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So, can / should I bid on their company name? It doesn't seem like anyone else is bidding on it.
If I DO bid on their company name, how can I get a good quality score without getting myself into copyright / trademark trouble?
AdWords policy is to make sure you don't use other advertisers trademarked terms.
That said we do bid on some competitor names and sometimes include a comparison of why people should come to us vs what the competitor offers.