A Study On Keyword Auctions With Budget Constraints

The auction also known as bidding is that an auction group determines the process of the price and distribution.Auction theory,as a branch of microeconomics,has been successfully applied in many fields.People are familiar with the auction form which includes Sotheby's and Christie's art auction,and the auction of the land management departments transferring land-use rights,etc.And with the rapid development of the Internet technology,search engines have become one of the main tools for people to obtain information.Because it is not subject to time and place restrictions,people can use the search engine to search information through the internet from anywhere at any time.The keyword advertising is the main source of income for search engines.And the turnover of keyword advertising is up to tens of billions of dollars annually.It is far higher than the traditional auction market.Therefore,many scholars have devoted themselves to the academic research of keyword auctions.After we had reviewed the existing literatures,the earlier research did not consider the advertisers' budget constraints.But in reality,advertisers must consider the budget constraints in order to make the advertising bidding strategy.In addition,from a business point of view,compared to high advertising costs of the traditional advertisements,many small and medium-sized enterprises prefer to choose to run the inexpensive and high-quality keyword advertisements,but they need to face the problem of the limited advertising budgets.From the economic point of view,large enterprises will set a reasonable budget strategy in order to improve their return on investment.If the budget is too high,it will easily result in the waste of resources;if the budget is too low,the online time of keyword advertising might be short which affect the promotion effect.In view of this,based on the related literatures and current research results of keyword auctions,and combined with the actual development of keyword advertising,this dissertation has deeply studied advertisers' equilibrium bidding condition with the budget constraints and the revenue of the search engine in keyword auctions.First of all,based on the Nash equilibrium condition,we analyze the problem that the advertisers' bidding vector reach the Nash equilibrium with budget constraints in the generalized second price auction mechanism.We define the Nash equilibrium with budget constraints.We give the upper and lower bounds of advertisers' bidding in the Nash equilibrium,and analyze the necessary and sufficient conditions for advertiser to increase his bidding profitlessly.The related conclusions are verified by the numerical examples.In addition,the results of numerical examples show that the different changes after advertiser j reduce her bidding.We give the necessary and sufficient conditions that the advertisers' bidding vector is the Nash equilibrium with budget constraints by upper bound of advertiser's bidding and budget.Finally,from the point of view of the search engine,we give the conditions when two advertisers to compete with an advertising position to reach the Nash equilibrium with budget constraints and revenue of search engine in equilibrium.The results show that the revenue of search engine considering the budget constraints is greater than or equal to the case without considering the budget constraints.Secondly,based on the VCG mechanism,we study the problem that the advertisings' bidding vector reach the global envy-free Nash equilibrium with the budget constraints.We give the upper and lower bounds of advertisers' payment in the global envy-free Nash equilibrium,and propose the concept of the global envy-free Nash equilibrium with budget constraints,and give the necessary and sufficient condition of advertiser's bidding upward deviation profitlessly,and verified the theorem by the numerical example.We also study the condition that the advertiser's upper bound of the bidding does not reach the global envy-free Nash equilibrium,and the vector formed by the upper bound of biddings is also the global envy-free Nash equilibrium.At last,we obtain the necessary and sufficient condition that advertisers' bidding vector is the global envy-free Nash equilibrium through these two lemmas.This result is an extension of the symmetric Nash equilibrium with the budget constraints.Finally,when the advertising display time is not fixed,we analyze the problem that the advertisers' bidding can reach the Nash equilibrium with budget constraints or not.In the keyword auctions with budget constraints,the condition for advertisers to drop out of the auction is that their budgets have been exhausted.In this case,the time for advertisers to participate in the auction is together determined by their own budget and bidding.When advertiser's budget is an integer multiple of her payment,she raise the bidding unprofitably,and reduce the bidding profitably.Therefore,under this assumption,the advertiser's bidding vector can not reach the Nash equilibrium with budget constraints.In addition,we establish a simple model to discuss the impact of the budget on the advertiser's revenue,and give the trend of the changes of advertiser's revenue in the auction respectively.When the two advertisers' budget exceeds the critical value,the advertier's bidding vector reach the Nash equilibrium with budget constraints.