Equilibrium Selection, Inefficiency, and Instability in Internet Advertising Auctions Tadashi Hashimoto Stanford Graduate School of Business 518 Memorial Way Stanford, CA 94305-5015

[email protected]

ABSTRACT This paper studies a model in which a non-strategic bidder named the noise bidder secretly participate in the generalized second price auction with a small probability. The main result is that the VCG equilbrium, which have been considered as the most plausible equilibrium in this literature, is no longer an equilibrium even in the limit where the noise bidder disappears. In this game, an efficient equilibrium is unique if exists. This unique equilibrium is further justified as a unique limit of trembling-hand perfect equilibria in which all bidders mistake in similar ways. However, this game may not have any efficient equilibrium, or more seriously any stable equilibrium. A simulation result suggests that in the parameter space the set of parameters such that no stable equilibrium exists is huge. It occupies more than 95% of the space if five or more positions are traded.

Categories and Subject Descriptors H.4.m [Information Systems Applications]: Auction Theory

General Terms Economics, Theory

Keywords Equilibrium Selection

1. INTRODUCTION In the internet, search engines such as Google, Yahoo!, and MSN are using variants of the generalized second price auction (GSP) to allocate advertising slots in which advertisers’ clickable advertisements are displayed. The GSP is the auction format in which the bidder with the k-th highest bid wins the k-th best ad position and pays the (k + 1)-highest bid for each click. After the two independent pioneering works by Edelman, Ostrovsky, and Schwarz (2007) (EOS, henceforth) and Varian (2007), the GSP and internet advertising auctions in general attract attentions from both

economists and computer scientists. An obstacle in analyzing the GSP is that it intrinsically has a continuum of Nash equilibria when valuations are common knowledge. Therefore, it is unavoidable to refine the equilibrium concept in order to make sharp predictions. EOS and Varian independently proposed two equivalent equilibrium refinements, locally envy-free equilibrium and symmetric Nash equilibrium, respectively, in which advertisers never envy any other advertiser’s position and payment. Although these refinements significantly ease analyses, still GSP suffers from a continuum of equilibria. EOS and Varian paid special attentions to equilibria which generate completely the same allocations and payments as the VCG mechanism, and to justify such equilibria EOS invented the dynamic version of GSP, the generalized English auction. To the extent of the author’s knowledge, there have been no equilibrium selection technique proposed in this literature other than EOS and Varian’s. Their approaches are so influential that they are applied in several studies such as Athey and Ellison (2007), B¨ orgers et al. (2007), Lahaie and Pennock (2007), Cary et al. (2008), Hafalir et al. (2009), Kominers (2009), and Yenmez (2009). The objective of this paper is to propose alternative approaches for the selection of equilibrium. The main finding of this paper is that our equilibrium refinements always eliminate VCG equilibria, which have been commonly considered as the most plausible equlibria. We first construct a slightly modified model in which an additional bidder, the noise bidder, enters the auction with a small probability and bids purely stochastically.1 This model incorporates the realistic feature that an entrant, or the noise bidder, may appear suddenly and randomly. In the modified game, the bids of the winners, the bidders who win some position, are uniquely determined in any efficient Nash equilibrium. These bids {b∗i } are completely independent of probabilistic characteristics of the noise bidder and always different from the VCG equilibrium bids. It is a corollary of this fact that any sequence of equilibria never converges to the VCG equilibrium even in the limit where the probability that the noise bidder is present has gone to zero. This paper further justify {b∗i } by considering a variant of trembling-hand perfect equilibrium in which all players mistake in similar ways. As the number of advertisers goes to 1

Leme and Tardos (2009) independently consider this situation to exclude the case that bids exceeding the bidder’s valuation.

infinity, all efficient equilibria uniformly converges to {b∗i }. The noise bidder model, however, may fail to have an efficient Nash equilibrium. More seriously, there may not be any stable equilibrium, in which the ranking of advertisers (other than the noise bidder) is deterministic. Theoretically, for any fixed click-through rates, there must be valuations such that there is no stable equilibrium, and the set of such valuations has a positive mass. To computationally estimate how large the set of parameters such that there is no stable equilibrium, we consider a Monte Carlo simulation in which click-through rates and valuations are randomly generated. According to the result of the simulation, the unstable set occupies more than 50%, and if five or more positions are auctioned, then its share becomes more than 95% of the whole parameter space. This is a highly undesirable result for search engines. In practice, to implement randomization, automated bid robots are used. That is, if the equilibrium involves randomization, then someone is using a robot which is attacking the search engine. This instability result explains the evidence found by Edelman and Ostovsky (2007) that in reality rankings of advertisers change very frequently in the GSP. To the best of the author’s knowledge, no paper have successfully explained this phenomenon yet, and this paper is the first paper providing a formal explanation. Introducing small amount of noise or irrationality is a traditional approach to eliminate implausible equilibria. Several equilibrium refinements such as trembling-hand perfect equilibrium, proper equilibrium (Myerson 1978) and quasiperfect equilibrium (van Damme 1984) are based on the idea that the players of a game may mistake with some tiny probability. Evolutionary dynamics (see e.g. Weibull 1995 and Fudenberg and Levine 1999) involve optimization failures and imprecise predictions and eliminate unstable equilibria. Finally, in global games, an infinitesimally small amount of uncertainty is added to the original game to select plausible equilibria (see Carlsson and van Damme 1993 and Morris and Shin 1998). Although this paper is the first study eliminating bidders’ implausible behavior by adding small amount of uncertainty, there are a few recent independent studies incorporating randomness in bids to the basic model by EOS and Varian (2007). Athey and Nekipelov (2010) construct a structural econometric model in which quality scores and the set of present bidders are uncertain. Gomes and Sweeney (2009) study the incomplete information version of the basic model and find that an efficient symmetric equilibrium is unique if it exists. Leme and Tardos (2009) temporarily add a nonstrategic stochastic bid to their model in order to justify their equilibrium concept, conservative bidder equilibrium, in which all bidders set their bids lower than or equal to their valuations. This paper is organized as follows. Section 2 formalizes the model with the noise bidder. In section 3, equilibria of the noise bidder model are analyzed. In section 4, theoretical and computational instability results are shown. Section 5 is devoted to an extension of the noise bidder model in which equilibrium bids are fully characterized asymptotically. In

section 6, we consider a variant of trembling hand perfect equilibrium of the model without the noise bidder in order to further justify the results of sections 3 and 5. In section 7, we come back to the noise bidder model and analyze a special case. In section 8, relationship of this paper’s analysis to locally envy-free equilibrium (EOS) and symmetric Nash equilibrium (Varian 2007) is briefly discussed. Section 9 concludes.

2.

