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An Overview of some Basic Properties of Sponsored Search
Auctions
S´ebastien Lahaie
Yahoo Research
April 25, 2008
1 Single Keyword, Multiple Slots
There are m positions to be allocated among n bidders, where n > m. We assume that the (expected) click-through rate of bidder i in position j is of the form αiγj , i.e. separable into an advertiser effect αi ∈ [0, 1] and position effect γj ∈ [0, 1]. We assume that γ 1 > γ 2 >... > γk > 0 and let γj = 0 for j > k. We will sometimes refer to αi as the relevance of bidder i. It is useful to interpret γj as the probability that an ad in position j will be noticed, and αi as the probability that it will be clicked on if noticed. Bidder i has value vi for each click. Bidders have quasi-linear utility, so that the utility to bidder i of obtaining position j at a price of p per click is
αiγj (vi − p).
The auctioneer observes the advertiser effects, but the bidders’ values remain private. A weight wi is associated with agent i, and agents bid for position. If agent i bids bi, his corresponding reported score, or simply his score, is si = wibi. His true score is ri = wivi. Agents are ranked by score, so that the agent with highest score is ranked first, and so on. Note that the weights may depend on the advertiser effects, but not on the bidder values, because the latter remain unobservable. We also disallow weights that depend on the agent bids. Throughout, agents are numbered such that agent i obtains position i, unless mentioned oth- erwise. An agent pays per click the lowest bid necessary to retain his position, so that the agent in position j pays w wj+1j bj+1. We refer to this payment rule as “second pricing.”
2 Incentives
2.1 Dominant Strategies
The second-price payment rule is reminiscent of the second-price (Vickrey) auction used for selling a single item. Recall that in a Vickrey auction it is a dominant strategy for a bidder to reveal his true value for the item [18]. However, using a second-price rule in a position auction does not yield an incentive-compatible mechanism, either in dominant strategies or ex post Nash equilibrium.^1 (^1) Unless of course there is only a single position available, since this is the single-item case. With a single position and a second-price payment rule, a position auction is dominant-strategy incentive-compatible for any weighting scheme.
With a second-price rule there is no incentive for a bidder to bid higher than his true value per click: this either leads to no change in the outcome, or a situation in which he will have to pay more than his value per click for each click received, resulting in a negative payoff. However, there may be an incentive to shade true values with second pricing.
Claim 1 With second pricing and m ≥ 2 , truthful bidding is not a dominant strategy nor an ex post Nash equilibrium in a position auction.
Example. There are two agents and two positions. The advertiser effects are α 1 = α 2 = 1 whereas the position effects are γ 1 = 1 and γ 2 = 1/2. Agent 1 has a value of v 1 = 6/w 1 per click, and agent 2 has a value of v 2 = 4/w 2 per click (recall that weights cannot depend on bidder values). Suppose agent 2 bids truthfully. If agent 1 also bids truthfully, he wins the first position and obtains a payoff of 6/w 1 − 4 /w 1 = 2/w 1. However, if he shades his bid below 4/w 1 , he obtains the second position at a cost of 0 per click yielding a payoff of 12 (6/w 1 − 0) = 3/w 1. Hence truthful bidding is not a dominant strategy, and neither is it an ex post Nash equilibrium.
Given that second pricing is not strategy-proof, we may ask whether there exists a payment rule that, together with a given weighting scheme, makes it a dominant strategy for the agents to bid their true values. To answer this question, we need to temporarily augment the notation just introduced to cast the problem in the framework of mechanism design. Agent i’s value is his type, and we denote it by ti rather than vi. Each agent has a value function parametrized by its type, which gives the utility derived from a position before any payments are issued:
vi(j; ti) = αiγj ti.
Agent i’s utility for position j at a total price of q (price per click times clickthrough rate), now parametrized by type, is again quasi-linear:
ui(j, p; tj ) = vi(j; ti) − q.
Let zi(b) be the position allocated to i when the vector of bids is b = (bi)i∈N , and similarly let pi(b) be agent i’s total payment. Holmstrom’s Lemma [7] gives a necessary condition on the structure of the payment rule p if we wish to implement allocation rule z in dominant strategies. Under the restriction that a bidder with value 0 per click does not pay anything, the lemma states that there is a unique candidate payment rule that achieves dominant-strategy incentive compatibility for a given allocation rule. Let Vi(ti, b−i) be agent i’s full information maximum value when the others are bidding b−i and his own value per click is ti: Vi(ti, b−i) = max bi≥ 0 ui(zi(b), pi(b); ti).
The following version of Holmstrom’s lemma is provided in Milgrom [13], who shows that it is a consequence of a variant of the envelope theorem [14].
