Pure and Bayes-Nash Price of Anarchy for Generalized Second Price Auction

Renato Paes Leme Eva´ Tardos Department of Computer Science Department of Computer Science Cornell University, Ithaca, NY Cornell University, Ithaca, NY [email protected] [email protected]

Abstract—The Generalized Second Price Auction has for advertisements and slots higher on the page are been the main mechanism used by search companies more valuable (clicked on by more users). The bids to auction positions for advertisements on search pages. are used to determine both the assignment of bidders In this paper we study the social welfare of the Nash equilibria of this game in various models. In the full to slots, and the fees charged. In the simplest model, information setting, socially optimal Nash equilibria are the bidders are assigned to slots in order of bids, and known to exist (i.e., the Price of Stability is 1). This paper the fee for each click is the bid occupying the next is the first to prove bounds on the price of anarchy, and slot. This auction is called the Generalized Second Price to give any bounds in the Bayesian setting. Auction (GSP). More generally, positions and payments Our main result is to show that the price of anarchy is small assuming that all bidders play un-dominated in the Generalized Second Price Auction depend also on strategies. In the full information setting we prove a bound the click-through rates associated with the bidders, the of 1.618 for the price of anarchy for pure Nash equilibria, probability that the advertisement will get clicked on by and a bound of 4 for mixed Nash equilibria. We also prove the users if assigned to the best slot. This is the version a bound of 8 for the price of anarchy in the Bayesian of the Generalized Second Price Auction mechanism setting, when valuations are drawn independently, and the valuation is known only to the bidder and only the adopted by all search companies. Here we will focus on distributions used are common knowledge. the basic model for simplicity of presentation, but our Our proof exhibits a combinatorial structure of Nash results extend to the standard model of separable click- equilibria and uses this structure to bound the price of though rates (see the full version of our paper [11]). anarchy. While establishing the structure is simple in the case of pure and mixed Nash equilibria, the extension to The Generalized Second Price Auction is a sim- the Bayesian setting requires the use of novel combinatorial ple and natural generalization of the Vickrey auction techniques that can be of independent interest. [15] for a single slot (or single item). The Vickrey Keywords-game theory; price of anarchy; GSP; Spon- auction [15] for a single item, and its generalization, sored Search Auction the Vickrey-Clarke-Groves Mechanism (VCG) [2], [5], make truthful behavior (when the advertisers reveal I.INTRODUCTION their true valuation) a dominant strategy, and make the Search engines and other online information sources resulting outcome maximize the social welfare. How- use Sponsored Search Auctions, or AdWord auctions, to ever, the Generalized Second Price Auction is neither monetize their services via advertisements sold. These truthful nor maximizes social welfare. In this paper we auctions allocate advertisement slots to companies, and will consider the social welfare of the GSP auction companies are charged per click, that is, they are outcomes. Our goal is to show that the intuition based charged a fee for any user that clicks on the link on the similarity of GSP to the Vickrey auction is not associated with the advertisement. There has been much so far from truth: we prove that the social welfare is work in understanding various aspect of the auctions within a small constant factor of the optimal in any Nash used in this context, see the survey of Lahaie et al. [8]. equilibrium under the mild assumption that the players Here we consider Sponsored Search Auctions in a use un-dominated strategies. game theoretic context: consider the game played by We consider both full information games when player advertisers in bidding for an advertisement slot. For valuations are fixed, and also consider the Bayesian set- each search word, advertisers can bid for showing their ting when the values are independent random variables, ad next to the search results. There are multiple slots the valuation is known only to the bidder, and only the distributions used are public knowledge. Supported in part by NSF grants CCF-0729006. In the case of the full information game Edelman Supported in part by NSF grants CCF-0910940 and CCF-0729006, ONR grant N00014-98-1-0589, and a Yahoo! Research Alliance et al. [3] and Varian [14] show that there exists Grant. Nash equilibria that are socially optimal (for both our simple model and the case of separable click-through Such pairs seem hard to define in the mixed case. rates). But there are Nash equilibria where the social Instead, we will consider bidder i and his optimal welfare is arbitrarily smaller than the optimum even slot i, and get the following condition for mixed for the special case of the single item Vickrey auction. Nash equilibria

