The president, who eked out a 12,000-vote victory in Georgia, received a small but potentially important boost from the state’s conservative areas if at least one local Democrat was running in a down-ballot race, according to a new study by Run for Something, an organization dedicated to recruiting and supporting liberal candidates. That finding extended even to the state’s reddest districts. The phenomenon appeared to hold nationally. Mr. Biden performed 0.3 percent to 1.5 percent better last year in conservative state legislative districts where Democrats put forward challengers than in districts where Republicans ran unopposed, the study found. The analysis was carried out using available precinct-level data in eight states — Florida, Ohio, North Carolina, Arizona, Georgia, Texas, Kansas and New York — and controlling for factors like education to create a comparison between contested and uncontested districts. The study showed a reverse coattails effect: It was lower-level candidates running in nearly hopeless situations — red districts that Democrats had traditionally considered no-win, low-to-no-investment territory — who helped the national or statewide figures atop the ballot, instead of down-ballot candidates benefiting from a popular national candidate of the same party.