# Position Auctions and Non-uniform Conversion Rates

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6 preserving, if for every two bidders i, i 2, if v i > v i2 then b i b i2 Lea. Consider a profile of valuations v,..., v n for which an equilibriu exists, denoted by b (b,..., b n). Then, there also exists an order preserving equilibriu b. Moreover, b achieves at least the sae social welfare as b. Proof. Notation: Let u i(v i,b) denote the expected utility of bidder i with value v i when the bidders bid b. That is, the expected utility is (let n i(b) denote the nuber of bidders bidding exactly b i and let the consecutive slots that they are assigned be t,..., t). u i(v X c j a j v i ( n i(b) )b i b i A n i(b) t j t Pjt,...,t+n h (b) cj a j where b i denotes the highest bid in b which is saller than b i. We will now prove the lea. Consider an equilibriu b (b,..., b n). Consider two bidders h and l such that b h > b l but h > l. We will show that the bid profile where bidder l bids b k and bidder k bids b l is also an equilibriu. It follows that we can continue swapping bidders until reaching an order-preserving equilibriu. Let b be a profile of bids identical to b except bidder h bids b l. Since b is an equilibriu, clearly u h (v h,b ) u h (v h,b ). Note that both u h (v h,b ) and u h (v h,b ) are linear functions of v h. The slope of u h (v h,b ) is n h (b) (the first slot assigned to bidders that bid b h is t) and the Pjt,...,t +s cj a j slope of u h (v h,b ) is n h (where s is the nuber of slots assigned to bidders who bid b l and t is the first (b) such slot), which is clearly saller since we assue that a j c j is decreasing in j. It follows, that if we increase v h to be v l, we will have that u h (v l,b ) u h (v l,b ). That is, for the bid profile b swap which is siilar to b except h bids v l and l bids v h, bidder l will not be willing to bid b l. Since b was an equilibriu, bidder l will not be willing to bid any other bid level in b swap for the sae reason. Siilar arguents show that h will not be willing to change his bid as well. 5. The Social Welfare Loss Might be Substantial We now give our first bound on the welfare loss in GSP with non-unifor conversion rates. For given paraeters (CTR s and conversion rates), we construct a profile of valuation such that all equilibria incur a large welfare loss. We show that as the paraeters of the proble becoe extree (but still agree with all the assuptions in our odel), the welfare loss can be arbitrarily close to of the total welfare. The reason for GSP inefficiency is that advertisers with distinct values bid in pools. To analyze the equilibria we distinguish between the pooling outcoes of the three advertisers with the highest values. Definition. An order preserving equilibriu is non-pooling if the top three advertisers have distinct bids, i.e., b > b 2 > b, Identifying pooling and separating equilibria plays a central role in icroeconoics, ainly in the context of signaling gaes (see [4]). -pooling if the top three advertisers bid the sae, i.e., b b 2 b, (, 2)-pooling if the top advertiser beats the second two who pool, i.e., b > b 2 b, and (2, )-pooling if the top two advertisers pool and beat the third, i.e., b b 2 > b. The above definitions include the case where the fourth and lower valued advertisers pool with the third highest value advertiser. For 2-slot auctions, we siplify notation and denote c /c 2 x and a /a 2 y. Notice that by our previous onotonicity assuptions x and y. We also denote to be the nuber of players whose values v i are at least the third highest value b ( n). Theore. For every ǫ >, and given any set of clickthrough rates and conversion rates, there exist valuations profile for which the social welfare achieved by every equilibriu of GSP is at ost a xy+y xy+ xy+y xy+ +ǫ fraction of the optial welfare. When xy and y, this fraction approaches 2 + ǫ. Proof. Consider the following set of bidders in a 2- slot auction. We first denote β c a +c 2 a 2, and note that c +c βv is exactly the price where bidders with 2 value v are indifferent between losing and winning the lottery of slot and 2. Bidder has value v, and all the other n bidders have the sae value v 2, such that a v < βv 2, or equivalently v 2 > xy+y xy+y v. Denote δ v2 v, and note xy+ xy+ that δ can be ade arbitrarily sall and that it is defined independently of. Figure 5 in the appendix describes utility functions for this construction. Assuption: the exact bid level βv 2 is not in the bid space for any discretization. Denote the highest winning bid by p. For the above valuations, every -pooling equilibriu will gain social welfare which is arbitrarily close to 2 of the social welfare in the optial slot allocation when xy and y. (a + b)(v + ( )v2) av + bv 2 (xy + ) v + ( ) xy+y xy+ v + ( )δ (xy + y)v xyv + xy+y xy+ v + δ + xy+ v + xyv + xy+y xy+ v + δ (xy + )δ (xy + y)v xyv + xy+y xy+ v + δ < xy + y + O(ax{ xy + xy+y, δ}) xy+ Clais 5, below, establish that the only equilibria in the above setting are -pooling. Finally, if the paraeters are chosen such that xy and y approaches zero, then clearly xy+y xy+ xy+y xy+ approaches 2. Clai. For the above valuations, GSP adits no nonpooling equilibriu.

