The flattery was as obvious as it was effective. Seated at a dinner table in the Blue Room of the White House on Monday, Prime Minister Benjamin Netanyahu of Israel handed President Trump a piece of paper.

“I want to present to you, Mr. President, the letter I sent to the Nobel Prize committee,” Mr. Netanyahu said, with the news cameras rolling to capture the moment. “It’s nominating you for the Peace Prize. It’s well deserved, and you should get it.”

Mr. Trump declared the gesture from the prime minister “very meaningful,” though he has long said he believes the Nobel committee would never give him the prize.

The effort to curry favor was the latest evidence that many of the world’s leaders have figured Mr. Trump out. Heaping praise on the American president is the best way to manage him — even if it’s not entirely clear that the schmoozing leads to concrete benefits for their countries.

After once calling Mr. Trump a “bully,” Mark Carney, the prime minister of Canada, last month gushed over “your personal leadership of the United States.” Mark Rutte, the Secretary General of NATO, sent a private text to Mr. Trump hailing “your decisive action in Iran, that was truly extraordinary.” Mr. Trump made it public the next day.

And even President Volodymyr Zelensky of Ukraine, who endured an angry Oval Office shouting match in February, has worked to repair his frayed relationship with the American leader. A meeting of the pair at the Vatican in April went much more smoothly, with no blowups, according to both sides.

On Monday, Mr. Trump agreed to resume shipments of some weapons to Ukraine that had been paused just a week ago. The reversal came amid the improved relations with Mr. Zelensky and after a phone call with Mr. Putin that Mr. Trump called disappointing.

Sam Edwards, an associate professor of modern political history at Loughborough University in Britain said Mr. Trump has reshaped global diplomacy.

“We’ve often thought of diplomacy as this big, broad endeavor. It’s about institutions connecting with one another,” he said. “In this instance, they’re playing the man.”

“This is how it works,” he added. “You come with gifts, you offer homage of sorts, in order to gain the respect, the support, the favor of the head of that court.”

And yet, political analysts, diplomats and others who follow international interactions say that “playing the man” with public declarations of admiration does not always work, especially with a leader like Mr. Trump, whose decision-making is often fickle.

Mr. Netanyahu declared Monday that Mr. Trump is “forging peace as we speak, in one country, in one region after the other.”

But stroking the president’s ego has not produced an end to the war in Gaza, which rages on even amid a resumption of cease-fire talks. In Europe, the war in Ukraine continues with no sign of the peace that Mr. Trump once promised would take him only 24 hours to implement. Some tariffs remain in place on British exports to America even after Keir Starmer, Britain’s prime minister, sealed a deal with Mr. Trump in part by delivering a royal invitation.

Yolanda Spies, a former South African diplomat and the director of the Oxford University Diplomatic Studies Program, said that flattery has long been built into the art of diplomacy.

But she said that the most personal interactions between leaders used to happen privately, not in front of cameras.

“One of the driving thoughts of the profession of diplomacy is do all the hard work behind the scenes, where no one is watching,” she said. “Now, you have to be really careful, because anything you send to him will be public. It means a new step in the flattery game.”

Mr. Zelensky may appreciate the need for flattery more than most other world leaders, Ms. Spies said. After the humiliation of his Oval Office meeting earlier this year, she said, Mr. Zelensky has dramatically changed the way he interacts with Mr. Trump.

“He has avoided those kinds of scenarios where he ends up having an argument with Donald Trump,” she said. “He now prefaces every statement with how grateful he is to America. There, a lesson was learned.”

In Mr. Netanyahu’s case, the appeal to Mr. Trump’s desire to be awarded a Nobel Peace Prize was not unique. Pakistan’s government formally nominated Mr. Trump in June, citing the president’s “decisive diplomatic intervention” during an outbreak of violence between India and Pakistan.

Mr. Trump has repeatedly complained, in public and in private, that he has not yet won the peace prize. He once posted on social media that “I won’t get a Nobel Peace Prize no matter what I do.”

And it’s not clear that a nomination from Mr. Netanyahu — who has been accused of war crimes and crimes against humanity in the Gaza Strip by the International Criminal Court — will help Mr. Trump’s case.

Medea Benjamin, the founder of Code Pink, an antiwar group, posted her thoughts on social media.

“I don’t know whether to laugh or cry,” she wrote. “Surreal.”