Characterizing Optimal Keyword Auctions ∗ Garud Iyengar
Anuj Kumar
Industrial Engineering and Operations Research Department, Columbia University New York, NY-10027
Industrial Engineering and Operations Research Department, Columbia University New York, NY-10027
[email protected]
[email protected]
ABSTRACT We present a variety of models for keyword auctions used for pricing advertising slots on the search engines (Google Inc., Yahoo! Inc. etc.). First, we formulate the general problem that allows the privately known valuation per click to depend both on the identity of merchant and the slot and presents a compact characterization of the set of all deterministic incentive compatible direct mechanisms. In particular, we show that there are incentive compatible mechanisms which are not affine maximizers in this multidimensional type model. Next, we study two interesting cases of this model: slot independent valuation and privately known constant valuations (i.e. slot independent) up to a privately known slot and zero thereafter. We characterize revenue maximizing as well as efficiency maximizing incentive compatible (IC) and individually rational (IR) mechanisms. These mechanism are computationally efficient with time complexity O(n2 m2 ) where n is no. of bidders and m is no. of slots. We also characterize the optimal mechanism when the mechanism is restricted to be rank-based. We use dominant strategy equilibrium as the solution concept.
Keywords Pay-per-click Keyword Auctions, Dominant Strategy Equilibrium, Assignment Problem.
1.
INTRODUCTION
Online advertising is a major source of revenue for internetbased companies. For example, Google’s total revenue reported online at Google’s website in the year 2005 was $6 billion, 98% of which came primarily through search keyword advertisements. Yahoo! reported a revenue of $5.258 billion for 2005, a 47% increase over 2004. It is believed that more than 50% of Yahoo!’s revenue were from keyword auctions. Search keyword advertisements work as follows. When an ∗Prepared for Second Workshop on Sponsored Search Auctions, June 11, 2006 Ann Arbor, Michigan in conjunction with the ACM Conference on Electronic Commerce (EC’06)
Internet user searches for a keyword in an online search engine, the results page displayed by the search engine contains the links to most relevant webpages and a number of sponsored links relevant to the keyword. For example - when we search for “Delhi” at Google.com, eight sponsored links appears on the results page which, for example, including links to websites of the hotels in Delhi. For every user click on any of these sponsored links, the search engine get paid a price from the advertiser. One would expect that higher the slot of an sponsored link the better a chance that a user would click on the link. Thus, the advertiser prefers higher slot. Similarly, higher the exogenous brand values of the advertiser, better the chance that the user would click on the link. Thus, the search engine prefers to allocate the slots according to the brand value of the advertisers. In conclusion, the search engine needs a pricing mechanism for allocating the slots to the advertisers. Since the number of merchants interested in displaying their advertisement on the results page typically is more than one - auctions for their revenue generation and allocation efficiency are a natural choice. We aim to design and analyze models for such keyword action for their incentives, revenue and efficiency. Since in the keyword auctions, the bidders can revise their bid at any time resulting in an updated winner pool, dominant strategy is the only appropriate solution concept under which the static one shot auction game is a good model for these dynamic auctions. The contributions of this paper can be summarized as follows, 1. We present a general model for “pay-per-click” keyword auctions in which the merchants have multidimensional private valuations per click. Specifically the valuation vij of merchant i depend on the slot j allocated. In this setting we characterize the set of all dominant strategy incentive compatible, individually rational allocation rules. Any deterministic, feasible IC and IR mechanism is equivalent to setting bidder dependent prices pi which depend on other bidder’s valuations so that each bidder self select the slot assign to him, maximizing it’s own surplus. From this results we also deduce that not all incentive compatible allocation rule are affine maximizers (see Example 1) in this setting. 2. If the value per click of each merchant is uniform across slots, we present a complete solution, based on the
mechanism design with single dimensional type [12], which include characterization of the set of all dominant strategy incentive compatible, individually rational allocation rules, the unique prices that implements it and revenue maximizing auction with a computationally efficient algorithm to implement it. In this model, we also analyze rank based allocations rules currently used in practice. An interesting observations about the rank based allocation rules is that if the click through rates are separable (which is assumed, for e.g. in [8, 5]) - the efficiency maximizing rank vector is different from a revenue maximizing1 rank vector and unconstrained revenue maximizing mechanism is not rank based (see Example 2). 3. We analyze a model in which the valuations are privately know constant up to a privately know slot and zero thereafter. We present a suboptimal (revenue maximizing) IC and IR mechanism for this model. The paper is organized as follows. We discuss the relevant literature in §1.1. In § 2 we describe the general multidimensional model. In § 3 we present the analysis of the uniform valuation model. In § 4 we present the model with privately know value up to a privately known slot. We conclude in § 5.
