Hybrid mechanisms for Vickrey–Clarke–Groves and generalized second-price bids

Abstract

This paper analyzes mechanisms for selling advertising opportunities for several different positions on a page that would enable some advertisers to bid using Vickrey–Clarke–Groves (VCG) pricing while other advertisers bid using generalized second-price (GSP) pricing. I focus on a setting in which the number of ads displayed may vary with the advertisers’ bids and showing fewer ads enables the remaining ads to obtain more clicks. I illustrate the types of mechanisms one can construct that would ensure that VCG bidders have an incentive to bid truthfully, GSP bidders cannot obtain the same number of clicks at a lower price by varying their bid, and if all bidders bid using VCG (GSP) pricing, then the outcome is the same as it would be under the VCG (GSP) mechanism.

Appendices

Appendix A: Inefficiency of GSP pricing

To illustrate that GSP pricing may result in inefficiencies in equilibrium, I consider a setting in which there are three bidders with values $v_H$, $v_M$, and $v_L$, where $v_H> v_M > v_L$. These bidders make bids of $b_H$, $b_M$, and $b_L$ in an auction with a maximum of two positions. Throughout I focus on a setting in which $x_{1,1} v_{H} < x_{1,2} v_{H} + x_{2,2} v_{M}$ so the efficient outcome is to show the advertiser with value $v_H$ in the first position and the advertiser with value $v_M$ in the second position. In this setting, I obtain the following result:

Theorem 6

Suppose that $x_{1,1} v_{H} < x_{1,2} v_{H} + x_{2,2} v_{M}$, $|\frac{x_{1,1} - x_{1,2}}{x_{1,2}} - \frac{1}{2}| < |\frac{x_{2,2}}{x_{1,2}} - \frac{1}{2}|$, and $x_{2,2}^{2} (v_H - v_L) > x_{1,2}(x_{1,2} + x_{2,2} - x_{1,1})v_H$. Then in any pure-strategy equilibrium in undominated strategies, the advertiser with value $v_H$ is placed in the second position and the advertiser with value $v_M$ is placed in the first position.

All proofs of the results in this section are in Appendix B. Under additional assumptions, we also have a guarantee that a pure-strategy equilibrium in undominated strategies exists:

Theorem 7

Suppose that $x_{1,1} v_{H} < x_{1,2} v_{H} + x_{2,2} v_{M}$, $x_{1,1} v_{M} < x_{1,2} v_{M} + x_{2,2} v_{L}$, and $x_{1,2}(v_H - v_M) \le x_{2,2}(v_H - v_L)$. Then there is a pure-strategy equilibrium in which $b_H = v_L$, $b_M = v_M$, and $b_L = v_L - \epsilon$ for some small $\epsilon > 0$.

The following result follows immediately from Theorems 6 and 7:

Corollary 1

Under the conditions in Theorems 6 and 7, the fractional efficiency loss from using GSP rather than VCG pricing is $\frac{(x_{1,2} - x_{2,2})(v_H - v_M)}{x_{1,2}v_H + x_{2,2}v_M}$ in any pure-strategy equilibrium.

This efficiency loss can be substantial, as the following example illustrates:

Example 2

If $v_H = 20$, $v_M = 14.1$, $v_L = 10$, $x_{1,1} = 1.421$, $x_{1,2} = 1$, and $x_{2,2} = 0.6$, then the conditions in Corollary 1 are satisfied, and using GSP rather than VCG pricing results in a $8.3\%$ efficiency loss.

Not only can GSP pricing lead to significant inefficiencies in equilibrium, but there is also no guarantee that a pure-strategy equilibrium will exist:

Theorem 8

Suppose there are two bidders with values $v_H > v_M$, and we have $x_{1,1} v_{H} < x_{1,2} v_{H} + x_{2,2} v_{M}$, $|\frac{x_{1,1} - x_{1,2}}{x_{1,2}} - \frac{1}{2}| < |\frac{x_{2,2}}{x_{1,2}} - \frac{1}{2}|$, and $x_{2,2}^{2} > x_{1,2}(x_{1,2}$ $+\, x_{2,2} - x_{1,1})$. Then there is no pure-strategy equilibrium in undominated strategies.