However these equilibria are unnatural, as some bid Eασ(i) Evπ(i) 1 exceeds the players valuations, and hence the player + ≥ , αi vi 2 takes unnecessary risk. We show that bidding above the We use this inequality to show that the social valuation is dominated strategy, and define conservative welfare of a mixed Nash equilibrium is at least bidders as bidders who won’t bid above their valuations. one-fourth of the optimal social welfare. Our results assume that players are conservative. • We prove a bound of 8 on the price of anarchy for the Bayesian setting, where the valuations v Our results: The main results of this paper are Price k are drawn independently at random. We do this of anarchy bounds for pure, mixed and Bayesian Nash via a slightly more complicated structural property, equilibria for the GSP game assuming conservative showing that an expression similar to the one used bidders. To motivate the conservative assumption, we in the case of mixed Nash must be at least 1/4th in observe that bidding above the player’s valuation is expectation. However, establishing this inequality dominated strategy in all settings. in the Bayesian setting in much harder. In the For each setting, we exhibit a combinatorial struc- context of pure and mixed Nash, the inequality ture of the Nash equilibria that can be of independent follows from the Nash property by considering a interest. To state this structure we need the following single deviation by a player, e.g., a player who notation. For an advertiser k let v be the value of k would be assigned to slot i in the optimum, may advertiser k for a click (a random variable in the want to try to bid high enough to take over slot Bayesian case). For a slot i, let π(i) be the advertiser i. In contrast, in the Bayesian case we obtain our assigned to slot i in an equilibrium (a random variable, structural result by considering many different bids, in the case of mixed Nash, or in the Bayesian setting). and combine the inequalities established by these • For the case of full information game, the social bids to show the structure. welfare in a pure Nash equilibrium with conserva- In the process we use a number of new techniques tive bidders is at most a factor of 1.618 above the of independent interest. The bids we use for player optimum. We achieve this bound via a structural i are twice the expected value of the minimal bid characterization of such equilibria: for any two that takes slot k conditioned both on the value v slots i and j, we show that in a Nash equilibrium i and the fact that the optimal position for bidder with conservative bidders, we must have that i is k. We show via an interesting combinatorial α v j + π(i) ≥ 1. argument using the max-flow min-cut theorem, that αi vπ(j) these bids decrease with k. Then we use a novel It is not hard to see that this structure implies averaging technique (using linear programming) to that the assignment cannot be too far from the combine the resulting inequalities. optimal: if two advertisers are assigned to positions Our results differ significantly from the existing work not in their order of values, then either (i) the two on the price of anarchy in a number of ways. Many of advertisers have similar values for a click; or (ii) the known results can be summarized via a smoothness the click-through rates of the two slots are not very argument as observed by Roughgarden [12]. In contrast, different, and hence in either case their relative it is easy to see that the GSP game is not smooth in the order doesn’t affect the social welfare very much. sense of [12] (see the full version of this paper [11] for • We also bound the quality of mixed Nash equilibria an example). Second, most known price of anarchy re- as a warm-up for the Bayesian setting. For a sults are for the case of full information games. The full mixed Nash equilibrium π(i) is a random variable, information setting makes the strong assumption that indicating the bidder assigned to slot i, and simi- all advertisers are aware of the valuations of all other larly let the random variable σ(i) denote the slot players. In contrast, the Bayesian setting requires only assigned to bidder i. For notational convenience the much weaker assumption that valuations are drawn we number players and slots in order of decreasing from independent distributions, and these distributions valuation and click-through rates respectively. By are known to all players. The Bayesian game is a better this notation, bidder i should be assigned to slot model for real AdWord Auctions, since players submit i in the optimum. The inequality for pure Nash a single bid that will be used in many auctions with equilibria is derived by thinking about a pair of different competitors, so players are, in fact, optimizing bidders that are assigned to slots in reverse order. for a distribution of other players. Related work: Sponsored search has been an active area of smoothness argument (see [12]) that if the greedy of research in the last several years. Mehta et al. [10] algorithm is a c-approximation algorithm, then the Price considered AdWord auctions in the algorithmic context. of Anarchy of the resulting mechanism is c + 1 - for Since the original models, there has been much work in pure and mixed Nash and for