This article is a follow-up to 'Optimal Bidding, Part 1. Behind the Scenes of 'Google AdWords Bidding Tutorial' 
There used to be this TV game show called “Card Sharks” where, in one portion, contestants were shown a playing card and then asked to guess whether the next card was going to be higher or lower in value. (This wasn’t high-brow entertainment.) The best strategy is obvious – when the card shown is low, guess that the next card will be higher, and when the card shown is high, guess that the next card will be lower.
In the new video 'Google AdWords Bidding Tutorial ', Google’s chief economist, Dr. Hal Varian, explains how advertisers can use Google’s Bid Simulator (GBS) to determine a bid that is “at or near the profit-maximizing level for each of your keywords.” His method involves calculating the value per click, and then comparing that value to the GBS’s estimated incremental cost per click (ICC), as seen in this screenshot:
- Figure 1. A screenshot from time 6:05 in 'Google AdWords Bidding Tutorial'
The specific example he gives is of an online retailer of digital cameras where each camera sells for $300 and costs the retailer $200 and where each click has a 5% chance of converting into a sale. (In this case, since each click has a 5% chance of bringing in $100, the value per click is $5.00.) Dr. Varian says, “Using data from Bid Simulator, or from your own experiments, we can see how many clicks we could have potentially received at different bids and how much those clicks would have cost.” These are the first three columns in the screenshot from time 7:33 below.
- Figure 2. A screenshot from time 7:33 in 'Google AdWords Bidding Tutorial'
Since each click brings the retailer $5 (before ad costs are considered), we can multiply the number of clicks by $5 to find the amount of money the retailer expects to make per week at each bid level (Revenue). Subtracting the ad Cost from that amount tells us the net Profit per week.
The incremental cost per click (ICC) is simply the change in cost between any two bids divided by the change in clicks between those bids. So, in Dr. Varian’s example, going from a bid of $3.50 to $4.00 costs an extra $97.29 (that is, $407.02 - $309.73) and brings in 21 more clicks (154 clicks – 133 clicks), for an ICC of $97.29 / 21 = $4.63. However, each click brings the retailer $5.00, so it’s in this advertiser’s best interest to raise the bid and get more clicks at (or just above) a price of $4.63. At time 5:52, Dr. Varian says, “Whenever your value per click is less than the incremental cost per click it will pay for you to lower your bid in order to reduce your cost. Conversely, if your value per click is higher than your incremental cost per click, you should increase your bid. You can see the ICC at our $4.00 bid is the closest value to our $5.00 value per click without going over. So it’s going to bring in the highest profit. Actually, in this example,” he says, “you probably want to bid a little bit more than $4.00.” How much higher? He doesn’t say.
I call this the ‘Card Sharks Approach to Bid Management’ and it might work well for people who have all the time and money in the world to ‘experiment’ with various bids, as Dr. Varian suggests, or for those who have the luxury to only know their best bid to the nearest $0.50 (or whichever other increment Google chooses to display) or who are willing to just guess some amount between the GBS’s tested values. My question is: Since Dr. Varian shows that we have all the information we need to determine an optimal (or near-optimal) bid, why test different bids at all? Why not just directly calculate (to the best extent possible) the single optimal bid and then just use that bid immediately, instead of nudging bids up and down until we hit some sort of observable sweet spot?
Personally, I think that showing AdWords’ users how to directly calculate a profit-maximizing bid is one of Dr. Varian’s ultimate goals and the simplified description he provides in this video is just a waypoint on that journey. So, rather than wait for him to get to it on his own, I am going to describe for you a simple, visual means for determining your optimal bid from Google’s Bid Simulator. Then, I’ll use some straightforward math to show you how to calculate the optimal bid (and CPA and ROI) for Dr. Varian’s example.
The screenshot from time 7:33 (Figure 2, above) contains some of the information found in Google’s Bid Simulator, but it also lacks a key component. If you actually look at the Bid Simulator for a word in your account, you’ll notice that for high-traffic words, in addition to the columns of numbers, the dialog box also contains a graph that looks something like:
Each green point on this graph comes from a row on the spreadsheet - the point furthest to the right from the highest bid (the top row of numbers) and each next point to the left from the next row down. (I’ve added the bid labels to each point for clarity, but they are not shown in Google’s actual Bid Simulator.) Since it is obvious that at a bid of $0.00 the word will get 0 clicks at $0 cost, I have also added that point to the graph.
A recent blog post by Google which states that Hal Varian’s research indicates conversion rate (CR) does not vary much with position  is very interesting, in part because this also implies that CRs do not change as a result of changing the bid. So, each word has a value per click (in effect, a rate at which an advertiser is willing to trade dollars for clicks) that does not depend on the bid, position, or number of clicks already obtained. Therefore, we can draw a straight line on the ‘Cost vs. Clicks’ graph whose slope is the value per click and slide that line until it just barely touches the Bid Simulator’s estimates. The point on the Bid Simulator’s estimates where the two lines meet (in this case, just over $4.00) is the optimal bid.
For bids lower than (that is, to the left of) this point, the advertiser should be willing to increase the bid because the slope of the ‘Cost vs. Clicks’ curve is less than the rate at which the advertiser is willing to trade dollars for clicks. For bids higher than this level, the advertiser should be willing to forgo (too-expensive) clicks to save those dollars. The point where the straight line crosses 0 clicks (in this case, about -$370) is the negative value of the expected Profit per week, which you can confirm in Figure 2 above.
