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We analyze the generalized first-price auction under incomplete information setting. Without setting a reserve price, the efficient symmetrical Bayes-Nash equilibrium is characterized and found to be increasing as the number of bidders is sufficiently large. Then, the explicit expression for the expected revenue of the search engine is found and the effect of the click rates of all the positions on the expected revenue is obtained. Finally, with setting of the reserve price, we have found the optimal reserve price and examine how the difference of the search engine’s revenues with setting reserve price and without setting reserve price varies with the reserve price.

With the increasing popularity of the Internet, sponsored search advertisements become one of important commercial advertisements and main revenue of the search engine. Practical importance of the sponsored search auctions has attracted much research interest of many researchers [

The advertisement positions are sold mostly through the generalized second-price auction (GSP) by the search engine such as Google, Yahoo!, and MSN. Some search engines like Overture sell the positions through the generalized first-price auction (GFP). The selling process is as follows. An Internet user performs a query using a search engine which shows a set of search results. These results show the links the search engine has deemed relevant to the search but it also includes a list of sponsored links (paid advertisements). Each time the user clicks on a sponsored link the advertiser pays the search engine. In GSP auctions, bidders submit simultaneous bids to the seller. The highest bidder wins and pays the value of his bid. In GSP auctions, bidders submit simultaneous bids to the sellers. Again, the highest bidder wins but in this case, he pays the value of the second-highest bid.

To our best knowledge, these are a few research papers which pay attention to GFP. Szymanski and Lee [

The remainder of this paper is organized as follows. In Section

In this section, we introduce some notations. We model GFP auction by the following assumptions:

We consider a single round static game in incomplete information setting.

There are

Bidders and the search engine are risk neutral. That is, both want to maximize their expected profits.

The bidders are symmetric. That is, the valuations of all the bidders are independently and identically distributed.

Let

Let

Let

Let

Let

The above assumptions are common knowledge for all bidders.

In setting of complete information where all information including all the valuations is common knowledge, bidders can get their desired positions. In the setting of incomplete information, since the valuations of bidders are private information, each bidder maximizes his total expected profit by treating all other bidders’ valuations as private information (random variables) in order to obtain his bidding strategy.

The efficient symmetric Bayes-Nash equilibrium bidding strategy of bidder

Suppose that all but bidder

Now, (

Proposition

Next we analyze how the number of bidders

As the number of bidders

First, we rewrite (

Proposition

After we get the expression of the equilibrium bidding strategy, we can find the expected revenue of the search engine paid by the bidders and make comparative static analysis of it.

The expected equilibrium revenue of the search engine is given by

Using the expected payment of a bidder, we can compute the search engine’s expected revenue. Notice that the expected number of clicks for bidder

If there is only one advertising position, the expected equilibrium revenue degenerates to the revenue of single good. From

The click-through rates are vital parameters. Next we analyze their effects on the expected revenue of the search engine.

The expected equilibrium revenue of the search engine is

increasing in the click-through rate

increasing in the click-through rate

Differentiating (

Because the top position has the largest probability of being clicked among all the positions and then brings more revenues to the search engine, the more the click rate of top position, the more the revenues which the search engine receives from it. Thus, Proposition

This section considers GSP auction with a uniform reserve price

An allocation of positions to bidders is quasi-efficient allocation, if the following conditions are satisfied [

The bidders with higher value are assigned higher positions.

No position is left vacant when there is a bidder with valuation

We start our analysis of GFP with reserve price

With the existence of the uniform reserve price

See Appendix.

If there is only one advertising position, the equilibrium bid strategy degenerates to the bid strategy of single good. From

Proposition

With the existence of the uniform reserve price

See Appendix.

For the search engine, setting the uniform reserve price is double-edged sword. That is, smaller reserve price decreases the probability of the inefficient event in which no sale occurs; larger reserve price eliminates more bidders and then reduces the competition of bidders. Next, we will probe the optimal reserve price. Remarkably, the optimal reserve price depends neither on the number of bidders nor on the number of available positions.

If the distribution

Differentiating (

Proposition

The difference between the search engine’s revenues under setting the reserve price

From (

Proposition

The games of incomplete information are the most central to the sponsored search auction. Under incomplete information we analyze the first-price auction. We have found the explicit expression for the equilibrium bidding strategy without setting the reserve price by the search engine. We make some comparative static analyses. It is concluded that the bidder’s equilibrium bidding strategy is approaching his valuation as the number of bidders is sufficiently larger. This implies that sufficiently larger number of bidders benefits the search engine. We have found that the expected revenue of the search engine is increasing in the click rate of the top position. We also give a necessary and sufficient condition under which the expected revenue of the search engine is increasing in the click rates of other positions. Given that the search engine sets the reserve price, we have found the expression for determining the optimal reserve price. We have not made comparisons between our results of GFP and that of GSP by Gomes and Sweeney [

Similar to the proof of Proposition

Therefore, in an efficient equilibrium, bidder

Similar to the proof of Proposition

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported by the National Science Foundation of China under Grant no. 71171052.