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4 sometimes winning higher slots than in the standard model. 3.1 The GSP Mechanism Theorem. In the externality model for keyword auctions, under the GSP mechanism, there is a pure Nash equilibrium. Proof. Our proof is by reduction to the proofs found in [18, 1]. First sort all players by v iri and label the highest k 1 of them as 1,... k 1. Let R = r r k 1. Sort the remaining n (k 1) players by decreasing value of v iri /( R + r i) and label them as k,..., n. Define w i = v iri /( R + r k ) if i k and w i = v iri /( R + r i) otherwise. Note that w 1 w... w k and that for i k + 1, w i = v iri /( R + r i) w k. Let R S = r r k. We have { viri /R w i = S if 1 i k, v iri /(R S r k + r i) if k + 1 i n. By definition of our ordering, the w i s are monotone decreasing. For our reduction, consider the standard valuation model for k slots with player values w 1 w... w n under GSP. By the results of [18, 1] there exist bids c i such that the allocation which gives slot i to player i is a GSP equilibrium and matches the VCG allocation (defined later). Since the VCG allocation orders players by their values, the c i s must be monotone decreasing. The expected utility of player i is θ i(w i c i+1). By the equilibrium conditions, we have: θ i(w i c i+1) θ j(w i c j+1) i < j k θ i(w i c i+1) θ j(w i c j) j < i k θ i(w i c i+1) i k θ j(w k+1 c j) j k Going back to our valuation model, let player i bid b i = c i/r i. The GSP mechanism sorts the players by decreasing value of b ir i. In other words, by decreasing value of c i, so we obtain the allocation where player i is in slot i. The price per click assigned to this player is going to be b i+1r i+1/r i, and the expected utility of player i is θ ir i ( riv i R bi+1ri+1 r i ) = θ i(w i b i+1r i+1) = θ i(w i c i+1). By definition of b i, the equilibrium constraints of our model reduce to the equilibrium constraints of the Varian model [18], therefore this is an equilibrium in our extended model. Our direct reduction to the regular GSP mechanism allows us not only to prove the existence of Nash equilibria in our setting but also provides a direct way to find a range of equilibria, as any equilibrium of GSP under the standard model can be converted to an equilibrium of our model. Among the most interesting of these equilibria is the envy-free equilibrium. The envy free equilibrium of a particular auction mechanism must satisfy all the constraints of being an equilibrium for that mechanism. Additionally, it must be the case that no player who is assigned a slot prefers to trade his current slot for a new slot given the current prices. Thus the envy-free constraint provides a natural fairness criterion. However, the precise definition of what an envy free equilibrium entails depends on the players valuation model. Under the standard valuation model, where the conversion rate for advertisers is assumed to be independent of the other sponsored ads which are displayed, [18, 1] showed that the GSP mechanism has a unique envy-free equilibrium which corresponds to the VCG equilibrium and can be obtained by setting the players bids according to a specific recursive rule. Under our valuation model the players not only care about the price of the slot they are allocated, but also about the winning set of players who are allocated slots. Thus, one natural way to define a GSP envy-free equilibrium in our model is the following: Definition 3. Given an allocation such that slot i is assigned to player i at price p i, the allocation is envy-free if it satisfies the following conditions: Players who are assigned a slot have non-negative utility: For 1 i k and 1 j k, ( ) vir i θ ir i p i R S No losing player i prefers to enter the set of winning players: For k + 1 i n and 1 j k, ( ) v ir i θ jr i p j R S r k + r i For 1 i k and 1 j k, the player in slot i does not prefer slot j at current prices: ( ) ( ) vir i vir i θ ir i p i θ jr i p j. R S R S The reduction we provided in the proof of Theorem above allows us to use the bidding rule of [18, 1] to directly compute an envy-free equilibrium in our setting. Theorem 4. In the externality model for keyword auctions, under the GSP mechanism, there is a pure Nash equilibrium which is envy-free. Proof. Define the ordering of the players as in the proof of Theorem, and set the losing players k + 1,..., n bids as b i = v iri /(R S r k + r i). Observe that this corresponds to a bid of c i = b ir i = w i per the definitions in Theorem. We can now set the bids of the winning players by recursively setting them from player k up to player. The bidding rule that derives the envy-free equilibrium can be expressed in the original GSP mechanism by θ i(w i c i+1) = θ i 1(w i c i) or equivalently c i = ( ) 1 θi w i + θ i 1 θi θ i 1 c i+1. Using this rule we can recursively obtain the following bids for the winning players,..., k. ( ) b i = 1 θi viri + θi b i+1r i+1. θ i 1 R S θ i 1 Finally, player 1 can bid arbitrarily high to ensure the necessary conditions. It has been shown in [18, 1] that the resulting bids and prices of this bidding rule define the envyfree equilibrium. But the equilibrium constraints are the same in our model, therefore we get an envy-free equilibrium under our model.

