Search engines are essential actors for web browsing. We analyze here the economic competition between search engines earning money from adword auctions. We develop a twolevel game where at the largest time scale search engines decide which allocation rule to implement, between revenue-based and bid-based; and at the lowest time-scale advertisers decide how to split their advertising budget between the two search engines, depending on the benefits this will bring to them. The game at the largest time scale is solved using backward induction, the search engines anticipating the reactions of advertisers. We describe the advertisers best strategies and show how to determine, depending on parameters, an equilibrium on the ranking rule strategy for search engines; this may explain Yahoo!'s move to switch from bid-based to revenue-based ranking to follow Google's strategy.