How much credibility will the United States lose if it doesn’t respond forcefully to Russian action? This, too, is a problem of game theory.

A commitment by a sovereign state is credible only when that state’s self-interest dictates honoring it. Previous American pledges to help or protect Ukraine were not all that credible to begin with, given the greater power and historical influence of Russia in the region. Failing to protect Crimea therefore doesn’t automatically lead to a big shift in the world’s perception of American willingness to honor commitments where the nation’s loyalties and interests are more certain. Daryl G. Press, a professor of government at Dartmouth, articulates a general version of that argument in his book “Calculating Credibility.”

Still, there may be a net loss of credibility, perhaps a serious one, when the world is uncertain where American self-interest lies. For instance, how dedicated is the United States to protecting various disputed small Asian islands from Chinese domination or conquest? How much does America care about the de facto independence of Taiwan these days, or about limiting China’s influence in the South China Sea? The answers may not be obvious, especially in a diverse democracy like ours.