5 Number of Slots Number of Slots Revenue (Uncorrelated) Revenue (Correlated) Average PPC Buying Probability Average number of clicks Sum of Relevancies Figure : Varying the number of available slots. 1 players, 1-9 slots, Values: Uniform distribution [1-1], Relevances: Uniform distribution (,1). 3. Efficiency and the VCG mechanism A fundamental property of a mechanism is the efficiency it achieves in equilibrium. As usual, the efficiency of an allocation π is defined as the total value it gives to the players who are allocated a slot which, in the externalities model, is: eff(π) = 1 i k θiv π(i)r π(i) 1 j k r π(i) The Vickrey-Clarke-Groves or VCG mechanism [19, 9, 14] is the mechanism which chooses the allocation so as to maximize efficiency. With VCG, it is in the best interest of participating advertisers to bid their true valuation so the input consists of the true values of the n players. (We assume that the relevances are public information.) VCG then charges each player an amount equal to how much his presence affects the maximum efficiency achievable in the system. Specifically, let π i be the maximum efficiency allocation when player i does not participate in the auction. The VCG price for slot s, allocated to player i = π(s), is p(s) = eff(π i) eff(π) + θ s v i r i 1 j k r π(j) (). (3) Since GSP does not maximize efficiency, a basic question one can ask is how much worse can it be. Unfortunately, it turns out that GSP has equilibrium allocations with arbitrarily worse efficiency than VCG. This result is perhaps not surprising given the fact that there exist VCG allocations which are not achievable as an equilibrium in GSP. (See Claim 8 in the appendix.) Claim 5. GSP equilibria can have efficiency arbitrarily lower than that of the efficiency-maximizing allocation. Proof. We show this for a simple example with slots and 3 players. Consider the players below, Players v i r i v iri A 1/x x 1 B C The envy free GSP equilibrium (defined above) always picks allocation AB: player A is picked for slot 1 as A maximizes v ir i, player B is picked for slot as v b r b /(r a + r b ) > v cr c/(r a+r c). When x > 1 (sufficiently), then the efficiencymaximizing allocation will be BC. Thus we have eff(opt) eff(gsp) = eff(bc) eff(ab) = (999θ θ )/ (1θ θ )/(x + 1) x Setting x arbitrarily proves the claim. In the experiments described in Section 4 we see that for random instances, the efficiency of the GSP equilibrium is usually more than 75% of the VCG efficiency. We now delve into VCG under the externalities model in a bit more detail. We begin by studying the implementability of VCG. Note that our model is not a standard unit-demand combinatorial auction, since the players values depend on the set of winners. Nonetheless, the following theorem indicates that it is possible to implement VCG efficiently even in our model. Theorem 6. Assume that all input parameters are integers. Then the VCG mechanism can be run in polynomial time. Proof. See Appendix. Finally, we observe that VCG in this setting has some undesirable properties. Claim 7. VCG has the following three unappealing properties: 1. The VCG prices may be negative.

