# Advertising in Google Search Deriving a bidding strategy in the biggest auction on earth.

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1 Advertising in Google Search Deriving a bidding strategy in the biggest auction on earth. Anne Buijsrogge Anton Dijkstra Luuk Frankena Inge Tensen Bachelor Thesis Stochastic Operations Research Applied Mathematics University of Twente June 8, 22

2 Contents Preface Introduction Contributions Related Work Analytical Model Notations and Assumptions Framework of the Model Optimal Position Optimal Bid Data Analysis CTR versus Average Position Average CPC versus Average Position Model Analysis Uniform Distribution Exponential Distribution Normal Distribution Simulation Uniform Distribution Exponential Distribution Normal Distribution An Algorithm to Determine the AdRanks The algorithm Optimal Bidding Practicability Vindictive Bidding Conclusion Discussion References Appendix

3 Abstract This research aims to derive an optimal bidding strategy for keyword auctions in Google Search from the perspective of the advertiser. It presents a model for the expected profit per view, depending on either the bid or the obtained position of the advertiser. In a subsequent analysis, the position and bid are optimized to reach the maximum profit per view. In the model the AdRanks are assumed to be distributed by a general probability distribution. The analysis is restricted to the uniform, exponential and normal distributions. In addition an approximation of the expected profit per view is derived and subsequently tested by Monte Carlo simulation for different distributions and parameters. Finally, this research presents an algorithm to determine the AdRanks of the other advertisers, which may be used for vindictive bidding.

9 3.3 Optimal Position First, the expected profit per view is conditioned on the position Y = i. E [P P V comp Y = i = E [(RP C comp CP C comp )CT R Y = i = (RP C comp E [CP C comp Y = i)ρ(i) (3.2) The last equation follows because the expectation of the CTR given Y = i results in the vector ρ(i). To calculate E [CP C comp Y = i formula (3.) is used. For the first line, a distribution of r (i) is needed. r (i) is the i th order statistic of r. E [CP C comp Y = i = E[r (i) if i < s m if i = s if i > s (3.3) In section 5 this formula will be used to calculate the maximum profit per view. Bare in mind that when RP C comp < E[CP C comp Y = i the expected profit becomes negative, so the company should not participate in the auction. 3.4 Optimal Bid In this part the expected profit will be conditioned on the bid B comp. E [P P V comp B comp = = E [(RP C comp CP C comp )CT R B comp = (3.4) To get more insight in the values of the AdRanks, several distributions are examined. First a general distribution of the AdRanks is implemented. The probability for the company of getting position i can be determined as a function of the AdRank R comp. P (r (i) R comp r (i ) B comp ) = P (r (i) R comp, r (i ) R comp B comp ) If r (i ) R comp there are i advertisers with an AdRank greater than R comp. Similarly, if r (i) R comp the remaining a i + advertisers have an AdRank between and R comp. If the distribution of the AdRanks is known, the probabilities of these events can be calculated. These probabilities have to be muliplied by the number of ways the i advertisers can be chosen. This results in the following: P (r (i) R comp, r (i ) R comp B comp ) = ( ) a (p(b comp )) i ( p(b comp )) a i+, (3.5) i with p(b comp ) = P (r(i) R comp B comp ). Of course the exact p(b comp ) depends on the distribution of the AdRanks. To calculate the expected profit per view, the full expectation formula, conditioning on Y = i is used. Thus from (3.4) and (3.5) the following formula gives 6

10 the total expected profit per view. E[P P V comp B comp = a+ = E[P P V comp B comp =, Y = ip (Y = i B comp = ) i= a+ = E[P P V comp B comp =, Y = i i= P (r (i) R comp, r (i ) R comp B comp = ) a+ ( ) a = (p( )) i ( p( )) a i+ i i= E [(RP C comp CP C comp )CT R B comp =, Y = i (3.6) Formula (3.) cannot be computed with the order statistics, because the knowledge of the bid gives extra information about the probabilities that the other advertisers are above or beneath the company. This is why the simplifying assumption is made that CP C comp = B comp. This is a reasonable worst-case scenario, because a company always pays less than their bid. Thus the lower boundary of P P V comp, P P Vcomp, is obtained in this way. E[P P Vcomp B comp = = a+ ( ) a (p( )) i ( p( )) a i+ ((RP C comp ) ρ(i) (3.7) i i= In this formula the binomial distribution can be recognized. Hence, a short-hand notation of formula (3.7) is obtained as follows: E[P P V comp B comp = = E[(RP C comp )ρ(x + ), (3.8) with X B(a, p( )). The AdRank of the company can be changed by varying the bid of the company. The expected profit per view belonging to these bids can be calculated by (3.7) or (3.8) and a bidding strategy can be derived. As said before there is a critical treshold, namely, when RP C comp < B comp the company should not participate in the auction, because of the negative expected profit. In section 5 various possible distributions of the AdRanks will be studied, followed by conclusions. 7