1.1
Previous Literature
The online keyword auctions came to attention of the academic community only recently – after all, search engine advertizement and the Internet itself is a recent phenomena considering the long tradition of academic research in Auction Theory. Among the papers that specifically address the keyword auction problem from the prospective of Auction Theory, are by Aggarwal, Goel and Motwani [1], Edelman, Ostrovsky and Schwartz [5] and Lahaie [8]. [5] presents a keyword auction model with separable click-though-rate and generalized second price payment rule and observe that truth-telling is not a dominant strategy under generalized second price rule. In [1], the authors present a payment rule under which any rank based allocation rule can be made dominant strategy incentive compatible and show that if the click-though-rate are separable, in which case the VickeryClark-Groves (VCG) allocation rule is rank based, there exist an ex-post equilibrium which results in same pointwise revenue as the generalized second price auction. This equilibrium is s simple adaptation of the one derived in [5] (Theorem 8) who assume separable click through rate. As we demonstrate, the revenue equivalence holds with uniform valuations as direct consequence of single dimensional private information. Lahaie [8] consider a uniform valuations model with separable click-through-rate and computes the equilibrium bidding strategies in rank based mechanisms with first price and second price payment schemes in both complete and incomplete information setting. Varian [16] characterize the Nash Equilibrium in the first price version of the adword auction problem in which m highest bids are allocated the slots. In his model, the value per click only depend on the agent and the click through 1
within the class of rank based allocation rules
rate only depend on the slot. [16] also reports results of comparing the Nash Equilibrium predictions with the empirical prices. Shapley and Shumik [15] describe an assignment game in which agents are assigned objects with at most one object being assigned to an agent. Several papers have developed auctions that yield efficient equilibria for the assignment game; see [3] for a recent survey. Leonard [10] show that optimal dual prices of the slots of the matching Linear programming problem implement efficiency maximizing allocation (optimal matching) in dominant strategy. The results in Aggarwal and Hartline [2] are related to the slotted mechanism described in this paper, where merchants having a commonly known size (a.k.a. demand) have a privately know value of being included in the knapsack. The mechanism designer seeks a static (not contingent on the bids) vector of prices, which induce truth-telling. The competitive ratio is used a selection criteria among the feasible mechanisms. In contrast, in our model the demands (the ki ’s) themselves are privately known, but we allow prices to be contingent on the bids. These results are primarily based on Iyengar and Kumar [7], where the author studies one-sided incentives in a reverse auction model. Vohra and Malakhov [17] also consider a similar but restrictive model where the social surplus does not depend on the component of private information with one sided incentive which is not adequate in the setting considered in this paper as we discuss in §4.
Notation We denote vectors by boldface lowercase letters, e.g. v. A vector indexed by −i, (for example x−i ) denotes the vector x with the i-th component excluded. We use the convention x = (xi , x−i ). Scalar (resp. vector) functions are denoted by lowercase letters, e.g. Xij (vi , v−i ) (resp. X(vi , v−i )). The possible misreport of the true parameters are represented ˆ ). with a hat over the same variable, e.g. v
2.
KEYWORD AUCTION MODEL
There are n merchants bidding for m(≤ n) slots on a specific keyword. Let cij denote the click-through-rate when merchant i is assigned to slot j. For convenience, we will set ci,m+1 = 0 for all i = 1, . . . , n. Assumption 1. The click through rates {cij } satisfy the following conditions. (i) For all merchants i, the rate cij is decreasing in j. (ii) All the rates {cij : i = 1, . . . , n, j = 1, . . . , m} are known to the auctioneer. (iii) The rates {cij : j = 1, . . . , m} are known to merchant i only. The true expected value2 vij of slot j to merchant i when assigned to slot j is private information. We assume Inde2 For multi-dimensional model we absorb the click-though rate inside the bid so that the true value of slot j can be vij t though of as vij = cij
pendent Private Values (IPV) with a commonly known prior distribution function that is continuously differentiable with density fi : Rm + 7→ R++ . It is worth emphasizing that even through we use dominant strategy as a solution concept, we do need the prior distribution to be able to select the optimal mechanism. We restrict attention to direct mechanisms – the revelation principle ensures that this does not introduce any loss of generality. Rn×m +
Let b ∈ denote the bids of the n merchants. An auction mechanism for this problem consists of the following two components. 1. An allocation rule X : Rn×m 7→ {0, 1}n×m that satis+ fies Pn i=1 Xij (b) = 1, Pm j=1
Xij (b) ≤ 1,
j = 1, . . . , m, i = 1, . . . , n.
Thus, X(b) is a matching that matches merchants to slots as a function of the bid b. Henceforth we denote the set of all possible matchings of n merchants to m slots by Mnm . 2. A payment function T : Rn×m 7→ Rn that specifies + what each of the n bodders pay the auctioneer. We normalized the payment of the bidder who is not allocated any slot to zero which is without loss of generality as we explain later. Thus, we can define the per click payment of merchant i as
Also let ui (vi ; (X, T), v−i ) = max u ˆi (b, vi ; (X, T), v−i ) m b∈R+
(4)
denote the utility attainable by merchant i under the mechanism (X, T). If the mechanism (X, T) is IC and IR then clearly, ui (vi ; (X, T), v−i ) = u ˆi (vi , vi ; (X, T), v−i )
(5)
Next, we develop an alternate characterization of IC when the private information is multi-dimensional. Fix v−i and consider the optimization problem of agent i, ui (vi , v−i ) = maxm ˆ i ∈R+ v
m X
(vij − Ti (ˆ vi , v−i ))Xij (ˆ vi , v−i )
j=1
Clearly ui is convex in vi since it is a maximum of a collection of linear functions and by envelope conditions its gradient (under truth-telling) is given by ∇vi ui (vi , v−i ) = (ci1 Xi1 (vi , v−i ), . . . , cim Xim (vi , v−i ))T
a.e.