The fact that any equilibrium may be in mixed strategies can further lead to inefficiencies from sometimes choosing a suboptimal allocation since the bids will again generally not equal the advertisers’ values in such an equilibrium.

Appendix B: Proofs of main results

Proof of Remark 1

Note that if a bidder makes a bid so that the bidder maintains the jth position, then the bidder obtains a payoff of $x_{j,k} v_{(j)} - R_{j, k^{\prime }(j)} + R_{j, k} - x_{j,k^{\prime }(j)} b_{(j+1)}$. Now if a bidder restricts attention to bids such that the bidder’s position never varies, then the terms $R_{j, k^{\prime }(j)}$ and $x_{j,k^{\prime }(j)} b_{(j+1)}$ will be constants that are independent of the bidder’s bid, and the bidder’s payoff can be written as a constant plus the term $x_{j,k} v_{(j)} + R_{j, k}$.

Thus if a bidder changes its bid in such a way that the bidder receives a different number of clicks, then the bidder’s payoff changes by an amount equal to the sum of the value that he obtains from these additional clicks plus the total change in value for the other bidders. But this is the same change in a bidder’s payoff that would result in the VCG mechanism, a mechanism in which truthful bidding is a dominant strategy. Thus if a bidder wishes to maintain its current position, this bidder can do no better than bid truthfully. $\square$

Proof of Remark 2

Note that the difference between the seller’s revenue under VCG and GSP is $S_{j,K} - (R_{j,k^{\prime }(j)} + x_{j,k^{\prime }(j)} b_{(j+1)})$. Now $S_{j,K}$ equals the value of $\sum _{i=1}^{k} x_{i,k} b_{(i)}$ if advertiser j did not bid in the mechanism, while $R_{j,k^{\prime }(j)} + x_{j,k^{\prime }(j)} b_{(j+1)}$ denotes the total value of this sum that would result if advertiser j made a bid of $b_{(j+1)}$. If we had $S_{j,K} > R_{j,k^{\prime }(j)} + x_{j,k^{\prime }(j)} b_{(j+1)}$, this would imply that the allocation selected under VCG when advertiser j does not bid results in a higher value of $\sum _{i=1}^{k} x_{i,k} b_{(i)}$ than the allocation selected under GSP when advertiser j makes a bid of $b_{(j+1)}$. But the allocation selected under VCG when advertiser j does not bid is also available to the GSP mechanism when advertiser j makes a bid of $b_{(j+1)}$, contradicting the fact that the GSP mechanism chooses the allocation that results in the highest possible value of $\sum _{i=1}^{k} x_{i,k} b_{(i)}$. Thus $S_{j,K} \le R_{j,k^{\prime }(j)} + x_{j,k^{\prime }(j)} b_{(j+1)}$, and revenue is at least as high under GSP as it is under VCG for any fixed bids of the advertisers. $\square$

Proof of Theorem 1

Since a VCG (GSP) bidder’s allocation and price for a given bid is the same as it would be in the standard VCG (GSP) mechanism (given the bids of the other bidders), and the VCG and GSP mechanisms satisfy the (truthful) and (CPC-invariant) properties, this mechanism also satisfies the (truthful) and (CPC-invariant) properties. Furthermore, if all bidders submit VCG (GSP) bids, the final outcome is the same as it would be under VCG (GSP), so this mechanism satisfies (VCG-equivalence) and (GSP-equivalence). Finally, for fixed bids of the advertisers, the allocation of ads is the same regardless of which of these bids are VCG bids or GSP bids since the allocation is chosen to maximize $\sum _{i=1}^{k} x_{i,k} b_{(i)}$, but we know from Remark 2 that an advertiser’s cost under the VCG mechanism is never greater than the advertiser’s cost under the GSP mechanism. Thus an advertiser will always do at least as well by making a VCG bid of b than by making a GSP bid of b and the (no switching) property is satisfied. $\square$

Lemma 1

Under the GSP mechanism, the CPC of the bidder in position j falls between $b_{(j)}$ and $b_{(j+1)}$.