11 4 Data Analysis In order to evaluate the parameters of the analytical model, real world data has to be used. In this section the real world data is being examined. 4. CTR versus Average Position Since an advertisement with a higher position receives on average more clicks than one with a lower position, the CTR depends on the position. Recall that the CTR is the Click Through Rate, equal to the number of clicks on the advertisement divided by the number of views. This dependency between the CTR and the position is examined in this subsection. Comparable data often follow Zipf s law. The probability mass function of Zipf s law is: f(k; s, N) = k s N n= n s, (4.) where f(k; s, N) is in this case the CTR, k =,.., N is the position, s > is a parameter to characterize the distribution and N > is the number of positions. The Zipf s law is observed by plotting the data on the log-log scale, because the plot then becomes close to a straight line. In order to test whether the data really follow the Zipf s law, two plots are made. Figure (2) is a plot of the average position against f(k;, 8), so s = and N = 8 in formula (4.). Only the first eight positions have been taken into account, because for lower positions the CTR is zero most of the time. Figure (3) is the log-log plot of figure (2). Figure 2: Zipf s law 8

12 Figure 3: Zipf s law log-log The figures show that the data approximately follows the Zipf s law. When taking s = 2 the average position differs more from the Zipf s law. Also for higher values of s the data do not fit better with Zipf s law. The conclusion is that Zipf s law with s = is in this case the best approximation for the CTR, that is for ρ. 4.2 Average CPC versus Average Position To analyse the dependency of the average CPC and the average position, several samples were taken from international campaigns. These samples have been examined per day and per month. The results are shown in the following figures. Each of the four keywords shirtjes kinder, legging, babymutsjes and polo korte mouw are examined in the same period of time. (a) Shirtjes kinder per day (b) Shirtjes kinder per month Figure 4: Shirtjes kinder 9

13 (a) Legging per day (b) Legging per month Figure 5: Legging (a) Babymutsjes per day (b) Babymutsjes per month Figure 6: Babymutsjes (a) Polo korte mouw per day (b) Polo korte mouw per month Figure 7: Polo korte mouw Figures (4a) and (4b) seem to show that for this keyword the average position is more or less equal no matter what the average CPC is. Some of the figures, like figures (5a) and (5b), seem to show that there is no influence of CPC on the position at all. Other samples seem to have a slightly upward or downward trend. According to formula (3.), if the CPC increases, the number of the position is decreasing, assuming that the quality stays the same. If the position number is

15 If the assumption is made that ρ(i) follows Zipf s law, formula (5.2) can be completed. E[P P V comp Y = i ( = RP C comp β ) i a (β α) = a+ n= n i a+ n= n ( RP Ccomp β β α a i ) + β α a This formula is decreasing in i and, since i is positive, the maximum value for E[P P V comp Y = i occurs for small values of i. Hence, in this case the best strategy is to obtain the highest position under the assumption that RP C comp β i a (β α) Optimal Bid, which is equal to E[CP C comp Y = i. Now the optimal bid that maximizes P P Vcomp is computed. Since p(b comp ) is needed to substitute in formula (3.7) it has to be specified. p(b comp ) = β Bcomp β α if α B comp β otherwise The following formula is used to give more insight in the behaviour of E[P P Vcomp B comp =. If the assumption is made that ρ(i) follows Zipf s law, formula (3.8) becomes as follows: E[P P Vcomp B comp = [ = E = X+ a+ n= n a+ E n= n [ (RP C comp ) X + (RP C comp ), (5.3) with X B(a, p( )). Unfortunately, no conclusions can be drawn from this formula. This is why the analysis continues with formula (3.7). Taking the derivative of the following formula equal to zero, the optimal bid is obtained. E[P P Vcomp B comp = = a+ ( a i= i ) ( β β α (RP C comp ) i a+ n= n ) i ( ) a i+ bcomp α β α otherwise if α β (5.4) 2

