“There’s a certain contradiction here,” said Itamar Rabinovich, a former Israeli diplomat and university president. “That’s what you need mediators for — you find that magic formula, constructive ambiguity, that enables both parties to claim achievement.

“Right now, those actors are not there.”

Washington, which has helped broker previous cease-fires, is consumed with other crises, and has diminished credibility in the Middle East. Egypt, which during the brief presidency of Mohamed Morsi strongly supported Hamas, now treats the group as an enemy, and is loath to let its rivals Qatar and Turkey play a significant diplomatic role to aid residents of Gaza.

That leaves President Mahmoud Abbas of the Palestinian Authority, an adversary of both Israel and Hamas, as the primary Palestinian interlocutor. Weak at home but increasingly active on the international stage, he shuttled from Cairo to Istanbul on Friday for what were described as cease-fire negotiations.

“The fact that Abbas is involved this time, unlike all previous cases, could mean something,” said Khalil Shikaki, a Palestinian pollster and analyst. “He does not have a lot of leverage here, but the little he has might allow for a three-way deal — Hamas, Abbas and the Israelis. That’s the way things might be smoother than to wait for the battlefield itself to determine the outcome, which could take a very, very long time, and a great deal of bloodshed.”