2D. Proof. First, recall that the only possible equilibrium outcome is S, so that the revenue of GSP 2D is max(b max 2, k θ ib i+1). We give lower-bounds for both terms and then show that the revenue of V CG 2D cannot be larger than the sum of the lower bounds; therefore, the revenue of V CG 2D cannot be more than twice of the revenue of GSP 2D. First we assume max [1]. We have b max 2 v max 2 v [1] θ 1v [1], since v i v i, and losers bid at least their true values. For the other term, we know that all bidders except max are losers in outcome S. Therefore, b i v i for every i max. So we get θ i b i+1 θ i 1 v [i] + k+2 +1 θ i 2 v [i] where j = min(max, k + 2). On the other hand, the revenue of V CG 2D is l 1 max(v max 2, θ i v [i] + k+1 θ i 1 v [i]) where l = min(max, k + 1). To finish the proof, we need to show that the sum of the lower bounds we have for GSP 2D is greater than or equal to revenue of V CG 2D : v max 2 + θ i 1 v [i] + l 1 max(v max 2, θ i v [i] + k+2 +1 k+1 θ i 2 v [i] θ i 1 v [i]). If the first term in the V CG 2D revenue is the dominant term, the inequality obviously holds. Otherwise, we need to show that v max 2 + θ i 1 v [i]+ k+2 +1 i.e., it is enough to show that θ 1v [1] + θ i 1v [i] + l 1 θ i 2 v [i] θ i v [i]+ k+2 +1 l 1 θ 1v [1] + θ iv [i] + k+1 θ i 2v [i] θ i 1v [i], k+1 θ i 1 v [i],

5 but this inequality clearly holds using term-by-term comparison. It remains to prove the the theorem for the case where max = [1]. In this case, b max 2 v max 2 v [2], and the revenue of GSP 2D is max(b max 2, θ i b i+1) max(b max 2, θ i v [i+2]) while the revenue of V CG 2D is max(v max 2, k θ iv [i+1]). As before, if the dominant term in revenue of V CG 2D is v max 2 we are done. Otherwise, the sum of the two terms of the GSP 2D revenue is at least θ 1v [2] + k θiv [i+2] which dominates the second term in the V CG 2D revenue term by term. A simple modification to Example 3.1 at the end this section shows that this factor of 2 is tight (set v 1 = 3 so that the efficient outcome is S). The following additive bound on revenue follows immediately from the previous proof: R GSP2D R V CG2D θ 1 v [1] + θ k v [k+2]. Theorem 3.3. Suppose the efficient outcome is M. Any envy-free equilibrium of GSP 2D with outcome M has revenue greater than or equal to that of V CG 2D. Proof. First, note that since the equilibrium is envyfree, the ordering of M-bids is the same as ordering of M- values ([20]), i.e., v i = v [i] for any i k + 1. We show that the payment of advertiser i (for i k) in GSP 2D is at least as much as his payment in V CG 2D. Recall from 2.1 that the payment for advertiser i in V CG 2D is p i = max( (θ j θ j+1)v j+1, v max i j=i θ jv j). First we prove that GSP 2D payment of bidder i, θ ib i+1, is at least v max i k j i θ jv j. We will prove this by contradiction: if not, we show that there is a bidder with a profitable deviation to S. Let l be the bidder with the highest S-value excluding i, i.e. v l = v max i. By the contradiction hypothesis, θ ib i+1 < v l k j i θjv j. If l is not a winner, he has a profitable deviation by bidding (v l, b l) which changes the