For your next competitive bid, ask the firms to submit two proposals. The second one should be blinded, the name of the firm and all information identifying the firm being concealed.
I hear you muttering, “You are crazy, Rees!” But think of the gross impressions and prejudices lawyers have of some firms.
Certainly some important information, such as lawyer bios and matters handled, cannot be redacted. But significant parts of a proposal could be submitted anonymously and the reviewers would be less likely to be improperly influenced by the aura of a particular firm. If the response reviewers don’t know whether the proposal comes from Cravath, Swaine & Moore or from Afew, Lawyers & Paralegals, they won’t fall prey to the affect heuristic whereby preconceived value judgments interfere with assessments (See my post of March 15 ,2009: seven cognitive biases; May 14, 2006: fundamental attribution error; July 10, 2007: fundamental attribution bias; and Nov. 21, 2008: value attribution distorts perceptions.).