First note that A2 now pays only $0.50 for a click in the number 2 position because A2’s quality score is now also twice that of A3, so he has to now pay only half as much as before or half of $1.00. 

Now the result changes for A2 as below: 

“Point of Position Indifference” for A2 

Daily Expected Net Profits from the number 1 position

Clicks per Day * Expected Net Profits per Click = 

Impressions per Day * Click Through Rate * (Expected Gross Profits per Click – Cost per Click) = 

100 * 0.20 * (0.04*$75 – Y) where Y is the click cost that A2 pays for number 1 position 

Daily Expected Net Profits daily from number 2 position = 

Clicks per Day * Expected Net Profits per Click = 

Impressions per Day * Click Through Rate * (Expected Gross Profits per Click – Cost per Click) = 

100 * 0.12 * (0.04*$75 - $0.50) 

“Point of Position Indifference” occurs when we equate these: 

100 * 0.20 * (0.04*$75 – Y) = 100 * 0.12 * (0.04*$75 - $0.50) 

Solving for Y; Y = 1.50 

So if A2 is willing to go all the way until $1.50 for the number 1 position, then the only way that A1 will get the number 1 position is if she bids till $3.00 for the number 1 position (2 * $1.50) because her quality score is half of A2.  But A1 is going to stop at $2.60 thus giving A2 the top position here. 

(As an aside, note that the reason Y went from $1.80 to $1.50 is that after A2’s Quality Score doubled, he was paying only $0.50 rather than $1.00 for the number 2 position). 

So when Quality Scores are taken into account, expected profits alone will not determine optimal ad position.  Rather, one must take into account the fact that an advertiser with a lower Expected Gross Profit may still have the number 1 position as their optimal position because they will be allowed to bid less for that top spot due to a higher quality score.