outcome to S because v l > k j i θ jv j + θ i b i+1 k θ jb j+1. (Of course, bidding (v l, 0) is a more profitable deviation, but is unnecessary for the argument.) So suppose that l is a winner. Adding and subtracting θ l b l+1 and rearranging we get θ l (v l b l+1) < v l (θ i b i+1 + θ l b l+1 + j i θ j v j). j i,l Note that the term in parentheses on the right hand side is an upper-bound on the price that l has to pay for S if he deviates and bids (v l, b l): the price for S is at most k θ jb j+1 (since the outcome with the original vector of bids was M, b max k θ jb j+1, so the price for S is always dominated by this term). Since b j+1 v j (the original vector of bids was in equilibrium), the price for S is upper-bounded by (θ i b i+1 + θ l b l+1 + k j i,l θ jv j) as claimed, showing that l can deviate profitably. (Note that as before, this bid does change the outcome to S.) The fact that θ ib i+1 k j=i (θj θj+1)v j+1 follows from the lower bound on bids in envy-free equilibria in [20], which also holds for envy-free equilibria in outcome M of GSP 2D. Theorem 3.4. Suppose the efficient outcome is M. The revenue in any equilibrium of GSP 2D with outcome S where losers bid at least their true values is at least half of the revenue of V CG 2D. Proof. The proof, unfortunately, proceeds by considering cases. The revenue of GSP 2D is max(b max 2, k θib i+1). For ease of notation let p 1 i = k j=i (θ j θ j+1 )v[j+1], and p 2 i = v max i k j i θ jv [j]. The revenue of V CG 2D is k p i, where p i = max(p 1 i, p 2 i ). First note that p 1 i θ iv [i+1]. Also, from individual rationality we have p i θ i v [i]. We consider the following three cases, and will prove for each case that b max 2 + k θ ib i+1 k p i. Therefore, the revenue of GSP 2D is at least half the revenue of V CG 2D. 1. If max / {[1],..., [k]}: Each bid b i, i k + 1, is at least v i in this case, so the revenue of GSP 2D is at least max(v max 2, k θiv [i+1]). The revenue of V CG 2D is at most p 1 + k θiv [i]. Since v max 2 v [1] v [1] θ 1 v [1], we have v max 2 p 1, and hence v max 2 + θ iv [i+1] p 1 + θ iv [i], which implies that the revenue of GSP 2D is at least half the revenue of V CG 2D. 2. If max {[2],..., [k]}: The revenue of GSP 2D in this case is at least max(v max 2, max 2 θ j v [j+1] + k j=max 1 θjv [j+2]) because all losers bid at least their true values. We first consider the case where p 1 i p 2 i for every i. The revenue of V CG 2D cannot be more than k p1 j k θ jv [j+1]. Since v max 2 v [1] v [max] θ max v [max], max 2 v max 2+ θ j v [j+1]+ j=max 1 θ j v [j+2] θ j v [j+1], which shows the revenue of V CG 2D cannot be more than twice the revenue of GSP 2D in this case. For the other case, let l be some index for which p 1 l < p 2 l. We consider two cases depending on whether p 1 max > p 2 max or p 2 max p 1 max. For both cases, we upper-bound the V CG 2D payment of bidder i (for i max and i l) by θ i v [i]. First, if p 2 max p 1 max, the revenue of V CG 2D is at most p 2 l + p 2 max + θ j v [j] = v max θ j v [j] j max,j l + v max 2 = j max ( v max θ jv [j] + j l j max,j l θ jv [j] ) θ j v [j] + v max 2.