Bayes-Nash equilibria. the area, see the survey of Lahaie et al. [8]. Here we The Generalized Second Price mechanism is a type of use the game theoretic model of the AdWord auctions greedy mechanism, but is not a combinatorial auction, of Edelman et al. [3] and Varian [14]. and hence it does not fit the framework of Lucier and In the full information setting Edelman et al. [3] and Borodin. The key to proving the c + 1 bound of Lucier Varian [14] show that the price of Stability for this game and Borodin [9] is to consider possible deviating bids, is l. More precisely, they consider a restricted class of such as a single minded bid for the slot in the optimal Nash equilibria called Envy-free equilibria or Symmet- solution, or modifying a bid by changing it only on a ric Nash Equilibria, and show that such equilibria exists, single slot (the one allocated in the optimal solution). and all such equilibria are socially optimal. In this class The combinatorial auction framework allows such com- of equilibria, an advertiser wouldn’t be better off after plex bids; in contrast, the bids in GSP have limited switching his bids with the advertiser just above him. expressivity, since a bid is a single number, and hence Note that this is a stronger requirement than Nash, as an bidders cannot make single-minded declarations for a advertiser cannot unilaterally switch to a position with certain slot, or modify their bid only on one of the slots. higher click-through rate by simply increasing his bid. Like the GSP game, many natural bidding languages Edelman et al. [3] claim that if the bids eventually con- have limited expressivity, since typically allowing ar- verge, they will converge to an envy-free equilibrium; bitrary complex bids makes the optimization problem otherwise some advertiser could increase his bid making hard. The limited expressivity of the bidding language the slot just above more expensive and therefore making can increase the set of Nash equilibria (since there are the advertiser occupying it underbid him. They do not fewer deviating bids to consider), so it is important to provide a formal game theoretical model that selects understand if such natural bidding languages result in such equilibria. greatly increased price of anarchy. Gomes and Sweeney [4] study the Generalized Sec- II.PRELIMINARIES ond Price Auction in the Bayesian context. They show that, unlike the full information case, there may not exist We consider an auction with n advertisers and n slots symmetric or socially optimal equilibria in this model, (if there are fewer slots, add virtual slots with click- and obtain sufficient conditions on click-through rates through rate zero). We model this auction as a game that guarantee the existence of a symmetric and efficient with n players, where each advertiser is one player. In equilibrium. the simple model the type of the advertisers is given by Lahaie [7] considers the problem of quantifying the their valuation vi, their value for one click. The strategy ∈ ∞ social efficiency of an equilibrium. He makes the strong for each advertiser is a bid bi [0, ) which expresses assumption that the click-through rates αi decay expo- the maximum he or she is willing to pay for a click. 1 The auction decides where to allocate each advertiser nentially along the slots with a factor of δ , and proves a { 1 − 1 } based on the bids. In the simple model, being assigned price of anarchy bound of min δ , 1 δ . We make no assumptions on the click-through rates. Thompson and to the k-th slot results in αk clicks and αk is a monotone ≥ ≥ ≥ Leyton-Brown [13] study the efficiency loss of equilibria non-increasing sequence, i.e., α1 α2 ... αn. The empirically in various models. simple game proceeds as follows: We assume that bidders are conservative, in the sense 1) each advertiser submits a bid bi ≥ 0, which is the that no bidder is bidding above their own valuation. maximum he is willing to pay for a click We can justify this assumption by noting that bidding 2) the advertisers are sorted by their bids (ties are above the valuation is a dominated strategy. Lucier and broken arbitrarily). Call π(k) the advertiser with Borodin [9] and Christodoulou at al. [1] also use the the k-th highest bid conservative assumption to establish price-of-anarchy 3) advertiser π(k) is placed on slot k and therefore results in the context of combinatorial auctions. received αk clicks The paper by Lucier and Borodin [9] on greedy 4) for each click, advertiser k pays bπ(k+1), which auctions is also closely related to our work. They is the next highest bid analyze the Price of Anarchy of the auction game The vector π is a permutation that indicates to which induced by a Greedy Algorithm. They consider a general slot each player is assigned - it is determined by the combinatorial auction setting, where a greedy algorithm set of bids (up to ties). We define the utility of a user is used for determining the allocation with payments i when occupying slot j as given by ui(b) = αj(vi − computed using the critical price. They show via a type bπ(j+1)). We define the social welfare of this game as the total value that the bidders∑ and the auctioneer get Mixed Nash equilibrium: The valuations vi are still from