Proof

Since the mechanism displays k ads when the advertiser in position j bids $b_{(j)}$, we know that $R_{j,k} + x_{j,k} b_{(j)} \ge R_{j,k^{\prime }(j)} + x_{j,k^{\prime }(j)} b_{(j)}$, where $k^{\prime }(j)$ denotes the number of ads the mechanism would display if the advertiser in position j bid $b_{(j+1)}$. Since $b_{(j)} \ge b_{(j+1)}$, it further follows that $R_{j,k} + x_{j,k} b_{(j)} \ge R_{j,k^{\prime }(j)} + x_{j,k^{\prime }(j)} b_{(j+1)}$. This in turn implies that $x_{j,k} b_{(j)} \ge R_{j,k^{\prime }(j)} - R_{j,k} + x_{j,k^{\prime }(j)} b_{(j+1)}$, meaning $b_{(j)} \ge \frac{1}{x_{j,k}} (R_{j,k^{\prime }(j)} - R_{j,k} + x_{j,k^{\prime }(j)} b_{(j+1)})$, and the CPC of the bidder in position j is no greater than $b_{(j)}$.

Similarly, since the mechanism would display $k^{\prime }(j)$ ads if the advertiser in position j bid $b_{(j+1)}$, it follows that $R_{j,k^{\prime }(j)} + x_{j,k^{\prime }(j)} b_{(j+1)} \ge R_{j,k} + x_{j,k} b_{(j+1)}$. This in turn implies that $R_{j,k^{\prime }(j)} - R_{j,k} + x_{j,k^{\prime }(j)} b_{(j+1)} \ge x_{j,k} b_{(j+1)}$, meaning $\frac{1}{x_{j,k}} (R_{j,k^{\prime }(j)}$ $- R_{j,k} + x_{j,k^{\prime }(j)} b_{(j+1)}) \ge b_{(j+1)}$. Thus the bidder in position j always pays a CPC of at least $b_{(j+1)}$. $\square$

Proof of Theorem 2

Suppose that two advertisers compete in the auction, advertiser 1 makes a VCG bid of $v_{1}$, and advertiser 2 makes a VCG bid of $v_{2} < v_{1}$. Note that $\frac{x_{1,1} - x_{2,2}}{x_{1,2}}< 1 < \frac{x_{2,2}}{x_{1,1} - x_{1,2}}$ because $x_{1,1} - x_{2,2} < x_{1,2}$ and $x_{1,1} - x_{1,2} < x_{2,2}$. From this it follows that $\frac{x_{2,2}}{x_{1,1} - x_{1,2}} > \frac{x_{2,2}(x_{1,1} - x_{2,2})}{x_{1,2}(x_{1,1} - x_{1,2})}$, which in turn means that there exist values of $v_{1}$ and $v_{2}$ such that $\frac{x_{2,2}}{x_{1,1} - x_{1,2}}> \frac{v_1}{v_2} > \frac{x_{2,2}(x_{1,1} - x_{2,2})}{x_{1,2}(x_{1,1} - x_{1,2})}$.

Note that when $v_{1}$ and $v_{2}$ satisfy these inequalities, we have $(x_{1,1} - x_{1,2})v_{1}$ $< x_{2,2} v_{2}$, so two ads will be displayed in the VCG mechanism. Under VCG pricing, advertiser 1 pays a total cost of $(x_{1,1} - x_{2,2})v_{2}$ and advertiser 2 pays a total cost of $(x_{1,1} - x_{1,2})v_{1}$, meaning advertiser 1 pays a CPC of $\frac{x_{1,1} - x_{2,2}}{x_{1,2}} v_{2}$ and advertiser 2 pays a CPC of $\frac{x_{1,1} - x_{1,2}}{x_{2,2}} v_{1}$.