Thus, an incentive compatible allocation rule is always a sub-gradient of some convex function and hence integrable and monotone3 , as was first observed by Rochet [13]. Several authors have characterized the set of incentive compatible allocation rule in quasi-linear environments (see [9, 6, 4, 14] ) in terms of no 2-negative cycles condition (also called weak monotonicity): m X
cij (vij Xij (v) + v˜ij Xij (˜ vi , v−i ))
j=1
Ti (v) . j=1 cij Xij (v)
Pm
≥
m X
cij (vij Xij (˜ vi , v−i ) + v˜ij Xij (v))
˜ i , v−i ∀vi , v
j=1
For v ∈ Rn×m and i = 1, . . . , n, let + u ˆi (b, v; (X, T), v−i ) =
m X
vij − Ti (b, v−i ) Xij (b, v−i )
j=1
(1) denote the utility the merchant i of type vi who bids b. When the mechanism (X, T) is clear by context, we will write the utility as u ˆi (b, v; v−i ). We want the mechanism to (X, T) to satisfy the following two properties. n×m 1. Incentive compatibility (IC): For all v ∈ R+ and all i = 1, . . . , n,
vi ∈ argmax ui (b; (X, T), v−i ) ,
(2)
b∈Rm +
i.e. truth telling is ex-post dominant. 2. Individual rationality (IR): For all v ∈ Rn×m and all + i = 1, . . . , n,
argmax ui (b; (X, T), v−i ) ≥ 0,
(3)
b∈Rm +
i.e. we implicitly assume that the outside alternative is worth zero.
which says that the sum of utility allocated to merchant i at ˜ i under truthful bidding is greater than the value vi and v ˜ i at sum of utility allocated to merchant i if he lie and bid v ˜ i . This condition for a convex domain implies vi and vi at v that the allocation X is integrable and the transfer payment implementing X are well defined. On the other hand [4] proposes an implementability condition called quasi-efficiency for general quasi-linear preferences, which says that an allocation rule X is IC if and only if, for all i, θ there exist functions gi : Θn−1 × A 7→ R such that X(θ) = argmax vi (θi , a) + gi (θ−i , a) a∈A
where θ is the private information and A is the set of allocations. Note that for efficient allocations which are always implementable the functions gi is precisely the sum of utilities of all bidders other than i at their respective type θ−i of allocation a. One drawback of these characterizations is that they do not throw any light on the transfer payments except to guarantee existence. Our characterization precisely characterizes the implementable allocation rules in terms of existence of bidder dependent object prices. Lemma 1. An allocation rule X : Rn×m 7→ Mnm is in+ centive compatible if and only if for all 1 ≤ i ≤ n and 3
m The function X : Rm is monotone if for every y, z ∈ + 7→ R m T R+ , (y − z) (X(y) − X(z)) ≥ 0
(n−1)×m
v−i ∈ R+ , there exists prices pi ∈ (R ∪ {∞})m and pi0 ∈ R ∪ {∞} such that, Xij (v) = 1 ⇒ vij − pij ≥ max(vik − pik , −pi0 )
∀j, k, j 6= k (1,0)
∇ui (vi ) = ej for some j ∈ {0, 1, . . . , m}
ci1(vi1-a)=ci2(vi2-b) vi2 →
Proof: Fix i, v−i and suppress the dependence v−i . X is IC iff there exists convex functions (i.e. the indirect utilities) u i : Rm + 7→ R such that a.e.
(1,0)
where ej is the j-th unit vector and e0 = 0. The absolute continuity of convex functions then implies that ui is piecewise linear. A piecewise linear function is convex if and only if it is the pointwise maximum of each pieces, thus giving n
ui (vi ) = max
0≤j≤m
ej T vi − pij
(0,0)
o
∀vi .
vi1→
(6)
On the other hand we have ui (vi ) =
Figure 1: IC Allocation Rules
m X
(vij − Ti (vi ))Xij (vi )
(n−1)×m
j=1
Now for j = 1, . . . , m define the set Sj = {x ∈ Rm + |Xij (x) = 1} and S0 = {x ∈ Rm + |Xij (x) = 0 ∀1 ≤ j ≤ m}. We claim that Ti (vi ) = Tij for all vi ∈ Sj i.e. the payment for merchant i does not change with its bid as long as he gets the same slot4 . Suppose this is not the case and there exist vi1 6= vi2 ∈ Sj such that Ti (vi1 ) > Ti (vi2 ) (w.l.o.g.), clearly bidder with valuations vi2 would lie and bid vi1 . Thus we can write ui (vi ) =
m X
(vij − Tij )Xij (vi )
(7)
j=1
Comparing (6) and (7), and noting that Xij (v) = 1 iff ∇ui = ej we get Tij = pij and Xi (vi ) ∈ argmax{vij − pij }
is uniquely identified by the pricing rules pi : R+ (R ∪ {∞}m .
7→
To further understand the relationship between the characterization in Lemma 1 and integrability and monotonicity of the IC allocation rule. Fix v−i . Lemma 1 implies an allocation rule is IC if and only if the following two conditions are satisfied. 1. If we increase vij , keeping the rest of the component of vi constant, there exist a threshold value pij such that for all vij ≤ pij merchant i is not allocated a slot j and for all vij > pij merchant i is allocated slot j. 2. The normal to the hyperplane separating the region in which merchant i is allocated slot j (i.e. Sj ) and the region in which merchant i allocated slot k (i.e. Sk ) is parallel to (0, . . . , cij , . . . , −cik , . . . , 0).