THE MODEL

A search engine offers P advertisement positions on its website, and N advertisers and one non-strategic bidder named the noise bidder compete for them. We assume 1 < P < N . From every one click on its advertisement, advertiser i receives a payoff of vi > 0 in expectation, which is called i’s per-click value and is common knowledge. The click-through rate of position p is αp , i.e., an advertisement shown in position p is clicked αp times in expectation. Thus, if advertiser i wins position p and its per-click payment is qp then i’s payoff is αs (vi − qp ). The advertisers and positions are ordered so that v1 > v2 > v3 > · · · > vN and α1 > α2 > · · · > αP > 0. For p > P , define αp = 0. The GSP is the auction format described as follows. Each advertiser i simultaneously submits a bid bi ∈ [0, ∞). Basically, the bidder with the i-th highest bid wins the position i and her per-click payment is the (i + 1)-th highest bid. Formally, for p = 1, . . . , P , advertiser ι(p) wins position p and her per-click payment is set to be bι(p+1) . Here, ι is a permutation of {1, . . . , N } such that bι(p) > bι(p′ ) implies p < p′ . If there are multiple candidates of ι, it is randomly chosen from them with an equal probability. The noise bidder secretly participates in the auction with probability ε > 0. It announce bid b0 randomly according to a density function f0 , which is continuous and positive on [0, ∞). The advertisers are present in the auction for sure, and they choose their bids strategically. Let Γε be the game which represents the above situation. We also write Γε (α, v) when we need to clarify parameters α = (α1 , . . . , αP ) and v = (v1 , . . . , vN ). A mixed strategy of advertiser i is a distribution function σi on [0, ∞), and the set of σi is denoted by Σi . A mixed strategy profile is a N tuple σ = (σ1 , . . . , σN ) ∈ Σ, where Σ = Σ1 × · · · × ΣN . Let bσ1 , . . . , bσN denote the independent random variables with distributions σ1 , . . . , σN , respectively. An assignment is a one-to-one mapping π : {1, . . . , P } → {1, . . . , N }. Definition 1. Let π be an assignment. A mixed strategy profile σ is π-stable if { } bσπ(1) > bσπ(2) > · · · > bσπ(P ) > max bσi : i ∈ ̸ Im π (1) almost surely. If π(p) = p for all p ∈ {1, . . . , P }, σ is said to be efficient. A mixed strategy profile is stable if it is π ′ -stable with some matching π ′ . Proposition 1. For all ε ∈ [0, 1], the following hold in the game Γε :

(i) There exists a Nash equilibrium.

This is a contradiction. Therefore, any σ in the δ-neighborhood cannot be an equilibrium regardless of the choice of ε > 0. This is the end of the proof of Theorem 1.3

(ii) Any pure strategy Nash equilibrium is stable. Proof. The first statement follows from Dasgupta and Maskin’s (1986) existence theorem. Let us prove the second. Suppose that there exists a pure strategy Nash equilibrium (b1 , . . . , bN ) such that bi = bj , i ̸= j, and i and j win some position with a positive probability. Since vi ̸= vj , at least one of them is better off by slightly increasing or decreasing its bid. This contradicts the assumption that (b1 , . . . , bN ) is an equilibrium.

Theorem 2 claims that efficient equilibria are almost uniquely characterized in Γε , and more generally, for each matching π, π-stable equilibria are almost uniquely characterized. Define recursively ) αp+1 ( vπ(p) − bππ(p+1) (5) bππ(p) = vπ(p) − αp for p = 1, . . . , P − 1, and bππ(p) = vπ(p)

(6) b∗i

3. THE VCG BIDS AND STABLE NASH EQUILIBRIA

for p = P + 1, . . . , N . When π(p) = p, we write of bπi .

In the literature of the GSP, equilibria whose outcomes are exactly the same as that of the truth-telling equilibrium of the VCG mechanism have attracted special attensions. Let (bVCG , . . . , bVCG ) be the solution of the following system: 2 N

Theorem 2. If σ is a π-stable Nash equilibrium of Γε , then

bVCG i

αp = vi − (vi − bVCG i+1 ) αp−1

= vi bVCG i

instead

bσπ(p) = bππ(p)

for 2 ≤ i ≤ P ,

(2)

for i > P .

(3)

almost surely for p = 1, . . . , P . Proof. See Appendix.

Let bVCG be some bid greater than bVCG .2 EOS show that 1 2 VCG b is an efficient Nash equilibrium of the original game Γ, and its equilibrium outcome is equivalent to that of the VCG mechanism. This motivates us to call bVCG the VCG bid profile and bVCG the VCG bid of advertiser i. EOS propose i the VCG as the most compelling equilibrium of the GSP. The next theorem is the main result of this paper: In this game Γε , the VCG bid profile bVCG is no longer an equilibrium even in an approximate sense. More formally, any sequence of Nash equilibria never converges to the VCG bid profile even in the limit ε → 0. To measure distance, we use a norm equivalent to the L∞ -norm. The δ-neighborhood of bVCG is the set of σ ∈ Σ such that |bσi − bVCG | < δ almost i surely for all i = 2, . . . , N . Recall that each bσi is the random variable induced by σi .

Theorem 1. There exists δ > 0 such that, for all ε > 0, there exists no Nash equilibrium of Γε in the δ-neighborhood of bVCG .

This is proved in the following way. Take sufficiently small δ > 0 so that { { VCG } } bi − bVCG i+1 δ ≤ min min , vP − bVP CG . i=1,...,P 2

(7)

Table 1: Equilibrium Bids b1 b2 b3 b4 VCG b 67.5 45 30 b∗ 87.5 75 60

Example 1. Suppose P = 3, N = 4, (α1 , α2 , α3 ) = (4, 2, 1), and (v1 , v2 , v3 , v4 ) = (100, 90, 60, 30). Table 1 reports the values of (bVCG , bVCG , bVCG ) and (b∗1 , b∗2 , b∗3 ). The VCG bid 2 3 4 VCG b3 of advertiser 3 is less than its valuation 60, whereas advertiser 3’s equilibrium bid b∗3 must equal 60 in any efficient Nash equilibrium of Γε . The bid of advertiser 2 is also different: The VCG bid is bVCG = 90 + 0.5(90 − 45) = 67.5 2 while the equilibrium bid must be b∗2 = 90+0.5(90−60) = 75 when the equilibrium is efficient.

Theorem 2 is easily proved when equilibrium bids are strictly monotonic, i.e., b1 > b2 > · · · > bN . Let BN = [0, bN −1 ), Although just the difference between b∗P and bVCG is used P in the proof of Theorem 1, generically b∗i differs from bVCG i for i = 2, . . . , P . The equality b∗i = bVCG holds if and only i if } P { ∑ αj αj+1 − (vj − vj+1 ) = 0, αi αi−1 j=i 3

Then any σ in the δ-neighborhood must be efficient. Suppose that there exists a Nash equlibrium σ in the δ-neighborhood. It is a corollary of the next theorem, Theorem 2, that bσP = which occurs only in degenerate cases. For example, if vP almost surely. This implies α2 /α1 = · · · = αP /αP −1 , the left-hand side is σ VCG VCG αP δ > bP − bP = vP − bP ≥ δ. (4) − (vP − vP +1 ) < 0. αi−1 2

In EOS, bVCG is simply set to be v1 . 1

Thus, in this special case, b∗i ̸= bVCG for i = 2, . . . , P . i

BN −1 = (bN , bN −2 ), . . . , B2 = (b3 , b1 ), and B1 = (b2 , ∞). Given (b1 , . . . , bN ), the payoff of i from bidding βi ∈ Bi is {∫ βi

Ui (βi ) = ε

αi (vi − (x ∨ bi+1 ))f0 (x)dx

0

∫

∞

+

}

an efficient Nash equilibrium σ. Then, (bσ1 , bσ2 ) = (b∗1 , b∗2 ) = (98, 90) and bσ3 ≤ 80. Advertiser 1 is better off by dropping to position 2, because the continuation payoff is at most 1 · (100 − 90) = 10 while the payoff from position 2 is at least (1 − ε) · 0.8(100 − 50) = 40(1 − ε) > 10. Therefore, such σ cannot exist.