None of this should be surprising to anyone who has read Dr. Varian’s article called ‘Position Auctions ’ (International Journal of Industrial Organization, vol. 25, iss. 6, Dec 2007, p. 1163-1178) and all of my description from above is taken directly from that article. It seems perfectly reasonable to me that Google might add this functionality to their Bid Simulator at some point. (In fact, it surprises me that they haven't done this already.) The advertiser could simply enter the ‘value per click’ in an input box, and the GBS could plot the line, find the optimal bid and determine the estimated profit per week in the blink of an eye.
However, if you know the relationships for ‘Clicks vs bid’ and ‘avg CPC vs bid’, it is also possible to just calculate the optimal bid directly on your own, without fiddling with the Bid Simulator and taking the Card Sharks Approach. For Dr. Varian’s example, we can plot ‘Clicks vs bid’ and ‘avg CPC vs bid’ and find that, for this simple demonstration case, they are both basically straight lines:
That is, Clicks follows the line ‘m bid + b’, and avg CPC follows the line ‘n bid + g’, where m, n, b and g are parameters that can be found by least-squares fitting (i.e., the ‘trendline’ feature in Microsoft Excel). For this case, m is about 47.865, n about 0.676, b about -33.514, and g about -0.0357. Oddly, the points corresponding to a bid of $4.50/click seem to have been adjusted upwards from a linear relationship, perhaps to make the results in Dr. Varian’s demonstration clearer. (It’s mildly disturbing that he might have fiddled with the numbers, even for demonstration purposes, since it makes one wonder to what extent the estimates provided by the Bid Simulator itself might be manipulated.)
Nevertheless, our goal as advertisers is to maximize the amount of net profit made per week (after ad costs are considered). Since net profit = Revenue – COGS – AdCost, our goal is simply to find the bid where d(net profit)/d(bid) = 0.
Revenue = Clicks x CR x RevPerConv
COGS = Clicks x CR x COGSPerConv
AdCost = Clicks x CPC
We know from Dr. Varian’s research on conversion rates that CR is not a function of bid, and d(Clicks)/d(bid) = m, so if we say ProfitPerConv = RevPerConv – COGSPerConv, then the equation reduces to:
Multiplying the equations for Clicks and CPC together and differentiating with respect to the bid gives:
The first two terms, ProfitPerConv x CR, is simply the value per click (VPC), so:
(Note: this equation is only true for the specific ‘Clicks vs bid’ and ‘CPC vs bid’ relationships used in Dr. Varian’s example.) That is, for the specific example where Clicks and CPC are linear functions of the bid and have parameters equal to the values listed above, the optimal bid (which Dr. Varian called “a little bit more than $4.00”) is actually a little more than $4.07. (If the Click and CPC relationships are assumed to be 2nd-order polynomial, rather than linear, the optimal bid turns out to be essentially the same, $4.08.) So, there’s no need to use the Card Sharks Approach when you can just calculate the optimal value directly.
What is the CPA at which profitability is maximized? If you recall, Dr. Varian calculated the ‘maximum profitable CPA’ in his video, but the ‘maximum profitable CPA’ is the CPA above which the advertiser’s profit is negative. In other words, it is the CPA at which the expected profit is equal to zero. When bidding, our target CPA is the ‘CPA of maximum profitability’, not the ‘maximum profitable CPA’. We can see the difference between the two in the diagram below, which plots net profit vs. CPA. Profit reaches a peak at a bid a little bit higher than $4.00 and declines from there until reaching 0 at a CPA of $100, when the bid is nearly $7.50 ($7.45, actually). The ‘maximum profitable CPA’ therefore is $100, but the CPA of maximum profitability is much less.
(Notice that the position of the point corresponding to a bid of $4.50 has perhaps been moved by the possible adjustment made to the Bid Simulator’s numbers at that point.)
We can actually find the CPA of maximum profitability (that is, the target CPA) quite easily from what we already know. CPA = Cost / Conversions, therefore:
It’s simple algebra to multiply the optimal bid, shown in an equation above, by ‘n’ and then add ‘g’. Thus:
(Again: this equation is only true for the specific ‘Clicks vs bid’ and ‘CPC vs bid’ relationships used in Dr. Varian’s example.) For the particular parameters that fit the sample data best, this optimal CPA is approximately $54.38.
It's remarkable to see how easy in Dr. Varian's example it appears to be to make a profit on AdWords. In his example, any bid in the range of $0.70 to $7.54 turns a profit. A bid of about $4.07 yields the most, but any bid from about $3.32 to $4.82 gives an expected profit that's within 95% of the maximum. In other words, even though the purpose of Dr. Varian's video was to demonstrate how to determine an optimal (or near-optimal) bid using Google's Bid Simulator, one of the interesting lessons of the specific example he crafted is that (for this particular example only) you can bid anywhere in a $1.50-wide range surrounding the optimal bid and still basically be maximizing your profit.
Many account managers say that they would like to push down their CPAs as low as possible. But another interesting lesson from this example is that in addition to a maximum profitable CPA ($100, where the advertiser makes no profit), there is also a minimum profitable CPA (in this case, about $8.75, corresponding to a bid of about $0.70, below which the advertiser also makes no profit). So, account managers who are too successful in pushing down their CPAs might also be pushing down their profits, perhaps without even realizing it!
Unfortunately, determining the conversion rate, revenue per conversion, COGS per conversion, the relationships for ‘Clicks vs bid’ and ‘CPC vs bid’, and the CPA (or ROI) of maximum profitability in most real-world examples is not as simple as Dr. Varian’s example. Therefore, The Search Agency  (and the AdMax online marketing platform) are here to help you maximize the return on all of your online marketing efforts. Please don't hesitate to contact us if you need assistance with your online marketing efforts.
Thanks to Eric Sodomka of Brown University for examining the 2nd-order polynomial Click and CPC models.