0≤j≤m
with vi0 = 0 ∀i for notational ease. Thus Xij = 1 implies that the argmax above contains j, proving the result. Note that IR implies that pi0 ≤ 0 since −pi0 is the surplus of bidder i when no slot is allocated to him. For any given IC and IR mechanism and a fixed v−i , let p = ˜ i = pi − pe also satisfies IC. To min0≤j≤m (pij ) < 0 then p ˜ also satisfy IR, let j ∗ ∈ argmax0≤j≤m {vij − pij }, see that p then vij ∗ − pij ∗ ≥ vik − pik ≥ −pik ≥ −p
Condition 1 above implies monotonicity of X (convexity of the surplus function) and condition 2 implies integrability of X. Figure 1 shows an allocation rule with two advertising slots illustrating the above conditions. For fixed v−i , there exist pi1 ≥ 0 and pi2 ≥ 0 such that 8 > : (0, 1)
vi1 ≤ pi1 , vi2 ≤ pi2 vi1 ≥ p1 , ci2 vi2 ≤ pi2 + ci1 (vi1 − pi1 ) otherwise
General Pricing Rules vs. Affine Maximizers
It is well known that truth telling can be implemented in dominant strategies by the Vickery-Clark-Groves mechanism. The social surplus maximization problem, given by Φ(v, n) =
max
X∈Mnm
n X m X
vij xij
(8)
i=1 j=1
can be solved in O(nm) time as min-cost flow problem on a bipartite graph (as well as any affine maximization problem). In particular the set of optimal solution of the LP relaxation always contains an integral solution.
Let the optimal allocations function in the above matching problem be given by Xe . Define the per click V CG prices as follows, pei (v) =
m X 1 n j=1
cij
o
vij − (Φ(v, n) − Φ(v−i , n − 1)) Xij (v) (9)
Using LP duality, it can be easily verified pei (v) is equal to dual price of the slot allocated to merchant i in the optimal matching problem and zero if no slot is allocated to merchant i (see [3]). In terms of the Lemma 1, the bidder dependent slot prices can be computed as follows. Fix v−i , chose an ¯ i and for each j = 1, 2, . . . , m increase vij high arbitrary v enough (say vij = kv−i k1 + k¯ vi k1 ), set pij equal to the resulting V CG slot price νj . Note that the dual price νj in the LP are piecewise linear functions of the valuations of bidders other than i. This is true for any affine maximizers5 , w,r
X
1
v12 ) 1+γ = −p22 (v11 , v12 ) which are strictly non-linear functions of v2 and v1 respectively. Thus this is not an example affine maximizing allocation rule.
(v) ∈ argmax X∈Mnm
n X m X
[wi vij + rij ]Xij
i=1 j=1
specifically the prices pij would be of the form pij = Kij +
It is easy to see that for n = 16 , The optimal deterministic selling mechanism is to set slot prices, p∗ ∈ argmax p∈Rm +
pj Pv j ∈ argmax(vk − pk ), vj ≥ pj 1≤k≤m
j=1
(10) For n > 1, The optimal mechanism design problem is rather involved and we leave it as par of ongoing research. We conclude this section with the following theorem. Theorem 1. There exist individually rational and incentive compatible deterministic direct mechanism which are not affine maximizers.
3.
XX
m X
UNIFORM VALUATIONS ACROSS SLOTS
Suppose that for all merchant, the true valuations per click does not depend on the slot they get the click at i.e. for all i = 1, . . . , n and for all j = 1, . . . , m we have vij = vi .
(wi0 vi0 j − wi0 vi0 k )
i0 6=i j,k
Dual prices are equal to the node potential in the equivalent min-cost network flow problem, which are equal to the length of the shortest path in the residual network with arc costs as negative of valuations and hence linear function of the valuations.
3.1
Incentive compatible mechanisms
The following Lemma characterize the set of all ex-post incentive compatible (IC) mechanisms.
A natural question asked in mechanism design problems with multi-dimensional types and quasi-linear preferences is: “Does there exist incentive compatible allocation rules which are not affine maximizer?” Lavi et al. [9] ask this (Open Problem 2, page 36) for a matching problem which does not satisfy their condition of conflicting preferences. Since Lemma 1 allows the dependence of the prices pi on v−i in a general fashion, and the above discussion shows that with the affine maximizers this dependence is linear. If we could choose the pricing rules of the form,
Lemma 2. An feasible allocation rule X is incentive compatible if, and only if, for all i and v−i there exist thresholds 0 ≤ am ≤ am−1 ≤ · · · ≤ a1 ≤ ∞ such that slot j is allocated to merchant i iff vi ∈ (aj , aj−1 ] and the corresponding prices per click, pij , if slot j is allocated to bidder i, that implement X are given by,
pij (v−i ) = hij (· · · , ±(vi0 j − vi0 k ), · · · )
Proof: Based on [11], it is straightforward to observe that X is incentive compatible if and only if the total click-through P rate m j=1 cij Xij (vi , v−i ) for merchant i is a non-decreasing n−1 function of vi for all 1 ≤ i ≤ n and v−i ∈ R+ . Since Xij ∈ {0, 1} and cijP≥ ci,j+1 by Assumption 1(i) the total clickthrough-rate m j=1 cij Xij (vi , v−i ) is non-decreasing with vi iff there exist thresholds 0 ≤ am ≤ am−1 ≤ · · · ≤ a1 ≤ ∞ such that X allocate slot j to merchant i iff vi ∈ (aj , aj−1 ]. This is illustrated in Figure 3.1.