αi+1 (vi − bi+1 )f0 (x)dx

βi

+ (1 − ε)αi (vi − bi+1 ).

(8)

Since βi = bi is optimal, the following first order condition must be satisfied: αi (vi − bi ) = αi+1 (vi − bi+1 ).

(9)

The above condition holds for i = 1, . . . , P . We therefore obtain bi = b∗i for i = 1, . . . , P . Equation (9), or more generally αp (vπ(p) − bπ(p) ) = αp+1 (vπ(p) − bπ(p+1) ),

(10)

is the indifference condition for advertiser i, or π(p), who is facing the risk of a price increase or a position decline due to the noise bidder’s random bid. In this model, the noise bidder hurts the payoff of bidder i = π(p) in two ways: First, the noise bidder lowers i’s position by one by announcing a higher bid than i’s, and second, it increases the price of i’s position by slightly underbidding i’s bid. These two channels create a trade-off. By increasing its bid, advertiser i can decrease the risk of dropping out, but at the same time increases the risk of a price rise. These two effects are balanced at bππ(p) , which satisfies (10). If the left-hand side of (10) is larger, losing position p is costly for i, so i should at least slightly increase its bid. If the right-hand side is larger, position p’s price bπ(p) at is too expensive for i compared to position (p + 1)’s price bπ(p+1) , which means i is better off by lowering its bid. In this section, equilibrium bids in π-stable Nash equilibria are almost fully characterized for each π. In particular, if we restrict attention to efficient pure-strategy Nash equilibria as in EOS and Varian (2007), for i = 1, . . . , P , advertiser i’s equilibrium bid must be αi+1 b∗i = vi + (vi − b∗i+1 ). (11) αi As a corollary of this characterization result, we have seen that there is no equilibrium around the VCG bid profile. If there exists an efficient equilibrium in this game, {b∗i } is a natural prediction of the winners’ bids. Unfortunately, there may not exist any efficient equilibrium, or more seriously, any stable equilibrium. Furthermore, in the parameter space of α and v, the set of such unstable cases is huge. We investigate this point in the next section.

4. INEFFICIENCY AND INSTABILITY In the previous section, we have seen that, if Γε has an efficient equilibrium, in that equilibrium the bids of the winners 1, . . . , P must be b∗1 , . . . , b∗P , respectively. However, in general, Γε fails to have an efficient Nash equilibrium. Example 2. Assume P = 2, N > P , (v1 , v2 , v3 ) = (100, 90, 50), (α1 , α2 ) = (1, 0.8) and ε < 0.1. Suppose that Γε has

Moreover, in the above example, there is no stable equilibrium. In any stable equilibrium, advertisers i ∈ {1, 2} must win some position, because otherwise it increases its payoff by raising its bid. However, (π(1), π(2)) = (2, 1) is impossible, because if so advertiser 2’s equilibrium payoff is negative. All possible assignments π are rejected, so stable equilibrium cannot exist. In general, the following negative result hold. Let A be the set of α = (α1 , . . . , αP ) such that α1 > · · · > αP > 0. For v > 0, define V(v) as the set of v = (v1 , . . . , vN ) such that v ≥ v1 > . . . > vN > 0. Let Vεus (v, α) be the set of v ∈ V(v) such that Γε (α, v) has no stable Nash equilibrium. Theorem 3. For any ε ∈ (0, 1), α ∈ A, and v > 0, the Lebesgue measure of Vεus (v, α) is positive.4 Proof. Suppose Γε (α, v) has a π-stable equilibrium Advertisers i > P cannot win any position, so Im π {1, . . . , P }. Also, π(P ) cannot be 1, because, if so, bσ1 bσ2 ≤ v2 < v1 = bσ1 . Advertiser 1’s equilibrium payoff is most U 1 = α1 (v1 − vP ),

σ. = < at

(12)

while payoffs from position P are at least U 2 = (1 − ε) · αP (v1 − vP +1 ).

(13)

For fixed ε, α, and v, the set of v ∈ V(v) such that U1 < U2 is open. This completes the proof.

We have seen that stable equilibria fail to exist, so the next question is how often this happens. To answer it, we computationally estimate the size of the set of α and v such that no stable equilibrium exists. Formally, we use a Monte Carlo method to evaluate the Lebesgue measure of set S(ε, P, N ) from below. The set S(ε, P, N ) is the set of α ˜ ∈ (0, 1)P and v˜ ∈ (0, 1)N such that Γε (α, v) has no stable Nash equilibrium for any ε ∈ (0, ε], where α and v are generated by reordering α ˜ and v˜, respectively, in descending order. By the Monte Carlo simulation, we estimate lowerbounds of S(ε, P, N ). The following is a rough description of the Monte Carlo simulation. Details of the simulation are described in the Appendix. The upperbound of ε, ε, is set to be 1%. For each P and N , α ˜ and v˜ are generated n-times (n = 10,000) randomly and independently from the uniform distribution on [0, 1]. Only if Γε (α, v) has no stable Nash equilibrium for all ε ∈ (0, ε], binary variable X is set to be 1. Finally, ¯ for each P and N , the mean X(P, N ) of X is calculated. ¯ This X(P, N ) is the computationally estimated lowerbound 4 If we allow v = ∞, the Lebesgue measure of Vεus (v, α) is infinity.

Proposition 2. For all ε ∈ [0, 1] and q ∈ [0, 1), the following hold in the game Γε,q :

The estimated measure evaluated from below

1 0.9

(i) There exists a Nash equilibrium.

0.8 0.7

(ii) Any pure strategy Nash equilibrium is stable.

0.6

N = 10 N = 20 N = 30

0.5

For all i ̸∈ Im π, we define

0.4

bπi = vi .

(14)

0.3 0.2 0.1 0

1

2

3

4

5

6

7

8

P (the number of positions)

Figure 1: The estimated Lebesgue measure of the set S(ε, P, N ) evaluated from below.