Since $\frac{v_1}{v_2} > \frac{x_{2,2}(x_{1,1} - x_{2,2})}{x_{1,2}(x_{1,1} - x_{1,2})}$, it is also the case that $\frac{x_{1,1} - x_{2,2}}{x_{1,2}} v_{2} < \frac{x_{1,1} - x_{1,2}}{x_{2,2}} v_{1}$, so these last expressions for the CPC imply that advertiser 1 pays a lower CPC than advertiser 2 under VCG pricing. However, under GSP pricing, it must be the case that the advertiser in top position pays at least as high a CPC as the advertiser in the second position since we know from Lemma 1 that the CPC of the bidder in position 1 is greater than or equal to $b_{(2)}$, while the CPC of the bidder in position 2 is no greater than $b_{(2)}$. Thus there do not exist bids for the advertisers under GSP such that the advertisers would pay the same CPC under GSP as they pay under VCG.

But if it were possible to design a mechanism that satisfied the properties in the statement of the theorem, then it would be necessary to ensure that if all advertisers submitted VCG bids, then the mechanism can impute GSP bids for the advertisers such that the cost advertisers are charged under GSP using their imputed bids is the same as the cost they would be charged under VCG. Since it is not possible for a mechanism to ensure this, there does not exist a mechanism that satisfies these properties. $\square$

Proof of Theorem 3

Suppose by means of contradiction that there exists some such mechanism for which there exist some bids of the VCG and GSP bidders, $v = (v_{1}, \ldots , v_{m})$ and $b = (b_{1}, \ldots , b_{l})$, for which the imputed bid of some bidder differs from the bidder’s given bid and for which either the resulting allocations or prices for the VCG bidders differ from those that would result if $v_{j}(v,b) = b_{j}$ held for all GSP bidders j. Let m denote the smallest number of VCG bidders for which there exist bids such that $v_{j}(v,b) \ne b_{j}$ holds for some GSP bidder j and for which either the resulting allocations or prices for the VCG bidders differ from those that would result if $v_{j}(v,b) = b_{j}$ held for all GSP bidders j.

Now suppose that there are exactly m VCG bidders, and consider some bids of the VCG and GSP bidders, $v = (v_{1}, \ldots , v_{m})$ and $b = (b_{1}, \ldots , b_{l})$, for which $v_{j}(v,b)$ $\ne b_{j}$ for some GSP bidder j and for which either the resulting allocations or prices for the VCG bidders differ from those that would result if $v_{j}(v,b) = b_{j}$ held for all GSP bidders j. I first seek to show that this implies that there exist some VCG bids $v^{\prime } = (v_{1}^{\prime }, \ldots , v_{m}^{\prime })$ and GSP bids $b^{\prime } = (b_{1}^{\prime }, \ldots , b_{l}^{\prime })$ such that there is some VCG bidder i and some GSP bidder j for which $v_{j}(v^{\prime }, b^{\prime })> v_{i}^{\prime }> b_{j}^{\prime } > b_{i}(v^{\prime }, b^{\prime })$.

Suppose first that under the VCG and GSP bids $v = (v_{1}, \ldots , v_{m})$ and $b = (b_{1}, \ldots , b_{l})$ that there is some VCG bidder whose allocation is different than it would be if $v_{j}(v,b) = b_{j}$ held for all GSP bidders j. In this case, there must be some GSP bidder j whose ad is shown for which $v_{j}(v,b) > b_{j}$. And there must also be some VCG bidder i whose ad would be shown if $v_{j}(v,b) = b_{j}$ held for all GSP bidders j and for which $v_{i} > b_{i}(v,b)$ because if we had $v_{i} = b_{i}(v,b)$ for all such VCG bidders i, then we would not select the same allocation under the imputed and given VCG bids as we would under the imputed and given GSP bids.

Now if $v_{i} \in (b_{j}, v_{j}(v,b))$, then the VCG and GSP bids already satisfy the desired condition. If $v_{i} \le b_{j}$, then consider the VCG bids $v^{\prime }$ that differ from v only in that under $v^{\prime }$, bidder i makes a bid $v_{i}^{\prime } \in (b_{j}, v_{j}(v,b))$. Under the bids $v^{\prime }$ and b it must be the case that $v_{j}(v^{\prime },b) = v_{j}(v,b) > v_{i}^{\prime }$, and in order for the allocation that is selected for ads i and j to be the same under the VCG and GSP bids, it must also be the case that $b_{i}(v^{\prime },b) < b_{j}$. Thus the VCG and GSP bids $v^{\prime }$ and b satisfy $v_{j}(v^{\prime },b)> v_{i}^{\prime }> b_{j} > b_{i}(v^{\prime },b)$.

