Obama's gay marriage stance could hurt him A New York Times/CBS News poll released Monday night shows the possible political perils of President Obama's recent announcement that he supports same-sex marriage. Most of those polled say the president's position will not impact how they vote. But among those who say it will influence their choice, 26 percent said they are less likely to vote for Obama as a result, while 16 percent say they are more likely to. And in what is expected to be a tight race, "even a small shift in swing states could be costly." Also troubling for the president, is that a majority of voters suspect that his decision was politically motivated. Sixty-seven percent said they thought Obama's announcement was made "mostly for political reasons," while 24 percent said it was "mostly because he thinks it is right." In another potentially damaging sign, 70 percent of Independents attribute the president's move to politics, along with nearly half of Democrats. Overall, Romney now has a three-point edge over Obama â a lead that is within the poll's margin of error â despite improving views of the economy. The Times/CBS poll had the race dead even a month ago. Romney now leads Obama among Independents, and has a 1-point advantage (again within the margin of error) among women. The economy remains by far the most important issue to voters, with 62 percent naming it their top concern, followed by the budget deficit at 11 percent and health care at 9 percent. Just 7 percent pick same-sex marriage as the most important issue in the election. The controversial issue is not without risks for Romney, either. A "well-known, openly gay" Romney supporter withdrew his backing for the former Massachusetts governor on Monday, citing his stance on same-sex marriage. "You have chosen to be on the wrong side of history and I do not support your run for president any longer," wrote Bill White, the chairman and CEO of a New York-based consulting firm, in a letter obtained by CNN.