7 This above clai follows iediately fro Theore 2. Clai 2. For the above valuations, GSP adits no (2,)- pooling equilibriu. Proof. If p > βv 2, then bidder 2 prefers losing over his current payoff. If p < βv 2, then all the other losing bidders will gain positive utility fro bidding p. Due to the above assuption, p βv 2 cannot be an equilibriu since it is not in the bid space. Clai. For the above valuations, GSP adits no (,2)- pooling equilibriu. Proof. First, p a v, otherwise bidder will prefer losing. Since we selected v 2 such that a v < βv 2, it follows that each one of the bidders with the value v 2 would earn soe positive utility by bidding p and sharing Slots and 2 with bidder. Since this positive utility is independent of the nuber of bidders, for sufficiently large bidder 2 will prefer bidding p over sharing Slot 2 with the other 2 bidders, in contradiction to the existence of an equilibriu. (Note that the bidders that win together Slot 2 gain positive utility, thus if such an equilibriu had existed, then all the bidders except bidder would have shared Slot 2.) Clai 4. For the above valuations, GSP adits a -pooling equilibriu. Proof. It is easy to see that bidding the highest available bid p such that p a v by all bidders is an equilibriu. Since a v < β v 2, every bidder with value v 2 gains positive utility, and thus bidder also gains positive utility. However, all bidders would gain zero utility by under-bidding p and gain negative utility by over-bidding. Clai 5. For the above valuations, GSP adits no nonorder-preserving equilibria (for the top three advertisers). Proof. This clai is a direct corollary of Lea. A - pooling equilibria is the only order-preserving equilibria that exists and it cannot be a result of a swapping process starting fro other equilibria. The other order-preserving equilibria do not exist and therefore neither to non-order-preserving equilibria that can be converted to the via Lea. 5.2 Existence of Equilibria with a Bounded Loss In the proof of Theore we constructed a faily of bidder profiles for which all the equilibria achieve social welfare which is arbitrarily close to 2 of the optiu. In this section we show that this bound is alost tight, at least for 2-slot auctions. We show that there always exists an equilibriu that achieves in{ 2 a,/2 + } of the optial social welfare; this does not rule out the existence of other equilibria a 2 with worse perforance. This kind of results is coonly referred to as the price of stability [2], and it is siilar in spirit to the work of [8, 6] that showed the existence of at least one welfare-axiizing equilibriu in GSP. We were able to prove this result only for 2-slot auctions, and we conjecture that a siilar constant lower bound on the suboptiality of GSP ay be proved for any nuber of slots; however, we leave this proble open. Note that it is reasonable in this odel to study a sall constant nuber of slot, rather than asyptotically growing nuber of slots, therefore the 2-slot case can give us good intuition on the ore general case. We will start by proving that pure Nash equilibria always exist, even under the assuption of non-unifor conversion rates. Again, we were only able to prove this for the 2-slot case. The ain reason is that our current proof relies on a case analysis that does not extend to a general nuber of slots. A new technique should probably be used for generalizing this lea. Lea 2. With a discrete bid space, GSP for two slots always adits a (pure Nash) equilibriu, even with nonunifor conversion rates. Proof. (sketch) Consider the following iterative ascendingprice process. All bids are initialized to zero. Each bidder in his turn (i,..., n) raises his bid by ǫ (the inial increent allowed) if it iproves his utility. Also, losing players will raise their bid in their turn as long as they do not decrease their utility by doing so. We clai that this process will always end up in an equilibriu. Consider the three bidders with the highest values (bidders,2,), and consider the first iteration where one of the will not increase his bid. If the three bidders stop bidding at the sae iteration, then it is clearly an equilibriu (they do not to bid higher bids since they stopped, but they do earn positive utilities; if they bid a lower bid they will get zero). If player 2 is the first to stop bidding, this is an