playing it, which is: j αjvπ(j). The goal of this fixed and we can assume (without loss of generality) paper is to bound the social welfare of the equilibria that v1 ≥ ... ≥ vn, but players pick a distribution over relative to the optimum. This measure is called the Price strategies. In a Mixed Nash equilibrium, each player of Anarchy. We analyze the Price of Anarchy in three chooses a random variable bi for his bid such that the different settings of increasing complexity. chosen random variable maximizes the expected utility In the full version of our paper [11] we extend for each player. In other words: the results to the more general model of separable ′ ′ E[u (b , b− )] ≥ E[u (b , b− )], ∀b , ∀i click-through rates, where the probability of clicking i i i i i i i on an advertisement i displayed in slot j is αjγi. Now where expectation is with respect to the distribution advertisers are assigned in order of the products biγi of bids. Now, the assignment π is a random variable (the expected total willingness of the bidder to pay: determined by b and therefore the social welfare is γi clicks at the rate of bi), and the fee for a click also a random variable (even though the optimal is the critical value of the bid needed to keep the ∑welfare is fixed).∑ The Price of Anarchy is the ratio: E advertiser in his current slot. We get the simple model j αjvj/ [ j αjvπ(j)]. as a special case by assuming that γi = 1 for all bidders. Bayes-Nash equilibrium: The partial information set- Pure Nash equilibrium: The valuation of each player ting, using the framework of Harsanyi [6], provides a is a fixed value vi. We number the bidders (without loss more realistic setting than the full information game. In of generality) so that v1 ≥ v2 ≥ ... ≥ vn. Each player this model the valuations vi are drawn from indepen- chooses a pure strategy, i.e., a deterministic bid bi. The dent distributions. The distributions used are common bids b = (b1, . . . , bn) is a Pure Nash Equilibrium if no knowledge, but only player i is aware of his valuation bidder can change his bid to increase his utility, i.e.: vi. (No assumptions are made about the distributions beyond independence). Now the strategy of a player i ≥ ′ ∀ ′ ∈ ∞ ui(bi, b−i) ui(bi, b−i), bi [0, ) is to choose a bid (possibly at random) based on his own valuation vi. Therefore, the strategy of player i is a ̸ where b−i denotes the vector of bids for bidders j = i. bidding function bi(vi) that associates for each valuation To gain some intuition, suppose advertiser i is cur- vi a distribution of bids. A set of bidding functions is a ′ rently bidding bi and occupying slot j. Changing his bid Bayes-Nash equilibrium if for all i, vi, bi(vi): to something between bπ(j−1) and bπ(j+1) won’t change ′ E[u (b (v ), b− (v− ))|v ] ≥ E[u (b (v ), b− (v− )|v ] the permutation π nor his payment. So, he could try to i i i i i i i i i i i i increase his utility by doing one of two things: where expectations are taken over values and random- • increasing his bid to get a slot with a better click- ness used by players. through rate. If he wants to get a slot k < j he The Nash assignment π is a random variable, since it needs to overbid advertiser π(k), say by bidding is dependent on the bids, which are random. The optimal allocation is also a random variable, let ν(k) be the bπ(k) + ϵ. This way he gets slot k for the price b per click, getting utility α (v − b ). slot occupied by player k in the optimal assignment. π(k) k i π(k) ⇒ • decreasing his bid to get a worse but cheaper slot. Therefore, ν is a random variable such that vi > vj If he wants to get slot k > j he needs to bid below ∑ν(i) < ν(j). The optimal social welfare is therefore advertiser π(k). This way he would get slot k for j αν(j)vj. In this setting the quantity we want to bound is the Bayes-Nash price of Anarchy, which we the price bπ(k+1) per click, getting utility αk(vi − ∑ ∑ define as the ratio: E[ αν(j)vj]/E[ αjvπ(j)]. bπ(k+1)). j j Note the asymmetry between the two options. The A. Equilibria with Low Social Welfare and Conservative symmetric (or envy free) equilibria studied by Edelman Bidders et al. [3] and Varian [14] satisfy the stronger symmetric Even for two slots the gap between the best and condition that αj(vi − bπ(j+1)) ≥ αk(vi − bπ(k+1)) for the worse Nash equilibrium can be arbitrarily large. all k. Edelman et al. [3] and Varian [14] show that For example, consider two slots with click-through-rates symmetric equilibria exist and have optimal welfare, α1 = 1 and α2 = 0 and two advertisers with valuations hence the Price of Stability for this game is 1. v1 = 1 and v2 = 0. It is easy to check that the bids We are interested in bounding∑ the Pure∑ Price of b1 = 0 and b2 = 1 are a Nash equilibrium where Anarchy, which is the ratio j αjvj/ j αjvπ(j), advertiser 1 gets the second slot and advertiser 2 gets between the social welfare in the optimum and in the the first slot. The social welfare in this equilibrium is 0 worst Nash equilibrium. while the optimum is 1. The price of anarchy is therefore unbounded. Notice, however, that this Nash equilibrium seems Inspired by the last lemma, given parameters α, v very artificial: the special case of GSP with α1 = 1 and we say that permutation