7 4. NP 2D : A NEXT PRICE AUCTION The current GSP auction, GSP M, is a next price auction every winner pays the next price, i.e., the minimum bid necessary in order to maintain her position, which in GSP M is the bid of the next highest bidder. In our twodimensional setting, where there are two types of outcomes in addition to multiple slots, maintaining one s position consists of two things for a winner in outcome M: first, the outcome must remain M and not switch to S; second, the bid must enable the bidder to maintain her position amongst the k slots. In a next price auction for our more expressive setting, therefore, the payment of a winner in slot i of outcome M is the larger of two terms the first being the minimum value at which the outcome still remains M, and the second being the bid of the next bidder, b i+1, as in GSP M. The auction is formally defined below. Definition 4.1 (The NP 2D auction). Bidders submit bids (b i, b i). Assume max = j, i.e., the bidder corresponding to b max has the jth largest M-bid, and let Γ = k θ ib i. If b max Γ, the outcome is S with payment max(b max 2, θ i b i + θ i b i+1). If b max Γ, the outcome is M and the bidder winning slot i max pays θ i p i = max(θ i b i+1, b max Γ + θ i b i) while the bidder max winning slot j pays θ j p j = max(θ j b j+1, b max 2 Γ + θ j b j) Note that in computing the price for outcome S, the second term is smaller than Γ. In the next two subsections, we will analyze the efficiency and revenue respectively in the equilibria of NP 2D. As before, we will prove guarantees for the revenue and efficiency of good equilibria, where losers bid at least their true value (such equilibria always exist, as we show in Proposition 4.1). Some proofs have been removed for want of space, and can be found in the full version of the paper [10]. 4.1 Efficiency As before, we consider two cases corresponding to the efficient outcome being S or M. We first start with the following lemma, which allows us to prove the efficiency result for S. Lemma 4.1. Assume that bidder max is bidding truthfully. If the outcome of NP 2D for a given vector of bids is S, then the winner max cannot benefit from any deviation that changes the outcome to M. Proof. Assume max = j, i.e., the bidder max has the jth largest M-bid for the given M-bids b i from the remaining bidders. By assumption that max bids truthfully, b j = v j and b max = v max. We need to show that bidder max = j prefers outcome S to any position in outcome M. Consider an M-bid b of bidder j with b l b < b l 1, i.e., targeting slot l, and assume the deviation changes the outcome to M. First notice that if b > b, outcome M gives bidder j negative utility because her payment would be at least b > b j in this case. So without loss of generality we may assume b b, i.e., l j. We have to show v max max(b max 2, θ i b i + θ i b i+1) θ l v j max(θ l b l+1, b max 2 Γ + θ l b ). We know b j b i for (i j), therefore, l 1 l 1 (θ j θ l )b j = (θ i θ i+1 )b j (θ i θ i+1 )b i+1 and since b j = v j we can write By adding l 1 θ j b j (θ i θ i+1 )b i+1 θ l v j 0. l 1 θ i b i + θ i b i+1 + θ l v j + to both sides of the inequality we get l 1 θ i b i θ i b i + θ i b i+1 + θ l b j + θ i b i θ i b i. Since the outcome is S, we have v max k θib i, and therefore, using the inequality above, l 1 v max θ i b i + θ i b i+1 + θ l v j + θ i b i. (1) We consider two cases, based on whether the dominant term for the price of max in outcome S is b max 2 or not. 1. Assume the dominant term is not b max 2, i.e., max(b max 2, θib i + k θib i+1) = θib i + k θ ib i+1. Then, by inequality (1), and since b i+1 b i for any i, we have l 1 v max θ ib i + θ ib i+1 + θ l v j + θ ib i+1 and, by adding and subtracting θ l b l+1 to the right hand side of the inequality we get l v max θ ib i+ θ ib i+1+θ l (v j b l+1)+ The final inequality can be written as v max ( θ i b i + which implies θ i b i+1) θ l (v j b l+1) v max max(b max 2, θ i b i + θ i b i+1) θ l v j max(θ l b l+1, b max 2 Γ + θ l b ). θ ib i+1.