equilibriu as well, since bidder one preferred the higher bid, and player and 2 preferred not to raise their bid. Finally, if bidder stopped increasing his bid, it is not necessarily an equilibriu since the iterative process ay proceed, but since we require that the losing bidders will continue raising their bids in the algorith, all we have to show is that bidder will not prefer decreasing his bid when he shares the highest bid with bidder 2. This is clear, since bidder preferred oving to the current bid over sharing slot 2 with the others, and thus gain non-negative value fro slot. He will therefore prefer the lottery of slots,2 over having slot 2 with probability of at ost /2 (since losers will always continue bidding). Corollary. GSP with 2 slots always adits an order preserving (pure) equilibriu. Proof. Iediate fro Leas 2 and. The following theore shows that the equilibriu social welfare is not fatally low. As long as the conversion rates in the slots are not extreely far fro each other (i.e., a 2 a and this is copatible with what we observed fro the data), GSP achieves at least 2 of the optial welfare (with 2 slots). Theore 4. In GSP for 2 slots and any nuber of bidders there always exists an equilibriu that achieves at least a fraction of in{ 2, + a } of the optial social welfare. 2 a 2 The theore is iediately concluded fro the following lea proved in Appendix 4. This is proved by a case analysis that bounds the social welfare loss in each equilibria for each possible pooling. Recall that the loss in social value in order-preserving equilibria is due to the randoized assignents in case of ties. An iportant part of the analysis is the use of the paraeterized constraints on the values of the bidders that are derived by the equilibriu constraints. We forulate these constraints to show that the values of the bidders that bid

9 7. REFERENCES [] Z. Abras, A. Ghosh, and E. Vee. Cost of conciseness in sponsored search auctions. In Proc. rd International Workshop on Internet and Network Econoics, 27. [2] E. Anshelevich, A. Dasgupta, J. Kleinberg, E. Tardos, T. Wexler, and T. Roughgarden. The price of stability for network design with fair cost allocation. In FOCS 4, pages 295 4, 24. [] S. Muthukrishnan Gagan Aggarwal, Jon Feldan. Bidding to the top: Vcg and equilibria of position-based auctions. In Proceedings of the Fourth Workshop on Approxiation and Online Algoriths, 26. [4] A. Mas-Collel, W. Whinston, and J. Green. Microeconoic Theory. Oxford university press, 995. [5] Paul Milgro. Siplified echaniss with applications to sponsored search and package auctions, 27. Working paper, Stanford University. [6] Michael Ostrovsky, Benjain Edelan, and Michael Schwarz. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. Forthcoing in Aerican Econoic Review, 26. [7] Ti Roughgarden and Eva Tardos. In Noa Nisan, Ti Roughgarden, Eva Tardos and Vijay Vazirani (Editors), Algorithic Gae Theory. Chapter 7. Introduction to the Inefficiency of Equillibria. Cabridge University Press., 27. [8] Hal Varian. Position auctions. To appear in International Journal of Industrial Organization, 26. [9] W. Vickrey. Counterspeculation, auctions and copetitive sealed tenders. Journal of Finance, pages 8 7, 96. APPENDIX A. PROOF OF LEMMA Notations: recall that denotes be the nuber of bidders that have at least the rd-highest value (i.e., where v i v ) and that a c a, b a 2 c 2, x c and y a, and let p c 2 a be the lowest winning bid. In the following three 2 subsections we prove the ites in Lea. A. A bound on the welfare loss in -pooling equilibria This subsection proves the following lea. Lea 4. Every -pooling equilibriu achieves at least 2 of the optial social welfare (if (2-)-pooling equilibria do not exist). Note that the optial welfare is av + bv 2, and the welfare when > 2 bidders bid the highest bid is (a + b) (v + v 2 + ( 2)v ). In this proof, we will denote the ratio between the as WR (for Welfare Ratio), and we will bound it fro below. WR (a + b) (v + v2 + ( 2)v) () av + bv 2 We first observe that the inefficiency ratio is either always increasing or always decreasing in v 2. It is easy to see (by first-order conditions) that the function is increasing in v 2 if v < ( )b v a b and decreasing otherwise. We will treat these two cases separately. Case : The function in Equation is decreasing in v 2. We can now change