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Search Engine Marketing (Paid Search) algorithm and modeling

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- 1. SEM analytics: A model for PPC bidding and budget management Index: 1. introduction 2. efficiency curve and optimization 3. a time series model of max CPC 4. budget allocation 5. best practices and experiment learning Abstract: Search marketers face a range of challenges managing PPC advertising campaigns: rising keyword price, measurement of impact on and offline, ROI tracking, expansion of keyword, management of multiple engines, click fraud, bid tool functionality, declining ROI, etc. Determining the maximum CPC or willingness to pay for a given keyword phrase is perhaps the most challenging and most rewarding task. On one hand, Maximum CPC value is dynamically depending on ad conversion rate, profit margins, advertising budget and other factors. Marketers have to adopt different bidding strategies at various grains such as keyword level, keyword group level, match type level and search engine level. On the other hand, a powerful bidding strategy will deliver sustainable ROI or other efficiency goals. Marketers need to take advantage of all the data that’s available through adaptive learning algorithms. This paper demonstrates how to use simple time series modeling to determine the max CPC. In addition, this paper reports a method to optimize the advertising budget allocated to a keyword portfolio. The method is adopted through a variation of the Dorfman-steiner theorem (1954). Introduction: PPC bidding management is both science and art to improve bidding and budget allocation efficiency for PPC efforts. It is a science because marketers heavily rely on quantitative analyses and routinely uses various data mining techniques such as regression, segmentation, time series model, etc. It is also an art because marketers need to find the perfect mix of max CPC, bidding frequency, positioning, day parting, budget allocation and more. Creativity is required to combine both science and art together to improve ROI or other efficiency goals. This paper focuses on the science part by introducing a simple time series model for bidding management and also a budget allocation method through marginal analysis. First, a simple framework for PPC ROI management is set up through efficiency curve. Second, efficiency is then defined as the optimal strategies based on efficiency curve. Third, max CPC is modeled to reach the optimal. Fourth, we make recommendations on how to allocate budget which is consistent with the framework and optimal strategies. Finally, a few best practices are proposed.
- 2. Efficiency Curve (EC) and optimization: Efficiency curve, which is similar to the production curve in economics, is defined as following: y=f(x) ---(1) y is the target revenue; x is the investment or spending. Fig. 1 shows a plot of daily spend and daily revenue for a keyword portfolio over a period of time. Even though there is a lot of daily variations of both spend and revenue because of weekly trend, positioning, competition, and other factors, we still see a nice curve with diminishing returns to scale. In addition, we have observed that EC for a keyword may be much more fluid with even more variations, especially true for lower volume keywords. However, the diminishing nature of the efficiency curve is still present. The following proposed methodologies should still apply at keyword level. Fig. 1. Typical daily spend level vs. daily revenue for a keyword portfolio on a main search engine. A power curve is fitted: y is the expected revenue and x is the spending. There are many candidate functional forms as long as a few constraints could be met. When x is zero, y needs to be zero. We also need diminishing returns to scale. We have found that the functional form for SEM is best described as a power curve as following: y=a*xe ---(2) y = 30.909x0.5375 R² = 0.8297 Revenue Spend Efficiency Curve
