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Some analysts said the incident will play into Mr. Ivanishvili’s hands in the polls.

“The prison torture videos have dealt a serious blow to the ruling party’s authority,” said Irakly Menagarishvili, a former Georgian foreign minister who now heads the Center for Strategic Research, an independent think tank.

He said Mr. Saakashvili needs to act quickly to save his United National Movement party from being beaten, and he added that the government has failed to contain the fallout.

Alexander Rondeli, an independent political expert in Tbilisi, agreed that the scandal would take a toll on Mr. Saakashvili’s party.

Mr. Saakashvili may have quelled some of the anger by giving the penitentiaries minister’s job to Giorgy Tugushi, the ombudsman who long has criticized Georgia’s prisons.

“Now I have a chance to completely reorganize the system and ensure the protection of inmates’ rights,” Mr. Tugushi said.

Gigi Tsereteli, a deputy speaker of parliament from Mr. Saakashvili’s ruling party, sought to play down the prison videos’ impact. He said the “intolerable and outrageous” prison videos have taught a “hard lesson to the government,” but he insisted it wouldn’t lead to the ruling party’s defeat at the polls.

“The government has reacted quickly: A minister has stepped down, and the perpetrators of these crimes have been arrested,” he told the Associated Press. “I don’t think it will affect our electorate. People have long made their choice in our favor.”