But Kenneth Arrow, the economics Nobel laureate, showed in his 1951 doctoral thesis that the problem runs far deeper than anyone had imagined. Mr. Arrow’s “impossibility theorem” says that there is no mechanism that can coherently speak for the will of the people.

Loosely speaking, this extraordinary result says that any mechanism that aims to speak for the will of the people — that is, not a dictatorship — will be susceptible to at least one of three defects.

The first possible defect is the problem the marquis illustrated — the problem of preference cycles. The second possible defect is that voters will make choices that suggest that the addition of irrelevant alternatives leads them to change their minds. This is the equivalent of choosing chocolate over vanilla, only to reverse course and choose vanilla instead, once strawberry has been added to the menu.

The third possibility is that even when each voter individually prefers chocolate ice cream to vanilla, somehow collectively the voters will choose vanilla instead. (I’m grateful to the economist Alex Tabarrok for this ice cream analogy.)

The news isn’t all bad. The fact that any voting rule is susceptible to one of these defects doesn’t mean that the voters must make incoherent choices, merely that it could happen. (And perhaps it just did.)

Mr. Arrow’s impossibility theorem suggests that maybe the Republican primary results say less about the desires of Republican voters than they do about tensions inherent in groups of people collectively deciding what to do.

Economic theorists have also pointed to a reason that the modern G.O.P. may be particularly susceptible to making strange choices. If disagreements between voters are simple enough — such as when some want more liberal policies and others more conservative policies — simple majority rule won’t result in any of the defects that concerned Mr. Arrow. Perhaps disagreements between Democrats are this simple.