- 3. Power curve has many very suitable features to serve as the production function for a KW portfolio. A power curve is called effective for optimization if e is in (0,1) such that we have diminishing returns. Otherwise, it is called ineffective for optimization. Average return (AR) or ROI is then calculated as: AR = y/x=a*x(e-1) ---(3) Note the relationship between Revenue per Click (RPC) and Cost per Click (CPC) is obtained through a simple transformation: AR = (y/clicks) / (x/clicks) = RPC/CPC ---(4) Marginal return (MR) could be easily calculated as following: µ=MR =dy/dx= a*e*x^(e-1) =e * y / x ---(5) Another term of interest to the bidding management of a keyword portfolio is the elasticity of revenue to spend is: Є=∆y/∆x= (dy/y)/(dx/x) = MR/AR=e ---(6) Elasticity (Є) basically describes that revenue will increase only e*100% if spending is increased by 100%. Efficiency is simply defined as the maximum obtainable revenue (or other conversion metrics) for a given ad spending level for certain time window (day, week, month, etc) on the same efficiency curve. We assume external factors such as competition or Search Engine algorithmic changes are uncontrollable. In addition, we assume that internal controllable factors such as keyword portfolio size, keyword grouping, CR and LP are optimal and static. Hence, the only tasks at hand are to determine the max CPC for keyword queries and how much to spend on a given KW portfolio. If we relax these assumptions, we could define another term called Efficiency Gain. Efficiency Gain is the percentage increase in revenue for a given spend level after we shifted the efficiency curve upwards. Proposed methodologies in this paper would still work after we relax these assumptions for various test/control scenarios. Efficiency gain could be used to measure the power of any enhancement of managing a keyword portfolio. We will provide usage of EG in the end of the paper. Similarly, efficiency loss is the percentage decrease in revenue for a given spend level if the curve is shifted downwards. We will discuss a few methodologies based on value of keyword to reach efficiency on static efficiency curve. For simplicity, let’s first assume there is a linear relationship between max CPC and actual average CPC as following: Max CPC = s * Avg CPC ---(7) s factor depends on uncontrollable external factors such as competition, search engine algorithmic change, etc. It is normally quite stable but different by keyword and search engines. However, the exact value doesn't impact following proposed algorithm. We could even assume it is equal to 1.
- 4. First methodology for estimating max CPC is to identify break-even point such that the historical average revenue is equal to actual average cost in a certain time window: Max CPC = s* Avg CPC = s* RPC ---(8) By setting keyword bidding amount at average revenue, we can ensure profitability or ROI goals. It is simple yet effective. Most practitioners recommend this methodology as a starter bidding strategy. However, optimal efficiency is not reached because this method is essentially AR=y/x=1. Optimal efficiency could be reached only by setting MR=1 which is the solution to maximizing profitability (P=y-x): MR=e*y/x = 1 ---(9) Transforming by adding clicks into the equation, we get following: e*RPC =CPC ---(10) Then we end up with: Max CPC = s* avg CPC = s* e* RPC ---(11) For the easy of usage, let’s define scaling factor (SF) as following. SF = s*e ---(12) This second methodology says that we should set maximum bidding amount based on the value of the keyword and also the elasticity of the keyword. This method obtained through marginal analysis is basically the Dorfman-steiner theorem which states “a firm which can influence the demand for its product by advertising will, in order to maximize its profits, choose the advertising budget and price such that the increase in gross revenue resulting from a one dollar increase in advertising expenditures is equal to the ordinary elasticity of demand for the firm’s product”. Dorfman-steiner theorem essentially says that advertising spend should be at the level at which an additional dollar of spending just produces additional e dollar of revenue. This is also our foundation for budget allocation in later part of the paper where each additional dollar spent will bring in the same additional revenue regardless where it is spent. After establishing the optimal solution to our optimization problem, we need an algorithm to model the process. Both AR and MR methodologies are similar where we need figure out the RPC and SF for a keyword or a keyword portfolio. RPC and SF are dynamic in nature. RPC has very big variance if the # of clicks is small for a keyword. For example, keyword ‘mp3 player 8gb’ has only 50 clicks so far since we started buying this keyword at some max CPC. The RPC is zero still. If a conversion could happen at click number 51. Then the RPC jumps to around 20 cents if the revenue per conversion is $10. It is apparent that simple calculation of RPC won’t be effectively. We need to balance the variance and stability of our max CPC if we want to apply our revenue based methodologies because identifying a steady and optimal state of RPC is the foundation of our