8 2. Assume max(b max 2, θ ib i + k θ ib i+1) = b max 2. Since after deviation, l 1 Γ = θ i b i + θ i b i+1 + θ l b + θ i b i, we can rewrite inequality (1) as v max θ l v j + Γ θ l b. Now, by just subtracting b max 2 from both sides we get which implies v max b max 2 θ l v j b max 2 + Γ θ l b v max max(b max 2, θ i b i + θ i b i+1) θ l v j max(θ l b l+1, b max 2 Γ + θ l b ). This allows for an easy proof of the following result: Theorem 4.1. Suppose the underlying valuations are such that the efficient outcome is S, i.e., v max > k θ iv [i]. There exists an equilibrium with outcome S where losers bid at least their true values. Further, there is no inefficient equilibrium where all bidders play undominated strategies. Unlike in GSP 2D, inefficient equilibria with outcome M (with arbitrarily large inefficiency) can occur in NP 2D when the efficient outcome is S. However, all such equilibria are bullying equilibria (such equilibria occur in GSP M as well) where some bidder bids above her true value, which (Lemma 7.1 in the full version of the paper) is a weakly dominated strategy in NP 2D. Next, suppose the efficient outcome is M. Here, similar to GSP 2D, inefficiency can occur in NP 2D as well; however, the extent of inefficiency is less than that in GSP 2D, as the following multiplicative and additive bounds show. Theorem 4.2. Suppose the efficient outcome is M. Then the efficiency in any good equilibrium of NP 2D with outcome S is at least 1/2 of the optimal efficiency. Corollary 4.1. Suppose the efficient outcome is M. Then the welfare in any good equilibrium of NP 2D with outcome S is at least OP T k θi(v [i] v [i+1]), where OP T is the optimal welfare and j is the rank of M-value of bidder max, i.e. max = [j]. Note that if the marketplace is competitive, i.e., the v i s are not very different, the additive bound shows that the loss in welfare will be small even when the inefficient outcome occurs. Finally, suppose the efficient outcome is M and the equilibrium outcome is M as well. A result similar to that [8, 20] stating that all envy free equilibria are efficient holds for NP 2D as well. However, before we state the result, we need to extend the notion of envy free equilibria to NP 2D; we will then prove, via Lemma 4.2, that the set of envy-free equilibria are all efficient. The notion of envy free equilibria in [8] can be thought of as restricting the set of bid vectors that are Nash equilibria to those that also generate envy free prices [13], i.e., a price for each slot such that no bidder envies the allocation of another bidder at this price. We use exactly this idea to define envy free equilibria for outcome M in the NP 2D auction: A vector of bids leading to outcome M in NP 2D is an envy free equilibrium if for any i and j (1 i, j n) θ i(v i p i) θ j(v i p j) where p i and p j are the prices bidders i and j are currently paying for slots i and j respectively. (Recall that θ i = 0 for i > k.) Lemma 4.2. If v a > v b for bidders a and b, and θ p > θ q for slots p and q, then any allocation A that assigns bidder a to slot q and bidder b to slot p is not envy-free. Proof. Assume for sake of contradiction that A is envyfree and suppose that the prices for slots p and q are p p and p q respectively. For A being envy-free we must have θ p (v a p p ) θ q (v a p q ) and θ p (v b p p ) θ q (v b p q ). Subtracting the second inequality from the first one we get θ p (v a v b) θ q (v a v b), but this last inequality contradicts v a > v b and θ q < θ p. The theorem below follows immediately from this lemma, since it implies that in any envy-free allocation (not necessarily even an equilibrium) of NP 2D with outcome M, the bidders must be allocated to the slots in decreasing order of their M-values. Theorem 4.3. Suppose the efficient outcome is M. Then any envy free equilibrium of NP 2D with outcome M is efficient. 