π is weakly feasible if in- α2 = 0 is the Vickrey auction, where truthful bidding equality (1) holds for each i, j. The main result of of bi = vi is a dominant strategy, yet in the equilibrium this section∑ follow∑ from analyzing the price of anarchy above the bids are not truthful. Advertiser 2 above is ratio j αjvj/ j αjvπ(j) over all weakly feasible exposed to the risk of negative utility with no benefit: if permutations π. advertiser 1 (or a new advertiser) adds a bid somewhere B. Price of Anarchy Bound in the interval (0, 1) this imposes a negative utility on advertiser 2. Here we present the bound on the price of anarchy More generally, for any bidder i, bidding above the for weakly feasible permutations, and hence for GSP for valuation vi (with any probability) is dominated by conservative bidders. bidding vi in any of the above models. We state the lemma here in the more general model of Bayesian Theorem III.3 For conservative bidders, the price of games. anarchy for pure Nash√ equilibria of GSP is bounded by 1+ 5 ≈ the golden ratio 2 1.618. Lemma II.1 Given a bidding function bi(vi), a strategy Proof: We will prove the bound by induction for in which P (bi(vi) > vi) > 0 for some vi is dominated all weakly feasible permutations. As a warm-up we will ′ { } by playing bi(vi) = min vi, bi(vi) . prove that the price of anarchy is bounded by 2, since We say that a player is conservative if he doesn’t the proof is easier and captures the main ideas. We use induction on n. The case n = 1 is obvious. overbid, i.e., P (bi(vi) ≤ vi) = 1. We assume throughout the paper that players are conservative. Consider parameters v, α and a weakly feasible permutation π. Let i = π−1(1) be the slot occupied by the III.PURE NASH EQUILIBRIUM advertiser with maximum valuation and j = π(1) be the advertiser occupying the first slot. If i = j = 1 then Theorem III.1 For 2 slots, if all advertisers are con- we can apply the induction hypothesis right away. If not, servative, then the price of anarchy is exactly 1.25. v inequality (1) tells us that αi ≥ 1 or j ≥ 1 . Suppose α1 2 v1 2 αi ≥ 1 Proof: To see that the price of anarchy is achievable α 2 and consider an input with slot i and advertiser 1 − consider two slots with α1 = 1 and α2 = 1/2, and 1 deleted. The permutation π restricted to these n 1 − two bidders with valuations v1 = 1 and v2 = 1/2, and advertisers and n 1 slots is still weakly feasible, so note that the bids b = 0 and b = 1/2 form a Nash by the induction hypothesis: 1 2 ∑ equilibrium. It is not hard to see that this is the worst α v case. See the full version [11] for more details. k π(k) k≠ i A. Weakly Feasible Assignments 1 ≥ (α v + ... + α − v + α v + ... + α v ) 2 1 2 i 1 i i+1 i+1 n n Next we show that equilibria with conservative bid- 1 ders satisfy the property mentioned in the introduction. ≥ (α v + ... + α v + α v + ... + α v ) 2 2 2 i i i+1 i+1 n n We will call the assignments satisfying this property weakly feasible. In the next subsection we analyze the and therefore, ∑ ∑ 1 1 ∑ welfare properties of weakly feasible assignments. α v = α v + α v ≥ α v + α v k π(k) i 1 k π(k) 2 1 1 2 k k k k≠ i k>1 Lemma III.2 For any valuation v, click-through rates If vj ≥ 1 we just do the same but deleting slot 1 and α and a Nash permutation π we have v1 2 advertiser j from the input. This proves the bound of 2. α v j + π(i) ≥ 1; (1) Next we sketch the proof of the improved bound. α v i π(j) See the full version [11] for more details. As before, in particular, αj ≥ 1 or vπ(i) ≥ 1 . we prove the conclusion for all weakly feasible per- αi 2 vπ(j) 2 mutations. Let rk be the worst price of anarchy for Proof: If j ≤ i the inequality is obviously true. feasible permutations in a k slots auction. By the proof Otherwise consider the bidder π(j) in slot j. Since it is of Theorem III.1 we know that r2 = 1.25. We will a Nash equilibrium, the bidder in slot j is happy with generate a recursion to bound rk and then prove√ that 1+ 5 his outcome and doesn’t want to increase his bid to the bound converges to the desired bound of 2 . take slot i, so: αj(vπ(j) − bπ(j+1)) ≥ αi(vπ(j) − bπ(i)) Consider parameter α, v, a weakly feasible permuta- −1 since bπ(j+1) ≥ 0 and bπ(i) ≤ vπ(i) then: αjvπ(j) ≥ tion π and let’s assume i = π (1) and j = π(1) . If αi(vπ(j) − vπ(i)) i = j = 1, the price of anarchy is bounded by rn−1. If not, assume without loss of generality that i ≤ j (since Now the allocation, represented by the permutation π, α1 inequality (1) is symmetric in α and v). Let β = α is also a random variable. For notational convenience, i − and γ = v1 . We know that 1 + 1 ≥ 1. Following the let σ = π 1. We begin by proving a bound similar vj β γ outline of the previous proof we have: to Lemma III.2 for mixed Nash and then using that to ∑ ∑ bound the price of anarchy. Note that by our notational αkvπ(k) = αiv1 + αkvπ(k) assumption the position of bidder i in the optimal k k≠ i allocation is position i. The new inequality is different 1 as it involves a bidder i and its location i in the optimal The first term is bounded by β α1v1. We bound the remaining terms as allocation, rather than two bidders that are allocated to ( ) “wrong relative positions”. ∑ 1 ∑i ∑n αkvπ(k) ≥ αk−1vk + αkvk rn−1 Lemma IV.1 If the random vector b is a mixed Nash k≠ i k=2 k=i+1 [ ] equilibrium for GSP then for each player i: ∑i ∑ 1 E E = (αk−1 − αk)vk + αkvk ασ(i) vπ(i) 1 rn−1 + ≥ (2) k=2 k>1 αi vi 2 1 1 ∑ ≥ (α1 − αi)vi + αkvk Proof: We will consider whether player i benefits rn−1 rn−1 ′ k>1 E by deviating to the deterministic bi = min(vi, 2 bπ(i)). 