Lemma 1 Suppose vi is continuously differentiable in type. If truthful reporting is a dominant strategy for agent i, his payment must satisfy
pi(bi, b−i) = −Vi(0, b−i) + vi(z(bi, b−i); bi) −
∫ (^) bi
0
∂vi ∂ti
(z(τ, b−i), τ ) dτ. (1)
For the remainder of this section, we assume that there are as many positions as bidders.^3 In a Nash equilibrium, each agent prefers his own position to the others given the bids, so the following inequalities must be satisfied:
πi = αiγi(ri − si+1) (i ∈ N ) (3) πi ≥ αiγj (ri − sj+1) (i ∈ N, j > i) (4) πi ≥ αiγj (ri − sj ) (i ∈ N, j < i) (5)
Here πi can be interpreted as the agent in position i’s weighted utility. Agent i’s weighted utility for position j at price b per click is defined simply as wiui(j, b). Given the rules of the auction, for the allocation where agent i gets position i to actually arise, we also need the following conditions:
si ≥ si+1 (1 ≤ i ≤ N − 1) (6) si ≥ 0 (i ∈ N ) (7)
We should also require πi ≥ 0 for individual rationality, because a bidder always has the option of not participating in the auction, but this is in fact implied. For each bidder i we have
πi ≥ αiγn(ri − sn+1) = αiγnri ≥ 0
and so ri ≥ si+1, i.e. the price paid by an agent in equilibrium is never greater than his value. We ask the following question: given an allocation of positions to bidders together with bidder values, does there exist a vector of scores s such that inequalities (3)–(7) are satisfied? This gives us an idea of the allocations that can arise in equilibrium. Indeed, B¨orgers et al. [2] show that there can be a multitude of pure-strategy Nash equilibria in a position auction. The question can be answered using linear programming methods to test for the feasibility of the inequalities (3)–(7). Here we give some simple necessary conditions for the answer to be affirmative. The following lemma shows that inequality (6) is not too restrictive.
Lemma 2 If (π, s) satisfy inequalities (3)–(5), there exists (π′, s′) that satisfy inequalities (3)–(6) such that s′^ ≤ s. Furthermore if (π, s) satisfy (7), then (π′, s′) satisfy (7).
Proof. Assume we have a vector (π, s) which satisfies inequalities (3)–(5) but not (6). Then there is some i for which si < si+1. Construct a new vector (π, s′) identical to the original except with s′ i+1 = si. We now have s′ i = s′ i+1. An agent in position j > i sees the price of position i decrease from si+1/wj to s′ i+1/wj = si/wj , but this does not make i more preferred than j to this agent because we have πj ≥ γi− 1 (rj − si) ≥ γi(rj − si) = γi(rj − s′ i+1)
(i.e. because the agent in position j did not originally prefer position i − 1 at price si/wj , he will not prefer position i at price si/wj ). A similar argument applies for agents in positions j < i. Meanwhile, the agent in position i sees the price of its position go down, which only makes the position more preferred. Hence, if we set π′ i = αiγi(ri − s′ i+1) and leave the remaining components of π unchanged, inequalities (3)–(5) remain valid for (π′, s′). Repeating this process, we eventually
game. (^3) If there are more bidders than positions, we can add dummy positions with effects 0. If there are more positions than bidders, we can add dummy bidders each with weight 1 and value 0 per click. The analyses then proceed correctly.
obtain a vector that satisfies inequalities (3)–(6). It is clear by construction that if the initial vector is non-negative, the final vector is as well.
We can now derive necessary conditions for equilibrium allocations.
Proposition 1 Taking π and s as the variables, inequalities (3)–(7) can be satisfied only if ( 1 −
γi γj+
ri ≤ rj
for 1 ≤ j ≤ N − 2 and i ≥ j + 2.
Proof. We safely ignore inequalities (6) in light of Lemma 2. For simplicity, we can also redefine πi as πi/αi and the αi drop out of the system of inequalities. By the Farkas lemma, the remaining inequalities can be satisfied if and only if there is no vector x such that ∑
i,j
(γj ri) xij > 0 ∑
i>j
γj xij +
i<j
γj− 1 xij− 1 ≤ 0 (j ∈ N ) (8) ∑
j
xij ≤ 0 (i ∈ N ) (9)
xij ≥ 0 (j 6 = i) xii free Now consider the following inequality, where i > j. γj ri γj
γj− 1 rj− 1 γj− 1
γiri γj
If the inequality does not hold, then setting xij = 1/γj , xj− 1 j− 1 = − 1 /γj− 1 , xii = − 1 /γj , and all other components of x to 0, we obtain a feasible solution with positive objective value. Hence inequality (10) must hold for inequalities (3)–(7) to be satisfied. By a slight reindexing, inequali- ties (10) for all i > j yield the statement of the theorem.
