“China is deliberately doing these things to demonstrate the unsustainability of the American position of having a good relationship with China and maintaining its alliances in Asia, which constitute the leadership of the United States in Asia,” Mr. White said.

China is betting that America, tired and looking inward, will back off, he said, eroding its traditional place of influence in Asia and enhancing China’s power.

But even as Mr. Hagel and the United States have adopted a public posture that backs Japan — and, to a lesser extent, the Philippines, Vietnam and any other country that finds itself at odds with China — some administration officials have privately expressed frustration that the countries are all engaged in a game of chicken that could lead to war.

“None of those countries are helping matters,” a senior administration official said. The official, who spoke on the condition of anonymity in order to talk candidly about American policy, said that the United States would publicly back Japan and that treaty obligations mean that if Japan and China go to war, the United States will almost certainly be dragged into it. But, he added, administration officials have privately prodded their Japanese counterparts to think carefully before acting, and to refrain from backing China into a corner.

“If these are kids in the schoolyard, they are running around with scissors,” said Vikram J. Singh, who until February was the United States deputy assistant secretary of defense for South and Southeast Asia and is now the vice president for national security at the Center for American Progress. “Wars start from small things, often by accident and miscalculation — like dangerous maneuvers by aircraft that result in a collision or aggressive moves that lead to an unexpected military response.”

Speaking at the opening session of the conference on Friday, Japan’s prime minister, Shinzo Abe, who has had a role in stirring tensions in the region by embracing a more assertive military stance, bypassed a question about whether he was willing to go to war with China over the disputed islands in the East China Sea, which Japan calls the Senkaku and China calls the Diaoyu. Instead, he said cryptically that it was “important that we all make efforts” so that certain “contingencies can be prevented.”