4.2 Revenue When the equilibrium outcome in NP 2D is S, the revenue is high, in the following sense: Theorem 4.4. Any good equilibrium of NP 2D with outcome S has at least the same revenue as V CG 2D. Note that here NP 2D does better than GSP 2D : the revenue of GSP 2D can be as small as half the V CG 2D revenue when the equilibrium outcome is S and the efficient outcome is either S or M (Theorems 3.2 and 3.4). However, while NP 2D leads to better revenue guarantees when the outcome is S, the same is not true for M: unlike GSP 2D, where every envy free equilibrium with outcome M revenuedominates V CG 2D, the revenue in an envy free equilibrium of NP 2D can be arbitrarily smaller than that of V CG 2D, as the following example shows. Example 4.1. Suppose that there are two slots with θ 1 = 1 and θ 2 = 1 ɛ/3. There are two bidders with values (v 1, v 1) = (3, 3) and (v 2, v 2) = (4, 2). The efficient outcome is M. If the bidders bid (b 1, b 1) = (3, 3) and (b 2, b 2) = (2ɛ, ɛ), the outcome is M; it is an envy-free equilibrium; and the revenue is ɛ. However, revenue of V CG 2D on this example is 2. That is, we cannot obtain a result similar to the previous revenue results bounding the revenue loss with respect to V CG 2D by a multiplicative constant. However, as the next three theorems will show, the situation is not quite as bleak as the previous example might suggest: first, Theorem 4.5 shows that the revenue of NP 2D in any envy-free equilibrium with outcome M is at least the V CG M revenue. Second, and more importantly, Proposition 4.1 shows that there always exists an equilibrium of

9 NP 2D with this revenue note that this is not the case with GSP 2D, where there exist values such that every equilibrium of GSP 2D has revenue strictly less than the revenue of V CG M (Example 3.1). The revenue comparison with V CG M is important for the following reason the V CG M revenue can be thought of as a proxy for the GSP M revenue, since there always exists an equilibrium of GSP M with this revenue [8], and further, this is a likely equilibrium in the sense that if bidders update their bids according to reasonable greedy bidding strategies, the bids converge to this equilibrium of GSP M [6]. Therefore, unlike GSP 2D, there always exists an equilibrium of NP 2D with revenue at least as much as in GSP M, i.e., the transition to the richer outcome space does not lead to revenue loss. Finally, Theorem 4.6 shows that the NP 2D auction also retains all the high revenue equilibria with outcome M of GSP 2D : the reason for the nonexistence of a multiplicative bound with respect to V CG 2D is simply that NP 2D has a larger set of equilibria, some of which have poor revenue; however, no high revenue M-equilibria of GSP 2D are lost in using the NP 2D auction. Theorem 4.5. Every envy-free outcome (not necessarily an equilibrium) of NP 2D with outcome M has revenue at least as much as V CG M. We point out that the proof of this result is independent of how payments p i are calculated. In fact, any allocation and pricing (even not restricted to our two-dimensional setting) which is envy-free and efficient satisfies the conditions needed for the above proof, and hence has revenue at least as much as V CG M. Proposition 4.1. There always exists a good equilibrium of NP 2D with revenue greater than or equal to that of V CG M. The revenue of the equilibria constructed in Proposition 4.1 are at least k θ iv [i+1]. Therefore, assuming bidders do not play weakly dominated strategies in GSP M (specifically, bidders do not overstate their values), no equilibrium of GSP M can have revenue higher than the equilibrium of NP 2D constructed in Proposition 4.1. Finally, we show that NP 2D retains all the high revenue M-equilibria of GSP 2D. Theorem 4.6. Every equilibrium of GSP 2D with outcome M is an equilibrium with outcome M of NP 2D with equal revenue. That is, while there is no multiplicative bound on the revenue of an envy-free equilibrium of NP 2D with outcome M, all the high revenue M-equilibria of GSP 2D, which dominate the V CG 2D revenue, are also equilibria of NP 2D Revenue Non-monotonicity We point out an interesting property of the NP 2D auction: when bids go up, the revenue can actually decrease. The following example illustrates this non-monotonicity in revenue as a function of the bids: Example 4.2. Suppose there are two slots with θ 1 = θ 2 = 1 and three bidders with bids (10, 0), (9, 9) and (2, 2). The outcome is M, and the prices for bidders 2 and 3 are 8 and 1 respectively, so the revenue is 