16 The derivative of the previous formula for α β is as follows: E[P P Vcomp B comp = = ( a i [ a+ ) [ ( i β bcomp a+ (i ) i= n= β α n ( β bcomp (RP C comp ) + (a i + ) β α ( β bcomp (RP C comp ) β α ) i ( β β α ) i 2 ( β β α ) i ( β β α ) a i+ otherwise it is zero. The roots of this derivative cannot be found analytically, that is why the maximum E[P P Vcomp B comp = in formula (5.4) is determined numerically. The following parameters are used: a =, RP C comp = 4, =, α =.5, β = 3.5., ) a i+ ) a i In each of the following figures one of the parameters varies. = a = a = 2 a = 3 a = 4 a = 5 a = 6 a = 7 a = 8 a = 9 a = =.5.5 = =.5 = 2 = 2.5 = 3 = 3.5 = 4 = 4.5 = (a) Number of other advertisers a (b) Revenue per click RP C comp 3

17 = = = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = = β = β =.5 β = 2 β = 2.5 β = 3 β = 3.5 β = 4 β = 4.5 β = (c) Quality (d) Right boundary of uniform distribution β Figure 8: Variation of the parameters for uniformly distributed AdRanks β, These figures show that E[P P Vcomp B comp = = if since p( ) = in this case. Figure (8a) shows that with an increasing number of advertisers, the expected profit per view decreases. The explanation for this could be that if less advertisers participate in an auction, it is more likely that the company obtains a higher position, resulting in a higher click through rate. If there is only one other advertiser, the company is already certain that they will get either slot or 2, so it is not necessary to place a high bid. If the number of advertisers increases, there are more competitors for the top position and the company has to make a higher bid in order to maximize their profit. Figure (8b) displays that varying RP C comp makes a big difference. If RP C comp is lower then CP C comp, which is equal to, the expected profit per view becomes negative. For example, if RP C comp is 3 the expected profit per view stays positive for bids lower than 3 and becomes negative for higher bids. So the best strategy is to bid as high is possible if RP C comp is bigger than, otherwise it is optimal to bid the minimum bid. In figure (8c) is varied. Because the expected profit per view is zero in case that β, the value of for which E[P P Vcomp B comp = becomes zero depends on. If increases, the company can make a lower bid to receive a high position, so the optimal bid will be lower. Also in figure (8d) the expected profit per view becomes zero if β and in this case this depends on β. For values of β higher than 4, the expected profit per view becomes negative because RP C comp is equal to 4. For values of β lower than RP C comp the conclusion can be drawn that the optimal bid is β, so it is best to bid as high as possible in this case. If this is not the case then it is optimal to bid the minimum bid. Recall that the conclusion from the optimal position was that it is optimal to get the first position, so to bid as high as possible when RP C comp This follows the conclusion for the optimal bid. i β a (β α). 4

18 5.2 Exponential Distribution Now the AdRanks of the other advertisers are assumed to be exponentially distributed with mean λ. Like in the case of the uniform distribution a distinction is made between the optimal position and the optimal bid. Optimal Position To determine the optimal position, the expected profit per view will be conditioned on the position Y = i with formula (3.2). To do so E[CP C comp Y = i of an exponential distribution has to be computed. This can be done with (3.3) and the expected order statistics of the exponential distribution from [9. This results in: E [CP C comp Y = i = a i+ k= λ a k+ if i < a + m if i = a +. (5.5) Because the summation is too difficult to substitute, boundaries of the sum of (5.5) can be derived. The sum is bounded by two integrals: a i+2 a i+ a t + 2 dt k= a i+2 a k + The integrals of the previous equation can be easily computed. a i+2 a i+2 dt = ln(a + ) ln(i) a t + 2 dt = ln(a) ln(i ) a t + dt. (5.6) a t + First the boundaries are substituted in (5.5) and then the resulting expression is substituted in (3.2). The lower boundary is substituted in formula (3.2) first and results in the following formula. E[P P V comp Y = i = ( RP C comp ) λ(ln(a + ) ln(i)) i q a+ comp n= n (5.7) In this formula the Zipf s law for ρ(i) is also substituted. Note that, since the lower boundary is subtracted from the RP C comp, it now becomes an upper boundary for E[P P V comp Y = i. To find the optimum of this formula, the derivative is computed and set equal to zero: E[P P V comp Y = i i = λ RP C comp + λln(a + ) λln(i) i 2 a+ n= n Thus the optimal position is the solution for i of the following equation: =. (5.8) λ RP C comp + λln(a + ) λln(i) =. (5.9) 5