- 5. optimization methodology. We will demonstrate how we can model the max CPC through a simple time series model of RPC and SF in the following sections. A time series model for estimating keyword level RPC based on the EC framework: Let’s continue using keyword ‘mp3 player 8gb’ as an example to illustrate our time series model. Assume we have tracking and correlation in place to tell us the cost and revenue for each and every click. For simplicity, we target Max CPC divided by SF for a keyword since SF is relatively independent of the RPC estimation. We will propose methodology for SF estimation in the budget allocation section. After we have SF estimation, we could easily put both estimated RPC and estimated SF together to reach optimal bidding amount. At time zero, we start buying the keyword at a default aRPC_0 (exact value doesn’t matter, a portfolio average # is a good starting point) which is sufficient for this keyword to be on the first page and generate some clicks: Time 0: Max CPC_0/SF =aRPC_0 ---(13) At time 1, we will be able to evaluate the effectiveness of this prior belief of the value of this keyword since we can calculate the actual revenue per click (aRPC_1) for whatever clicks came in so far. Based on the error or gap, we can make linear or non-linear adjustment to our max CPC so that it could minimize the error going forward: Time 1: error = aRPC_1 – aRPC_0 Max CPC_1/SF = aRPC_0 + % * error = aRPC_0 + % * (aRPC_1 – aRPC_0) = aRPC_0 * (1-%) + aRPC_1 * % ---(14) % is also called decaying factor because it reduces the impact of initial value aRPC_0 over time. At time t, after t iterations, we end up with following max CPC formula: Time t: error =aRPC_t – max CPC_t-1 Max CPC_t/SF = max CPC_t-1 (1-%) +aRPC_t * % =∑ (aRPCn ∗ (1 − %)t−n ∗ %n )𝑡−1 𝑛=0 + aRPC_t *% ---(15) Essentially, our scaled max CPC is an Auto Regression (AR) time series model. To simplify the matter, we have found the AR(1) is sufficient for our optimization problem with a linear adjustment factor based on # of clicks at time t. We propose following formula for max CPC at any given time t: Max CPC_t/SF = aRPC_t * % + aRPC_0 * (1-%) ---(16) Where % is defined as following if we have k clicks so far: % = max(k, c) / c ---(17)
- 6. Here c is the cut-off point (# of clicks) for calculating the decay factor. For example, if the cut off clicks is 200, then ‘mp3 player 8gb’ with 50 clicks would have a decaying factor of 50/200 = 0.25. Cut-off point largely depends on the nature of the keyword portfolio and target metrics. Normally it can be estimated by looking at the portfolio average conversion rates and CPC. The above algorithm works well because it learns the dynamics of the bidding over time and makes attempt to reach the optimal. However, we are not optimal without applying the elasticity or scaling factor. The next question is how to estimate the scaling factor. So far, we have simply created a learning algorithm based on the revenue. The interaction between revenue and cost is assumed away. Naturally, cost perspective should be in our algorithm. This intuition goes like following: if a keyword is too expensive relatively speaking, we should reduce its bidding amount. Otherwise, we should increase its bidding amount. Following section will introduce a method to do so. Budget allocation based on marginal analysis at subgroup level of a keyword portfolio: Target objective of PPC campaigns is normally one or more of following: pre- defined budget (B), pre-defined return on ads spend (ROAS), and/or cost per order. Regardless of the target, the proposed methodology should still apply with minor modification of the algorithm. Here we will use the Dorfman-Steiner theorem to optimally allocation B and recommend scaling factor. let's split the keyword portfolio (S) into a mutually exclusive subgroups (S1, S2,...,SN). Each subgroup has its own elasticity (e) and intercept (a) by fitting a power curve on historical spend and revenue pairs. In case a subgroup has abnormal power curve with e<0 or e>1, we need combine it with some other group until we end up with a smooth and effective power curve such that the marginal analysis could be conducted. Based on the D-S theorem, optimal spend allocation for budget B is reached if following is true: µ = MR1=MR2.... = MRN ---(18) Since we already know elasticity’s (e1, e2, ..., eN) and intercepts (a1, a2, ...., aN), it is quite easy to search for a µ such that each subgroup will get a spend allocation of Bi which is the solution to following system functions: ∑ BiN n=1 = B Where Bi = (µ/ai*ei)^(1/(ei-1) for i in (1,…,N) ---(19) See fig. 2 for an illustration of budget allocation.
