7. A2’s
“optimal” position (profit maximizing position) for ad is the number
2 position
Let’s say that we have an initial situation
where A1 is bidding $1.30 per click and her ad is in the number 1 position.
A1 pays a little less, say for example, $1.25 per click because A2 is bidding
$1.24 for a click. A2 is in the number 2 position paying $1.00 per
click as assumed above in i) due to A3 bidding $0.99
Summary of the different parameters and
their current assumed values

Conversion Rate 
Margin per Sale 
Position 1 CTR 
Position 2 CTR 
Impressions per Day 
Cost per Click 
A1 
5% 
$100 
10% 
6% 
100 
$1.25 
A2 
4% 
$75 
10% 
6% 
100 
$1.00 
A1’s daily Expected Net Profits are
Clicks per Day * Expected Net Profits per Click =
Impressions per Day * Click Through Rate * (Expected Gross Profits per Click – Cost per Click) =
100 * 0.10 * (5%*$100  $1.25) = $37.50
A2’s daily Expected Net Profits are:
Clicks per day * Expected Net Profits per Click =
Impressions per Day * Click Through Rate * (Expected Gross Profits per Click – Cost per Click)
100 * 0.06 * (4%*$75  $1.00) = $12.00.
What if A2 was to now increase his bid from $1.24 to $1.35 and get the top position for $1.31 per click? (1c + A1’s bid of $1.30). A1 would then get the second position for $1.00 per click as assumed in i) above due to A3 bidding $0.99
In tabular form, again, here are the parameters and (new) values

Conversion Rate 
Margin per Sale 
Position 1 CTR 
Position 2 CTR 
Impressions per Day 
Cost per Click 
A1 
5% 
$100 
10% 
6% 
100 
$1.00 
A2 
4% 
$75 
10% 
6% 
100 
$1.31 
A1’s daily Expected Net Profits now are
Clicks per Day * Expected Net Profits per Click =
Impressions per Day * Click Through Rate * (Expected Gross Profits per Click – Cost per Click) =
100 * 0.06 * (5%*$100  $1.00) = $24.00, i.e. a decrease from the earlier $37.50
A2’s daily expected profits now are
Clicks per Day * Expected Net Profits per Click =
Impressions per Day * Click Through Rate * (Expected Gross Profits per Click – Cost per Click) =
100 * 0.10 * (4%*$75  $1.31) = $16.90 an increase from the earlier $12.00.
In order to increase her profits, A1 would now increase her bid to get the number 1 position again and A2 would respond in kind. This jostling for the number 1 position would continue until one of the advertisers expected net profits actually reduce in going from the number 2 position to the number 1 position. So which advertiser would drop out first from this rat race to the top? To answer that, let’s calculate the “break even point” for each advertiser, i.e. the bid amount at which the advertiser is indifferent between the number 1 position and the number 2 position.
In tabular form:

Conversion Rate 
Margin per Sale 
Position 1 CTR 
Position 2 CTR 
Impressions per day 
Cost per Click in Pos. 1 
Cost per Click in Pos. 2 
A1 
5% 
$100 
10% 
6% 
100 
X 
$1.00 
A2 
4% 
$75 
10% 
6% 
100 
Y 
$1.00 
For A1, this “Point of Position Indifference” is when her daily Expected Net Profits from the number 1 position = daily Expected Net Profits from the number 2 position.
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 * 0.10 * (5%*$100 – X) where X is the click cost that A1 pays for the number 1 position (1c + A2’s bid)
Daily Expected Net Profits from the 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 * 0.06 * (5%*$100  $1.00)
“Point of Position Indifference” occurs when we equate these:
100 * 0.10 * (5%*$100 – X) = 100 * 0.06 * (5%*$100  $1.00)
Solving for X; X = $2.60
Thus A1 will continue to increase her bid for the number 1 position until she is paying $2.60; after which she would prefer the number 2 position.
For A2 as well, the “Point of Position Indifference” is when his daily Expected Net Profits from the number 1 position = daily Expected Net Profits from the number 2 position.
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 * 0.10 * (4%*$75 – Y) where Y is the click cost that A2 pays for the number 1 position (1c + A1’s bid)
Daily Expected Net Profits from the 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 * 0.06 * (4%*$75  $1.00)
“Point of Position Indifference” occurs when we equate these:
100 * 0.10 * (4%*$75 – Y) = 100 * 0.06 * (4%*$75  $1.00)
Solving for Y; Y = $1.80
Thus A2 will continue to increase his bid for the number 1 position until he is paying $1.80; after which he would prefer the number 2 position.
Thus A2, who has the lower Expected Gross Profits per click will drop out from the “race” first and will therefore settle for the number 2 position. Hence, the optimal or profit maximizing position for A2 is in the number 2 position.