9. However, if the third bidder increases her bid to (3, 3), the outcome remains M, but the payments change to 7 and 1 for bidder 2 and 3 respectively. Therefore, the revenue decreases to 8. When the bids are such that b max = θ ib i, the total revenue is exactly θ ib i irrespective of which outcome is chosen. If the tie is broken in favor of outcome M, every bidder must pay exactly his bid, since for any bid below this the outcome will switch to S. Now suppose all bidders bid b i + ɛ (and don t change their S- bids). The outcome remains M, but every bidders payment decreases, since the minimum amount needed to maintain outcome M given the other bids has decreased. So the revenue decreases even though the bids increase. Note that this revenue non-monotonicity occurs in the VCG auction as well, for the same reason. However, GSP 2D does not have this property in the sense that when the bids increase the total revenue cannot decrease. Revenue monotonicity is often considered a desirable property in practice, and could influence the choice between which of the two auctions, GSP 2D or NP 2D is actually used in practice. 5. DISCUSSION In this paper, we designed two expressive GSP-like auctions for exclusivity-based valuations, and showed that they have good revenue and efficiency properties in equilibria. While the NP 2D auction has, roughly speaking, better worst-case efficiency properties, and does better than GSP 2D in several cases in terms of revenue, it does have envy-free equilibria with poor revenues, and shares VCG s revenue non-monotonicity problem. Choosing between the two auctions will require an empirical assessment of the marketplace parameters as well as an understanding of bidder valuations and behavior, to predict which equilibria are actually likely to arise in practice. (Note, as an aside, that there is no way to infer S-values from the current GSP M auction, since multiple ads are always shown.) There are many directions for further work. The first obvious direction is to combine the more expressive bidding language with more complex cascade-like models for CTRs, and analyze a model that incorporates both attention and conversion based externalities. Second, the bidding language we use can be thought of as a one way to succinctly represent a general decreasing k-dimensional vector valuation, where an advertiser specifies the first entry in the vector (v i), and uses v i as a proxy for all the remaining k 1 entries. An alternative language (also discussed in [16]) which also solicits only two bids from an advertiser, is one where an advertiser specifies that he has value v i provided no more than n i advertisers are shown in all (and zero if any more are shown). Which of these is a better representation of actual advertiser valuations, and which is it possible to design better mechanisms for? In general, the question of designing succinct mechanisms that achieve high efficiency in the presence of underlying high-dimensional valuations is an interesting open question. While the auctions we design have pleasant properties with respect to revenue and efficiency in their equilibria, the comparison of these GSP-like auctions to VCG does not remain quite as starkly positive as in the original onedimensional setting. Specifically, unlike [8, 20], where all envy-free equilibria of GSP M are efficient and have at least the revenue of V CG M, both the NP 2D and GSP 2D auctions

11 utility, or will stay in her current place for the same price. Therefore, she cannot benefit from overstating her value in this case. Now, suppose the outcome is S. In this case, if bidder i changes the outcome to M by overstating her value, she has to pay more than her value because of the pricing policy. Therefore, she cannot benefit from overstating her M-value in this case either. Proof (Theorem 3.1). Assume for the sake of contradiction that (b, b ) is an equilibrium vector of bids in GSP 2D with outcome M and recall that indices 1,..., n are such that b 1... b n. First, note that by Lemma 3.1, since (b, b ) is an equilibrium, v max > k θiv i k θib i+1. Therefore, if the bidder with highest S value, max, bids his true value, the outcome will change to S. Consequently, max