Theorem 4. For all δ > 0, there exists q δ > 0 such that the following holds for all q ∈ (0, q δ ): If σ is a π-stable Nash equilibrium of Γq,ε , then σ bi − bπi < δ (15) for all i. Proof. See Appendix.

of the Lebesgue measure of S(ε, P, N ) with the 95% confi¯ ¯ dence interval X(P, N ) ± E, where E = 1.96{X(P, N )(1 − √ ¯ X(P, N ))/n}1/2 < 1/ n = 0.01. Figure 1 reports the result of the Monte Carlo simulation. If α and v are randomly chosen in the above sense, there exists no stable Nash equilibrium with probability 50% if P = 2, 75% if P = 3, 90% if P = 4, and more than 95% if P = 5 or more, approximately, and this result is almost independent of the choice of N . In this section, we have seen that instability is unavoidable in the GSP. First, for any fixed click-through rates, there must be valuations which cause instability, and the Lebesgue measure of such valuations is always positive. Second, more strikingly, in the parameter space of α and v, the set S(ε, P, N ) is huge, especially when more than three positions are auctioned. These instability results are undesirable for search engines, because unstable equilibria must involve randomization of bids, which means in practice that bidders frequently change their bids, probably by using automated bidding robots.

Now that in this game bids for i ̸∈ π(N ) is no longer indeterminant. Suppose that bidder i’s bid is the (P + k)-th highest. If k of the bidders with higher bids exit the auction, bidder i have a chance to win position P depending on the bid of the noise bidder. Although bidder i also has a chance to win a better position, this is much less likely to happen because at least k + 1 bidders must be absent. We may therefore ignore this possibility asymptotically, and hence in the limit bidder i’s bid is solely determined by the case that exactly k bidders except the noise bidder are absent in the auction. The corresponding first order necessary condition is αP (vi − bi ) ≃ 0. To demonstrate the above argument a little bit more formally, consider the simple case again in which σ is a pure strategy equilibrium b = (b1 , . . . , bN ) with b1 > b2 > · · · > bN . Assume i = P + k with k > 0. The first order condition is written as   ∑ ∑ ∑ ϕP +k−ℓ ϕP αP (vi − bi ) + X(S) = 0, (16) S∈Sk

ℓ>k

S∈Sℓ

where

5. A NOISE-BIDDER MODEL WITH RANDOM MANDATORY EXITS In the game Γε , equilibrium bids for the bidders who do not win any position are left completely undetermined. This is because advertisers i = P + 1, . . . , N have no chance of winning any position regardless of the presence of the noise bidder. To obtain bi = b∗i for these advertisers, we allow advertisers i = 1, . . . , N to quit the market in privacy with a tiny probability. Formally, we define Γε,q as the game which is basically Γε but all of advertisers i = 1, . . . , P secretly quit the game randomly and independently with probability q > 0. For exactly the same reason as before, the following counterpart of Proposition 1 holds:

{ } Sk = S ⊆ {1, . . . , i − 1} : |S| = k , ϕp = (1 − q)p−1 q i−p , and X(S) is a term independent of q. Since limq→0 ϕP −m /ϕP = 0 for m > 0, lim αP (vi − bi ) = 0.

q→0

(17)

Therefore, in the limit, bi converges to vi . In the case of i ≤ P , the convergence is proved more simply. The first order condition is written as (1 − Qi )αi (vi − bi ) + Qi Xi = (1 − Qi )αi+1 (vi − bi+1 ) + Qi Yi ,

(18)

where 1−Qi = (1−q)i is the probability that advertisers j = 1, . . . , i − 1, i + 1 are present, and Xi and Yi are independent of q. Since Qi → 0 as q → 0, { } lim αi (vi − bi ) − αi+1 (vi − bi+1 ) = 0. (19) q→0

Therefore, bi converges to vi − 0.

αi+1 αi

Theorem 5. Fix α1 > · · · > αP > 0, v1 > v2 > · · · > 0, and M > 0. For all δ > 0, there exists N0 > 0 such that, for all integers N ≥ N0 , if (b1 , . . . , bN ) is a π-stable equally trembling-hand perfect equilibrium with bound M in the game ΓN (α, v1 , . . . , vN ), then

· (vi − bi+1 ) as q goes to

|bi − bπi | < δ

(20)

for all i.

6. ANOTHER INTERPRETATION: TREMBLINGHAND PERFECT EQUILIBRIUM Proof. In this section, we propose an alternative justification of bπ by using a variant of trembling-hand perfect equilibrium. Trembling-hand perfect equilibrium, or shortly perfect equlibrium, (Selten 1975) is a classical refinement of Nash equilibrium. This concept is originally defined on extensive form games with finite nodes, while this model has the continuum action space. Thus we need to extend the definition of trembling-hand perfect equilibrium to this setting. Let ΓN (α, v), or simply ΓN , denote the game without the noise bidder or mandatory exits. Let F be the set of continuous functions ∫ ∞ f : [0, ∞) → [0, ∞) such that f (x) > 0 for all x > 0 and 0 f (x)dx = 1. Definition 2. A pure strategy Nash equilibrium (b1 , . . . , bN ) of game ΓN is a trembling-hand perfect equilibrium if ∞ there exist {(ε1,k , . . . , εN,k )}∞ k=K , {(f1,k , . . . , fN,k )}k=K and ∞ {(b1,k , . . . , bN,k )}k=K satisfying the following: (i) εi,k > 0 and limk→∞ εi,k = 0. (ii) fi,k ∈ F . (iii) limk→∞ bi,k = bi , and bi,k is a best response to σk = (σ1,k , . . . , σN,k ). Here, σi,k is the mixed strategy in which advertiser i’s bid is bi with probability ε and is randomly drawn with the density function fi,k . Further, (b1 , . . . , bN ) is said to be an equally trembling-hand perfect equilibrium with bound M > 0, if, in addition to the above conditions, the following are satisfied: (iv) fi,k is independent of k. (v) εi,k is independent of i. (vi) fi,k (x)/fj,k (x) ≤ M for all i, j and k. An alternative extension of perfect equilibrium is proposed in M´endez-Naya et al. (1995). The difference arises from two different but equivalent interpretations of perfect equilibrium. Roughly, a perfect equilibrium is defined as a limit of equilibria of perturbed games, which is the interpretation on which M´endez-Naya et al. (1995) rely. A perfect equilibrium can be also seen as a “robust” Nash equilibrium in the sense that players’ best responses do not change even when players make mistakes with small probability. This paper’s extension is based on this second interpretation. Roughly, the next theorem claims that any π-stable equally trembling-hand perfect equilibrium with a fixed bound converges to {bπi } as N goes to infinity.

See Appendix.

While bids of agents i = 1, . . . , P have small indeterminancy, the equilibrium bids of agents i = P + 1, . . . , N are completely determined.

Proposition 3. Suppose that (b1 , . . . , bN ) is a π-stable equally trembling-hand perfect equilibrium with bound M > 0 in ΓN . Then bi = vi ( = bπi ) for all i ̸∈ Im π. Proof. See Appendix.