where hij (.) is a monotone with each arguments; other arguments fixed, then it would imply that there are IC mechanisms which are not affine maximizers. Based on the above discussion the following example gives an allocation rule which is not an affine maximizer. Example 1. Consider a situation with two slots and two bidders. Let object 1 is allocated to bidder 1 iff v11 − v12 ≥ sign(v21 − v22 ) · (kv21 − v22 k)1+γ for some 0 < γ < 1 and to bidder 2 otherwise. Since bidder 2 get object whenever,
pij (v−i ) =
m 1 X (ak − ak+1 )(cij − ci,k+1 ) cij
(11)
k=j
Furthermore, the expected payment Ti (vi ; v−i ) of merchant i under X must be of the form Ti (vi ; v−i ) =
m X
cij vi Xij (vi , v−i )
j=1
Z
1
v21 − v12 ≥ sign(v11 − v12 ) · (kv11 − v12 k) 1+γ − this allocation rule is IC with the pricing rules p11 (v21 , v22 ) = (v21 − v22 )1+γ = −p12 (v21 , v22 ) and p21 (v11 , v12 ) = (v11 − 5
The biases only works as reservation prices
0
vi
m X
cij Xij (u, v−i ) du − ui (0, v−i ).
j=1
6 An application of n = 1 model is the problem of optimal lease pricing for slots on the public webpages
in same revenue pointwise to the truth-telling equilibrium in the direct mechanism.
Allcated Click-through Rate
↑
Define the pointwise maximizing allocation,
payment_i(v)
(
X∗ (v) ∈ argmax
cij
X∈Mnm
ci ,m−3
ci ,m−1 cim
aim
ai ,m −1
aij v
vi →
Figure 2: Surplus as function of allocated clickthough-rate The payment above is precisely the area of the shaded region in Figure 3.1. Without loss of generality, we set the surplus of loosing merchants, ui (0; v−i ) = 0. Given the thresholds a the payment per click, pij , if slot j is allocated to bidder i, is equal to Ti (vi , v−i ) divided by cij , thus pij (v−i ) =
1 n (am − 0)(cij − 0) + (am−1 − am )(cij − ci,m−1 ) cij o
+ · · · + (aj − aj+1 )(cij − ci,j+1 ) =
m 1 X (ak − ak+1 )(cij − ci,k+1 ) cij k=j
where for convenience, we used ci0 = 0 ∀i. We remark that the price in the lemma above, alternatively can also be written as, m 1 X (cik − ci,k+1 )aij pij (v−i ) = cij
(12)
k=j
by summing vertical the area of “horizontal” triangles (see Figure 3.1). Based on [12], it is straightforward to note that the expected profit of the online service provider under any dominant strategy incentive compatible allocation rule X is give by, E v
" n m XX i=1 j=1
cij
1 − Fi (vi ) vi − fi (vi )
Xij (v) +
n X
#
ui (0, v−i )
i=1
Thus, any two mechanisms (direct or indirect) which at a7 dominant strategy equilibrium agree on the pointwise assignment X(v) for every v and on the equilibrium utilities8 ui (0, v−i ) for all i, v−i leads to identical expected revenues. Note that in [1, 5] the authors explicitly construct one equilibrium for the generalized second price auction which results 7 In case of multiple equilibria, the revenues may be dominated 8 For revenue (or efficiency) maximization, we can restrict the mechanisms to ui (0, v−i ) = 0 ∀i
n X m X
cij
vi −
i=1 j=1
1 − Fi (vi ) fi (vi )
)
Xij
It is easy to see that the slot allocated to bidder i at the optimal solution, X∗ is non-increasing in vi (remember slot 1 has highest click-through rate), under the regularity condition i (vi ) that the virtual valuations per click µi (vi ) = vi − 1−F fi (vi ) 9 are non-decreasing in vi . Thus the total click-through rate allocated to bidder i is non-decreasing in vi . Hence X∗ is incentive compatible (Lemma 2). Since pointwise maximum is an upper bound on any expected revenue maximizing allocation, (X∗ , T∗ ) is expected revenue maximizing, dominant strategy incentive compatible, individually rational allocation rule with payments per click given by equation (11). Next, we present an algorithm to compute the payment implementing the allocation rule X∗ efficiently. Consider the allocation rules of the form, ψ
X (v) ∈ argmax X∈Mnm
( n m XX
)
cij ψi (vi )Xij
i=1 j=1
for any set of ψi : R+ 7→ R+ , ψi ∈ C[R] and ψi nondecreasing. We call Xψ monotone maximizer with respect to the set of monotone transformations ψi , i = 1, . . . , n. Given v, Xψ (v) can computed efficiently in O(mn) time as a solution to optimal assignment problem. It is straight forward to observe that Xψ is incentive compatible and hence has the form presented in Figure 3.1. Our next result shows that the prices implementing Xψ given by equation (11) can be computed efficiently. Lemma 3. Given v, c and a routine OPTMATCH(m, n, z, c) that solves the optimal bi-partite matching of n merchants to m slots with valuations z and click-through rate c in O(mn) time, the Algorithm 1 correctly compute the prices pij implementing Xψ in O(n2 m2 ) time. Proof: To show that Algorithm 1 is corect, w.l.o.g. consider the computations for bidder 1 and slot p, 1 ≤ p ≤ m. The call OPTMATCH(n − 1, m − 1, z−1 , c−1,−p ) at step (4) returns the optimal matching by excluding bidder 1 and slot p. Let (r, s) denotes the optimal dual vector at this solution. For i = 2, . . . , n, ri ≥ 0 is the surplus of a generic bidder i and for j 6= p, sj is the price of generic slot j in this optimal dual vector. The Algorithm 1 computes the minimum z1 so that it is optimal to allocate slot p to bidder 1. Given z1 it is optimal to allocate slot p to bidder 1 iff there exist (r1 , sp ) such that ((r1 , r), (sp , s)) satisfy dual feasibility and complementary slackness at z1 . Dual feasibility requires, ri + sp ≥ cip zi
∀i 6= 1 ⇐⇒ sp ≥ max{cip zi − ri } i6=1
(13)
9 In general, we can iron the virtual valuations easily to get non-decreasing virtual valuations νi (vi ), for details see [12]
and r1 + sj ≥ c1j z1
∀j 6= p ⇐⇒ r1 ≥ max(0, max{c1j z1 − sj })
vector w ∈ Rn if it allocate the slot j to a merchant in the j th decreasing order statistics of {wi bi }n i=1 .