1 1 ≤ ≥ ≥ We claim that with probability at least 2 , this bid By( the assumption) that i j we have vi vj = γ v1 ′ gets one of the slots of {1, . . . , i}. If b = vi then 1 − 1 v , and we get i β 1 this happens for sure, as our conservative assumption [ ( ) ] ∑ 2 guarantees that only the previous i − 1 players can 1 1 1 ′ E αkvπ(k) ≥ + 1 − α1v1+ bid more. If bi = 2 bπ(i) then by Markov’s inequality E β rn−1 β ′ bπ(i) k ≥ ≤ 1 P (bπ(i) bi) b′ = 2 . Therefore we have that 1 ∑ i + αkvk rn−1 k>1 E ≥ E ≥ E ′ ≥ 1 − ′ ≥ ασ(i)vi ui(b) ui(bi, b−i) αi(vi bi) Symmetrically, we can remove slot 1 and advertiser 2 1 1 j in the inductive step and get a similar equation. The ≥ α (v − 2Eb ) ≥ α (v − 2Ev ) 2 i i π(i) 2 i i π(i) bound for rn is the maximum of the two. Finally, to get bound for rn valid for all β we need to use the Now it is just a matter of rearranging the expression. value of β that minimizes the resulting bound. We get the following recursion for rn Theorem IV.2 The Price of Anarchy for the mixed Nash equilibria of GSP with conservative bidders is at  ( )−1  rn−1 4 most 4.  1 − , r − < 4 n 1 3 ( √ )− Proof: The proof is a simple application of Lemma rn =  1  2 4  r − − r − r − , r − ≥ IV.1 and some algebraic manipulation: n 1 n−1 n 1 n 1 3 [ ] √ ∑ 1 ∑ ∑ To show that the sequence is bounded by φ = 1+ 5 , E[ u (b)] = E α v + E α v = 2 i 2 σ(i) i i π(i) note that if rn−1 ≤ φ then rn ≤ φ. i ( i )i 1 ∑ Eα Ev 1 ∑ = α v σ(i) + π(i) ≥ α v Remark: Proving matching upper and lower bounds 2 i i α v 4 i i i i i i for this problem remains an interesting open problem. The worse example of the Price of Anarchy the authors are aware of (in any of the models) is 1.259 (and it is V. BAYES-NASHEQUILIBRIUM for a pure Nash equilibrium for 3 players). Recall that in the Bayesian setting, the values vi IV. MIXED NASHEQUILIBRIUM are independent random variables, their distributions are As before, we assume that players are numbered such common knowledge, but the value vi is only known to that v1 ≥ ... ≥ vn and slots with click-through rates bidder i. A strategy for a player i is a bidding function α1 ≥ ... ≥ αn. In a mixed Nash equilibrium the bi(vi) (or a probability distribution of such functions) strategy of player i is a probability distribution on [0, vi] where bi(vi) is the player’s bid when his value is vi. As represented by a random variable bi, and we assume that before, we will assume that P (bi(vi) ≤ vi) = 1, since bidders are conservative: P (bi ≤ vi) = 1. overbidding is dominated strategy. We will use π and σ = π−1 to denote the permutation inequality is not established by a single deviating bid. representing the allocation, and we will use ν to denote The structural inequalities of Lemmas III.2 and IV.1 in the random permutation (defined by v) such that player i the full information setting were obtained by consider- occupies slot ν(i) in∑ the optimal solution.∑ The expected ing a single deviation, e.g., for mixed Nash equilibria E E we considered a single bid just above 2Eb , as by social welfare is [ i αivπ(i)]∑ = [ i ασ(i)vi] and the π(ν(i)) E Markov’s inequality this value is above b with social optimum is given by [ i αν(i)vi]. The goal of π(ν(i)) this section is to bound the price of anarchy, the ratio probability at least 1/2. In contrast, in the Bayesian of these two expectations. setting, we obtain our structural result by considering deviations to different bids and then combining them

Theorem V.1 If a set of functions b1, . . . , bn are a using a novel averaging argument. Bayes-Nash equilibrium in conservative strategies then: To define the deviating bids, consider the following [ ] [ ] i ∑ ∑ notation: let π (k) be the bidder occupying slot k in 1 i E α v ≥ E α v the case i didn’t participate in the auction, i.e., π (k) = i π(i) 8 ν(i) i i i i π(k) if σ(i) > σ(k) and π (k) = π(k + 1) otherwise. Note the following property of πi(k) that is, the Bayes-Nash Price of Anarchy in conservative strategies for GSP is bounded by 8. Lemma V.3 A deviating bid B by player i gets a slot The proof of the theorem is based on a structural k or above if and only if B > bπi(k). characterization analogous to the one used for Pure For mixed equilibria in the full information setting, and Mixed Nash equilibria in previous sections, but we considered the bid 2Eb . To extend this to the much harder to prove. The structural characterization π(ν(i)) Bayesian setting, we will consider a sequence of bids, for Mixed Nash (Lemma IV.1) can be written as E E ≥ 1 conditioned on the value of ν(i) defined as vi ασ(i) + αi vπ(i) 2 αivi. The Bayesian structural characterization is obtained by taking expectation of Bk = min{vi, 2E[bπi(k)|vi; ν(i) = k]}. this inequality (and losing a factor of 2). In the full information model, bidder i is assigned to slot i in the Notice that Bk is defined as a conditional expectation, optimum by notation, and the inequality above uses so it is