Finally if $v_{i} \ge v_{j}(v,b)$ (and thus $b_{i}(v,b) \ge b_{j}$), then consider GSP bids $b^{\prime }$ that differ from b only in that under $b^{\prime }$, bidder j makes a bid $b_{j}^{\prime } \in (b_{i}(v,b), v_{i})$. Under the bids v and $b^{\prime }$ it must be the case that $b_{i}(v,b^{\prime }) = b_{i}(v,b) < v_{i}$, and in order for the allocation that is selected for ads i and j to be the same under the VCG and GSP bids, it must also be the case that $v_{i} < v_{j}(v,b^{\prime })$. Thus the VCG and GSP bids v and $b^{\prime }$ satisfy $v_{j}(v,b^{\prime })> v_{i}> b_{j}^{\prime } > b_{i}(v,b^{\prime })$. Thus if there is some VCG bidder whose allocation is different than it would be if $v_{j}(v,b) = b_{j}$ held for all GSP bidders j under the bids v and b, then there exist some VCG bids $v^{\prime } = (v_{1}^{\prime }, \ldots , v_{m}^{\prime })$ and GSP bids $b^{\prime } = (b_{1}^{\prime }, \ldots , b_{l}^{\prime })$ such that there is some VCG bidder i and some GSP bidder j for which $v_{j}(v^{\prime }, b^{\prime })> v_{i}^{\prime }> b_{j}^{\prime } > b_{i}(v^{\prime }, b^{\prime })$.

Next suppose that under the VCG and GSP bids $v = (v_{1}, \ldots , v_{m})$ and $b = (b_{1}, \ldots , b_{l})$ that there is some VCG bidder whose price (but not its allocation) is different than it would be if $v_{j}(v,b) = b_{j}$ held for all GSP bidders j. Let i denote one of these VCG bidders, and let j denote one of the GSP bidders whose imputed VCG bid differs from its given GSP bid and affects the price paid by bidder i. Consider the VCG bids $v^{\prime }$ that differ from v only in that under $v^{\prime }$ bidder i had changed its VCG bid to some $v_{i}^{\prime } \in (b_{j}, v_{j}(v,b))$. Under these alternative bids, bidder j has a higher imputed VCG bid than bidder i’s given VCG bid, so in order to ensure that the order of the ads is the same under the GSP and VCG bids, it must be the case that $b_{i}(v^{\prime },b)< b_{j}< v_{i}^{\prime } < v_{j}(v^{\prime }, b)$. Thus in this case there again exist some VCG bids $v^{\prime } = (v_{1}^{\prime }, \ldots , v_{m}^{\prime })$ and GSP bids $b^{\prime } = (b_{1}^{\prime }, \ldots , b_{l}^{\prime })$ such that there is some VCG bidder i and some GSP bidder j for which $v_{j}(v^{\prime }, b^{\prime })> v_{i}^{\prime }> b_{j}^{\prime } > b_{i}(v^{\prime },b^{\prime })$.