10 [4] Animesh, A., V. S., and Agarwal, R. t. [5] Arbatskaya, M. Ordered search. 4. [6] Blumrosen, L., Hartline, J., and Nong, S. Conversion rates in sponsored search auctions. Preliminary version in Dagstuhl workshop on Computational Social Systems and the Internet, 7. [7] Broder, A. Private Communication, 8. [8] Cary, M., Das, A., Edelman, B., Giotis, I., Heimerl, K., Karlin, A., Mathieu, C., and Schwarz, M. Greedy bidding strategies for keyword auctions. In EC 7: Proceedings of the 8th ACM conference on Electronic commerce (New York, NY, USA, 7), ACM, pp [9] Clarke, E. H. Multipart pricing of public goods. Public Choice 11 (1971), [1] Edelman, B., Ostrovsky, M., and Schwarz, M. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. To appear in American Economic Review, 6. [11] Feng, J., Bhargava, H. K., and D.M., P. Implementing sponsored search in web search engines: Computational evaluation of alternative mechanisms. INFORMS Journal on Computing (6). Forthcoming. [1] Ghosh, A., and Mahdian, M. Externalities in online advertising. In Proceedings of the 17th InternationalWorld Wide Web Conference (8). [13] Granka, L. Eye-r: Eye tracking analysis of user behavior in online search. Masters Thesis, Cornell University Library Press, 4. [14] Groves, T. Incentives in teams. Econometrica 41 (1973), [15] Lahaie, S. An analysis of alternative slot auction designs for sponsored search. In EC 6: Proceedings of the 7th ACM conference on Electronic commerce (New York, NY, USA, 6), ACM Press, pp [16] Prepared by Hotchkiss, G. Eye tracking report: Google, msn and yahoo! compared, November 6. [17] Schwartz, B. The paradox of choice: Why more is less. HarperCollins Publishers. [18] Varian, H. Position auctions. To appear in International Journal of Industrial Organization, 6. [19] Vickery, W. Counterspeculation, auctions and competitive sealed tenders. Journal of Finance (1961), [] Zhang, X. Finding edgeworth cycles in online advertising auctions. MIT Sloan School of Management, Working Paper, 5. APPENDIX Claim 8. There exist a VCG allocation which cannot be realized in any GSP equilibrium. Proof. We will give an example for slots and 3 bidders. Suppose the VCG allocation is to allocate the bidders in order ABC. As θ 1 > θ this implies that v ara > v b rb as otherwise VCG would reverse AB in the allocation. What conditions must be satisfied for this allocation to be a GSP equilibrium? WLOG we can assume that player A will get the top slot by bidding infinity so that players B and C will not target slot 1. Let R = r a +r b +r c, and let the prices of slot 1 and be p 1 = b b r b, p = b cr c. In order for this allocation to be an equilibrium in GSP, it is necessary and sufficient to satisfy the following constraints: p p 1 p 1 v ara/(r r c) p v b r b /(R r c) p 1 (θ /θ 1)p (1 (θ /θ 1))v ar a/(r r c) v cr c/(r r b ) p 1 The second inequality follows from the third and fourth inequalities (as v b r b v ar a) and can be eliminated. Observe that p is upper bounded in two equations and lower bounded on only one equation, so for any feasible solution (p 1, p ) there exists a feasible solution (p 1, p ) such that the lower bound is tight. In other words, without loss of generality we can assume that p = (θ 1/θ )p 1 (θ 1/θ 1)v ar a/(r r c) Substituting in for p, the constraints above are equivalent to the following equation which must be satisfied to get allocation ABC as a GSP equilibrium: v cr c R r b θ θ 1 v b r b R r c + ( 1 θ θ 1 ) var a R r c. (4) Any VCG allocation (for slots, 3 bidders) which violates equation 4 is not achievable as a GSP equilibrium. The example given in Claim 7, part (), the VCG allocation is ABC. For click-through-rates θ 1 = 1 and θ =.9) the right hand side of equation 4 is (.95(39) +.5(48))/3 = and the left hand side is 8/6 = Thus Equation 4 is violated implying that the ordering A, B, C cannot be an equilibrium in GSP. A. PROOF OF RESULTS OF SECTION 3. Proof of Theorem 6. For any set of values v 1,... v n and relevances r 1,..., r n, we will prove that the allocation of maximum efficiency assigning k slots can be found in polynomial time. Thus, for any given set of players, the allocation of maximum efficiency can be found in polynomial time. This implies the Theorem, as the VCG prices can be calculated from the maximum efficiency allocations for two sets of players per slot. First consider the decision version of the problem: Does there exist an allocation σ with efficiency greater than