must be in {1,..., k}, since otherwise, he has utility 0 in the outcome M while he can make his utility positive by bidding truthfully. Next, we show that even if max {1,..., k}, he still has an incentive to deviate and bid truthfully to change the outcome to S. More precisely, we show that the (following lower bound on the) payoff of max in outcome S is more than his payoff in outcome M: v max (θ 1 b θ k b k+1) > θ j (v max b j+1), where index j is such that j = max. That is, we want to show v j (θ 1b θ k b k+1) > θ j(v j b j+1). Rearranging, and using the fact that max {1,..., k}, that is, j k, it suffices to prove: v j > (θ 1 b θ b j + θ j v j + θ j+1 b j θ k b k+1). To show this, we start with: v max > θ 1v [1] θ k v [k]. Since v [1]... v [n], and θ i s are decreasing, we get: v max > θ 1 v θ k v k. By Lemma 3.1, we can replace v i by b i+1 for all i j, which gives us v j > (θ 1 b θ b j + θ j v j + θ j+1 b j θ k b k+1), which is what we needed to show. Proof (Proposition 3.1). For brevity and clarity, we describe the proof when bidder max is bidder [1], i.e., has the highest M-value (the proof for [j] is very similar). Since the outcome of GSP 2D is S, she must prefer outcome S to every slot; specifically, to the first slot in outcome M. Therefore, v max θ ib i+1 θ 1(v [1] b 2). Since losers bid at least their true values b i v [i+1], so we get v max θ 1v [1] + θ iv [i+2]. Our goal is to show 3v max k θ iv [i] which follows from the above inequality because v max θ 1v [1], v max θ 2v [2], and θ 1v [1] + k θiv [i+2] k i=3 θiv [i]. The efficiency of an envy-free equilibrium of GSP 2D follows directly from the arguments in [20]. Proof (Theorem 4.1). We show that the vector of bids where all bidders except max = [j] bid truthfully and max bids (v max, 0), is an equilibrium of NP 2D. First, note that these bids lead to outcome S because v max > θ iv [i] θ iv [i] + θ iv [i+1] = Γ. Next, we show that no bidder i max has an incentive to deviate: If loser i wants to change the outcome to M by increasing her current bid to b i > v i, we show that her payment becomes larger than her value. By the pricing rule we have θ ip i v max l i θ lb l; furthermore, since the initial outcome is S we have v max l i θ lb l > θ i v i. Therefore, θ ip i > θ iv i. It only remains to show bidder max = j has no incentive to deviate, i.e., she prefers outcome S to any position in outcome M; but this is directly implied by Lemma 4.1. Therefore, the vector of bids in which every bidder except max bids truthfully and max bids (v max, 0) is a good equilibrium of NP 2D. We show in Lemma 7.1 in the appendix that bidding above one s true value is a weakly dominated strategy in NP 2D. Suppose all bidders play undominated strategies, then no bidder is bidding higher than her true value. If max bids truthfully and every other bidder bids less equal her true value, the outcome will be S (since the efficient outcome is S). By Lemma 4.1, we know that bidder max prefers this outcome to winning any slot in outcome M. Therefore, for any outcome M, the deviation to truthfulness is always profitable for max. Proof (Theorem 4.2). Since the equilibrium with outcome S is a good equilibrium, the winner is max (since otherwise max would be a loser and hence must bid at least her true S-value leading to negative utility for the winner, which contradicts the equilibrium assumption). So, the welfare in this equilibrium is v max. Since the efficient outcome is M we have k θiv [i] v max. Since the NP 2D equilibrium is a good equilibrium we have v max θ iv [i] + k θ iv [i+1], where j is the rank of bidder max among all M-values, i.e. max = [j]. Therefore, the loss in efficiency of any good equilibrium with outcome S of NP 2D is at most k θi(v [i] v [i+1]) = θ jv [j] k (θi θ i+1 )v [i+1] θ j v [j]. Finally, since v max = v [j] v [j] θ j v [j], 2v max v max + θ jv [j] ( θ iv [i] + +( θ iv [i+1]) θ i (v [i] v [i+1])) = θ i v [i]. Proof (Theorem 4.4). If the outcome of V CG 2D is S, then since all losers bid at least their true values, the payment of the winner in NP 2D is at least as much as the win-