Next I seek to show that there exist some VCG bids $v^{\prime } = (v_{1}^{\prime }, \ldots , v_{m}^{\prime })$ and GSP bids $b^{\prime } = (b_{1}^{\prime }, \ldots , b_{l}^{\prime })$ such that there is some VCG bidder i and some GSP bidder j for which $v_{j}(v^{\prime }, b^{\prime })> v_{i}^{\prime }> b_{j}^{\prime } > b_{i}(v^{\prime },b^{\prime })$ and all other bidders make a bid of zero. To see this, consider some VCG bids $v = (v_{1}, \ldots , v_{m})$ and GSP bids $b = (b_{1}, \ldots , b_{l})$ such that there is some VCG bidder i and some GSP bidder j for which $v_{j}(v,b)> v_{i}> b_{j} > b_{i}(v,b)$. Note that if the VCG bids $v^{\prime }$ differ from v only in that under $v^{\prime }$, some VCG bidder $k \ne i$ made a bid $v_{k}^{\prime } = 0$, then $v_{j}(v^{\prime },b)> v_{i}> b_{j} > b_{i}(v^{\prime },b)$ would still hold because $v_{j}(v^{\prime },b) = v_{j}(v,b) > v_{i}$, which in turn implies that $b_{j} > b_{i}(v^{\prime },b)$. Similarly, if the GSP bids $b^{\prime }$ differ from b only in that under $b^{\prime }$, some GSP bidder $k \ne j$ made a bid $b_{k}^{\prime } = 0$, then $v_{j}(v,b^{\prime })> v_{i}> b_{j} > b_{i}(v,b^{\prime })$ would still hold because $b_{i}(v,b^{\prime }) = b_{i}(v,b) < b_{j}$, which in turn implies that $v_{i} < v_{j}(v,b^{\prime })$.

By repeatedly applying this logic, it follows that if the VCG bids $v^{\prime }$ differ from v only in that under $v^{\prime }$, all VCG bidders $k \ne i$ make a bid $v_{k}^{\prime } = 0$ and the GSP bids $b^{\prime }$ differ from b only in that under $b^{\prime }$, all GSP bidders $k \ne j$ make a bid $b_{k}^{\prime } = 0$, then it is still the case that $v_{j}(v^{\prime }, b^{\prime })> v_{i}^{\prime }> b_{j}^{\prime } > b_{i}(v^{\prime },b^{\prime })$. Thus there exist some VCG bids $v^{\prime } = (v_{1}^{\prime }, \ldots , v_{m}^{\prime })$ and GSP bids $b^{\prime } = (b_{1}^{\prime }, \ldots , b_{l}^{\prime })$ such that there is some VCG bidder i and some GSP bidder j for which $v_{j}(v^{\prime }, b^{\prime })> v_{i}^{\prime }> b_{j}^{\prime } > b_{i}(v^{\prime },b^{\prime })$ and all other VCG and GSP bidders make a bid of zero.

Now if the VCG bidder i receives zero clicks when the bidders make the bids $v^{\prime }$ and $b^{\prime }$, then this bidder can improve its payoff by making a GSP bid equal to $v_{i}^{\prime }$, which would place the bidder in the top position and ensure that the bidder obtains a positive payoff. And if this VCG bidder receives a positive number of clicks when the bidders make the bids $v^{\prime }$ and $b^{\prime }$, then this bidder can receive the same number of clicks by instead making a GSP bid equal to its imputed GSP bid, $b_{i}(v^{\prime }, b^{\prime })$, while paying a lower cost because the bidder faces lower competing bids. In either case, the VCG bidder i can improve its payoff by making a GSP bid rather than a VCG bid. This contradiction proves the result. $\square$

Proof of Theorem 5

Note that this is a feasible mechanism since $\alpha > 1$ implies the imputed VCG (GSP) bids are always greater (less) than the given GSP (VCG) bids. Also, since the VCG bids are all exactly $\alpha$ times the GSP bids, the resulting allocations are the same regardless of whether the allocations are selected using VCG bids or GSP bids. And the imputed VCG (GSP) bids of all the GSP (VCG) bidders are independent of the bids submitted by the VCG (GSP) bidders. The result then follows. $\square$

Proof of Theorem 6

Note that $b_L \le v_L$, $b_M \le v_M$, and $b_H \le v_H$ in any undominated strategy. Also note that the bidders with values $v_M$ and $v_H$ must obtain a positive number of clicks in equilibrium because these bidders can obtain a positive payoff by making bids slightly lower than their values, so any bid that would result in a zero payoff is suboptimal. Thus it suffices to prove that there is no equilibrium in which the advertiser with value $v_H$ is placed in the first position and the advertiser with value $v_M$ is placed in the second position.