equal to W? Going back to Equation (), this is equivalent to asking whether there exists an allocation σ such that θ irσ(i)v σ(i) W r σ(i) (5) 1 i k To answer this question, define a complete bipartite graph G with n vertices on the right side (representing the players), k vertices on the left side (the slots), such that the edge between player j and slot s has weight θ sr j v j W r j + B. Here B = W max i r i is a value to ensure that all weights are positive. Compute a maximum weight matching on G. Since all edge weights are positive, the maximum weight matching of G has k edges, hence corresponds to an allocation of the k slots; and it has weight at least kb iff there exists an allocation σ which assigns k slots and has efficiency at least W. As maximum weight matching in G can be found in

11 O(n 3 + k ) steps, we can answer the decision version of the problem in poly-time. The optimization problem can be solved by doing binary search along a bounded interval of possible efficiency values, using the decision version algorithm to check for the existence of allocations with particular efficiencies. The inequality (a a n)/(b b n) max a i/b i can be used on Equation () to derive an upper bound for the efficiency of any allocation of max 1 j n v jr j. Thus the initial binary search interval is [, max(v jr j)]. Now, in order to bound the number of steps of the binary search, we compute a lower bound on the difference in efficiencies of two allocations π and σ. Each input parameter is an integer. If π and σ have distinct efficiencies, then eff(π) eff(σ) 1 ( 1 j k r π(j))( 1 j k r σ(j)) 1 R, where R denotes the sum of k highest relevances in the input. Thus we can stop the binary search when the interval is of size 1/R, and return the last allocation where the decision version algorithm returned a positive answer. Going from an interval of size max v jr j to one of size 1/R using binary search can be done by solving O(log(R max v ir i)) decision problems, giving a polynomial time algorithm for finding the allocation of maximum efficiency. Proof of Claim 7, part (1). We demonstrate this using a simple example with slots having click through rates θ 1 = 1 and θ = 1/. Consider the following players: 3/4 which is the highest. Now let us calculate the VCG prices for slot 1 and : p 1 = eff(cb) eff(ab) + v ar a/(r a + r b ) = 11.6 p = eff(ca) eff(ab) + θ v b r b /(r a + r b ) = 4.9 This is not envy free as A would rather have slot at price \$4.9 then his current slot at price \$11.6. Indeed, A s utility at his current slot is θ 1(v ar a/(r r c) p 1) = 4.94 whereas his utility in slot at price p would be θ (v ar a/(r r c) p b ) = Proof of Claim 7, part (3). Suppose not and let σ a more efficient allocation containing k slots and let S = k i=1 r σ(i). The efficiency of σ is eff(σ) = (θ 1v σ(1) r σ(1) θ k v σ(k) rσ(k))/s. Let m be the player who maximizes v ir i so that v mr m v σ(i) r σ(i). Using this and the fact that θ 1 > θ s for all slots s > 1, we get eff(σ) < θ 1v mr m(r σ(1) r σ(k) )/S = θ 1v mr m The right hand side of the inequality is the efficiency of allocation the top slot is allocated to player m. Value Relevance Player A 1 1 Player B 1-ε 1 Player C 1/8 It is easy to check that allocation AB has the highest efficiency of (θ 1v ara + θ v b rb )/(r a + r b ) 3/4. Now let us examine the price of the second slot p. If player B was not participating in the auction, the most efficient allocation of the remaining players would be AC, with efficiency approximately 5. Finally the value B obtains from allocation ABC is θ v r/(r 1 + r ) 1/4. Therefore using 1 the vcg pricing equation 3, the total price for slot is: p = It is not hard to generalize this example to larger instances where the VCG prices are negative. Since VCG is forced to fill k slots, this situation may occur when a low relevance high valued bidder (such as B) is part of the VCG allocation and his next best replacement is a bidder of with much higher relevance. Proof of Claim 7, part (). Once again we demonstrate this with a simple slots example, with click through rates θ 1 = 1 θ =.95. Consider the following set of bidders 5 Player Value Relevance A 1 B 39 1 C 5 4 Examining all possible allocations it is easy to check that the allocation AB has efficiency of (θ 1v ar a +θ v b r b )/(r a +r b ) 5 We can always scale all relevance in this example to be 1 and the same results would hold.

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