Proposition 1 only gives necessary conditions for equilibrium allocations, not sufficient condi- tions, so it cannot be used to prove the existence of a Nash equilibrium. A pure-strategy Nash equilibrium exists in a position auction for any weighting scheme, but the proof of this fact is deferred to the next section. The import of Proposition 1 is the restriction it gives on possible equilibrium allocations; we apply the proposition in Section 3 to give a lower bound on equilibrium efficiency.
2.3 Symmetric Equilibrium
Varian [17] introduced a refinement of the Nash equilibrium concept for position auctions which he called “symmetric equilibrium.” Edelman et al. [3] independently introduced this refinement and called it “locally envy-free equilibrium.” With a slight modification we can make the Nash equi- librium inequalities above resemble those that arise in the assignment problem. In inequalities (5), we replace sj by sj+1. For clarity of notation we let pi = si+1. A symmetric NE then satisfies
πi = αiγi(ri − pi) (i ∈ N ) (11) πi ≥ αiγj (ri − pj ) (i ∈ N, j 6 = i) (12)
payments, which are given by Holmstrom’s lemma. As a result, minimal symmetric equilibria and strategy-proof position auctions are revenue-equivalent, a point first made by Aggarwal et al. [1]. This observation will allow us to formulate the optimal position auction problem—where the objective is to maximize revenue, as opposed to efficiency—as a mathematical program in Section 4.
3 Efficiency
To maximize total value, we need to order the agents according to some permutation σ such that the inner product of the vectors (ασ(j)vσ(j))j∈K and (γj )j∈K is maximized. As explained in the previous section, standard results on rearrangements then state that it is efficient to order the agents so that ασ(1)γ 1 vσ(1) ≥... ≥ ασ(n)γnvσ(n). When using the strategy-proof payment rule 2, agents reveal their true values and therefore are ranked by true score. Hence, if we take wi = αi, the resulting equilibrium allocation is efficient. This fact applies to the symmetric equilibrium concept as well, because we saw in the previous section that in symmetric equilibirum, agents are ranked by true score. According to Lemma 1, a ranking that results from a Nash equilibrium profile can only deviate from the allocation where agents are ranked by true score by having agents with relatively similar scores switch places. That is, if ri > rj , then agent j can be ranked higher than i only if the ratio rj /ri is sufficiently large. This suggests that the value of an equilibrium allocation cannot differ too much from the value obtained when agents are ranked by true score. The next result confirms this. We denote the total value of an allocation σ of positions to agents by f (σ) =
∑k j=1 γj^ rσ(j). Let
L = min j=1,...,k− 1 min
γj+ γj
γj+ γj+
(where by default γk+1 = 0). Let η be the permutation such that rη(1) ≥... ≥ rη(k). Lahaie [10] gives the following bound.
Proposition 2 For an allocation σ that results from a pure-strategy Nash equilibrium of a position auction, we have f (σ) ≥ Lf (η).
Proof. We number the agents so that agent i has the ith^ highest revenue, so r 1 ≥ r 2 ≥... ≥ rN. Hence the standard allocation has value f (ηr) =
∑N
i=1 γiri. To prove the theorem, we will make repeated use of the fact that
P P^ t^ at t bt^ ≥^ mint
at bt when the^ at^ and^ bt^ are positive. Note that according
to Proposition 1, if agent i lies at least two positions below position j, then rσ(j) ≥ ri
1 − γ γjj+2+
It may be the case that for some position i, we have σ(i) > i and for positions j > i + 1 we have σ(j) > i. We then say that position i is inverted. Let S be the set of agents with indices at least i + 1; there are n − i of these. If position i is inverted, it is occupied by some agent from S. Also all positions strictly lower than i + 1 must be occupied by the remaining agents from S, since σ(j) > i for j ≥ i + 2. The agent in position i + 1 must then have an index σ(i + 1) ≤ i (note this means position i + 1 cannot be inverted). Now there are two cases. In the first case we have
σ(i) = i + 1. Then
γirσ(i) + γi+1rσ(i+1) γiri + γi+1ri+
γi+1ri + γiri+ γiri + γi+1ri+
≥ min
γi+ γi
γi γi+
γi+ γi
In the second case we have σ(i) > i + 1. Then since all agents in S except the one in position i lie strictly below position i + 1, and the agent in position i is not agent i + 1, it must be that agent i + 1 is in a position strictly below position i + 1. This means that it is at least two positions below the
agent that actually occupies position i, and by Proposition 1 we then have rσ(i) ≥ ri+
1 − γ γii+2+
Thus,
γirσ(i) + γi+1rσ(i+1) γiri + γi+1ri+
γi+1ri + γirσ(i) γiri + γi+1ri+
≥ min
γi+ γi
γi+ γi+
If position i is not inverted, then on one hand we may have σ(i) ≤ i, in which case rσ(i)/ri ≥ 1. On the other hand we may have σ(i) > i but there is some agent with index j ≤ i that lies at least
two positions below position i. Then by Proposition 1, rσ(i) ≥ rj
1 − γ γii+2+
≥ ri
1 − γ γii+2+
We write i ∈ I if position i is inverted, and i 6 ∈ I if neither i nor i − 1 are inverted. By our arguments above two consecutive positions cannot be inverted, so we can write
f (σ) f (γr)
i∈I
γirσ(i) + γi+1rσ(i+1)
∑^ i^6 ∈I^ γirσ(i) i∈I (γiri^ +^ γi+1ri+1) +^
i 6 ∈I γiri ≥ min
min i∈I
γirσ(i) + γi+1rσ(i+1) γiri + γi+1ri+
, min i 6 ∈I
γirσ(i) γiri
≥ L
and this completes the proof.
