Four legally married gay couples filed a federal civil rights lawsuit seeking a court order to force Ohio to recognize same-sex marriages on birth certificates despite a statewide ban. Broughton is the birth mother of the twins. (Al Behrman/AP)

In a crucial swing state during the 2004 election, Ohio voters passed a constitutional amendment defining marriage as a union between one man and one woman.

A decade later, a flurry of court decisions has struck down similar bans on same-sex marriage across the country, and polls show Americans rapidly abandoning their opposition to gay people marrying each other. That dramatic change in climate had persuaded some same-sex marriage activists here that this was the year they could get voters to overturn the state’s same-sex marriage ban.

But there will be no same-sex marriage question on the Ohio ballot in 2014, even though advocates collected more than 650,000 petition signatures to put the question to voters, many more than the 385,000 required.

The reasons? Raw political calculation and a more disciplined approach to strategy by the leaders of the gay rights movement.

Marriage-equality advocates say winning nationwide is a game of chess, not checkers, and a vote on same-sex marriage this year in Ohio may be the absolutely wrong move.

“We did not support . . . the idea of just rolling the dice and throwing it on the ballot before we built the support, campaign and war chest that would be likely to win,” said Evan Wolfson, executive director of Freedom to Marry, a national marriage-equality group and one of the chief architects behind the national strategy that has seen a rapid series of victories in states where same-sex marriage was previously banned.

That strategy involves winning court decisions in favor of legalizing same-sex marriage — even in states where the electorate remains torn on the issue — and only pursuing ballot measures in states where they are likely to win. The end goal, they say, is a Supreme Court decision.

“It’s not about winning every single battle,” Wolfson said. “But if you know there’s a chance you’re going to lose, you have to ask whether now is the time to fight it.”

For Ian James, a political consultant and same-sex marriage activist here, it took some convincing. James began circulating petitions, mobilizing volunteers through his FreedomOhio organization to collect the 385,000 signatures needed to put a same-sex marriage question on the ballot.

He amassed a lot more — upward of 650,000 signatures. But James isn’t going to turn them in, although he initially vowed to get on the ballot this year. After several well-publicized clashes with other marriage-equality groups, James backed down, ending an insurrection that some activists feared could jeopardize efforts in Ohio and hand a major victory to opponents of same-sex marriage.

The decision to wait — despite the latest poll, conducted by Quinnipiac University in February, finding 50 percent of Ohio voters in support of same-sex marriage — is indicative of a new discipline, cohesion and tactical savvy that seems to define the current gay rights movement, one that has carefully charted its path forward with great success and is now diligently trying to avoid major missteps.

“We can’t just be tilting at windmills. We have to go when the environment is conducive to a victory,” said Michael Premo, campaign manager for Why Marriage Matters Ohio, a pro-same-sex marriage organization that had opposed efforts to get a measure on the ballot in 2014. “We’ve already lost at the ballot box once in Ohio. To lose again would be a huge boost of momentum to the opponents of equality who have not had a victory in a long time.”

Strategists on both sides of the marriage fight agree that a ballot measure effort would awaken the relatively dormant defenders of traditional marriage — a 10,000-church network that drove thousands to Ohio polls in 2004, securing 62 percent of the vote in favor of the law banning same-sex marriage and simultaneously propelling President George W. Bush to victory in the state and securing his reelection.

“I don’t see where 10 years has changed anything,” said Phil Burress, who led the 2004 effort to get the amendment banning same-sex marriage on the Ohio ballot and runs the Columbus-based Citizens for Community Values, a conservative activist group. “We’re not paying a whole lot of attention until they submit the signatures and put this on a ballot. But if they do, we’ll beat them again.”

The preferred strategy, marriage-equality advocates say, is in courtrooms, where judges have handed down several decisions in recent months legalizing same-sex marriage. Since last June, there have been 14 federal court rulings against state bans on same-sex marriage — with the most recent judge’s ruling coming earlier this month in Wisconsin.

“The reason that there is such great success is that there is no legal argument to continue the ban,” Premo said. “That’s why in state after state, these bans are starting to fall.”

Judges have ruled in favor of same-sex marriage advocates in the past two court cases dealing with marriage, ruling that Ohio must recognize same-sex marriages performed in other states. In April, lawyers filed a federal lawsuit on behalf of six same-sex couples in Cincinnati, urging a federal judge to strike down the state’s current ban.

“It’s our hope that courts take care of it this year or next,” Premo said.

The court method falls into the broader national political strategy being employed by same-sex-marriage groups. The goal, they say, is to win same-sex marriage in as many states as possible while simultaneously continuing to dominate the national dialogue and eroding the opposing sentiments among a majority of voters. At that point, they say, the stage will be set for a Supreme Court decision that can, in one final blow, end the debate over same-sex marriage once and for all.

“We don’t have to win within the four corners of every single state. It’s a national strategy,” Wolfson said. “It also means that we have to win a critical mass of states and a critical mass of public support throughout the country that together create the climate that will encourage the courts and, ultimately, the Supreme Court to do the right thing.”

In the meantime, they plan on pursuing a ballot measure only in states where it is politically favorable — such as Maine, Maryland and Washington state, where voters legalized same-sex marriage in 2012, and Oregon, where advocates were pursuing a ballot measure this year until a court decision tossed the state’s ban on same-sex marriage. National traditional-marriage groups are appealing the Oregon ruling.

Even if a Circuit Court or Supreme Court ruling does not come soon, same-sex marriage groups in Ohio say the wiser move remains waiting for 2016.

Political strategists note that, even given the state’s monumental shift in attitudes toward same-sex marriage in the decade since it was banned, the 2014 electoral turnout is not conducive to passing the measure in Ohio.

With a Democratic incumbent in the White House, Republican turnout is expected to exceed Democratic turnout in much of the country this year. Polling consistently shows voter dissatisfaction with Democrats, and many observers believe that sentiment will drive Republican gains in Congress this year. Meanwhile, Ohio Gov. John Kasich, a Republican, remains relatively popular in the state and is widely considered to be in strong position to defeat his Democratic challenger in 2014.

If same-sex marriage is to be on the ballot in Ohio, LGBT operatives say they want it voted on by the 2016 presidential electorate — not the right-leaning 2014 voters.

“In Ohio, in 2014, the electorate tilts more Republican and probably older,” said Neil Newhouse, a Republican pollster who has worked for Kasich and was Mitt Romney’s lead pollster in 2012.

While most say they hope to see a judge strike down Ohio’s same-sex marriage ban, advocates say that they are beginning to prepare for the chance that, if no decision is rendered next year, they will put a measure on the ballot in 2016.

Freedom to Marry and other national groups are already working with local grass-roots organizations in Ohio, Arizona, Colorado, Michigan and Nevada to lay the groundwork. If there is no Supreme Court decision legalizing same-sex marriage nationwide by 2016, same-sex marriage proponents hope to be prepared to launch ballot initiatives in each of those states.

“The litigation landscape is so dynamic and the progress is so clear that we may see victory nationwide before we have to go to the ballot in Ohio,” Wolfson said. “Nobody knows today what timeline we’re on, so we have to work with the urgency of a potential national ruling soon and at the same time with the tenacity of being ready to win these battles at the ballot if that is what’s required.”