The following is the idea of Theorem 5 and Proposition 3. Let us consider an efficient equilibrium. Consider the situation in which advertisers make mistakes randomly and independently with probability ε. The distribution function of advertiser i’s mistake bid is Fi , which has a continuous density function fi . Let ES be the event that the members of S make mistakes but the others do not. First, consider the case of i = 1, . . . , P . For |S| ≥ 2, the probability of the event ES is at most ε2 , which is vanishingly small compared to ε, so that we may ignore these events. Also, when N is huge, we can ignore mistakes by j = 1, . . . , i + 1 because their fraction in the whole population is approximately 0. By the above argument, the only unignorable part of i’s payoff is: {∫ N [ ] bi ∑ N −2 fj (x) αi (vi − (x ∨ bi+1 )) dx ε(1 − ε) j=i+2

0

} ] ( )[ + 1 − Fj (bi ) αi+1 (vi − bi+1 ) . (21)

The corresponding first order condition is αi (vi − bi ) = αi+1 (vi − bi+1 ),

(22)

and its unique solution is bi = v i −

αi+1 (vi − bi+1 ). αi

(23)

Next, we consider advertisers i > P . Here N can be a small number. For i to obtain some position, at least (i − P ) advertisers with values higher than i’s are needed to make mistakes. We can ignore the case that more than (i − P ) advertisers make mistakes, because its probability is much smaller than the probability that (i − P ) advertisers make

is not beneficial for advertiser i.

mistakes. Thus, the unignorable part of i’s payoff is εi−P (1 − ε)N −1−(i−P ) ∫ bi [ ] ∑ · fS (x) αP (vi − (x ∨ bi+1 )) dx, S⊆{1,...,P } |S|=i−P

αi (vi − b∗i+1 ) − αj (vi − b∗j+1 ) =

(24)

P −1 ∑ k=i

0

j P −1 ∑ ∑ αi αj αj γk − γk − γk αi+1 αk+1 αj+1

j−1 {

FS′ (x)

and FS (x) = where fS (x) = it is maximized at bi = vi .

∏ j∈S

= Fj (x). Obviously,

∑ k=i

k=i

k=j+1

}

αi αj − γk αi+1 αk+1 { } P∑ −1 αi αj + − γk , αi+1 αj+1

(30)

k=j

7. A SPECIAL CASE: EXPONENTIAL CLICKTHROUGH RATES We back to the game Γε . In the special case in which αp = Aλp , where A > 0 and λ ∈ (0, 1), oftentimes the game Γε have an efficient Nash equilibrium.5 Without loss of generality, we may focus on b∗ = (b∗1 , . . . , b∗N ), because if there exists an efficient Nash equilibrium then b∗ is also an efficient Nash equilibrium. The bid profile b∗ is a Nash equilibrium if and only if no advertiser i < P has an incentive to descend to position P . First note that advertisers i > P have no incentive to deviate, because the minimum per-click price is b∗P = vP∗ , which is too expensive for them. We consider deviations by the other advertisers i ≤ P . Let us show that any advertiser i ≤ P is worse off by raising their bids from b∗P when ε is sufficiently small. This is obvious when i = P , because it must pay more than its valuation for each click to occupy a better position. Assume i < P . Write γi = αi+1 (vi − vi+1 ). Then b∗i is written ∑P −1 as6 b∗i = vi − α−1 i j=i γj . By using this expression, for i, ℓ ∈ {1, . . . , P }, we can write vi − b∗ℓ =

ℓ−1 P −1 ∑ ∑ γk γk + αk+1 αℓ k=i

vi − b∗ℓ = −

i−1 ∑ k=ℓ

if ℓ > i,

(25)

which is positive because αi /αi+1 > 1 ≥ αj /αk+1 for i ≤ k < j and αi /αi+1 ≥ αj /αj+1 . Therefore, descending to position j < P is not beneficial for advertisers i < j for sufficiently small ε. Since losing the current position is not an option to advertisers i ≤ P , the only possible deviations are those in which some advertiser i < P descends to position P . A sufficient condition to prevent advertiser i’s deviation (i < P ) is (1 − ε)αi (vi − b∗i+1 ) ≥ αP (vi − vP +1 ) ⇔ ⇔

γk + αk+1

k=ℓ

k=i

>

P −1 ∑

γk αℓ

if ℓ < i.

k=2

for i ≤ P − 2, ψp is maximized at p = P − 1 if λ ≤ is maximized at p = P − 1.

k=i

k=j

(26)

(27)

γk +

k=j

γk −

(28)

k=j

P −1 ∑

(33)

(34) √

γk = 0.

(29)

k=j

Therefore, if ε is sufficiently small, advertisers lose payoffs by raising their bids. Note that so far we have not use the assumption on the click-through rates. Next, we show that, if i < j < P , descending to position j 5 The author is grateful to anonymous reviewers of the Sixth Ad Auction Workshop for suggesting this analysis. 6 If i ≥ P , the summation is set to be 0.

1 − ε.

Similarly, (35)

for i < P is a necessary condition for the existence of an efficient equilibrium. This condition is equivalent to vP +1 ≥ vP −1 −

i−1 P −1 ∑ ∑ αi αj γk + γk − γk αi+1 αk+1 i−1 ∑

λk (vk − vk+1 )

αi (vi − b∗i+1 ) ≥ (1 − ε)αP (vi − vP +1 )

αi (vi − b∗i+1 ) − αj (vi − b∗j ) P −1 ∑

vP +1 ≥ vi − (1 − ε) · λ−P

k=i

P −1 ∑

αk+1 (vk − vk+1 ) (32) αP

Let ψi be the right-hand side of (33). Since ) P −i ( ∑ 1−ε ψP −1 − ψi = − 1 (vP −k − vP −k−1 ) λk

Hence, for i, j ∈ {1, . . . , P − 1} with j < i,

=

αi αi+1

(31)

k=i

k=ℓ

P −1 ∑

vP +1 ≥ vi − (1 − ε) ·

P −1 ∑

vP −1 − vP (1 − ε)λ

(36)

without any additional assumption. The following is the summary of the above analysis: Proposition 4. Suppose αp = Aλp , where √ A > 0 and λ ∈ (0, 1). Also assume either P = 2 or λ ≤ 1 − ε. Then, (b∗1 , . . . , b∗N ) is a Nash equlibrium if 1−ε · (vP −1 − vP ). (37) λ There exists an efficient Nash equilibrium only if (35) holds. vP +1 ≥ vP −1 −

8.

RELATION TO LOCALLY ENVY-FREE EQUILIBRIA AND SYMMETRIC EQUILIBRIA

Finally, we briefly discuss two equivalent concepts independently introduced by EOS and Varian (2007), respectively.

