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Voters on Tuesday selected two candidates whose disagreements on the issues could not be sharper. There is potential for a battle of ideas over the next few weeks between Paul Vallas and Brandon Johnson that augurs well for a city that needs answers.

Mayor Lori Lightfoot had her ideas too. Plus a mayoral record to run on. But voters rejected her Tuesday, and as Johnson and Vallas launch into a campaign that concludes with the April 4 runoff, we would do well to learn a lesson from Lightfoot: Ideas are only the start. Just as important are the ability and temperament to govern.

The two finalists clearly are ready to play hardball, not beanbag, over the next four weeks. “We’re about to get real,” Johnson told jubilant supporters Tuesday night, as he launched into an indictment of Vallas’ track record. His opponent resumed campaigning Wednesday by responding in kind.

If that’s the way this contest plays out — and if Vallas and Johnson can stick to the issues — Chicago will be better for it.

Consider their contrasting approaches to policing.

Vallas offers a law-and-order approach that would start with filling some 1,500 vacant Chicago Police Department jobs and implement “true community policing.” Tuesday’s top vote-getter, he is backed by the Fraternal Order of Police and its president, John Catanzara, an acolyte of Donald Trump.

Johnson, who placed a respectable but distant second to Vallas on Tuesday, once talked about defunding the police. He has amended that now to propose redirecting money from cops on the streets toward addressing the inputs of crime, such as social and economic inequities and mental health disorders. He would stop using CPD’s ShotSpotter technology, which is prone to false alarms that send swarms of cops into poor neighborhoods, prompting anger and charges of inequity among residents.

Go up and down the list of the city’s major concerns, and the contrasts continue. Education, economic inequities, taxes: You name it, and they disagree on how to approach it.

Johnson is backed by the Chicago Teachers Union; Vallas, a former Chicago Public Schools CEO, believes in school vouchers and charter schools, which CTU abhors. On economics, Vallas talks about creating a community investment authority and a city-run land bank. Johnson would tax his way toward funding more equitable policies: an employers’ head tax and a commuter tax. He proposes $800 million in new taxes altogether.

Oh yes, it’s worth noting too: Johnson, a Cook County commissioner, is Black. Vallas is white. Racial tribalism still factors strongly in the city’s politics and in the governing of this city.

Four years ago, Lightfoot laid out her own policy agenda. Voters embraced it, and her reformist ideals, and propelled her into office. And make no mistake: Despite her deficiencies as a leader, Lightfoot got a lot done.

Structurally improving the city’s finances, directing more than $1 billion in new investment toward the South and West sides, effectively fighting the COVID-19 pandemic and launching major new projects such as a downtown casino and an expanded O’Hare International Airport: Lightfoot made progress. Even the schools improved, though not without difficulty. The city is better for her time in office.

On only one major issue — the rise in violent crime in neighborhoods across the city — did Lightfoot seem completely stymied. The resignation of police Superintendent David Brown on Wednesday, the day after the election, marked his final gasp of exasperated ineptitude.

But Lightfoot will be a one-term mayor, Chicago’s first in four decades, in part because in her time as mayor, she consistently revealed a temperament that exposed an inability to govern. Although she often was able to persist and grind her way to success, she found ways to be her own worst enemy. And that tendency was a key factor that helped cost her the job.

Lightfoot’s impatience, foul language and temper, lack of gratitude toward political allies and community supporters, and a seeming delight in political enmity ultimately left her isolated and ineffective as a leader. A series of broken campaign promises alienated progressives in particular, a much-needed voting bloc for Lightfoot’s reelection hopes.

It’s telling that at least seven aldermen who had served as Lightfoot’s City Council leadership team endorsed someone else among the field of nine candidates. None of the four people Lightfoot appointed to fill open council seats in the last year endorsed her. As at least a dozen aldermen called it quits ahead of this year’s election, some made clear that the acrimony that brewed in City Hall during her tenure was a factor.

Lightfoot also shed large blocs of voters. After winning all 50 wards in the 2019 runoff against a seasoned politician, Cook County Board President Toni Preckwinkle, Lightfoot on Tuesday won only 16. All were on the South and West sides, and only two — the 29th and 37th Wards on the West Side — have neighborhoods north of North Avenue.

The lakefront wards that had catapulted Lightfoot into the runoff against Preckwinkle in 2019 threw their support toward Vallas and Johnson instead.

There is a lesson here for Vallas and Johnson, as well as the voters who will choose between them next month. To earn the job, one of the candidates will need to muster the ideas and political skill to win with voters. To hold on to it, the new mayor must also find a winning way of governing.

David Greising is president and CEO of the Better Government Association.

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