12 ner s payment in V CG 2D. Therefore, the revenue of NP 2D is greater than or equal to the revenue of V CG 2D. If the outcome of V CG 2D is M, we consider two cases, depending on whether v max 2 k,i j θiv [i] is the dominant term of the payment for bidder max = [j] in V CG 2D, or k (θ i θ i+1 )v [i+1] is the dominant term. In both cases, by individual rationality, we use k,i j θ iv [i] as an upperbound on the sum of the payments of all other bidders in V CG 2D. In the first case, the revenue of V CG 2D is upperbounded by v max 2, and in the second, it is upper-bounded by θ iv [i] + k θ iv [i+1]. But, by the NP 2D pricing rule, we know that the revenue of any good equilibrium of NP 2D with outcome S is at least max(v max 2, θ iv [i] + k θiv [i+1]). Proof (Theorem 4.5). Pick an arbitrary envy-free allocation of NP 2D with outcome M and suppose that the price of the i-th slot is p i. By theorem 4.3 we know v i = v [i]. Moreover, the bidder occupying the i-th slot must not envy the bidder occupying the i 1-st slot, i.e., θ i 1(v i p i 1) θ i(v i p i). By writing this inequality for every i (2 i k + 1), then multiplying both sides of the inequality for bidder i by a factor i, and finally summing up all inequalities (and canceling out repeated terms) we get i(θ i θ i+1)v i+1 θ ip i. The LHS of the above inequality is revenue of V CG M while the RHS is the sum of payments of all bidders in NP 2D, or in other words, the revenue of NP 2D. Proof (Proposition 4.1). If the efficient outcome is S, we proved in Theorem 4.1 the good equilibrium with outcome S exists. The revenue of such equilibrium is at least θ iv [i] + k θ iv [i+1] k θ iv [i+1] k i(θ i θ i+1)v [i+1],which is greater than or equal to the revenue of V CG M. Now suppose that the efficient outcome is M, or equivalently for NP 2D, the outcome with truthful bids is M. Consider the utility maximizing deviation for bidder max keeping the bids of the remaining bidders fixed, with respect to NP 2D pricing and allocation. If the deviation leads to outcome S, then max bidding (, 0) and all other bidders bidding truthfully is an equilibrium with outcome S since no bidder i max can profitably deviate. If the deviation leads to outcome M, then just increase b max to just ɛ below the threshold where the outcome of NP 2D switches to S, i.e., set b max = k θ ib i ɛ where b is the vector of bids in which every bidder except max is bidding truthfully, and max is M-bidding according to her profitable deviation. Setting b max to this value does not affect the price of max herself after deviation, therefore, max does not have any profitable deviation with the current vector of bids. For any bidder i max, decreasing M-bid changes the outcome to S nonprofitably. Moreover, since bidder i is currently M-bidding v i, increasing the M-bid cannot be profitable either, since it can only result in negative utility for her. In either case, we have constructed an equilibrium of NP 2D. Both of these bid vectors have revenue at least θ iv [i] + k θ iv [i+1] k θiv [i+1] k i(θi θi+1)v [i+1], which is at least as much as revenue of V CG M. Proof (Theorem 4.6). Suppose (b i, b i) is an equilibrium vector of bids with outcome M in GSP 2D. Since k θ ib i k θ ib i+1, the outcome of NP 2D for the same vector of bids is M too. First we argue that no bidder wants to deviate to obtain a different slot in M. Since b max k θib i+1, we have θ jb j+1 b max k i j θib i+1 b max k θ ib i + θ j b j, so the dominant term for the price of the bidder in slot j (for 1 j k) is θ j b j+1. Therefore, the outcome of NP 2D has the same allocation and pricing as GSP 2D. Since deviating to slot l was not profitable for any bidder j in GSP 2D, it is not profitable in NP 2D either (note that the price upon deviation in NP 2D for any slot is at least as much as the price in GSP 2D for the same deviation). Next, we show that deviating and changing the outcome to S cannot be profitable for any bidder. The price in NP 2D for bidder j for outcome S is at least θib i+ k θib i+1. The price for outcome S for bidder j in GSP 2D is at most k θ ib i+1 θ ib i + k θ ib i+1. Therefore, since bidder j has no incentive deviate to outcome S in GSP 2D, she has no incentive to deviate to outcome S in NP 2D either.

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