Mr. Brzezinski notes that President Obama has “failed to speak directly to the American people about America’s changing role in the world, its implications, and its demands,” but this book curiously lacks any detailed analysis of Mr. Obama’s policies so far — nothing remotely approaching the acute assessments of Presidents George W. Bush, Bill Clinton and George H. W. Bush contained in this author’s 2007 book, “Second Chance,” which charted the opportunities he considers missed at the end of the cold war (like using the victory in the first Gulf war strategically to press for an Israeli-Palestinian accord).

Also missing from this book are any substantive discussions of how the United States might overcome “its staggering domestic challenges and reorient its drifting foreign policy” and how the current European debt crisis might affect the United States and the future fortunes of the West.

What Mr. Brzezinski does do here — lucidly, and for the most part with great persuasiveness — is explore the consequences that a steady slide by America into impotence and irrelevance might have on the rest of the world. Such a development, he argues, would probably not result in the “ ‘coronation’ of an effective global successor” like China, but would likely lead to a “protracted phase of rather inconclusive and somewhat chaotic realignments of both global and regional power, with no grand winners and many more losers.”

An America “in serious decline for domestic and/or external reasons,” he says, would lead to a breakdown in the ability of the international system to prevent conflict once it became evident that “America is unwilling or unable to protect states it once considered, for national interest and/ or doctrinal reasons, worthy of its engagement.” As he sees it, a more Darwinian world of tumbling dominoes would most likely result: there would be little to prohibit regional powers (like Russia) from exerting claims on neighbors falling within traditional or claimed spheres of influence (like Georgia, Belarus and Ukraine). Taiwan would become increasingly vulnerable, and so too would Israel.

In the case of Afghanistan, Mr. Brzezinski says, a failure to sustain United States-sponsored international involvement in the region could turn that country into a haven again for international terrorism, while a decline in American power and aid could lead to a worst-case outcome in which Pakistan devolved into “some variation of nuclear warlordism” or became “a militant-Islamic and anti-Western government similar to Iran.”

For that matter, Mr. Brzezinski suggests, a weakened America would increase the dangers of nuclear proliferation around the world. Were doubts to be raised about the United States’ nuclear umbrella, he says, countries like South Korea, Taiwan, Japan, Turkey and Israel would have to seek security elsewhere — that elsewhere meaning “nuclear weapons of one’s own or from the extended deterrence of another power — most likely Russia, China or India.”

Global environmental issues — including climate change and growing water shortages — would be similarly affected. In a gloomy conclusion to this insightful book Mr. Brzezinski writes that without a revitalized America helping to manage the international commons, “progress on the issues of central importance to social well-being and ultimately to human survival would stall.”