To see this, I first show that in any pure-strategy equilibrium, it must be the case that $x_{1,1} b_{2} < x_{1,2} b_{2} + x_{2,2} b_{3}$, where $b_{j}$ denotes the bid of the advertiser in the jth position. If this did not hold and the advertiser in the top position deviated by making a bid that placed this advertiser in the second position, then this advertiser would pay a total cost of $(x_{1,1} - x_{1,2}) b_{2}$ and obtain a total payoff of $x_{2,2} v - (x_{1,1} - x_{1,2}) b_{2}$, where v denotes the advertiser’s value for a click. Since the advertiser obtains a payoff $x_{1,2} (v - b_{2})$ in the top position, in order for this to be an equilibrium, we must have $x_{1,2} (v - b_{2}) \ge x_{2,2} v - (x_{1,1} - x_{1,2}) b_{2}$, meaning $(x_{1,2} - x_{2,2})v \ge (2x_{1,2} - x_{1,1}) b_{2}$ and $b_{2} \le \frac{x_{1,2} - x_{2,2}}{2x_{1,2} - x_{1,1}} v$.

Now we know the bidder in the top position bids no more than its value in any undominated strategy. Furthermore, from Remark 1, we know this bidder can do no better than bid truthfully conditional on remaining in the first position. Finally, we know that the bidder in the second position must obtain a positive number of clicks in equilibrium. By combining these facts, it follows that $x_{1,1} v \le x_{1,2} v + x_{2,2} b_{2}$ because otherwise the bidder in the second position would not obtain a positive number of clicks if the bidder in the top position bid truthfully. This in turn implies that $b_{2} \ge \frac{x_{1,1} - x_{1,2}}{x_{2,2}} v$.

By combining the results in the previous two paragraphs, we see that if $x_{1,1} b_{2} < x_{1,2} b_{2} + x_{2,2} b_{3}$ did not hold, then it must be the case that $\frac{x_{1,1} - x_{1,2}}{x_{2,2}} v \le b_{2} \le \frac{x_{1,2} - x_{2,2}}{2x_{1,2} - x_{1,1}} v$, meaning $(x_{1,1} - x_{1,2})(2x_{1,2} - x_{1,1}) \le x_{2,2}(x_{1,2} - x_{2,2})$. This expression in turn holds if and only if $(x_{1,1} - x_{1,2})(x_{1,2} - (x_{1,1} - x_{1,2})) \le x_{2,2}(x_{1,2} - x_{2,2}) \Leftrightarrow \frac{x_{1,1} - x_{1,2}}{x_{1,2}} (1 - \frac{x_{1,1} - x_{1,2}}{x_{1,2}}) \le \frac{x_{2,2}}{x_{1,2}} (1 - \frac{x_{2,2}}{x_{1,2}}) \Leftrightarrow |\frac{x_{1,1} - x_{1,2}}{x_{1,2}} - \frac{1}{2}| \ge |\frac{x_{2,2}}{x_{1,2}} - \frac{1}{2}|$. Since a condition of the theorem states that $|\frac{x_{1,1} - x_{1,2}}{x_{1,2}} - \frac{1}{2}| < |\frac{x_{2,2}}{x_{1,2}} - \frac{1}{2}|$, it follows that $x_{1,1} b_{2} < x_{1,2} b_{2} + x_{2,2} b_{3}$ must hold in any pure-strategy equilibrium.

Now if there is an equilibrium in which the advertiser with value $v_H$ is placed in first position and the advertiser with value $v_M$ is placed in second position, then it must be the case that $x_{1,1} v_H \le x_{1,2} v_H + x_{2,2} b_M$ or $b_M \ge \frac{x_{1,1} - x_{1,2}}{x_{2,2}} v_H$ in order for the bidder with value $v_M$ to obtain a positive number of clicks. Also note that it must be the case that $x_{1,2} (v_H - b_M) \ge x_{2,2} (v_H - b_L)$ because the fact that $x_{1,1} b_{2} < x_{1,2} b_{2} + x_{2,2} b_{3}$ means the advertiser with value $v_H$ would pay a cost of $x_{2,2} b_{L}$ if this advertiser made a bid that placed this advertiser in second position, so $x_{1,2} (v_H - b_M) \ge x_{2,2} (v_H - b_L)$ must hold in order for making such a bid to not be a profitable deviation. This in turn implies that $b_M \le (1 - \frac{x_{2,2}}{x_{1,2}}) v_H + \frac{x_{2,2}}{x_{1,2}} b_L \le (1 - \frac{x_{2,2}}{x_{1,2}}) v_H + \frac{x_{2,2}}{x_{1,2}} v_L$.