19 Note that the denominater of formula (5.8) is never equal to zero. Solving (5.9) for i this becomes: RP Ccompqcomp i = e λ +ln(a ). This is the optimal position for the upper boundary of E[P P V comp Y = i. This does not always has to be an integer, but then i and i have to be calculated to see which of the two has the highest E[P P V comp Y = i. This is probably a good approximation for the optimal position. Secondly, the upper boundary from formula (5.5) is substituted. Also this boundary is substituted in formula (3.2) and the derivative is set equal to zero. Note that this will result in a lower boundary for E[P P V comp Y = i. E[P P V comp Y = i = The derivative is as follows: E[P P V comp Y = i = i λ a+ n= n ( RP C comp ) λ(ln(a) ln(i )) i q a+ comp n= n ( RP Ccomp λ + ln(a) ln(i ) i 2 (i ) ) (i ) + i. So the optimal position is the solution of the following formula: ( RP C ) comp + ln(a) ln(i ) (i ) + i =. λ This equation is not solvable analytically, so the boundaries as well as the original formula are implemented numerically. In the following figure these parameters are used: a =, RP C comp = 4, =, λ =.75. 6

20 .9.8 E[PPV Y=i Upper boundary Lower boundary Original i Figure 9: E[P P V comp Y = i with the boundaries. The figure shows that the upper and lower boundaries result in a good approximation for the original problem and it shows that with these parameters the first position is the best one. Next, the optimal bid will be analysed and later on it will be compared to the optimal position. Optimal Bid Now the maximum P P V comp is calculated for the optimal bid. For the exponential distribution the probability p(b comp ) in formula (3.7) becomes: p(b comp ) = e λ Bcompqcomp. With this probability the maximum P P Vcomp can be calculated in the same way as in the previous subsection. Again a binomial distribution can be recognized, the only difference is p( ). So (5.3) holds true for the exponential distribution as well. The derivative of P P Vcomp is also taken in this case. E[P P Vcomp B comp = = [ a+ i= i a+ n= n ( e bcompqcomp λ (e bcompqcomp λ ( ) [ a i (i )e bcompqcomp λ λ ) (e bcompqcomp i 2 λ ) a i+ (RP Ccomp ) + (a i + )e bcompqcomp λ ) i ( e bcompqcomp λ ) ( e bcompqcomp a i+ λ ) a i (RP Ccomp ) (e bcompqcomp λ λ ) i 7

21 Since the roots of this formula cannot be found analytically, formula (3.7) is implemented numerically. The parameters are varied to determine the influence of each parameter. The standard parameters are as follows: a =, RP C comp = 4, =, λ =.75. =.5.5 a = a = 2 a = 3 a = 4 a = 5 a = 6 a = 7 a = 8 a = 9 a = = = =.5 = 2 = 2.5 = 3 = 3.5 = 4 = 4.5 = (a) Number of other advertisers a (b) Revenue per click RP C comp = = = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = = λ =.5 λ = λ =.5 λ = 2 λ = 2.5 λ = 3 λ = 3.5 λ = 4 λ = 4.5 λ = (c) Quality (d) Mean of exponential distribution λ Figure : Variation of the parameters for exponentially distributed AdRanks In figures (a), (c) and (d) RP C comp is equal to 4. Remember that CP C comp is chosen to be equal to. That is why all lines in these figures intersect at bid 4, where the expected profit per view equals zero. Similar as for the uniform distribution, figure (a) shows that with a lower number of advertisers, the company can make a lower bid. From figure (b) it can be concluded that a higher RP C comp results in a higher expected profit per view and the optimal bid becomes higher. 8

22 As for the uniform distribution, figure (c) shows that a higher quality results in a higher expected profit per view and a lower optimal bid. If the mean λ of the AdRanks of the other advertisers becomes higher in figure (d), the company has to make a higher bid to reach the same position. Recall that it still has to be discussed whether the conclusion of the position follows the conclusion of the bid. With the standard parameters the optimal position is the first position, so to bid as high as possible. With the same parameters the optimal bid is between 2 and 2.5. The difference between these results may occur because for the optimal bid E[P P Vcomp B comp = is used. 5.3 Normal Distribution Next, the assumption is made that the AdRanks of the a advertisers are normally N (µ, σ 2 ) distributed. For the normal distribution only the optimal bid is determined and the optimal position is skipped, since the order statistics are not known analytically. Optimal Bid Because the optimal bid is difficult to determine analytically, this is done numerically. To do so, p(b comp ) is taken as: ( ) qcomp B comp µ p(b comp ) = Φ. σ This is substituted in formula (3.7). As done for the uniform and exponential distribution, the outcomes for different values of the parameters are compared to each other. The standard parameters are chosen as follows: a =, RP C comp = 4, =, µ =.75, σ =.6. 9