v 2 to be v and the ratio will not increase. WR > (a + b)(v + ( )v) (2) av + bv Since this is a -pooling equilibriu, we will now write constraints derived by the equilibriu definition. These constraints will help us bound the WR function. If bidding p is an equilibriu bid for the top bidders, then bidder will not benefit fro deviating and bid p + ǫ. Therefore, c (a v p) + c2 (a 2 v p) c (a v p) Since bidder 4 gains non negative utility, c (a v p) + c2 (a 2 v p) () The two above inequalities give an upper bound and a lower bound on p; taken together, we get that v tv where t (( )a b) `c + c 2 (( )c c 2 ) (a + b) (( )xy ) (x + ) (4) (( )x ) (xy + ) Since Equation 2 is increasing in v (assuing b < and > 2), we get that: WR > (a + b) (v + ( )tv) av + btv (a + b) ( + ( )t) a + bt (xy + ) ( + ( )t) xy + t (5) It is easy to see that in this case WR xy+ xy+2 a+b a+2b > 2.6 Case 2: The function in Equation is increasing in v 2. We can now change the v 2 in equation to be v and the ratio will not increase. WR > (a + b)(2v2 + ( 2)v) (7) av 2 + bv 2 6 Fro Equation 4 clearly t ( )xy. Substituting into ( )(xy+) Equation 5 we get: (xy + ) ( + ( )t) xy + t xy( )(xy + ) ( )(x 2 y 2 + xy) + ( )xy xy(xy + ) (xy + ) (x 2 y 2 + xy) + xy xy + 2 (6)

10 In the analysis of this part of the proof we will actually need a stronger constraints than the one that follows fro the -pooling equilibriu properties. We will assue that there is no (2,)-pooling equilibriu. (We can do this, since part of Lea shows that (2,)-pooling equilibria achieve at least of the social welfare.) It follows that 4 bidder 2 prefers sharing the second slot at price p with the bidders with the value v over sharing the highest bid with bidder : 7 `c2 (a 2 v 2 p) 2 c (a v 2 (p + ǫ)) + 2 c2 (a 2 v 2 p) After rearranging the ters and together with Equation we get: v > (a + (a + b) 2 c + b)(c + c 2 ) > 2 )c2 v 2 And we denote v > tv 2, and note that t > a+( 2 )b a+b v 2. Using Equation 7 we get the following bound on the inefficiency: WR > (a + b)(2v2 + ( 2)tv2) av 2 + bv 2 (2 + ( 2)t) > a + ( 2) > 5 6 ( 2)2b ( )(a + b) 2 a + b b A Where the last equality holds since 2( 2) ( ) < for > 2 and since b < a. A.2 A bound on the loss in (,2)-pooling equilibria Lea 5. Every (,2)-pooling equilibriu achieves at least + a of the optial social welfare. 2 a 2 Proof. First, bidder ust have non zero utility fro bidding p, where he receives Slot 2 with a positive probability and pays p. Thus, p a 2 v. On the other hand, bidder 2 prefers bidding p to bidding p + ǫ. That is, he prefers no slot (zero utility) to Slot at price p+ǫ. Thus, p+ǫ > a v 2. It follows that a 2 v + ǫ > a v 2, v > yv 2 ǫ a 2 (8) 7 If such an equilibriu does not exists, bidder 2 ust want to deviate. The bidders with the value v will not bid above p since p ust be exactly the price for which bidder is willing to get Slot and Slot 2 at rando, but this lottery will gain hi negative utility for price p + ǫ. Also, if bidder will deviate, necessarily bidder 2 will deviate too. WR av + 2 bv2 + bv av + bv 2 av + bv y O(ǫ) (9) av + bv 2 av + av + 2 y O(ǫ) av + av + ( 2)y + O(ǫ) 2 2( ) > 2 + y O(ǫ) () 2 Where the 2nd inequality is due to Equation 8, and the rd inequality is due to the fact that Equation 9 is decreasing in the value of bv 2 (since + 2 y < ), which is bounded by av. A. A bound on the loss in (2,)-pooling equilibriu 4 Lea 6. Every (2,)-pooling equilibriu achieves at least of the optial social welfare. Bidders and 2 win by bidding p, and bidder bids p ǫ. Again, we will consider constraints on the valuations of the bidders that follow fro the equilibriu properties. Bidder 2 ust gain non-negative utility by bidding p. 2 c `a v 2 p + 2 c2 `a 2 v 2 (p ǫ) Therefore: p a + b c + c 2 v2 + ǫc2 () c + c 2 Bidder will not increase his utility fro bidding above p and winning Slot alone: 2 c `a v p + 2 c2 `a 2 v (p ǫ) c (a v p) Therefore: p Fro Inequalities and 2 we get: a b c v (2) c2 v 2 (a b)(c + c 2 ) (a + b)(c c 2 v O(ǫ) ) > xy v O(ǫ) xy + Now we can bound the social-welfare loss. Note that the expected welfare in (2,)-pooling equilibriu is 2 (a+b)(v+ v 2), and recall that the optial welfare is given by av +bv 2. We will use the above constraints to prove the following lower bound on the ratio between the equilibriu welfare and the optial welfare.

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