For the common exponential decay model of γi = δ^1 −i^ for δ > 1, the factor becomes L = min{ (^1) δ , 1 − (^1) δ }. Feng et al. [4] report that an exponential decay model with δ = 1.428 fits their Overture click-through rate data well. In this case, L ≈ 1 / 3 .34. Though being a factor of more than 3 away from the efficient value may seem unacceptable, we stress that this is a worst-case bound, and we would expect actual deviations to be much less than this in practice.
4 Revenue
Because multiple different allocations can arise in Nash equilibrium, it is difficult to give any bounds on equilibrium revenue. With the refinement of symmetric equilibrium, however, much more can be said. Here we provide a mathematical program whose solution gives the revenue-optimal ranking rule for a position auction given a distribution over bidder values are relevance, assuming
0
0
qi(αi, vi; w)vi −
∫ (^) vi
0
qi(αi, τi; w) dτi
fi(αi, vi) dvi dαi
0
0
qi(αi, vi; w)vifi(αi, vi) dvi dαi −
0
0
∫ (^) vi
0
qi(αi, τi; w)fi(αi, vi) dτi dvi dαi
0
0
qi(αi, vi; w)vifi(αi, vi) dvi dαi −
0
0
τi
qi(αi, τi; w)fi(αi, vi) dvi dτi dαi
0
0
qi(αi, vi; w)vifi(vi|αi)f (^) iα (αi) dvi dαi
0
0
(1 − Fi(τi|αi))qi(αi, τi; w)f (^) iα (αi) dτi dαi
0
0
vi − 1 − Fi(vi|αi) fi(vi|αi)
qi(αi, vi; w)fi(αi, vi) dvi dαi
[0,1]n
[0,∞]n
∑^ n
j=
αiγj
vi − 1 − Fi(vi|αi) fi(vi|αi)
Qij (α, v; w)f (α, v) dv dα
where the second equality follows by interchanging the order of integration in the second term, the third by the fact that fi(αi, vi) = fi(vi|αi)f (^) iα (αi), and the fifth by the definition of qi(αi, vi; w). We define agent i’s “virtual valuation” by
ψi(αi, vi) = vi − 1 − Fi(vi|αi) fi(vi|αi)
Summing over each agent’s ex ante expected payment, the expected revenue is then
∫
[0,1]n
[0,∞]n
∑^ n
i=
∑^ n
j=
αiγj ψi(αi, vi)Qij (α, v; w)f (α, v) dv dα
According to this analysis, we should rank bidders by “virtual score” αiψ(αi, vi) to optimize rev- enue (and exclude any bidders with negative virtual score). However, unlike in the incomplete information setting, here we are constrained to ranking rules that correspond to a certain weight- ing scheme wi ≡ g(αi). Lahaie and Pennock [11] show that there is be no function g such that αiψ(αi, v) = g(αi)v, for any distribution f ; i.e. the virtual score is never a linear function of value, for fixed effect αi. Of course, to rank bidders by virtual score, we only need g(αi)vi = h(αiψ(αi, vi)) for some monotonically increasing transformation h. (A necessary condition for this is that ψ(αi, vi) be increasing in vi for all αi.) Still, it is unclear how to do this in general. Lahaie and Pennock [11] study the restricted family of weights wi = αqi for q ∈ (−∞, +∞) and note that smaller q imply greater revenue when value and relevance are correlated. This kind of approach, where restricted families of weights are considered rather than arbitrary functions g, appears to be a fruitful avenue for progress towards the revenue-optimal mechanism. Our analysis does allow for an analytical solution in one special case. Suppose that value is independent of relevance, and that the distribution over value is uniform on [0, 1]. Then the virtual
score of a bidder is 2αivi − αi. We can recreate this by giving each bidder a weight of 2αi, and introducing a discount of αi. (The discount can be construed as a kind of reserve score, because a bidder whose score αivi does not exceed the discount should not be shown.) The symmetric equilbrium analysis generalizes to the case where additive discounts are included in the score, so this ranking scheme is indeed revenue-optimal when value is uniformly distributed.
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