a function of vi, and not a constant. We will this notational convenience. In the Bayesian setting, drop the dependence on vi from the notation as we are the optimal slot for a bidder is a random variable, so focusing on a single value vi throughout the proof. we cannot deterministically order bidders by valuation; The proof of Lemma V.2, depends on two combi- instead we need to use a random variable ν(i) to denote natorial results. The first is a structural property: we the slot bidder i is assigned to in the optimum. claim that the bids Bk are monotone in k for any fixed value of vi. Showing this will allow us to argue that bid Bk not only has a good chance of taking slot k when Lemma V.2 If {bi(·)}i is a Bayes-Nash equilibrium of ν(i) = k, but also has a good chance of taking any other the GSP then for all i and for all vi: ′ ′ slot k > k when ν(i) = k , since B ≥ B ′ . 1 k k v E[α |v ] + E[α v |v ] ≥ v E[α |v ] i σ(i) i ν(i) π(ν(i)) i 4 i ν(i) i Lemma V.4 The expectation E[bπiν(i)|vi, ν(i) = k] is The price of anarchy bound follows from the lemma. non-increasing in k for any fixed value vi. Proof of Theorem V.1 : We will prove the lemma above using flows and the ∑ 1 B SW = E (α v + α v ) = max-flow min-cut theorem. The value k is defined as a 2 i π(i) σ(i) i conditional expectation assuming ν(i) = k, while B i k+1 1 ∑ is defined as a conditional expectation conditioning on = E (α v + α v ) = 2 ν(i) π(ν(i)) σ(i) i a disjoint part of the probability space: assuming ν(i) = [i ] k+1. To relate the two expectations we define a flow of 1 ∑ probabilities from the probability space where ν(i) = k = E E[α v |v ] + v E[α |v ] ≥ 2 ν(i) π(ν(i)) i i σ(i) i to the space where ν(i) = k + 1 that transfers the mass i [ ] b i ∑ of probability with the property that the value π (ν(i)) 1 is non-increasing along the flow lines. This will prove ≥ E viαν(i) 8 that B , the expectation of b i on the source side, i k π (ν(i)) is no bigger than Bk+1, the expectation of the same value on the sink side. The hard part of the proof is proving Lemma V.2. We combine the inequalities obtained by considering The main difficulty in the Bayesian setting is that the the different bids Bk using a novel ”dual averaging argument”, finding an average that will simultaneously Now we use the Lemma V.5 applied with Bk and guarantee that one average is not too low, and a different γk = pkαk. We can interpret xk from the lemma average is not to high. We combine the bids Bk via a as probabilities, and consider the deviating strategy of probability distribution x (bidding Bk with probability bidding Bk with probability xk. xk). The two inequalities of the lemma will guarantee Combining the above inequalities with the coeffi- that the resulting randomized bid, on one hand, gets a cients xk from the Lemma, we get that high enough number of clicks, and on the other hand, the resulting payment is not too large. ∑ ∑ 1 v E[α |v ] ≥ x p α (v − B ) ≥ i σ(i) i k 2 j j i k Lemma V.5 Given any nonnegative values γ ,B there k j≥k ∑ k k ∑ ∑ is a probability distribution x ≥ 0, x = 1 such 1 1 k k k ≥ v α p − p α B ≥ that 4 i j j 2 j j j ∑ ∑n 1 ∑n j j xk γj ≥ γj 1 2 ≥ viE[αν(i)|vi] − E[αν(i)b i |vi]. k j=k j=1 4 π (ν(i)) ∑ ∑n ∑n To get the claimed inequality, note that bπi(k) ≤ x B γ ≤ γ B k k j j j bπ(k) ≤ vπ(k). k j=k j=1

Before we prove these key lemmas, we show how to A. Proving that bids Bk are non-increasing use them for proving the main Lemma V.2: We will prove Lemma V.4 in several steps. First we Proof of Lemma V.2 : As outlined above, we will prove bounds assuming all but a single player has a consider n deviations for a player i at bids Bk for all deterministic value, and we take expectations to get a possible slots k. Since the bidding functions are a Nash conditional version. We define a probability flow from equilibrium, player i can’t benefit from changing his the probability space where ν(i) = k to the space where strategy, and so each deviation will give us an inequality ν(i) = k + 1 that transfers the mass of probability on the utility of player i. We will use Lemma V.5 to so that only a single value is changing along the flow average the inequalities and get the claimed inequality. edges, and hence by the first claim the value bπi(ν(i)) is Suppose bidder i deviates to Bk = non-increasing along the flow lines. In transferring the { E | } ′ min vi, 2 [bπi(k) vi; ν(i) = k] . Let αk be the probability mass we take advantage of the fact that the random variable that means the click-through rate of valuations are drawn from independent distributions. the slot he occupies by bidding B . First we estimate k Proof of Lemma V.4 : We want to prove that the probability that by bidding Bk the player gets the slot k or better when ν(i) = k. In the case Bk = vi this E[bπi(k)|vi, ν(i) = k] ≥ E[bπi(k+1)|vi, ν(i) = k + 1]. is trivially guaranteed, since only ν(i) − 1 players have The value vi is in position k in the optimum if exactly values above vi and only these players can bid above n − k values are below vi. Consider such a set S of v . If B = 2E[b i |v ; ν(i) = k], we use Lemma V.3, i k π k i agents, i∈ / S, and the corresponding event: and Markov’s inequality to get: ′ ≥ | A = {v ≤ v ; ∀j ∈ S, v > v ; ∀j∈ / S}. P (αk αk vi, ν(i) = k) = S j i j i 1 The event ν(i) = k can now be stated as ∪|S|=n−kAS, = P (Bk ≥ bπi(k)|vi, ν(i) = k) ≥ . 