Search Engine's Profit Function

To compute the search engine's profits, we first determine the fraction of users that will visit the search engine and the fraction of content providers that will choose paid placement. Let c be the threshold value desired by users before they use the search engine. This may represent an opportunity cost or effort in using the search engine, or the value provided by a competing search engine. Hence the users who use the search engine's service are 20#20. Since 3#3 is uniformly distributed in [1, 2], the search engine's market coverage is

 21#21 (1)

To make the problem meaningful, we require that at least the highest valuation user ( 22#22) will use the search engine when there is no paid placement, i.e., that 23#23. The search engine benefits content providers by directing users to their sites. Content providers are heterogeneous in their profit expectations, which is a function of the search engine's market coverage 8#8. Providers have a choice between regular placement (which provides a value 24#24 at no cost) and paid placement (which provides value 18#18 at cost 12#12). Rational providers will choose paid placement if and only if 25#25 i.e., 26#26. Hence the fraction of providers who choose paid placement is

 27#27 (2)

The search engine obtains revenues from two sources, third party firms and paid placement. The first type of revenue 28#28 is a function of the search engine's market coverage, 8#8, and profit rate brought by each user, a. If 8#8 is interpreted as the number of queries to the search engine, a may be considered as the rate per impression. Hence

29#29

The search engine's placement revenue 30#30 is 31#31. Substituting for 8#8 and rearranging terms, we get

32#32

The search engine's total profits are 33#33, and it aims to choose the optimal fraction of paid placement 1 - x - alternately, the optimal degree of independence, x - in order to maximize 34#34.