By combining the results in the previous paragraph, we see that $\frac{x_{1,1} - x_{1,2}}{x_{2,2}} v_H$ $\le b_M \le (1 - \frac{x_{2,2}}{x_{1,2}}) v_H + \frac{x_{2,2}}{x_{1,2}} v_L$ must hold in order for there to be an equilibrium in which the advertiser with value $v_H$ is placed in first position and the advertiser with value $v_M$ is placed in second position. This in turn requires that $\frac{x_{1,1} - x_{1,2}}{x_{2,2}} v_H \le (1 - \frac{x_{2,2}}{x_{1,2}}) v_H$ $+ \frac{x_{2,2}}{x_{1,2}} v_L$, which holds if and only if $x_{1,2}(x_{1,1} - x_{1,2}) v_H \le x_{2,2}(x_{1,2} - x_{2,2}) v_H + x_{2,2}^2 v_L \Leftrightarrow x_{2,2}^{2} (v_H - v_L) \le x_{1,2}(x_{1,2} + x_{2,2} - x_{1,1})v_H$. Since this contradicts a condition of the theorem, there is no equilibrium in which the advertiser with value $v_H$ is placed in first position and the advertiser with value $v_M$ is placed in second position. $\square$

Proof of Theorem 7

Note that if the bidders follow these strategies, then the bidder with value $v_L$ cannot profitably deviate, as this bidder would have to bid more than its value to obtain a positive number of clicks. The bidder with value $v_M$ cannot profitably deviate for sufficiently small $\epsilon > 0$ because we know from Remark 1 that bidding truthfully is optimal if the bidder wishes to maintain its current position and if the bidder reduced its bid to obtain the second position, then the bidder would obtain fewer clicks for a CPC that is only $\epsilon$ lower. This would not be a profitable deviation for sufficiently small $\epsilon > 0$.

Finally, the bidder with value $v_H$ cannot profitably deviate: If this bidder deviates and occupies the top position, then the best this bidder can do is bid truthfully, and thereby obtain $x_{1,2}$ clicks at a cost of $x_{1,2} v_{M}$, resulting in a payoff of $x_{1,2}(v_H - v_M)$. By contrast, the bidder obtains a payoff of $x_{2,2}(v_H - b_L) > x_{2,2}(v_H - v_L)$ in its current position. Since $x_{1,2}(v_H - v_M) \le x_{2,2}(v_H - v_L)$, the bidder with value $v_H$ cannot profitably deviate. $\square$

Proof of Theorem 8

We have seen in the proof of Theorem 6 that under the above conditions, $x_{1,1} b_{2} < x_{1,2} b_{2} + x_{2,2} b_{3}$ must hold in any pure-strategy equilibrium, where $b_{j}$ denotes the bid of the advertiser in the jth position. Since $b_{3} = 0$ when there are two advertisers, a necessary condition for there to exist a pure-strategy equilibrium in undominated strategies is that $x_{1,1} b_{2} < x_{1,2} b_{2}$. But a consequence of the condition that $x_{2,2}^{2} > x_{1,2}(x_{1,2} + x_{2,2} - x_{1,1})$ is that $x_{1,1} > x_{1,2}$ because if $x_{1,1} \le x_{1,2}$, then $x_{1,2}(x_{1,2} + x_{2,2} - x_{1,1}) \ge x_{1,2} x_{2,2} \ge x_{2,2}^2$. Since $x_{1,1} b_{2} < x_{1,2} b_{2}$ can never hold when $x_{1,1} > x_{1,2}$, the result then follows. $\square$