2 and so what we are trying to prove is: | Let pj = P (ν(i) = j vi). Recall that by Lemma E | ∪ ≥ E | ∪ [bπi(k) vi, |S|=n−kAS ] [bπi(k+1) vi, |S′|=n−k−1AS′ ] V.4 we have that B1 ≥ B2 ≥ ... ≥ Bn, and ′ ′ hence the probability of bid Bk taking a slot j or Consider a pair of sets S ⊆ S, i.e., S = S ∪ {t} for better when ν(i) = j is also at least 1/2 whenever some agent t ≠ i. The first claim is the following. j ≥ k. The expected value of bidding Bk is at least E ′ − | ′ ′ ∪ { } ̸ [αk(vi Bk) vi], and the value for player i in the Claim V.6 For sets S and S = S t for t = i, current solution is at most viE[ασ(i)|vi]. This leads to E[b i |v ,A ] ≥ E[b i |v ,A ′ ] the following inequality. π (k) i S π (k+1) i S v E[α |v ] ≥ E[α′ (v − B )|v ] = To see this, notice that i σ(i) i ∑ k i k i ′ E | { } ≥ E | { } E − | ≥ [b i vi,AS , vj j≠ i,t] [b i vi,AS′ , vj j≠ i,t] = pj [αk(vi Bk) vi, ν(i) = j] π (k) π (k+1) j The conditioning on the two sides differs only by the ∑ 1 ≥ p α (v − B ). value of bidder t. In identical conditioning and identical 2 j j i k j≥k bids, the bid of position k is clearly higher than the bid of position k + 1, and by letting one bidder (bidder t) that needs to leave the set, is at most the demand that change, we can’t violate the above inequality. Taking is available at the neighbors of the set: the expectation over the valuations {vj}j≠ i,t and the bids used (if the strategies are randomized) we get the ∑p ′ ∑ ∑ϕ(Ai) ≤ ∑ϕ(A) inequality of Claim V.6. ′ ′ ϕ(S ) ϕ(S) i=1 S ′ ⊆ | | − S To finish the proof of Lemma V.4, we would like to Ai A, A =n k ′ add the inequalities for different set pairs (S, S ). The which can be rewritten as next combinatorial lemma states that if the values v ∑ ∑ ∑ ∑ i · ′ ≤ · ′ are drawn from independent distributions, then there is ϕ(S) ϕ(Ai) ϕ(A) ϕ(S ) S i A′ ⊆A,|A|=n−k S′ a “probability flow” λS,S′ that transfers the probability i mass from ∪|S|=n−kAS to ∪|S′|=n−k−1AS′ along the Notice that both sides have sums of products of 2(n − ′ ⊆ pairs S S. More formally, Lemma V.7 will show that k) − 1 terms of type ϕj. If we can prove that all terms ′ there are coefficients λS,S′ ≥ 0 for S ⊆ S such that in the LHS appear in the RHS with at least the same ∑ multiplicity we are done. We prove this based on a ′ ′ | ∪ ′ ′ λS,S = P (AS vi, |T |=n−k−1AT ) combinatorial construction. S ∑ The left-hand side consists of products of ϕ values for pairs of sets (S, A′ ). The right-hand side contains the λ ′ = P (A |v , ∪| | − A ) i S,S S i T =n k T − ′ S′ products of ϕ values for pairs of the form (S j, Ai +j) ∈ \ ′ ′ for some j S Ai. We want to map each pair (S, Ai) Taking linear combination of the inequalities (V.6) for − ′ ′ to a set (S j, Ai + j) without collisions. If we can set pair (S, S ) with coefficients λS,S′ gives the claimed 1 ′ do this, it proves the claim. We say the pairs (S ,Ai) bound. 2 ′ 1 ∪ 2 ∪ and (S ,Aj) are equivalent if S Ai and S Aj are the same (including multiplicities of the elements). Lemma V.7 If valuations are drawn from independent Now we just need to map each equivalence class of elements in a collision-free manner. Lemma V.8 below distributions, then there exists a probability flow λS,S′ ≥ 0 for set pairs S′ ⊆ S with |S′| = n − k − 1 and shows a construction that satisfies the property for t = 1 | ∪ ′ | − | ∩ ′ | − ∪ ′ \ ∩ ′ |S| = n − k such that the equations above hold. 2 ( S Ai S Ai 1): identify (S Ai) (S Ai) ′ \ \ ′ with [2t + 1] and choose( ) j = ft(Ai S) Ai, where { } S { ⊆ | | } Proof: We will use the max-flow min-cut theorem [n] = 1, . . . , n and t = T S; T = t . to prove that the λS,S′ values exist. We characterize the probabilities P (AS|vi, ∪|T |=n−kAT ) using the indepen- Lemma( ) V.8( For all) t there is a bijective function ft : [2t+1] → [2t+1] ⊆ dence assumption. Let qj = P (vj ≥ vi), then we can t t+1 such that S ft(S). write ∏ ∏ Proof:( Consider) a bipartite graph where( the) left − [2t+1] [2t+1] j∈S qj j∈ /S+i(1 qj ) P (A |v , ∪ A ) = ∑ ∏ ∏ nodes are t and the right nodes are t+1 and S i |T |=n−k T − . |T |=n−k j∈T qj j∈ /T +i(1 qj ) there is an (A, B) edge if A ⊆ B. Notice this is a ∏ qj regular k + 1-graph. Since all regular bipartite graphs If we define ϕj = − and ϕ(S) = ∈ ϕj then we 1 qj j S have perfect matchings, the claim is proved. can rewrite the above equation as B. Proving the dual averaging Lemma ∑ ϕ(S) P (AS|vi, ∪|T |=n−kAT ) = . |T |=n−k ϕ(T ) Proof of Lemma V.5 : We want to prove that the following linear inequality system is feasible: The existence of the λS,S′ values is equivalent to the existence a flow of value 1 in the following network: ∑ ∑n 1 ∑n − x γ ≤ − γ consider a bipartite graph where the left nodes are k j 2 j sources corresponding to sets S′ with |S′| = n − k − 1 k j=k j=1 ′ ∑ϕ(S ) ∑ ∑n ∑n ′ with supply ′ ϕ(T ) and the right nodes are sinks T x B γ ≤ γ B corresponding to sets S of |S| = n − k with demand k k j j j k j=k j=1 ∑ϕ(S) . We add an edge (S′,S) with capacity ∞ if ∑ T ϕ(T ) S′ ⊆ S. We need to prove that the max-flow in this xk = 1 graph has flow value 1 (and then the flow values define k xk ≥ 0 λS′,S). 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