j6=p
(14) and complementary slackness at the proposed solution requires, r1 + sp = c1p z1
w Let w ∈ Rn + be any vector. Let X denote the rank based allocation rule with ranking vector w. Define γ = [w1 b1 , . . . , wn bn ]. Then under Xw , the total click-through rate for merchant i is given by
(15)
By (13) and (14), (15) is feasible if and only if z1 satisfy
m X
max(0, max{c1j z1 − sj }) + max{cip zi − ri } − c1p z1 ≤ 0 j6=p
i6=1
The LHS above is piecewise linear convex function of z1 ≥ 0, hence it’s level set S1p at level 0 must be an interval or empty (This is infect already known by the implementability of Xψ ). Hence a ˜1p = inf z1≥0 {z1 ∈ Sip } is precisely the threshold in z-space at which bidder 1 gets slot p. Thus10 a1p = ψi−1 (˜ a1p ) is the threshold in v-space. Step 10-12 in the algorithm make sure to reset the threshold so that if slot p is never allocated to bidder 1 and some next higher slot is allocated at a1k < ∞, we set a1p = a1k (just a convention for price formula in (11)). Step 14-16 simply use (11) to compute the prices p1p given the thresholds. The dominating computation inside the two FOR loops is the call to OPTMATCH, thus giving a O(n · m · mn) = O(n2 m2 ) time complexity.
w cij Xij (b) =
j=1
m X
−i (cij − ci,j+1 )1(γi ≥ γ[j] ),
j=1
−i γ[j]
where denotes the j-th largest term in the vector γ−i and 1(·) is the indicator function that takes the value 1 when its argument is true, and zero otherwise. Since γi is increasing in bi and , it follows that the total Pcmij ≥ ci,j+1 w click-through rate j=1 cij Xij (b) is non-decreasing in bi . Thus, Xw is incentive compatible (Lemma 2). The following lemma presents the unique prices per click which implement the rank based allocation rule giving a closed form expression. Lemma 4. The unique price per click pw ik to merchant i at rank k, implementing the allocation rule, Xw defined above are given by, pw ij (v−i ) =
m X cik − ci,k+1 γ[k+1]
cij
k=j
Algorithm 1 Compute Prices: Monotone Maximizer Rule. 1: z ← (ψ1 (v1 ), . . . , ψn (vn )), ci,m+1 ← 0, ai,m+1 ← 0 ∀i. 2: for i=1 to n do 3: for j = 1, . . . , m do 4: (r, s) ← OPTMATCH(n − 1, m − 1, z−i , c(−i,−j) ). 5: γ ← maxk6=i {ckj zk − rk }. 6: a ˜ij ← inf zi ≥0 {zi | max(0, maxl6=j (zi cil − sl )) + γ ≤ zi cij } 7: aij ← ψi−1 (˜ aij ) 8: end for 9: for j=1 to m do 10: if aij = ∞ then 11: aij ← minj γ[j+1] . Thus in terms of Lemma 2, for j = 1, . . . , m the γ threshold aij is precisely equal to [j+1] . Hence using (12) wi corresponding prices per click, pw ij (v−i ) =
m X cij − ci,j+1 γ[j+1]
cik
j=k
wi
.
The payment derived in Lemma 4 is precisely the payment derived in [1] using ad-hoc arguments. In the proof above we simply computed to the surplus of bidder i as the area under total click-through rate ladder as a function of vi (See Figure 3.1). Each term in the sum is the area of the shaded rectangular blocks in Figure 3.1. The simplicity of the rank based mechanism make them a very attractive choice for application. But it is not at all clear which rank vector should be used? In particular, if the click-though-rate are not observable then both Google rank vector (w = ci1 ) and Yahoo! rank vector (w = 1) neither maximize efficiency nor maximize revenue. In the following we describe, a methodology for optimizing the rank vector. We can maximize the expected profit (or efficiency), thus selecting the best static rank based allocation, i.e. ∗ wSP ∈ argmax
w∈Rn + ,w1 =1
n X i=1
E v
m X
(c[j],j − c[j],j+1 )
j=i
γ[j+1] wj
The above stochastic programs is not a convex optimization problem. The indicator function is not a convex function and hence it’s expectation may not be either. It’s hard to simplify the expectation anyway! So it can be hard optimization problem to solve in practice. It is also true if we want to maximize the statistical average of the revenue generated with recorded historical bids of the merchants. We can still probably solve it using Non-linear Programming algorithms.