- 7. Fig. 2. Efficiency curves for 4 subgroups for a keyword portfolio. Slope on EC is the marginal returns. Optimal spend is achieved by allocating budget such that MR for all subgroups are the same. Now that we have gained insights into the marginal returns based on the budget allocated for each subgroup, we could easily calculate the scaling factor by transforming the MR back into CPC and RPC: µ = (ai*ei*X^(ei-1)) =ei*y/x (20) Hence, x=ei*y/ µ ---(21) After plugging in clicks, we get following: CPC = ei*/ µ *RPC ---(22) As a result, we have just identified our max CPC formula based on marginal analysis and elasticity of our EC for a sub group of a keyword portfolio: max CPC = s * ei/ µ * RPC ---(23) Our new scaling factor SFi for a subgroup is following: SFi = s*ei/ µ ---(24) After plugging in the steady state of optimal RPC results from the learning algorithm, we end up with the final formula for max CPC at any given time t for keywords in subgroup i: max CPC_t = s* ei/ µ * [aRPC_t * % + aRPC_0 * (1-%)] ---(25) This formula employs subgroup level elasticity (ei), marginal return (µ) and keyword level conversion rate (RPC) at time t to identify the optimal bidding
- 8. amount for a keyword. Efficiency is reached overtime through this bidding management tool giventhe current status of the keyword portfolio set up of CR/LP, QS and competition. Best practices: A few best practices based on this theoretical framework are provided here: 1. We get lower marginal return as we get more budget. Naturally, we tend to increase spending cross the board. However, the right thing to do is to look at the elasticity of different subgroup of our keyword portfolio and scale up accordingly. 2. Head keywords may not have higher elasticity. Tail keywords may not have lower elasticity. If we use volume as a dimension for segmenting keyword portfolio, we still need overlay elasticity on top of volume as an additional dimension. The common practice of scaling up head keywords is not necessarily optimal. Normally, we get more clicks while efficiency suffers. 3. A longer time window (t) is more beneficial than a shorter time window for estimating the conversion rate. The learning algorithm needs enough data to learn the optimal steady state. However, longer time window hides the seasonal or spiking impact of certain keywords. One solution is to decay the RPC again with a short window like recent 30 days’ data to include recency management in the formula. 4. Simple test/control could be used to isolate or identify the impact of various PPC campaign management factors. Improving CR/LP could shift the EC upwards and result in efficiency gains. In addition, keyword addition and keyword pruning could lead to the same shift and hence efficiency gains. This framework is not only easy to use but also a powerful PPC management tool. Fig. 3 demonstrates the results of one experiment. Efficiency gain is easily demonstrated as the upward shifting of the EC. On the contrary, efficiency loss could be easily identified as well if there was a downward shifting of the curve as the result of the test. 5. Positioning management is simplified after adapting this value based method. There is essentially no need to manage the dynamics of the position of a keyword. If a keyword is valued high enough to be on the first page of SERP, this methodology will ensure max CPC is sufficiently high enough. Pruning of keywords out of a portfolio is also simplified because the max CPC will be driven down to be lower than the minimum bid for first page SERP or even zero. However, we need balance the model results with minimum bid to ensure we get enough data. At least, we need be mindful of the minimum bid while evaluating the outcome of the model. 6. SF management is mostly an art in practice. There are many candidate factors for grouping of keywords such as time on site, category, volume of clicks and conversions, # of tokens in keywords, semantics of the keywords, etc. As long as we are constantly identifying factors where differentiation in elasticity and margin exist, we could leverage the learning ability of the algorithm to reach optimal.
- 9. Fig. 3. Improvement in CR/LP is illustrated as the shifting upwards of the efficiency curve. Reference: R. Dorfman and P. O. Steiner: Optimal Advertising and Optimal. Quality. The American Economic Review 44 (1954), 826--836. y = 30.909x0.5375 R² = 0.8297 y = 7.7191x0.6855 R² = 0.8312 Revenue Spend Efficiency Gain Efficiency Pre Efficiency Post Power (Efficiency Pre) Power (Efficiency Post) Increase in Efficiency!

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