Definition 3. A π-stable pure strategy Nash equilibrium (b1 , . . . , bN ) is locally envy-free if αp (vπ(p) − cp ) ≥ αp−1 (vπ(p) − cp−1 )

(38) ′

for p = 2, 3, . . . , P , where cp′ = bπ(p′ +1) for p < P and cP = max{bi : i ̸∈ Im π}. (b1 , . . . , bN ) is symmetric if αp (vπ(p) − cp ) ≥ αp′ (vπ(p) − cp′ )

(39)

′

for all p, p ∈ {1, . . . , P }. EOS prove that these two concepts are equivalent and imply efficiency: Proposition 5. Suppose b is a π-stable pure strategy Nash equilibrium. Then, b is locally envy-free if and only if it is symmetric. If b is locally envy-free, then b is efficient. Proposition 6. (b∗1 , . . . , b∗N ) is locally envy-free if and only if R1 ≤ · · · ≤ RP −1 , where Ri = αi /αi+1 . Proof. For i = P , (38) is satisfied because bj = vj for j ≥ P . If i = 2, . . . , P − 1, then (38) is equivalent to αi (vi − vi+1 ) + Ri

P ∑

P ∑

αj+1 (vj − vj+1 )

(40)

j=i

⇔

(Ri − Ri−1 )

P ∑

αj+1 (vj − vj+1 ) ≥ 0

(41)

j=i

⇔

Ri−1 ≤ Ri . (b∗1 , . . . , b∗N )

Therefore, · · · ≤ RP −1 .

The main message of this paper is that, if advertisers realize the existence of a possible entrants, or a noise bidder, their behavior will be very different from the VCG bids, and that it is highly likely that some unstable equilibrium is realized. A natural next question is how unstable “unstable equilibria” are. Unfortunately, the nature of unstable equilibria is completely unknown. Characterizing them would be an important future work.

10.

(42) is locally envy-free if and only if R1 ≤

9. CONCLUSION The GSP has a continuum of equilibria in the complete information setting. In the literature of the GSP, Nash equilibria whose outcome is equivalent to the VCG mechanism have been attracted special attentions since two pioneering work by EOS and Varian (2007). This paper constructs a model which is basically the same as that of EOS and Varian (2007) but contains a non-strategic bidder called the noise bidder whose bid is purely stochastic. In this model, efficient equilibria, and other stable equilibria, are almost fully characterized, and as a corollary of this result, it is shown that any bid profile equivalent to the VCG mechanism have no Nash equilibrium in its neighborhood even in the limit where the noise bidder never appears. Even without the noise bidder, a similar characterization result is achieved if a variant of trembling-hand perfect equilibrium is used as the equilibrium concept. The noise bidder model may not have any stable Nash equilibrium. For any fixed click-through rates, there exists a positive mass of valuations such that there is no stable Nash equilibrium. According to the simulation conducted in this paper, in the space of click-through rates and valuations, the

ACKNOWLEDGMENTS

The author wishes to thank three anonymous reviewers for insightful comments. The author is also grateful to Mike Ostrovsky, Michael Schwartz, Andy Skrzypacz, Bob Wilson, Fuhito Kojima, Michihiro Kandori, Ben Golub, Songzi Du, Yosuke Yasuda, Daisuke Hirata, Takeshi Murooka, Yuichiro Kamada, and the seminar participants at Stanford University and the University of Tokyo for helpful discussions and comments.

11.

αj+1 (vj − vj+1 )

j=i+1

≥ Ri−1

subset in which no stable Nash equilibrium exists is huge. When five or more positions are auctioned, that unstable subset occupies more than 95% of the space. This instability result can explain the fact found by Edelman and Ostrovsky (2007) that in reality rankings of advertisers change very frequently.

REFERENCES

ogers T., I. Cox, M. Pesendorfer, and V. Petricek. [1] B¨ 2008. “Equilibrium Bids in Sponsored Search Auctions: Theory and Evidence.” Mimeo. [2] Carlsson, H., and E. van Damme. 1993. “Global Games and Equilibrium Selection.” Econometrica, 61(5): 989-1018. [3] Cary, M., A. Das, B. Edelman, I. Giotis, K. Heimerl, A. R. Karlin, C. Mathieu, and M. Schwarz. 2007. “On best-response bidding in GSP auctions.” In EC: Proceedings of the ACM Conference on Electoronic Commerce, 262-271. [4] Edelman, B., and M. Ostrovsky. 2007. “Strategic Bidder Behavior in Sponsored Search Auctions.” Decision Support Systems, 43(1): 192-198. [5] Edelman, B., M. Ostrovsky, and M. Schwarz. 2007. “Internet Advertising and the Generalized Second Price Auction: Selling Billions of Dollars Worth of Keywords”, American Economic Review, 97(1): 242-259. [6] Fudenberg, D., and D. K. Levine. 1999. The Theory of Learning in Games. Cambridge, MA: MIT Press. [7] Hafalir, I., R. Ravi, and A. Sayedi. 2009. “Sort-Cut: A Pareto-Optimal and Semi-Truthful Mechanism for Multi-Unit Auctions with Budget-Constrained Bidders.” In the Fifth Ad Auction Workshop. [8] Kominers, S. D. 2009. “Dynamic Position Auctions with Consumer Search.” Proceedings of The 5th International Conference on Algorithmic Aspects in Information and Management (AAIM), Lecture Notes in Computer Science, 5564: 240-250. [9] Lahaie, S., and D. M. Pennock. 2007. “Revenue analysis of a family of ranking rules for keyword auctions.” In: EC: Proceedings of the ACM Conference on Electronic Commerce. [10] Leme, R. P., and E. Tardos. 2009. “Sponsored search

[11]

[12]

[13]

[14]

[15] [16]

[17] [18]

equilibria for conservative bidders.” In the Fifth Ad Auctions Workshop. M´endez-Naya, L., I. Garcia-Jurado, and J. C. Ceso, 1995. “Perfection of Nash equilibria in continuous games.” Mathematical Social Sciences, 29, 225-237. Morris, S., and H. S. Shin. 1998. “Unique equilibrium in a Model of Self-Fulfilling Currency Attacks,” American Economic Review, 88: 587-597. Myerson, R. B. 1978. “Refinements of the Nash equilibrium concept.” International Journal of Game Theory, 15: 133-154. Selten, R. 1975. “A reexamination of the perfectness concept for equilibrium points in extensive games.” International Journal of Game Theory, 4: 25-55. Varian, H. R. 2007. “Position Auctions.” International Journal of Industrial Organization, 25: 1163-1178. van Damme, E. 1984. “A relationship between perfect equilibria in extensive form games and proper equilibria in normal form games.” International Journal of Game Theory 13: 1-13. Weibull, J. W. 1995. Evolutionary game theory, MIT Press. Yenmez, M. B. 2009. “Pricing in Position Auctions and Online Advertising.” Mimeo.