The robust optimization approach which maximizes the worst case revenues when the valuation are realized from uncertainty set can be used to select a good ranking function. Thus, approach selects the ranking function ∗ wRO ∈ argmax
Example 2. Consider two merchants bidding for two slots with valuations vi uniformly distributed between 0 and 1. Suppose a rank based auction mechanism awards the slot 1 to merchant 1 if v1 ≥ αv2 and to merchant 2 otherwise. Define A = c11 − c12 and B = c21 − c22 . Let us compute the expected profit of the auctioneer, h
Π(α) =
E (v1 ,v2 )
v1 v1 i Aαv2 1(v1 ≥ αv2 ) + B 1(v2 ≥ ) α α
After algebraic computations, we get Π(α) =
8 1 1
aij then ai,j+1 ← aij end if end for for j=1 to mPdo pij ← c1ij m k=j (aik − ai,k+1 )(cij − ci,k+1 ) end for end for n (vn ) + 1 (v1 ) + using z = ((v1 − 1−F ) , . . . , (vn − 1−F ) ) for rankf1 (v1 ) fn (vn ) ing instead of v and inverting the respective thresholds back
to v-space in Algorithm 2, customized rank based allocation rule can approximate optimal auction as well. In general Customized Rank Based Allocation would out perform any rank based allocation.
Lemma 5. A feasible allocation rule X is incentive compatible if, and only if, the total click-through rate ki X
cij Xij (b, ki ), (v−i , k−i )
j=1
3.3
Separable Click-though Rate
Suppose the click-through rate cij is separable i.e. cij = φi µj and the value per click of each agent is uniform across the slots, i.e. for j = 1, . . . , m, vij = vi . In this setting
for merchant i is a non-decreasing function of b for all (v−i , k). Furthermore, the utility ui (vi ; v−i ) of merchant i under any incentive compatible allocation X is of the form ui (vi , ki ; v−i , k−i ) =ui ((0, ki ); (v−i , k−i ))
1. The solution to the social (virtual) surplus maximization problem is rank based, i.e. the slot j is awarded to the j th order statistics of {φi bi }n i=1 (respectively {φi νi (bi )}n i=1 . 2. V CG price-per-click given by (9) implementing the above allocation are e (v) = T[i]
=
φ[i+1] 1 e (µ[i] − µ[i+1] ) b[j+1] + T[i+1] µ[i] φ[i] 1 µ[i]
m X
(µ[j] − µ[j+1] )
j=i
φ[j+1] b[j+1] φ[j]
(18)
Similar formula can be derived for prices implementing virtual surplus maximizing allocation rule. where φ[k] and b[k] represent the φ and bid of merchant ranked k. We remark that in the case of separable click through rate, customized rank based allocation rule is same as Google rank based (i.e. with rank vector ci1 ) and optimal for efficiency maximization.
4.
vij =
vi 0
E
" m n X X i=1
i=1
vi cij Xij ki < j ≤ m, 1 ≤ i ≤ n,
j=1
Xij = 0,
1 − Fi (vi |ki ) vi − fi (vi |ki )
cij
j=1
#
Xij (v) − ui (ki , v−i )
subject to m X
cij Xij (vi , v−i )
j=1
being non-decreasing with vi for all i, v−i , k and Z
vi
0
The social welfare maximization problem is given by maxX∈Mnm subject to
j=1
Using the Lemma 5 we can easily deduce using the change of integral technique (refer [7], Theorem 1) that the revenue maximization problem is to choose a feasible allocation rule X : Rn × {1, . . . , m}n 7→ Mnm with Xij (v) = 0 for all ki < j ≤ m, 1 ≤ i ≤ n, v and a set of side payments ui :
ui (ki , v−i ) + Pm
cij Xij ((u, ki ), (v−i , k−i )) du.
Lemma 5 above implies that any feasible mechanism which is incentive compatible with respect to the value bid can be made incentive compatible with respect to the slot bid using appropriately chosen side payments ui (ki , v−i ) = ui ((0, ki ); (v−i , k−i )) which does not depend on vi , so that the total surplus of each bidder is increasing with ki . Similar to Lemma 2, we also have the IC characterization that for all i, v−i and k, there exists thresholds 0 ≤ ai,m ≤ ai,m−1 ≤ · · · ≤ ai,ki ≤ ∞ such that merchant i is allocated slot j iff vi ∈ (aij , ai,j−1 ].
if j ≤ ki otherwise
where the tuple (vi , ki ) is the private information (i.e. type) of the merchant i. Pn
m X
where ui ((0, ki ); (v−i , k−i )) is non-negative and nondecreasing functions of ki appropriately chose so that ui (vi , ki ; v−i , k−i ) is non-decreasing in ki for all (v, k−i ).
X(v,k) (v,k)
In this section we consider the valuations per bid of the following form,
vi
0
max
SLOTTED MECHANISM (
Z
+
(19)
The merchants are restricted to lie about ki only downward. Thus it is not a standard mechanism design problem. But because of the specific structure of the problem - the total surplus in non-decreasing in ki for all i, v which provide incentive compatibility with respect to ki at V CG payments. Also note that if the demands ki are common knowledge, then the solution presented in § 3 can be directly applied by pointwise maximizing the virtual surplus. The following Lemma characterize the set of all incentive compatible allocation rules. The proof is a simple adaptation of the proof of Lemma 1 in [7].
m X
cij Xij ((u, ki ), (v−i , k−i )) du.
j=1
being non-decreasing with ki for all i, v, k−i . We claim there exist no regularity conditions on the prior distribution fi (vi , ki ) such that we can set ui = 0 without loss of generality and optimize pointwise. This is true because the social surplus itself depend on the capacity dimension of the bid (k) resulting in non-monotone (with ki ) total click through rate for any prior distribution. We present a suboptimal mechanism. Given an IC allocation rule X, define the nominal surplus of player i as, Z
πi0 (v, k) = 0
vi
m X
ˆi ), (v−i , k−i )) du cij Xij ((u, k
j=1
The sub-optimal mechanism is described as follows.
i (vi |ki ) • If the virtual valuations νi (vi , ki ) = vi − 1−F are fi (vi |ki ) not non-decreasing in vi , use the ironing procedure detailed in [7] (section 3.2) to get ironed out virtual valuations νˆi (vi , ki ).
