A Study On Keyword Auctions With Risk-Averse Advertisers

Keyword auctions,as an important form of advertising service developed in the last decade,is the main profit model of Google,Bing,Baidu and other internet enterprises.Most researchers study keyword auctions from the aspects of equilibrium analysis,mechanism design,advertisers’ equilibrium bidding strategy and so on,but they assume that bidders are risk neutral.Based on the relevant literature and development direction of traditional auctions and keyword auctions,combined with the specific situation in practice,we assume that bidders are risk averse and study the basic theory of keyword auctions without budget constraint,keyword auctions with budget constraint,optimal reserve price and so on,and compare with those obtained in the risk-neutral case.First of all,we study the keyword auctions without budget constraint under the assumption of static and complete information.Assuming bidders share a general concave utility function,we establish the existence of Nash equilibria（NE）and symmetrical Nash equilibria（SNE）.We then prove that symmetric Nash equilibrium has the same property in the two cases of risk aversion and risk neutrality,and exemplify which property does not hold for Nash equilibrium.Furthermore,we obtain analytical solutions for the smallest and largest SNE bids and prove that the smallest and largest SNE bids and the associated revenues all increase in bidders’ degrees of risk aversion.In particular,the smallest and largest SNE bids are greater when the bidders are risk averse than when they are risk neutral,implying that the smallest and largest SNE revenues are higher when bidders are risk averse.These results imply that search engines prefer more risk-averse bidders,which is consistent with the case of the first-price sealed-bid auction when bidders are assumed to be risk averse.Then we extend all the results to a case where each bidder is assigned to a quality effect.Secondly,we analyse the keyword auctions with budget constraint under the assumption of static and complete information.In a symmetric Nash equilibrium,we provide the sufficient and necessary conditions,under which there is no profit upward deviation for advertisers.We obtain a budget threshold for each advertiser,under which a symmetric Nash equilibrium may be sustained as an equilibrium with budget constraints.We then consider downward deviation and obtain another budget threshold for each advertiser,under which a symmetric Nash equilibrium must be sustained as an equilibrium with budget constraints.If each advertiser’s budget is no less than this threshold,then there is no profit upward deviation or downward deviation for advertisers in a symmetric Nash equilibrium.We find that the two budget thresholds are greater when the bidders are risk averse than when they are risk neutral.Finally,we study the keyword auctions with reserve price under the assumption of static and incomplete information.When the reserve price does not exceed the expectation of advertisers’ Sth highest value,E(V_（N-S+1）),we prove that the expected revenue of search engines increases with reserve price,where there are N risk-averse bidders,and S advertising slots with N > S.We then show numerically that the expected revenue of search engines is lower when the reserve price is greater than E(V_（N-S+1）)when the reserve price is equal to E(V_（N-S+1）),which implies that the expected revenue of search engines in the former case is not necessarily greater than that in the latter case.Thus,the optimal reserve price for search engines is equal to E(V_（N-S+1）),which makes sure that all the advertising slot just sold out.This result is identical to that obtained in the risk-neutral case.