APPENDIX A. DETAILS OF THE SIMULATION For each P and N , n samples are generated. Each sample (k) (k) (k) (k) k consists of α ˜1 , . . . , α ˜ P and v˜1 , . . . , v˜N , which are independently drawn from the uniform distribution on [0, 1]. (k) (k) (k) (k) By reordering α ˜1 , . . . , α ˜ P and v˜1 , . . . , v˜N in descending (k) (k) (k) (k) order, we generate α1 , . . . , αP and v1 , . . . , vN , respectively. In our noise bidder model, if there exists a π-stable Nash equilibrium, π must satisfy Im π = {1, . . . , P } and π(P ) = P , so in the simulation all of such π are considered. For each π, the following inequalities are checked: bππ(1) > bππ(2) > · · · > bππ(P ) , αp (vπ(p) −

bππ(p+1) )

(43)

≥ (1 − ε)αp′ (vπ(p) −

′

bππ(p′ ) )

(44)

′

for p, p ∈ {1, . . . , P } such that p < p, and αp (vπ(p) − bππ(p+1) ) ≥ (1 − ε)αp′′ (vπ(p) − bππ(p′′ +1) ) ′′

(45)

′′

for p, p ∈ {1, . . . , P } such that p < p , where π(P + 1) = P + 1. In this simulation, if some of the above inequalities is violated in sample k, then Xkπ is set to be 1. Otherwise, Xkπ is 0. Mathematically, if any one of the conditions (43)(45) fails to be satisfied, then there exists no π-stable Nash equilibrium in Γε (α(k) , v (k) ) for all ε ∈ (0, ε). The outcome Xk of sample k is 1 if Xkπ = 1 for all π. Otherwise Xk = 0. If Xk = 0, there exists no stable equilibrium in Γε (α(k) , v (k) ) for all ε ∈ (0, ε). The result of the simulation ∑ (k) /n. is the mean of X (k) , i.e., n k=1 X

B.

PROOFS

We introduce a few notations. Let N = {1, . . . , N } and Ni = N \ {i}. Let g = (g1 , . . . , gN ) : [0, ∞) → RN be the payoff function of GSP. The probability measure over [0, ∞)Ni induced by σ−i = (σj )j̸=i is denoted by P(·|σ−i ), or P−i .

B.1

Proof of Theorem 2

Let y = max{bi : i ̸∈ Im π}, and, for each i, define bi = ess sup bσi , bi = ess inf bσi , and y = ess sup y. Let B1 = [bπ(2) , ∞), Bp = [bπ(p+1) , bπ(p−1) ] for p = 2, . . . , P − 1, and BP = [y, bπ(P −1) ]. Since σ is π-stable, bσπ(p) ∈ Bp almost surely. Let Ui (βi ) denote the payoff of i from bidding βi . For all p ∈ {1, . . . , P }, σπ(p) is a pure strategy. This is obvious when Bp is a singleton, so we assume Bp is a nondegenerate interval. Let i = π(p). For βi ∈ int Bi , {∫ ( βi [ ]) Ui (βi ) = ε · αp vi − E x ∨ b+ f0 (x)dx 0

∫

∞

+

(

[

αp+1 vi − E b+

])

} f0 (x)dx

βi

( [ ]) + (1 − ε) · αp vi − E b+ ,

(46)

where b+ = bπ(p+1) if p < P and b+ = y if p = P . Its first order derivative is { Ui′ (βi ) = αp (vi − βi ) ( [ ])} · εf0 (βi ). (47) − αp+1 vi − E b+

This equals zero only if βi equals ( [ ]) ˆbi = vi − αp+1 vi − E b+ , αp

Next, let us evaluate bσπ(p) − bππ(p) from above. (48)

so βi ∈ int Bp \ {ˆbi } cannot be chosen as equilibrium bids with positive probability, that is, σi (int Bp \ {ˆbi }) = 0. Therefore: [19] (i) If ˆbi ∈ int Bp , then bσi = ˆbi almost surely. (ii) If ˆbi ≥ max Bp , then bσi = max Bp almost surely. (iii) If ˆbi ≤ min Bp , then bσi = min Bp almost surely.

Qp

} ) αp+1 {( σ ≤ bπ(p+1) − bππ(p+1) + Ap (q) . αp

In every case, σi is a pure strategy. Finally we show bσπ(p) = ˆbπ(p) for p = 1, . . . , P , which completes the proof. Let i = π(p). When p = P , bσi = vi = ˆbi because Ui′ (x) > 0 for x < vi . Assume p = 1, . . . , P − 1. Since bσπ(1) , . . . , bσπ(P ) are deterministic, bσi ∈ int Bp . The equilibrium bid bσi must be an optimal bid, so it satisfies the first order condition: U ′ (bσi ) = 0. Its solution is bσi = ˆbi .

B.2 Proof of Proposition ?? and Theorem 4 Define Uπ(p) and Bp as in the proof of Theorem 2. If Bp has a non-empty interior, p { } ∑ Ui′ (βi ) = Qs αs (vi − βi ) − αs+1 (vi − Xp ) , εf0 (βi ) s=1 ( ) p−1 Qs = (1 − q)s−1 q p−s , s−1 ∑ { Xp = (1 − q)|S| q N −p−|S| S⊆Np

( )} · max {bσj : j ∈ S} ∪ {0}

bσπ(p) − bππ(p) ) αp+1 ( ≤ vπ(p) − bππ(p+1) αp Qp αp+1 (vπ(p) − Xp ) − Qp αp + (1 − Qp )α1 ) αp+1 ( Xp − bππ(p+1) = αp αp+1 1 − Qp α ( 1 ) + · · · (vπ(p) − Xp ) 1−Qp αp Qp αp + α1

(49)

(54)

Therefore bσπ(p) − bππ(p) ≤

B.2.2

P 1 ∑ αs+1 As (q). αp s=p

(55)

Proof of Theorem 4

Without loss of generality, we assume δ < min{|vi − vi+1 | : i = 1, . . . , N }, where vN +1 = 0. Let Σπ (q) be the set of π-stable Nash equilibria of Γε,q . Define ι(k) so that ι(1) < · · · < ι(N − P ) and the disjoint union of Im ι and Im π is {1, . . . , N }. Define, for convenience, ι(0) = π(P ) and ι(N − P ) = N + 1. First, we show that vι(0) > vι(1) when q is sufficiently small:

(50)

Lemma 1. Suppose vπ(P ) < vi with some i ̸∈ Im π. There exists q¯0′ such that, for all q < q¯0′ there exist no π-stable Nash equilibrium of Γε,q .

(51)

Proof. Suppose that such an equilibrium exists. The payoff of advertiser i from bidding b′i = min{vi +vπ(P ) , bσπ(P −1) + bσπ(P ) }/2 is at least ( ) vi + vπ(P ) V (q) = (1 − q)P −1 (1 − ε)αP vi − 2 ( ) vi − vπ(P ) P −1 = (1 − q) (1 − ε)αP (56) 2

for βi ∈ int Bp , where i = π(p) and Np = {1, . . . , N } \ {π(1), . . . , π(p)}. Thus, in the interior of Bp , Ui′ (βi ) = 0 only if βi = βˆi , where ∑p s=1 Qs αs+1 ˆbi = vi − ∑ (vi − Xp ). (52) p s=1 Qs αs By the same argument as in the proof of Theorem 2, σπ(1) , . . . , σπ(P ) are pure strategies and bσπ(p) = ˆbπ(p) for p = 1, . . . , P − 1. For p = P , bσπ(P ) = max{ˆbπ(P ) , y}.