• Set X(v, k) ∈ argmax
n X m X
cij νˆi (vi , ki )Xij (v)
i=1 j=1
This allocation rule is incentive compatible since νˆi is non-decreasing in vi . • Set transfer payment Ti (v, k) =
ki X
cij vi Xij (v, k) − πi (v, k)
where, πi (v, k)
−
+ max πi0 ((ˆ vi , ki ), t−i ) − πi ((ˆ vi , ki−1 ), t−i ) v ˆi
Finally, we remark that the problem of finding optimal mechanism is hard because we need to select both an allocation rule and a set of side payments independently. This is because the transfer payment implementing the allocation rule are not unique in contrast to both sided incentives where they are (even with multi-dimensional types).
5.
CONCLUSION
Most of the classical results in single object IPV auctions can be easily applied to the adword auction problem when the private information is one dimensional, in particular revenue equivalence holds. If the click through is rate not separable and we want to restrict to simple rank based mechanisms the Google choice (w = ci1 ) and Yahoo! choice (w = 1) are arbitrary and rank vector can be efficiently optimized for revenue (or efficiency) maximization using the history of bids available for each keyword. If no prior information is available, customized rank based allocation rule should be a very attractive choice if click-though rate are not separable since they are simple to implement, computationally efficient and generate higher revenues than rank based. Finally, because of the specific structure (i.e. {0, 1}) of the allocation space, the incentive compatibility with multidimensional private values has compact characterization in terms of bidder dependent slot prices at which the bidder should self-select the slot assigned to them. With multidimensional types, we also showed that there are IC mechanisms which are not affine maximizers. In the slotted auction model even though the set of IC mechanism is easy to characterize the optimal mechanism problem is a complex stochastic program due to non-zero side payments used to screen the slot information.
6.
[3] S. Bikhchandani and J. M. Ostroy. From the assignment model to combinatorial auctions. In Y. S. Peter Cramton and R. Steinberg, editors, Combinatorial Auctions. MIT Press, Cambridge, MA, 2006. [4] K.-S. Chung and J. C. Ely. Ex-post incentive compatible mechanisms. Working Paper, Department of Economics, Northwestern University, 2002. [5] B. Edelman, M. Ostrovsky, and M. Schwarz. Internet advertising and the generalized second price auction: Selling billions of dollars worth of keywords. NBER Working paper 11765, November 2005.
j=1
=πi0 (v, k)
[2] G. Aggarwal and J. D. Hartline. Knapsack auctions. In Proceedings of 17th Annual ACM-SIAM Symposium on Discrete Algorithms, Miami, FL, 2006.
REFERENCES
[1] G. Aggarwal, A. Goel, and R. Motvani. Truthful auctions for pricing search keywords. In Proceedings of the 2006 ACM Conference on Electronic Commerce, Ann Arbor, MI, 2006.
[6] G. Hongwei, R. Muller, and R. V. Vohra. Dominant strategy mechanisms with multidimensional types. Working Paper, 2004. [7] G. Iyengar and A. Kumar. Optimal procurement auctions for divisible goods with capacitated suppliers. Technical report, 2006. CORC Tech Report TR-2006-01, Available at http://www.corc.ieor.columbia.edu/reports/ techreports/tr-2006-01.pdf. [8] S. Lahaie. An analysis of alternative slot auction designs for sponsored search. In Proceedings of the 2006 ACM Conference on Electronic Commerce, Ann Arbor, MI, 2006. [9] R. Lavi, A. Mu’alem, and N. Nisan. Towards a characterization of truthful combinatorial auctions. Working Paper, 2004. [10] H. Leonard. Elicitation of honest preferences for the assignment of individuals to positions. Journal of Political Economy, 91(3), 1983. [11] P. R. Milgrom. Putting Auction Theory to Work. Cambridge University Press, Cambridge, UK, 2004. [12] R. B. Myerson. Optimal auction design. Mathematics of Operations Research, 6(1):58–73, 1981. [13] J. C. Rochet. A condition for rationalizability in a quasi-linear context. Journal of Mathematical Economics, 16:191–200, 1987. [14] M. Saks and L. Yu. Weak monotonicity suffices for truthfulness on convex domains. Working Paper, Available at http://www.math.rutgers.edu/~saks/ PUBS/truthful.ecsub.pdf, 2005. [15] L. Shapley and M. Shubik. The assignment game-I: the core. Inter-national Journal of Game Theory, 1:111–130, 1972. [16] H. R. Varian. Position auctions. working paper, 2006. [17] R. V. Vohra and A. Malakhov. Optimal auction for capacitated bidders - a network approach. Working Paper, Managerial Economics and Decision Sciences, Kellogg School of Management, Northwestern University, December 2005.