23 = a = a = 2 a = 3 a = 4 a = 5 a = 6 a = 7 a = 8 a = 9 a = =.5.5 = =.5 = 2 = 2.5 = 3 = 3.5 = 4 = 4.5 = (a) Number of other advertisers a (b) Revenue per click RP C comp = = = 2 = 3 = 4 = 5 = 6 = 7 = 8 = 9 = = µ =.5 µ = µ =.5 µ = 2 µ = 2.5 µ = 3 µ = 3.5 µ = 4 µ = 4.5 µ = (c) Quality (d) Mean of normal distribution µ = σ =. σ =.2 σ =.3 σ =.4 σ =.5 σ =.6 σ =.7 σ =.8 σ =.9 σ = σ =. σ = (e) Standard deviation of normal distribution σ Figure : Variation of the parameters for normally distributed AdRanks Just as with the exponential distribution, RP C comp is equal to 4 in figures (a), (c), (d) and (e). Remember that CP C comp is chosen to be equal to. That is why all curves in these figures have a root at bid 4. 2

25 6 Simulation In C++ a simulation program is written, that simulates auctions and determines the expected profit per view by the Monte Carlo method. The code of this program can be found in appendix A. In this simulation program AdRanks of the other advertisers are uniformly, exponentially or normally distributed. These distributions are input parameters of the program. The number of other advertisers, which is equal to the number of slots minus one, is also an input value, as are the click-through-probabilities, the revenue per click of the company and the quality of the advertisement. For several input parameters this simulation is compared to formula (3.7). This formula was used as an estimation for the profit per view for given input parameters. For each distribution, figures are shown for several standard parameters and for a few variations of the parameters. The standard parameters are as follows: 6. Uniform Distribution a =, RP C comp = 4, =. The AdRanks of the other advertisers are Uniform[α, β distributed, with α is.5 and β is = Outcome of simulation E[PPV E[PPV = Outcome of simulation (a) Standard parameters (b) Number of other advertisers a = 5 22

26 .8.7 = Outcome of simulation.2. E[PPV E[PPV = Outcome of simulation (c) Right boundary of uniform distribution β = 2 (d) Revenue per click RP C comp = E[PPV = Outcome of simulation (e) Quality = 2 Figure 2: Comparison of simulation to E[P P V comp B comp = with Ad- Ranks uniformly distributed These figures show that E[P P Vcomp B comp = = if, since p( ) = in this case. As the figures show, E[P P Vcomp B comp = is a pretty good estimate of the expected profit per view except for figure (2b). By the definition of E[P P Vcomp B comp = it is always lower than the expected profit per view. For higher bids the approximation gets less accurate. The reason for this is that in the simulation, when is increasing, the company pays the AdRank of the company beneath divided by their own quality, which is bounded. On the contrary, the cost used in E[P P Vcomp B comp = keeps increasing. β 23

27 6.2 Exponential Distribution Next, the AdRanks of the other advertisers are exponentially distributed with mean λ = E[PPV.5. E[PPV = Outcome of simulation = Outcome of simulation (a) Standard parameters (b) Number of other advertisers a = = Outcome of simulation.5 E[PPV (c) Mean of exponential distribution λ =.9 24

28 = Outcome of simulation.5 E[PPV..2 E[PPV = Outcome of simulation (d) Revenue per click RP C comp = 2 (e) Quality = 2 Figure 3: Comparison of simulation to E[P P V comp B comp = with Ad- Ranks exponentially distributed For the exponential distribution the figures show that for small values of, E[P P Vcomp B comp = is an accurate lower boundary for the expected profit per view. For the same reason as for the uniform distribution, the approximation gets less accurate when increases. 6.3 Normal Distribution Next, the AdRanks of the other advertisers are N(µ, σ 2 ), with µ =.75 and σ = = Outcome of simulation.7.6 = Outcome of simulation E[PPV E[PPV (a) Standard parameters (b) Number of other advertisers a = 5 25

29 E[PPV.4 E[PPV =.6 = Outcome of simulation Outcome of simulation (c) Mean of normal distribution µ =.9 (d) Revenue per click RP C comp = E[PPV = Outcome of simulation (e) Quality = 2 E[PPV = Outcome of simulation (f) Standard deviation of normal distribution σ =.2 Figure 4: Comparison of simulation to E[P P V comp B comp = with Ad- Ranks normally distributed In figure (4a), (4c) and (4f) there is a spike in the graph of the simulated P P V comp at =.5. The reason for this is that normal random variates lower than.5 are adjusted to.5 by the simulation program. If µ decreases or σ increases the probability that an AdRank is smaller than.5 increases which results in a bigger spike at =.5. As was the case for the uniform and exponential distribution, E[P P Vcomp B comp = gives a good approximation for the expected profit per view for low values of. 26

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