B.2.1 Proof of Proposition ?? First we evaluate bσπ(p) − bππ(p) from below. bσπ(p) − bππ(p) ) αp+1 ( ≥ vπ(p) − bππ(p+1) αp Qp αp+1 + (1 − Qp )α2 (vπ(p) − Xp ) − Qp αp ) 1 − Qp α2 αp+1 ( = Xp − bππ(p+1) − · · (vπ(p) − Xp ) αp Qp αp αp+1 1 − Qp α2 ≥ ((1 − q) − 1)bππ(p+1) − · · vπ(p) αp Qp αp { } αp+1 1 − Qp α 2 ≥− q· · vπ(p+1) + · · vπ(p) . (53) αp Qp αp

while its equilibrium payoff is at most ( ) W (q) = 1 − (1 − q)P −1 α1 vi .

(57)

Since limq→∞ {V (q) − W (q)} > 0, there exists q¯0 > 0 such that, for all q ∈ (0, q¯0′ ), V (q) > W (q). This completes the proof. Let q¯0′ as in the above lemma. By Proposition ??, there exists q¯0 ∈ (0, q¯0′ ) such that, for all q ∈ (0, q¯0 ) and σ ∈ Σπ (q), σ bπ(p) − bππ(p) < δ for all p ∈ {1, . . . , P }. We construct q¯k in the following procedure. Suppose that q¯k−1 is already defined and that, for all q ∈ (0, q¯k−1 ) and σ ∈ Σπ (q), bσι(k−1) > vι(k−1) − δ

for all ℓ ∈ {0, . . . , k − 1}. Let x = max{bσι(ℓ) : ℓ > k} and Fx be the distribution function of x. The payoff of advertiser ι(k) from bidding vι(k) is at least ∫ vι(k) { k W1 = Qk (1 − ε)αP (vι(k) − x) 0

∫

vι(k)

+ε

} ( ) αP vι(k) − (x ∨ b0 ) f0 (b0 )db0 dFx (x).

0

(58)

0 β

+ε

}

( ) αP vι(k) − (x ∨ b0 ) f0 (b0 )db0 dFx (x)

0

+ Qk+ · α1 vι(k) .

˜bj ≥ vi > ˜bj ′

(63)

for all j and j ′ such that j < i < j ′ . This is guaranteed if i = P + 1. If i > P + 1, we obtain this condition after proving ˜bj = vj for j = P + 1, . . . , i − 1. Let Si be the set of S ⊆ {1, . . . , i − 1} such that |S| = i − P . Then, for all b′i , b′′i ∈ [0, vi ], Ui,k (b′′i ) − Ui,k (b′i ) εi−P k } ∑{ Πi,S (b′′i ) − Πi,S (b′i ) − 2N V¯ · εk . ≥

The payoff of bidding β ≤ vι(k) − δ is at most ∫ β{ k W2 = Qk (1 − ε)αP (vι(k) − x) ∫

Proof of Proposition 3. Let i > P . We assume

(59)

(64)

S∈Si

The next lemma completes the proof because it implies that ˜bi must equal vi . Define bi = maxj>i ˜bj .

By evaluating the difference of them from below, W1k − W2k ∫ ≥ Qk · ε

vι(k) {

} ( ) αP vι(k) − vι(k+1) f0 (b0 ) db0 dFx (x)

β

− Qk+ · α1 vι(k) { } ≥ Qk · εαP δ F0 (vι(k) ) − F0 (vι(k) − δ) − Qk+ · α1 vι(k)

(60) (61)

Note that (61) is independent of the choice of equilibria σ. Since Qk /Qk+ → ∞ as q → 0, there exists q¯k ∈ (0, q¯k−1 ) such that bσι(k)

Lemma 2. Let z ∈ (0, vi − bi ). There exists Kz such that k > Kz implies Ui,k (bi ) < Ui,k (vi ) for all bi < vi − z. Proof. Let y = z/2 and FS (x) = Si and bi < vi − z,

∫ ≥

for all q ∈ (0, q¯k ) and σ ∈ Σ (q).

Given σ−i,k , the payoff of i from bidding bi is written as } ∑ { |S| Ui,k (bi ) = εk (1 − εk )|Ni \S| πi,S (bi ), (62) S⊆Ni

where Πi,S is the expected payoff of i conditional on the event that the set of all the people making mistakes is S. Note |Πki,S (bi )| ≤ M for all bi ∈ [0, vi ], because in this game payoffs are at most α1 v1 and per-click payments are at most vi ≤ v1 .

(65)

(bi ,vi )

αP δ ′ dFS (x)

(66)

(vi −2y,vi −y)

(67)

which is positive and independent of bi . Find Kz such that { } ∑ 2N V¯ εk < αP y FS (vi − y) − FS (vi − 2y)

B.3 Proofs of Theorem 5 and Proposition 3

First note that ˜bi ≤ vi , because bidding bi > vi is dominated by bidding bi = vi and the event that the former bid is strictly better than the latter occurs with positive probability. Also, ˜bπ(p) ≥ vP +1 must hold for all p ∈ {1, . . . , P } because ˜b is a Nash equilibrium.

Fj (x). For S ∈

{ } = αP y FS (vi − y) − FS (vi − 2y) ,

π

Suppose that ˜b = (˜b1 , . . . , ˜bN ) is a π-stable equally tremblinghand perfect equilibrium with bound M > 0. Pick {εk }∞ k=K , ˜ (f1 , . . . , fN ), and {(b1,k , . . . , bN,k )}∞ k=K such that b satisfies the definition of strong trembling-hand perfect equilibrium Let σi,k be as in Definition 2, and Fi the distribution function of density f . Define V¯ = α1 v1 , Ni = {1, . . . , N } \ {i}, ˜b0 = ∞, π(0) = 0, and π(P + 1) = P + 1.

j∈S

Πi,S (vi ) − Πi,S (bi ) ∫ ≥ αP (vi − (x ∨ bi )) dFS (x)

> vι(k) − δ

Finally set q¯ = q¯N −P . Then the proof is completed.

∏

S∈Si

for all k > Kz . Then, Ui,k (bi ) < Ui,k (vi ) for all k > Kz and bi < vi − z.

Proof of Theorem 5. Let p ≤ P and i = π(p). Set Bi = (˜bπ(p−1) , ˜bπ(p+1) ). Define Mi = {j : π(j) > p + 1 or j ̸∈ Im π} if p < P , and Mi = {P + 2, . . . , N } if p = P . Let ˆbi = vi − αp+1 (vi − ˜bπ(p+1) ), αp f¯ = maxj f , and f = minj fj . For bi ∈ Bi , ′ Ui,k (bi ) =

∑

{ } fj (bi ) αp (vi − bi ) − αp+1 (vi − ˜bπ(p+1) )

j∈Mi

+ X(bi ) + Y (bi ), where X and Y are some functions whose bounds are given by |X(bi )| ≤ pV¯ · f¯(bi ), |Y (bi )| ≤ 22N V¯ f¯(bi ) · εk .

′ Since Ui,k (bi,k ) = 0, { } M V¯ p + 22N · εk . bi,k − ˆbi ≤ (N − P − 2)αP

In the limit k → ∞, ˜ bi − ˆbi ≤

M V¯ p . (N − P − 2)αP

(68)

(69)

By letting N sufficiently large, we can make ˜bi arbitrarily close to ˆbi and thus to b∗i .